8/10/2019 9910093

1/21

8/10/2019 9910093

2/21

where M(t) = 8GM(t)/3c2H2(t) is due to ordinary matter (viz. matter for which M(t) =

A/Rn(t) where A > 0 and 3 n 4) and where (t) = 8G/3cH2(t) is due to acosmological constant , has revealed that not only must the current era (t0) actuallybe non-zero today, it is even explicitly required to be of order one. Typically, the allowedparameter space compatible with the available data is found to be centered on the line

(t0) = M(t0) + 1/2 or so with (the presumed positive) M(t0) being found to be limitedto the range (0, 1) and (t0) to the range (1/2, 3/2) or so, with the current (n=3) eradeceleration parameter q(t0) = (n/2 1)M(t0) (t0) thus having to approximately liewithin the (1/2,1) interval. Thus, not only do we find that the universe is currentlyaccelerating, but additionally we see that with there being no allowed (t0) = 0 solutionat all (unless M(t0) could somehow be allowed to go negative), the longstanding problemof trying to find some way by which (t0) could be quenched by many orders of magnitudefrom both its quantum gravity and particle physics expectations (perhaps by making itvanish altogether) has now been replaced by the need to find a specific such mechanismwhich in practice (rather than just in principle) would explicitly put (t0) into this very

narrow (1/2, 3/2) box. Not only is it not currently known how it might be possible toactually do this, up to the present time no mechanism has been identified which might evenfix the sign of the standard model (t0) let alone its magnitude.

As such, the new supernovae data pose problems for both quantum and classical gravi-tational physics, but since the problems (familiar as they are) are somewhat different in thetwo cases it is useful to discuss them independently. As regards first the quantum gravityand elementary physics cosmological constant problem, we note that while it actually pre-ceded the new high zdata (see e.g. [3,4] for a review of the prior situation) the new datahave actually compounded the problem. Specifically, with quantum gravity being associatedwith the Planck temperaturekTPL= (hc

5/G)1/2 and with elementary particle physics being

associated with a hadronic physics vacuum breaking scale,

1

the expectation of fundamentalphysics is that the ratio (t)/M(t) should be absolutely enormous. With the insertionof this ratio into the Friedmann cosmological evolution equation of Eq. (1) then leading toviolent conflict with observation (even prior to the new high zdata in fact), we see that asit stands, the standard theory is explicitly in disagreement with data, and stress the pointnow to emphasize that this is not merely a fine tuning problem (such as the one we willencounter below when we discuss the classical gravitational physics cosmological constantproblem), but rather it constitutes the actual observational failure of an explicit predictionof the standard theory (with a possible 120 orders of magnitude discrepancy potentiallybeing the largest disagreement ever recorded by any known theory). Now prior to the newhigh zdata it was widely hoped that might somehow be quenched, with some symmetry

principle perhaps bringing it down to zero identically, or with some dynamical mechanismperhaps making it negligibly small. While neither of these possibilities has yet been achievedin any convincing, universally accepted way, the situation was suddenly made all the more

1We can represent such a vacuum breaking scale by a typical effective temperature TV, a temper-

ature which is then overwhelmingly larger than the effective temperature (of order a factor of ten

or so times the current temperature T(t0) of the universe) that would be associated with a black

body with critical density C(t0) = 3c2H2(t0)/8G.

2

8/10/2019 9910093

3/21

complicated by the new high zdata since this yet to be established mechanism would nownot only have to quench both the quantum gravity and the particle physics contributionsto (i.e. not just either but both), but also it would have to actually leave the quenched with a very small but nonetheless non-vanishing component so that the current value ofthe (t0)/M(t0) ratio might then be of order one instead.

Beyond the fact that such a quenching has yet to be achieved, it is important to notethat cosmology actually puts additional explicit demands on any such mechanism above andbeyond simply requiring the quenched to be such that the current value of (t0)/M(t0)comes out correctly. Specifically, suppose that some explicit quenching does take place inthe very early Planck temperature dominated quantum gravity universe (some candidatemechanisms such as wormhole effects [5] which might be able to do this are discussed in[3,4]), so that at the end of that era (or at the end of a subsequent but still early inflationaryuniverse era [6]) takes some particular value (this value might even be due to a fundamentalor anthropically induced cosmological constant which simply appears in the fundamentalgravitational action as an a priori fundamental constant). As the ensuing universe then

expands and cools it will potentially go through a whole sequence of elementary particlephysics phase transitions, in each one of which the vacuum energy would be lowered (thisbeing the definition of a phase transition). The residual from the early universe quenchingwould then have to be such that it would just almost (but not quite completely) cancel thenet drop in vacuum energy due to all the particle physics phase transitions (transitions thatwould occur only after the early universe quenching had already taken place - unless eachsuch phase transition is to be accompanied by its own quenching that is), so that just today,i.e. conveniently just for our own particular epoch, (t0) would then be of order one.Difficult as this is to even conceive of let alone demonstrate in an explicit dynamical model,we note, however, that even if such a delicate balancing were to actually take place, the

resulting universe would then be one in which there could not have been any such delicatecancellation prior (or near) to the very last phase transition. Thus in epochs prior or near tothe very last phase transition there could well have been substantial cosmological constantcontributions, to thus potentially give the universe a history and cosmology very differentfrom the standard one.

As an explicit example of how potentially severe a problem this might be, we note thata standard electroweak or grand-unified symmetry breaking phase transition, for instance,can generically be described by an effective Ginzburg-Landau theory with potential

V(, T) =g4/2 2(T)2, (2)

where is the relevant order parameter and where 2

(T) is typically given [7] by a formsuch as2(T) =g(T2V T2) in a convenient normalization. When the temperature T is less

than the transition temperature TV, the potential V(, T) possesses a non-trivial minimumaway from the origin in which it takes the temperature dependent value

Vmin(T < TV) = g(T2V T

2)2/2, (3)

a value which is expressly negative (with respect to the zero value which Vmin(T > TV)takes above the critical temperature), with the zero temperature Vmin(T= 0) taking theconvenient negative value gT4V/2 in our normalization. To this potential we must now addon an extra residual potential from the early universe of the form

3

8/10/2019 9910093

4/21

Vres=g(T2V T

2(t0))2/2 + T4(t0), (4)

to thus give us a total potential

Vtot(T < TV) =Vmin(T < TV) + Vres =gT2V(T

2 T2(t0)) g(T

4 T4(t0))/2 + T

4(t0) (5)

and a current eraVtot(T(t0)) which would then nicely be of order the energy density in ordermatter (generically given as T4(t0)) today.

