orfeu bertolami- dark energy - dark matter unification: generalized chaplygin gas model

8/3/2019 Orfeu Bertolami- Dark Energy - Dark Matter Unification: Generalized Chaplygin Gas Model

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Dark Energy - Dark Matter Unification:

Generalized Chaplygin Gas Model

Type Ia Supernovae and Accelerated Expansion

Quintessence Quintessence and the Brane Generalized Chaplygin Gas ModelCMBR Constraints

Supernovae and Gravitational Lensing Constraints

Structure Formation

M.C. Bento, O. B., A.A. Sen

Phys. Rev. D66 (2002) 043507; D67 (2003) 063003; D70 (2004) 083519

Phys. Lett. B575 (2003) 172; Gen. Rel. Grav. 35 (2003) 2063

O. B., A.A. Sen, S. Sen, P.T. Silva

Mon. Not. Roy. Astron. Soc. 353 (2004) 329P.T. Silva, O. B.

Astrophys. J. 599 (2003) 829

O. B.

astro-ph/0403310; astro-ph/0504275

M.C. Bento, O. B., N.M. Santos, A.A. Sen

Phys. Rev. D71 (2005) 063501

Orfeu Bertolami

Instituto Superior Tecnico, Depto. Fsica, Lisbon

KIAS-APCTP-DMRC Workshop on The Dark Side of the Universe

24-26 May, 2005, KIAS, Seoul, Korea


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Type Ia Supernovae and Accelerated Expansion

Study of recently discovered Type Ia Supernovae with z 0.35 indicatesthat the deceleration parameter

q0 a aa2 ,where a(t) is the scale factor, is negative

1


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Cosmological Constraints

Observational Parameters

M

0.30

,Q 0.70

k 0

H0 = 100 h km s1 M pc1 , h = 0.71

Big-Bang Nucleosynthesis (BBN)

V() = V0 exp()

< 0.045 (2 )

> 9

[Bean, Hansen, Melchiorri 2001]

Cosmic Microwave Background

6[Efstathiou 1999]


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Figure 1: Concordance Model (latest)


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Some Ideas

- V0 exp() (Troublesome on the brane)[Ratra, Peebles 1988; Wetterich 1988; Ferreira, Joyce 1998]

- V0 , > 0 (Fine on the brane for 2 < < 6)

[Ratra, Peebles 1988]

- V0 exp(2) , > 0 [Brax, Martin 1999, 2000]

- V0 [exp(Mp/) 1] [Zlatev, Wang, Steinhardt 1999]

- V0(cosh 1)p [Sahni, Wang 2000]

- V0 sinh ()

[Sahni, Starobinsky 2000; Urena-Lopez, Matos 2000]

- V0[exp() + exp ()] [Barreiro, Copeland, Nunes 2000]

- Scalar-Tensor Theories of Gravity

[Uzan 1999; Amendola 1999; O.B., Martins 2000; Fujii 2000; ...]

- V0 exp()[A + ( B)2] [Albrecht, Skordis 2000]

- V0 exp()[a + ( 0)2 + b ( 0)2 + c ( 0)2 + d ( 0)2][Bento, O.B., Santos 2002]


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Quintessence and the Brane

Brane-World Scenarios

[L. Randall, R. Sundrum 1999, ...]

- 5-dim AdS spacetime in the bulk with matter confined on a 3-brane 4-dimensional Einstein Eqs.

G = g + 8M2P

T +

8

M35

2S E .

[Shiromizu, Maeda, Sasaki 2000]

IfT is the energy-momentum of a perfect fluid on the brane, then

S =1

22uu +

1

12( + 2p)h ,

, p are the energy density and isotropic pressure of a fluid with 4-velocity

u, h = g + uu,

E = 6k2

[(uu +1

3h) + P + Qu + Qu] ,

so that k2 8/M2P (GR limit 1 0) and the tensors P and Qcorrespond to non-local contributions to pressure and flux of energy. For a

perfect fluid P = Q = 0 and = 0a4.

