Dark energy and dark matter unification from dynamical space time:
observational constraints and cosmological implications
A recently proposed Dynamical Space-time Cosmology (DSC) that unifies dark energy and dark matter is studied. The general action of this scenario includes a Lagrange multiplier, which is coupled to the energy momentum tensor and a scalar field which is different than quintessence. First for various types of potentials we implement a critical point analysis and we find solutions which lead to cosmic acceleration and under certain conditions to stable late-time attractors. Then the DSC cosmology is confronted with the latest cosmological data from low-redshift probes, namely measurements of the Hubble parameter and standard candles (Pantheon SnIa, Quasi-stellar objects).
Performing an overall likelihood analysis and using the appropriate information criteria we find that the explored DSC models are in very good agreement with the data. We also find that one of the DSC models shows a small but non-zero deviation from cosmology, nevertheless the confidence level is close to .
Almost twenty years after the observational evidence of cosmic acceleration the cause of this phenomenon, labeled as "dark energy" (hereafter DE), remains an open question which challenges the foundations of theoretical physics. The simplest model of DE is the so called CDM model that contains non-relativistic matter and cosmological constant. Although, this model fits accurately the present cosmological data it suffers from two fundamental problems, namely the tiny value of the cosmological constant and the coincidence problem, Bull:2015stt (). Furthermore, there is criticism on the conceptual foundations of the current view of the cosmos, in a sense that there are too many ad hoc features (e. g dark Energy, dark matter) needed for "explaining the phenomena", e. g Merritt:2017xeh (). Therefore, it is imperative to develop alternative cosmological models which can provide a deeper insight regarding the nature of the dark sector.
Unifications between dark energy and dark matter from an action principle were obtained from scalar fields Scherrer:2004au ()-Fadragas:2013ina (), by a complex scalar field Arbey:2006it () or other models Chen:2008ft ()-Leon:2013qh () including Galileon cosmology Leon:2012mt () or Telleparallel gravity Kofinas:2014aka ()-Skugoreva:2014ena (). Beyond those approaches, a unification of Dark Energy and Dark Matter using a new measure of integration (the so-called Two Measure Theories) has been formulated Guendelman:2015jii ()-Guendelman:2016kwj (). A diffusive interacting of dark energy and dark matter models was introduced in Koutsoumbas:2017fxp ()-Haba:2016swv () and it has been found that diffusive interacting dark energy - dark matter models can be formulated in the context of an action principle based on a generalization of those Two Measures Theories in the context of quintessential scalar fields Benisty:2017eqh ()-Benisty:2017rbw (), although these models are not equivalent to the previous diffusive interacting dark energy - dark matter models. In order to overcome the coincidence problem, Gao, Kunz, Liddle and Parkinson Gao:2009me () suggested a unification of dark energy and dark matter resulting from a single scalar field. Unlike usual quintessence model, here the scalar field behaves either as dark matter or dark energy. Within this framework the unified picture of dark sector introduces a number of modifications in the equations of motion of the aforementioned scalar field. Recently, a Lagrangian formulation was introduced in Benisty:2018qed () (see also Guendelman:2009ck ()) toward building the so called Dynamical Space-time Cosmological (hereafter DSC) model. In this scenario the gravitational field is described not only by the metric tensor but also by a Lagrange multiplier that is coupled to the energy momentum tensor, a scalar field potential and another potential that describes the interactions between DE and DM. The scalar field plays an important role in the description of the dynamics, since the kinetic term of behaves as DM and the potential is responsible for DE. Therefore, the DSC model provides an elegant alternative in describing the DM and DE dominated eras respectively.
In the current paper we attempt to continue our previous work of Benisty:2018qed () in the sense that we study both dynamically and observationally the DSC scenario for a large family of potentials. Specifically, the manuscript is organized as follows. In Sec. II we briefly present the theoretical framework of the Dynamical Space-time Cosmological model and provide the basic cosmological equations. In section III we use a dynamical analysis by studying the critical points of the field equations in the dimensionless variables for a large family of potentials. In Sec. IV we provide the likelihood analysis and the observational data sets that we utilize in order to constraint the free parameters of the DSC model and we compare it with the CDM model. Finally, in Sec. V we draw our conclusions.
