Observational constraints on one-parameter dynamical dark-energy parametrizations and the tension
The phenomenological parametrizations of dark-energy (DE) equation of state can be very helpful, since they allow for the investigation of its cosmological behavior despite the fact that its underlying theory is unknown. However, although in the literature there has been a large amount of research on DE parametrizations which involve two or more free parameters, the one-parameter parametrizations seem to be underestimated. We perform a detailed observational confrontation of five one-parameter DE models, with observational data from cosmic microwave background (CMB), Joint light-curve analysis sample from Supernovae Type Ia observations (JLA), baryon acoustic oscillations (BAO) distance measurements, and cosmic chronometers (CC). We find that all models favor a phantom DE equation of state at present time, while they lead to values in perfect agreement with its direct measurements and therefore they offer an alleviation to the -tension. Finally, performing a Bayesian analysis we show that although CDM cosmology is still favored, one-parameter DE models have similar or better efficiency in fitting the data comparing to two-parameter DE parametrizations, and thus they deserve a thorough investigation.
pacs:98.80.-k, 95.36.+x, 98.80.Es
The remarkable journey of modern cosmology started in 1998, when the observational evidences showed that we are living in an accelerating universe and that the previous physical scenarios needed to be retraced. The introduction of dark energy (DE) concept was just a need in order for the observational predictions to acquire a solid theoretical formulation. The dark energy is a component with high negative pressure that drives the universe acceleration, nevertheless its nature has remained to be a mysterious chapter in the scientific history after a series of investigations by a large number of researchers. The cosmological constant is the simplest DE fluid with the above features, however the “cosmological constant problem” Weinberg:1988cp and the possibility that DE sector could be dynamical led to a a number of explanations, mainly in two directions. The first way is to consider that the DE sector corresponds to a peculiar extra fluid that fills the universe in the framework of general relativity Copeland:2006wr ; Cai:2009zp . The second direction is to consider that the DE fluid is an effective one, arising from a modification of the gravitational sector itself modgrav1 ; Capozziello:2011et ; Cai:2015emx .
Independently of the underlying nature and the micro-physical theory of DE, one can introduce phenomenological parametrizations of the DE equation-of-state parameter , where and are respectively the pressure and energy density of the (effective) DE perfect fluid, which is considered to have a dynamical character in general. Since for the moment we do not have any fundamental rule in favor of some specific equation-of-state parameters, we may consider various functional forms for . For a literature survey of various DE parametrizations we refer to the works Chevallier:2000qy ; Linder:2002et ; Cooray:1999da ; Efstathiou:1999tm ; Astier:2000as ; Weller:2001gf ; Jassal:2005qc ; Linder:2005ne ; Gong:2005de ; Nesseris:2005ur ; Feng:2004ff ; Xia:2006rr ; Basilakos:2006us ; Nojiri:2006ww ; Saridakis:2008fy ; Barboza:2008rh ; Saridakis:2009pj ; Dutta:2009yb ; Saridakis:2009ej ; Ma:2011nc ; Feng:2011zzo ; Feng:2012gf ; DeFelice:2012vd ; Chen:2011cy ; Basilakos:2013vya ; DiValentino:2016hlg ; Chavez:2016epc ; DiValentino:2017zyq ; DiValentino:2017gzb ; Zhao:2017cud ; Yang:2017amu ; Marcondes:2017vjw ; Yang:2017alx ; Pan:2017zoh ; Vagnozzi:2018jhn .
In general, the well known DE parametrizations have two free parameters, usually denoted by CDM models, where marks the present value of and characterizes the dynamical nature of the DE sector. However, apart from the CDM parametrizations, one-parameter dynamical DE parametrizations, as well as models with more than two parameters, have also been introduced and investigated in the last years. Nevertheless, the one-parameter dynamical DE parametrizations are much neglected in the literature compared to the DE parametrizations having two or more parameters. In principle we do not find any strong reason behind this underestimation, and thus in this work we take an initiation to investigate the features of this particular class of DE parametrizations, and explore its cosmological viabilities with the recent observational evidences, taking into account their advantage that they are more economical and have one free parameter less.
