Will there be future deceleration during the dark energy evolution of the universe?
In this work, we have proposed a general dark energy density parametrization to examine whether the dark energy evolves with time or not. We have also constrained the model parameters using the combination of Type Ia supernova (SNIa), baryonic acoustic oscillations (BAO), cosmic microwave background radiation (CMB) and observational datasets. For the dataset, we have used the direct observations of the Hubble rate, from the radial BAO size and the cosmic chronometer methods. Our result indicates that the SNIa++BAO/CMB dataset does not favour the CDM model at more than confidence level. Furthermore, we have also measured the percentage deviation in the evolution of the normalized Hubble parameter for the present model compared to a CDM model, and the corresponding deviation is found to be at low redshifts (). Finally, we have also investigated whether the deceleration parameter may have more than one transition during the evolution of the universe. The present model shows a transient accelerating phase, in which the universe was decelerated in the past and is presently accelerating, but will return to a decelerating phase in the near future. This result is in great contrast to the CDM scenario, which predicts that the cosmic acceleration must remain forever.
Keywords: Dark energy density, Future deceleration
Various independent cosmic observations SN (); Riess:1998cb (); LSS (); Seljak:2004xh (); Planck:2015xua (); planckr (); Ade:2014xna (); Komatsu:2010fb (); Hinshaw:2012aka (); dvbc () have strongly confirmed that the present universe experiences an accelerated expansion. The exotic matter content responsible for such a certain phase of evolution of the universe is popularly referred to as “dark energy”. Many dark energy models were proposed in the literatures, for a recent review, one can look into Refs. 3 (); 4 (); 4a (). In the context of dark energy, the Einstein cosmological constant is the simplest way to explain the observed expansion measurements. The so-called concordance CDM () model is the model that best agrees with cosmological data Planck:2015xua (). Despite a very good agreement with data, the CDM model can not escape from the cosmological coincidence and the fine tuning problems sw (); Steinhardt () and is still a challenging problem in cosmology.
Going beyond the cosmological constant where the dark energy density is constant throughout
the evolution of the universe, there are several approaches to model the dark energy evolution 4 (). One best way is to construct parametrizations of the dark energy equation of state parameter cpl1 (); cpl2 (); lin1 (); lin2 () or the dark energy density dedywang (); dedmaor (); dedwang (); dedaam () as a function of scale factor or redshift, and then confront such parametrizations to the cosmological data. However, such models are more consistent with the present observational constraints for some restrictions on model parameters and search is still on for finding a suitable cosmologically viable model of dark energy. Recently, Zhao et al. zhao () reported that the dynamical dark energy is preferred over the cosmological constant model from recent observations at the confidence level, although the Bayesian evidence for the dynamical dark energy is insufficient to favour it over constant dark energy. This clearly motivates theoreticians to put further constraint on dark energy behaviour. Motivated by the above facts, in the present work, we have studied the case whether the dark energy is dynamical or not. We have considered a spatially flat FRW universe where the dark energy and the cold dark matter evolve independently. Specifically, we have considered a general dark energy density parametrization which varies with the cosmic evolution. The nature of this parametrization is characterized by dimensionless real parameters and . For different choices of and , one can recovers other popular dark energy density parametrizations (see section II). In this paper, we have used the recent cosmic chronometers dataset along with the estimation of the local Hubble parameter value as well as the standard dark energy probes, namely the SNIa, BAO and CMB measurements to study the different properties of this model extensively. Under this scenario, we also made an attempt to explain not only the present accelerated expansion phase but also the past decelerated phase of the universe and further made a prediction about the future evolution of the universe. Our analysis shows the evolution of the universe from an early decelerated to the late-time accelerated phase and it also predicts future decelerating phase. In addition, the present study also indicates that the CDM model is not compatible at confidence level for the SNIa++BAO/CMB dataset.
The paper is organized as follows. In the next section, we have discussed the present cosmological model. In Section III, we have described the observational dataset and analysis methodology, while in section IV we have presented the results of this analysis. Finally, the summary of the work is presented in section V.