2 However, as we immediately see, in the nucle-osynthesis era where TV T T(t0), the total vacuum energy densityVtot(T) would thenbe given bygT2VT

2 in leading order and (withg being related to the standard Higgs and vec-tor boson masses asg e2M2H/M

2W) would thus be substantially larger than the black body

energy density at the same temperature T. Since the temperature dependence of a standardk = 0 radiation era cosmology with non-zero is given by T2/T2 = (8G/3c2)(T4 + c),we see that the very validity of standard big bang nucleosynthesis is now contingent on anexplicit demonstration that vacuum terms such as gT2VT

2 are not in fact of relevance in thenucleosynthesis era.3 It thus not sufficient to simply find a mechanism which quenches the

cosmological constant once (in some particular chosen epoch). Rather one needs renewed,temperature dependent, quenching each and every time there is phase transition or new con-tribution to the vacuum energy.4 This then is quantum gravitational cosmological constantproblem.

Further, even independent of any of the above quantum considerations, the new highz data pose a problem for Eq. (1) even when considered purely from the viewpoint ofphenomenological classical physics alone. Specifically, even if we ignore the above phasetransition issue, so that the temperature T(t) then uninterruptedly evolves adiabatically as1/R(t), the current closeness to one of the ratio (t)/M(t) entails that in the early universethis same ratio would have had to have been fantastically small, with a standard Friedmannuniverse then only being able to evolve into its current state if this ratio had been extremelyfine tuned in the early universe (this is the fine tuning problem to which we referred above,being one of having to adjust initial conditions to incredible accuracy, rather than one ofhaving to remove an explicit conflict with data). Moreover, this particular fine tuning wouldhave to be above and beyond that imposed by the flat (k(t) = kc

2/ R2(t) 0) inflationaryuniverse model since inflation only constrains the sum of M(t) and (t) to be one anddoes not fix their ratio.5 This then is the classical gravitational cosmological constant

2Why exactly the residual early universe Vres should explicitly depend on the current temperature

T(t0) so that just our particular epoch of observers would see only the quenched energy density

T4(t0) remains of course to be explained.

3While it had generally been recognized [8] thatVtot(T > TV) Vres would not compete with the

black body energy density at temperatures above the phase transition temperature TV, it appears

not to have been appreciated that there could still be a problem below it.

4While Vres can cancel the leading term in Vmin(T < TV), because it is temperature independent

it leaves the next to leading term untouched.

5While there is still some question as to the extent of the region in the (M(t0), (t0)) parameter

4

8/10/2019 9910093

5/21

problem, and as we see, at the present time neither inflationary nor quantum cosmologycan readily accommodate the new high z data at all (i.e. neither a very early quantumcosmology phase nor an early universe inflationary phase have yet been shown capable ofproducing a subsequent Robertson-Walker phase whose onset value of /M would thenbe the particular fantastically small one now required by data), with the early universe

now needing to be fine tuned all over again. Now while a solution to both the classicaland the quantum mechanical cosmological constant problems might yet be found withinstandard gravity, the above described situation is so disquieting (with the cosmologicalconstant problem having resisted solution for such a very long time now) as to suggest thatin fact there might actually be something basically wrong with the whole standard picture,a viewpoint to which we shall return below. Moreover, since the elementary particle physicscosmological constant problem is a clash between two different branches of physics, gravityand particle physics, we should not immediately assume that it is the particle physics sidewhich needs addressing. Rather, the indications of particle physics could well be correct,with its contribution to actually being as big as it would appear to be, with the problem

then having to lie on the gravitational side instead. And indeed, in the following we shallexplicitly explore the implications for cosmology of actually being a very big rather thana very small quantity. Thus, with attempts to quench not having been successful thus far,we instead turn the issue around and explore below whether it is possible for cosmology toaccommodate a large instead.

In order to try to isolate the primary cause of all the above problems (so as to then knowexactly where it is that the standard theory is running into problems, with a view to thenidentifying what explicitly needs to be done - either in standard gravity itself or beyond),we shall now present as general a diagnosis of the problem as would appear possible. Thuswe note that in any cosmology with a big bang, the early universe R(t = 0) would have

to be divergent (or at least be extremely large), with Eq. (1) then requiring the quantity(M(t = 0) + (t = 0)) to be equal to one no matter what the value of the spatialcurvature k. Thus, given the radically different temporal behaviors of M(t) and (t),in standard gravity we see directly that no cosmology, flat or non-flat, could ever evolveinto one in which (t0) M(t0) O(1) today without extreme fine tuning. Thus itis actually the very fact of the big bang itself which is bringing standard cosmology to sosevere a current impasse. Further, ifR(t= 0) does start off divergent, it must diminish asthe universe begins to evolve, with the early universe thus decelerating. Since the currentuniverse now appears to be accelerating, the Friedmann universe fine tuning problem can

space which is allowed by the supernovae data, the one thing in the data that appears to be definitive

is that the point (1, 0) is overwhelmingly excluded. Since this is the only point which standard

classical cosmology is capable of reaching today without a fine tuning of Eq. ( 1), no change in the

supernovae data would appear likely to lead to a standard cosmology without some form or other

of fine tuning problem. Moreover, even a change in the data that might actually eliminate any

need for an explicit non-vanishing (t0) would still leave us with an M(t0) which would need to

be fine tuned to some value less than one, as well as with an k(t0) which could then no longer be

negligible. And even in that case we would of course still need to explain why would then have

been quenched to zero.

5

8/10/2019 9910093

6/21

be viewed as the need to adjust parameters in such a way that the cosmology can exhibitdiametrically opposite deceleration and acceleration behaviors in differing epochs. Since thebig bang singularity itself derives from the fact that standard gravity is always attractive(since G controls standard gravity on all distance scales including those much larger thanthe solar system one on which standard gravity was first established) while acceleration is

more naturally associated with repulsion (cf. the highzdata solution in which M(t0) istaken to be negative), it is thus suggestive that we might be able to more readily balancethe early and current universes if there were no initial singularity at all, and if cosmologicalgravity in fact got to be repulsive in all epochs, with the universe then expanding from someinitial (but still very hot) state characterized by R(t = 0) = 0 instead. To achieve sucha singularity free cosmology, and to hope to decouple locally attractive gravity from suchcosmological repulsion would thus appear to require the removal ofG from the fundamentalgravitational action.6 Moreover, ifR(t = 0) were indeed to vanish, the initial values of(t = 0) and M(t = 0) would then both be infinite, and thus potentially never requirefine tuning at all. Thus we identify the very presence ofGin the fundamental gravitational

action as a primary cause of the cosmological constant problem, an issue whose implicationswe shall address in detail below.

