Recent observational constraints on EoS parameters of a class of emergent Universe

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1 Praana J. Phys. (2017) 89:27 DOI 1007/s Indian Acadey of Sciences Recent observational constraints on EoS paraeters of a class of eergent Universe P THAKUR Physics Departent, Alipurduar College, Alipurduar , India E-ail: prasenjit_thakur1@yahoo.co.in MS received 13 October 2016; revised 19 Deceber 2016; accepted 25 January 2017; published online 20 July 2017 Abstract. Eergent Universe (EU) odel is investigated here using the recent observational data of the background tests. The background test data coprise observed Hubble data, baryon acoustic oscillation data, cosic icrowave background shift data and Union copilation(2.1) data. The flat EU odel obtained by Mukherjee et al is peritted with a non-linear equation of state (in short, EoS) (p = Aρ Bρ 1/2 ), where A and B are constants. The best-fit values and peritted range of values of the EoS paraeters are deterined in general EU odel and in specific EU odel (A = 0) by using chi-square iniiation technique. The best-fit values of the EoS paraeters are used to study the evolution of the squared adiabatic sound speed cs 2, state paraeter ω and deceleration paraeter q for different red-shifts. The present age of the Universe t 0 has been deterined in general EU odel as well as for EU odel with A = 0. The late accelerating phase of the Universe in the EU odel is accoodated satisfactorily. Keywords. PACS Nos Cosology; dark energy; dark atter; large-scale structures k; x; d; Dx 1. Introduction After the discovery of cosic icrowave background radiation [1,2], the Big-Bang odel becae the Standard Model of the Universe which has a beginning at soe finite past. However, Big-Bang odel based on perfect fluid assuptions fails to account for soe of the observed facts of the Universe. Further, it is observed that while probing the early Universe, a nuber of probles naely, the horion proble, flatness proble and singularity proble, cropped up. In order to resolve those issues of the early Universe, the concept of inflation [3 6] was introduced in cosology. These inflationary odels could not solve the initial singularity proble. Another ystery is the observational prediction of accelerating Universe [7 10]. This phase of acceleration which is a late-tie phenoenon of the Universe, can be accoodated in the Standard Model with the introduction of a positive cosological constant. However, the physics of the inflation and introduction of a sall cosological constant for latetie acceleration, is not clearly understood [11,12]. In the literature, the late accelerating phase of the Universe is obtained with exotic atter or with a odification of the Einstein gravity. A non-linear equation of state is also considered in the literature to construct cosologies [13], and eergent Universe (EU) odel is one such odel. The EU odel obtained by Mukherjee et al in the flat Universe perits an accelerating phase. The EU starts fro a static Einstein Universe thereby avoiding the essy situation of the initial singularity [14,15] and afterward successfully accoodates the accelerating phase. In the EU odel, the late-tie de-sitter phase exists which naturally incorporates the late-tie accelerating phase as well. EU scenario has been explored with quinto atter [16] and the realiation of the scenario with a non-conventional ferion field was further investigated to obtain a scale-invariant perturbation [17]. It has been shown that the EU scenario can be ipleented successfully in the fraework of general relativity [13] in addition to Gauss Bonnet gravity [18]. The odified Gauss Bonnet gravity as gravitational alternative for dark energy is however considered by Nojiri and Oddintsov [19]. It is also shown recently that EU odel can be successfully ipleented in brane-world gravity [20,21], Brans Dicke theory [22]. A nuber of cosological odels are obtained with different cosological fluids and fields [14,23 25]