The 4-dimensional cosmological constant is related to the 5-dimensional

one and the 3-brane tension, :

=4

M35

5 +

4

3M352

while the Planck scale is given by

MP = 3

4

M35

.


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Generalized Chaplygin Gas

Radical new idea: change of behaviour of the missing energy densitymight be controlled by the change in the equation of state of the back-

ground fluid.

Interesting case: Chaplygin gas, described by the Eq. of state

p = A

,

with = 1 and A a positive constant.

From the relativistic energy conservation Eq., within the framework of aFriedmann-Robertson-Walker cosmology,

=

A +

B

a6,

where B is an integration constant.

Smooth interpolation between a dust dominated phase where, Baand a De Sitter phase where p , through an intermediate regime de-scribed by the equation of state for stiff matter, p = .

[Kamenshchik, Moschella, Pasquier 2001]

This setup admits a brane interpretation via a parametrization invariant

Nambu-Goto d-brane action in a (d + 1, 1) spacetime. This action leads, in

the light-cone parametrization, to the Poincare-invariant Born-Infeld actionin a (d, 1) spacetime. The Chaplygin is the only known gas to admit a

supersymmetric generalization.

[Jackiw 2000]


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Bearing on the observed accelerated expansion of the Universe: Eq. ofstate is asymptotically dominated by a cosmological constant, 8G

A.

Inhomogeneous generalization can be regarded as a dark energy - darkmatter unification.

[Bilic, Tupper, Viollier 2001]

[Bento, O.B., Sen 2002]


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A Model (0 < 1)

Lagrangian density for a massive complex scalar field, :

L= g,,

V(

|

|2) .

Writing = ( 2m

)exp(im) in terms of its mass, m:

L = 12

g

2,, +1

m2,,

V(2/2) .

Scale of inhomogeneities arises from the assumption:

, 0, that is

U =g,

V,

so that, on the mass shell, UU = 1.

Energy-momentum tensor takes the form of a perfect fluid:

=2

2V + V ,

p =2

2V

V .


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Covariant conservation of the energy-momentum tensor

+ 3H(p + ) = 0 ,

where H = a/a, leads for the generalized Chaplygin gas

=

A +

B

a3(1+)

11+

.

Furthermore

d ln 2 =d( p)

+ p,

which, together with the Eq. of state implies that:

2() = (1+ A)11+ . Generalized Born-Infeld theory:

LGBI = A 11+

1 (g,,)1+2

1+,

which reproduces the Born-Infeld Lagrangian density for = 1.

LGBI can be regarded as a d-brane plus soft correcting terms as can beseen from the expansion around = 1:

1 X1+2

1+

=

1 X + Xlog(X) + (1 X) log(1 X)4

1 X (1 )

+

E+ F + G

32(1 X)3/2(1 )2

+ O((1 )3

) ,

where X g,, and

E = X(X 2)log2(X) ,

F = 2X(X 1) log(X)[log(1 X) 2] ,

G = (X 1)2[log(1 X) 4] log(1 X) .


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Potential arising from the model

V =1+ + A

2

=1

22/ + A

2 ,

where B(1/1+)a3(1)2, which reduces to the duality invariant,2 A/2, and scale-factor independent potential for the Chaplygin gas. Intermediate regime between the dust dominated phase and the De Sitterphase:

A1

1+ + 1

1 + B

A

1+ a3(1+) ,

p A 11+ +

1 +

B

A

1+a3(1+) ,

which corresponds to a mixture of vacuum energy density A1

1+ and matter

described by the soft equation of state:

p = .


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p = A

( = 1: dbrane)

p


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=

A +

B

(1+)

11+

.