To continue with the above analysis, we note further that if the apparently currentlyaccelerating universe continues to accelerate indefinitely, then, no matter what may or maynot have occurred in the early universe, in the very late universe R(t) will actually becomearbitrarily large, with Eq. (1) then requiring the quantity (M(t) + (t)) to tend to oneat very late times, again independent of the value ofk. However, because of their differingtime behaviors, we see that in the very late universe it would precisely be (t) which wouldthen have to tend to one no matter what its early universe value. Thus at very late timesthe cosmological constant problem would actually get solved, and would in fact get solved

by cosmology itself (i.e. no matter how big might actually be, in universes whose fateit is to accelerate at late times, there will eventually come a time in which the measurableconsequence of (t) will be that it will make a contribution to the expansion of the universewhich will be of order one). Thus even while the discovery of cosmic acceleration makesthe cosmological constant problem more acute, nonetheless, its very existence also suggestsa possible resolution of the issue. We shall thus explore this option, first (Sec. II) as ageneric cosmological effect, and then (in Sec. III) use it to obtain an explicit solution to thecosmological constant problem in an explicitly solvable cosmological model.

II. GENERIC SOLUTION TO THE COSMOLOGICAL CONSTANT PROBLEM

In order to see how we might be able to capitalize on the behavior of universes whichaccelerate at late times, it is very instructive [9] to analyze de Sitter geometry in a purelykinematic way which requires no commitment to any particular dynamical equation of mo-

6IfG is not to be a fundamental parameter, the G as measured in a Cavendish experiment would

then only be an effective, low energy parameter, with the very high temperature universe then

being controlled by some altogether different (and possibly even repulsive) scale instead.

6

8/10/2019 9910093

7/21

tion. Specifically, suppose we know only that a given geometry is de Sitter, i.e. that itsRiemann tensor is given by

R=(gg gg). (6)

For such a geometry contraction then yields the kinematic relation

R gR/2 = 3g, (7)

a relation which reduces to

R2(t) + kc2 =c2R2(t) (8)

when expressed in Robertson-Walker coordinates. On defining (t) = c2R2(t)/R2(t) weobtain q(t) = (t) = 1 + kc2/R2(t), withq(t) and (t) then being found [9] to be givenby

(t, 0) =coth2(1/2ct) (9)

in the various allowed cases. As we thus see, when the parameter is positive, each asso-ciated solution corresponds (because of the absence of any explicit M matter term) to apermanently accelerating universe, and that in each such universe (t, > 0) will even-

tually reach one no matter how big the parameter might be, and independent in fact ofwhether or not G even appears in the cosmological evolution equations at all.7 Moreover,while (t, >0, k >0) will only come down to one at very late times, quite remarkably, thenegative spatial curvature (t, >0, k 0) is then equal to one - for instance in the familiar flat k = 0 case

R(t)/R(t) = 1/2c at all times.

7

8/10/2019 9910093

8/21

cosmology rapidly quenches k(t) = 1 (t) with (t) rapidly becoming one. Nowwhile this quenching of k(t) is the major achievement of inflation, inflationary cosmologywas always faced with the (still not fully resolved) task of then disposing of this now large(equal to unity in fact) (t) some time during the inflation exit transition to a subsequentRobertson-Walker era, an era which would (ideally) then acquire an M(t) with the desired

value of one. In addition to this still not fully understood task, we now see that given thenew highzdata, this exit transition must now not quite quench (t) completely, but mustinstead make it fantastically small, so that it could then build itself back up to order onetoday. Inflationary cosmology thus requires an (t) which first rises to one, then fine tunesitself almost to zero, and then rises back to one once again. Thus even while an epoch ofinflationary acceleration nicely provides for an (t) equal to one, it is naturally only ableto do so at what might now possibly be the wrong time (unless inflation can naturally do ittwice that is), since independent of whether or not one should have had an (t) of orderone in the early universe, it is precisely an (t0) of order one today which is what appearsto be needed now.

Thus we propose that no matter what may or may not have happened in the earlyuniverse (standard inflation or whatever), it is late universe acceleration which can naturallyresolve the cosmological constant problem, with a late universe de Sitter phase (either inaddition to or instead of an early universe one) then being able to naturally solve thecosmological constant problem. Moreover, in such a situation need not itself be small,and thus even if its associated temperature TVreally is as big as suggested by elementaryparticle physics, even so, (t) will still eventually approach one. Thus given these remarks,it would appear that the key task then is to find a cosmology in which the current erais already sufficiently late. As regards finding any such cosmology we are immediatelyconfronted with the fact that M(t0) is not zero today. Thus, in order to able to implement

our above ideas, the cosmological constant problem has to be converted from being one ofneeding to quench into being one of needing to find a way in which M(t0) can be naturallyquenched instead. In order to actually try to achieve this alternate quenching, we note firstthat the ratio M(t)/(t) is actually independent of Newtons constant G, with its currentvalue being the gravity independent quantityM/c T

4(t0)/T4V. Moreover, not only is thisratio independent ofG, the only place whereGdoes appear in cosmology is in the Friedmannevolution equation where it fixes the overall normalization of both (t) and M(t), i.e. itdetermines exactly just how big a contribution to cosmological evolution is to be made byany given source of energy density. Thus we see that if we want to dominate cosmologytoday with an effective contribution to cosmological evolution which is to be of order one,we must in turn suppress the contribution ofM to current cosmology, something we can

(drastic as it may initially seem) do if we can remove G from the cosmological evolutionequation and replace it by an effective coupling constant which is altogether smaller. Thuswe need not modify or M themselves. Rather, we need only change their effect oncosmology. Thus just as in our discussion of the implications of the existence of a big bang,we are again led to consider removing G from cosmology, and are again led to a G which isto only be an effective low energy parameter, with this then enabling itself to actually beas big as elementary particle physics seems to imply, while still providing us with a current

8

8/10/2019 9910093

9/21

era (t0) of order one.8

With these remarks we have now concluded our diagnosis of the cosmological constantproblem and provided what we believe to be useful pointers for future attacks on the problem,pointers that could prove helpful in trying to find a solution even within standard gravityitself (aG which evolves with temperature is certainly conceivable within standard gravity -

though perhaps not one which might also change sign as it evolves). However, since we arenot currently aware of how to actually naturally implement the above ideas within standardgravity, in the following, we shall instead turn to an alternate gravitational theory and showthat the above (t) bounding mechanism will precisely be found to occur in the conformalgravity theory, a theory which has recently been advanced as an alternative to standardgravity and its familiar dark matter paradigm, a theory in which G does not in fact set thescale for cosmology. Beyond being of interest in and of itself (with it being quite remarkablethat there even exists any theory at all in which (t) and M(t) can both be naturallyquenched), our study can also be viewed simply as an existence theorem which shows thatit is actually possible to construct a model which explicitly realizes all the ideas we have

presented above.