2 27 Page 2 of 9 Praana J. Phys. (2017) 89:27 where initial singularity proble is addressed. Mukherjee et al [13] obtained an EU odel in the fraework of GTR with a polytropic equation of state (EoS) given by p = Aρ Bρ 1/2, (1) where A, B are constants with B > 0. This EU odel is interesting as it avoids the initial singularity and accoodates the late-accelerating phase. The EoS paraeters in the odel play iportant roles which decide the coposition of atter in the Universe. In ref. [13], it is shown that for a discrete set of values of A, naely, A ( = 0, 1 3, 1 3, 1), one obtains Universe with a ixture of three different kinds of cosic fluids. Each of the above EU odel has dark energy as one of its constituents. A study on EU odel has been perfored to deterine the paraeters fro cosological observations. The analysis adopted here consists of four ain background tests. (1) The differential age of old galaxies, given by H(). (2) The peak position of the baryonic acoustic oscillations (BAO). (3) The CMB shift paraeter. (4) The SN Ia data. H() data are given in table 1. The supernovae data are taken fro the union copilation data (Union2.1) [26]. The paper is presented as follows: In 2, relevant field equations obtained fro Einstein field equations are given. In 3, constraints on the EoS paraeters obtained fro background tests are presented. In 4, ageineu odels are calculated. In 5, a suary of the results are noted. Finally, in 6, a brief discussion is given. 2. Field equations The Einstein field equation is given by R μν 1 2 g μν R = 8πGT μν, (2) where R μν represents Ricci tensor, R represents Ricci scalar, T μν represents energy oentu tensor and g μν represents the etric tensor in four diensions. Here, Robertson Walker etric is considered which is given by ds 2 = dt 2 [ dr + a 2 2 ] (t) 1 kr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ), (3) where k = 0, +1( 1) is the curvature paraeter 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 diensionless co-oving coordinates. Using etric (3) in the Einstein field equation (2), the following equations are obtained: (ȧ2 3 a 2 + k ) a 2 = 8πGρ, (4) 2ä a + ȧ2 a 2 + k = 8πGp, (5) a2 where ρ and p represent the energy density and pressure respectively. The conservation equation is given by dρ + 3H(ρ + p) = 0, (6) dt where H = ȧ/a is the Hubble paraeter. Using EoS given by eq. (1) ineq.(6), and integrating once, the expression for energy density is obtained as [ B ρ eu = 1 + A + 1 ] K 2 A + 1 a 3(A+1)/2, (7) where K is a positive integration constant. For convenience, eq. (7) is rewritten as [ ρ eu = ρ eu0 A s + 1 A ] 2 s a 3(A+1)/2, (8) where A s = B 1 + A 1 ρ 1/2 eu0 and K A + 1 = ρ1/2 eu0 B A + 1. The scale factor of the Universe can be expressed as a/a 0 = 1/(1 + ), where is the red-shift paraeter and the present scale factor of the Universe is chosen to be a 0 = 1. Therefore, the Hubble paraeter in ters of red-shift paraeter can be rewritten using the field eq. (4) as [ H() = H 0 As + (1 A s )(1 + ) 3(A+1)/2], (9) where H 0 represents the present Hubble paraeter. Using the present atter density of the Universe = (1 A s ) 2 [27], Hubble paraeter can be expressed as [ H() = H 0 (1 ) + (1 + ) 3(A+1)/2]. (10) The square of the speed of sound is given by cs 2 = δp δρ = ṗ ρ (11)

3 Praana J. Phys. (2017) 89:27 Page 3 of 9 27 which reduces to cs 2 = A (1 )(1 + A) 2(1 + (1 + ) 3(A+1)/2 ). (12) In ters of the state paraeter, it reduces to cs 2 = ω + A. (13) 2 Fro the above equation, the inequality takes the for ( ) A 1 < 2 (14) A + 1 for a realistic solution which adits stable perturbation [28]. Again, positivity of sound speed leads to an upper bound on cs 2 1 which arises fro the causality condition. The deceleration paraeter is given by q(a) = eu(a)[1 + 3ω(a)], (15) 2[ eu (a)] where eu (a) = eu0 [1 ] 2 +. (16) 3. Observational constraints a 3(A+1)/2 The EoS for the EU contains three unknown paraeters, naely H 0, and A, which are deterined fro nuerical analysis. For this, the Einstein field equation is rewritten in ters of a diensionless Hubble paraeter and a suitable chi-square function is defined in different cases. Case I: For OHD The observed Hubble data (OHD) is then taken fro the table given below: To analyse, first chi-square χ 2 H function is defined as χ 2 OHD (H 0,, A) = (H(H 0,, A, ) H obs ()) 2 σ 2, (17) where H obs () is the observed Hubble paraeter at redshift and σ is the error associated with that particular observation as shown in table 1. Case II: For BAO data A odel-independent BAO peak paraeter for low red-shift 1 easureents in a flat Universe is given by [29] A(, A, 1 ) = E( 1 ) 1/3 ( 1 0 d/e() 1 ) 2/3, (18) where is the present atter density paraeter for the Universe. The chi-square function in this case is defined as χbao 2 (, A) = (A(, A, 1 ) A obs ( 1 )) 2 (σ A ) 2. The BAO data are given in table 2. (19) Table 1. Observed Hubble data (OHD). H() σ References H() σ References ±19.68 [31] ±13.0 [34] ±12.0 [32] ±6.1 [38] ±26.2 [31] ±8.0 [34] ±8.0 [33] ±7.0 [38] ±4.0 [34] ±12.0 [34] ±5.0 [34] ±17.0 [34] ±29.6 [31] ±40.0 [39] ±2.65 [35] ±23.0 [33] ±14.0 [33] ±20.0 [34] ±36.6 [31] ±17.0 [33] , 4.9 [36] ±33.6 [41] ±7.0 [37] ±18.0 [33] ± 14.0 [34] ±14.0 [33] ±17.0 [33] ±40.0 [33] ±3.68 [35] ±5 [41] ±7.80 [38] ±8.0 [42] ±62.0 [39] ±7.0 [43] ±4.5 [40] ±8.0 [44]

