Observational constraints on EoS parameters of emergent universe

Astrophys Space Sci (2017) 362:73 DOI 10.1007/s10509-017-3019-x

O R I G I N A L A RT I C L E

Observational constraints on EoS parameters of emergent universe Bikash Chandra Paul1 · Prasenjit Thakur2

Received: 13 June 2016 / Accepted: 17 January 2017 © Springer Science+Business Media Dordrecht 2017

Abstract We investigate emergent universe model using recent observational data of the background as well as the growth tests. The flat emergent universe model obtained by Mukherjee et al. is permitted with a non-linear equation 1 of state (in short, EoS) (p = Aρ − Bρ 2 ), where A and B are constants (here in our analysis A = 0 is considered). We carried out analysis considering the Wang–Steinhardt ansatz for growth index (γ ) and growth function (f defined γ as f = Ωm (a)). The best-fit values of the EoS and growth parameters are determined making use of chi-square minimization technique. Here we specifically determined the best-fit value and the range of value of the present matter density (Ωm ) and Hubble parameter (H0 ). The best-fit values of the EoS parameters are used to study the evolution of the growth function f , growth index γ , state parameter ω and deceleration parameter (q) for different red shift parameter z. The late accelerating phase of the universe in the EU model is accommodated satisfactorily. Keywords Emergent universe · Accelerating universe · Cosmology

1 Introduction The recent cosmological and astronomical observations predict that the present universe is expanding. It is also believed

B B.C. Paul

[email protected] P. Thakur [email protected]

1

Physics Department, North Bengal University, Siliguri, Darjeeling, 734 013, West Bengal, India

2

Physics Department, Alipurduar College, Alipurduar, 736 122, West Bengal, India

that the present universe might have emerged out of an inflationary phase in the past. After the discovery of Cosmic Microwave Background Radiation (Penzias and Wilson 1965; Dicke et al. 1965) the big-bang model become the standard model of the universe which has a beginning of the Universe at some finite past. However, big-bang model based on perfect fluid fails to account some of the observed facts of the universe. Further it is observed that while probing the early universe a number of problems namely, the horizon problem, flatness problem, singularity problem, large scale structure formation problem cropped up. In order to resolve those issues of the early universe the concept of inflation (Guth 1981; Sato 1981; Linde 1982; Albrecht and Steinhardt 1982) in cosmology was introduced. A number of inflationary models are proposed in the last thirty years. The recent cosmological observations predict that the present universe is passing through a phase of acceleration (Riess et al. 1998; Tonry et al. 2003; Perlmutter et al. 1998, 1999) another mystery of the universe. This phase of acceleration is believed to be a late time expansion phase of the universe which can be accommodated in the standard model with the help of a positive cosmological constant. However, the physics of the inflation and introduction of a small cosmological constant for late time acceleration, is not clearly understood (Albrecht 2000; Carroll 2001). In the literature the late accelerating phase of the universe is obtained with exotic matter or with a modification of the Einstein gravity. A nonlinear equation of state is also considered in the literature to construct cosmologies (Mukherjee et al. 2006), emergent universe model is one such model. The Emergent universe (EU) model obtained by Mukherjee et al. in the flat universe permits an accelerating phase. Emergent universe scenario was introduced mainly to avoid the initial singularity. It replaces the initial singularity by an Einstein static phase in which the scale factor of the Friedmann-Robertson-Walker

