Variable Chaplygin Gas: Constraints from Supernovae,BAO, Look Back Time, and GRBs

Bhuvan Agrawal∗a, Geetanjali Sethi†b, and Shruti Thakurb

aDepartment of Applied Physics, Delhi Technological UniversitybDepartment of Physics, St. Stephen’s College, University of Delhi

May 31, 2019

Abstract

We examine the variable Chaplygin gas (VCG) model with the aim of establishing tight constraintsby using type-Ia supernovae, lookback time measurements, Baryon Acoustic Oscillations (BAO) and theGamma Ray Bursts. We report the parameter constraints obtained via statistical analysis of cosmicobservables namely, luminosity distance, BAO distance and lookback time measurement.

1 IntroductionBolstering our understanding of the Universe, observations also have the potential to revolutionize the waywe perceive our reality. Two decades back one such observation discovered that the Universe is expanding.From the observations of distant supernovae type-Ia (SNe-Ia) by two independent groups [1, 2] it has beenestablished that the expansion is accelerating. This fact is corroborated by a number of separate observationssuch as Baryon Acoustic Oscillations (BAO), Cosmic Microwave Background, growth of structures andGamma-ray Bursts (GRBs).

The dynamics of the Universe is governed by the Einstein’s general theory of relativity, but it is not pos-sible to explain the accelerated expansion if only the observable (normal) matter is taken into consideration.Different cosmological models have been proposed to explain our observations. Among the various explana-tions, the one that is prevalent and most favourable is the Λ-Cold Dark Matter (ΛCDM) model, where Λrepresents the cosmological constant. In this model, Λ accounts for the vacuum energy or the energy densityof space. However this model suffers from serious fine tuning problems [5, 6]. Hence many more modelshave been explored in the past and this remains an active area of research. Different models explored canbe broadly categorized into two classes. One class of models involve reforming the geometry part of the Ein-stein’s equations. This includes generalisation of gravity action and higher dimension spacetime. The otherclass of models alter the matter component of the Universe in the Einstein’s equations. In this approachexotic matter with negative pressure is added to the mass distribution of the Universe. Some of the modelsbased on this are quintessence, k-essence, tachyons, barotropic fluid etc. These are collectively known asDark Energy models. Recently, an alternative class of models have been proposed which involve a slowlyevolving and spatially homogenous scalar field [7, 8] or two coupled fields [9]. However, these "Quintessence”models also suffer from fine tuning problem.

Quintessence, k-essence are scalar field models, while Dark Energy models include barotropic fluids whosepressure is a function of energy density, P = f(ρ). The relationship between pressure and energy densitydetermines the dynamics of the fluid. One such example of a barotropic fluid is the Chaplygin gas [10, 11].By introducing a cosmic fluid in lieu of the cosmological constant and quintessence one can unify the CDMand the Λ models’ features into a single component with an exotic equation of state. This versatility of∗abhuvan053@gmail.com†getsethi@gmail.com

1

arX

iv:1

409.

8521

v3 [

astr

o-ph

.CO

] 2

9 M

ay 2

019

the models built on the Chaplygin gas becomes an attractive feature as these models can explain both darkenergy and dark matter with a single component. The equation of state for the Chaplygin gas is P = −A/ρ,where A is a positive constant. A more generalized model of Chaplygin gas is characterised by an equationof state

Pch = − A

ραch(1)

where α is a constant such that 0 < α ≤ 1. Generalised Chaplygin gas automatically leads to anasymptotic accelerated expanding Universe. We see that the normal Chaplygin gas model is assumed forα = 1. By using the energy-momentum conservation, d(ρa3) = −pd(a3), we can see that the equation ofstate evolves as [12]

ρch =

(A+

B

a3(1+α)