4 27 Page 4 of 9 Praana J. Phys. (2017) 89:27 Table 2. BAO data. 1 A σ A References [45] [45] [45] [46] [45,47] [46] [46] A Case III: For CMB The CMB shift paraeter (R)isgivenby[30] R = ls d 0 H(, (20) )/H 0 where ls is the at the surface of last scattering. The WMAP7 data predict R = 1.726±0.018 at = Now, chi-square function takes the for H χcmb 2 ( (R 1.726)2, A) = (0.018) 2. (21) Case IV: For supernovae data The distance odulus function (μ)isdefinedinters of luinosity distance (d )as μ(h 0,, A, ) = M = 5log 10 (d ) + 25, (22) where c(1 + ) d d = H 0 0 E( ). (23) In this case, the chi-square χμ 2 function is defined as χμ 2 (H 0,, A)= (μ(h 0,, A, ) μ obs ()) 2, (24) where μ obs () is the observed distance odulus at redshift and σ is the corresponding error for the 580 observed data [26]. Finally, the chi-square function for the background tests is defined as χback 2 (H 0,, A) σ 2 = χ 2 OHD (H 0,, A) + χ 2 BAO (, A) +χcmb 2 (, A) + χμ 2 (H 0,, A). (25) The chi-square function for the background test is utilied for the deterination of EoS paraeters, H 0, A and. 3.1 Observational constraints fro background tests ikelihood function () is related to the chi-square for the background test as exp( χback 2 /2). Togetthe A (iii) H Figure 1. Contours of H 0 A, H 0 and (iii) A fro OU data for the general EU odel at 68.3% (yellow), 95.4% (blue) and 99.7% (green) confidence liits, where OU = OHD + BAO + CMB + Union2.1. best-fit values of the EU odel, likelihood function can be axiied or the chi-square function can be iniied. In this case, the best-fit values are obtained by iniiing the chi-square function. Since chi-square function depends on H 0, A and, it is possible to draw contours at different confidence liits. The contours aong the paraeters H 0, A and for background tests are shown in figure 1. The plot of the axiu likelihood functions with the paraeters H 0, A and for background test are shown in figure 2. The plot of

5 Praana J. Phys. (2017) 89:27 Page 5 of 9 27 ax ax H H 0 ax ax ax Figure 3. Plot of axiu likelihood functions with H 0 and in EU odel with A = 0 fro OU data. (iii) A Figure 2. Plot of axiu likelihood functions with H 0, and (iii) A in the general EU odel fro OU data, where OU = OHD + BAO + CMB + Union2.1. axiu likelihood functions with the paraeters H 0 and for the background test for A = 0areshownin figure Age of the Universe in the EU odel The present age of the Universe is given as 1 [ ] da t 0 =, (26) ah(a) 0 where (a/a 0 ) = 1/(1 + ).IntersofH() it is given in the EU odel as t 0 = 1 inf [ ] d (27) H 0 0 (1 + ) f (, A, ) with f (, A, ) = H() (28) H 0 and H() is given by eq. (10). 5. Results In this analysis, the present Hubble paraeter is also taken as a free paraeter. So here H 0, A and are the three paraeters whose values are deterined at different confidence levels. The best-fit values of the odel obtained with background data are H 0 = 67.90, A = and = 56 (shown in table 3). The peritted ranges at 68.3% confidence level (1σ ) of the odel obtained with background data are H 0 = , A = and =