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B.C. Paul, P. Thakur

(FRW) metric does not vanish. As a result the energy density, pressure do not diverge. In this description the universe started expanding from the initial phase, smoothly joins a stage of exponential inflation followed by standard reheating then subsequently it approaches the classical thermal radiation dominated era of the conventional big bang cosmology (Ellis and Maartens 2004). Universe in this model can stay large enough to avoid quantum gravitational effects even in the very early universe. The emergent universe (EU) scenario is considered to begin from a static Einstein universe forever which after wards successfully accommodates the early inflationary phase and avoid the messy situation of the initial singularity (Ellis and Maartens 2004; Harrison 1967). In EU model late time deSitter phase exists which naturally incorporates the late time accelerating phase as well. EU scenario has been explored with quintom matter (Cai et al. 2012) and further investigated the realization of the scenario with a non-conventional fermion field to obtain a scale invariant perturbation (Cai et al. 2014). It has been shown that the EU scenario can be implemented successfully in the framework of general relativity (Mukherjee et al. 2006) in addition to GaussBonnet gravity (Paul and Ghose 2010). The modified GaussBonnet gravity as gravitational alternative for dark energy is however considered by Nojiri and Oddintsov (2005a). It is also shown recently that EU model can be successfully implemented in Brane world gravity (Banerjee et al. 2008; Debnath 2008), Brans-Dicke theory (del Campo et al. 2007). A number of cosmological models are obtained with different cosmological fluids and fields (Ellis and Maartens 2004; Nojiri and Oddintsov 2005b; Astashenok et al. 2012; Bag et al. 2014) where initial singularity problem is addressed. Mukherjee et al. (2006) obtained an emergent universe model in the framework of GTR with a polytropic equation of state (EoS) given by 1

p = Aρ − Bρ 2

(1)

where A, B are constants with B > 0. It is interesting as it avoids the initial singularity problem and the initial size of the universe is large. EU model also accommodates the late accelerating phase. It may be pointed out here that the EoS state parameters in the model play an important role which decide the composition of matter in the universe. In Mukherjee et al. (2006), it is shown that for discrete set of values of A namely, A (= 0, −1/3, 1/3, 1), one obtains universe with a mixture of three different kinds of cosmic fluids. The dark energy is one of the prime constituents of the mixture. Each of the above EU model have dark energy as one of its constituent fluid. Here the parameter B is arbitrary (as A = 0 is considered). It may be mentioned here that A = 0 corresponds to a universe with a composition of fluid namely, dust, exotic matter and dark energy (Mukherjee et al. 2006). As the present universe is mater dominated and one of the

component is dust we numerically analyzed cosmological model here with A = 0. Further the constraints imposed on the EU model parameters are determined making use of cosmological observations. The analysis we adopt here consists of both the background test and the growth test. Case A: Analysis using background tests: There are four main background tests for a cosmological model: • The differential age of old galaxies, given by H (z). • The peak position of the baryonic acoustic oscillations (BAO). • The CMB shift parameter. • The SN Ia data. We use H (z) − z data given in Table 1. The supernovae data is taken from the union compilation data (Union 2.1) (Suzuki et al. 2012). Case B: Analysis using growth data: The growth of the large scale structures derived from linm ear matter density contrast δ(z) ≡ δρ ρm of the universe is considered to be an important tool to constrain cosmological model parameters. In this case one parametrizes the log δ growth function f = dd log a in terms of growth index γ to describe the evolution of the inhomogeneous energy density. Initially, Peebles (1980) and later Wang and Steinhardt (1998) parametrized δ in terms of γ . The above parametrizations in cosmology have been used in different contexts in the literature (Linder 2005; Laszlo and Bean 2008; Jain and Zhang 2008; Hu and Sawicki 2007; Lue et al. 2004; Acquaviva et al. 2008; Koyama and Maartens 2006; Koivisto and Mota 2006; Daniel et al. 2008; Ishak et al. 2006). The growth data set given by Table 3 corresponds to a value for growth function f at a given red-shift. In Table 4 rms mass fluctuations (σ8 (z)) data displayed corresponds to various sources such as: the red-shift distortion of galaxy power spectra (Hawkins et al. 2003), root mean square (rms) mass fluctuation (σ8 (z)) obtained from galaxy and Ly-α surveys at various red-shifts (Viel et al. 2004; Viel and Haehnelt 2006), weak lensing statistics (Kaiser 1998), baryon acoustic oscillations (Eisenstein et al. 2005), X-ray luminous galaxy clusters (Mantz et al. 2008), Integrated Sachs-Wolfs (ISW) Effect (Rees and Sciama 1968; Amendola et al. 2007; Hoekstra et al. 2006; Crittenden and Turok 1996; Pogosian et al. 2005). It is known that red-shift distortions are caused by velocity flow induced by gravitational potential gradient which evolved both due to the growth of the universe under gravitational attraction and dilution of the potentials due to the cosmic expansion. The gravitational growth index γ is also related to red-shift distortions (Linder 2005). The cluster abundance evolution, however, strongly depends on rms