) 11+α

(2)

where a is the scale factor, and B is the constant of integration. From the above equation it can beseen that at early times, a 1, we have ρ ∝ a−3 and it behaves as CDM. For late times, a 1, and weget p = −ρ = constant, i.e. the case of the cosmological constant which leads to the observed acceleratedexpansion. The models based on Chaplygin gas have garnered great interest as they have shown promise inthe past. These models have been found to be consistent with the SNe Ia data [13], CMB peak locations[14] and other observational tests like gravitational lensing, cosmic age of old high redshift objects etc. [15],as also with some combination of some of them [16]. Chaplygin gas models can also be accommodatedwithin the standard structure formation scenarios [11, 12, 17]. Therefore, the Chaplygin gas model seemsto be a good alternative to explain the accelerated expansion of the universe. However the Chaplygin gasmodel produces oscillations or an exponential blowup of matter power spectrums that are inconsistent withobservations [18].

A modification of Chaplygin gas recently proposed [19] is variable Chaplygin gas. The model was con-strained using SNe Ia ’gold’ data [21]. In this paper, we obtain better constrains for the variable Chaplygingas model parameters using statistical analysis of SNe Ia in Union 2.1 compilation from the SupernovaCosmology Project [20], Baryon Acoustic Oscillation (BAO) from the Sloan Digital Sky Survey (SDSS),lookback time measurements, and the GRBs. The basic formalism of the model is reviewed in Section 2.Section 3 gives the general theory for cosmic observables employed for the statistical analysis - luminositydistance, look back time and baryonic acoustic oscillations. Analysis of datasets and results are presentedin Section 4. We discuss the results and conclusions in Section 5.

2 Variable Chaplygin Gas Model

2.1 TheoryIn this section, we consider the variable Chaplygin Gas (VCG) [19], which is characterised by the equationof state:

P ch = −A(a)

ρch(3)

where A(a) = A0a−n is a positive function of the cosmological scale factor a. A0 and n are constants.

Using the energy conservation equation, d(ρa3) = −pd(a3), in a flat Friedmann-Robertson-Walker universeand equation (3), the variable Chaplygin gas density evolves as:

ρch =

√6

6− nA0

an+B

a6(4)

where B is a constant of integration. For n = 0, the original Chaplygin gas behaviour is restored. The gasinitially behaves as dust-like matter (ρch ∝ a−3) and later as a cosmological constant (p = −ρ = constant).However, in the present case the Chaplygin gas evolves from dust dominated epoch to cosmological constantin present times (see [19]).

2

The Friedmann equation gives the expansion rate of the Universe in terms of matter and radiation density,ρ, curvature, k, and the cosmological constant, Λ, as

H2 ≡(a

a

)2

=8πG

3ρ− k

a2+

Λ

3(5)

After reducing it to the case of the spatially flat Universe,

H2 =8πG

3ρ (6)

where H ≡ a/a is the Hubble parameter. Therefore the acceleration condition a > 0 is equivalent to(3− 6

6− n

)a6−n >

B

A0(7)

In order to incorporate accelerated expansion of the Universe, the necessary condition is n < 4. Thisgives the present value of energy density of the variable Chaplygin gas

ρcho =

√6

6− nA0 +B (8)

where a0 = 1. Defining a parameter, Ωm,

Ωm =B

6A0/(6− n) +B(9)

the energy density becomes

ρch(a) = ρch0

[Ωma6

+1− Ωman

]1/2

(10)

We also use the deceleration parameter, q = −(aa/a2). It is a measure of the cosmic accelerationand expansion of space, for an accelerated expansion q < 0. Now, the general expression for decelerationparameter in terms of redshift, z, and Hubble parameter, H, is given by

q(z) =d lnH(z)

d ln(1 + z)− 1 (11)

In addition to deceleration parameter, calculation of the age of the Universe is also employed in orderto draw a comparison between the evolution of the Universe with redshift in the cases of LCDM and VCGmodels. The age of the universe is given by

t(z) =

∫ ∞z

dz′

(1 + z′)H(z′)(12)

2.2 ModelThe Friedmann equation, using equation (10) for a variable Chalygin gas, becomes

H2 =8πG

3

ρr0(1 + z)4 + ρb0(1 + z)3 + ρch0

[Ωm(1 + z)6 + (1− Ωm)(1 + z)n

]1/2(13)

where ρr0 and ρb0 are the present values of energy densities of radiation and baryons, respectively. Using1

ρr0ρch0

=Ωr0Ωch0

=Ωr0

1− Ωr0 − Ωb0(14)

1We have used the fact that for a flat Universe, Ωb0 + Ωr0 + Ωch0 = 1, i.e the total matter density sums up to unity.