Ii Dynamical Space-time Cosmology
In previous publications two of us Benisty and Guendelman Benisty:2018qed () proposed the Dynamical Space-time Cosmological model (DSC) via a space time vector field, and demonstrated the behavior of this scenario toward unifying the dark sector. In this section we briefly present the main features of the DSC model based on first principles. The action that describes the gravitational field equations and unifies the dark sector was first introduced by Benisty and Guendelman Benisty:2018qed ():
where is the scalar field and is the Ricci scalar. Notice that we have set . The vector field is the so called dynamical space time vector, hence the corresponding covariance derivative is , where are the Christoffel coefficients. In this context, denotes the stress energy tensor which was first introduced by Gao and colleagues Gao:2009me ()
Obviously the action integral contains two different potentials, namely which is coupled to the stress energy momentum tensor , and which is minimally coupled into the action. Moreover, the action depends on three different quantities: the scalar field the dynamical space time vector and the metric . For simplicity we consider a homogeneous scalar field , while the dynamical vector is given by the following formula , where is also just a function of time.
Varying the action with respect to the dynamical space time vector field we obtain the modified ”Klein-Gordon” equation
where the prime denotes derivative with respect to . The second variation with respect to the scalar field gives a non-conserved current:
and the derivatives of the potentials are the source of the current. For constant potentials the source becomes zero, and we get a covariant conservation of this current. For homogeneous and Eq.(5) becomes
hence for FRWL metric we obatin
and the source term yields:
Therefore, the equation of motion takes the form:
Lastly, varying the action integral with respect to the metric, we derive the gravitational field equations
Companing the stress energy tensor with equations (10), we provide the functional forms of the energy density and pressure respectively. These are:
Unlike usual DE models, quintessence and the like, here the vector field and the potential modify the density and the pressure of the cosmic fluid. In the case of and constant equations (11) (12) reduce to those of the quintessence model. For more details regarding the aforementioned analysis we refer the reader the work of Benisty:2018qed ().
In order to proceed with the analysis we need to know the forms of and . Below, we consider special forms of the potentials and study the performance of the models at the expansion level.
ii.0.1 Coupled Constant potential into the Lagrange multiplier
Similar to Benisty:2018qed () we consider DSC models for which the potential that is coupled to the stress energy momentum tensor is constant
The general study of varying will appear in a forthcoming paper. Substituting the potential into Eq.(3), using the definition of and performing the integration we find
where is the integration constant which can be viewed as the effective dark matter density parameter. Introducing the new variable
In this context, the energy density and the pressure of the scalar field are given by
Furthermore, if we assume then the solution of Eq. (16) is
hence with the aid of (14) we obtain
where is a constant of integration and we have set . In such a case it is trivial to show that the Hubble parameter is given by
where is the Hubble constant. It is easy to observe that the above Hubble rate will diverge at high redshifts. Î¤hus, this model can not describe the whole cosmic history. Instead could be thought as an approximation a of general U,V potential that is very slowly varying near today. For a barotropic cosmic fluid whose the corresponding equation of state parameter is given by one can easily recognize three "dark fluids", namely cosmological constant [, ], dark matter (), and another fluid with .
Of course for the current solution boils down to CDM model as it should. Lastly, it is interesting to mention that for we can get bouncing solutions, as discussed in Benisty:2018qed ().
ii.0.2 Dynamical DM-DE
|whith the Hubble parameter|
Suppose that we know the functional form of the potential . First we evaluate Eq.(21c) at which means that . Second, the fact that prior to the present time together with the cosmic sum imply , hence the form of obeys , where at . Concerning the types of potentials involved in the present analysis, we consider the following three cases: exponential with , cosine with and thawing potential with , Clemson:2008ua (). This family of models has CDM as an asymptotic solution. Notice that the initial condition for is chosen appropriately to be compliant with the aforementioned constrain, that is . Once steps (i) and (ii) are accomplished, we numerically solve the system (21a), (21b).
