research article dark energy from the gas of wormholes · 2017. 7. 17. · is called as the...

Hindawi Publishing CorporationJournal of AstrophysicsVolume 2013 Article ID 543717 14 pageshttpdxdoiorg1011552013543717

Research ArticleDark Energy from the Gas of Wormholes

A A Kirillov and E P Savelova

Dubna International University of Nature Society and Man Universitetskaya Street 19 Dubna 141980 Russia

Correspondence should be addressed to A A Kirillov ka98mailru

Received 21 March 2013 Accepted 29 May 2013

Academic Editors C Ghag M Jamil and S Profumo

Copyright copy 2013 A A Kirillov and E P Savelova This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We assume the space-time foam picture in which the vacuum is filled with a gas of virtual wormholes It is shown that virtualwormholes form a finite (of the Planckian order) value of the energy density of zero-point fluctuations However such a hugevalue is compensated by the contribution of virtual wormholes to the mean curvature and the observed value of the cosmologicalconstant is close to zero A nonvanishing value appears due to the polarization of vacuum in external classical fields In the earlyUniverse some virtual wormholes may form actual ones We show that in the case of actual wormholes vacuum polarization effectsare negligible while their contribution to the mean curvature is apt to form the observed dark energy phenomenon Using thecontribution of wormholes to dark matter and dark energy we find estimates for characteristic parameters of the gas of wormholes

1 Introduction

As is well known modern astrophysics (and even moregenerally theoretical physics) faces two key problems Thoseare the nature of dark matter and dark energy Recall thatmore than 90 of matter of the Universe has a nonbaryonicdark (to say mysterious) form while lab experiments stillshow no evidence for the existence of such matter Bothdark components are intrinsically incorporated in the mostsuccessful ΛCDM (Lambda cold dark matter) model whichreproduces correctly properties of the Universe at very largescales (eg see [1] and references therein) We point out thatΛCDM predicts also the presence of cusps (120588DM sim 1119903)

in centers of galaxies [2] and a too large number of galaxysatellites Therefore other models are proposed for examplelike axions [3] which may avoid these To be successfulsuch models should involve a periodic self-interaction andtherefore require a fine tuning while in general the presenceof standard nonbaryonic particles cannot solve the problemof cusps Indeed if we admit the existence of a self-interactionin the dark matter component or some coupling to baryonicmatter (which should be sufficiently strong to remove cusps)then we completely change properties of the dark mattercomponent at the moment of recombination and destroy allsuccessful predictions at very large scales Recall that both

warm and self-interacting darkmatter candidates are rejectedby the observing Δ119879119879 spectrum [1] In other words the twokey observational phenomena (cores of darkmatter in centersof galaxies [4ndash6] andΔ119879119879 spectrum) give a very narrow gapfor dark matter particles which seems to require attractingsome exotic objects in addition to standard nonbaryonicparticles

As it was demonstrated recently [7] the problem of cuspscan be cured if some part of nonbaryon particles is replacedby wormholes Wormholes represent extremely heavy (incomparison to particles) objects which at very large scalesbehave exactly like nonbaryon cold particles while at smallerscales (in galaxies) they strongly interact with baryons andform the observed [4ndash6] cored (120588DM sim const) distributionWe note that stable wormholes violate necessarily the aver-aged null energy conditions which gives the basic argumentagainst the existence of such objects Without exotic matterstable wormholes may however exist in modified theoriesfor example see [8] and references therein In the casewhen the energy conditions hold a wormhole collapses intoa couple of conjugated (of equal masses) blackholes whichalmost impossible to distinguish from standard primordialblackholes However the topological nontriviality of suchobjects retains and gravitational effects of a gas of wormholesconsidered in [9] and some results of [7] still remain valid

2 Journal of Astrophysics

which means that nontraversable wormholes can be used tosmooth cusps in centers of galaxies Thus it worth expectingthatwormholesmay play an important role in the explanationof the dark matter phenomenon

Saving the dark matter component ΛCDM requiresthe presence (sim70) of dark energy (of the cosmologicalconstant) Moreover there is evidence for the start of anacceleration phase in the evolution of the Universe [10ndash14]In the present paper we use virtual wormholes to estimatethe contribution of zero-point fluctuations in the value of thecosmological constant The idea to relate virtual wormholes(or baby universes) and the cosmological constant is not newin somewhat different context it was used by Coleman in [15]and developed in [16] Our basic aim is to demonstrate thatvirtual wormholes form a finite value of the energy density ofzero-point fluctuations

It is necessary to point here out to the principle differ-ence between actual and virtual wormholes The principledifference is that a virtual wormhole exists only for a verysmall period of time and at very small scales and doesnot necessarily obey to the Einstein equations It representstunnelling event and therefore the averaged null energycondition (ANEC) cannot forbid the origin of such an objectFor the future we also note that a set of virtual wormholesmay work as an actual wormhole opening thus the way foran artificial construction of wormhole-type objects in labexperiments

In the present paper we describe a virtual wormholeas follows From the very beginning we use the Euclideanapproach (eg see [17] and the standard textbooks [18])Thenthe simplest virtual wormhole is described by themetric (120572 =

1 2 3 4)

1198891199042= ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (1)

where

ℎ (119903) = 1 + 120579 (119886 minus 119903) (1198862

1199032minus 1) (2)

and 120579(119909) is the step function Such a wormhole has vanishingthroat length while the step function at the junction maycause a problem in Einsteinrsquos equation or when a topologicalEuler term is involved A more careful analysis needs to con-sider distributional curvature and so forth see [19] To avoidthese difficulties we may consider from the very beginninga wormhole of a finite throat length sim1120573 where the stepfunction is replaced with a smooth function (eg 120579(119909 120573) =

(exp (120573119909)+1)minus1)Thenwhere it is necessary onemay considerthe limit 120573 rarr infin only in final expressions This insuresthat the Bianchi identity holds and that the above metricremains inside of the domain of usual gravity (Unexpectedlyour approach (the use of step function) seems to irritate anessential part of physicists working with wormholes (egPRD reviewers moreover there is a claim that we studysome other exotic objects which are not wormholes [20])Therefore it is necessary to clarify our position here We arequite aware that at present state the physics of wormholes iscertainly on the most speculative side Therefore we do not

see the difference between specific forms of a smooth metricthat one may use For example the simplest choice ℎ = (1 +

1198862119903

2) gives the well-known metric which in 3-dimensions

is called as the Bronnikov-Ellis metric or the metric ℎ =

(1 + (119887119903) + (1198862119903

2)) which includes an additional parameter

(an arbitrary length of the handle) which is not called by anyname but is not less trivial In the case of actual wormholesthe exact and correct form of metric may be establishedonly upon understanding the nature and properties of exoticmatter which may support such a metric as a stable solutionto the Einstein equations For this time has not come yet Inthe case of virtual wormholes even this is not important for inthe complete theory one has to sum over all possible metricswhich formally may be included in 120593 in (5) Our basic aimis to present sufficiently clear and simple model (let it be farfrom realistic) which retains basic qualitative features relatedto a nontrivial topology)

In the region 119903 gt 119886 ℎ = 1 and the metric (2) is flatwhile the region 119903 lt 119886 with the obvious transformation119910120572= (119886

2119903

2)119909

120572 is also flat for 119910 gt 119886 Therefore the regions119903 gt 119886 and 119903 lt 119886 represent two Euclidean spaces glued at thesurface of a sphere 1198783 with the centre at the origin 119903 = 0 andradius 119903 = 119886 Such a space can be described with the ordinarydouble-valued flat metric in the region 119903

plusmngt 119886 by

1198891199042= 120575

120572120573119889119909

120572

plusmn119889119909

120573

plusmn (3)

where the coordinates 119909120572

plusmndescribe two different sheets of

space Now identifying the inner and outer regions of thesphere 1198783 allows the construction of a wormhole which con-nects regions in the same space (instead of two independentspaces) This is achieved by gluing the two spaces in (3) bymotions of the Euclidean space (the Poincare motions) If 119877

plusmn

is the position of the sphere in coordinates 119909120583

plusmn then the gluing

is the rule

119909120583

+= 119877

120583

++ Λ

120583

] (119909]minusminus 119877

]minus) (4)

where Λ120583

] isin 119874 (4) which represents the composition of atranslation and a rotation of the Euclidean space (Lorentztransformation) In terms of common coordinates such awormhole represents the standard flat space in which the twospheres 1198783

plusmn(with centers at positions 119877

plusmn) are glued by the rule

(4)We point out that the physical region is the outer region ofthe two spheres Thus in general the wormhole is describedby a set of parameters the throat radius 119886 positions of throats119877plusmn and rotation matrix Λ120583

] isin 119874 (4)In the present paper we assume the space-time foam

picture in which the vacuum is filled with a gas of virtualwormholes We show that virtual wormholes form a finite (ofthe Planckian order) value of the energy density of zero-pointfluctuations However such a huge value is compensated bythe contribution of virtual wormholes to the mean curvatureand the observed value of the cosmological constant shouldbe close to zero

To achieve our aimwe in Section 2 present the construc-tion of the generating functional in quantum field theoryThe main idea is that the partition function includes thesum over field configurations and the sum over topologies

Journal of Astrophysics 3

Where the sum over topologies is the sum over virtualwormholes described above Such an approach gives a rathergood leading approximation for calculation of the partitionfunction and corresponds to the standard methods (egRitza method etc) In Section 3 we investigate properties ofthe two-point Green function We show that the presenceof the gas of virtual wormholes can be described by thetopological bias exactly as it happens in the presence of actualwormholes [7 9] For limiting topologies when the densityof virtual wormholes becomes infinite the Green functionshows a good ultraviolet behavior which means that thereexists a class of such systems when quantum field theoriesare free of divergencies We demonstrate how the sum overtopologies defines themean value for the bias which takes thesense of a cutoff function in the space of modes In Section 4we explicitly demonstrate that for a particular set of virtualwormholes the bias defines not more than the projectionoperator on the subspace of functions obeying to the properboundary conditions at wormhole throats The projectivenature of the bias means that wormholes merely cut someportion of degrees of freedom (modes) Phenomenologicallyit means that wormholes can be described by the presence ofghost fields which compensate the extra (cut by wormholes)modes In Section 5 we show how the cutoff expresses viasome dynamic parameters of wormholesThe exact definitionof such parameters we leave it for the future investigationIn Section 6 we consider the origin of the cosmologicalconstant We demonstrate that the cosmological constantis determined by the contribution of the energy density ofzero-point fluctuations and by the contribution of virtualwormholes to the mean curvature We estimate contributionof virtual wormholes to the mean curvature and show alsothat wormholes lead to a finite (of the planckian order)value of ⟨119879

120583]⟩ which requires considering the contributionfrom the smaller and smaller wormholes with divergentdensity 119899 rarr infin We also present arguments of why inthe absence of external classical fields the total value ofthe cosmological constant is exactly zero while it acquiresa nonvanishing value due to vacuum polarization effects(ie due to an additional distribution of virtual wormholes)in external fields We also speculate the possibility of theformation of actual wormholes and in Section 7 we estimatetheir contribution to the dark energy Finally in Section 8 werepeat basic results an discuss some perspectives

2 Generating Function

The basic aim of this section is to construct the generatingfunctional which can be used to get all possible correlationfunctions Consider the partition functionwhich includes thesum over topologies and the sum over field configurations

119885total = sum

120591

sum

120593

119890minus119878 (5)

For the sake of simplicity we use from the very beginning theEuclidean approach The action has the form

119878 = minus1

2(120593119860120593) + (119869120593) (6)

and we use the notions (119869120593) = int 119869(119909)120593(119909)1198894119909 If we fix

the topology of space by placing a set of wormholes withparameters 120585

119894 then the sum over field configurations 120593 gives

the well-known result

119885lowast(119869) = 119885

0(119860) 119890

minus(12)(119869119860minus1

119869) (7)

where 1198850(119860) = int[119863120593]119890

(12)(120593119860120593) is the standard expressionand 119860

minus1= 119860

minus1(120585) is the Green function for a fixed topology

that is for a fixed set of wormholes 1205851 120585

119873

Consider now the sum over topologies 120591 To this end werestrict the sum over the number of wormholes and integralsover parameters of wormholes

sum

120591

997888rarr sum

119873

int

119873

prod

119894=1

119889120585119894= int [119863119865] (8)

where

119865 (120585119873) =1

119873

119873

sum

119894=1

120575 (120585 minus 120585119894) (9)

and 119873119865 is the density of wormholes in the configurationspace 120585 We also point out that in general the integration overparameters is not free (eg it obeys the obvious restriction|

+

119894minus

minus

119894| ge 2119886

119894) This defines the generating function as

119885total (119869) = int [119863119865]1198850(119860) 119890

minus(12)(119869119860minus1

119869) (10)

The sum over topologies assumes an additional averaging outfor all mean values with the measure 119889120583

119873= 120588(120585119873)119889

119873120585

where

120588 (120585119873) =

1198850(119860 (120585119873))

119885total (0) (11)

which obey the obvious normalization conditionsum119873int119889120583

119873=

sum119873120588119873

= 1 The averaging out over topologies assumes thetwo stages First we fix the total number of wormholes 119873and average over the parameters of wormholes 120585 (ie overparameters of a static gas of wormholes in 119877

4) Then we sumover the number of wormholes119873 (the so-called big canonicalensemble)

The basic difficulty of the standard field theory is thatthe perturbation scheme based on (7) leads to divergentexpressions This remains true for every particular topologyof space (ie for any particular finite set of wormholes) sincethere always exists a scale below which the space looks likethe ordinary Euclidean spaceWhat we expect is that the sumover all possible topologies will remove such a difficulty

And indeed the above measure (11) has the structure

1198850(119860 (120585119873)) = exp(minusintΛ (120585119873) 119889

4119909) (12)

where Λ(120585119873) is the cosmological constant related to theenergy density of zero-point fluctuations calculated for aparticular distribution of wormholes (We recall that the total


cosmological constant should include also the contributionfrom the mean curvature (55)) Any finite distribution ofwormholes leads to the divergent expression Λ(120585119873) rarr

infin and is suppressed (ie 120588 (120585119873) rarr 0) However thesum over all possible topologies assumes also the limitingtopologies 119899 rarr infin where 119899 = 119873119881 is the density ofwormholes In this limit wormhole throats degenerate intopoints and the minimal scale below which the space lookslike the Euclidean space is merely absent We point out thatfrom the rigorous mathematical standpoint such limitingtopologies cannot be described in terms of smoothmanifoldssince they are not locally Euclidean and does not possess afinite set of maps In mathematics similar objects are wellknown for example fractal sets However if a fractal set isobtained by cutting (by means of a specific rule or iterations)portions of space our limiting topologies are obtained bygluing (identifying) some portions (or in the limit couplesof points) of the Euclidean space The basic feature of suchtopologies is that QFT becomes finite on such a set Indeedas we shall see a particular infinite distribution of wormholescan always be chosen in such a way that the energy of zero-point fluctuations becomes a finite 0 le Λ

infin(120585) lt infin (eg

see the next section or the second term in (67)) In the sumover topologies only such limiting topologies do survive (ie120588infin(120585) = 0)

3 The Two-Point Green Function

From (7) we see that the very basic role in QFT plays the twopoint Green function Such a Green function can be foundfrom the equation

119860119866(119909 1199091015840) = minus120575 (119909 minus 119909

1015840) (13)

with proper boundary conditions at wormholes which gives119866 = 119860

minus1 Now let us introduce the bias function119873(119909 1199091015840) as

119866 (119909 119910) = int1198660(119909 119909

1015840)119873 (119909

1015840 119910) 119889119909

1015840 (14)

where 1198660(119909 119909

1015840) is the ballistic (or the standard Euclidean

Green function) and the bias can be presented as

119873(119909 1199091015840) = 120575 (119909 minus 119909

1015840) +sum

119894

119887119894120575 (119909 minus 119909

119894) (15)

where 119887119894are fictitious sources at positions 119909

119894which should

be added to obey the proper boundary conditions We pointout that the bias can be explicitly expressed via parametersof wormholes that is 119873(119909 119909

1015840) = 119873(119909 119909

1015840 120585

1 120585

119873) For the

sake of illustration we consider first a particular example

31 The Bias for a Particular Distribution of Wormholes(Rarefied Gas Approximation) Consider now the bias for aparticular set of wormholes For the sake of simplicity weconsider the case when119898 = 0 The Green function obeys theLaplace equation

minusΔ119866 (119909 1199091015840) = 120575 (119909 minus 119909

1015840) (16)

with proper boundary conditions at throats (we require119866 and120597119866120597119899 to be continual at throats)The Green function for theEuclidean space is merely 119866

0(119909 119909

1015840) = 14120587

2(119909 minus 119909

1015840)2 (and

1198660(119896) = 1119896

2 for the Fourier transform) In the presence ofa single wormhole which connects two Euclidean spaces thisequation admits the exact solution For outer region of thethroat 1198783 the source 120575(119909 minus 119909

1015840) generates a set of multipoles

placed in the center of sphere which gives the corrections tothe Green function 119866

0in the form (we suppose the center of

the sphere at the origin)

120575119866 = minus1

412058721199092

infin

sum

119899=1

1

119899 + 1(119886

1199091015840)

2119899

(1199091015840

119909)

119899minus1

119876119899 (17)

where 119876119899= (4120587

22119899)sum

119899minus1

119897=0sum

119897

119898=minus119897119876

lowast1015840

119899119897119898119876

119899119897119898and 119876

119899119897119898(Ω) are

four-dimensional spherical harmonics for example see [21]In the present section we shall consider a dilute gas approx-imation and therefore it is sufficient to retain the lowest(monopole) term only A single wormhole which connectstwo regions in the same space is a couple of conjugatedspheres 1198783

plusmnof the radius 119886with a distance =

+minus

minusbetween

centers of spheres So the parameters of the wormhole are(The additional parameter (rotation matrix Λ) is importantonly for multipoles of higher orders) 120585 = (119886 119877

+ 119877

minus) The

interior of the spheres is removed and surfaces are gluedtogether Then the proper boundary conditions (the actualtopology) can be accounted for by adding the bias of thesource

120575 (119909 minus 1199091015840) 997888rarr 120575 (119909 minus 119909

1015840) + 119887 (119909 119909

1015840) (18)

In the approximation 119886119883 ≪ 1 (eg see for details [9]) thebias for a single wormhole takes the form

1198871(119909 119909

1015840 120585) =

1198862

2(

1

(119877minusminus 1199091015840)

2minus

1

(119877+minus 1199091015840)

2)

times [120575 ( minus +) minus 120575 ( minus

minus)]

(19)

This form for the bias is convenient when constructing thetrue Green function and considering the long-wave limithowever it is not acceptable in considering the short-wavebehavior and vacuum polarization effects Indeed the posi-tions of additional sources are in the physically nonadmissibleregion of space (the interior of spheres 1198783

plusmn) To account for

the finite value of the throat size we should replace in (19)the point-like source with the surface density (induced on thethroat) that is

120575 ( minus plusmn) 997888rarr

1

212058721198863120575 (

10038161003816100381610038161003816 minus

plusmn

10038161003816100381610038161003816minus 119886) (20)

Such a replacement does not change the value of the trueGreen function however now all extra sources are in thephysically admissible region of space

In the rarefied gas approximation the total bias is additivethat is

119887total (119909 1199091015840) = sum119887

1(119909 119909

1015840 120585

119894) = 119873int119887

1(119909 119909

1015840 120585) 119865 (120585) 119889120585

(21)


where 119873119865 is given by (9) For a homogeneous and isotropicdistribution 119865(120585) = 119865(119886119883) and then for the bias we find

119887total (119909 minus 1199091015840)

= int1

21205872119886(

1

1198772

minus

minus1

1198772

+

)120575 (10038161003816100381610038161003816 minus

1015840minus

+

10038161003816100381610038161003816minus 119886)119873119865 (120585) 119889120585

(22)

Consider the Fourier transform 119865(119886119883) = int119865(119886 119896)119890minus119894119896119883

sdot

(1198894119896(2120587)

4) and using the integral 11199092

= int(41205872119896

2)119890

minus119894119896119909sdot

(1198894119896(2120587)

4) we find for 119887(119896) = int 119887(119909)119890

1198941198961199091198894119909 the expression

119887total (119896) = 119873int1198862 4120587

2

1198962(119865 (119886 119896) minus 119865 (119886 0))

1198691(119896119886)

1198961198862119889119886 (23)

(1) Example of a Finite Density of Wormholes Consider nowa particular (of a finite density) distribution of wormholes119865(119886119883) for example

119873119865 (119886119883) =119899

212058721199033

0

120575 (119886 minus 1198860) 120575 (119883 minus 119903

0) (24)

where 119899 = 119873119881 is the density of wormholes In the case119873 = 1 this function corresponds to a single wormhole withthe throat size 119886

0and the distance between throats 119903

0=

|119877+minus119877

minus|We recall that the action (6) remains invariant under

translations and rotations which straightforwardly leads tothe above function Then 119873119865(119886 119896) = int119873119865(119886119883)119890

1198941198961199091198894119909 re-

duces to 119873119865(119886 119896) = 119899(1198691(119896119903

0)(119896119903

02))120575(119886 minus 119886

0) Thus from

(23) we find

119887 (119896) = minus1198991198862 4120587

2

1198962(1 minus

1198691(119896119903

0)

11989611990302

)1198691(119896119886

0)

11989611988602

(25)

And for the true Green function we get

119866true = 1198660(119896)119873 (119896) = 119866

0(119896) (1 + 119887 (119896)) (26)

In the short-wave limit (119896119886 1198961199030≫ 1) 119887(119896) rarr 0 and

therefore 119873(119896) rarr 1 This means that at very small scalesthe space filled with a finite density of wormholes looks likethe ordinary Euclidean space In the long-wave limit 119896 rarr 0

we get 1198691(119896119903

0)(119896119903

02) asymp 1 minus (12) (119896119903

02)