3

andρb0ρch0

=Ωb0Ωch0

=Ωb0

1− Ωr0 − Ωb0, (15)

Equation (13) becomes,H2 = Ωch0H

20a−4X2(a), (16)

where

X2(a) =Ωr0

1− Ωr0 − Ωb0+

Ωb0a

1− Ωr0 − Ωb0+ a4

(Ωma6

+1− Ωman

)1/2

. (17)

3 Cosmological ObservablesIn this work, we analyse three different cosmological observables - Luminosity Distance, Lookback Time, andBAO distance measurement. As we will see in this section, these observables show an explicit dependenceon the cosmological model under consideration. These parameters are chosen in order to compare theexperimentally obtained and the theoretical values of the respective model parameters.

3.1 Luminosity DistanceWe have used the SNe-Ia to constrain the parameters of the variable Chaplygin gas model. Using theFriedmann equation, in a flat Universe, the luminosity distance is expressed as

dL(z,p) = c(1 + z)

∫ z

0

dz′

H(z′,p)(18)

where, p denotes the set of all parameters describing the cosmological model and H(z,p) denotesthe Hubble parameter as defined in the model chosen. While calculating, we have taken into considerationthe contributions from radiation and baryons in addition to the Chaplygin gas. The distance modulus isobtained from the above relation using

µth = 5 logH0dLch

+ 42.38 (19)

H0 being the value of Hubble parameter at present, c is the speed of light and h ≡ H0/100kms−1Mpc−1 isthe dimensionless Hubble parameter. The theoretical distance modulus from the equation above is comparedto the observed values from the dataset. To determine the best fit parameters we minimise

χ2 =∑i

[µith − µiobs

σi

]2

− C1

C2

(C1 +

2

5ln 10

)− 2 ln h, (20)

here,

C1 ≡∑i

µith − µiobsσ2i

, (21)

C2 ≡∑i

1

σ2i

. (22)

In conjunction with the Union 2.1 compilation [20], we also incorporate GRBs data sample as describedin ref. [37]. Gamma-ray bursts are short-lived violent explosions that release high intensity gamma radiation.These are the most energetic events in the cosmos, they have the capacity to deliver more energy in a fewseconds than that our Sun will emit in its entire 10 billion year lifetime. For this reason, they are detectableupto high redshifts and are used to study the nature of the Universe and thus the dark energy.

4

Figure 1: χ2 contours for Union 2.1 compilation on Ωm − n parameter space.

5

Figure 2: χ2 contours for GRB dataset on Ωm − n parameter space.

6

3.2 Look Back TimeWe consider a homogeneous, isotropic, spatially flat universe described by the Friedmann-Robertson-Walkermetric with the line element ds2 = dt2 − a2(t)(dx2 + dy2 + dz2), where a(t) is the cosmological scale factor.

The lookback time is defined as the difference between the age of the Universe today, t0, and its age, tz,at redshift z. The dependence of lookback time on redshift z can be written as

tL(z,p) = H−10

∫ z

0

dz′

(1 + z′)H (z′,p)(23)

where H (z,p) = H(z)/H0 denotes the dimensionless Hubble parameter as defined in the model chosen.In order to calculate the age of the Universe, we use equation (23) with one small change - the upper limitof integration is increased from z to infinity.