6 27 Page 6 of 9 Praana J. Phys. (2017) 89:27 Table 3. Best-fit values of the EoS paraeters fro OU data in the EU odel. Data H 0 A χ 2 /d.o.f. OU Table 8. Best-fit values of the EoS paraeters fro OU data for A = 0 odel. Data H 0 χ 2 /d.o.f. OU Table 4. Range of values of H 0 and A fro OU data in the EU odel. Data C (%) H 0 A OU 68.3 (67.57, 68.23) ( 0.033, 0.023) OU 95.4 (67.36, 68.46) ( 0.036, 0.020) OU 99.7 (67.16, 68.66) ( 0.038, 0.017) Table 5. Range of values of H 0 and fro OU data in the general EU odel. Data C (%) H 0 OU 68.3 (67.53, 68.26) (45, 68) OU 95.4 (67.30, 68.50) (38, 76) OU 99.7 (67.09, 68.72) (31, 83) Table 6. Range of values of and A fro OU data in the general EU odel. Data C (%) A OU 68.3 (46, 67) ( 0.033, 0.023) OU 95.4 (40, 74) ( 0.036, 0.012) OU 99.7 (34, 80) ( 0.038, 0.017) Table 7. Best-fit values of EoS paraeters in the general EU odel in OU data. Model ω 0 q EU (tables 4 6). The acceptable ranges in the background test for the paraeters H 0, A and at 99.7% confidence level (3σ )are(67.09, 68.72), ( 0.038, 0.017) and (31, 83) (which are shown in figure 1 and in tables 4 6). The present values of the paraeters ω 0 and q in this EU odel are and 71 respectively (shown in table 7). The best-fit values of the EU odel with A = 0 obtained with background data are H 0 = and = 67 (shown in table 8). The acceptable ranges in the background for the paraeters H 0 and at 99.7% confidence level (3σ )are(66.64, 68.29) and (42, 94) respectively (shown in figure 4 and table 9). The present values of the paraeters ω 0 and q in this EU odel are 85 and 26 respectively H 0 Figure 4. Contours of H 0 in the EU odel with A = 0 fro OU data at 68.3% (yellow), 95.4% (blue) and 99.7% (green) confidence liit. Table 9. Range of values of H 0 and fro OU data in the EU odel with A = 0. Data C (%) H 0 OU 68.3 (67.09, 67.80) (55, 79) OU 95.4 (66.86, 68.06) (48, 86) OU 99.7 (66.64, 68.29) (42, 94) Table 10. Best-fit values of EoS paraeters fro OU data in the EU odel with A = 0. Model ω 0 q EU (shown in table 10). The plot of likelihood functions in both the scenario are plotted in figures 2 and Discussion In this paper, an analysis of the flat EU odel [13] and the EU odel with A = 0 with observational data has been perfored. The following points are noted: The best-fit values and range of values obtained with the background data for the EU odels considered are very close to each other. The best-fit values obtained with background data

7 Praana J. Phys. (2017) 89:27 Page 7 of 9 27 c^2_ s c^2_ s ω ω q q (iii) Figure 5. Evolution of the squared adiabatic sound speed cs 2, state paraeter ω and (iii) deceleration paraeter q with red shift in the general EU odel fro OU data. (iii) Figure 6. Plot of the squared adiabatic sound speed c 2 s, EoS paraeter ω and (iii) deceleration paraeter q with redshift for the EU odel with A = 0 in OU data. are H 0 = 67.90, A = and = 56 in the general EU odel and H 0 = , = 67 in a particular EU odel (A = 0). Here, in the general EU odel, the present value of Hubble paraeter is slightly higher and the present atter density is slightly lower than the odel with A = 0. The present Hubble values predicted by our analysis are close to Planck 15 data [48]. The present observed data perit an EU scenario with A 0andA = 0 which is evident fro the plot of the state paraeter (figures 5 and 6). The plot of deceleration paraeter q (shown in figure 5) shows the presence of dark energy in this odel with A 0. (iii) In figures 5 and 6, the plot of the state paraeter with shows that ω 1 3 at the present