Zeldovich method for considering inhomogeneities can be implemented

through the deformation tensor:

Dij = a(t)

ij b(t)

2(q)

qiqj

,

where q are generalized Lagrangian coordinates and

ij = mnDmi D

nj ,

h being a perturbation

h = 2b(t),ii ,

with b(t) parametrizing the time evolution of the inhomogeneities and

(1 + ) , p A

(1 ) , being given by the evolution Eq. of the energy density and the density

contrast, , by

=h

2(1 + w) ,

with

w p

= A1+

.

The induced metric leads to the (0 0) component of the Einstein Eqs:

3aa

+1

2h + Hh = 4G[(1 + 3w) + (1 3w)] ,

where the unperturbed part corresponds to the Raychaudhuri Eq.

3a

a = 4G(1 + 3w) .


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It follows from the Friedmann Eq. for a flat space section

H2 =8G

3 ,

that the Einsteins Eqs. can be written as a differential equation for b(a):

23

a2b + (1 w)ab (1 + w)(1 3w)b = 0 ,where the primes denote derivatives with respect to the scale-factor.

From the observational constraints

w(a) = a3(1+)

1 + a3(1+).

We numerically integrate the Eq. for b for different values of usingaeq = 10

4 for matter-radiation equilibrium, a0 = 1 at present and b(aeq) =0 as initial condition.

Generalized Chaplygin scenarios start differing from the CDM onlyrecently (z

< 1) and yield a density contrast that closely resembles, for

any value of = 0, the standard CDM before the present.It can be seen that for any value of , b(a) saturates as in the CDM.


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Ratio between in the Chaplygin and the CDM scenarios is given by:Chap

CDM=

bChapbCDM

1 + a31 + a3(1+)

,

meaning that their difference diminishes as a evolves.

Evolution of as a function ofa can be obtained from numerical integra-

tion. We find that for any value of the density contrast decays for large a

(as the = 1 case).

[Bilic, Tupper, Viollier 2001]

[Fabris, Goncalves, Souza 2001]

The difference in behaviour of the density contrast between a Universe

filled with matter with a soft or stiff Eqs. of state can be seen in the

Figure. The former resembles more closely the CDM.


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-10000

0

10000

20000

30000

40000

50000

60000

0.0001 0.001 0.01 0.1 1 10

b/b(aeq

)

a

CDM

CDM

=2/3

=1/3

=1

=1/4

Figure 3: Evolution ofb(a)/b(aeq) for the generalized Chaplygin gas model, for different values of, as compared with CDMand CDM.

Location of CMBR peaks for the generalized Chaplygin gas

CMBR peaks arise from acoustic oscillations of the primeval plasma justbefore the Universe becomes transparent. The angular momentum scale of

the oscillations is set by the acoustic scale lA, which for a flat Universe is

given by

lA = 0 ls

csls,

where 0 and ls are the conformal time at present and at the last scatteringand cs is the average sound speed before decoupling.

The assumptions in our subsequent calculations are as follows:

Scale factor at present a0 = 1, scale factor at last scattering als = 11001,

h = 0.65, density parameter for radiation and baryons at present r0 =

9.89 105, b0 = 0.05, average sound velocity cs = 0.52, and spectralindex for the initial energy density perturbations, n = 1.

To compute lA we rewrite the Chaplygin equation in the form


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0

2000

4000

6000

8000

10000

12000

0.0001 0.001 0.01 0.1 1 10 100 1000

b/b(aeq

)

a

CDM

=2/3

=1/3

=1

=1/4

=1/2

Figure 4: Evolution ofb(a)/b(aeq) for the generalized Chaplygin gas model, for different values of, as compared with CDM.

0

500

1000

1500

2000

2500

3000

3500

0.0001 0.001 0.01 0.1 1 10

a

CDM

=1=2/3

=1/3

=1/2

Figure 5: Density contrast for different values of, as compared with bCDM.


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ch = ch0

As +

(1 As)a3(1+)

1/1+,

where As A/1+ch0 and ch0 = (A+B)1/1+. The Friedmann eq. becomes

H2 = 8G3

r0a4 + b0

a3 + ch0

As + (1 As)a3(1+)1/1+

,

where we have included the contribution of radiation and baryons.