III. EXPLICIT SOLUTION TO THE COSMOLOGICAL CONSTANT PROBLEM

In attempting to go beyond standard second order gravity, even within the confines ofcovariant, pure metric based theories of gravity, we immediately realize that the choice isvast, since we can in principle consider covariant theories based on derivative functions of themetric of arbitrarily high order. However, given our previous remarks, within this infinitefamily of higher order derivative gravitational theories, one of them is immediately singledout, namely conformal gravity,9 a theory which can immediately lead to a cosmology free of

intrinsic scales at sufficiently high enough temperatures precisely because its gravitationalaction possesses no fundamental scale at all. As such, conformal gravity emerges as apotential gravitational analog of the Weinberg-Salam-Glashow electroweak theory, with itimmediately being suggested [11,12] that in it Newtons constant G might be generated as a

8An additional virtue of not having G be a fundamental parameter which is to control gravity

on all distance scales, is that then the Planck length no longer need control quantum gravity.

Consequently, as long as the mechanism which is to remove G from cosmology does so without

inducing any new quantum gravity scale itself, the quantum gravity contribution to the cosmological

constant would then be under control, leaving us with only the need to have to deal with theelementary particle physics contributionTV.

9Conformal gravity is a locally conformal invariant theory of gravity (i.e. one which is invariant

under any and all local conformal stretchings g2(x)gof the geometry), which has as its

uniquely allowed gravitational action the Weyl action IW = g

d4x(g)1/2CC where

C is the conformal Weyl tensor [10] and where g is a purely dimensionless gravitational

coupling constant, and which consequently has gravitational field equations which are fourth order

derivative equations of motion.

9

8/10/2019 9910093

10/21

low energy effective parameter in much the same manner as Fermis constantGFis generatedin the electroweak theory, with G as measured in a low energy Cavendish experiment thenindeed nicely being decoupled from the hot early universe. However, it turns out that the lowenergy limit of conformal gravity need not emerge in precisely this fashion since [13] it is notin fact necessary to spontaneously break the conformal gravity action down to the Einstein-

Hilbert action (i.e. down to the standard theory equations of motion). Rather [13] it isonly necessary to obtain the solutions to those equations in the kinematic region (viz. solarsystem distance scales) where those standard solutions have been tested. Thus, as had beennoted by Eddington [14] already in the very early days of relativity, the standard gravityvacuum Schwarzschild solution is just as equally a vacuum solution to higher derivativegravity theories as well, since the vanishing of the Ricci tensor entails the vanishing of itsderivatives as well. And indeed, with variation of the conformal gravity action leading tothe equation of motion [15]

(g)1/2IW/g= 2gW = T/2 (10)

where W is given by

W =g(R);;/2 + R

;;R;

;R;

; 2RR+ g

RR/2

2g(R);;/3 + 2(R

)

;;/3 + 2RR/3 g(R)

2/6, (11)

and whereT is the associated energy-momentum tensor, we confirm immediately that theSchwarzschild solution is indeed a vacuum solution to conformal gravity despite the totalabsence of the Einstein-Hilbert action in the purely gravitational piece IWof the conformalaction. Standard gravity is thus seen to be only sufficient to give the standard Schwarzschildmetric phenomenology but not at all necessary, with it thus being possible to bypass the

Einstein-Hilbert action altogether as far as low energy phenomena are concerned.10

As weshall show in detail below, it is precisely this aspect of the theory which will lead us toa demarcation between the high and low energy regions which is very different from thatpresent in the electroweak case.

While conformal gravity itself is indeed an old idea, almost as old as General Relativityitself in fact, it is only recently that its potential role in cosmology and astrophysics appearsto have been emphasized, with it having been found capable of addressing so many of theproblems (such as dark matter) which currently afflict standard gravity (see [9] and referencestherein). As noted above, it has as a motivation the desire to give gravity a dimensionlesscoupling constant and a local invariance structure, to thereby make it analogous to the three

10Since higher derivative theories have different continuations to larger distances [13], we see that

the standard Schwarzschild solar system distance scale wisdom is compatible with many differing

extrapolations to larger distances, with standard gravity giving only one particular possible such

extrapolation. And indeed, it has been argued [16,13] that this is actually the origin of the dark

matter problem, with standard gravity simply giving an unsatisfactory extrapolation to galactic

distance scales and beyond. Interestingly, the conformal gravity extrapolation [17] has been found

to provide for a satisfactory explanation of galactic rotation curve systematics without the need to

introduce any galactic dark matter.

10

8/10/2019 9910093

11/21

other fundamental interactions. And indeed as stressed in [18], the local conformal symmetryinvoked to do this then not only excludes the existence of any fundamental mass scales suchas a fundamental cosmological constant, even after mass scales are induced by spontaneousbreakdown of the conformal symmetry, the (still) traceless energy-momentum tensor thenconstrains any induced cosmological constant term to be of the same order of magnitude

as all the other terms in T, neither smaller nor larger. Thus, unlike standard gravity,precisely because of its additional symmetry, conformal gravity has a great deal of controlover the cosmological constant (essentially, with all mass scales - of gravity and particlephysics both - being jointly generated by spontaneous breakdown of the scale symmetry,conformal gravity knows exactly where the zero of energy is), and it is our purpose nowto show that it is this very control which then provides for both a natural solution to thecosmological constant problem and for a complete accounting of the new high zdata.

The cosmology associated with conformal gravity was first presented in [19] where itwas shown, well in advance of the recent high zdata, to be a cosmology with an effectivenegative G, with the associated repulsion yielding a cosmology which was then found to

possess no flatness problem and thus be free of the copious amounts of cosmological darkmatter required in the standard theory. Subsequently [9], the cosmology was shown topossess no horizon problem or universe age problem, and to also be capable of producingcosmic repulsion through negative spatial curvature. To discuss conformal cosmology it isconvenient to consider the conformal matter action

IM= h

d4x(g)1/2[SS/2 + S4 S2R/12 + i

(x)(+ (x)) gS] (12)

for generic massless scalar and fermionic fields, a matter action which contains up to onlysecond order derivative functions of the matter fields.11 In this matter action we haveintroduced a dimensionful scalar field S(x) which is to serve to spontaneously break theconformal symmetry, and have, for simplicity, taken it to be a fundamental, conformallycoupled, purely macroscopic field. Nonetheless, this scalar field could just as easily begenerated microscopically as the dynamical expectation value of some elementary particlemultilinear condensate, withS(x) then being an effective Ginzburg-Landau field. Moreover,in such a case, not only would there be the standard T= 0 elementary particle physicscontribution toS(x) due to a condensation of the filled negative energy modes of the relevantelementary particles, but at non-zero temperatures there could also be a condensation of the

11The same conformal invariance which forces the pure gravity Weyl action IW

to be fourth order

also obliges the matter action IMto be a familiar second order quantity. In fact since actions such

as IM =

d4x(g)1/2(FF)2 are both coordinate invariant and locally gauge invariant, they

are not in fact excluded by these particular invariances, with it being only scale invariance which

actually ensures their (otherwise simply assumed) absence in the standard S U(3) SU(2)U(1)

model of quarks and leptons. Hence it is actually scale invariance which obliges the standard

model to actually be second order, and thus renormalizable, in the first place. And moreover, if

the spontaneous breakdown of the standard model is generated by a gauge mediated interaction

such as hypercolor rather than by a fundamental tachyonic scalar field, the entire standard particle

physicsIMwould then actually possess no fundamental scale at all.