Observational constraints on EoS parameters of emergent universe Table 1 Observed Hubble data (OHD) z

H (z)

Page 3 of 10

73

2 Field equations

σ

Ref.

The Einstein field equation is given by 0.0708

79

± 19.68

Zhang et al. (2014)

0.09

69.0

± 12.0

Jimenez et al. (2003)

0.12

68.6

± 26.2

Zhang et al. (2014)

0.17

83.0

± 8.0

Simon et al. (2005)

0.179

75.0

± 4.0

Moresco et al. (2012)

0.199

75.0

± 5.0

Moresco et al. (2012)

0.20

72.9

± 29.6

Zhang et al. (2014)

0.240

79.69

± 2.65

Gaztanaga et al. (2009)

0.27

77.0

± 14.0

Simon et al. (2005)

0.28

88.8

± 36.6

Zhang et al. (2014)

0.35

82.1

+4.8, −4.9

Chuang et al. (2012)

0.35

84.4

± 7.0

Xu et al. (2013)

0.352

83.0

± 14.0

Moresco et al. (2012)

0.4

95.0

± 17.0

Simon et al. (2005)

0.43

86.45

± 3.68

Gaztanaga et al. (2009)

0.44

82.6

± 7.80

Blake et al. (2012)

0.48

97.0

± 62.0

Stern et al. (2010)

0.57

92.4

± 4.5

Samushia et al. (2012)

0.593

104.0

± 13.0

Moresco et al. (2012)

0.6

87.9

± 6.1

Blake et al. (2012)

0.68

92.0

± 8.0

Moresco et al. (2012)

0.73

97.3

± 7.0

Blake et al. (2012)

0.781

105.0

± 12.0

Moresco et al. (2012)

0.875

125.0

± 17.0

Moresco et al. (2012)

0.88

90.0

± 40.0

Stern et al. (2010)

0.9

117.0

± 23.0

Simon et al. (2005)

1.037

154.0

± 20.0

Moresco et al. (2012)

1.3

168.0

± 17.0

Simon et al. (2005)

1.363

160.0

± 33.6

Moresco (2015)

1.43

177.0

± 18.0

Simon et al. (2005)

1.53

140.0

± 14.0

Simon et al. (2005)

1.75

202.0

± 40.0

Simon et al. (2005)

1.965

186.5

± 50.4

Moresco (2015)

2.3

224.0

± 8.0

Busca et al. (2013)

2.34

222.0

± 7.0

Delubac et al. (2015)

2.36

226.0

± 8.0

Font-Ribera et al. (2014)

mass fluctuations (σ8 (z)) (Wang and Steinhardt 1998) which will be also considered in the present analysis. The paper is presented as follows: In Sect. 2, relevant field equations obtained from Einstein field equations are given. In Sect. 3, constraint on the EoS parameters obtained from background test are presented. In Sect. 4, growth index parametrization in terms of EoS parameters is studied. In Sect. 5, constraint on the EoS parameters obtained from background test and growth test are determined. In Sect. 6, a summary of the results analyzed are tabulated. Finally, in Sect. 7, we give a brief discussion.