Iii Dynamical system method
In this section we provide a dynamical analysis by studying the fixed points of the field equations, so that we can investigate the various phases of the current cosmological models. Specifically, for a general potential we introduce the new dimensionless variables
In the new system of variables the field equations form an autonomous system which is given by
These are the basic variables that we use for mapping the dynamical system. In this case the equation of motion with respect to the metric is written as:
Notice that for , which means , the phase plane of the system takes the form of a complete circle, where the points and correspond to dark matter and dark energy dominated eras respectively.
Bellow we provide the results of the dynamical analysis for different types of potentials. The corresponding critical points of the system (23a), (23b) and (23c) are listed in Tables I, II and III. In all cases point A with coordinates is ruled out from the constrain (25).
iii.1 Exponential potential- ()
We continue our work by using the exponential potential. In this case the new variable (see the last term in Eq.22) becomes constant. The dynamical system includes four critical points, among which one point is stable. Point B with coordinates corresponds to the matter epoch and it is stable when . Point C with coordinates describes the dark energy dominated era, while point D with coordinates is unstable.
|B||stable for||Dark Matter|
|C||asymptotically stable||Dark Energy|
|D ()||unstable saddle p.||unified DE-DM|
iii.2 Cosine potential- ()
Now we proceed with the cosine potential . Inserting this formula into we find
Therefore, for the dynamical analysis we utilize the aforementioned equation together with Eqs. (23a)-(23b). In this case we find three critical points which are not affected by . As expected, points and describes the dark matter and dark energy dominates eras respectively. Here is always unstable, while is asymptotically stable.
|C||asymptotically stable||Dark Energy|
iii.3 Thawing potential-()
Using the thawing potential (23c) becomes:
In this case point is unstable when and it is saddle for . The dark energy point is always stable. Lastly, point D with coordinates is stable when and it is saddle when .
|B||all||unstable, saddle point.||Dark Matter|
|D||stable focus, saddle||unified DE-DM|
Iv Observational Constraints
In the following we describe the observational data sets along with the relevant statistics in constraining the DSC models presented in Sect. III.
iv.0.1 Direct measurements of the Hubble expansion
We use the latest data set compilation, that can be found in YuRatra2018 (). This set contains measurements of the Hubble expansion in the following redshift range . Out of these, there are 5 measurements based on Baryonic Acoustic Oscillations (BAOs), while for the rest, the Hubble parameter is measured via the differential age of passive evolving galaxies.
Here, the corresponding function reads
where and are the observed Hubble rates at redshift (). Notice, that the statistical vector contains the parameters that we want to fit. The matrix denotes the covariance matrix. Further considerations regarding the statistical analysis and the corresponding covariances can be found in Ref. AA () and references therein.
iv.0.2 Standard Candles
The second data-set that we include in our analysis is the binned Pantheon sample of Scolnic et al. Scolnic:2017caz () and the binned sample of Quasi-Stellar Objects (QSOs), RisalitiLusso:2015 (). We would like to note the importance of using the Pantheon SnIa data along with those of QSOs. The latter allows to trace the cosmic history to the redshift range . It is important to utilize alternative probes at higher redshifts in order to verify the SnIa results and test any possible evolution of the DE equation of state Plionis2011 (). Following standard lines, the chi-square function of the standard candles is given by
where and the subscript s denotes SnIa and QSOs. For the SnIa data the covariance matrix is not diagonal and the distance modulus is given as , where is the apparent magnitude at maximum in the rest frame for redshift and is treated as a universal free parameter, Scolnic:2017caz (), quantifying various observational uncertainties. It is apparent that and parameters are intrinsically degenerate in the context of the Pantheon data set, so we can not extract any information regarding from SnIa data alone. In the case of QSOs, is the observed distant modulus at redshift and the covariance matrix is diagonal.In all cases, the theoretical form of the distance modulus reads
is the luminosity distance, for spatially flat FRWL cosmology.