11

8/10/2019 9910093

12/21

Noccupied positive energy modes of these same particles as well. This latter condensationwould then lead to an effective, temperature dependent order parameter which would beof order N in magnitude (the magnetization of a ferromagnet is, for example, proportionalto the number of positive energy spins present in a crystal), as well as to an effectiveVmin(T < TV) which would, according to Eq. (3), explicitly be negative.

12 In such a case the

effective would be negative, and would necessarily have to be so in fact in conformal gravity,since in a theory with an underlying conformal symmetry, the (dimensionful) vacuum energydensity has to be zero identically in the scale invariant S(x) = 0 high temperature regimeabove all scale generating phase transitions since an exact conformal symmetry ensures theabsence of any fundamental cosmological constant.13 Consequently, we shall incorporatethis aspect of conformal gravity in the following by considering cosmologies based on afundamental scalar field whose quartic self coupling coefficient is preferentially taken toactually be negative, and find that it is precisely such negative which leads to cosmicacceleration in the conformal theory.

For the above matter action, when the scalar field acquires a non-zero vacuum expectation

value (an expectation value which can always be rotated into a spacetime constant S0 byan appropriate local conformal transformation), the entire energy-momentum tensor of thetheory is found (for a perfect matter fluid Tkin of the fermions) to take the form

T=Tkin hS20(R gR/2)/6 g

hS40 ; (13)

12Since the free energy density of a ferromagnet is a non-extensive function of the number of spins

which is dependent only on the particle number density and the mean magnetization per spin, the

identification ofS(x) with a cosmological analog of the total magnetization of a ferromagnet and

the simulation of the analog of the large quantity Vmin(T < TV) by the equally large chS4

(x)together then entail that a positive frequency mode cosmological condensation would generate an

effective coupling constant which itself would be of order 1/N4.

13In passing we note that this particular aspect of the theory is even manifest in the event of

dynamical symmetry breaking in scale invariant theories such as QED with no fundamental electron

mass, theories which also enjoy the scale invariance that is being considered here for gravity. In

such theories, it was found [2025] that even though the scale symmetry is destroyed by the bad

ultraviolet behavior of perturbation theory, nonetheless it gets restored non-perturbatively by a

Gell-Mann Low eigenvalue for the coupling constant, at which the Greens functions of the theory

then scale with anomalous dimensions. Additionally, it was noted [26] that in such theories the

same anomalous dimensions which serve to soften the short distance behavior of the massless

theory then cause the massless theory to become more badly behaved in the infrared, with such

infrared behavior then being found to lead to a dynamical double well effective potential of the

form (m) = m2[ln(m2/M2) 1], and to drive the theory to the non-trivial minimum at m= M

where the potential takes the expressly negative value min(m = M) = M2, with the fermion

then acquiring a non-zero dynamical mass M. Since the unbroken(m = 0) is zero, we see that

non-perturbative dynamical symmetry breaking in scale invariant theories leads to an explicitly

negative vacuum energy density (and even to an explicit Ginzburg-Landau structure [27]), just as

desired.

12

8/10/2019 9910093

13/21

with the complete solution to the scalar field, fermionic field, and gravitational field equationsof motion in a background Robertson-Walker geometry (viz. a geometry in which the Weyltensor and W both vanish) then reducing [9] to just one relevant equation, namely

T= 0, (14)

a remarkably simple condition which immediately fixes the zero of energy. On rewriting thiscondition in the form

hS20(R gR/2)/6 =T

kin g

hS40 , (15)

we thus see that the evolution equation of conformal cosmology looks identical to that ofstandard gravity save only that the quantity hS20/12 has replaced the familiar c

3/16G.With this homogeneous and isotropic, global scalar field S0 filling all space and actingcosmologically, we see that this change in sign compared with standard gravity leads toa cosmology in which gravity is globally repulsive rather than attractive. Because of this

change in sign, conformal cosmology thus has no initial singularity (i.e. it expands from afinite minimum radius), and is thus precisely released from the standard big bang modelconstraints described earlier. Similarly, because of this change in sign the contribution ofM(t) to the expansion of the universe is now effectively repulsive, to nicely mesh with thephenomenological highzdata fits in which M(t) was allowed to go negative. Apart froma change in sign, we see that through S0 there is also a change in the strength of gravitycompared to the standard theory. It is this feature which will prove central to the solutionto the cosmological constant problem which we present below.

Despite the fact that conformal gravity has now been found to be globally repulsive,nonetheless, it is important to note that in the conformal theory local solar system gravity

can still be attractive; with it having been specifically found [13] that for a static, sphericallysymmetric source such as a star, the conformal gravity field equation of Eq. (10) reduces to afourth order (i.e. not a second order) Poisson equation4g00 = 3(T

00Trr)/4gg00 f(r),

with solution g00(r) = 12/r + rwhere =

drf(r)r4/12 and =

drf(r)r2/2.

With the coupling constant g in the Weyl action IW simply making no contribution inhighly symmetric cosmologically relevant geometries where C and W vanish, andwith the sign of being directly given by the sign of this thus cosmologically irrelevantg, we see that locally attractive and globally repulsive gravity are now decoupled andthus able to coexist. Local gravity is thus fixed by local sources alone, sources which areonly gravitational inhomogeneities in the otherwise homogeneous global cosmological back-

ground, i.e. sources which are characterized by small, local variations in the backgroundscalar field S(x), variations which themselves are completely decoupled from the homoge-neous, constant, cosmological background field S0 itself. It is thus the distinction betweenhomogeneity and inhomogeneity which provides the demarcation between local and globalgravity, to thus now enable us to consider repulsive cosmologies which are not incompatiblewith the attractive gravity observed on solar system distance scales.