1 (2) Rμν − gμν R = 8πG Tμν 2 where Rμν represents Ricci tensor, R represents Ricci scalar, Tμν represents energy momentum tensor and gμν represents the metric tensor in 4-dimensions. We consider a Robertson-Walker metric which is given by    2  dr 2 2 2 2 2 2 2 (3) ds = −dt + a (t) + r dθ + sin θ dφ 1 − kr 2 where k = 0, +1(−1) is the curvature parameter in the spatial section representing flat, closed (open) universe respectively and a(t) is the scale factor of the universe with r, θ , φ the dimensionless co-moving co-ordinates. Using metric (3) in the Einstein field Eq. (2), we obtain the following equations:   2 k a˙ (4) 3 2 + 2 = 8πG ρ, a a a¨ a˙ 2 k 2 + 2 + 2 = −8πG p, (5) a a a where ρ and p represent the energy density and pressure respectively. The conservation equation is given by dρ + 3H (ρ + p) = 0, dt

(6)

where H = aa˙ is Hubble parameter. Using EoS given by Eq. (1) in Eq. (6), and integrating once we obtain energy density which is (for A = 0 model) given by   K 2 (7) ρeu = B + 3 a2 where K is a positive integration constant. For convenience we rewrite Eq. (7) as   1 − As 2 (8) ρeu = ρeu0 As + 3 a2 where As =

B 1 2 ρeu0

1

2 and K = ρeu0 − B. The scale factor of the

1 universe can be expressed as aa0 = 1+z , where z is the redshift parameter and we choose the present scale factor of the universe a0 = 1. Therefore the Hubble parameter in terms of red-shift parameter can be rewritten using the field Eq. (4) as  3 (9) H (z) = H0 As + (1 − As )(1 + z) 2

where H0 represents the present Hubble parameter. Using the present matter density of the universe Ωm = (1 − As )2 (Li et al. 2012) Hubble parameter can be expressed as



 3 H (z) = H0 (1 − Ωm ) + Ωm (1 + z) 2 . (10)

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B.C. Paul, P. Thakur Table 2 BAO data

The square of the speed of sound is given by cs2 =

δp p˙ = δρ ρ˙

z1

A

σA

Ref.

0.106

0.526

0.028

Hinshaw et al. (2013)

0.20

0.488

0.016

Hinshaw et al. (2013)

0.35

0.484

0.016

Hinshaw et al. (2013)

0.44

0.474

0.034

Blake et al. (2011b)

0.57

0.436

0.017

Hinshaw et al. (2013)

(11)

which reduces to

√ (1 − Ωm ) 2 cs = − . √ √ 3 2(1 − Ωm + Ωm (1 + z) 2 )

(12)

In terms of state parameter it reduces to cs2 =

ω . 2

Chuang et al. (2013)

(13)

From the above equation we obtain the inequality

(1 − Ωm ) < 2

(14)

for a realistic solution which admits stable perturbation (Lixin et al. 2012). Again positivity of sound speed leads to a upper bound on cs2 ≤ 1 which arises from the causality condition. The deceleration parameter is given by

0.60

0.442

0.020

Blake et al. (2011b)

0.73

0.424

0.021

Blake et al. (2011b)

Case II: For BAO Data A model independent BAO (Baryon Acoustic Oscillation) peak parameter for low red shift z1 measurements in a flat universe is given by Eisenstein et al. (2005): √  z1 dz 2/3 Ωm 0 E(z) A(Ωm , z1 ) = (18) z1 E(z1 )1/3

(16)

where Ωm is the present matter density parameter for the Universe. The chi square function in this case is defined as:  (A(Ωm , z1 ) − Aobs (z1 ))2 2 χBAO (Ωm ) = . (19) (σA )2 The BAO data is given in Table 2. Case III: For CMB The CMB shift parameter (R) is given by Komatsu et al. (2011):

zls

dz R = Ωm (20) H (z )/H0 0

We consider first the background tests from observed cosmological data for analyzing cosmological models.

where zls is the z at the surface of last scattering. The WMAP7 data predicts R = 1.726 ± 0.018 at z = 1091.3. We now define chi-square function as:

3.1 Observational constraints

2 χCMB (Ωm ) =

Ωeu (a)[1 + 3ω(a)] q(a) = 2[Ωeu (a)]