iv.0.3 Joint analysis and model selection
In order to perform a joint statistical analysis of cosmological probes (in our case ), we need to use the total likelihood function
Consequently the expression is given by
where the statistical vector has dimension , which is the sum of the parameters of the model at hand plus the number of hyper-parameters of the data sets used, that is .The distinction between the hyper-parameters quantifying uncertainties in a data set and the free parameters of the cosmological model is purely conceptual. Regarding the problem of likelihood maximization, we use an affine-invariant Markov Chain Monte Carlo sampler, as described in Ref. AffineInvMCMC (). We utilize the open-source python package emcee, emcee (), using 1000 "walkers" and 1500 "states". The latter setup corresponds to calls of the total likelihood function. In each call, we need to numerically solve the system of Eqs. (21) for the redshift range [0.0,5.93] and also calculate the luminocity distance. This procedure became practical by optimizing critical parts of the calculations using C++ code from Ref. Mpampis (). The convergence of the MCMC algorithm is checked with autocorrelation time considerations.
iv.1 Statistical Results
In order to test the performance of the cosmological models in fitting the data it is imperative to utilize the Akaike Information Criterion (AIC), Akaike1974 (), and Bayesian Information Criterion (BIC), Schwarz1978 (). The AIC criterion is an asymptotically unbiased estimator of the Kullback-Leibler information, measuring the loss of information during the fit. Within the standard assumption of Gaussian errors, the AIC estimator is given by Ann2002 ()
where is the maximum likelihood of the data set(s) under consideration and is the total number of data. It is easy to observe that for large , this expression reduces to , that is the standard form of the AIC criterion. Following the previous point, it is considered good practice to use the modified AIC criterion in all cases, Liddle:2007fy ().
The AIC and BIC criteria employ only the likelihood value at maximum. In principle, due to to the Bayesian nature of our analysis, the accuracy of the is reduced, meaning that the AIC and BIC values are meant to be used just for illustrative purposes. In practice, however, by using long chains, we obtain values with enough accuracy to use them in order to calculate AIC and BIC. Furthermore, we also compute the Deviance Information Criterion (DIC), that provides all the information obtained from the likelihood calls during the maximization procedure. The DIC estimator is defined as, (see Liddle:2007fy (), Spiegelhalter2002 ())
where is the so called Bayesian complexity that measures the power of data to constrain the parameter space compared to the predictivity of the model which is provided by the prior. In particular, , where the overline denotes the usual mean value. Also, is the Bayesian Deviation, where in our case it boils down to .
To proceed with the model selection we need to assign a "probability" to each model following the classical treatment of Jeffreys, Jeffreys (), that is by using the relative difference of the IC value for a number of models, , where the is the minimum IC value in the set of competing models. Following the notations of Ref. KassRaftery1995 (), , means that the model under consideration is statistically compatible with the “best” model, while the condition indicates a middle tension between the two models and the condition suggests a strong tension.
For the model with constant potentials, we find with . Interestingly, the value which corresponds to the limit is outside the area.
Regarding the exponential potential (), we find with . Furthermore, the cosmological parameters for the cosine potential () are and the relevant . Lastly, for the potential we obtain and .
In most of the cases, the best fir values of the matter energy density are in good agreement with those of Planck 2018, Aghanim:2018eyx (). Considering the result for the flat , we observe compatibility for the potentials, while the result for the cosmological model with constant potentials is within limits. It is important to note that for and potentials we set . However, we have tested that the likelihood analysis provides very similar results for .
We deem interesting to discuss our results with respect to the Hubble constant problem, that is a discrepancy between the Hubble constant measured by Riess et al. , RiessHo (), () and the relevant value from Planck collaboration, ( Km/s/Mpc), Aghanim:2018eyx (), see Fig. 2. Our results are in agreement (within ) with those provided by the team of Planck, while there is compatibility at level with Riess et al. results. As a consistency check we compare our results with the result from the model-independent assessment of the cosmic history obtained by Haridasu et al, Haridasu2018 (), namely and we report compatibility for all considered potentials.
Concerning AIC, BIC and DIC and we present the relevant values at the Table 5. In the context of BIC, all models considered are in mild to strong tension with CDM. As we used binned datasets, we do not anticipate that an information criterion with an explicit dependence from the dataset length could estimate reliably the relative quality of the fits. Further, the BIC criterion is just an asymptotic approximation that is valid while the dataset length tends to infinity.