To use the lookback time measurements, consider an object i at redshift z. The age of the object isdefined as the difference between the age of the Universe when the object was born, i.e. at the formationredshift zF , and the one at z. That is,

ti(z) =

∫ ∞z

dz′

(1 + z′)H (z′,p)−∫ ∞zF

dz′

(1 + z′)H (z′,p)

=

∫ zF

z

dz′

(1 + z′)H (z′,p)

= tL(zF )− tL(z)

(24)

here, we have used the equation (23) for lookback time. This relation is extended to all the N objects.Using this relation, we calculate the lookback time tobsL (zi) as

tobsL (zi) = tL(zF )− ti(z)= [tobs0 − ti(z)]− [tobs0 − tL(zF )]

= tobs0 − ti(z)− df(25)

where tobs0 is the estimated age of the Universe today, and a delay factor can be defined as

df = tobs0 − tL(zF ) (26)

We introduce an incubation time or delay factor in order to account for the formation redshift zF , i.e.the amount of time since the beginning of the structure formation in the Universe until the formation timeof the object. In this work, we have used the dataset as described in [23] in addition with [24]. In this caseas well, we minimize

χ2 =

N∑i=1

[tthL (zi,p)− tobsL (zi)√

σ2i + σ2

t

]2

+

(tth0 (p)− tobs0

σtobs0

)2

+

(h − hobs

σh

)2

, (27)

Now, to analytically obtain marginalised chi square, we define a log-likelihood function as

χ2 = −2ln

∫ ∞0

dτ exp

(− 1

2χ2age

)(28)

where τ represents the delay factor, σt is the uncertainty on observed age of the Universe, tobs0 , and σi isthat on tobsL (zi). The modified log-likelihood function can be rewritten as,

χ2 = A− B2

C+D − 2ln

[√π

2Cerfc

(B

2C

)](29)

here,

A =

N∑i=1

∆2

σ2i + σ2

t

, B =

N∑i=1

∆

σ2i + σ2

t

, C =

N∑i=1

1

σ2i + σ2

t

, (30)

∆ = tthL − [tobs0 − tobsL ], and D is the sum of last two terms in equation (27). We use an estimate oftobs0 , for which we choose (tobs0 , σt) = (13.6, 0.35)Gyr [24]. Also, hobs is the estimate value of h with σh itsuncertainty, (h, σh) = (0.72, 0.08) following from the Hubble space telescope key project results [26].

7

Figure 3: χ2 contours from lookback time measurement.

8

Survey z dz(z) Reference

6dFGS 0.106 0.3360± 0.0150 [30]MGS 0.15 0.2239± 0.0084 [31]

BOSSLOWZ 0.32 0.1181± 0.0024 [32]SDSS(R) 0.35 0.1126± 0.0022 [33]

BOSSCMAS 0.57 0.0726± 0.0007 [32]WiggleZ 0.44 0.073 [34]WiggleZ 0.6 0.0726 [34]WiggleZ 0.73 0.0592 [34]

Table 1: BAO distance measurements

3.3 Baryonic Acoustic OscillationsPhotons in the early Universe were trapped in the hot, dense plasma of electrons and baryons (protons andneutrons). Being scattered by plasma via Compton scattering, photons were restricted to a considerablylow mean-free path, up until the Universe cooled down to low enough temperatures that the electrons andprotons could combine to form Hydrogen (Recombination). Prior to recombination, photons and matterare remain coupled together. In an overdense region, the photon and matter oscillate together under theinfluence of both gravitational attraction and outward radiation pressure. Once the photons decouple fromthe neutral matter, its density distribution gets imprinted on the radiation spectrum. These, now free,photons are observed in the Cosmic Microwave Background (CMB).

These density fluctuations are the result of acoustic density waves in the ionised matter in the earlyUniverse. The perturbations of baryons starts to grow and interact with the dark matter perturbationsleaving an impression in the large-scale structure formation at late time [27]. BAO matter clustering canthus be used as’Standard Ruler’ [28]. Table 1 contains the BAO distance measurements used.