8 27 Page 8 of 9 Praana J. Phys. (2017) 89:27 epoch which confirs an accelerating phase of the Universe. (iv) Figures 5 and 6 show the variation of deceleration paraeter with red-shift. It is evident that in the recent past the Universe transits fro deceleration phase to accelerating phase. (v) In the general EU odel, values of ω and q are ore negative than in the A = 0 odel indicating the possibility of higher acceleration in the forer case. (vi) In the general EU odel, the present age of the Universe t 0 is Gyr and in the EU odel with A = 0, t 0 = Gyr. (vii) Plot of squared adiabatic sound speed cs 2 in the general EU odel (figure 5) and in the specific EU odel (A = 0) (figure 6) showsa tendency towards positivity at higher red-shift thereby hinting at structure foration. This tendency is ore in specific EU odel (A = 0). The analysis perits an EU odel with A 0, accoodating dust and dark energy as its constituents. The EU odel with A = 0 is equally valid and in soe aspects (t 0, cs 2 ) it gives better results than general EU odel. So, an EU odel is viable in accordance with the background tests conducted. The CDM odel works rearkably well at large scale but it has soe probles in describing structures (cusp/core (CC) proble, issing satellite proble (MSP)) at sall scales. The probles at sall scales coe fro the vanishing pressure of dark atter (DM). EU odel in standard gravity is one of the odels that unify DM and dark energy and its EoS is different fro CDM. This EU starts fro the static phase in the past and hence avoids the probles at sall scales. In this study, it is found that the deterined values of EoS paraeters go with the observations. So, EU odel is worthy of being treated as an alternative odel of CDM odel. Acknowledgeents The author would like to thank IUCAA Reference Centre at North Bengal University for extending necessary research facilities to initiate the work. References [1] A A Penias and R W Wilson, Astrophys. J. ett. 142, 419 (1965) [2] R H Dicke, P J E Peebles, P J Roll and D T Wilkinson, Astrophys. J. ett. 142, 414 (1965) [3] A H Guth, Phys. Rev. D 23, 347 (1981) [4] K Sato, Mon. Not. R. Aston. Soc. 195, 467 (1981) [5] A inde, Phys. ett. B 108, 389 (1982) [6] A Albrecht and P Steinhardt, Phys. Rev. ett. 48, 1220 (1982) [7] AGRiesset al, Astron. J. 116, 1009 (1998) [8] J Tonry et al, Astrophys. J. 594, 1 (2003) [9] S Perlutter et al, Nature 391, 51 (1998) [10] S Perlutter et al, Astrophys. J. 517, 565 (1999) [11] A Albrecht, arxiv:astro-ph/ (2000) [12] S M Carroll, iving Rev. Rel. 4, 1 (2001) [13] S Mukherjee, B C Paul, N K Dadhich, S D Maharaj and A Beesha, Class. Quantu Grav. 23, 6927 (2006) [14] G F R Ellis and R Maartens, Class. Quantu Grav. 21, 223 (2004) [15] E R Harrison, Mon. Not. R. Astron. Soc. 69, 137 (1967) [16] Y F Cai, M i and X Zhang, Phys. ett. B 718, 248 (2012) [17] Y F Cai, Y Wan and X Zhang, Phys. ett. B 731, 217 (2014) [18] B C Paul and S Ghose, Gen. Relativ. Gravit. 42, 795 (2010) [19] S Nojiri and S D Oddintsov, Phys. ett. B 631, 1 (2005) [20] A Banerjee, T Bandyopadhyay and S Chakraborty, Gen. Relativ. Gravit. 40, 1603 (2008) [21] U Debnath, Class. Quantu Grav. 25, (2008) [22] S del Capo, R Herrera and P abrana, J. Cosol. Astropart. Phys. 30, 0711 (2007) [23] S Nojiri and S D Oddintsov, Phys. Rev. D 71, (2005) [24] A V Astashenok, S Nojiri, S D Oddintsov and A V Yurov, Phys. ett. B 709, 396 (2012) [25] S Bag, V Sahni, Y Shtanov and S Unnikrishnan, arxiv: (2014) [26] N Suuki et al, Astrophys. J. 746, 85 (2012) [27] Z i, W Puxun and Y Hongwei, Astrophys. J. 744, 176 (2012) [28] X ixin, W Yuting and N Hyeri, Eur. Phys. J. C 72, 1931 (2012) [29] D J Eisenstein et al, Astrophys. J. 633, 560 (2005) [30] E Koatsu et al, Astrophys. J. Suppl. 192, 18 (2011) [31] C Zhang et al, Res. Astron. Astrophys. 14, 1221 (2014) [32] R Jiene et al, Astrophys. J. 593, 622 (2003) [33] J Sion et al, Phys. Rev. D 71, (2005) [34] M Moresco et al, J. Cosol. Astropart. Phys. 8, 6 (2012) [35] E Gatanaga et al, Mon. Not. R. Astron. Soc. 399, 1663 (2009) [36] C-H Chuang et al, Mon. Not. R. Astron. Soc. 426, 226 (2012) [37] X Xu et al, Mon. Not. R. Astron. Soc. 431, 2834 (2013) [38] C Blake et al, Mon. Not. R. Astron. Soc. 425, 405 (2012) [39] D Stern et al, J. Cosol. Astropart. Phys. 1002, 008 (2010) [40] Saushia et al, Phys. Rev. D 86, (2012) [41] M Moresco, Mon. Not. R. Astron. Soc. 450, 16 (2015)

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