2+ sdot sdot sdot which gives

119887(119896) asymp minus1205872119899119886

21199032

02 while in a more general case we find

119887(119896) asymp minus int(12058722)119886

21199032

0119899(119886 119903

0)119889119886 119889119903

0 where 119899(119886 119903

0) is the den-

sity of wormholes with a particular values of 119886 and 1199030 and for

the bias function (15) we get

119873(119896) 997888rarr 1 minus1205872

2int 119886

4119899 (119886 119903

0)1199032

0

1198862119889119886 119889119903

0le 1 (27)

In other words in the long-wave limit (119896119886 1198961199030≪ 1) the

presence of a particular set of virtual wormholes diminishesmerely the value of the charge values

(2) Limiting Topologies or Infinite Densities of WormholesConsider now the limiting distribution when the density of

wormholes 119899 rarr infin Since every throat cuts the finite portionof the volume (120587

22)119886

4 this case requires considering thelimit 119886 rarr 0 We assume that in this limit 1198862119873119865(119886119883) rarr

120575(119886)](119883) where ](119883) is a finite specific distribution Then(23) reduces to 119887total(119896) = (4120587

2119896

2) (](119896) minus ](0)) where ](119896) =

int ](119883)119890119894119896119883

1198894119883 and the bias (15) (18)119873(119896) becomes

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (28)

It is important that this limit still agrees with the rarefiedgas approximation for the basic gas parameter (ie theportion of volume cut by wormholes) tends to 120585 =

int(12058722)119886

4119865(119886119883)119889

4119883119889119886 rarr 0 The above expression is ob-

tained in the linear approximation only Taking into accountnext orders (eg see [22] where the case of a dense gas is alsoconsidered) we find 119873(119896) = 1 + 119887total(119896) + 119887

2

total(119896) + sdot sdot sdot andthe true Green function in a gas of wormholes becomes

119866true = 1198660(119896)119873 (119896) =

1

1198962 + 41205872(] (0) minus ] (119896))

(29)

In the long-wave limit 119896 rarr 0 the function ](119896) can beexpanded as ](119896) asymp ](0) + (12)]10158401015840(0)1198962 which also definesa renormalization of charge values119873(119896) rarr 1(1minus2120587

2]10158401015840(0))which coincides with (27) However in this limiting case wehave some freedom in the choice of ](119896) which we can useto assign 119873(119896) an arbitrary function of 119896 In other wordswe may get here a class of limiting topologies where Greenfunctions119866true = 119866

0(119896)119873(119896) have a good ultraviolet behavior

and quantumfield theories in such spaces turn out to be finiteIn particular this will certainly be the case when

119866true (119909 minus 1199091015840= 0) = int

1

1198962 + 41205872(] (0) minus ] (119896))

1198894119896

(2120587)4lt infin

(30)

This possibility however requires the further and more deepinvestigation

32 Green Function General Consideration The action (6)remains invariant under translations

1015840= + 119888 with an

arbitrary 119888 which means that the measure (11) does notactually depend on the position of the center of mass of thegas of wormholes and therefore we may restrict ourself withhomogeneous distributions 119865(120585) of wormholes in space onlyIndeed we may define 119889119873

120585 = 11988911987312058510158401198894119888 while the integration

over 1198894119888 gives the volume of1198774 that is int1198894

119888 = 1198714= 119881which

disappears from (11) due to the denominator (Technically wemay first restrict a portion of1198774 in (6) to a finite volume119881 andthen in final expressions consider the limit 119881 rarr infin (whichrepresents the standard tool in thermodynamics and QFT))In what follows we shall omit the prime from 120585

1015840Let us consider the Fourier representation119873(119909 119909

1015840 120585) rarr

119873(119896 1198961015840 120585) which in the case of a homogeneous distribution

of wormholes gives119873(119896 1198961015840) = 119873(119896 120585) 120575(119896 minus 119896

1015840) then we find

119866 (119896) = 1198660(119896)119873 (119896 120585) (31)


and the Green function can be taken as

119866 =119873 (119896 120585)

1198962 + 1198982 (32)

Then for the total partition function we find

119885total (119869)

= int [119863119873 (119896)] 119890minus119868(119873)

119890minus(12)(119871

4

(2120587)4

) int((119873(119896)(1198962

+1198982

))|119869119896

|2

)119889119896

(33)

where [119863119873] = prod119896119889119873

119896and 120590(119873) comes from the integration

measure (ie from the Jacobian of transformation from 119865(120585)

to119873(119896))

119890minus119868(119873)

= int [119863119865]1198850(119873 (119896 120585)) 120575 (119873 (119896) minus 119873 (119896 120585)) (34)

We point out that 119868(119873) can be changed by means of addingto the action (6) of an arbitrary ldquonondynamicalrdquo constantterm which depends only on topology 119878 rarr 119878 + Δ119878 (119873(119896))

(eg a topological Euler term) The multiplier 1198850(119873) defines

the simplest measure for topologies Now by means of usingthe expression (32) and (33) we find the two-point Greenfunction in the form

119866 (119896) =119873 (119896)

1198962 + 1198982 (35)

where 119873(119896) is the cutoff function (the mean bias) whichis given by (We recall that in this integral contribute onlylimiting topologies in which density of wormholes diverges(119899 rarr infin))

119873(119896) =1

119885total (0)int [119863119873] 119890

minus119868(119873)119873(119896) (36)

At the present stage we still cannot evaluate the exactform for the cutoff function 119873(119896) in virtue of the ambiguityof Δ119878(119873(119896)) pointed out Such a term may include twoparts First part Δ

1119878 describes the proper dynamics of

wormholes and should be considered separately Indeedin general wormholes are dynamical self-gravitating objectswhich require considering the gravitational contribution tothe action Some part of such a contribution (mean curvatureinduced by wormholes) is discussed in Section 6 Howeversince a wormhole represents an extended nonlocal objectit possesses a rather complex dynamics and this problemrequires the further investigation The second part Δ

2119878 may

describe ldquoexternal conditionsrdquo (eg an external classicalfield in (33)) for the mean topology Actually the last termcan be used to prescribe an arbitrary particular value forthe cutoff function 119873(119896) = 119891(119896) Indeed the ldquoexternalconditionsrdquo can be accounted for by adding the term Δ

2119878 =

(120582119873) = int 120582(119896)119873(119896)1198894119896 where 120582(119896) plays the role of a

specific chemical potential which implicitly depends on 119891(119896)

through the equation

119891 (119896) =1

119885total (120582 0)int [119863119873] 119890

(120582119873)minus119868(119873)119873(119896) (37)

From (33) we see that the role of such a chemical potentialmay play the external current 120582(119896) = minus(12)(119871

4(2120587)

4)sdot

1198660(119896)|119869

ext119896|2 or equivalently an external classical field 120593

ext=

1198660(119896)119869

ext119896 In quantum field theory such a term leads merely

to a renormalization of the cosmological constant By otherwords the mean topology (ie the cutoff function or meandistribution of wormholes) is driven by the cosmologicalconstant Λ and vice versa

4 Topological Bias as a Projection Operator

By the construction the topological bias 119873(119909 1199091015840) plays the

role of a projection operator onto the space of functions (asubspace of functions on 119877

4) which obey the proper bound-ary conditions at throats of wormholes This means thatfor any particular topology (for a set of wormholes) thereexists the basis 119891

119894(119909) in which it takes the diagonal form

119873(119909 1199091015840) = sum119873

119894119891119894(119909)119891

lowast

119894(119909

1015840) with eigenvalues 119873

119894= 0 1

(since 1198732

119894= 119873

119894) In this section we illustrate this simple fact

(which is probably not obvious for readers) by the explicitconstruction of the reference system for a single wormholewhen physical functions become (due to the boundaryconditions) periodic functions of one of coordinates

Indeed consider a single wormhole with parameters 120585(ie 120585 = (119886 119877

+ 119877

minus) where 119886 is the throat radius and 119877

plusmn

are positions of throats in space (In general there existsan additional parameter Λ120572

120573which defines a rotation of one

of throats before gluing However it does not change thesubsequent constructionThere always exists a diffeomorphicmap of coordinates 1199091015840

= ℎ(119909) which sets such a matrix tounity)) Consider now a particular solution120601

0to the equation

Δ1206010= 0 (harmonic function) for 1198774 in the presence of the

wormhole which corresponds to the situation when throatspossess a unit chargemass but those have the opposite signsNow define the family of lines of force 119909(119904 119909

0) which obey

the equation 119889119909119889119904 = minusnabla1206010(119909) with initial conditions 119909(0) =

1199090 Physically such lines correspond to lines of force for a

two charged particles in positions 119877plusmn with charges plusmn1 Wenote that all points which lay on the trajectory 119909(119904 119909

0) may

be taken as initial conditions and they define the same lineof force with the obvious redefinition 119904 rarr 119904 minus 119904

0 By

other words we may take as a new coordinates the parameter119904 and portion of the coordinates orthogonal to the familyof lines 119909

perp

0 Coordinates 119909

perp

0can be taken as laying in the

hyperplane 1198773 which is orthogonal to the vector 119889 = minusminus

+

and goes through the point 0

= (minus+

+)2 (Instead

of the construction used here one may use also anothermethod Indeed consider two point charges and then thefunction 120601

0= 1(119909 minus 119909

+)2minus 1(119909 minus 119909

minus)2 can be taken as a

new coordinate Wormhole appears when we identify (glue)surfaces 120601

0= plusmn120596 We point out that such surfaces are not

spheres though they reduce to spheres in the limit |119909+minus

119909minus| rarr infin or 120596 rarr infin)Let 119904plusmn(119909perp

0) be the values of the parameter 119904 at which

the line intersects the throats 119877plusmn Then instead of 119904 we mayconsider a new parameter 120579 as 119904(120579) = 119904

minus+ (119904

+minus 119904

minus)1205792120587

so that when 120579 = 0 2120587 the parameter 119904 takes the values


119904 = 119904minus 119904+ respectivelyThe gluing procedure at throatsmeans

merely that we identify points at 120579 = 0 and 120579 = 2120587 and allphysical functions in the space 119877

4 with a single wormhole120585 become periodic functions of 120579 Thus the coordinatetransformation 119909 = 119909(120579 119909

perp

0) gives the map of the above space

onto the cylinder with a specific metric 1198891198972 = (119889(120579 119909perp

0))

2

=

119892120572120573119889119910

120572119889119910

120573 (where 119910 = (120579 119909perp

0)) whose components are also

periodic in terms of 120579 Now we can continue the coordinatesto thewhole space1198774 (to construct a cover of the fundamentalregion 120579 isin [0 2120587]) simply admitting all valuesminusinfin lt 120579 lt +infin

this however requires to introduce the bias

1

radic119892120575 (120579 minus 120579

1015840) 997888rarr 119873(120579 minus 120579

1015840) =

+infin

sum

119899=minusinfin

1

radic119892120575 (120579 minus 120579

1015840+ 2120587119899)

(38)

since every point and every source in the fundamental regionacquires a countable set of images in the nonphysical region(inside of wormhole throats) Considering now the Fouriertransforms for 120579 we find

119873(119896 1198961015840) =

+infin

sum

119899=minusinfin

120575 (119896 minus 119899) 120575 (119896 minus 1198961015840) (39)

We point out that the above bias gives the unit operator in thespace of periodic functions of 120579 From the standpoint of allpossible functions on 1198774 it represents the projection operator

2= (120585) (taking an arbitrary function 119891 we find that upon

the projection 119891119873= 119891 119891

119873becomes a periodic function of

120579 that is only periodic functions survive)The above construction can be easily generalized to the

presence of a set of wormholes In the approximation of adilute gas of wormholes we may neglect the influence ofwormholes on each other (at least there always exists a suf-ficiently smooth map which transforms the family of linesof force for ldquoindependentrdquo wormholes onto the actual lines)Then the total bias (projection) may be considered as theproduct

119873total (119909 1199091015840) = int(prod

119894

radic1198921198941198894119910119894)119873(120585

1 119909 119910

1)

times 119873 (1205852 119910

1 119910

2) sdot sdot sdot 119873 (120585

119873 119910

119873minus1 119909

1015840)

(40)

where 119873(120585119894 119909 119909

1015840) is the bias for a single wormhole with

parameters 120585119894 Every such a particular bias119873(120585

119894 119909 119909

1015840) realizes

projection on a subspace of functions which are periodic withrespect to a particular coordinate 120579

119894(119909) while the total bias

gives the projection onto the intersection of such particularsubspaces (functions which are periodic with respect to everyparameter 120579

119894)

5 Cutoff

The projective nature of the bias operator 119873(119909 1199091015840) allows us

to express the cutoff function 119873(119896) via dynamic parametersof wormholes Indeed consider a box 1198714 in 119877

4 and periodic

boundary conditions which gives 119896 = 2120587119899119871 (in final expres-sions we consider the limit 119871 rarr infin which gives sum

119896rarr

(1198714(2120587)

4) int 119889

4119896) And let us consider the decomposition for

the integration measure in (33) as

119868 = 1198680+sum120582

1(119896)119873 (119896) +

1

2sum120582

2(119896 119896

1015840)119873 (119896)119873 (119896

1015840) + sdot sdot sdot

(41)

where 1205821(119896) includes also the contribution from 119885

0(119896) We

point out that this measure plays the role of the action for thebias119873(119896) Indeed the variation of the above expression givesthe equation of motions for the bias in the form

sum

1198961015840

1205822(119896 119896

1015840)119873 (119896

1015840) = minus120582

1(119896) (42)

which can be found by considering the proper dynamics ofwormholes We however do not consider the problem of thedynamic description of wormholes here and leave this forthe future research Moreover we may expect that in the firstapproximation one may retain the linear term only Indeedthis takes place when119873(119896) is not a dynamic variable or if wetake into account that 119873(119896) is a collective variable Then theprojective nature of the bias 119873(119896) = 0 1 means that it canbe phenomenologically expressed via some Fermionic ghostfield Ψ(119896) (eg 119873(119896 119896

1015840) = Ψ(119896)Ψ

+(119896

1015840)) where the negative

and positive frequency parts of the operator Ψ(119896) obey theanticommutation relationsΨ+

(119896)Ψ(1198961015840)+Ψ(119896

1015840) Ψ

+(119896) = 120575(119896minus

1198961015840) In the absence of ghost particles Ψ(119896)|0⟩ = 0 we get

119873(119896 1198961015840) = 120575(119896 minus 119896

1015840) that is 119873(119896) = 1 and wormholes

are absent In the term of the ghost field the action becomes119868(119873) = 119868

0+ (Ψ

1Ψ) + sdot sdot sdot Therefore in the leading approx-

imation equations of motion take the linear form 1Ψ = 0

Thus taken into account that 119873(119896) = 0 1 (1198732= 119873) we

find

119873(119896) =1

119885total (119896)sum

119873=01

119890minus1205821

(119896)119873(119896)119873(119896) =

119890minus1205821

(119896)

1 + 119890minus1205821(119896) (43)

The simplest choice gives merely 1205821(119896) = minussum ln119885

0(119896)

where the sum is taken over the number of fields and 1198850(119896)

is given by 1198850(119896) = radic120587(1198962 + 1198982) In the case of a set

of massless fields we find 119873(119896) = 119885(119896)(1 + 119885(119896)) where119885(119896) = (radic120587119896)

120572 and 120572 is the effective number of degreesof freedom (the number of boson minus fermion fields) Toensure the absence of divergencies one has to consider thenumber of fields 120572 gt 4 [23] However such a choice givesthe simplest estimate which in general cannot be correctIndeed while its behavior at very small scales (ie whenexceeding the Planckian scales 119885(119896) ≲ 1 and 119873(119896) = 119885(119896))may be physically accepted since it produces some kind of acutoff on the mass-shell 1198962 + 119898

2rarr 0 it gives the behavior

119873(119896) rarr 1 which is merely incorrect (eg from (27) we seethat the true behavior should be119873(119896) rarr const lt 1)

One may expect that the true cutoff function has a muchmore complex behavior Indeed some theoretical models inparticle physics (eg string theory) have the property tobe lower-dimensional at very small scales The mean cutoff


119873(119896) gives the natural tool to describe a scale-dependantdimensional reduction [24 25] In fact this function definesthe spectral number of modes in the interval between 119896 and119896 + 119889119896 as

int119873 (119896)1198894119896

(2120587)4= int

119873 (119896) 1198964

(2120587)2

119889119896

119896 (44)

Hence we can define the effective spectral dimension 119863 ofspace as follows

1198964119873(119896) sim 119896

119863 (45)

From the empirical standpoint the dimension 119863 = 4 isverified at laboratory scales only while the rigorous tool todefine the spectral density of states (or the mean cutoff) cangive the lattice quantum gravity for example see [26 27]and references therein And indeed the spectral dimensionfor nonperturbative quantum gravity defined via Euclideandynamical triangulations was calculated recently in [28] Itturns out that it runs from a value of119863 = 32 at short distanceto 119863 = 4 at large distance scales We also point out that allobserved dark matter phenomena can be explained by thefractal dimension119863 asymp 2 starting from scales 119871 ≳ (1 divide 5)Kpcfor example [29ndash33]

6 Cosmological Constant

Let us consider the total Euclidean action [17]

119868119864= minus

1

16120587119866int (119877 minus 2Λ

0)radic119892119889

4119909 minus int119871

119898radic1198921198894119909 (46)

The variation of the above action leads to the Einstein equa-tions

119877119886119887minus1

2119892119886119887119877 + 119892

119886119887Λ

0= 8120587119866119879

119886119887 (47)

where 119879119886119887= (12)(119892)

minus12(120575119871

119898120575119892

119886119887) is the stress energy ten-

sor and Λ0is a naked cosmological constant In cosmology

such equations are considered from the classical standpointwhich means that they involve characteristic scales ℓ ≫ ℓplHowever the presence of virtual wormholes at Planckianscales defines some additional contribution in both parts ofthese equations which can be adsorbed into the cosmologicalconstant Therefore the total cosmological constant can bedefined as

Λ tot = Λ0+ Λ

119898+ Λ

119877= Λ

0+ 2120587119866 ⟨119879⟩ +

1

4⟨119877⟩119908 (48)

where ⟨119879⟩ is the energy of zero-point fluctuations (It includesalso the contribution of zero-point fluctuations of gravitons)That is themean vacuum value (we recall that in the standardQFTΛ

119898is infinite while wormholes form a finite value) and

⟨119877⟩119908

= Λ119877is a contribution of wormholes into the mean

curvature due to gluing (1)

61 Contribution of Virtual Wormholes into Mean CurvatureConsider a single wormhole whose metric is given by (1)

1198891199042= ℎ

2(119903)120575

120572120573119889119909

120572119889119909

120573 then the components of the Riccitensor are (We are much obliged to the Referee who pointedout to the subtleties when working with a step function in themetric)

119877120572120573

= 7 minus (ℎ1015840

ℎ)

1015840

(120575120572120573

+ (119873 minus 2) 119899120572119899120573)

minus1

119903

ℎ1015840

ℎ((119873 minus 2) Δ

120572120573+ 120575

120572120573(119873 minus 1))

minus (ℎ1015840

ℎ)

2

(119873 minus 2) Δ120572120573

(49)

where 119899]= 119909

]119886 is the unite normal vector to the throat

surface Δ120572120573

= 120575120572120573

minus 119899120572119899120573 ℎ1015840 = 120597ℎ120597119903 and119873 is the number

of dimensions The curvature scalar is

119877 = minus(119873 minus 1)

ℎ2[2

ℎ10158401015840

ℎ+ 2

1

119903

ℎ1015840

ℎ(119873 minus 1) + (

ℎ1015840

ℎ)

2

(119873 minus 4)]

(50)

Then substituting ℎ = 1 + (1198862119903

2minus 1)120579 in the above equation

we find

minusℎ4

119873 minus 1119877 = 4ℎ

1198862

1199033120575 +

2ℎ

119903119873minus1(119903

119873minus1120582)

1015840

+ (119873 minus 4)(1205822minus 4120579

1198862

1199033120582)

+ 4 (119873 minus 4)1198862

1199034120579 (120579 minus 1)

(51)

where 120579 is a smooth function which only in a limit becomesa step function 120582 = (1 minus 119886

2119903

2)120575 and 120575 = minus120579

1015840 In the limit120579 rarr 120579(119886 minus 119903) we find 120575 rarr 120575(119903 minus 119886) and the last two terms in(51) are negligible as compared to the first two terms while infour dimensions the last two terms vanishThen the curvatureis concentrated on the throat of the wormhole where 120579 differsfrom the step function In the limit of a vanishing throat sizewe have ℎ rarr 1 at the throat and we get

minus119877 =4 (119873 minus 1)

119886120575 (119903 minus 119886) +

2 (119873 minus 1)

119903119873minus1(119903

119873minus3(119903

2minus 119886

2) 120575)

1015840

(52)

In general all the terms together in (51) and analogous termsin the Ricci tensor provide that the Bianchi identity holdsand the energy is conserved [19] Therefore none of themcan be dropped out However it is easy to see the leadingcontribution to the integral over space comes from the firstterm only and in the limit of a vanishing throat size we findfor a single wormhole

1

4int119877radic119892119889

4119909 = minus6120587

21198862 (53)


In the case of a set of wormholes (24) we find

1

4int119877radic119892119889

4119909 = minus6120587

2sum

119895

1198862

119895

= minus121205872int119899 (119886) 119886

2119889119886 119889

4119909 = intΛ

1198771198894119909

(54)

where 2119899(119886) = int 119899(119886 1199030)119889119903

0is the density of wormhole

throats with a fixed value of the throat size 119886 This definesthe contribution to the cosmological constant from the meancurvature as

Λ119877= minus12120587

2int119899 (119886) 119886

2119889119886 lt 0 (55)

We see that this quantity is always negative

62 Stress Energy Tensor In this section we consider thecontribution from matter fields In the case of a scalar fieldthe stress energy tensor has the form

minus119879120572120573

(119909) = 120597120572120593120597

120573120593 minus

1

2119892120572120573

(120597120583120593120597

120583120593 + 119898

21205932) (56)