Ii Cosmological Model
In this section, we have provided the basic equations of a general cosmological scenario. Throughout the work, we have considered the spatially flat FRW space-time of the form
where, is the scale factor of the universe, which is set to at the present epoch for simplicity and is the cosmic time. For a spatially flat FRW universe, the Einstein field equations can be written as,
where is the Hubble function and an dot implies differentiation with respect to the cosmic time . In the above equation, represents the energy density of the dust matter while and represent the energy density and pressure of the dark energy component respectively. It is noteworthy that we have chosen the natural units () throughout this paper.
One can now write the conservation equation of the dark energy sector and the one of the matter sector as
Solving the above equation, we have found the evolution of as
where denotes the present matter energy density and is the redshift parameter.
In the present work, we have proposed a simple dark energy density parametrization that exhibits dynamical behaviour with the evolution of the universe. Our primary goal is to investigate this model with current cosmological data. To examine the nature of dark energy, we have proposed the following functional form for the evolution of given by
where , and denote the present dark energy density and free parameters of the model. One important advantage of this choice (as given in equation (7)) is that it reduces to the flat CDM ( constant) model for . So, the free parameter is a good indicator of deviation of the present dark energy model from the CDM model. Note that the above form of is similar to the parametrization of for dedwang (). Hence, the new parametrization of covers a wide range of other dark energy models and also shows a bounded behaviour in the redshift range, . However, for our analysis in section III, we have considered .
which is equivalent to the CDM model for . In the above equation, , , and denote the present values of , and respectively. In particular, we have considered that the universe consists of dark matter and dark energy, and therefore the total density parameter of the universe is
For this choice of , the deceleration parameter evolves as
For the present model, the dark energy equation of state (EoS) parameter becomes
which is independent of the present matter density parameter . Another interesting point regarding the above expression of is that for , this behaves exactly like the standard CDM () model. So, the estimated value of will indicate whether a cosmological constant or a time evolving dark energy is preferred by cosmological observations.
In the next section, we shall try to extract the values of the model parameters using the latest cosmological dataset.
Iii Observational data and fitting method
In this section, we have explained the datasets and their analysis method employed to constrain the proposed theoretical model. We have used datasets from the following probes:
data: We have used observational dataset consisting 41 data points, to probe the nature of dark energy. Among them, 36 data points (10 data points are deduced from the radial BAO size method and 26 data points are obtained from the galaxy differential age method) are compiled by Meng et al. hzdataMeng () and 5 new data points of H(z) are obtained from the differential age method by Moresco et al. hzdataMore (). For the dataset, the is defined as
where, is any model parameter, the superscript “th” refers to theoretical quantities and superscript “obs” is for the corresponding observational ones. Also, the uncertainty for normalized is given by sigmah ()
where and are the uncertainties in and respectively. In addition, the present value of is taken from Ref. H0 ().
SNIa data: Next, we have incorporated the Union2.1 compilation u2.1 () dataset of total 580 data points with redshift ranging from 0.015 to 1.414. This observations directly measure the distance modulus of a supernova and its redshift. The relevant for the SNIa dataset is defined as chisn ()
where , and are given by
BAO/CMB data: We have also used BAO and CMB measurements data to obtain the BAO/CMB constraints on the model parameters. For BAO data, the results from the 6dFGS Survey measurement at beutler11 (), the WiggleZ Dark Energy Survey measurement at and blake11 (), the SDSS DR7 Survey measurement at padmanabhan12 () and BOSS CMASS Survey measurement at anderson14 () have been used. In addition, we have also used the CMB data derived from the Planck 2015 observations Planck:2015xua () for the combined analysis TT, TE, EE+lowP+lensing. In this case, the function is defined as
where, and are the transformation matrix and the inverse covariance matrix, respectively chibc1 (). For this dataset, the details of the methodology for obtaining the constraints on model
parameters are described in Refs. chibc1 (); chibc2 ().