The distance-redshift relation for BAO measurement is given by

dz =rs(zdrag)

DV (z), (31)

wherers(zdrag) =

c√3

∫ ∞zdrag

dz

H(z)√

1 + (3Ωb0/4Ωr0(1 + z)−1(32)

is the radius of the comoving sound horizon at the drag epoch zdrag i.e. when photons and baryonsdecouple [29]. Also,

DV (z) =

[czd2

C(z)

H(z)

]1/3

(33)

is the volume-averaged distance [35] and dC(z) = dL(z)/(1+z) is the comoving angular diameter distance,with dL(z) being the luminosity distance given by equation (18). Ωb0 and Ωr0 in equation (32) are the presentvalues of baryon and photon density parameters receptively. Throughout our calculations, the dimensionlessHubble parameter is chosen to be h = 0.71, also we have taken Ωb0 = 0.05 and Ωγ0 = 9.89× 10−5 as in [36].

The three measurements from the WiggleZ survey in table 1 are correlated. The covariance matrixincluded while using these data points is as follows.

C−1 =

1040.3 −807.5 336.8−807.5 3720.3 −1551.9336.8 −1551.9 2914.9

For each of the survey in table 1, except the WiggleZ survey, chi-square is given as

χ2 =

[dthz − dobsz (z,p)

σ

]2

(34)

9

Figure 4: χ2 contours from BAO distance measurement

For the WiggleZ survey, we use the correlation matrix and the chi-square is written as

χ2WiggleZ = [dthz,i − dobsz (p)]TC−1[dthz,i − dobsz (p)]. (35)

The calculated χ2 values are then added together in order to obtain the joint chi-square value for thecomplete BAO distance measurement.

4 Statistical Analysis and ResultsIn order to constrain the parameters of the variable Chaplygin model we subject the model to statisticalanalysis using the four data sets and parameters described in the previous section. From figure 5, it is clearthat contours of the three observables result in an overlap of the 1σ confidence levels.

To estimate the best fit to the parameters the likelihood function is defined as

L ∝ exp[−χ2(z;p)/2], (36)

where the χ2 functions for all the datasets are calculated using equations described in their respectivesubsections. The parameter set, p, contains all the parameter including thenuisance parameters. Theseparameters are the dimensionless Hubble parameter, h, the observed age of the Universe, tobs0 , and the delayfactor, df , and the observed age of the Universe, tobs0 are marginalised over. We maximise the marginalisedlikelihood given by

Lpi ∝∫dp1...

∫dpi−1

∫dpi+1...

∫dpnL (p) (37)

10

Figure 5: χ2 contours from Union + Lookback time + BAO distance + GRBs

11

Figure 6: Age of the Universe, H0t(z)

After marginalisation these likelihood values are normalised and we obtain the contour plots. From theUnion 2.1 compilation, χ2

min = 563.0114 with 579 degrees of freedom, at Ωm = 0.087 and n = 1.14. As shownin figure 1, the 1σ constraints are Ωm ∈ [0.0265, 0.1415] and n ∈ [0.085, 2.01]. In addition to these, lookbacktime measurements result in χ2

min = 32.0827 with 45 degrees of freedom, and the best fit parameters areΩm = 0.01 and n = 3.29. Figure 3 gives the range of Ωm ∈ [0, 0.0592] and n ∈ [−2.01, 3.84]. Whereas,measurements from the BAO give χ2

min = 7.8819 with 7 degrees of freedom, at Ωm = 0.171 and n = −1.21.Figure 4 gives the parameter constraints as Ωm ∈ [0.117, 0.2485] and n ∈ [−2.586, 0.265]. The GRBs wereused to obtain the contour plot as shown in figure 2. We report that χ2

min = 153.5349 with 161 degrees offreedom, at Ωm = 0.0 and n = 4.27. While figure 5 overlays the three plots. 1σ(68.3%), 2σ(95.4%), and3σ(99.7%) confidence regions are shown. We see that the best fit parameters obtained in this work are inagreement with those reported in [19] which are Ωm = 0.25 and n = −3.4 with χ2

min = 174.54.Using the best fit parameters obtained we plot the evolution of age of the universe and deceleration

parameter, the best fit values are listed in table 2. For the ΛCDM model, we use Ωb0 = 0.3 and ΩΛ0 = 0.7.Figure 6 shows the evolution of the age of the universe with redshift, z. Figure 7 shows the evolution ofdeceleration parameter with redshift. Table 3 shows the bounds obtained for the variable Chaplygin gasmodel obtained from the four datasets.