Then the mean vacuum value of the stress energy tensor canbe obtained directly from the two-point green function (32)(35) as

minus ⟨119879120572120573

(119909)⟩

= lim1199091015840

rarr119909

(1205971205721205971015840

120573minus1

2119892120572120573

(1205971205831205971015840

120583+ 119898

2)) ⟨119866 (119909 119909

1015840 120585)⟩

(57)

By means of using the Fourier transform 119866(119909 1199091015840 120585) =

int 119890minus119894119896(119909minus119909

1015840

)119866(119896 120585) (119889

4119896(2120587)

4) and the expressions (32) (35)

we arrive at

⟨119879120572120573

(119909)⟩ =1

4119892120572120573

int(1 +119898

2

1198962 + 1198982)119873 (119896)

1198894119896

(2120587)4 (58)

where the property int 119896120572119896120573119891(119896

2)119889

4119896 = (14)119892

120572120573int 119896

2119891(119896

2)119889

4119896

has been used and 119873(119896) = ⟨119873(119896 120585)⟩ is the cutoff function(36)

For the sake of simplicity we consider the massless caseThen by the use of the cutoff 119873(119896) = 120587

1205722(120587

1205722+ 119896

120572) from

the previous section we get the finite estimate (120572 gt 4 is theeffective number of the field helicity states)

Λ119898=2120587119866sumint

1205871205722

1205871205722+119896120572

1198894119896

(2120587)4=120587119866

4Γ (

120572 minus 4

120572) Γ (

4

120572) sim 1

(59)

where the sum is taken over the number of fields Since theleading contribution comes here from very small scales wemay hope that this value will not essentially change if thetrue cutoff function changes the behavior on the mass-shellas 119896 rarr 0 (eg if we take 120582

1(119896) = minussum ln119885

0(119896) + 120575120582(119896) with

120575120582(119896) ≪ ln1198850(119896) as 119896 ≫ 1)

To understand how wormholes remove divergencies itwill be convenient to split the bias function into two parts119873(119896 120585) = 1 + 119887(119896 120585) where 1 corresponds to the standardEuclidean contribution while 119887(119896 120585) is the contribution ofwormholes The first part gives the well-known divergentcontribution of vacuumfield fluctuations 8120587119866⟨1198790

120572120573⟩ = Λ

lowast119892120572120573

with Λlowast

rarr +infin while the second part remains finite forany finite number of wormholes and due to the projectivenature of the bias described in the previous section itpartially compensates (reduces) the value of the cosmologicalconstant that is 8120587119866⟨Δ119879

120572120573⟩ = 120575Λ119892

120572120573 where 120575Λ =

sum119873120588119873120575Λ(119873) and 120575Λ(119873) is a negative finite contribution of

a finite set of wormholesConsider now the particular distribution of virtual

wormholes (24) and evaluate their contribution tothe cosmological constant which is given by 120575Λ(119873) =

2120587119866int 119887(119896) (1198894119896(2120587)

4) = 2120587119866119887total(0) Then from the ex-

pressions (22) and (24) we get

119887total (0) = minus119899

4120587411988631199033

0

int(1 minus1198862

1198772

minus

)120575 (119877+minus 119886)

times 120575 (1003816100381610038161003816119877+

minus 119877minus

1003816100381610038161003816 minus 1199030) 119889

4119877minus1198894119877+

(60)

which gives

119887total (0) = minus119899(1 minus 119891(119886

1199030

)) (61)

where

119891(119886

1199030

) =2

120587int

120587

0

1198862sin2

120579119889120579

1198862 + 21198861199030cos 120579 + 119903

2

0

(62)

For 1198861199030≪ 1 (we recall that by the construction 119886119903

0le 12)

this function has the value 119891(1198861199030) asymp 119886

2119903

2

0 Thus for the

contribution of wormholes we find

120575Λ119898= minus2120587119866int119899 (119886 119903

0) (1 minus 119891(

119886

1199030

))119889119886 1198891199030

= minus2120587119866119899 (1 minus ⟨119891⟩)

(63)

63 Vacuum Value of the Cosmological Constant From theabove expression we see that to get the finite value of thecosmological constant Λ

119898= Λ

lowast+ 120575Λ

119898lt infin one should

consider the limit 119899 rarr infin (infinite density of virtualwormholes) which requires considering the smaller andsmaller wormholes From the other handwe have the obviousrestriction int 2119899(119886 119903

0)(120587

22)119886

4119889119886 119889119903

0lt 1 where (120587

22)119886

4

is the volume of one throat (wormholes cannot cut morethan the total volume of space) (We also point out that inremoving divergencies the leading role plays the zero-pointenergy Indeed 120575Λ

119898sim minus2120587119866119899 while the mean curvature has

the order Λ119877sim minus119886

2119899 and for 119886 ≪ ℓpl we have 120575Λ119898

≫ Λ119877

Moreover in the limit 119899 rarr infin one gets 119886 rarr 0 and thereforeΛ

119877120575Λ

119898rarr 0)Therefore in the leading order it seems to be

sufficient to retain point-like wormholes only (ie consider


the limit 119886 rarr 0) Then instead of (24) we may assume thevacuum distribution of virtual wormholes in the form

119873119865 (119886119883) =1

1198862120575 (119886) ] (119883) (64)

where ](119883) = int 1198862119873119865(119886119883)119889119886 and int(1119886

2)](119883)119889

4119883 =

119899 rarr infin has the meaning of the infinite density of point-like wormholes while ] sim 119886

2119899 remains a finite In this case

the volume cut by wormholes vanishes int 2119899(12058722)119886

4119889119886 119889119903

0=

1198862int ](119883)119889

4119883 rarr 0 and the rarefied gas approximationworks

wellThis defines the bias and themean cutoff (here we definethe Fourier transform ](119896) = int ](119883)119890

1198941198961198831198894119883) as

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (65)

The contribution to themean curvature (55) can be expressedvia the same function ](119896) as

Λ119877= minus12120587

2int119899 (119886 119903

0) 119886

2119889119886 119889119903

0

= minus121205872int ] (119883) 119889

4119883 = minus12120587

2] (0)

(66)

Thus for the total cosmological constant we get the expres-sion

Λ tot = Λ0

+ 2120587119866int(1minus4120587

2

1198962(] (0)minus] (119896)))

1198894119896

(2120587)4minus12120587

2] (0)

(67)

We stress that all these terms should be finite Indeed all dis-tributions of virtual wormholes ](119896) which lead to an infinitevalue of Λ tot are suppressed in (5) by the factor sim119890minusintΛ tot119889

4

119909while the minimal value is reached when wormholes cut allof the volume of space and the action is merely 119878 = 0(Frankly speaking this statement is not rigorous At first lookthe two last terms in (67) are independent and one may tryto take ](0) an arbitrary big If this were the case then theaction would not possess the minimum at all However ](0)cannot be arbitrary big since it will violate the rarefied gasapproximation and the linear expression (67) brakes downMoreover fermions give here a contribution of the oppositesign The rigorous investigation of this problem requires thefurther studying and we present it elsewhere) We also pointout that here we considered the real scalar field as the mattersource while in the general case the stress energy tensorshould include all existing Bose and Fermi fields (Fermi fieldsgive a negative contribution to Λ

119898)

The value of Λ0looks like a free parameter which in

quantum gravity runs with scales [27] However at largescales its asymptotic value may be uniquely fixed by thesimple arguments as follows Indeed in quantum field theoryproperties of the ground state (vacuum) change when weimply an external classical fields The same is true for thedistribution of virtualwormholes ](119896 119869) for example see (33)and (37) and therefore Λ tot = Λ tot(119869) which we describe

in the next subsection We recall that in gravity the role ofthe external current plays the stress energy tensor of matterfields 119869 = 119879

119886119887 However one believes that in the absence

of all classical fields the vacuum state should represent themost symmetric (Lorentz invariant) state which in our casecorresponds to the Euclidean space In order to be consistentwith the Einstein equations this requires Λ tot(119869 = 0) = 0

which uniquely fixes the value of Λ0in (67) We point out

that from somewhat different considerations such a choicewas advocated earlier in [15 34 35]

64 Vacuum Polarization in an External Field Consider nowtopology fluctuations in the presence of an external currentIn the presence of an external current 119869ext the distribution ofvirtual wormholes changes ](119896 119869) = ](119896) + 120575](119896 119869) Indeedin (33) for the case of a weak external field the contributionof the external current into the action can be expanded asexp(minus119881(119869)) ≃ 1 minus 119881 where

119881 = minus1

2int 119869 (119909) 119866 (119909 119910) 119869 (119910) 119889

4119909 119889

4119910

= minus1

2

1198714

(2120587)4int119866

0(119896)119873 (119896)

10038161003816100381610038161198691198961003816100381610038161003816

2

1198894119896

(68)

Then using (36) we find119873(119896 119869) = 119873(119896 0) + 120575119873(119896 119869) where

120575119873 (119896 119869) = 120575119887 (119869) ≃ minus1

2

1198714

(2120587)4int120590

2(119896 119901) 119866

0(119901)

10038161003816100381610038161003816119869119901

10038161003816100381610038161003816

2

1198894119901

(69)

is the bias related to an additional distribution of virtualwormholes and 120590

2(119896 119901) = Δ119873lowast(119896)Δ119873(119901)

1205902(119896 119901) =

1

119885total (0)int [119863119873] 119890

minus119868(119873)Δ119873

lowast(119896) Δ119873 (119901) (70)

defines the dispersion of vacuum topology fluctuations (hereΔ119873 = 119873 minus 119873) The exact definition of 1205902

(119896 119901) requires thefurther development of a fundamental theory In particular itcan be numerically calculated in lattice quantum gravity [28]However it can be shown that at scales 119896 119901 ≫ 119896pl it reducesto 1205902

(119896 119901) rarr 1205902

119896120575(119896 minus 119901) and therefore

120575119887 (119869) = minus1205902

119896

41205872

21198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(71)

Now comparing this function with (23) we relate the addi-tional distribution of virtual wormholes and the externalclassical field as

41205872

1198962(120575] (0 119869) minus 120575] (119896 119869)) =

1

21205902

119896

41205872

1198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(72)

where 120575](119896 119869) = int 1198862120575119873119865(119886 119896)119889119886 We point out that the

above expression does not define the value 120575](0 119869) whichrequires an additional considerationMoreover in general theexternal field 119869 does not possess a symmetry and therefore thecorrection ⟨120575119879

120572120573(119909)⟩ does not reduce to a single cosmological


constant However such corrections always violate the aver-aged null energy condition [36 37] and may be considered assome kind of dark energy or by other words it represents anexotic matter Some portion of dark energy still has the formof the cosmological constant which defines a nonvanishingpresent day value (we recall that fermions give a contributionof the opposite sign)

120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)

where ⟨120575](119896 119869)⟩ denotes an averaging over rotationsThe only unknown parameter in (72) is the dispersion

1205902

119896which defines the intensity of topology fluctuations in

the vacuum It has also the sense of the efficiency coefficientwhich defines the portion of the energy of the external fieldspent on the formation of additional wormholes Thoughthe evaluation of 1205902

119896requires the further development of a

fundamental theory one may expect that 1205902

119896= 119873(119896)(1 minus

119873(119896)) where 119873(119896) is the mean cutoff It is expected that119873(119896) rarr 0 as 119896 ≫ 119896pl and119873(119896) rarr 119873

0le 1 This means that

120590 rarr 0 as 119896 ≫ 119896pl and 120590 rarr 1205900≪ 1 as 119896 ≪ 119896pl while it takes

the maximum value 120590max sim 1 at Planckian scales 119896 sim 119896pl Byotherwords themost efficient transmission of the energy intowormholes takes place for wormholes of the Planckian sizeIn the case when external classical fields have characteristicscales 120582 = 2120587119896 ≫ ℓpl in (72) the efficiency coefficient 1205902

119896and

the cutoff 119873(119896) become constant 120590 ≃ 1205900 119873(119896) ≃ 119873

0 while

their ratiomay be estimated as 1205721205902

0119873

0= ΩDEΩ119887

where 120572 isthe effective number of fundamental fields which contributeto 120575Λ tot and ΩDE Ω119887

are dark energy and baryon energydensities respectively According to the modern picture thisratio gives ΩDEΩ119887

asymp 075005 = 15 while 1205902

0119873

0sim 1

(as 1198730≪ 1) and therefore this defines the estimate for the

effective number of fundamental fields (helicity states) as 120572 sim

15

65 Speculations on the Formation of Actual WormholesAs we already pointed out the additional distribution ofvirtual wormholes (72) reflects the symmetry of externalclassical fields and therefore it forms a homogeneous andisotropic background and perturbations We recall that vir-tual wormholes represent an exotic form of matter In theearly Universe such perturbations start to develop and mayform actual wormholes The rigorous description of sucha process represents an extremely complex and interestingproblemwhich requires the further study Some aspects of thebehavior of the exotic density perturbations were consideredin [7] while the simplest example of the formation of awormhole-type object was discussed recently by us in [38]Therefore we may expect that some portion of such a form ofdark energy is reserved now in actual wormholes which weconsider in the next section

7 Dark Energy from Actual Wormholes

Consider now the contribution to the dark energy fromthe gas of actual wormholes Unlike the virtual wormholesactual wormholes do exist at all times and therefore a singlewormhole can be viewed as a couple of conjugated cylinders1198793

plusmn= 119878

2

plusmntimes 119877

1 So that the number of parameters of an actualwormhole is less 120578 = (119886 119903

+ 119903

minus) where 119886 is the radius of 1198782

plusmn

and 119903plusmnisin 119877

3 is a spatial part of 119877plusmn

Actual wormholes also produce two kinds of contributionto the dark energy One comes from their contribution tothe mean curvature which corresponds to an exotic stressenergy momentum tensor Such a stress energy momentumtensor reflects the dark energy reserved by additional virtualwormholes discussed in the previous section Such energyis necessary to support actual wormholes as a solution tothe Einstein equations The second part comes from vacuumpolarization effects by actual wormholes The considerationin the previous section shows that for macroscopic worm-holes the second part has the order ⟨Δ119879

120572120573⟩ sim 8120587119866119899 and

is negligible as compared to the curvature 119877 sim 1198862119899 (since

macroscopic wormholes have throats 119886 ≫ ℓpl) Howeverfor the sake of completeness and for methodological aims wedescribe it as well

For rigorous evaluation of dark energy of the secondtype we first have to find the bias 119887

1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3

minus) There are many papers

treating different wormholes in this respect (eg see [36 37]and references therein) However in the present paper for anestimation we shall use a more simple trick

71 Beads of Virtual Wormholes (Quantum Wormhole)Indeed instead of the cylinders 1198793

plusmnwe consider a couple of

chains (beads of virtual wormholes 1198793

plusmnrarr cup

1198991198783

plusmn119899) Such an

idea was first suggested in [39] and one may call such anobject as quantum wormhole Then the bias can be writtenstraightforwardly

1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)

where 119877plusmn119899

= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the

step We may expect that upon averaging over the position1199050isin [minusℓ ℓ] the bias for the beads will reproduce the bias for

cylinders1198793

plusmn(at least it looks like a very good approximation)

We point out that the averaging out (12ℓ) intℓ

minusℓ119889119905

0and the

sum sum+infin

119899=minusinfinreduces to a single integral (12ℓ) intinfin

minusinfin119889119905 of the

zero term in (74) And moreover the resulting total biascorresponds merely to a specific choice of the distributionfunction 119865(120585) in (21) Namely we may take

119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)


where119877plusmn= (119905

plusmn 119903

plusmn) and119891(119904 119886) is the distribution of cylinders

which can be taken as (119899 is 3-dimensional density)

119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)

Using the normalization condition int119873119865(120585)119889120585 = 119873 we findthe relation 119873 = (12ℓ)119899119881 = 119899119881 where 119899 is a 4-dimensional density of wormholes and 1(2ℓ) is the effectivenumber of wormholes on the unit length of the cylinder (iethe frequency with which the virtual wormhole appears atthe positions 119903

plusmn) This frequency is uniquely fixed by the

requirement that the volume which cuts the bead is equal tothat which cuts the cylinder (43)1205871198863 = (120587

22)119886

4(12ℓ) (ie

2ℓ = (31205878)119886 and 119899 = (83120587119886)119899) Thus we can use directlyexpression (23) and find (compare to (25))

119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)

where 119896 = (1198960 k) Here the first term merely coincides with

that in (25) and therefore it gives the contribution to thecosmological constant 120575Λ(8120587119866) = minus1198994 = minus2119899(3120587119886)while the second term describes a correction which does notreduce to the cosmological constant and requires a separateconsideration

72 Stress Energy Tensor From (57) we find that the stressenergy tensor

minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)

reduces to the two functions

11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582

2= 120583 and these functions are

1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)

By means of the use of the spherical coordinates 11989620119896

2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)

Thus finally we find

120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)

which upon the continuation to the Minkowsky space givesthe equation of state in the form (An arbitrary gas ofwormholes splits in fractions with a fixed 119886 and 119903

0)

119901 = minus(1 minus4

3(119886

1199030

)

2

) 120576 (85)

which in the case when 1198861199030≪ 1 behaves like a cosmological

constant However when 119886 ≫ ℓpl such a constant is extremelysmall and can be neglected while the leading contributioncomes from the mean curvature

73 Mean Curvature In this subsection we consider theMinkowsky spaceThen the simplest actual wormhole can bedescribed by the metric analogous to (1) for example see [7]

1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903

2minus 1) To avoid problems

with the Bianchi identity and the conservation of energy thestep function should be also smoothed as in (1) The stressenergy tensor which produces such a wormhole can be foundfrom the Einstein equation 8120587119866119879

120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both

regions 119903 gt 119886 and 119903 lt 119886 represent portions of the ordinaryflat Minkowsky space and therefore the curvature is 119877119896

119894equiv 0

However on the boundary 119903 = 119886 it has the singularity Sincethe metric (86) does not depend on time we find

1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909

120572119903 is the outer normal to the throat 1198782

and 120582120573

120572are additional terms (eg see (49) and (51)) which

in the leading order are negligible upon averaging over someportion of space Δ119881 ≳ 119886

3 In the case of a set of wormholesthis gives in the leading order

1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)

where 119886119894is the radius of a throat and 119903

119894is the position of the

center of the throat in space and 119899120572

119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|

In the case of a homogeneous and isotropic distribution ofsuch throats we find 119877

120573

120572= (13)119877120575

120573

120572(averaging over spatial

directions gives ⟨119899120572119899120573⟩ = (13)120575

120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)


where 119879 stands for the trace of the stress energy momentumtensor which one has to add to the Einstein equations tosupport such awormhole It is clear that such a source violatesthe weak energy condition and therefore it reproduces theformof dark energy (ie119879 = 120576+3119901 lt 0) If the density of suchsources (and resp the density of wormholes) is sufficientlyhigh then this results in the observed [10ndash14] accelerationof the scale factor for the Friedmann space as sim 119905

120572 with120572 = 21205763(120576 + 119901) = 2120576(2120576 + (120576 + 3119901)) gt 1 for examplesee also [40ndash42] In terms of the 4-dimensional density ofwormholes 119899 = (83120587119886)119899 we get 119877 sim 119886

2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl

and therefore the leading contribution indeed comes fromthe mean curvature

8 Estimates and Concluding Remarks

Now consider the simplest estimates Actual wormholes seemto be responsible for the dark matter [7 9] Therefore to getthe estimate to the number density of wormholes is ratherstraightforward First wormholes appear at scales when darkmatter effects start to display themselves that is at scales ofthe order 119871 sim (1 divide 5)Kpc which gives in that range thenumber density

119899 sim1

1198713sim (3 divide 0024) times 10

minus65cmminus3 (90)

The characteristic size of throats can be estimated from (89)120576DE sim (119866)

minus1119899119886 Since the density of dark energy is 120576DE1205760 =

ΩDE sim 075 where 1205760is the critical density then we find the

estimate

119886 sim2

3(1 divide 125) times 10

minus3119877⊙ΩDEℎ

2

75 (91)

where119877⊙is the Solar radius ℎ

75= 119867(75 km(secMpc)) and

119867 is the Hubble constant We also recall that the backgrounddensity of baryons 120576

119887generates a nonvanishingwormhole rest

mass119872119908= (43)120587119886

31198773120576119887(where119877(119905) is the scale factor of the

Universe and therefore 119872119908remains constant) for example

see [7] It produces the dark matter density related to thewormholes as 120576DM ≃ 119872

119908119899 The typical mass of a wormhole

119872119908is estimated as

119872119908sim 1 7 times (1 divide 125) times 10

2119872

⊙ΩDMℎ

2

75 (92)

where 119872⊙is the Solar mass We point out that this mass

has not the direct relation to the parameters of the gas ofwormholes However it defines the moment when wormholethroats separated from the cosmological expansion Theabove estimate shows that if wormholes form due to thedevelopment of perturbations in the exotic matter then thisprocess should start much earlier than the formation ofgalaxies

Thus we see that virtual wormholes should indeed leadto the regularization of all divergencies in QFT which agreeswith recent results [28] Therefore they form the local finitevalue of the cosmological constant In the absence of externalclassical fields such a value should be exactly zero at macro-scopic scales A some nonvanishing value for the cosmolog-ical constant appears as the result of vacuum polarization

effects in external fields Indeed external fields form anadditional distribution of virtual wormholes which possessan exotic stress energy tensor (some kind of dark energy)Only some part of it forms the cosmological constant whilethe rest reflects the symmetry of external fields and possessesinhomogeneitiesWe assume that during the evolution of ourUniverse inhomogeneities in the exotic matter develop andmay form actual wormholes Although this problem requiresthe further and more deep investigation we refer to [38]where the formation of a simplest wormhole-like object hasbeen considered In other words such polarization energyis reserved now in a gas of actual wormholes We estimatedparameters of such a gas and believe that such a gas mayindeed be responsible for both dark matter and dark energyphenomena

References

[1] J R Primack ldquoThe nature ofdark matterrdquo httparxivorgabsastro-ph0112255