One can now use the maximum likelihood method and take the total likelihood function as
where, . The best-fit corresponds to the model parameters for which the (likelihood function) is minimized (maximized). In this analysis, for simplicity, we have fixed the free parameter and have minimized the function (say, ) with respect to the model parameters to obtain their best fit values.
In this section, we have discussed the results obtained from the analysis method using the SNIa, , BAO and CMB datasets. The and confidence level contours in plane is shown in figure 1 for . The corresponding best-fit values for the model parameters are obtained as , with (for data) and , with (for SNIa++BAO/CMB data). It should be noted that the best-fit value of obtained in this work is slightly lower than the value obtained by the Planck observation planckr ().
As discussed earlier, the model parameters is a good indicator of deviation of our model from cosmological constant as for , the model behaves like the CDM model. It is observed from figure 1 that the CDM model is ruled out at more than confidence level by the combined (SNIa++BAO/CMB) dataset, but it is still in agreement with the dataset at the confidence level. Figure 2 shows the evolution of the deceleration parameter for the best-fit values of and arising from the analysis of the (black curve) and SNIa++BAO/CMB (red curve) datasets. It is seen from figure 2 that the universe was decelerated () in the past, began to accelerate at (for data) and (for SNIa++BAO/CMB data), is presently accelerated () but will return to a decelerating phase in the near future. This results are consistent with the results obtained by several authors from different cosmological scenarios fudp1 (); fudp2 (); fudp3 (); fudp4 ().
In the upper panel of figure 3, we have shown the best-fit evolution of the dark energy EoS parameter as a function of z for different datasets. It has been found that for each dataset, resembles a CDM () model at the present epoch (i.e., ), but, finally, will become positive in the near future. This result is also consistent with the recent works as given in Refs. arman (); liw (); magana (), where authors have shown that the cosmic acceleration is currently witnessing its slowing down by using a distinct method. For the sake of completeness, in the lower panel of figure 3, we have plotted the percentage deviation for the above model as compared to a CDM model, and the corresponding deviation is observed to be at low redshifts ().
Therefore, the overall dynamic behaviour of our model supports the claims of Valentino et al. nlcdm1 (), Sahni et al. nlcdm2 () and Ding et al. nlcdm3 () that the CDM model may not be the best description of our universe and also seems to be in agreement with the requirements of String theory st1 (); st2 (); st3 ().
In this work, we have studied the dynamics of accelerating scenario of the universe by considering one specific parameterization of the dark energy density and from this we have obtained analytical solutions for different cosmological parameters. As we have mentioned before, the new parametrization of , given by equation (7), covers a wide range of other popular dark energy models for different choices of and . We have used the recent Hubble parameter dataset along with the estimation of the local Hubble parameter value as well as the standard dark energy probes, such as the SNIa, BAO and CMB measurements to constrain different parameters of our model.
In summary, our analysis predicts a transient accelerating phase, in which the universe was decelerated () in the past, began to accelerate at redshift , is presently accelerated (), but will return to a decelerating phase in the near future. This overall dynamic behaviour is much different from the standard CDM scenario. Hence, the present model supports the claims of several authors nlcdm1 (); nlcdm2 (); nlcdm3 () that the CDM model may not be the best description of our universe. Therefore, this specific dark energy model, with a transient accelerating phase and , can be considered as an alternative for the CDM model and is in agreement with the observational datasets analyzed here.
The author acknowledges the financial support from SERB, Government of India through National Post-Doctoral Fellowship Scheme (File No: PDF/2017/000308).
- (1) S. Perlmutter et al., Astrophys. J., 517, 565 (1999).
- (2) A. G. Riess et al., Astron. J., 116, 1009 (1998).
- (3) M. Tegmark et al., Phys. Rev. D 69, 103501 (2004).
- (4) U. Seljak et al., Phys. Rev. D, 71, 103515 (2005).
- (5) P. A. R. Ade et al., AA, 594, A13 (2016).
- (6) Ade P A R et al., Astron. Astrophys., 571, A16 (2014).