12

Figure 7: Deceleration Parameter, q(z)

Method t0(Gyr) qUnion2.1 13.243 −0.554GRBs 12.291 0.089

Lookbacktime 13.608 −0.137BAO 12.568 −0.919

LCDMmodel 13.277 −0.55

Table 2: Best fit parameter values of age of the Universe and deceleration parameter

Dataset Ωm n

Union2.1 0.087+0.0365,+0.071−0.039,−0.082 1.14+0.586,+1.117

−0.676,−1.434

GRBs 0+0.191−0 4.27+0.73

−1.77

LookbackT ime 0.01+0.0268,+0.0786−0.01,−0.01 3.29+0.455,+0.636

−0.68,−2.356

BAO 0.171+0.0479,+0.109−0.0382,−0.0675 −1.21+0.96,+1.98

−0.92,−1.81

Table 3: 1 and 2− σ likelihood bounds on Ωm and n.

13

5 DiscussionWe see that the variable Chaplygin gas model is able to account for the observations regarding the evolutionof the universe. Initially the gas behaves like non-relativistic matter and later is able to account for theaccelerated expansion of the universe. With our analysis of the Union 2.1 compilation, BAO distance, look-back time measurements and the GRBs, we see that the ranges obtained for Ωm and n not only lie withinthe confidence levels obtained by Zong-Kuan Guo et al. but we have further constrained the possible valuesof Ωm and n by a joint analysis of the four datasets and have obtained tight limits on the parameters.

6 AcknowledgementThe authors would like to thank Dr. Sampurnanand for their comments and suggestions and The Centrefor Theoretical Physics, St. Stephen’s College, University of Delhi for facility and support. Dr. GeetanjaliSethi and Dr. Shruti Thakur are grateful to the Principal, St. Stephen’s College, University of Delhi for hissupport.

References[1] S. Perlmutter et al., Astrophys. J.517 (1999), 565; R. A. Knop et al., Astrophys. J.598 (2003), 102.

[2] A. G. Riess et al., Astronom. J.116 (1998), 1009; J. L. Tonry et al., Astrophys. J.594 (2003), 1.

[3] P. deBernardis et al., Nature(London) 404 (2000), 955; C. B. Netterfield et al., Astrophys. J.571 (2002),604.

[4] R. Stompor et al., Astrophys. J.561 (2001), L7.

[5] P. J. E. Peebles, & B. Ratra, Rev. Mod. Phys. 432 (2003), 559; V. Sahni, & A. Starobinsky, Int. J.Mod. Phys. D 9 (2000), 373; T. Padmanabhan, Phys. Rept. 380 (2003), 235.

[6] S. Weinberg, Rev. Mod. Phys. 61 (1989), 1.

[7] B. Ratra, & P. J. E. Peebles, Phys. Rev. D37 (1988), 3406; C. Wetterich, Nuc. Phy. B302 (1988), 668.

[8] I. Zlatev, L. M. Wang, & P. J. Steinhardt, Phys. Rev. Lett.82 (1999), 896; P. J. Steinhardt, L. Wang,& I. Zlatev, Phys. Rev. D59 (1999), 123504; W. Lee, & K. W. Ng, Phys. Rev. D67 (2003), 107302; D.S. Lee, W. Lee, & K. W. Ng, Int. J. Mod. Phys. D14 (2005), 335.

[9] M. C. Bento, O. Bertolami, & N. C. Santos, Phys. Rev. D65 (2002), 067301.