[2] J Diemand M Zemp B Moore J Stadel and M CarolloldquoCusps in cold dark matter haloesrdquoMonthly Notices of the RoyalAstronomical Society vol 364 no 2 pp 665ndash673 2005

[3] E W Mielke and J A V Velez Perez ldquoAxion condensate as amodel for dark matter halosrdquo Physics Letters B vol 671 no 1pp 174ndash178 2009

[4] G Gentile P Salucci U Klein D Vergani and P Kalberla ldquoThecored distribution of dark matter in spiral galaxiesrdquo MonthlyNotices of the Royal Astronomical Society vol 351 no 3 pp 903ndash922 2004

[5] D T F Weldrake W J G de Blok and F Walter ldquoA high-resolution rotation curve of NGC 6822 a test-case for cold darkmatterrdquo Monthly Notices of the Royal Astronomical Society vol340 no 1 pp 12ndash28 2003

[6] W J G de Blok andA Bosma ldquoHigh-resolution rotation curvesof low surface brightness galaxiesrdquoAstronomy and Astrophysicsvol 385 pp 816ndash846 2002

[7] A Kirillov and E P Savelova ldquoDensity perturbations in a gas ofwormholesrdquoMonthly Notices of the Royal Astronomical Societyvol 412 no 3 pp 1710ndash1720 2011

[8] P Kanti B Kleihaus and J Kunz ldquoStable Lorentzianwormholesin dilatonic Einstein-Gauss-Bonnet theoryrdquo Physical Review Dvol 85 no 4 Article ID 044007 2012

[9] A A Kirillov and E P Savelova ldquoDark matter from a gas ofwormholesrdquo Physics Letters B vol 660 no 3 pp 93ndash99 2008

[10] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[11] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 p 565 1999

[12] J L Tonry B P Schmidt B Barris et al ldquoCosmological resultsfromhigh-z supernovaerdquoTheAstrophysical Journal vol 594 no1 p 594 2003

[13] N W Halverson E M Leitch C Pryke et al ldquoDegreeangular scale interferometer first results a measurement of thecosmic microwave background angular power spectrumrdquo TheAstrophysical Journal vol 568 no 1 p 38 2002


[14] C B Netterfield P A R Ade J J Bock et al ldquoA measurementby BOOMERANGofmultiple peaks in the angular power spec-trum of the cosmic microwave backgroundrdquo The AstrophysicalJournal vol 571 no 2 p 604 2002

[15] S Coleman ldquoWhy there is nothing rather than something atheory of the cosmological constantrdquoNuclear Physics B vol 310no 3-4 pp 643ndash668 1988

[16] I Klebanov L Susskind and T Banks ldquoWormholes and thecosmological constantrdquo Nuclear Physics B vol 317 no 3 pp665ndash692 1989

[17] S W Hawking ldquoSpacetime foamrdquoNuclear Physics B vol 114 p349 1978

[18] T P Cheng and L F Li Gauge Theory of Elementary ParticlesClarendon Press Oxford UK 1984

[19] A H Taub ldquoSpace-times with distribution valued curvaturetensorsrdquo Journal ofMathematical Physics vol 21 no 6 pp 1423ndash1431 1980

[20] A Shatsky privite communications[21] V A Fock ldquoZur Theorie des Wasserstoffatomsrdquo Zeitschrift fur

Physik vol 98 no 3-4 pp 145ndash154 1935[22] E P Savelova ldquoGas of wormholes in Euclidean quantum field

theoryrdquo httparxivorgabs12115106[23] A A Kirillov and E P Savelova ldquoOn the behavior of

bosonic systems in the presence of topology fluctuationsrdquohttparxivorgabs08082478

[24] A A Kirillov ldquoViolation of the Pauli principle and dimensionof the Universe at very large distancesrdquo Physics Letters B vol555 no 1-2 pp 13ndash21 2003

[25] A A Kirillov ldquoDark matter dark charge and the fractalstructure of the Universerdquo Physics Letters B vol 535 no 1ndash4pp 22ndash24 2002

[26] J Ambjorn J Jurkiewicz and R Loll ldquoThe spectral dimensionof the universe is scale dependentrdquo Physical Review Letters vol95 no 17 Article ID 171301 4 pages 2005

[27] E Manrique S Rechenberger and F Saueressig ldquoAsymptoti-cally safe Lorentzian gravityrdquo Physical Review Letters vol 106no 25 Article ID 251302 4 pages 2011

[28] J Laiho and D Coumbe ldquoEvidence for asymptotic safety fromlattice quantum gravityrdquo Physical Review Letters vol 107 no 16Article ID 161301 2011

[29] A A Kirillov and D Turaev ldquoOn modification of the Newtonrsquoslaw of gravity at very large distancesrdquo Physics Letters B vol 532no 3-4 pp 185ndash192 2002

[30] A A Kirillov and D Turaev ldquoThe universal rotation curveof spiral galaxiesrdquo Monthly Notices of the Royal AstronomicalSociety vol 371 no 1 pp L31ndashL35 2006

[31] A A Kirillov and D Turaev ldquoFoam-like structure of theuniverserdquo Physics Letters B vol 656 pp 1ndash8 2007

[32] A A Kirillov ldquoThe nature of dark matterrdquo Physics Letters B vol632 no 4 pp 453ndash462 2006

[33] A A Kirillov and E P Savelova ldquoAstrophysical effects of space-time foamrdquo Gravitation and Cosmology vol 14 no 3 pp 256ndash261 2008

[34] S W Hawking and N Turok ldquoOpen inflation without falsevacuardquo Physics Letters B vol 425 no 1-2 pp 25ndash32 1998

[35] Z C Wu ldquoThe cosmological constant is probably zero and aproof is possibly rightrdquo Physics Letters B vol 659 no 5 pp 891ndash893 2008

[36] R Garattini ldquoEntropy and the cosmological constant aspacetime-foam approachrdquo Nuclear Physics B vol 88 no 1ndash3Article ID 991003 pp 297ndash300 2000

[37] MVisserLorentzianWormholes SpringerNewYorkNYUSA1996

[38] A A Kirillov and E P Savelova ldquoArtificial wormholerdquohttparxivorgabs12040351

[39] E W Mielke ldquoKnot wormholes in geometrodynamicsrdquo Gen-eral Relativity and Gravitation vol 8 no 3 pp 175ndash196 1977

[40] E B Gliner ldquoAlgebraic properties of the energy-momentumtensor and vacuum-like states of matterrdquo Journal of Experimen-tal and Theoretical Physics vol 22 p 378 1966

[41] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[42] A A Linde ldquoA new inflationary universe scenario a possiblesolution of the horizon flatness homogeneity isotropy andprimordial monopole problemsrdquo Physics Letters B vol 108 no6 pp 389ndash393 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


which means that nontraversable wormholes can be used tosmooth cusps in centers of galaxies Thus it worth expectingthatwormholesmay play an important role in the explanationof the dark matter phenomenon

Saving the dark matter component ΛCDM requiresthe presence (sim70) of dark energy (of the cosmologicalconstant) Moreover there is evidence for the start of anacceleration phase in the evolution of the Universe [10ndash14]In the present paper we use virtual wormholes to estimatethe contribution of zero-point fluctuations in the value of thecosmological constant The idea to relate virtual wormholes(or baby universes) and the cosmological constant is not newin somewhat different context it was used by Coleman in [15]and developed in [16] Our basic aim is to demonstrate thatvirtual wormholes form a finite value of the energy density ofzero-point fluctuations

It is necessary to point here out to the principle differ-ence between actual and virtual wormholes The principledifference is that a virtual wormhole exists only for a verysmall period of time and at very small scales and doesnot necessarily obey to the Einstein equations It representstunnelling event and therefore the averaged null energycondition (ANEC) cannot forbid the origin of such an objectFor the future we also note that a set of virtual wormholesmay work as an actual wormhole opening thus the way foran artificial construction of wormhole-type objects in labexperiments

In the present paper we describe a virtual wormholeas follows From the very beginning we use the Euclideanapproach (eg see [17] and the standard textbooks [18])Thenthe simplest virtual wormhole is described by themetric (120572 =

1 2 3 4)

1198891199042= ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (1)

where

ℎ (119903) = 1 + 120579 (119886 minus 119903) (1198862

1199032minus 1) (2)

and 120579(119909) is the step function Such a wormhole has vanishingthroat length while the step function at the junction maycause a problem in Einsteinrsquos equation or when a topologicalEuler term is involved A more careful analysis needs to con-sider distributional curvature and so forth see [19] To avoidthese difficulties we may consider from the very beginninga wormhole of a finite throat length sim1120573 where the stepfunction is replaced with a smooth function (eg 120579(119909 120573) =

(exp (120573119909)+1)minus1)Thenwhere it is necessary onemay considerthe limit 120573 rarr infin only in final expressions This insuresthat the Bianchi identity holds and that the above metricremains inside of the domain of usual gravity (Unexpectedlyour approach (the use of step function) seems to irritate anessential part of physicists working with wormholes (egPRD reviewers moreover there is a claim that we studysome other exotic objects which are not wormholes [20])Therefore it is necessary to clarify our position here We arequite aware that at present state the physics of wormholes iscertainly on the most speculative side Therefore we do not

see the difference between specific forms of a smooth metricthat one may use For example the simplest choice ℎ = (1 +

1198862119903

2) gives the well-known metric which in 3-dimensions

is called as the Bronnikov-Ellis metric or the metric ℎ =

(1 + (119887119903) + (1198862119903

2)) which includes an additional parameter

(an arbitrary length of the handle) which is not called by anyname but is not less trivial In the case of actual wormholesthe exact and correct form of metric may be establishedonly upon understanding the nature and properties of exoticmatter which may support such a metric as a stable solutionto the Einstein equations For this time has not come yet Inthe case of virtual wormholes even this is not important for inthe complete theory one has to sum over all possible metricswhich formally may be included in 120593 in (5) Our basic aimis to present sufficiently clear and simple model (let it be farfrom realistic) which retains basic qualitative features relatedto a nontrivial topology)

In the region 119903 gt 119886 ℎ = 1 and the metric (2) is flatwhile the region 119903 lt 119886 with the obvious transformation119910120572= (119886

2119903

2)119909

120572 is also flat for 119910 gt 119886 Therefore the regions119903 gt 119886 and 119903 lt 119886 represent two Euclidean spaces glued at thesurface of a sphere 1198783 with the centre at the origin 119903 = 0 andradius 119903 = 119886 Such a space can be described with the ordinarydouble-valued flat metric in the region 119903

plusmngt 119886 by

1198891199042= 120575

120572120573119889119909

120572

plusmn119889119909

120573

plusmn (3)

where the coordinates 119909120572

plusmndescribe two different sheets of

space Now identifying the inner and outer regions of thesphere 1198783 allows the construction of a wormhole which con-nects regions in the same space (instead of two independentspaces) This is achieved by gluing the two spaces in (3) bymotions of the Euclidean space (the Poincare motions) If 119877

plusmn

is the position of the sphere in coordinates 119909120583

plusmn then the gluing

is the rule

119909120583

+= 119877

120583

++ Λ

120583

] (119909]minusminus 119877

]minus) (4)

where Λ120583

] isin 119874 (4) which represents the composition of atranslation and a rotation of the Euclidean space (Lorentztransformation) In terms of common coordinates such awormhole represents the standard flat space in which the twospheres 1198783

plusmn(with centers at positions 119877

plusmn) are glued by the rule

(4)We point out that the physical region is the outer region ofthe two spheres Thus in general the wormhole is describedby a set of parameters the throat radius 119886 positions of throats119877plusmn and rotation matrix Λ120583

] isin 119874 (4)In the present paper we assume the space-time foam

picture in which the vacuum is filled with a gas of virtualwormholes We show that virtual wormholes form a finite (ofthe Planckian order) value of the energy density of zero-pointfluctuations However such a huge value is compensated bythe contribution of virtual wormholes to the mean curvatureand the observed value of the cosmological constant shouldbe close to zero

To achieve our aimwe in Section 2 present the construc-tion of the generating functional in quantum field theoryThe main idea is that the partition function includes thesum over field configurations and the sum over topologies






119885total = sum

120591

sum

120593

119890minus119878 (5)


119878 = minus1

2(120593119860120593) + (119869120593) (6)





119885lowast(119869) = 119885

0(119860) 119890

minus(12)(119869119860minus1

119869) (7)

where 1198850(119860) = int[119863120593]119890


minus1= 119860



119873


sum

120591

997888rarr sum

119873

int

119873

prod

119894=1

119889120585119894= int [119863119865] (8)

where

119865 (120585119873) =1

119873

119873

sum

119894=1

120575 (120585 minus 120585119894) (9)


+

119894minus

minus

119894| ge 2119886


119885total (119869) = int [119863119865]1198850(119860) 119890

minus(12)(119869119860minus1

119869) (10)


119873= 120588(120585119873)119889

119873120585

where

120588 (120585119873) =

1198850(119860 (120585119873))

119885total (0) (11)


119873=

sum119873120588119873





1198850(119860 (120585119873)) = exp(minusintΛ (120585119873) 119889

4119909) (12)









119860119866(119909 1199091015840) = minus120575 (119909 minus 119909

1015840) (13)



119866 (119909 119910) = int1198660(119909 119909

1015840)119873 (119909

1015840 119910) 119889119909

1015840 (14)

where 1198660(119909 119909



119873(119909 1199091015840) = 120575 (119909 minus 119909

1015840) +sum

119894

119887119894120575 (119909 minus 119909

119894) (15)


119894which should


1015840) = 119873(119909 119909

1015840 120585

1 120585

119873) For the



minusΔ119866 (119909 1199091015840) = 120575 (119909 minus 119909

1015840) (16)


0(119909 119909

1015840) = 14120587

2(119909 minus 119909

1015840)2 (and

1198660(119896) = 1119896






120575119866 = minus1

412058721199092

infin

sum

119899=1

1

119899 + 1(119886

1199091015840)

2119899

(1199091015840

119909)

119899minus1

119876119899 (17)

where 119876119899= (4120587

22119899)sum

119899minus1

119897=0sum

119897

119898=minus119897119876

lowast1015840

119899119897119898119876

119899119897119898and 119876

119899119897119898(Ω) are



+minus

minusbetween


+ 119877

minus) The


120575 (119909 minus 1199091015840) 997888rarr 120575 (119909 minus 119909

1015840) + 119887 (119909 119909

1015840) (18)


1198871(119909 119909

1015840 120585) =

1198862

2(

1

(119877minusminus 1199091015840)

2minus

1

(119877+minus 1199091015840)

2)


minus)]

(19)





1

212058721198863120575 (

10038161003816100381610038161003816 minus

plusmn

10038161003816100381610038161003816minus 119886) (20)



119887total (119909 1199091015840) = sum119887

1(119909 119909

1015840 120585

119894) = 119873int119887

1(119909 119909

1015840 120585) 119865 (120585) 119889120585

(21)



119887total (119909 minus 1199091015840)

= int1

21205872119886(

1

1198772

minus

minus1

1198772

+

)120575 (10038161003816100381610038161003816 minus

1015840minus

+

10038161003816100381610038161003816minus 119886)119873119865 (120585) 119889120585

(22)


sdot

(1198894119896(2120587)


= int(41205872119896

2)119890

minus119894119896119909sdot

(1198894119896(2120587)

4) we find for 119887(119896) = int 119887(119909)119890

1198941198961199091198894119909 the expression

119887total (119896) = 119873int1198862 4120587

2

1198962(119865 (119886 119896) minus 119865 (119886 0))

1198691(119896119886)

1198961198862119889119886 (23)


119873119865 (119886119883) =119899

212058721199033

0

120575 (119886 minus 1198860) 120575 (119883 minus 119903

0) (24)



0=

|119877+minus119877



1198941198961199091198894119909 re-

duces to 119873119865(119886 119896) = 119899(1198691(119896119903

0)(119896119903

02))120575(119886 minus 119886

0) Thus from

(23) we find

119887 (119896) = minus1198991198862 4120587

2

1198962(1 minus

1198691(119896119903

0)

11989611990302

)1198691(119896119886

0)

11989611988602

(25)


119866true = 1198660(119896)119873 (119896) = 119866

0(119896) (1 + 119887 (119896)) (26)



we get 1198691(119896119903

0)(119896119903

02) asymp 1 minus (12) (119896119903

02)


119887(119896) asymp minus1205872119899119886

21199032


119887(119896) asymp minus int(12058722)119886

21199032

0119899(119886 119903

0)119889119886 119889119903

0 where 119899(119886 119903

0) is the den-



119873(119896) 997888rarr 1 minus1205872

2int 119886

4119899 (119886 119903

0)1199032

0

1198862119889119886 119889119903

0le 1 (27)





22)119886



2119896

2) (](119896) minus ](0)) where ](119896) =

int ](119883)119890119894119896119883


119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (28)


int(12058722)119886

4119865(119886119883)119889



2


119866true = 1198660(119896)119873 (119896) =

1

1198962 + 41205872(] (0) minus ] (119896))

(29)





119866true (119909 minus 1199091015840= 0) = int

1

1198962 + 41205872(] (0) minus ] (119896))

1198894119896

(2120587)4lt infin

(30)



1015840= + 119888 with an




119888 = 1198714= 119881which



1015840 120585) rarr




119866 (119896) = 1198660(119896)119873 (119896 120585) (31)



119866 =119873 (119896 120585)

1198962 + 1198982 (32)


119885total (119869)

= int [119863119873 (119896)] 119890minus119868(119873)

119890minus(12)(119871

4

(2120587)4

) int((119873(119896)(1198962

+1198982

))|119869119896

|2

)119889119896

(33)

where [119863119873] = prod119896119889119873



to119873(119896))

119890minus119868(119873)

= int [119863119865]1198850(119873 (119896 120585)) 120575 (119873 (119896) minus 119873 (119896 120585)) (34)




119866 (119896) =119873 (119896)

1198962 + 1198982 (35)


119873(119896) =1

119885total (0)int [119863119873] 119890

minus119868(119873)119873(119896) (36)




2119878 may


2119878 =




119891 (119896) =1

119885total (120582 0)int [119863119873] 119890

(120582119873)minus119868(119873)119873(119896) (37)


4(2120587)

4)sdot

1198660(119896)|119869


ext=

1198660(119896)119869








119873(119909 1199091015840) = sum119873

119894119891119894(119909)119891

lowast

119894(119909


119894= 0 1

(since 1198732

119894= 119873




+ 119877


plusmn





0to the equation



0) which obey




0) may


0 By


perp


perp



+


= (minus+

+)2 (Instead


0= 1(119909 minus 119909

+)2minus 1(119909 minus 119909








minus+ (119904

+minus 119904

minus)1205792120587






perp



0))

2

=

119892120572120573119889119910

120572119889119910

120573 (where 119910 = (120579 119909perp




1

radic119892120575 (120579 minus 120579

1015840) 997888rarr 119873(120579 minus 120579

1015840) =

+infin

sum

119899=minusinfin

1

radic119892120575 (120579 minus 120579

1015840+ 2120587119899)

(38)


119873(119896 1198961015840) =

+infin

sum

119899=minusinfin

120575 (119896 minus 119899) 120575 (119896 minus 1198961015840) (39)



the projection 119891119873= 119891 119891




119873total (119909 1199091015840) = int(prod

119894

radic1198921198941198894119910119894)119873(120585

1 119909 119910

1)

times 119873 (1205852 119910

1 119910

2) sdot sdot sdot 119873 (120585

119873 119910

119873minus1 119909

1015840)

(40)

where 119873(120585119894 119909 119909



119894 119909 119909

1015840) realizes




119894)

5 Cutoff



4 and periodic


119896rarr

(1198714(2120587)

4) int 119889



119868 = 1198680+sum120582

1(119896)119873 (119896) +

1

2sum120582

2(119896 119896

1015840)119873 (119896)119873 (119896


(41)


0(119896) We


sum

1198961015840

1205822(119896 119896

1015840)119873 (119896

1015840) = minus120582

1(119896) (42)


1015840) = Ψ(119896)Ψ

+(119896



(119896)Ψ(1198961015840)+Ψ(119896

1015840) Ψ

+(119896) = 120575(119896minus


119873(119896 1198961015840) = 120575(119896 minus 119896



0+ (Ψ




find

119873(119896) =1

119885total (119896)sum

119873=01

119890minus1205821

(119896)119873(119896)119873(119896) =

119890minus1205821

(119896)

1 + 119890minus1205821(119896) (43)


0(119896)










int119873 (119896)1198894119896

(2120587)4= int

119873 (119896) 1198964

(2120587)2

119889119896

119896 (44)


1198964119873(119896) sim 119896

119863 (45)




119868119864= minus

1

16120587119866int (119877 minus 2Λ

0)radic119892119889

4119909 minus int119871

119898radic1198921198894119909 (46)


119877119886119887minus1

2119892119886119887119877 + 119892

119886119887Λ

0= 8120587119866119879

119886119887 (47)

where 119879119886119887= (12)(119892)

minus12(120575119871

119898120575119892




Λ tot = Λ0+ Λ

119898+ Λ

119877= Λ

0+ 2120587119866 ⟨119879⟩ +

1

4⟨119877⟩119908 (48)



⟨119877⟩119908




1198891199042= ℎ

2(119903)120575

120572120573119889119909

120572119889119909


119877120572120573

= 7 minus (ℎ1015840

ℎ)

1015840

(120575120572120573

+ (119873 minus 2) 119899120572119899120573)

minus1

119903

ℎ1015840

ℎ((119873 minus 2) Δ

120572120573+ 120575

120572120573(119873 minus 1))

minus (ℎ1015840

ℎ)

2

(119873 minus 2) Δ120572120573

(49)

where 119899]= 119909


surface Δ120572120573

= 120575120572120573



119877 = minus(119873 minus 1)

ℎ2[2

ℎ10158401015840

ℎ+ 2

1

119903

ℎ1015840

ℎ(119873 minus 1) + (

ℎ1015840

ℎ)

2

(119873 minus 4)]