(15)

where

√  

Ωm 2 Ωeu (a) = Ωeu0 1 − Ωm + 3 a2

3 Background tests

The equation of state for emergent universe contains two unknown parameters namely H0 and Ωm which are determined from numerical analysis. For this the Einstein field equation is rewritten in terms of a dimensionless Hubble parameter and a suitable chi-square function is defined in different cases. Case I: For OH D The Observed Hubble Data (OHD) is then taken from the table given below. To analyze first we define chi-square χH2 −z function is given by 2 χOH D (H0 , Ωm ) =

 (H (H0 , Ωm , z) − Hobs (z))2 σz2

(17)

where Hobs (z) is the observed Hubble parameter at red shift z and σz is the error associated with that particular observation as shown in Table 1.

(R − 1.726)2 . (21) (0.018)2 Case IV: For Supernovae Data The distance modulus function (μ) is defined in terms of luminosity distance (dL ) as μ(H0 , Ωm , z) = m − M = 5 log10 (dL ) + 25

where

c(1 + z) z dz dL =  H0 0 E(z )

(22)

(23)

In this case the chi-square χμ2 function is defined as χμ2 (H0 , Ωm ) =

 (μ(H0 , Ωm , z) − μobs (z))2 σz2

(24)

where μobs (z) is the observed distance modulus at red shift z and σz is the corresponding error for the 580 observed data (Suzuki et al. 2012). Finally the chi-square function for background tests is defined as

Observational constraints on EoS parameters of emergent universe 2 2 2 χback (H0 , Ωm ) = χOH D (H0 , Ωm ) + χBAO (Ωm ) 2 + χCMB (Ωm ) + χμ2 (H0 , Ωm )

(25)

Page 5 of 10

The logarithmic growth factor f , according to Wang and Steinhardt (1998) is given by γ

The chi-square function for background test is minimized and best fit values of H0 and Ωm are determined.

f = Ωm (a)

γ=

a˙ δ¨ + 2 δ˙ − 4πGρm δ = 0. (26) a The field equations for the background cosmology in a flat Robertson–Walker metric are given below  2 a˙ 8πG (ρeu ), = (27) a 3  2 a¨ a˙ 2 + = −8πGωeu ρeu (28) a a where ρeu and ωeu represent energy density and the equation of state parameter for EU respectively. The equation of state parameter for EU (for A = 0) corresponding to the EoS given by Eq. (1) is √ (1 − Ωm ) ωeu = − . (29) √ √ 3 (1 − Ωm ) + ( Ωm )(1 + z) 2 Now we replace time variable (t) by a scale factor variable in the above to solve the equation. Consequently, we replace t variable to ln a in Eq. (26), and finally obtain     1 3 (ln δ) + (ln δ) 2 + (ln δ) − ωeu 1 − Ωm (a) 2 2 3 (30) = Ωm (a). 2 The effective matter density is given by Ωm (a) =

H02 Ωm a −3 H 2 (a)

(31)

where Ωm = (1 − As )2 is the present matter density of the universe (Li et al. 2012). Using the energy conservation Eq. (6) and changing ln a variable to Ωm (a) once again it is possible to rewrite the Eq. (30) in terms of the logarithmic log δ growth factor (f = dd log a ), which is given by df 3ωeu Ωm (1 − Ωm ) +f2 dΩm     3 1 3 − ωeu 1 − Ωm (a) = Ωm (a). +f 2 2 2

3(ω0 − 1) . 6ω0 − 5

(34)

6 For a ΛCDM model, it reduces to 11 (Linder 2005; Linder and Cahn 2007), however, for a matter dominated model one obtains γ = 47 (Fry 1985; Nesseries and Perivolaropoulos 2008). It is also convenient to express γ as a parametrized function of red shift parameter z in cosmology. One of the parametrized form is obtained from the Taylor expansion of the function about z = 0 keeping the first two terms only. Accordingly one obtains