On the other hand, AIC criterion provides a somewhat different view. The model with constant potentials has , hence it is nearly indistinguishable from CDM. The other models () are in mild tension with CDM since they have . In the context of our Bayesian treatment, both AIC and BIC values could only serve indicative purposes, as they employ only the value of the likelihood at maximum and not the full set of likelihood values obtained during the sampling procedure. The most interesting observation comes from the DIC criterion, which seems to prefer the cosmological model with constant potentials over the concordance model, as and the relevant (>2) indicates that the difference is rather significant. However, as we mentioned before, the constant potentials model is an approximation of a more general cosmological model, valid for late universe only. With respect to the other models under consideration, we observe mild - to - strong tension with each of them, with the to be in the second place. A general ascertainment regarding the somewhat similar results of the physically different potentials in the free parameters (e.g matter energy density and Hubble constant) is that is very small at late universe, so any is effectively . This is what someone could naively foresee as the field changes very smoothly across the cosmic history. We expect that a study of the model using the CMB spectrum could discriminate between the different DSC models.
We explored a large family of cosmological models in the context of Dynamical Space-time Cosmology (DSC). This scenario unifies naturally the dark sector and it provides an elegant theoretical platform toward describing the various phases of the cosmic expansion. Initially, we performed a standard dynamical analysis and we found that under certain circumstances DSC model includes stable late-time attractors. Then we tested the class of DSC models against the latest observational data and we placed constraints on the corresponding free parameters. In particular, our observational constraints regarding the Hubble constant are in agreement (within ) with those of Planck 2018. Moreover, our results are compatible at level with the measurement obtained from Cepheids
Using the most popular information criteria we found cases for which the DSC model is statistically equivalent with that of CDM and thus it can be viewed as a viable cosmological alternative. On top of that we found that one of the DSC models, that with and , shows a small but non-zero deviation from CDM, where the confidence level is close to . We argued that the theoretical formulation of Ref. Benisty:2018qed () could provide competitive cosmological models and thus it deserves further consideration. Finally, in a forthcoming paper we attempt to investigate DSC at the perturbation level for the general case of potentials . This will allow us to modify CAMB and thus to confront Dynamical Space-time Cosmology to the Cosmic Microwave Background (CMB) power spectrum from Planck.
Acknowledgements.FA wishes to thank Charalambos Kioses for a number of very interesting discussions and also for vast help at optimization of the code for this project. This article is supported by COST Action CA15117 "Cosmology and Astrophysics Network for Theoretical Advances and Training Action" (CANTATA) of the COST (European Cooperation in Science and Technology). SB would like to acknowledge support by the Research Center for Astronomy of the Academy of Athens in the context of the program "Tracing the Cosmic Acceleration".
- (1) P. Bull et al., Phys. Dark Univ. 12 (2016) 56 doi:10.1016/j.dark.2016.02.001 [arXiv:1512.05356 [astro-ph.CO]].
- (2) D. Merritt, Stud. Hist. Philos. Mod. Phys. 57 (2017) 41 doi:10.1016/j.shpsb.2016.12.002 [arXiv:1703.02389 [physics.hist-ph]].
- (3) R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998) doi:10.1103/PhysRevLett.80.1582 [astro-ph/9708069].
- (4) B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988). doi:10.1103/PhysRevD.37.3406
- (5) R. J. Scherrer, Phys. Rev. Lett. 93, 011301 (2004) doi:10.1103/PhysRevLett.93.011301 [astro-ph/0402316].
- (6) C. R. Fadragas, G. Leon and E. N. Saridakis, Class. Quant. Grav. 31 (2014) 075018 doi:10.1088/0264-9381/31/7/075018 [arXiv:1308.1658 [gr-qc]].
- (7) A. Arbey, Phys. Rev. D 74, 043516 (2006) doi:10.1103/PhysRevD.74.043516 [astro-ph/0601274].