(50)



we find

minusℎ4

119873 minus 1119877 = 4ℎ

1198862

1199033120575 +

2ℎ

119903119873minus1(119903

119873minus1120582)

1015840

+ (119873 minus 4)(1205822minus 4120579

1198862

1199033120582)

+ 4 (119873 minus 4)1198862

1199034120579 (120579 minus 1)

(51)


2119903

2)120575 and 120575 = minus120579


minus119877 =4 (119873 minus 1)

119886120575 (119903 minus 119886) +

2 (119873 minus 1)

119903119873minus1(119903

119873minus3(119903

2minus 119886

2) 120575)

1015840

(52)


1

4int119877radic119892119889

4119909 = minus6120587

21198862 (53)



1

4int119877radic119892119889

4119909 = minus6120587

2sum

119895

1198862

119895

= minus121205872int119899 (119886) 119886

2119889119886 119889

4119909 = intΛ

1198771198894119909

(54)

where 2119899(119886) = int 119899(119886 1199030)119889119903



Λ119877= minus12120587

2int119899 (119886) 119886

2119889119886 lt 0 (55)



minus119879120572120573

(119909) = 120597120572120593120597

120573120593 minus

1

2119892120572120573

(120597120583120593120597

120583120593 + 119898

21205932) (56)


minus ⟨119879120572120573

(119909)⟩

= lim1199091015840

rarr119909

(1205971205721205971015840

120573minus1

2119892120572120573

(1205971205831205971015840

120583+ 119898

2)) ⟨119866 (119909 119909

1015840 120585)⟩

(57)


int 119890minus119894119896(119909minus119909

1015840

)119866(119896 120585) (119889

4119896(2120587)


we arrive at

⟨119879120572120573

(119909)⟩ =1

4119892120572120573

int(1 +119898

2

1198962 + 1198982)119873 (119896)

1198894119896

(2120587)4 (58)


2)119889

4119896 = (14)119892

120572120573int 119896

2119891(119896

2)119889

4119896



1205722(120587

1205722+ 119896

120572) from


Λ119898=2120587119866sumint

1205871205722

1205871205722+119896120572

1198894119896

(2120587)4=120587119866

4Γ (

120572 minus 4

120572) Γ (

4

120572) sim 1

(59)


1(119896) = minussum ln119885

0(119896) + 120575120582(119896) with

120575120582(119896) ≪ ln1198850(119896) as 119896 ≫ 1)


120572120573⟩ = Λ

lowast119892120572120573

with Λlowast


120572120573⟩ = 120575Λ119892

120572120573 where 120575Λ =




2120587119866int 119887(119896) (1198894119896(2120587)



119887total (0) = minus119899

4120587411988631199033

0

int(1 minus1198862

1198772

minus

)120575 (119877+minus 119886)

times 120575 (1003816100381610038161003816119877+

minus 119877minus

1003816100381610038161003816 minus 1199030) 119889

4119877minus1198894119877+

(60)

which gives


1199030

)) (61)

where

119891(119886

1199030

) =2

120587int

120587

0

1198862sin2

120579119889120579

1198862 + 21198861199030cos 120579 + 119903

2

0

(62)


0le 12)


2119903

2

0 Thus for the


120575Λ119898= minus2120587119866int119899 (119886 119903

0) (1 minus 119891(

119886

1199030

))119889119886 1198891199030

= minus2120587119866119899 (1 minus ⟨119891⟩)

(63)


119898= Λ

lowast+ 120575Λ



0)(120587

22)119886

4119889119886 119889119903

0lt 1 where (120587

22)119886

4





≫ Λ119877


119877120575Λ





119873119865 (119886119883) =1

1198862120575 (119886) ] (119883) (64)

where ](119883) = int 1198862119873119865(119886119883)119889119886 and int(1119886

2)](119883)119889

4119883 =




4119889119886 119889119903

0=

1198862int ](119883)119889



1198941198961198831198894119883) as

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (65)


Λ119877= minus12120587

2int119899 (119886 119903

0) 119886

2119889119886 119889119903

0

= minus121205872int ] (119883) 119889

4119883 = minus12120587

2] (0)

(66)


Λ tot = Λ0

+ 2120587119866int(1minus4120587

2

1198962(] (0)minus] (119896)))

1198894119896

(2120587)4minus12120587

2] (0)

(67)


4


119898)









119881 = minus1

2int 119869 (119909) 119866 (119909 119910) 119869 (119910) 119889

4119909 119889

4119910

= minus1

2

1198714

(2120587)4int119866

0(119896)119873 (119896)

10038161003816100381610038161198691198961003816100381610038161003816

2

1198894119896

(68)


120575119873 (119896 119869) = 120575119887 (119869) ≃ minus1

2

1198714

(2120587)4int120590

2(119896 119901) 119866

0(119901)

10038161003816100381610038161003816119869119901

10038161003816100381610038161003816

2

1198894119901

(69)


2(119896 119901) = Δ119873lowast(119896)Δ119873(119901)

1205902(119896 119901) =

1

119885total (0)int [119863119873] 119890

minus119868(119873)Δ119873

lowast(119896) Δ119873 (119901) (70)



(119896 119901) rarr 1205902


120575119887 (119869) = minus1205902

119896

41205872

21198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(71)


41205872

1198962(120575] (0 119869) minus 120575] (119896 119869)) =

1

21205902

119896

41205872

1198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(72)






120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)


1205902





119896= 119873(119896)(1 minus





119896and


0 while


0119873

0= ΩDEΩ119887



asymp 075005 = 15 while 1205902

0119873

0sim 1



15




plusmn= 119878

2

plusmntimes 119877


+ 119903


plusmn




120572120573⟩ sim 8120587119866119899 and




1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3






plusmnrarr cup

1198991198783



1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)


= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the


cylinders1198793



minusℓ119889119905

0and the

sum sum+infin




119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)



plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








119885total = sum

120591

sum

120593

119890minus119878 (5)


119878 = minus1

2(120593119860120593) + (119869120593) (6)





119885lowast(119869) = 119885

0(119860) 119890

minus(12)(119869119860minus1

119869) (7)

where 1198850(119860) = int[119863120593]119890


minus1= 119860



119873


sum

120591

997888rarr sum

119873

int

119873

prod

119894=1

119889120585119894= int [119863119865] (8)

where

119865 (120585119873) =1

119873

119873

sum

119894=1

120575 (120585 minus 120585119894) (9)


+

119894minus

minus

119894| ge 2119886


119885total (119869) = int [119863119865]1198850(119860) 119890

minus(12)(119869119860minus1

119869) (10)


119873= 120588(120585119873)119889

119873120585

where

120588 (120585119873) =

1198850(119860 (120585119873))

119885total (0) (11)


119873=

sum119873120588119873





1198850(119860 (120585119873)) = exp(minusintΛ (120585119873) 119889

4119909) (12)









119860119866(119909 1199091015840) = minus120575 (119909 minus 119909

1015840) (13)



119866 (119909 119910) = int1198660(119909 119909

1015840)119873 (119909

1015840 119910) 119889119909

1015840 (14)

where 1198660(119909 119909



119873(119909 1199091015840) = 120575 (119909 minus 119909

1015840) +sum

119894

119887119894120575 (119909 minus 119909

119894) (15)


119894which should


1015840) = 119873(119909 119909

1015840 120585

1 120585

119873) For the



minusΔ119866 (119909 1199091015840) = 120575 (119909 minus 119909

1015840) (16)


0(119909 119909

1015840) = 14120587

2(119909 minus 119909

1015840)2 (and

1198660(119896) = 1119896






120575119866 = minus1

412058721199092

infin

sum

119899=1

1

119899 + 1(119886

1199091015840)

2119899

(1199091015840

119909)

119899minus1

119876119899 (17)

where 119876119899= (4120587

22119899)sum

119899minus1

119897=0sum

119897

119898=minus119897119876

lowast1015840

119899119897119898119876

119899119897119898and 119876

119899119897119898(Ω) are



+minus

minusbetween


+ 119877

minus) The


120575 (119909 minus 1199091015840) 997888rarr 120575 (119909 minus 119909

1015840) + 119887 (119909 119909

1015840) (18)


1198871(119909 119909

1015840 120585) =

1198862

2(

1

(119877minusminus 1199091015840)

2minus

1

(119877+minus 1199091015840)

2)


minus)]

(19)





1

212058721198863120575 (

10038161003816100381610038161003816 minus

plusmn

10038161003816100381610038161003816minus 119886) (20)



119887total (119909 1199091015840) = sum119887

1(119909 119909

1015840 120585

119894) = 119873int119887

1(119909 119909

1015840 120585) 119865 (120585) 119889120585

(21)



119887total (119909 minus 1199091015840)

= int1

21205872119886(

1

1198772

minus

minus1

1198772

+

)120575 (10038161003816100381610038161003816 minus

1015840minus

+

10038161003816100381610038161003816minus 119886)119873119865 (120585) 119889120585

(22)


sdot

(1198894119896(2120587)


= int(41205872119896

2)119890

minus119894119896119909sdot

(1198894119896(2120587)

4) we find for 119887(119896) = int 119887(119909)119890

1198941198961199091198894119909 the expression

119887total (119896) = 119873int1198862 4120587

2

1198962(119865 (119886 119896) minus 119865 (119886 0))

1198691(119896119886)

1198961198862119889119886 (23)


119873119865 (119886119883) =119899

212058721199033

0

120575 (119886 minus 1198860) 120575 (119883 minus 119903

0) (24)



0=

|119877+minus119877



1198941198961199091198894119909 re-

duces to 119873119865(119886 119896) = 119899(1198691(119896119903

0)(119896119903

02))120575(119886 minus 119886

0) Thus from

(23) we find

119887 (119896) = minus1198991198862 4120587

2

1198962(1 minus

1198691(119896119903

0)

11989611990302

)1198691(119896119886

0)

11989611988602

(25)


119866true = 1198660(119896)119873 (119896) = 119866

0(119896) (1 + 119887 (119896)) (26)



we get 1198691(119896119903

0)(119896119903

02) asymp 1 minus (12) (119896119903

02)


119887(119896) asymp minus1205872119899119886

21199032


119887(119896) asymp minus int(12058722)119886

21199032

0119899(119886 119903

0)119889119886 119889119903

0 where 119899(119886 119903

0) is the den-



119873(119896) 997888rarr 1 minus1205872

2int 119886

4119899 (119886 119903

0)1199032

0

1198862119889119886 119889119903

0le 1 (27)





22)119886



2119896

2) (](119896) minus ](0)) where ](119896) =

int ](119883)119890119894119896119883


119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (28)


int(12058722)119886

4119865(119886119883)119889



2


119866true = 1198660(119896)119873 (119896) =

1

1198962 + 41205872(] (0) minus ] (119896))

(29)





119866true (119909 minus 1199091015840= 0) = int

1

1198962 + 41205872(] (0) minus ] (119896))

1198894119896

(2120587)4lt infin

(30)



1015840= + 119888 with an




119888 = 1198714= 119881which



1015840 120585) rarr




119866 (119896) = 1198660(119896)119873 (119896 120585) (31)



119866 =119873 (119896 120585)

1198962 + 1198982 (32)


119885total (119869)

= int [119863119873 (119896)] 119890minus119868(119873)

119890minus(12)(119871

4

(2120587)4

) int((119873(119896)(1198962

+1198982

))|119869119896

|2

)119889119896

(33)

where [119863119873] = prod119896119889119873



to119873(119896))

119890minus119868(119873)

= int [119863119865]1198850(119873 (119896 120585)) 120575 (119873 (119896) minus 119873 (119896 120585)) (34)




119866 (119896) =119873 (119896)

1198962 + 1198982 (35)


119873(119896) =1

119885total (0)int [119863119873] 119890

minus119868(119873)119873(119896) (36)




2119878 may


2119878 =




119891 (119896) =1

119885total (120582 0)int [119863119873] 119890

(120582119873)minus119868(119873)119873(119896) (37)


4(2120587)

4)sdot

1198660(119896)|119869


ext=

1198660(119896)119869








119873(119909 1199091015840) = sum119873

119894119891119894(119909)119891

lowast

119894(119909


119894= 0 1

(since 1198732

119894= 119873




+ 119877


plusmn





0to the equation



0) which obey




0) may


0 By


perp


perp



+


= (minus+

+)2 (Instead


0= 1(119909 minus 119909

+)2minus 1(119909 minus 119909








minus+ (119904

+minus 119904

minus)1205792120587






perp



0))

2

=

119892120572120573119889119910

120572119889119910

120573 (where 119910 = (120579 119909perp




1

radic119892120575 (120579 minus 120579

1015840) 997888rarr 119873(120579 minus 120579

1015840) =

+infin

sum

119899=minusinfin

1

radic119892120575 (120579 minus 120579

1015840+ 2120587119899)

(38)


119873(119896 1198961015840) =

+infin

sum

119899=minusinfin

120575 (119896 minus 119899) 120575 (119896 minus 1198961015840) (39)



the projection 119891119873= 119891 119891




119873total (119909 1199091015840) = int(prod

119894

radic1198921198941198894119910119894)119873(120585

1 119909 119910

1)

times 119873 (1205852 119910

1 119910

2) sdot sdot sdot 119873 (120585

119873 119910

119873minus1 119909

1015840)

(40)

where 119873(120585119894 119909 119909



119894 119909 119909

1015840) realizes




119894)

5 Cutoff



4 and periodic


119896rarr

(1198714(2120587)

4) int 119889



119868 = 1198680+sum120582

1(119896)119873 (119896) +

1

2sum120582

2(119896 119896

1015840)119873 (119896)119873 (119896


(41)


0(119896) We


sum

1198961015840

1205822(119896 119896

1015840)119873 (119896

1015840) = minus120582

1(119896) (42)


1015840) = Ψ(119896)Ψ

+(119896



(119896)Ψ(1198961015840)+Ψ(119896

1015840) Ψ

+(119896) = 120575(119896minus


119873(119896 1198961015840) = 120575(119896 minus 119896



0+ (Ψ




find

119873(119896) =1

119885total (119896)sum

119873=01

119890minus1205821

(119896)119873(119896)119873(119896) =

119890minus1205821

(119896)

1 + 119890minus1205821(119896) (43)


0(119896)










int119873 (119896)1198894119896

(2120587)4= int

119873 (119896) 1198964

(2120587)2

119889119896

119896 (44)


1198964119873(119896) sim 119896

119863 (45)




119868119864= minus

1

16120587119866int (119877 minus 2Λ

0)radic119892119889

4119909 minus int119871

119898radic1198921198894119909 (46)


119877119886119887minus1

2119892119886119887119877 + 119892

119886119887Λ

0= 8120587119866119879

119886119887 (47)

where 119879119886119887= (12)(119892)

minus12(120575119871

119898120575119892




Λ tot = Λ0+ Λ

119898+ Λ

119877= Λ

0+ 2120587119866 ⟨119879⟩ +

1

4⟨119877⟩119908 (48)



⟨119877⟩119908




1198891199042= ℎ

2(119903)120575

120572120573119889119909

120572119889119909


119877120572120573

= 7 minus (ℎ1015840

ℎ)

1015840

(120575120572120573

+ (119873 minus 2) 119899120572119899120573)

minus1

119903

ℎ1015840

ℎ((119873 minus 2) Δ

120572120573+ 120575

120572120573(119873 minus 1))

minus (ℎ1015840

ℎ)

2

(119873 minus 2) Δ120572120573

(49)

where 119899]= 119909


surface Δ120572120573

= 120575120572120573



119877 = minus(119873 minus 1)

ℎ2[2

ℎ10158401015840

ℎ+ 2

1

119903

ℎ1015840

ℎ(119873 minus 1) + (

ℎ1015840

ℎ)

2

(119873 minus 4)]

(50)



we find

minusℎ4

119873 minus 1119877 = 4ℎ

1198862

1199033120575 +

2ℎ

119903119873minus1(119903

119873minus1120582)

1015840

+ (119873 minus 4)(1205822minus 4120579

1198862

1199033120582)

+ 4 (119873 minus 4)1198862

1199034120579 (120579 minus 1)

(51)


2119903

2)120575 and 120575 = minus120579


minus119877 =4 (119873 minus 1)

119886120575 (119903 minus 119886) +

2 (119873 minus 1)

119903119873minus1(119903

119873minus3(119903

2minus 119886

2) 120575)

1015840

(52)


1

4int119877radic119892119889

4119909 = minus6120587

21198862 (53)



1

4int119877radic119892119889

4119909 = minus6120587

2sum

119895

1198862

119895

= minus121205872int119899 (119886) 119886

2119889119886 119889

4119909 = intΛ

1198771198894119909

(54)

where 2119899(119886) = int 119899(119886 1199030)119889119903



Λ119877= minus12120587

2int119899 (119886) 119886

2119889119886 lt 0 (55)



minus119879120572120573

(119909) = 120597120572120593120597

120573120593 minus

1

2119892120572120573

(120597120583120593120597

120583120593 + 119898

21205932) (56)


minus ⟨119879120572120573

(119909)⟩

= lim1199091015840

rarr119909

(1205971205721205971015840

120573minus1

2119892120572120573

(1205971205831205971015840

120583+ 119898

2)) ⟨119866 (119909 119909

1015840 120585)⟩

(57)


int 119890minus119894119896(119909minus119909

1015840

)119866(119896 120585) (119889

4119896(2120587)


we arrive at

⟨119879120572120573

(119909)⟩ =1

4119892120572120573

int(1 +119898

2

1198962 + 1198982)119873 (119896)

1198894119896

(2120587)4 (58)


2)119889

4119896 = (14)119892

120572120573int 119896

2119891(119896

2)119889

4119896



1205722(120587

1205722+ 119896

120572) from


Λ119898=2120587119866sumint

1205871205722

1205871205722+119896120572

1198894119896

(2120587)4=120587119866

4Γ (

120572 minus 4

120572) Γ (

4

120572) sim 1

(59)


1(119896) = minussum ln119885

0(119896) + 120575120582(119896) with

120575120582(119896) ≪ ln1198850(119896) as 119896 ≫ 1)


120572120573⟩ = Λ

lowast119892120572120573

with Λlowast


120572120573⟩ = 120575Λ119892

120572120573 where 120575Λ =




2120587119866int 119887(119896) (1198894119896(2120587)



119887total (0) = minus119899

4120587411988631199033

0

int(1 minus1198862

1198772

minus

)120575 (119877+minus 119886)

times 120575 (1003816100381610038161003816119877+

minus 119877minus

1003816100381610038161003816 minus 1199030) 119889

4119877minus1198894119877+

(60)

which gives


1199030

)) (61)

where

119891(119886

1199030

) =2

120587int

120587

0

1198862sin2

120579119889120579

1198862 + 21198861199030cos 120579 + 119903

2

0

(62)


0le 12)


2119903

2

0 Thus for the


120575Λ119898= minus2120587119866int119899 (119886 119903

0) (1 minus 119891(

119886

1199030

))119889119886 1198891199030

= minus2120587119866119899 (1 minus ⟨119891⟩)

(63)


119898= Λ

lowast+ 120575Λ



0)(120587

22)119886

4119889119886 119889119903

0lt 1 where (120587

22)119886

4





≫ Λ119877


119877120575Λ





119873119865 (119886119883) =1

1198862120575 (119886) ] (119883) (64)

where ](119883) = int 1198862119873119865(119886119883)119889119886 and int(1119886

2)](119883)119889

4119883 =




4119889119886 119889119903

0=

1198862int ](119883)119889



1198941198961198831198894119883) as

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (65)


Λ119877= minus12120587

2int119899 (119886 119903

0) 119886

2119889119886 119889119903

0

= minus121205872int ] (119883) 119889

4119883 = minus12120587

2] (0)

(66)


Λ tot = Λ0

+ 2120587119866int(1minus4120587

2

1198962(] (0)minus] (119896)))

1198894119896

(2120587)4minus12120587

2] (0)

(67)


4


119898)









119881 = minus1

2int 119869 (119909) 119866 (119909 119910) 119869 (119910) 119889

4119909 119889

4119910

= minus1

2

1198714

(2120587)4int119866

0(119896)119873 (119896)

10038161003816100381610038161198691198961003816100381610038161003816

2

1198894119896

(68)


120575119873 (119896 119869) = 120575119887 (119869) ≃ minus1

2

1198714

(2120587)4int120590

2(119896 119901) 119866

0(119901)

10038161003816100381610038161003816119869119901

10038161003816100381610038161003816

2

1198894119901

(69)


2(119896 119901) = Δ119873lowast(119896)Δ119873(119901)

1205902(119896 119901) =

1

119885total (0)int [119863119873] 119890

minus119868(119873)Δ119873

lowast(119896) Δ119873 (119901) (70)



(119896 119901) rarr 1205902


120575119887 (119869) = minus1205902

119896

41205872

21198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(71)


41205872

1198962(120575] (0 119869) minus 120575] (119896 119869)) =

1

21205902

119896

41205872

1198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(72)






120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)


1205902





119896= 119873(119896)(1 minus





119896and


0 while


0119873

0= ΩDEΩ119887



asymp 075005 = 15 while 1205902

0119873

0sim 1



15




plusmn= 119878

2

plusmntimes 119877


+ 119903


plusmn




120572120573⟩ sim 8120587119866119899 and




1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3






plusmnrarr cup

1198991198783



1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)


= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the


cylinders1198793



minusℓ119889119905

0and the

sum sum+infin




119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)



plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










119860119866(119909 1199091015840) = minus120575 (119909 minus 119909

1015840) (13)



119866 (119909 119910) = int1198660(119909 119909

1015840)119873 (119909

1015840 119910) 119889119909

1015840 (14)

where 1198660(119909 119909



119873(119909 1199091015840) = 120575 (119909 minus 119909

1015840) +sum

119894

119887119894120575 (119909 minus 119909

119894) (15)


119894which should


1015840) = 119873(119909 119909

1015840 120585

1 120585

119873) For the



minusΔ119866 (119909 1199091015840) = 120575 (119909 minus 119909

1015840) (16)