Several important features are worth remarking:

(i) 0 As 1 (ii) For As = 0 the Chaplygin gas behaves as dust and, forAs = 1, it behaves like as a cosmological constant. For = 0, the Chap-

lygin gas corresponds to a CDM model. Hence, for the chosen range of

, the generalised Chaplygin gas is clearly different from CDM. Anotherrelevant issue is that the sound velocity of the fluid is given, at present, by

As and thus As 1. Moreover using thatr0ch0

=r0ch0

=r0

1 r0 b0 ,and

b0ch0

=b0ch0

=b0

1 r0 b0 ,we obtain

H2 = ch0H20a

4X2(a) ,

with

X(a) =r0

1 r0 b0 +b0 a

1 r0 b0 + a4

As +

(1 As)a3(1+)

1/1+.

From the fact that H2 = a4

dad

2, we get

d =da

1/2ch0H0X(a)

,


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so that

lA =

cs

0

1 da

X(a)

0

als da

X(a)

1 1

.

In an idealised model of the primeval plasma, there is a simple relation

between the location of the m-th peak and the acoustic scale, namely

lm mlA. However, the location of the peaks is slightly shifted by drivingeffects and this can be compensated by parameterising the location of the

m-th peak, lm as

lm

lA (m

m) .

It is not possible in general to analytically derive a relationship between

the cosmological parameters and the peak shifts, but one can use fitting

formulae. In particular, for n = 1 and b0h2 = 0.02 one finds that:

1

0.267

rls

0.30.1

,

where rls = r(zls)/m(zls).

[Doran, Lilley, Schwindt, Wetterich 2000]

[Hu, Fukugita, Zaldarriaga, Tegmark 2001]

According to the dark energy - dark matter unification hypothesis, ch will

behave as non-relativistic matter at the last scattering and hence

ch ch0a3

(1 As)1/1+ ,from which follows

rls =r0ch0

a1ls(1

As)1/1+

r0a1ls

(1

r0

b0)(1

As)1/1+

.


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We show l1 as a function of for different values of As, for the observa-

tional bounds on l1 as derived from BOOMERANG (dashed lines)

l1 = 221 14 .and Archeops data (full lines)

l1 = 220 6 .

Notice that, since As 1, for a specific value of As curves end wherethis relation gets saturated, As = 1.

It is very difficult to extract any constraints from the position of the sec-ond peak since it depends on too many parameters, hence it is disregarded.

As for the shift of the third peak, it turns out to be a relatively insensitive

quantity

3 0.341 .

[Doran, Lilley, Wetterich 2001]

We show l3 as a function of for different values of As, in relation to the

current lower and upper bounds on l3 as derived from BOOMERANG data

l3 = 825+1013 .

We see that l1 and l3 put rather tight constraints on the parameters of the

model, and As.


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Figure 6: WMAP Power Spectrum


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0 1 2 3 4 5

200

210

220

230

240

250

260

l1

As=0.4

As=0.5

As=0.6

As=0.7

As=0.8

As=0.9

l1

constraints from BOOMERANG

l1

constsraints from Archeops

Figure 7: Dependence of the position of the CMBR first peak, l1, as a function of for different values ofAS . Also shown arethe observational bounds on l1 from BOOMERANG (dashed lines), and Archeops (full lines).


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0.5 0.6 0.7 0.8 0.9 1A

s

0

0.5

1

1.5

Allowed region

As

range from supernova

As

lower bound from APM08279+5252

l

1

contour from Archeops

l3

contour from BOOMERANG

Figure 9: Contours in the (, AS) plane arising from Archeops constraints on l1 (full contour) and BOOMERANG constraintson l3 (dashed contour), supernova and APM 08279 + 5255 object. The allowed region of the model parameters lies in the

intersection between these regions.