γ (z) = γ (0) + γ  z, where γ  ≡ dγ dz |(z=0) (Polarski and Gannouji 2008; Gannouji and Polarski 2008). It has been shown recently (Ishak and Dossett 2009) that it smoothly interpolates a low and intermediate red shift range to a high red shift range up to the cosmic microwave background (CMB) scale. Similar parametrization technique is also used in cosmology in different contexts (Dosset et al. 2010) to study evolution. Here we parametrize γ in terms of EoS parameter Ωm for emergent universe. Therefore, we begin with the following ansatz which is given by γ (Ωm )

f = Ωm

(35)

(a)

where the growth index parameter is represented by γ (Ωm ). It can be expanded in Taylor series expansion around Ωm = 1 which leads to γ (Ωm ) = γ |(Ωm =1) + (Ωm − 1) + O(Ωm − 1)2 .

dγ |(Ω =1) dΩm m (36)

Now the Eq. (32) can be rewritten in terms of γ as   1 dγ γ + Ωm − 3ωeu Ωm γ − 3ωeu Ωm (1 − Ωm ) ln Ωm dΩm 2 3 1−γ 3 1 − Ωm + 3ωeu γ − ωeu + = 0. (37) 2 2 2 Differentiating once again the above equation around Ωm = 1, one obtains a zeroth order term in the expansion for γ given by γ=

(32)

(33)

where γ represents the growth index parameter. In the case of flat dark energy model with constant state parameter ω0 , the growth index γ is given by

4 Parametrization of the growth index In this section the growth rate of the large scale structures is m derived from matter density perturbation given by δ = δρ ρm (where δρm represents the fluctuation of matter density ρm ) in the linear regime which satisfies the following equation (Padmanabhan 2002; Liddle and Lyth 1999):

73

3(1 − ωeu ) , 5 − 6ωeu

(38)

which supports a dark energy model for a constant ω0 (Eq. (34)). Differentiating the above equation once again

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B.C. Paul, P. Thakur

Table 3 Data for the observed growth functions fobs used in our analysis

Table 4 Data for the rms mass fluctuations (σ8 ) at various red-shift z

σ8

σσ8

Ref.

2.125

0.95

0.17

(Viel et al. 2004)

z

fobs

σ

Ref.

0.15

0.51

0.11

(Hawkins et al. 2003; Verde et al. 2002)

2.72

0.92

0.17

0.22

0.60

0.10

(Blake et al. 2011a)

2.2

0.92

0.16

0.32

0.654

0.18

(Reyes et al. 2010)

2.4

0.89

0.11

0.35

0.70

0.18

(Tegmark et al. 2006)

2.6

0.98

0.13

0.41

0.70

0.07

(Blake et al. 2011a)

2.8

1.02

0.09

0.55

0.75

0.18

(Ross et al. 2006)

3.0

0.94

0.08

0.60

0.73

0.07

(Blake et al. 2011a)

3.2

0.88

0.09

0.77

0.91

0.36

(Guzzo et al. 2008)

3.4

0.87

0.12

0.78

0.70

0.08

(Blake et al. 2011a)

3.6

0.95

0.16

1.4

0.90

0.24

(da Angela et al. 2008)

3.8

0.90

0.17

3.0

1.46

0.29

(McDonald et al. 2005)

0.35

0.55

0.10

0.6

0.62

0.12

0.8

0.71

0.11

1.0

0.69

0.14

1.2

0.75

0.14

1.65

0.92

0.20

with respect to Ωm , the first order terms in the expansion at Ωm = 1, is given by 3(1 − ωeu )(1 − 3ω2eu ) dγ |(Ωm =1) = . dΩm 125(1 − 6ω5eu )3

(39)

Using the above equation in Eq. (36), γ is further determined. Now the zeroth and first order terms together give the following expression 3ω

γ=

3(1 − ωeu )(1 − 2eu ) 3(1 − ωeu ) + (1 − Ωm ) . 5 − 6ωeu 125(1 − 6ωeu )3

(40)