0(119909 119909

1015840) = 14120587

2(119909 minus 119909

1015840)2 (and

1198660(119896) = 1119896






120575119866 = minus1

412058721199092

infin

sum

119899=1

1

119899 + 1(119886

1199091015840)

2119899

(1199091015840

119909)

119899minus1

119876119899 (17)

where 119876119899= (4120587

22119899)sum

119899minus1

119897=0sum

119897

119898=minus119897119876

lowast1015840

119899119897119898119876

119899119897119898and 119876

119899119897119898(Ω) are



+minus

minusbetween


+ 119877

minus) The


120575 (119909 minus 1199091015840) 997888rarr 120575 (119909 minus 119909

1015840) + 119887 (119909 119909

1015840) (18)


1198871(119909 119909

1015840 120585) =

1198862

2(

1

(119877minusminus 1199091015840)

2minus

1

(119877+minus 1199091015840)

2)


minus)]

(19)





1

212058721198863120575 (

10038161003816100381610038161003816 minus

plusmn

10038161003816100381610038161003816minus 119886) (20)



119887total (119909 1199091015840) = sum119887

1(119909 119909

1015840 120585

119894) = 119873int119887

1(119909 119909

1015840 120585) 119865 (120585) 119889120585

(21)



119887total (119909 minus 1199091015840)

= int1

21205872119886(

1

1198772

minus

minus1

1198772

+

)120575 (10038161003816100381610038161003816 minus

1015840minus

+

10038161003816100381610038161003816minus 119886)119873119865 (120585) 119889120585

(22)


sdot

(1198894119896(2120587)


= int(41205872119896

2)119890

minus119894119896119909sdot

(1198894119896(2120587)

4) we find for 119887(119896) = int 119887(119909)119890

1198941198961199091198894119909 the expression

119887total (119896) = 119873int1198862 4120587

2

1198962(119865 (119886 119896) minus 119865 (119886 0))

1198691(119896119886)

1198961198862119889119886 (23)


119873119865 (119886119883) =119899

212058721199033

0

120575 (119886 minus 1198860) 120575 (119883 minus 119903

0) (24)



0=

|119877+minus119877



1198941198961199091198894119909 re-

duces to 119873119865(119886 119896) = 119899(1198691(119896119903

0)(119896119903

02))120575(119886 minus 119886

0) Thus from

(23) we find

119887 (119896) = minus1198991198862 4120587

2

1198962(1 minus

1198691(119896119903

0)

11989611990302

)1198691(119896119886

0)

11989611988602

(25)


119866true = 1198660(119896)119873 (119896) = 119866

0(119896) (1 + 119887 (119896)) (26)



we get 1198691(119896119903

0)(119896119903

02) asymp 1 minus (12) (119896119903

02)


119887(119896) asymp minus1205872119899119886

21199032


119887(119896) asymp minus int(12058722)119886

21199032

0119899(119886 119903

0)119889119886 119889119903

0 where 119899(119886 119903

0) is the den-



119873(119896) 997888rarr 1 minus1205872

2int 119886

4119899 (119886 119903

0)1199032

0

1198862119889119886 119889119903

0le 1 (27)





22)119886



2119896

2) (](119896) minus ](0)) where ](119896) =

int ](119883)119890119894119896119883


119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (28)


int(12058722)119886

4119865(119886119883)119889



2


119866true = 1198660(119896)119873 (119896) =

1

1198962 + 41205872(] (0) minus ] (119896))

(29)





119866true (119909 minus 1199091015840= 0) = int

1

1198962 + 41205872(] (0) minus ] (119896))

1198894119896

(2120587)4lt infin

(30)



1015840= + 119888 with an




119888 = 1198714= 119881which



1015840 120585) rarr




119866 (119896) = 1198660(119896)119873 (119896 120585) (31)



119866 =119873 (119896 120585)

1198962 + 1198982 (32)


119885total (119869)

= int [119863119873 (119896)] 119890minus119868(119873)

119890minus(12)(119871

4

(2120587)4

) int((119873(119896)(1198962

+1198982

))|119869119896

|2

)119889119896

(33)

where [119863119873] = prod119896119889119873



to119873(119896))

119890minus119868(119873)

= int [119863119865]1198850(119873 (119896 120585)) 120575 (119873 (119896) minus 119873 (119896 120585)) (34)




119866 (119896) =119873 (119896)

1198962 + 1198982 (35)


119873(119896) =1

119885total (0)int [119863119873] 119890

minus119868(119873)119873(119896) (36)




2119878 may


2119878 =




119891 (119896) =1

119885total (120582 0)int [119863119873] 119890

(120582119873)minus119868(119873)119873(119896) (37)


4(2120587)

4)sdot

1198660(119896)|119869


ext=

1198660(119896)119869








119873(119909 1199091015840) = sum119873

119894119891119894(119909)119891

lowast

119894(119909


119894= 0 1

(since 1198732

119894= 119873




+ 119877


plusmn





0to the equation



0) which obey




0) may


0 By


perp


perp



+


= (minus+

+)2 (Instead


0= 1(119909 minus 119909

+)2minus 1(119909 minus 119909








minus+ (119904

+minus 119904

minus)1205792120587






perp



0))

2

=

119892120572120573119889119910

120572119889119910

120573 (where 119910 = (120579 119909perp




1

radic119892120575 (120579 minus 120579

1015840) 997888rarr 119873(120579 minus 120579

1015840) =

+infin

sum

119899=minusinfin

1

radic119892120575 (120579 minus 120579

1015840+ 2120587119899)

(38)


119873(119896 1198961015840) =

+infin

sum

119899=minusinfin

120575 (119896 minus 119899) 120575 (119896 minus 1198961015840) (39)



the projection 119891119873= 119891 119891




119873total (119909 1199091015840) = int(prod

119894

radic1198921198941198894119910119894)119873(120585

1 119909 119910

1)

times 119873 (1205852 119910

1 119910

2) sdot sdot sdot 119873 (120585

119873 119910

119873minus1 119909

1015840)

(40)

where 119873(120585119894 119909 119909



119894 119909 119909

1015840) realizes




119894)

5 Cutoff



4 and periodic


119896rarr

(1198714(2120587)

4) int 119889



119868 = 1198680+sum120582

1(119896)119873 (119896) +

1

2sum120582

2(119896 119896

1015840)119873 (119896)119873 (119896


(41)


0(119896) We


sum

1198961015840

1205822(119896 119896

1015840)119873 (119896

1015840) = minus120582

1(119896) (42)


1015840) = Ψ(119896)Ψ

+(119896



(119896)Ψ(1198961015840)+Ψ(119896

1015840) Ψ

+(119896) = 120575(119896minus


119873(119896 1198961015840) = 120575(119896 minus 119896



0+ (Ψ




find

119873(119896) =1

119885total (119896)sum

119873=01

119890minus1205821

(119896)119873(119896)119873(119896) =

119890minus1205821

(119896)

1 + 119890minus1205821(119896) (43)


0(119896)










int119873 (119896)1198894119896

(2120587)4= int

119873 (119896) 1198964

(2120587)2

119889119896

119896 (44)


1198964119873(119896) sim 119896

119863 (45)




119868119864= minus

1

16120587119866int (119877 minus 2Λ

0)radic119892119889

4119909 minus int119871

119898radic1198921198894119909 (46)


119877119886119887minus1

2119892119886119887119877 + 119892

119886119887Λ

0= 8120587119866119879

119886119887 (47)

where 119879119886119887= (12)(119892)

minus12(120575119871

119898120575119892




Λ tot = Λ0+ Λ

119898+ Λ

119877= Λ

0+ 2120587119866 ⟨119879⟩ +

1

4⟨119877⟩119908 (48)



⟨119877⟩119908




1198891199042= ℎ

2(119903)120575

120572120573119889119909

120572119889119909


119877120572120573

= 7 minus (ℎ1015840

ℎ)

1015840

(120575120572120573

+ (119873 minus 2) 119899120572119899120573)

minus1

119903

ℎ1015840

ℎ((119873 minus 2) Δ

120572120573+ 120575

120572120573(119873 minus 1))

minus (ℎ1015840

ℎ)

2

(119873 minus 2) Δ120572120573

(49)

where 119899]= 119909


surface Δ120572120573

= 120575120572120573



119877 = minus(119873 minus 1)

ℎ2[2

ℎ10158401015840

ℎ+ 2

1

119903

ℎ1015840

ℎ(119873 minus 1) + (

ℎ1015840

ℎ)

2

(119873 minus 4)]

(50)



we find

minusℎ4

119873 minus 1119877 = 4ℎ

1198862

1199033120575 +

2ℎ

119903119873minus1(119903

119873minus1120582)

1015840

+ (119873 minus 4)(1205822minus 4120579

1198862

1199033120582)

+ 4 (119873 minus 4)1198862

1199034120579 (120579 minus 1)

(51)


2119903

2)120575 and 120575 = minus120579


minus119877 =4 (119873 minus 1)

119886120575 (119903 minus 119886) +

2 (119873 minus 1)

119903119873minus1(119903

119873minus3(119903

2minus 119886

2) 120575)

1015840

(52)


1

4int119877radic119892119889

4119909 = minus6120587

21198862 (53)



1

4int119877radic119892119889

4119909 = minus6120587

2sum

119895

1198862

119895

= minus121205872int119899 (119886) 119886

2119889119886 119889

4119909 = intΛ

1198771198894119909

(54)

where 2119899(119886) = int 119899(119886 1199030)119889119903



Λ119877= minus12120587

2int119899 (119886) 119886

2119889119886 lt 0 (55)



minus119879120572120573

(119909) = 120597120572120593120597

120573120593 minus

1

2119892120572120573

(120597120583120593120597

120583120593 + 119898

21205932) (56)


minus ⟨119879120572120573

(119909)⟩

= lim1199091015840

rarr119909

(1205971205721205971015840

120573minus1

2119892120572120573

(1205971205831205971015840

120583+ 119898

2)) ⟨119866 (119909 119909

1015840 120585)⟩

(57)


int 119890minus119894119896(119909minus119909

1015840

)119866(119896 120585) (119889

4119896(2120587)


we arrive at

⟨119879120572120573

(119909)⟩ =1

4119892120572120573

int(1 +119898

2

1198962 + 1198982)119873 (119896)

1198894119896

(2120587)4 (58)


2)119889

4119896 = (14)119892

120572120573int 119896

2119891(119896

2)119889

4119896



1205722(120587

1205722+ 119896

120572) from


Λ119898=2120587119866sumint

1205871205722

1205871205722+119896120572

1198894119896

(2120587)4=120587119866

4Γ (

120572 minus 4

120572) Γ (

4

120572) sim 1

(59)


1(119896) = minussum ln119885

0(119896) + 120575120582(119896) with

120575120582(119896) ≪ ln1198850(119896) as 119896 ≫ 1)


120572120573⟩ = Λ

lowast119892120572120573

with Λlowast


120572120573⟩ = 120575Λ119892

120572120573 where 120575Λ =




2120587119866int 119887(119896) (1198894119896(2120587)



119887total (0) = minus119899

4120587411988631199033

0

int(1 minus1198862

1198772

minus

)120575 (119877+minus 119886)

times 120575 (1003816100381610038161003816119877+

minus 119877minus

1003816100381610038161003816 minus 1199030) 119889

4119877minus1198894119877+

(60)

which gives


1199030

)) (61)

where

119891(119886

1199030

) =2

120587int

120587

0

1198862sin2

120579119889120579

1198862 + 21198861199030cos 120579 + 119903

2

0

(62)


0le 12)


2119903

2

0 Thus for the


120575Λ119898= minus2120587119866int119899 (119886 119903

0) (1 minus 119891(

119886

1199030

))119889119886 1198891199030

= minus2120587119866119899 (1 minus ⟨119891⟩)

(63)


119898= Λ

lowast+ 120575Λ



0)(120587

22)119886

4119889119886 119889119903

0lt 1 where (120587

22)119886

4





≫ Λ119877


119877120575Λ





119873119865 (119886119883) =1

1198862120575 (119886) ] (119883) (64)

where ](119883) = int 1198862119873119865(119886119883)119889119886 and int(1119886

2)](119883)119889

4119883 =




4119889119886 119889119903

0=

1198862int ](119883)119889



1198941198961198831198894119883) as

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (65)


Λ119877= minus12120587

2int119899 (119886 119903

0) 119886

2119889119886 119889119903

0

= minus121205872int ] (119883) 119889

4119883 = minus12120587

2] (0)

(66)


Λ tot = Λ0

+ 2120587119866int(1minus4120587

2

1198962(] (0)minus] (119896)))

1198894119896

(2120587)4minus12120587

2] (0)

(67)


4


119898)









119881 = minus1

2int 119869 (119909) 119866 (119909 119910) 119869 (119910) 119889

4119909 119889

4119910

= minus1

2

1198714

(2120587)4int119866

0(119896)119873 (119896)

10038161003816100381610038161198691198961003816100381610038161003816

2

1198894119896

(68)


120575119873 (119896 119869) = 120575119887 (119869) ≃ minus1

2

1198714

(2120587)4int120590

2(119896 119901) 119866

0(119901)

10038161003816100381610038161003816119869119901

10038161003816100381610038161003816

2

1198894119901

(69)


2(119896 119901) = Δ119873lowast(119896)Δ119873(119901)

1205902(119896 119901) =

1

119885total (0)int [119863119873] 119890

minus119868(119873)Δ119873

lowast(119896) Δ119873 (119901) (70)



(119896 119901) rarr 1205902


120575119887 (119869) = minus1205902

119896

41205872

21198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(71)


41205872

1198962(120575] (0 119869) minus 120575] (119896 119869)) =

1

21205902

119896

41205872

1198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(72)






120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)


1205902





119896= 119873(119896)(1 minus





119896and


0 while


0119873

0= ΩDEΩ119887



asymp 075005 = 15 while 1205902

0119873

0sim 1



15




plusmn= 119878

2

plusmntimes 119877


+ 119903


plusmn




120572120573⟩ sim 8120587119866119899 and




1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3






plusmnrarr cup

1198991198783



1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)


= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the


cylinders1198793



minusℓ119889119905

0and the

sum sum+infin




119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)



plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119887total (119909 minus 1199091015840)

= int1

21205872119886(

1

1198772

minus

minus1

1198772

+

)120575 (10038161003816100381610038161003816 minus

1015840minus

+

10038161003816100381610038161003816minus 119886)119873119865 (120585) 119889120585

(22)


sdot

(1198894119896(2120587)


= int(41205872119896

2)119890

minus119894119896119909sdot

(1198894119896(2120587)

4) we find for 119887(119896) = int 119887(119909)119890

1198941198961199091198894119909 the expression

119887total (119896) = 119873int1198862 4120587

2

1198962(119865 (119886 119896) minus 119865 (119886 0))

1198691(119896119886)

1198961198862119889119886 (23)


119873119865 (119886119883) =119899

212058721199033

0

120575 (119886 minus 1198860) 120575 (119883 minus 119903

0) (24)



0=

|119877+minus119877



1198941198961199091198894119909 re-

duces to 119873119865(119886 119896) = 119899(1198691(119896119903

0)(119896119903

02))120575(119886 minus 119886

0) Thus from

(23) we find

119887 (119896) = minus1198991198862 4120587

2

1198962(1 minus

1198691(119896119903

0)

11989611990302

)1198691(119896119886

0)

11989611988602

(25)


119866true = 1198660(119896)119873 (119896) = 119866

0(119896) (1 + 119887 (119896)) (26)



we get 1198691(119896119903

0)(119896119903

02) asymp 1 minus (12) (119896119903

02)


119887(119896) asymp minus1205872119899119886

21199032


119887(119896) asymp minus int(12058722)119886

21199032

0119899(119886 119903

0)119889119886 119889119903

0 where 119899(119886 119903

0) is the den-



119873(119896) 997888rarr 1 minus1205872

2int 119886

4119899 (119886 119903

0)1199032

0

1198862119889119886 119889119903

0le 1 (27)





22)119886



2119896

2) (](119896) minus ](0)) where ](119896) =

int ](119883)119890119894119896119883


119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (28)


int(12058722)119886

4119865(119886119883)119889



2


119866true = 1198660(119896)119873 (119896) =

1

1198962 + 41205872(] (0) minus ] (119896))

(29)





119866true (119909 minus 1199091015840= 0) = int

1

1198962 + 41205872(] (0) minus ] (119896))

1198894119896

(2120587)4lt infin

(30)



1015840= + 119888 with an




119888 = 1198714= 119881which



1015840 120585) rarr




119866 (119896) = 1198660(119896)119873 (119896 120585) (31)



119866 =119873 (119896 120585)

1198962 + 1198982 (32)


119885total (119869)

= int [119863119873 (119896)] 119890minus119868(119873)

119890minus(12)(119871

4

(2120587)4

) int((119873(119896)(1198962

+1198982

))|119869119896

|2

)119889119896

(33)

where [119863119873] = prod119896119889119873



to119873(119896))

119890minus119868(119873)

= int [119863119865]1198850(119873 (119896 120585)) 120575 (119873 (119896) minus 119873 (119896 120585)) (34)




119866 (119896) =119873 (119896)

1198962 + 1198982 (35)


119873(119896) =1

119885total (0)int [119863119873] 119890

minus119868(119873)119873(119896) (36)




2119878 may


2119878 =




119891 (119896) =1

119885total (120582 0)int [119863119873] 119890

(120582119873)minus119868(119873)119873(119896) (37)


4(2120587)

4)sdot

1198660(119896)|119869


ext=

1198660(119896)119869








119873(119909 1199091015840) = sum119873

119894119891119894(119909)119891

lowast

119894(119909


119894= 0 1

(since 1198732

119894= 119873




+ 119877


plusmn





0to the equation



0) which obey




0) may


0 By


perp


perp



+


= (minus+

+)2 (Instead


0= 1(119909 minus 119909

+)2minus 1(119909 minus 119909








minus+ (119904

+minus 119904

minus)1205792120587






perp



0))

2

=

119892120572120573119889119910

120572119889119910

120573 (where 119910 = (120579 119909perp




1

radic119892120575 (120579 minus 120579

1015840) 997888rarr 119873(120579 minus 120579

1015840) =

+infin

sum

119899=minusinfin

1

radic119892120575 (120579 minus 120579

1015840+ 2120587119899)

(38)


119873(119896 1198961015840) =

+infin

sum

119899=minusinfin

120575 (119896 minus 119899) 120575 (119896 minus 1198961015840) (39)



the projection 119891119873= 119891 119891




119873total (119909 1199091015840) = int(prod

119894

radic1198921198941198894119910119894)119873(120585

1 119909 119910

1)

times 119873 (1205852 119910

1 119910

2) sdot sdot sdot 119873 (120585

119873 119910

119873minus1 119909

1015840)

(40)

where 119873(120585119894 119909 119909



119894 119909 119909

1015840) realizes




119894)

5 Cutoff



4 and periodic


119896rarr

(1198714(2120587)

4) int 119889



119868 = 1198680+sum120582

1(119896)119873 (119896) +

1

2sum120582

2(119896 119896

1015840)119873 (119896)119873 (119896


(41)


0(119896) We


sum

1198961015840

1205822(119896 119896

1015840)119873 (119896

1015840) = minus120582

1(119896) (42)


1015840) = Ψ(119896)Ψ

+(119896



(119896)Ψ(1198961015840)+Ψ(119896

1015840) Ψ

+(119896) = 120575(119896minus


119873(119896 1198961015840) = 120575(119896 minus 119896



0+ (Ψ




find

119873(119896) =1

119885total (119896)sum

119873=01

119890minus1205821

(119896)119873(119896)119873(119896) =

119890minus1205821

(119896)

1 + 119890minus1205821(119896) (43)


0(119896)










int119873 (119896)1198894119896

(2120587)4= int

119873 (119896) 1198964

(2120587)2

119889119896

119896 (44)


1198964119873(119896) sim 119896

119863 (45)




119868119864= minus

1

16120587119866int (119877 minus 2Λ

0)radic119892119889

4119909 minus int119871

119898radic1198921198894119909 (46)


119877119886119887minus1

2119892119886119887119877 + 119892

119886119887Λ

0= 8120587119866119879

119886119887 (47)

where 119879119886119887= (12)(119892)

minus12(120575119871

119898120575119892




Λ tot = Λ0+ Λ

119898+ Λ

119877= Λ

0+ 2120587119866 ⟨119879⟩ +

1

4⟨119877⟩119908 (48)



⟨119877⟩119908




1198891199042= ℎ

2(119903)120575

120572120573119889119909

120572119889119909


119877120572120573

= 7 minus (ℎ1015840

ℎ)

1015840

(120575120572120573

+ (119873 minus 2) 119899120572119899120573)

minus1

119903

ℎ1015840

ℎ((119873 minus 2) Δ

120572120573+ 120575

120572120573(119873 minus 1))

minus (ℎ1015840

ℎ)

2

(119873 minus 2) Δ120572120573

(49)

where 119899]= 119909


surface Δ120572120573

= 120575120572120573



119877 = minus(119873 minus 1)

ℎ2[2

ℎ10158401015840

ℎ+ 2

1

119903

ℎ1015840

ℎ(119873 minus 1) + (

ℎ1015840

ℎ)

2

(119873 minus 4)]

(50)



we find

minusℎ4

119873 minus 1119877 = 4ℎ

1198862

1199033120575 +

2ℎ

119903119873minus1(119903

119873minus1120582)

1015840

+ (119873 minus 4)(1205822minus 4120579

1198862

1199033120582)

+ 4 (119873 minus 4)1198862

1199034120579 (120579 minus 1)

(51)


2119903

2)120575 and 120575 = minus120579


minus119877 =4 (119873 minus 1)

119886120575 (119903 minus 119886) +

2 (119873 minus 1)

119903119873minus1(119903

119873minus3(119903

2minus 119886

2) 120575)

1015840

(52)


1

4int119877radic119892119889

4119909 = minus6120587

21198862 (53)