[10] A. Kamenshchik, U. Moschella, & V. Pasquier, Phys. Lett. B511 (2001), 265; V. Gorini, A. Kamen-shchik, & U. Moschella, Phys. Rev. D67 (2003), 063509.

[11] N. Bilic, G. B. Tupper, & R. D. Viollier, Phys. Lett. B535 (2002), 17.

[12] M. C. Bento, O. Bertolami, & A. A. Sen, Phys. Rev. D66 (2002), 43507.

[13] M. Makler, S. Q. deOliveira, & I. Wang, Phys. Lett.B555 (2003), 1; J. C. Fabris, S. B. V. Goncalves& P. E. deSouza, astro-ph/0207430; Y. Gong, & C. K. Duan, Class. Quant. Grav. 21 (2004), 3655;Mon. Not. R. Astr. Soc.352 (2004), 847; Y. Gong, JCAP 0503 (2005), 007.

[14] M. C. Bento, O. Bertolami, & A. A. Sen, Phys. Rev. D67 (2003), 063003; D. Carturan, & F. Finelli,Phys. Rev. D68 (2003), 103501.

[15] A. Dev, J. S. Alcaniz, D. Jain, Phys. Rev. D67 (2003), 023515; J. S. Alcaniz, D. Jain, & A. Dev,Phys. Rev. D67 (2003), 043514; Z. H. Zhu, Astr. Astrophys.423 (2004), 421.

[16] R. Bean, & O. Dore, Phys. Rev. D68 (2003), 023515; L. Amendola, F. Finelli, C. Burigana, & D.Carturan, JCAP 0307 (2003), 005.

14

[17] J. C. Fabris, S. B. V. Goncalves, & P. E. deSouza, Gen. Rel. Grav.34 (2002), 53.

[18] H. Sandvik, M. Tegmark, M. Zaldarriaga, & I. Waga, Phys. Rev. D69 (2004), 123524.

[19] Z. K. Guo, & Y. Z. Zhang, astro-ph/0506091.

[20] N. Suzuki et al., Astrophys. J.746 (2012), 1.

[21] Adam G. Riess et al., Astrophys. J.607 (2004), 665.

[22] S. Capozziello, V. F. Cardone, M. Funaro, & S. Andreon, Phys. Rev. D70 (2004), 123501.

[23] L. Samushia, A. Dev, D. Jain, B. Ratra, Phys. Lett. B 693, 5 (2010), p. 509.

[24] M. A. Dantas, J. S. Alcaniz, D. Jain, and A. Dev, Astr. Astrophys.467, 2 (2007), p. 421-426.

[25] R. Rebolo et al., astro-ph/0402466.

[26] W. L. Freedman et al., Astrophys. J.553 (2001), 47.

[27] D. J. Eisenstein, W. Hu, Astrophys. J.496 (1998), 605.

[28] D. J. Eisenstein, New Astro. Rev. 49 (2005), 7-9, pg. 360-365.

[29] D. J. Eisenstein & W. Hu, Astrophys. J. 496 (1998), 605.

[30] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell et al., Mon. Not. Roy.Astron. Soc. 416 (2011), 3017.

[31] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden & M. Manera, Mon. Not. Roy. Astron.Soc. 449 (2015), 835.

[32] L. Anderson et al., Mon. Not. Roy. Astron. Soc. 441 (2014), 24.

[33] N. Padmanabhan, X. Xu, D. J. Eisenstein, R. Scalzo, A. J. Cuesta, K. T. Mehta et al., Mon. Not. Roy.Astron. Soc. 427 (2012), 2132.

[34] C. Blake et al., Mon. Not. Roy. Astron. Soc. 425 (2012), 405.

[35] D. J. Eisenstein et al., Astrophys. J. 633 (2005), 560.

[36] G. Sethi, S. K. Singh, P. Kumar, D. Jain & A. Dev, Int. J. Mod. Phy. D 15 (2006), 7.

[37] M. Demianski, E. Piedipalumbo, D. Sawant & L. Amati, A & A 598 (2017), A112.

15