Given the equation of motion T = 0, the conformal cosmology evolution equation isthen found to take the form (on setting = hS40)

R2(t) + kc2 = 3R2(t)(M(t) + (t))/4S20L

2PL

R2(t)(M(t) +(t)) (16)

13

8/10/2019 9910093

14/21

(Eq. (16) serves to define M(t) and (t)), with the deceleration parameter now beinggiven as q(t) = (n/2 1)M(t) (t). As we see, precisely because the underlyingconformal invariance forces the conformalT to be of the standard second order form, Eq.(16) is found to be remarkably similar in form to Eq. (1), with conformal cosmology thusonly containing familiar ingredients. As an alternate cosmology then, conformal gravity

thus gets about as close to standard gravity as it is possible for an alternative to get whilenonetheless still being different. Moreover, even though that had not been its intent, becauseof this similarity, we see that phenomenological fits in which M(t) and (t) are allowedto vary freely in Eq. (1) are thus also in fact phenomenological fits to Eq. (16), with thevarious (t) simply being replaced by their barred counterparts. In order to see whetherconformal gravity can thus fit into the relevant(t0) =M(t0)+1/2 window, it is necessaryto analyze the solutions to Eq. (16). Such solutions are readily obtained [9], and can beclassified according to the signs of and k. In the simpler to treat high temperature erawhere M(t) =A/R

4 =T4 the complete family of solutions is given as

R2

(t, 0) =k(1 + )/2 + ksinh2(1/2ct)/, (17)

where we have introduced the parameters = 2S20 and= (116A/k2hc)1/2. Similarly

the associated deceleration parameters take the form

q(t, 0) = coth2(1/2ct) 2(1 )cosh(21/2ct)/sinh2(21/2ct). (18)

Now while Eq. (17) yields a variety of temporal behaviors for R(t), it is of great interestto note that every single one of them begins with R(t= 0) being zero (rather than infinite)just as desired above, and that each one of the solutions in which is negative (viz. >0)is associated with a universe which permanently expands (only the > 0 solution canrecollapse, with conformal cosmology thus correlating the long time behavior ofR(t) with

the sign of rather than with the sign ofk). We thus need to determine the degree to whichthe permanently expanding universes have by now already become permanently accelerating.

To this end we note first from Eq. (18) that with being greater than one when isnegative, both the >0, k 0, k = 0 cosmologies are in fact permanentlyaccelerating ones no matter what the values of their parameters. To explore the degree towhich they have by now already become asymptotic, as well as to determine the accelerationproperties of the >0, k >0 cosmology, we note that since each of the solutions given inEq. (17) has a non-zero minimum radius, each associated > 0 cosmology has some verylarge but finite maximum temperature Tmax given by

14

8/10/2019 9910093

15/21

T2max( >0, k 0, k 0, k= 0)/T2(t, >0, k = 0) = cosh(21/2ct),

T2max( >0, k >0)/T2(t, >0, k >0) = 1 + 2sinh2(1/2ct)/(+ 1), (19)

with all the permanently expanding ones thus necessarily being way below their maximum

temperatures once given enough time. To obtain further insight into these solutions it isconvenient to introduce an effective temperature according to chS40 = T

4V. In terms of

this TVwe then find that in all the < 0 cosmologies the energy density terms take theform

(t) = (1 T2/T2max)

1(1 + T2T2max/T4V)

1,

M(t) = (T4/T4V)(t), (20)

where ( 1)/(+ 1) =T4V/T4max for thek 0 case. With being greater than one, we find that for the k > 0 case TVis greater than Tmax, for k = 0 TV is equal to Tmax, and for k < 0 TV is less than Tmax,

with the energy in curvature (viz. the energy in the gravitational field itself) thus makinga direct contribution to the maximum temperature of the universe. Hence, simply becausethe temperature Tmax is overwhelmingly larger than the current temperature T(t0) (i.e.simply because the universe has been expanding and cooling for such a long time now), wesee that, without any fine tuning at all, in both the k > 0 and k = 0 cases (i.e. caseswhere TV Tmax T(t0)), the quantity (t0) is already at its asymptotic limit of onetoday, that M(t0) is completely suppressed, and that the deceleration parameter is givenbyq(t0) = 1.

For the >0,k 0 case whereTVis less thanTmax) a very differentoutcome is possible however. Specifically, since in this case the quantity (1 + T2T2max/T

4V)

1

is always bounded between zero and one no matter what the relative magnitudes of TV,Tmax and T(t), we see that as long as Tmax is very much greater than T(t0), rather thanhaving had to have already reached its asymptotic limit of one by now, the quantity (t0)is instead only required to be bounded by it. With it thus being expressly bounded fromabove, the current value of(t0) thus has to lie somewhere between zero and one today nomatter how big or small TVmight be; with the simple additional requirement that TV alsobe very much greater than T(t0) then entailing that M(t0) will yet again be completelysuppressed in the current era. Moreover, (t0) will take a typical value of one half shouldthe value of the quantity T2(t0)T

2max/T

4V currently be close to one. Values of

(t0) notmerely less than one but even appreciably so are thus readily achievable in the k < 0 case

for a continuous range of temperature parameters which obey Tmax TV T(t0) withoutthe need for any fine-tuning at all.14 Noting from Eq. (19) that the temporal evolution ofthe >0, k 0,

k

8/10/2019 9910093

16/21

sinh2(1/2ct) = (T2max/T2 1)/(T4max/T

4V + 1), (21)

we see that in the Tmax TV T(t0) case, the current value of(t0) is then given bythe nicely bounded form tanh2(1/2ct0), i.e. given precisely by the form found in the modelindependent analysis of de Sitter space that was presented above. Additionally, in this

case k(t0) is then given by sech2

(1/2

ct0), with negative spatial curvature then explicitlycontributing to current era cosmology (and even doing so repulsively according to [9]).15 Inthek 0 cases the simple requirement thatTmax T(t0),TV T(t0) ensures that M(t0) is completely negligible at current temperatures (it canthus only be relevant in the early universe), with the current era Eq. ( 16) then reducing to

R2

(t) + kc2

= R2

(t)(t), (22)to thus not only yield as a current era conformal cosmology what in the standard theorycould only possibly occur as a very late one, but to also yield one which enjoys all the nicepurely kinematic properties of a de Sitter geometry which we identified above. Since studiesof galaxy counts indicate that the purely visible matter contribution to M(t0) is of orderone (actually of order 102 or so in theories in which dark matter is not considered), itfollows from Eq. (16) that current era suppression ofM(t0) will in fact be achieved if theconformal cosmology scale parameterS0 is altogether larger than the inverse Planck lengthL1PL, a condition which is naturally imposable (i.e. for a continuous, non-fine-tuned, rangeof parameters of the theory) and which is explicitly precisely compatible with a large rather

than a small TV, i.e. with a large rather than a small S0.16 Comparison with Eq. (1)

15For completeness, we note that in the case where > 0 and k < 0, the Hubble param-

eter obeys H(t) = 1/2c(1 T2(t)/T2max)/tanh(1/2ct), with its current value thus obeying

q(t0)H2(t0) = c

2, and with the current age of the universe then being given by H(t0)t0 =

arctanh[(q(t0))1/2]/(q(t0))1/2. Thus in general we see that t0 is greater than 1/H(t0) (t0 =

1/H(t0) in the k < 0 case in which = 0), with it taking the value t0 = 1.25/H(t0) when

q(t0) = 1/2. Thus as already noted in [9] conformal cosmologies have no universe age problem.