1

4int119877radic119892119889

4119909 = minus6120587

2sum

119895

1198862

119895

= minus121205872int119899 (119886) 119886

2119889119886 119889

4119909 = intΛ

1198771198894119909

(54)

where 2119899(119886) = int 119899(119886 1199030)119889119903



Λ119877= minus12120587

2int119899 (119886) 119886

2119889119886 lt 0 (55)



minus119879120572120573

(119909) = 120597120572120593120597

120573120593 minus

1

2119892120572120573

(120597120583120593120597

120583120593 + 119898

21205932) (56)


minus ⟨119879120572120573

(119909)⟩

= lim1199091015840

rarr119909

(1205971205721205971015840

120573minus1

2119892120572120573

(1205971205831205971015840

120583+ 119898

2)) ⟨119866 (119909 119909

1015840 120585)⟩

(57)


int 119890minus119894119896(119909minus119909

1015840

)119866(119896 120585) (119889

4119896(2120587)


we arrive at

⟨119879120572120573

(119909)⟩ =1

4119892120572120573

int(1 +119898

2

1198962 + 1198982)119873 (119896)

1198894119896

(2120587)4 (58)


2)119889

4119896 = (14)119892

120572120573int 119896

2119891(119896

2)119889

4119896



1205722(120587

1205722+ 119896

120572) from


Λ119898=2120587119866sumint

1205871205722

1205871205722+119896120572

1198894119896

(2120587)4=120587119866

4Γ (

120572 minus 4

120572) Γ (

4

120572) sim 1

(59)


1(119896) = minussum ln119885

0(119896) + 120575120582(119896) with

120575120582(119896) ≪ ln1198850(119896) as 119896 ≫ 1)


120572120573⟩ = Λ

lowast119892120572120573

with Λlowast


120572120573⟩ = 120575Λ119892

120572120573 where 120575Λ =




2120587119866int 119887(119896) (1198894119896(2120587)



119887total (0) = minus119899

4120587411988631199033

0

int(1 minus1198862

1198772

minus

)120575 (119877+minus 119886)

times 120575 (1003816100381610038161003816119877+

minus 119877minus

1003816100381610038161003816 minus 1199030) 119889

4119877minus1198894119877+

(60)

which gives


1199030

)) (61)

where

119891(119886

1199030

) =2

120587int

120587

0

1198862sin2

120579119889120579

1198862 + 21198861199030cos 120579 + 119903

2

0

(62)


0le 12)


2119903

2

0 Thus for the


120575Λ119898= minus2120587119866int119899 (119886 119903

0) (1 minus 119891(

119886

1199030

))119889119886 1198891199030

= minus2120587119866119899 (1 minus ⟨119891⟩)

(63)


119898= Λ

lowast+ 120575Λ



0)(120587

22)119886

4119889119886 119889119903

0lt 1 where (120587

22)119886

4





≫ Λ119877


119877120575Λ





119873119865 (119886119883) =1

1198862120575 (119886) ] (119883) (64)

where ](119883) = int 1198862119873119865(119886119883)119889119886 and int(1119886

2)](119883)119889

4119883 =




4119889119886 119889119903

0=

1198862int ](119883)119889



1198941198961198831198894119883) as

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (65)


Λ119877= minus12120587

2int119899 (119886 119903

0) 119886

2119889119886 119889119903

0

= minus121205872int ] (119883) 119889

4119883 = minus12120587

2] (0)

(66)


Λ tot = Λ0

+ 2120587119866int(1minus4120587

2

1198962(] (0)minus] (119896)))

1198894119896

(2120587)4minus12120587

2] (0)

(67)


4


119898)









119881 = minus1

2int 119869 (119909) 119866 (119909 119910) 119869 (119910) 119889

4119909 119889

4119910

= minus1

2

1198714

(2120587)4int119866

0(119896)119873 (119896)

10038161003816100381610038161198691198961003816100381610038161003816

2

1198894119896

(68)


120575119873 (119896 119869) = 120575119887 (119869) ≃ minus1

2

1198714

(2120587)4int120590

2(119896 119901) 119866

0(119901)

10038161003816100381610038161003816119869119901

10038161003816100381610038161003816

2

1198894119901

(69)


2(119896 119901) = Δ119873lowast(119896)Δ119873(119901)

1205902(119896 119901) =

1

119885total (0)int [119863119873] 119890

minus119868(119873)Δ119873

lowast(119896) Δ119873 (119901) (70)



(119896 119901) rarr 1205902


120575119887 (119869) = minus1205902

119896

41205872

21198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(71)


41205872

1198962(120575] (0 119869) minus 120575] (119896 119869)) =

1

21205902

119896

41205872

1198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(72)






120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)


1205902





119896= 119873(119896)(1 minus





119896and


0 while


0119873

0= ΩDEΩ119887



asymp 075005 = 15 while 1205902

0119873

0sim 1



15




plusmn= 119878

2

plusmntimes 119877


+ 119903


plusmn




120572120573⟩ sim 8120587119866119899 and




1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3






plusmnrarr cup

1198991198783



1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)


= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the


cylinders1198793



minusℓ119889119905

0and the

sum sum+infin




119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)



plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119866 =119873 (119896 120585)

1198962 + 1198982 (32)


119885total (119869)

= int [119863119873 (119896)] 119890minus119868(119873)

119890minus(12)(119871

4

(2120587)4

) int((119873(119896)(1198962

+1198982

))|119869119896

|2

)119889119896

(33)

where [119863119873] = prod119896119889119873



to119873(119896))

119890minus119868(119873)

= int [119863119865]1198850(119873 (119896 120585)) 120575 (119873 (119896) minus 119873 (119896 120585)) (34)




119866 (119896) =119873 (119896)

1198962 + 1198982 (35)


119873(119896) =1

119885total (0)int [119863119873] 119890

minus119868(119873)119873(119896) (36)




2119878 may


2119878 =




119891 (119896) =1

119885total (120582 0)int [119863119873] 119890

(120582119873)minus119868(119873)119873(119896) (37)


4(2120587)

4)sdot

1198660(119896)|119869


ext=

1198660(119896)119869








119873(119909 1199091015840) = sum119873

119894119891119894(119909)119891

lowast

119894(119909


119894= 0 1

(since 1198732

119894= 119873




+ 119877


plusmn





0to the equation



0) which obey




0) may


0 By


perp


perp



+


= (minus+

+)2 (Instead


0= 1(119909 minus 119909

+)2minus 1(119909 minus 119909








minus+ (119904

+minus 119904

minus)1205792120587






perp



0))

2

=

119892120572120573119889119910

120572119889119910

120573 (where 119910 = (120579 119909perp




1

radic119892120575 (120579 minus 120579

1015840) 997888rarr 119873(120579 minus 120579

1015840) =

+infin

sum

119899=minusinfin

1

radic119892120575 (120579 minus 120579

1015840+ 2120587119899)

(38)


119873(119896 1198961015840) =

+infin

sum

119899=minusinfin

120575 (119896 minus 119899) 120575 (119896 minus 1198961015840) (39)



the projection 119891119873= 119891 119891




119873total (119909 1199091015840) = int(prod

119894

radic1198921198941198894119910119894)119873(120585

1 119909 119910

1)

times 119873 (1205852 119910

1 119910

2) sdot sdot sdot 119873 (120585

119873 119910

119873minus1 119909

1015840)

(40)

where 119873(120585119894 119909 119909



119894 119909 119909

1015840) realizes




119894)

5 Cutoff



4 and periodic


119896rarr

(1198714(2120587)

4) int 119889



119868 = 1198680+sum120582

1(119896)119873 (119896) +

1

2sum120582

2(119896 119896

1015840)119873 (119896)119873 (119896


(41)


0(119896) We


sum

1198961015840

1205822(119896 119896

1015840)119873 (119896

1015840) = minus120582

1(119896) (42)


1015840) = Ψ(119896)Ψ

+(119896



(119896)Ψ(1198961015840)+Ψ(119896

1015840) Ψ

+(119896) = 120575(119896minus


119873(119896 1198961015840) = 120575(119896 minus 119896



0+ (Ψ




find

119873(119896) =1

119885total (119896)sum

119873=01

119890minus1205821

(119896)119873(119896)119873(119896) =

119890minus1205821

(119896)

1 + 119890minus1205821(119896) (43)


0(119896)










int119873 (119896)1198894119896

(2120587)4= int

119873 (119896) 1198964

(2120587)2

119889119896

119896 (44)


1198964119873(119896) sim 119896

119863 (45)




119868119864= minus

1

16120587119866int (119877 minus 2Λ

0)radic119892119889

4119909 minus int119871

119898radic1198921198894119909 (46)


119877119886119887minus1

2119892119886119887119877 + 119892

119886119887Λ

0= 8120587119866119879

119886119887 (47)

where 119879119886119887= (12)(119892)

minus12(120575119871

119898120575119892




Λ tot = Λ0+ Λ

119898+ Λ

119877= Λ

0+ 2120587119866 ⟨119879⟩ +

1

4⟨119877⟩119908 (48)



⟨119877⟩119908




1198891199042= ℎ

2(119903)120575

120572120573119889119909

120572119889119909


119877120572120573

= 7 minus (ℎ1015840

ℎ)

1015840

(120575120572120573

+ (119873 minus 2) 119899120572119899120573)

minus1

119903

ℎ1015840

ℎ((119873 minus 2) Δ

120572120573+ 120575

120572120573(119873 minus 1))

minus (ℎ1015840

ℎ)

2

(119873 minus 2) Δ120572120573

(49)

where 119899]= 119909


surface Δ120572120573

= 120575120572120573



119877 = minus(119873 minus 1)

ℎ2[2

ℎ10158401015840

ℎ+ 2

1

119903

ℎ1015840

ℎ(119873 minus 1) + (

ℎ1015840

ℎ)

2

(119873 minus 4)]

(50)



we find

minusℎ4

119873 minus 1119877 = 4ℎ

1198862

1199033120575 +

2ℎ

119903119873minus1(119903

119873minus1120582)

1015840

+ (119873 minus 4)(1205822minus 4120579

1198862

1199033120582)

+ 4 (119873 minus 4)1198862

1199034120579 (120579 minus 1)

(51)


2119903

2)120575 and 120575 = minus120579


minus119877 =4 (119873 minus 1)

119886120575 (119903 minus 119886) +

2 (119873 minus 1)

119903119873minus1(119903

119873minus3(119903

2minus 119886

2) 120575)

1015840

(52)


1

4int119877radic119892119889

4119909 = minus6120587

21198862 (53)



1

4int119877radic119892119889

4119909 = minus6120587

2sum

119895

1198862

119895

= minus121205872int119899 (119886) 119886

2119889119886 119889

4119909 = intΛ

1198771198894119909

(54)

where 2119899(119886) = int 119899(119886 1199030)119889119903



Λ119877= minus12120587

2int119899 (119886) 119886

2119889119886 lt 0 (55)



minus119879120572120573

(119909) = 120597120572120593120597

120573120593 minus

1

2119892120572120573

(120597120583120593120597

120583120593 + 119898

21205932) (56)


minus ⟨119879120572120573

(119909)⟩

= lim1199091015840

rarr119909

(1205971205721205971015840

120573minus1

2119892120572120573

(1205971205831205971015840

120583+ 119898

2)) ⟨119866 (119909 119909

1015840 120585)⟩

(57)


int 119890minus119894119896(119909minus119909

1015840

)119866(119896 120585) (119889

4119896(2120587)


we arrive at

⟨119879120572120573

(119909)⟩ =1

4119892120572120573

int(1 +119898

2

1198962 + 1198982)119873 (119896)

1198894119896

(2120587)4 (58)


2)119889

4119896 = (14)119892

120572120573int 119896

2119891(119896

2)119889

4119896



1205722(120587

1205722+ 119896

120572) from


Λ119898=2120587119866sumint

1205871205722

1205871205722+119896120572

1198894119896

(2120587)4=120587119866

4Γ (

120572 minus 4

120572) Γ (

4

120572) sim 1

(59)


1(119896) = minussum ln119885

0(119896) + 120575120582(119896) with

120575120582(119896) ≪ ln1198850(119896) as 119896 ≫ 1)


120572120573⟩ = Λ

lowast119892120572120573

with Λlowast


120572120573⟩ = 120575Λ119892

120572120573 where 120575Λ =




2120587119866int 119887(119896) (1198894119896(2120587)



119887total (0) = minus119899

4120587411988631199033

0

int(1 minus1198862

1198772

minus

)120575 (119877+minus 119886)

times 120575 (1003816100381610038161003816119877+

minus 119877minus

1003816100381610038161003816 minus 1199030) 119889

4119877minus1198894119877+

(60)

which gives


1199030

)) (61)

where

119891(119886

1199030

) =2

120587int

120587

0

1198862sin2

120579119889120579

1198862 + 21198861199030cos 120579 + 119903

2

0

(62)


0le 12)


2119903

2

0 Thus for the


120575Λ119898= minus2120587119866int119899 (119886 119903

0) (1 minus 119891(

119886

1199030

))119889119886 1198891199030

= minus2120587119866119899 (1 minus ⟨119891⟩)

(63)


119898= Λ

lowast+ 120575Λ



0)(120587

22)119886

4119889119886 119889119903

0lt 1 where (120587

22)119886

4





≫ Λ119877


119877120575Λ





119873119865 (119886119883) =1

1198862120575 (119886) ] (119883) (64)

where ](119883) = int 1198862119873119865(119886119883)119889119886 and int(1119886

2)](119883)119889

4119883 =




4119889119886 119889119903

0=

1198862int ](119883)119889



1198941198961198831198894119883) as

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (65)


Λ119877= minus12120587

2int119899 (119886 119903

0) 119886

2119889119886 119889119903

0

= minus121205872int ] (119883) 119889

4119883 = minus12120587

2] (0)

(66)


Λ tot = Λ0

+ 2120587119866int(1minus4120587

2

1198962(] (0)minus] (119896)))

1198894119896

(2120587)4minus12120587

2] (0)

(67)


4


119898)









119881 = minus1

2int 119869 (119909) 119866 (119909 119910) 119869 (119910) 119889

4119909 119889

4119910

= minus1

2

1198714

(2120587)4int119866

0(119896)119873 (119896)

10038161003816100381610038161198691198961003816100381610038161003816

2

1198894119896

(68)


120575119873 (119896 119869) = 120575119887 (119869) ≃ minus1

2

1198714

(2120587)4int120590

2(119896 119901) 119866

0(119901)

10038161003816100381610038161003816119869119901

10038161003816100381610038161003816

2

1198894119901

(69)


2(119896 119901) = Δ119873lowast(119896)Δ119873(119901)

1205902(119896 119901) =

1

119885total (0)int [119863119873] 119890

minus119868(119873)Δ119873

lowast(119896) Δ119873 (119901) (70)



(119896 119901) rarr 1205902


120575119887 (119869) = minus1205902

119896

41205872

21198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(71)


41205872

1198962(120575] (0 119869) minus 120575] (119896 119869)) =

1

21205902

119896

41205872

1198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(72)






120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)


1205902





119896= 119873(119896)(1 minus





119896and


0 while


0119873

0= ΩDEΩ119887



asymp 075005 = 15 while 1205902

0119873

0sim 1



15




plusmn= 119878

2

plusmntimes 119877


+ 119903


plusmn




120572120573⟩ sim 8120587119866119899 and




1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3






plusmnrarr cup

1198991198783



1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)


= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the


cylinders1198793



minusℓ119889119905

0and the

sum sum+infin




119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)



plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







perp



0))

2

=

119892120572120573119889119910

120572119889119910

120573 (where 119910 = (120579 119909perp




1

radic119892120575 (120579 minus 120579

1015840) 997888rarr 119873(120579 minus 120579

1015840) =

+infin

sum

119899=minusinfin

1

radic119892120575 (120579 minus 120579

1015840+ 2120587119899)

(38)


119873(119896 1198961015840) =

+infin

sum

119899=minusinfin

120575 (119896 minus 119899) 120575 (119896 minus 1198961015840) (39)



the projection 119891119873= 119891 119891




119873total (119909 1199091015840) = int(prod

119894

radic1198921198941198894119910119894)119873(120585

1 119909 119910

1)

times 119873 (1205852 119910

1 119910

2) sdot sdot sdot 119873 (120585

119873 119910

119873minus1 119909

1015840)

(40)

where 119873(120585119894 119909 119909



119894 119909 119909

1015840) realizes




119894)

5 Cutoff



4 and periodic


119896rarr

(1198714(2120587)

4) int 119889



119868 = 1198680+sum120582

1(119896)119873 (119896) +

1

2sum120582

2(119896 119896

1015840)119873 (119896)119873 (119896


(41)


0(119896) We


sum

1198961015840

1205822(119896 119896

1015840)119873 (119896

1015840) = minus120582

1(119896) (42)


1015840) = Ψ(119896)Ψ

+(119896



(119896)Ψ(1198961015840)+Ψ(119896

1015840) Ψ

+(119896) = 120575(119896minus


119873(119896 1198961015840) = 120575(119896 minus 119896



0+ (Ψ




find

119873(119896) =1

119885total (119896)sum

119873=01

119890minus1205821

(119896)119873(119896)119873(119896) =

119890minus1205821

(119896)

1 + 119890minus1205821(119896) (43)


0(119896)










int119873 (119896)1198894119896

(2120587)4= int

119873 (119896) 1198964

(2120587)2

119889119896

119896 (44)


1198964119873(119896) sim 119896

119863 (45)




119868119864= minus

1

16120587119866int (119877 minus 2Λ

0)radic119892119889

4119909 minus int119871

119898radic1198921198894119909 (46)


119877119886119887minus1

2119892119886119887119877 + 119892

119886119887Λ

0= 8120587119866119879

119886119887 (47)

where 119879119886119887= (12)(119892)

minus12(120575119871

119898120575119892




Λ tot = Λ0+ Λ

119898+ Λ

119877= Λ

0+ 2120587119866 ⟨119879⟩ +

1

4⟨119877⟩119908 (48)



⟨119877⟩119908




1198891199042= ℎ

2(119903)120575

120572120573119889119909

120572119889119909


119877120572120573

= 7 minus (ℎ1015840

ℎ)

1015840

(120575120572120573

+ (119873 minus 2) 119899120572119899120573)

minus1

119903

ℎ1015840

ℎ((119873 minus 2) Δ

120572120573+ 120575

120572120573(119873 minus 1))

minus (ℎ1015840

ℎ)

2

(119873 minus 2) Δ120572120573

(49)

where 119899]= 119909


surface Δ120572120573

= 120575120572120573



119877 = minus(119873 minus 1)

ℎ2[2

ℎ10158401015840

ℎ+ 2

1

119903

ℎ1015840

ℎ(119873 minus 1) + (

ℎ1015840

ℎ)

2

(119873 minus 4)]

(50)



we find

minusℎ4

119873 minus 1119877 = 4ℎ

1198862

1199033120575 +

2ℎ

119903119873minus1(119903

119873minus1120582)

1015840

+ (119873 minus 4)(1205822minus 4120579

1198862

1199033120582)

+ 4 (119873 minus 4)1198862

1199034120579 (120579 minus 1)

(51)


2119903

2)120575 and 120575 = minus120579


minus119877 =4 (119873 minus 1)

119886120575 (119903 minus 119886) +

2 (119873 minus 1)

119903119873minus1(119903

119873minus3(119903

2minus 119886

2) 120575)

1015840

(52)


1

4int119877radic119892119889

4119909 = minus6120587

21198862 (53)



1

4int119877radic119892119889

4119909 = minus6120587

2sum

119895

1198862

119895

= minus121205872int119899 (119886) 119886

2119889119886 119889

4119909 = intΛ

1198771198894119909

(54)

where 2119899(119886) = int 119899(119886 1199030)119889119903



Λ119877= minus12120587

2int119899 (119886) 119886

2119889119886 lt 0 (55)



minus119879120572120573

(119909) = 120597120572120593120597

120573120593 minus

1

2119892120572120573

(120597120583120593120597

120583120593 + 119898

21205932) (56)


minus ⟨119879120572120573

(119909)⟩

= lim1199091015840

rarr119909

(1205971205721205971015840

120573minus1

2119892120572120573

(1205971205831205971015840

120583+ 119898

2)) ⟨119866 (119909 119909

1015840 120585)⟩

(57)


int 119890minus119894119896(119909minus119909

1015840

)119866(119896 120585) (119889

4119896(2120587)


we arrive at

⟨119879120572120573

(119909)⟩ =1

4119892120572120573

int(1 +119898

2

1198962 + 1198982)119873 (119896)

1198894119896

(2120587)4 (58)


2)119889

4119896 = (14)119892

120572120573int 119896

2119891(119896

2)119889

4119896



1205722(120587

1205722+ 119896

120572) from


Λ119898=2120587119866sumint

1205871205722

1205871205722+119896120572

1198894119896

(2120587)4=120587119866

4Γ (

120572 minus 4

120572) Γ (

4

120572) sim 1

(59)


1(119896) = minussum ln119885

0(119896) + 120575120582(119896) with

120575120582(119896) ≪ ln1198850(119896) as 119896 ≫ 1)


120572120573⟩ = Λ

lowast119892120572120573

with Λlowast


120572120573⟩ = 120575Λ119892

120572120573 where 120575Λ =




2120587119866int 119887(119896) (1198894119896(2120587)



119887total (0) = minus119899

4120587411988631199033

0

int(1 minus1198862

1198772

minus

)120575 (119877+minus 119886)

times 120575 (1003816100381610038161003816119877+

minus 119877minus

1003816100381610038161003816 minus 1199030) 119889

4119877minus1198894119877+

(60)

which gives


1199030

)) (61)

where

119891(119886

1199030

) =2

120587int

120587

0

1198862sin2

120579119889120579

1198862 + 21198861199030cos 120579 + 119903

2

0

(62)


0le 12)


2119903

2

0 Thus for the


120575Λ119898= minus2120587119866int119899 (119886 119903

0) (1 minus 119891(

119886

1199030

))119889119886 1198891199030

= minus2120587119866119899 (1 minus ⟨119891⟩)

(63)