- (7) P. A. R. Ade et al., Phys. Rev. Lett., 112, 241101 (2014).
- (8) E. Komatsu et al., Astrophys. J. Suppl., 192, 18 (2011).
- (9) G. Hinshaw et al., Astrophys. J. Suppl., 208, 19 (2013).
- (10) D. J. Eisenstein et al., Astrophys. J., 633, 560 (2005).
- (11) V. Sahni, A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000).
- (12) E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006).
- (13) P. J. E. Peebles, B. Ratra, Rev. Mod. Phys., 75, 559 (2003).
- (14) S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
- (15) P. J. Steinhardt et al., Phys. Rev. Lett., 59, 123504 (1999).
- (16) M. Chevallier, D. Polarski, Int. J. Mod. Phys. D 10, 213 (2001).
- (17) E.V. Linder, Phys. Rev. Lett. 90, 091301 (2003).
- (18) D. Huterer, M.S. Turner, Phys. Rev. D 60, 081301 (1999).
- (19) J. Weller, A. Albrecht, Phys. Rev. Lett. 86, 1939 (2001).
- (20) Y. Wang, P.M. Garnavich, Astrophys. J. 552, 445 (2001).
- (21) I. Maor et al., Phys. Rev. D, 65, 123003 (2002).
- (22) D. Wang, X. -H. Meng, Phys. Rev. D 96, 103516 (2017).
- (23) A. A. Mamon, K. Bamba, S. Das, Eur. Phys. J. C, 77, 29 (2017).
- (24) G. B. Zhao et al., Nature Astronomy, 1, 627 (2017).
- (25) X.-L. Meng et al., arXiv:1507.02517 (2015).
- (26) M. Moresco et al., JCAP, 05, 014 (2016).
- (27) M. Seikel, S. Yahya, R. Maartens, C. Clarkson, Phys. Rev. D 86, 8083001 (2012).
- (28) A. G. Riess et al., ApJ, 826, 56 (2016).
- (29) N. Suzuki et al., Astrophy. J., 746, 85 (2012).
- (30) S. Nesseris, L. Perivolaropoulos, Phys. Rev. D, 72, 123519 (2005).
- (31) F. Beutler et al., Mon. Not. R. Astron. Soc., 416, 3017 (2011).
- (32) C. Blake et al., Mon. Not. R. Astron. Soc., 418, 1707 (2011).
- (33) N. Padmanabhan et al., Mon. Not. Roy. Astron. Soc., 427, 2132 (2012).
- (34) L. Anderson et al., Mon. Not. Roy. Astron. Soc. 441, 24 (2014).
- (35) M. V. dos Santos et al., JCAP, 02, 066 (2016).
- (36) A. A. Mamon, Int. J. Mod. Phys. D, 26, 1750136 (2017).
- (37) F. C. Carvalho et al. Phys. Rev. Lett. 97, 081301 (2006).
- (38) A. C. C. Guimaraes, J. A. S. Lima, Class. Quant. Grav. 28, 125026 (2011).
- (39) S. Pan, S. Chakraborty, Eur. Phys. J. C, 73, 2575 (2013).
- (40) S. Chakraborty, S. Pan, S. Saha, Physics Letters B 738, 424 (2014).
- (41) A. Shafieloo, V. Sahni, A. A. Starobinsky, Phys. Rev. D 80, 101301 (2009).
- (42) Z. Li, P. Wu, H. Yu, JCAP, 11, 31 (2010).
- (43) J. Magana et al., JCAP, 017, 1410 (2014).
- (44) E. D. Valentino et al., Phys. Rev. D 96, 023523 (2017).
- (45) V. Sahni, A. Shafieloo, A. A. Starobinsky, Astrophys.J. 793, L40 (2014).
- (46) X. Ding et al., ApJ 803, L22 (2015).
- (47) S. Hellerman et al., JHEP 3, 0106, (2001).
- (48) J.M. Cline, JHEP 0108, 35 (2001).
- (49) W. Fischler et al., JHEP 3, 0107 (2001).