- (8) X. m. Chen, Y. g. Gong and E. N. Saridakis, JCAP 0904, 001 (2009) doi:10.1088/1475-7516/2009/04/001 [arXiv:0812.1117 [gr-qc]].
- (9) G. Leon and E. N. Saridakis, JCAP 0911 (2009) 006 doi:10.1088/1475-7516/2009/11/006 [arXiv:0909.3571 [hep-th]].
- (10) G. Leon and E. N. Saridakis, Class. Quant. Grav. 28, 065008 (2011) doi:10.1088/0264-9381/28/6/065008 [arXiv:1007.3956 [gr-qc]].
- (11) G. Leon and E. N. Saridakis, JCAP 1303, 025 (2013) doi:10.1088/1475-7516/2013/03/025 [arXiv:1211.3088 [astro-ph.CO]].
- (12) G. Leon and E. N. Saridakis, JCAP 1504 (2015) no.04, 031 doi:10.1088/1475-7516/2015/04/031 [arXiv:1501.00488 [gr-qc]].
- (13) G. Leon, J. Saavedra and E. N. Saridakis, Class. Quant. Grav. 30, 135001 (2013) doi:10.1088/0264-9381/30/13/135001 [arXiv:1301.7419 [astro-ph.CO]].
- (14) G. Kofinas, G. Leon and E. N. Saridakis, Class. Quant. Grav. 31 (2014) 175011 doi:10.1088/0264-9381/31/17/175011 [arXiv:1404.7100 [gr-qc]].
- (15) M. A. Skugoreva, E. N. Saridakis and A. V. Toporensky, Phys. Rev. D 91 (2015) 044023 doi:10.1103/PhysRevD.91.044023 [arXiv:1412.1502 [gr-qc]].
- (16) S. Basilakos and M. Plionis, Astron. & Astrophysics 507, 47, (2009)
- (17) E. Guendelman, E. Nissimov and S. Pacheva, Eur. Phys. J. C 76, no. 2, 90 (2016) doi:10.1140/epjc/s10052-016-3938-7 [arXiv:1511.07071 [gr-qc]].
- (18) E. Guendelman, D. Singleton and N. Yongram, JCAP 1211, 044 (2012) doi:10.1088/1475-7516/2012/11/044 [arXiv:1205.1056 [gr-qc]].
- (19) E. Guendelman, E. Nissimov and S. Pacheva, Eur. Phys. J. C 75, no. 10, 472 (2015) doi:10.1140/epjc/s10052-015-3699-8 [arXiv:1508.02008 [gr-qc]].
- (20) S. Ansoldi and E. I. Guendelman, JCAP 1305, 036 (2013) doi:10.1088/1475-7516/2013/05/036 [arXiv:1209.4758 [gr-qc]].
- (21) E. Guendelman, E. Nissimov and S. Pacheva, Bulg. J. Phys. 44, 15 (2017) [arXiv:1609.06915 [gr-qc]].
- (22) G. Koutsoumbas, K. Ntrekis, E. Papantonopoulos and E. N. Saridakis, JCAP 1802, no. 02, 003 (2018) doi:10.1088/1475-7516/2018/02/003 [arXiv:1704.08640 [gr-qc]].
- (23) Z. Haba, A. Stachowski and M. SzydÅowski, JCAP 1607, no. 07, 024 (2016) doi:10.1088/1475-7516/2016/07/024 [arXiv:1603.07620 [gr-qc]].
- (24) D. Benisty and E. I. Guendelman, Eur. Phys. J. C 77, no. 6, 396 (2017) doi:10.1140/epjc/s10052-017-4939-x [arXiv:1701.08667 [gr-qc]].
- (25) D. Benisty and E. I. Guendelman, Int. J. Mod. Phys. D 26 (2017) no.12, 1743021. doi:10.1142/S0218271817430210 Eur. Phys. J. C 77, no. 6, 396 (2017) doi:10.1140/epjc/s10052-017-4939-x [arXiv:1701.08667 [gr-qc]].
- (26) C. Gao, M. Kunz, A. R. Liddle and D. Parkinson, Phys. Rev. D 81, 043520 (2010) doi:10.1103/PhysRevD.81.043520 [arXiv:0912.0949 [astro-ph.CO]].