119898= Λ

lowast+ 120575Λ



0)(120587

22)119886

4119889119886 119889119903

0lt 1 where (120587

22)119886

4





≫ Λ119877


119877120575Λ





119873119865 (119886119883) =1

1198862120575 (119886) ] (119883) (64)

where ](119883) = int 1198862119873119865(119886119883)119889119886 and int(1119886

2)](119883)119889

4119883 =




4119889119886 119889119903

0=

1198862int ](119883)119889



1198941198961198831198894119883) as

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (65)


Λ119877= minus12120587

2int119899 (119886 119903

0) 119886

2119889119886 119889119903

0

= minus121205872int ] (119883) 119889

4119883 = minus12120587

2] (0)

(66)


Λ tot = Λ0

+ 2120587119866int(1minus4120587

2

1198962(] (0)minus] (119896)))

1198894119896

(2120587)4minus12120587

2] (0)

(67)


4


119898)









119881 = minus1

2int 119869 (119909) 119866 (119909 119910) 119869 (119910) 119889

4119909 119889

4119910

= minus1

2

1198714

(2120587)4int119866

0(119896)119873 (119896)

10038161003816100381610038161198691198961003816100381610038161003816

2

1198894119896

(68)


120575119873 (119896 119869) = 120575119887 (119869) ≃ minus1

2

1198714

(2120587)4int120590

2(119896 119901) 119866

0(119901)

10038161003816100381610038161003816119869119901

10038161003816100381610038161003816

2

1198894119901

(69)


2(119896 119901) = Δ119873lowast(119896)Δ119873(119901)

1205902(119896 119901) =

1

119885total (0)int [119863119873] 119890

minus119868(119873)Δ119873

lowast(119896) Δ119873 (119901) (70)



(119896 119901) rarr 1205902


120575119887 (119869) = minus1205902

119896

41205872

21198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(71)


41205872

1198962(120575] (0 119869) minus 120575] (119896 119869)) =

1

21205902

119896

41205872

1198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(72)






120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)


1205902





119896= 119873(119896)(1 minus





119896and


0 while


0119873

0= ΩDEΩ119887



asymp 075005 = 15 while 1205902

0119873

0sim 1



15




plusmn= 119878

2

plusmntimes 119877


+ 119903


plusmn




120572120573⟩ sim 8120587119866119899 and




1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3






plusmnrarr cup

1198991198783



1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)


= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the


cylinders1198793



minusℓ119889119905

0and the

sum sum+infin




119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)



plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





int119873 (119896)1198894119896

(2120587)4= int

119873 (119896) 1198964

(2120587)2

119889119896

119896 (44)


1198964119873(119896) sim 119896

119863 (45)




119868119864= minus

1

16120587119866int (119877 minus 2Λ

0)radic119892119889

4119909 minus int119871

119898radic1198921198894119909 (46)


119877119886119887minus1

2119892119886119887119877 + 119892

119886119887Λ

0= 8120587119866119879

119886119887 (47)

where 119879119886119887= (12)(119892)

minus12(120575119871

119898120575119892




Λ tot = Λ0+ Λ

119898+ Λ

119877= Λ

0+ 2120587119866 ⟨119879⟩ +

1

4⟨119877⟩119908 (48)



⟨119877⟩119908




1198891199042= ℎ

2(119903)120575

120572120573119889119909

120572119889119909


119877120572120573

= 7 minus (ℎ1015840

ℎ)

1015840

(120575120572120573

+ (119873 minus 2) 119899120572119899120573)

minus1

119903

ℎ1015840

ℎ((119873 minus 2) Δ

120572120573+ 120575

120572120573(119873 minus 1))

minus (ℎ1015840

ℎ)

2

(119873 minus 2) Δ120572120573

(49)

where 119899]= 119909


surface Δ120572120573

= 120575120572120573



119877 = minus(119873 minus 1)

ℎ2[2

ℎ10158401015840

ℎ+ 2

1

119903

ℎ1015840

ℎ(119873 minus 1) + (

ℎ1015840

ℎ)

2

(119873 minus 4)]

(50)



we find

minusℎ4

119873 minus 1119877 = 4ℎ

1198862

1199033120575 +

2ℎ

119903119873minus1(119903

119873minus1120582)

1015840

+ (119873 minus 4)(1205822minus 4120579

1198862

1199033120582)

+ 4 (119873 minus 4)1198862

1199034120579 (120579 minus 1)

(51)


2119903

2)120575 and 120575 = minus120579


minus119877 =4 (119873 minus 1)

119886120575 (119903 minus 119886) +

2 (119873 minus 1)

119903119873minus1(119903

119873minus3(119903

2minus 119886

2) 120575)

1015840

(52)


1

4int119877radic119892119889

4119909 = minus6120587

21198862 (53)



1

4int119877radic119892119889

4119909 = minus6120587

2sum

119895

1198862

119895

= minus121205872int119899 (119886) 119886

2119889119886 119889

4119909 = intΛ

1198771198894119909

(54)

where 2119899(119886) = int 119899(119886 1199030)119889119903



Λ119877= minus12120587

2int119899 (119886) 119886

2119889119886 lt 0 (55)



minus119879120572120573

(119909) = 120597120572120593120597

120573120593 minus

1

2119892120572120573

(120597120583120593120597

120583120593 + 119898

21205932) (56)


minus ⟨119879120572120573

(119909)⟩

= lim1199091015840

rarr119909

(1205971205721205971015840

120573minus1

2119892120572120573

(1205971205831205971015840

120583+ 119898

2)) ⟨119866 (119909 119909

1015840 120585)⟩

(57)


int 119890minus119894119896(119909minus119909

1015840

)119866(119896 120585) (119889

4119896(2120587)


we arrive at

⟨119879120572120573

(119909)⟩ =1

4119892120572120573

int(1 +119898

2

1198962 + 1198982)119873 (119896)

1198894119896

(2120587)4 (58)


2)119889

4119896 = (14)119892

120572120573int 119896

2119891(119896

2)119889

4119896



1205722(120587

1205722+ 119896

120572) from


Λ119898=2120587119866sumint

1205871205722

1205871205722+119896120572

1198894119896

(2120587)4=120587119866

4Γ (

120572 minus 4

120572) Γ (

4

120572) sim 1

(59)


1(119896) = minussum ln119885

0(119896) + 120575120582(119896) with

120575120582(119896) ≪ ln1198850(119896) as 119896 ≫ 1)


120572120573⟩ = Λ

lowast119892120572120573

with Λlowast


120572120573⟩ = 120575Λ119892

120572120573 where 120575Λ =




2120587119866int 119887(119896) (1198894119896(2120587)



119887total (0) = minus119899

4120587411988631199033

0

int(1 minus1198862

1198772

minus

)120575 (119877+minus 119886)

times 120575 (1003816100381610038161003816119877+

minus 119877minus

1003816100381610038161003816 minus 1199030) 119889

4119877minus1198894119877+

(60)

which gives


1199030

)) (61)

where

119891(119886

1199030

) =2

120587int

120587

0

1198862sin2

120579119889120579

1198862 + 21198861199030cos 120579 + 119903

2

0

(62)


0le 12)


2119903

2

0 Thus for the


120575Λ119898= minus2120587119866int119899 (119886 119903

0) (1 minus 119891(

119886

1199030

))119889119886 1198891199030

= minus2120587119866119899 (1 minus ⟨119891⟩)

(63)


119898= Λ

lowast+ 120575Λ



0)(120587

22)119886

4119889119886 119889119903

0lt 1 where (120587

22)119886

4





≫ Λ119877


119877120575Λ





119873119865 (119886119883) =1

1198862120575 (119886) ] (119883) (64)

where ](119883) = int 1198862119873119865(119886119883)119889119886 and int(1119886

2)](119883)119889

4119883 =




4119889119886 119889119903

0=

1198862int ](119883)119889



1198941198961198831198894119883) as

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (65)


Λ119877= minus12120587

2int119899 (119886 119903

0) 119886

2119889119886 119889119903

0

= minus121205872int ] (119883) 119889

4119883 = minus12120587

2] (0)

(66)


Λ tot = Λ0

+ 2120587119866int(1minus4120587

2

1198962(] (0)minus] (119896)))

1198894119896

(2120587)4minus12120587

2] (0)

(67)


4


119898)









119881 = minus1

2int 119869 (119909) 119866 (119909 119910) 119869 (119910) 119889

4119909 119889

4119910

= minus1

2

1198714

(2120587)4int119866

0(119896)119873 (119896)

10038161003816100381610038161198691198961003816100381610038161003816

2

1198894119896

(68)


120575119873 (119896 119869) = 120575119887 (119869) ≃ minus1

2

1198714

(2120587)4int120590

2(119896 119901) 119866

0(119901)

10038161003816100381610038161003816119869119901

10038161003816100381610038161003816

2

1198894119901

(69)


2(119896 119901) = Δ119873lowast(119896)Δ119873(119901)

1205902(119896 119901) =

1

119885total (0)int [119863119873] 119890

minus119868(119873)Δ119873

lowast(119896) Δ119873 (119901) (70)



(119896 119901) rarr 1205902


120575119887 (119869) = minus1205902

119896

41205872

21198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(71)


41205872

1198962(120575] (0 119869) minus 120575] (119896 119869)) =

1

21205902

119896

41205872

1198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(72)






120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)


1205902





119896= 119873(119896)(1 minus





119896and


0 while


0119873

0= ΩDEΩ119887



asymp 075005 = 15 while 1205902

0119873

0sim 1



15




plusmn= 119878

2

plusmntimes 119877


+ 119903


plusmn




120572120573⟩ sim 8120587119866119899 and




1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3






plusmnrarr cup

1198991198783



1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)


= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the


cylinders1198793



minusℓ119889119905

0and the

sum sum+infin




119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)



plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1

4int119877radic119892119889

4119909 = minus6120587

2sum

119895

1198862

119895

= minus121205872int119899 (119886) 119886

2119889119886 119889

4119909 = intΛ

1198771198894119909

(54)

where 2119899(119886) = int 119899(119886 1199030)119889119903



Λ119877= minus12120587

2int119899 (119886) 119886

2119889119886 lt 0 (55)



minus119879120572120573

(119909) = 120597120572120593120597

120573120593 minus

1

2119892120572120573

(120597120583120593120597

120583120593 + 119898

21205932) (56)


minus ⟨119879120572120573

(119909)⟩

= lim1199091015840

rarr119909

(1205971205721205971015840

120573minus1

2119892120572120573

(1205971205831205971015840

120583+ 119898

2)) ⟨119866 (119909 119909

1015840 120585)⟩

(57)


int 119890minus119894119896(119909minus119909

1015840

)119866(119896 120585) (119889

4119896(2120587)


we arrive at

⟨119879120572120573

(119909)⟩ =1

4119892120572120573

int(1 +119898

2

1198962 + 1198982)119873 (119896)

1198894119896

(2120587)4 (58)


2)119889

4119896 = (14)119892

120572120573int 119896

2119891(119896

2)119889

4119896



1205722(120587

1205722+ 119896

120572) from


Λ119898=2120587119866sumint

1205871205722

1205871205722+119896120572

1198894119896

(2120587)4=120587119866

4Γ (

120572 minus 4

120572) Γ (

4

120572) sim 1

(59)


1(119896) = minussum ln119885

0(119896) + 120575120582(119896) with

120575120582(119896) ≪ ln1198850(119896) as 119896 ≫ 1)


120572120573⟩ = Λ

lowast119892120572120573

with Λlowast


120572120573⟩ = 120575Λ119892

120572120573 where 120575Λ =




2120587119866int 119887(119896) (1198894119896(2120587)



119887total (0) = minus119899

4120587411988631199033

0

int(1 minus1198862

1198772

minus

)120575 (119877+minus 119886)

times 120575 (1003816100381610038161003816119877+

minus 119877minus

1003816100381610038161003816 minus 1199030) 119889

4119877minus1198894119877+

(60)

which gives


1199030

)) (61)

where

119891(119886

1199030

) =2

120587int

120587

0

1198862sin2

120579119889120579

1198862 + 21198861199030cos 120579 + 119903

2

0

(62)


0le 12)


2119903

2

0 Thus for the


120575Λ119898= minus2120587119866int119899 (119886 119903

0) (1 minus 119891(

119886

1199030

))119889119886 1198891199030

= minus2120587119866119899 (1 minus ⟨119891⟩)

(63)


119898= Λ

lowast+ 120575Λ



0)(120587

22)119886

4119889119886 119889119903

0lt 1 where (120587

22)119886

4





≫ Λ119877


119877120575Λ





119873119865 (119886119883) =1

1198862120575 (119886) ] (119883) (64)

where ](119883) = int 1198862119873119865(119886119883)119889119886 and int(1119886

2)](119883)119889

4119883 =




4119889119886 119889119903

0=

1198862int ](119883)119889



1198941198961198831198894119883) as

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (65)


Λ119877= minus12120587

2int119899 (119886 119903

0) 119886

2119889119886 119889119903

0

= minus121205872int ] (119883) 119889

4119883 = minus12120587

2] (0)

(66)


Λ tot = Λ0

+ 2120587119866int(1minus4120587

2

1198962(] (0)minus] (119896)))

1198894119896

(2120587)4minus12120587

2] (0)

(67)


4


119898)









119881 = minus1

2int 119869 (119909) 119866 (119909 119910) 119869 (119910) 119889

4119909 119889

4119910

= minus1

2

1198714

(2120587)4int119866

0(119896)119873 (119896)

10038161003816100381610038161198691198961003816100381610038161003816

2

1198894119896

(68)


120575119873 (119896 119869) = 120575119887 (119869) ≃ minus1

2

1198714

(2120587)4int120590

2(119896 119901) 119866

0(119901)

10038161003816100381610038161003816119869119901

10038161003816100381610038161003816

2

1198894119901

(69)


2(119896 119901) = Δ119873lowast(119896)Δ119873(119901)

1205902(119896 119901) =

1

119885total (0)int [119863119873] 119890

minus119868(119873)Δ119873

lowast(119896) Δ119873 (119901) (70)



(119896 119901) rarr 1205902


120575119887 (119869) = minus1205902

119896

41205872

21198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(71)


41205872

1198962(120575] (0 119869) minus 120575] (119896 119869)) =

1

21205902

119896

41205872

1198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(72)






120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)


1205902





119896= 119873(119896)(1 minus





119896and


0 while


0119873

0= ΩDEΩ119887



asymp 075005 = 15 while 1205902

0119873

0sim 1



15




plusmn= 119878

2

plusmntimes 119877


+ 119903


plusmn




120572120573⟩ sim 8120587119866119899 and




1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3






plusmnrarr cup

1198991198783



1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)


= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the


cylinders1198793



minusℓ119889119905

0and the

sum sum+infin




119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)



plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119873119865 (119886119883) =1

1198862120575 (119886) ] (119883) (64)

where ](119883) = int 1198862119873119865(119886119883)119889119886 and int(1119886

2)](119883)119889

4119883 =




4119889119886 119889119903

0=

1198862int ](119883)119889



1198941198961198831198894119883) as

119873(119896) = 1 minus4120587

2

1198962(] (0) minus ] (119896)) (65)


Λ119877= minus12120587

2int119899 (119886 119903

0) 119886

2119889119886 119889119903

0

= minus121205872int ] (119883) 119889

4119883 = minus12120587

2] (0)

(66)


Λ tot = Λ0

+ 2120587119866int(1minus4120587

2

1198962(] (0)minus] (119896)))

1198894119896

(2120587)4minus12120587

2] (0)

(67)


4


119898)









119881 = minus1

2int 119869 (119909) 119866 (119909 119910) 119869 (119910) 119889

4119909 119889

4119910

= minus1

2

1198714

(2120587)4int119866

0(119896)119873 (119896)

10038161003816100381610038161198691198961003816100381610038161003816

2

1198894119896

(68)


120575119873 (119896 119869) = 120575119887 (119869) ≃ minus1

2

1198714

(2120587)4int120590

2(119896 119901) 119866

0(119901)

10038161003816100381610038161003816119869119901

10038161003816100381610038161003816

2

1198894119901

(69)


2(119896 119901) = Δ119873lowast(119896)Δ119873(119901)

1205902(119896 119901) =

1

119885total (0)int [119863119873] 119890

minus119868(119873)Δ119873

lowast(119896) Δ119873 (119901) (70)



(119896 119901) rarr 1205902


120575119887 (119869) = minus1205902

119896

41205872

21198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(71)


41205872

1198962(120575] (0 119869) minus 120575] (119896 119869)) =

1

21205902

119896

41205872

1198962

10038161003816100381610038161003816119869ext119896

10038161003816100381610038161003816

2

(72)






120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)


1205902





119896= 119873(119896)(1 minus





119896and


0 while


0119873

0= ΩDEΩ119887



asymp 075005 = 15 while 1205902

0119873

0sim 1



15




plusmn= 119878

2

plusmntimes 119877


+ 119903


plusmn




120572120573⟩ sim 8120587119866119899 and




1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3






plusmnrarr cup

1198991198783



1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)


= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the


cylinders1198793



minusℓ119889119905

0and the

sum sum+infin




119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)



plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120575Λ tot = minus2120587119866int4120587

2

1198962⟨120575] (0 119869) minus 120575] (119896 119869)⟩

1198894119896

(2120587)4

minus 121205872120575] (0 119869)

(73)


1205902





119896= 119873(119896)(1 minus





119896and


0 while


0119873

0= ΩDEΩ119887



asymp 075005 = 15 while 1205902

0119873

0sim 1



15




plusmn= 119878

2

plusmntimes 119877


+ 119903


plusmn




120572120573⟩ sim 8120587119866119899 and




1(119909 119909

1015840 120578) analogous to

(19) for the topology 1198774(119879

3

+cup 119879

3






plusmnrarr cup

1198991198783



1198871(119909 119909

1015840 120578) =

+infin

sum

119899=minusinfin

1

41205872119886(

1

(119877minus119899

minus 1199091015840)2minus

1

(119877+119899

minus 1199091015840)2)

times [120575 (10038161003816100381610038161003816 minus

+119899

10038161003816100381610038161003816minus 119886) minus 120575 (

10038161003816100381610038161003816 minus

minus119899

10038161003816100381610038161003816minus 119886)]

(74)


= (119905119899 119903

plusmn) with 119905

119899= 119905

0+ 2ℓ119899 and ℓ ge 119886 is the


cylinders1198793



minusℓ119889119905

0and the

sum sum+infin




119873119865 (120585) =1

2ℓ120575 (119905

+minus 119905

minus) 119891 (

1003816100381610038161003816119903+ minus 119903minus

1003816100381610038161003816 119886) (75)



plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





plusmn 119903



119891 (120578) =119899 (119886)

41205871199032

0

120575 (119904 minus 1199030) (76)




22)119886

4(12ℓ) (ie


119887 (119896) = minusint 119899 (119886) 1198862 4120587

2

1198962(1 minus

sin |k| 1199030|k| 1199030

)1198691(119896119886)

1198961198862119889119886 (77)




minus⟨Δ119879120572120573

(119909)⟩ = int

119896120573119896120572minus (12) 119892

1205721205731198962

1198962119887 (119896 120585)

1198894119896

(2120587)4

(78)


11987900

= 120576 = 1205821minus1

2120583

119879119894119895= 119901120575

119894119895 119901 =

1

31205822minus1

2120583

(79)

where 120576 + 3119901 = minus120583 and 1205821+ 120582


1205821= minusint

1198962

0

11989621198871198894119896

(2120587)4 120582

2= minusint

|k|2

11989621198871198894119896

(2120587)4 (80)


2=

cos2120579 |k|21198962 = sin2120579 and 119889

4119896 = 4120587sin2

1205791198963119889119896119889120579 we get

120582119894=119899 (119886 119903

0)

4120573119894

(1 minus 2120573119894(119886

1199030

)

2

119891119894(119886

1199030

)) (81)

where 1205731= 1 120573

2= 13 and 119891

119894is given by

119891(1

2)(119910)

=2

120587int

1

minus1

int

infin

0

sin (119909 sin 120579)1198691(119910119909)

1199101199092(cos2120579sin2

120579) 119889119909 119889cos 120579

(82)

For 1198861199030≪ 1 we find

11989112

(119886

1199030

) asymp (1 + 11990012

(119886

1199030

)) (83)


120576 ≃ minus119899

4= minus

2119899

3120587119886 119901 ≃ 120576(1 minus

4

3(119886

1199030

)

2

) (84)


0)


3(119886

1199030

)

2

) 120576 (85)




1198891199042= 119888

2119889119905

2minus ℎ

2(119903) 120575

120572120573119889119909

120572119889119909

120573 (86)

where ℎ(119903) = 1 + 120579(119886 minus 119903)(1198862119903



120573

120572= 119877

120573

120572minus (12)120575

120573

120572119877 Both


119894equiv 0


1198770

0= 119877

0

120572= 0 119877

120573

120572=2

119886120575 (119886 minus 119903) 119899

120572119899120573+ 120575

120573

120572 + 120582

120573

120572 (87)

where 119899120572 = 119899120572= 119909


and 120582120573




1198770

0= 119877

0

120572= 0 119877

120573

120572= sum

2

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119903

119894

1003816100381610038161003816) 119899119894120572119899120573

119894+ 120575

120573

120572

(88)




119894= (119909

120572minus 119903

120572

119894)|119903 minus 119903

119894|


120573

120572= (13)119877120575

120573



120573

120572) where

119877 = minus8120587119866119879 = sum8

119886119894

120575 (119886119894minus1003816100381610038161003816119903 minus 119877

119894

1003816100381610038161003816) = 32120587int 119886119899 (119886) 119889119886

(89)




2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






2119899 ≫ 8120587119866119899 as 119886 ≫ ℓpl




119899 sim1






estimate

119886 sim2



2

75 (91)


75= 119867(75 km(secMpc)) and



mass119872119908= (43)120587119886







2119872

⊙ΩDMℎ

2

75 (92)





References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article dark energy from the gas of wormholes · 2017. 7. 17. · is called as the...

Documents