16In passing we note that once M(t0) is suppressed by large S0 it no longer matters whetherM(t) is itself dominated by n = 3 matter or n = 4 radiation, since neither of them makes any

substantial contribution to the full current era conformal gravity energy-momentum tensor (with

this suppression, in our model M(t) is only of relevance in the early universe). As regards this

suppression we recall, as noted earlier, that if the scalar field is microscopic, this same suppression

ofM(t0) can be generated simply by virtue of the number, N, of occupied positive energy states

in the universe being very large, with the negative energy modes not themselves then needing to

generate a temperature way in excess ofTPL. Moreover, in such a case the parameter = 2S20

would be of order 1/N2, with the quantity tanh2(1/2ct0) not then having to be asymptotic in the

current epoch.

16

8/10/2019 9910093

17/21

shows that current era < 0 conformal cosmology looks exactly like a low mass standardmodel cosmology, except that instead of M(t0) being negligibly small (something difficultto understand in the standard theory) it is M(t0) = 3M(t0)/4S

20L

2PLwhich is negligibly

small instead (M(t0) itself need not actually be negligible in conformal gravity - rather, itis only the contribution ofM(t) to the evolution of the current universe which needs be

small). Hence, we see that the very essence of our work is that the same mechanism whichcauses (t0) to be of order one today, viz. a large rather than a small = hS

40 , serves at

the same time, and without any fine tuning, to cause M(t0) to decouple from current eracosmology.17 Thus to conclude we see that when is negative, that fact alone is sufficientto automatically lead us to M(t0) = 0 and to 0 (t0) 1, with (t0) coming closerto one half the more negative the spatial curvature of the universe gets to be.

Now while we have seen that all the three negative conformal cosmologies lead us to acurrent era(t0) of order one, in order to be able to choose between them it is necessary totry to determinek. To this end we appeal [17] to an at first highly unlikely source, namelygalactic rotation curve data. Recalling that in conformal gravity the metric outside of a static

spherically symmetric source such as a star is given by g00(r) = 1 2

/r+

r, we seethat in the conformal theory the departure from Newton is found to be given by a potentialthat actually grows (linearly) with distance. Hence, unlike the situation in standard gravity,in conformal gravity it is not possible to ever neglect the matter exterior to any region ofinterest, with the rest of the universe (viz. the Hubble flow) then also contributing to galacticmotions (i.e. a test particle in a galaxy not only samples the local galactic gravitational field,it also samples that of the global Hubble flow as well). And indeed, it was found [17] thatthe effect on galaxies of the global Hubble flow was to generate an additional linear potentialwith a universal coefficient given by 0/2 = (k)

1/2, i.e. one which is generated explicitlyby the negative scalar curvature of the universe18 (heuristically, the repulsion associated

with negative scalar curvature pushes galactic matter deeper into any given galaxy, an effectwhich an observer inside that galaxy interprets as attraction), with conformal gravity thenbeing found able to give an acceptable accounting of galactic rotation curve systematics (inthe data fitting0 is numerically found to be given by 3.061030 cm1, i.e. to be explicitlygiven by a cosmologically significant length scale) without recourse to dark matter at all.We thus identify an explicit imprint of cosmology on galactic rotation curves, recognizethat it is its neglect which may have led to the need for dark matter, and for our purposes

17Cosmologies withM(t0) = 0 are within the region allowed by the new high z data, and while

such a situation would be hard to understand within standard gravity where M(t0) = 0 wouldcorrespond to an empty universe, we see that conformal cosmology can contain ordinary matter

(M>0) and yet still have it decouple from its current evolution.

18Essentially, under the general coordinate transformation r = /(1 0/4)2, t =

d/R(),

a static, Schwarzschild coordinate observer in the rest frame of a given galaxy recognizes the

(conformally transformed) comoving Robertson-Walker metric ds2 = (, )[c2d2 R2()(d2 +

2d)/(1 220/16)2] (where (, ) = (1 +0/4)

2/R2()(1 0/4)2) as being conformally

equivalent to the metricds2 = (1 +0r)c2dt2 dr2/(1 +0r) r2d.

17

8/10/2019 9910093

18/21

here confirm that k is indeed negative.19 Conformal cosmology thus leads us directly toM(t0) = 0, (t0) =tanh

2(1/2ct0), and would thus appear to lead us naturally (providingonly that

8/10/2019 9910093

19/21

yielding the requisite amount of helium as well as the metallicity which is explicitly seen inpopulation II stars,20 with the inability [2931] of conformal cosmology to yield a sufficientamount of deuterium being its only outstanding nucleosynthesis problem. Now, as regardsthe production of deuterium, we note that while it is generally thought difficult to producepost-primordially, this is not quite the case, as it is actually fairly easy to both produce and

then retain deuterium by spallation or fragmentation of light nuclei [3234], particularly ifthe spallation is pre rather than post galactic.21 In fact the problem is then not one of anunderproduction of deuterium, but rather of an overproduction of the other light elements.However, as noted by Epstein [34], if the spallation is to also take place in the early universewith its onset occurring after the nucleosynthesis itself, then (i) in such a situation onlyhydrogen and helium interactions would be of any significance, with only Z 3 nuclei thenbeing producible, and (ii) that in such a case the high energies involved would serve to favordeuterium production over the lithium production which is favored at ordinary energies.In addition to this, the authors of [31], on having found the helium abundance in theirnucleosynthesis calculations to be a rather sensitive function of the baryon to entropy ratio,

have suggested that lithium production could also be suppressed if the spallation were to takeplace inhomogeneously with helium deficient clouds then spallating with helium rich ones.In such a case, deuterium would then be produced not during nucleosynthesis itself but sometime afterwards just as inhomogeneities first begin to form in the universe. 22 Since a theoryfor the growth of inhomogeneities in conformal cosmology has not yet been developed, it isnot possible to currently provide a detailed analysis of this issue or assess its implicationsfor conformal gravity. And while one should not understate the seriousness of the deuteriumproblem in conformal cosmology (indeed its viability as a cosmological model is contingentupon a successful resolution of this very issue), nonetheless, the relative ease with whichconformal gravity deals with cosmological constant problem, the most severe problem the

standard theory faces, would appear to entitle the conformal theory to further consideration.And even if the conformal gravity alternative were to fall by the wayside, nonetheless ouranalysis of the role that G plays in the standard model cosmological constant problem wouldstill remain valid.

20Even though the cosmology expands far more slowly than the standard cosmology, nonetheless,

this gets compensated for in the conformal case due to the fact that weak interactions are then found

[31] to remain in thermal equilibrium down to much lower temperatures than in the standard case,

with its metallicity predictions then being found [31] to actually outperform those of the standard

model.21Even though spallation models of deuterium production were never actually ruled out, the

models were quickly set aside once it became apparent that deuterium could be produced by

standard big bang nucleosynthesis.

22It could thus be of interest to measure the lithium to deuterium abundance ratio of high z quasar

absorbers, with the obtaining of a value for this ratio different from that expected in standard big

bang nucleosynthesis then possibly indicating the occurrence of inhomogeneous but still fairly early

universe spallation.