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WMAP Constraints

l1 = 220.1 0.8 l2 = 546 10 ld1 = 411.7 3.5

Main conclusions:

1. Assuming WMAP priors, the Chaplygin gas model, = 1 , is incom-

patible with the data and so are models with > 0.62. For = 0.6, consistency with data requires for the spectral tilt, ns >

0.97, and that, h

< 0.68

3. The CDM model barely fits the data for ns 1 (WMAP data yieldsns = 0.99 0.04) and for that h > 0.72. For low values of ns, CDMis preferred to the GCG models. For intermediate values of ns, the GCG

model is favoured only if 0.2These results are consistent with the ones obtained by Amendola et al.

2003 using the CMBFast code. Furthermore, we find:

4. In the (As, ) plane the variation of h within the bounds h = 0.71+0.040.03

does not lead to important changes in the allowed regions, as compared to

the value h = 0.71. However, these regions become slightly larger as they

shift up-wards for h < 0.71; the opposite trend is found for h > 0.71

5. Our results are consistent with bounds obtained using BOOMERanG

data for the third peak and Archeops data for the first peak as well as results

from SNe Ia and age bounds, namely 0.81 < As < 0.85 and 0.2 < < 0.66. If one abandons the constraint on h arising from WMAP, then the Chap-

lygin gas case = 1 is consistent with the peaks location, if h 0.64


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0.55 0.665 0.780.15

0.2

0.25

0.3

=0

m

h

0.55 0.665 0.780.15

0.2

0.25

0.3

=0.2

m

h

0.55 0.665 0.780.15

0.2

0.25

0.3

=0.6

m

h

0.55 0.665 0.780.15

0.2

0.25

0.3

=1

m

h

Figure 10: Contour plots of the first three Doppler peaks and first trough locations in the (m, h) plane for GCG model, with

ns = 0.97, for different values of. Full, dashed, dot-dashed and dotted contours correspond to observational bounds on,respectively, p1 , p2 , p3 and d1 . The box on the = 0 plot (corresponds to CDM model) indicates the bounds on h andmh

2 from a combination of WMAP, ACBAR, CBI, dFGRS and Ly.


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Supernovae Constraints

SNe Ia data sets:

Tonry at al. (2003) 230 points

Barris at al. (2004) 23 points

Riess at al. (2004) - Gold sample (HST) 143 (157) points

Apparent magnitude:

m(z) = M + 5 log10 DL(z)where DL =

H0c

dL(z), dL(z) = r(z)(1 + z) is the luminosity distance and

r(z) the comoving distance

r(z) = c

z0

dz

H(z)

Absolute magnitude (taken to be constant for all SNe Ia):

M = M + 5 log10 c/H01 Mpc + 25

We consider points with z > 0.01 and with host galaxy extinction Av >

0.5. This yields 194 points. The Gold and HST data sets were studied in

Bento et al., Phys. Rev. D71 (2005) 063501.

SNe Ia data points are listed in terms oflog10 dL(z) and the error log10 dL(z The best fit model is obtained by minimizing the quantity

2 =194i=1

log10 dLobs(zi) 0.2M log10 dLth(zi; c)

log10 dL(zi)

2

where M = MMobs denotes the difference between the actual M andthe assumed value Mobs in the data. Due to the uncertainty arising from thepeculiar motion at low redshift one adds v = 500 km s1 to 2log10 dL(z)

2log10 dL(z) 2log10 dL(z) +

1ln10 1DLvc

2


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

As

2

min= 199.74

Best fit values:

As

= 0.79

= 0.999

Figure 11: Confidence contours in the As parameter space for flat unified GCG model. The solid and dashed lines representthe 68% and 95% confidence regions, respectively. The best fit value used for M is 0.033.

1 2 3 4 5 6 7 8 9 100.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

As

2

min= 198.23

Best fit values:

As

= 0.936

= 3.75

Figure 12: Same as previous Figure, but with a wider range for .