5

Using the expression of ωeu in the above, γ can be parametrized in EU model in terms of the EoS parameters, namely, Ωm respectively and red shift parameter z. Let us now define normalized growth function g by δ(z) (41) δ(0) which is determined from the solution of Eq. (30). Thereafter the corresponding approximate normalized growth function is obtained from the parametrized form of f from Eq. (35). It is given by   1 1+z γ da gth (z) = exp (42) Ωm (a) a 1 g(z) ≡

which will be considered here to construct chi-square function in the next section. 4.1 Observational constraints We define chi-square of the growth function f as   fobs (zi ) − fth (zi , γ ) 2 2 χf (Ωm ) = Σ σfobs

(43)

where fobs and σfobs are obtained from Table 3. However, fth (zi , γ ) is obtained from Eqs. (35) and (40). Another observational probe for the matter density perturbation δ(z) is

(Viel and Haehnelt 2006)

(Marinoni et al. 2005)

derived from the red shift dependence of the rms mass fluctuation σ8 (z). The dispersion of the density field σ 2 (R, z) on a co-moving scale R is defined as

inf dk 2 (44) σ (R, z) = W 2 (kR)2 (k, z) k 0 where W (kR) = 3( sin(kR) − (kR)3 function, and

cos(kR) ), (kR)2

2 (kz) = 4πk 3 Pδ (k, z),

represents window (45)

where Pδ (k, z) ≡ is the mass power spectrum at redshift z. The rms mass fluctuation σ8 (z) is the σ 2 (R, z) at R = −1 8h Mpc. The function σ8 (z) is connected to δ(z) as δ(z) (46) σ8 (z) = σ8 |(z=0) δ(0) which implies (δk2 )

σ8 (z1 ) δ(z1 ) = σ8 (z2 ) δ(z2 ) 1+z1 exp[ 1 1 Ωm (a)γ da a ] = . 1+z1 2 exp[ 1 Ωm (a)γ da ] a

sth (z1 , z2 ) ≡

(47)

In Table 4, a systematic evolution of rms mass fluctuation σ8 (zi ) with observed red shift for flux power spectrum of Ly-α forest (Viel et al. 2004; Viel and Haehnelt 2006; Marinoni et al. 2005) are displayed. In this context we define a new chi-square function which is given by   sobs (zi , zi+1 ) − sth (zi , zi+1 ) 2 2 χs (Ωm ) = Σ . (48) σsobs,i

Observational constraints on EoS parameters of emergent universe

Page 7 of 10

73

Fig. 1 Contours of (i) H0 –Ωm in EU model with A = 0 from OU data, where OU = OHD + BAO + CMB + Union2.1 data at 68.3% (Yellow) 95.4% (Blue) and 99.7% (Green) confidence limit

Fig. 3 Plot of Maximum Likelihood functions with (i) H0 and (ii) Ωm from OHD + BAO + CMB + Union2.1 + growth + rms mass fluctuation data

Fig. 2 Contours of (i) H0 –Ωm in EU model with A = 0 from OS data, where OS = OHD + BAO + CMB + Union2.1 + Growth + σ8 at 68.3% (Yellow) 95.4% (Blue) and 99.7% (Green) confidence limit

Data for rms mass fluctuation at various red shift given in Table 4 will be considered here. Now considering growth function mentioned above, one can define chi-square function which is given by 2 χgrowth (Ωm ) = χf2 (Ωm ) + χs2 (Ωm ).

(49)

The chi-square functions defined above will be considered for the analysis in the next section.

5 Observational constraints from background test and growth test Using Eq. (25) and Eq. (49), we define total chi-square function as 2 2 2 χtotal (H0 , Ωm ) = χback (H0 , Ωm ) + χgrowth (Ωm )

(50)

2 where χgrowth (Ωm ) = χf2 (Ωm ) + χs2 (Ωm ). Likelihood function L are related with the chi-square for background test 2 /2) and L ∝ and in combined tests as L ∝ Exp(−χback 2 Exp(−χtotal /2). To get the best-fit values of the EU model likelihood function can be maximized or the chi-square function can be minimized. In this case the best fit values are obtained minimizing the chi-square function. Since chisquare function depends on H0 & Ωm , it is possible to draw contours at different confidence limit. The contours among the parameters H0 and Ωm for background test and in combined test are shown respectively in Figs. 1 and 2. The plot of maximum likelihood functions with the parameters H0 and Ωm in combined test are shown in Fig. 3.