19

8/10/2019 9910093

20/21

To conclude this paper, we note once again that spontaneous breakdown effects such asthose associated with a Goldstone boson pion or with massive intermediate vector bosonsseem to be very much in evidence in current era particle physics experiments, and arethus not quenched at all apparently. Hence all the evidence of particle physics is that itscontribution to should in fact be large rather than small today. However, since in such

a case it is nonetheless possible for (t0) to still be small today, we see that the standardgravity fine tuning problem associated with having M(t0) (t0) today can be viewed asbeing not so much one of trying to understand why it is (t0) which is of order one after15 or so billion years, but rather of trying to explain why the matter density contribution tocosmology should be of order one after that much time rather than a factor T4/T4V smaller.Since this latter problem is readily resolved if G does not in fact control cosmology, butif cosmology is instead controlled by some altogether smaller length squared scale such as1/S20 , we see that the origin of the entire cosmological constant problem can be directlytraced to the assumption that gravity is controlled by Newtons constant G on each andevery distance scale; with the very existence of the cosmological constant problem possibly

being an indicator that the extrapolation of standard gravity from its solar system originsall the way to cosmology might be a lot less reliable than is commonly believed.

The author wishes to thank Dr. D. Lohiya for useful discussions. This work has beensupported in part by the Department of Energy under grant No. DE-FG02-92ER40716.00.

Added Notes

(1.) In Sec. (I) we noted that the canceling of the cosmological constant term in thecurrent epoch does not in and of itself guarantee its irrelevance in all earlier ones. However,as regards the particular Ginzburg-Landau mean field model discussed in Sec. (I), theauthor is indebted to Dr. M. Sher for pointing out to him that, at least in the absenceof any chemical potential (such as that generatable by the condensation contemplated in

this paper of a large number of occupied positive energy modes), the one loop correctionto the classical tree approximation effective potential can actually serve to suppress thetemperature dependence of the cosmological constant contribution at temperatures belowthe critical temperature TV(see L. Dolan and R. Jackiw, Phys. Rev. D 9,3320 (1974); M.Sher, Phys. Rept. 179, 273 (1989)); with it thus being necessary to check for any possibletemperature dependence to the vacuum energy in general on a case by case basis.

(2.) It is of interest to note that within the general framework of our proposal of tryingto suppress the effective cosmologicalG, a new mechanism may have potentially just becomeavailable within standard gravity itself. Specifically, it was suggested recently (L. Randalland R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); ibid. 83, 4690 (1999)) that standard

four dimensional gravity might only be a brane-localized limit of gravity in some higher largeextra dimension with the effective four-dimensional G then being determined dynamicallyby the structure of the embedding in the higher dimensional space. It would thus be ofinterest to explore whether such embeddings could lead to a different effective G in thefour-dimensional high and low energy limits.

(3.) Since this paper was finished we have been able to show from a study of conformalcosmology at temperatures above all phase transitions that the sign of the spatial curvaturek of the universe is then uniquely fixed to be negative, just as desired in this paper. Thisresult as well as further related discussion of our work may found in P. D. Mannheim, Found.Phys. 30, (2000), in press (gr/qc 0001011).

20

8/10/2019 9910093

21/21

REFERENCES

[1] A. G. Riess et. al., Astronom. J. 116, 1009 (1998).[2] S. Perlmutter et. al., Astrophys. J. 517, 565 (1999).[3] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).

[4] Y. J. Ng, Int. J. Mod. Phys. D 1, 145 (1992).[5] S. Coleman, Nucl. Phys. B 310, 643 (1988).[6] A. H. Guth, Phys. Rev. D 23, 347 (1981).[7] S. Weinberg, Phys. Rev. D 9, 3357 (1974).[8] S. A. Bludman and M. A. Ruderman, Phys. Rev. Lett. 38, 255 (1977).[9] P. D. Mannheim, Phys. Rev. D 58, 103511 (1998).

[10] H. Weyl, Math. Zeit., 2, 384 (1918).[11] S. L. Adler, Rev. Mod. Phys. 54, 729 (1982).[12] A. Zee, Ann. Phys. 151, 431 (1983).[13] P. D. Mannheim and D. Kazanas, Gen. Relativ. Gravit. 26, 337 (1994).

[14] A. S. Eddington, The Mathematical Theory of Relativity, Eighth Edition (First Edition1922), Cambridge University Press, Cambridge, U. K. (1960).[15] B. S. DeWitt, Dynamical Theory of Groups and Fields, Gordon and Breach, New York

(1965).[16] P. D. Mannheim and D. Kazanas, Astrophys. J. 342, 635 (1989).[17] P. D. Mannheim, Astrophys. J. 479, 659 (1997).[18] P. D. Mannheim, Gen. Relativ. Gravit. 22, 289 (1990).[19] P. D. Mannheim, Astrophys. J. 391, 429 (1992).[20] K. Johnson, M. Baker and R. Willey, Phys. Rev. 136, B1111 (1964).[21] K. Johnson, R. Willey and M. Baker, Phys. Rev. 163, 1699 (1967).[22] M. Baker and K. Johnson, Phys. Rev. 183, 1292 (1969).[23] M. Baker and K. Johnson, Phys. Rev. D 3, 2516 (1971).[24] M. Baker and K. Johnson, Phys. Rev. D 3, 2541 (1971).[25] K. Johnson and M. Baker, Phys. Rev. D 8, 1110 (1973).[26] P. D. Mannheim, Phys. Rev. D 12, 1772 (1975).[27] P. D. Mannheim, Nucl. Phys. B 143, 285 (1978).[28] S. S. McGaugh, in Proceedings of Galaxy Dynamics, A Rutgers Symposium, August

1998. Astronomical Society of the Pacific Conference Series, Vol. 182, edited by D.Merritt, J. A. Sellwood, and M. Valluri, A. S. P., San Francisco (1999).

[29] L. Knox and A. Kosowsky, Primordial Nucleosynthesis in Conformal Weyl Gravity,Fermilab-Pub-93/322-A,astro-ph/9311006(1993).

[30] D. Elizondo and G. Yepes, Astrophys. J. 428, 17 (1994).[31] D. Lohiya, A. Batra, S. Mahajan, and A. Mukherjee, Nucleosynthesis in a Simmering

Universe, nucl-th/9902022 (1999); M. Sethi, A. Batra, and D. Lohiya, Phys. Rev. D60, 108301 (1999).

[32] F. Hoyle and W. A. Fowler, Nature 241, 384 (1973).[33] R. I. Epstein, J. M. Lattimer, and D. N. Schramm, Nature 263, 198 (1976).[34] R. I. Epstein, Astrophys. J. 212, 595 (1977).

21

http://xxx.unizar.es/abs/astro-ph/9311006http://xxx.unizar.es/abs/nucl-th/9902022http://xxx.unizar.es/abs/nucl-th/9902022http://xxx.unizar.es/abs/astro-ph/9311006