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Structure Formation

Unphysical oscillations or exponential blow-up in the matter power spec-trum at present in the unified model?

[Sandvik, Tegmark, Zaldarriaga, Waga 2004]

Solution: Decompose the energy density into a pressureless dark mattercomponent, dm, and a dark energy component, X, dropping the phantom

component

[M.C. Bento, O. B., A.A. Sen 2004]

Introduce the redshift dependence in the pressure and the energy density

(a0 = 1)

pch = AA + B(1 + z)3(1+)

1+

ch =

A + B(1 + z)3(1+) 11+

Equation of state:

w =pchch

=pX

dm + X=

wXXdm + X

.

where

X = dm1 + wX

1 + BA(1 + z)

3(1+)

As X 0 then wX 0 for early times (z 1) and wX 1 for future(z = 1). Hence, one concludes that wX 1 for the entire history of theuniverse. The case wX < 1 corresponds to the so-called phantom-likedark energy, which violates the dominant-energy condition and leads to an

ill defined sound velocity. If one excludes this possibility:


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= dm +

where

dm =B(1 + z)3(1+)

A + B(1 + z)3(1+)

1+

= p = AA + B(1 + z)3(1+)

1+

from which one obtains the scaling behaviour

dm

=B

A(1 + z)3(1+)

The entanglement of dark energy and dark matter is such that the energyexchange, which is described by

dm + 3Hdm = ,implies that the dominance of dark energy at z 0.2 is correlated with thegrowth of structure!

The linear perturbation eq. for dark matter in the Newtonian limit:2dm

t2+

2

a

a+

dm

dm

t

4Gdm 2a

a

dm

t

dm

dm = 0

where = 18G and = 8G. For = 0, i.e. no energy transfer,one recovers the standard equation for the dark matter perturbation in the

CDM case.

The study of evolution of dm allow obtaining the behaviour of the bias

parameter, b b/dm, of the linear growth function D(y) /0, wherey = ln(a) and of the so-called growth exponent m(y) = D

(y)/D(y)

Observational values obtained from the 2DF survey for the bias and the

distortion parameter, m/b, in the context of the CDM model

= 0.49 0.09 , b = 1.04 0.14im l that m = 0.51 0.11 and 0.1 0.15.


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0 2 4z

0

0.2

0.4

0.6

0.8

1

DensityParameter

dm

b

Figure 13: dm and and b as a function of redshift. We have assumed dm0 = .25, 0 = 0.7 and b0 = 0.05 and = 0.2.

0.001 0.01 0.1 1a

0.001

0.01

0.1

1

10

dm

Figure 14: dm as function of scale factor. The solid, dotted, dashed and dash-dot lines correspond to = 0, 0.2, 0.4, 0.6,respectively. We have assumed dm = 0.25,b = 0.05 and = 0.7.


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0.001 0.01 0.1 1a

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

m(y)

Figure 15: The growth factor m(y) as a function of scale factor a. The solid, dotted, dashed and dash-dot lines correspondto = 0, 0.2, 0.4, 0.6, respectively. We have assumed dm = 0.25,b = 0.05 and = 0.7.

0.001 0.01 0.1 1a

0.6

0.7

0.8

0.9

1

1.1

b(a)

Figure 16: The bias b as a function of the scale factor, a. The solid, dotted, dashed and dash-dot lines correspond to =0, 0.2, 0.4, 0.6, respectively. We have assumed dm = 0.25,b = 0.05 and = 0.7.


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0 0.05 0.1 0.15 0.2 0.25 0.30.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

m

Figure 17: Contours for parameters b and m in the m plane. Solid lines are for b whereas dashed lines are for m. For b,contour values are 0.98, 0.96,...,0.9 from left to right. For m, contour values are 0.6, 0.65, ...,0.8 from left to right.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

106sources

SNe

CMBR

orfeu bertolami- dark energy - dark matter unification: generalized chaplygin gas model

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