- (27) D. Benisty and E. I. Guendelman, Phys. Rev. D 98, no. 2, 023506 (2018) doi:10.1103/PhysRevD.98.023506 [arXiv:1802.07981 [gr-qc]].
- (28) E. I. Guendelman, Int. J. Mod. Phys. A 25, 4081 (2010) doi:10.1142/S0217751X10050317 [arXiv:0911.0178 [gr-qc]].
- (29) J. Goodman and J. Weare, Comm. App. Math. and Comp. Sci. 5, 65 (2010).
- (30) D. Foreman-Mackey, D. W. Hogg, D. Lang and J. Goodman, Publ. Astron. Soc. Pac. 125, 306 (2013) doi:10.1086/670067 [arXiv:1202.3665 [astro-ph.IM]].
- (31) C. Kioses, double-linked-list, (2018), GitHub repository: \(https://github.com/mpambisk\)
- (32) N. Aghanim et al. [Planck Collaboration], arXiv:1807.06209 [astro-ph.CO].
- (33) T. G. Clemson and A. R. Liddle, Mon. Not. Roy. Astron. Soc. 395, 1585 (2009) doi:10.1111/j.1365-2966.2009.14641.x [arXiv:0811.4676 [astro-ph]].
- (34) H. Yu, B. Ratra and F. Y. Wang, Astrophys. J. 856, no. 1, 3 (20l8) doi:10.3847/1538-4357/aab0a2 [arXiv:1711.03437 [astro-ph.CO]].
- (35) F. K. Anagnostopoulos, S. Basilakos, G. Kofinas and V. Zarikas, arXiv:1806.10580 [astro-ph.CO].
- (36) F. K. Anagnostopoulos and S. Basilakos, Phys. Rev. D 97, no. 6, 063503 (2018) doi:10.1103/PhysRevD.97.063503 [arXiv:1709.02356 [astro-ph.CO]]; S. Basilakos, S. Nesseris, F. K. Anagnostopoulos and E. N. Saridakis, arXiv:1803.09278 [astro-ph.CO].
- (37) D. M. Scolnic et al., Astrophys. J. 859 (2018) no.2, 101 doi:10.3847/1538-4357/aab9bb [arXiv:1710.00845 [astro-ph.CO]].
- (38) G. Risaliti and E. Lusso, Astrophys. J. 815, 33 (2015) doi:10.1088/0004-637X/815/1/33 [arXiv:1505.07118 [astro-ph.CO]]; C. Roberts, K. Horne, A. O. Hodson and A. D. Leggat, arXiv:1711.10369 [astro-ph.CO].
- (39) . M. Plionis, R. Terlevich, S. Basilakos, F. Bresolin, E. Terlevich, J. Melnick, R. Chavez, MNRAS, 416, 2981 (2011)
- (40) H. Akaike, IEEE Transactions on Automatic Control 19, 716 (1974).
- (41) G. Schwarz, Ann. Statist., 6, 2 (1978), 461-464.
- (42) A. R. Liddle, Mon. Not. Roy. Astron. Soc. 377, L74 (2007) doi:10.1111/j.1745-3933.2007.00306.x [astro-ph/0701113].
- (43) K. Anderson, Model selection and multimodel inference: a practical information-theoretic approach, 2nd edn. Springer, New York (2002); K. Anderson, Sociological Methods and Research 33, 261 (2004).
- (44) Spiegelhalter, David J., et al. Jour. of the R. Stat. Soc., 64 4 (2002): 583-639.
- (45) Jeffreys, H. (1998). The theory of probability. OUP Oxford.
- (46) Kass, Robert E., and Adrian E. Raftery, Journal of the american statistical association 90.430 (1995): 773-795.
- (47) A. G. Riess, et al., Astrophys. J. 855, 18 (2018) doi:10.3847/1538-4357/aaadb7 [arXiv:1801.01120 [astro-ph.SR]].
- (48) B. S. Haridasu, V. V. Lukovic, M. Moresco and N. Vittorio, arXiv:1805.03595 [astro-ph.CO].