6 Results In this analysis present Hubble parameter is also taken as a free parameter. So here H0 and Ωm are the two parameters whose value are determined at different confidence level.

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B.C. Paul, P. Thakur Table 5 Range of values of H0 & Ωm from OU data, where OU = OHD + BAO + CMB + Union2.1 in EU model with A = 0 Data

CL

H0

Ωm

OU

68.3%

(67.09, 67.80)

(0.255, 0.279)

OU

95.4%

(66.86, 68.06)

(0.248, 0.286)

OU

99.7%

(66.64, 68.29)

(0.242, 0.294)

Table 6 Range of values of H0 & Ωm from OS data, where OS = OHD + BAO + CMB + Union2.1 + Growth + σ8 in EU model with A=0 Data

CL

H0

Ωm

OS

68.3%

(67.18, 67.91)

(0.250, 0.272)

OS

95.4%

(66.96, 68.14)

(0.243, 0.279)

OS

99.7%

(66.73, 68.37)

(0.237, 0.286)

Table 7 Values of the best-fit EoS parameters at present for EU model with A = 0 Model

f

γ

ω0

q

EU

0.461

0.574

−0.489

−0.242

The best-fit values of the EU model with A = 0 obtained with background data are H0 = 67.46, Ωm = 0.267 and in the combined test are H0 = 67.55, Ωm = 0.260 (Table 5). In the combined test determined values of the parameters +0.012 are H0 = 67.55+0.36 −0.37 and Ωm = 0.260−0.010 (Table 6) at 1σ level. The acceptable range in the combined test for the parameters H0 & Ωm are at 99.7% confidence level (3σ ) are (66.73, 68.37) & (0.237, 0.286) respectively (shown in Table 6). The present value of the parameters f , γ , ω0 & q in this EU model are 0.461, 0.574, −0.489 & −0.242 respectively (shown in Table 7).

7 Discussion In this paper we present an analysis of flat emergent universe model (Mukherjee et al. 2006) with observational data. We note the following:

Fig. 4 Plot of (i) growth function f , (ii) growth index γ , (iii) EoS parameter ω and (iv) deceleration parameter q with red shift for the EU model with A = 0 in OS data

(i) The best-fit values and range of values obtained with the background data and background+growth data are very close to each other. The best-fit values of the model obtained with background+growth data are H0 = 67.55, Ωm = 0.260. The present Hubble value predicted by our analysis are close to Planck 15 data (Ade et al. 2015). (ii) From the plot of growth function (f ) and growth index (γ ) with z in Fig. 4 it is evident that both f and γ attains a maximum value in the early universe showing signs of structure formation in that era.

Observational constraints on EoS parameters of emergent universe

(iii) From the plot of state parameter ω with z in Fig. 4 it is evident that the condition ω ≤ − 13 is maintained at the present epoch which points towards an accelerating phase of the universe at recent past. (iv) The last plot in Fig. 4, showing the variation of deceleration parameter with red shift (z) it also hints that in the recent past the universe transits from deceleration phase to accelerating phase. So, the best-fit values of the EoS parameters of this EU model of the universe satisfactorily accommodates accelerating phase in the recent past and a structure formation era in past. Acknowledgements The authors would like to thank IUCAA Reference Centre at North Bengal University for extending necessary research facilities to initiate the work. BCP would like to acknowledge the University Grants Commission (UGC), New Delhi for a Major Research Project Grant (No. F.42-783/(SR) 2013). BCP would like to thank TWAS-UNESCO for awarding Associateship to visit ITP, Chinese Academy of Sciences, Beijing, China and Institute of Mathematical Sciences, Chennai for hospitality.

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