Hence, we introduce various one-parameter dynamical DE parametrizations that are primarily motivated from the phenomenological ground, and we perform a detailed observational confrontation. In particular, we use data from cosmic microwave background (CMB) observations, from Joint light-curve analysis sample from Supernovae Type Ia observations (JLA), from baryon acoustic oscillations (BAO) distance measurements, as well as from cosmic chronometers Hubble parameter measurements (CC), performing additionally various combined analyses.
The manuscript is organized as follows. In Section II we present the basic equations for a general dark-energy scenario at both the background and perturbation level, and we display the five one-parameter DE parametrizations that are going to be investigated. In Section III we describe the observational data sets that will be used. In Section IV we perform the observational confrontation, extracting the observational constraints on the various cosmological quantities. Finally, we close the present work in Section VI with a brief summary.
Ii One-parameter parametrizations at background and perturbation levels
In this section we present the basic equations that determine the universe evolution at both the background and perturbation levels, and we introduce various one-parameter parametrizations for the dark-energy equation-of-state parameter. Throughout the work we consider the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) geometry, with metric
where is the scale factor and determines spatial curvature, with values , and corresponding to spatially flat, open and closed universe, respectively.
We consider a universe filled with radiation, baryons, cold dark matter, as well as of the (effective) DE fluid. In this case the Friedmann equations, that determine the universe evolution at the background level, read as
with the Newton’s constant and the Hubble function, with dots denoting derivatives with respect to cosmic time. In the above expressions we have introduced the total energy density and pressure and respectively, with the symbols denoting radiation, baryon, cold dark matter and dark energy fluids. Finally, for simplicity in the following we will focus on the spatially flat case () since it is favored by observations.
As usual we assume that the above sectors do not have any mutual interaction, and thus the conservation equation of each fluid is
where and . Note that out of equations (2), (3) and (4), only two are independent. Hence, using the known equation-of-state parameters , , equations (4) give , and , with the present value of and where we have set the present scale factor to 1. Similarly, concerning the dark energy sector, equation (4) leads to
Thus, the evolution equation (5) implies that the dynamics of DE can be determined as long as a specific parametrization of the DE equation of state is given.
Having presented the equations that determine the universe evolution at the background level, we now proceed to the investigation of its evolution at the perturbation level, since this is related to the observed structure formation. In order to study the perturbation equations, one needs to consider the perturbed FLRW metric either in synchronous or in conformal Newtonian gauge. In the following we consider the former choice, in which the perturbed metric takes the form
where is the conformal time, and , are respectively the unperturbed and the perturbated metric tensors. Perturbing additionally the fluid sectors, one can result to the following equations for a mode with wavenumber Mukhanov ; Ma:1995ey ; Malik:2008im :
where primes mark derivatives with respect to conformal time and is the conformal Hubble parameter. Furthermore, is the density perturbation for the -th fluid, denotes the divergence of the -th fluid velocity, stands for the trace of the metric perturbations , and is the anisotropic stress related to the -th fluid. Finally, the quantity denotes the adiabatic speed of sound of the -th fluid, and it is given by in the case where we set the sound speed to . In the following analysis we neglect the anisotropic stress for simplicity.
In this work we are interested in investigating one-parameter DE equation-of-state parametrizations. In particular, we consider five such parametrizations given by:
where is the only free parameter, corresponding to the dark energy equation-of-state parameter at present. In order to provide a more transparent picture of the behavior of the above parametrizations, in Fig. 1 we depict , taking two cases for , namely one lying in the quintessence and one lying in the phantom regime. As we can see, in all models presents a decreasing behavior.
Iii Observational data
In this section we proceed to a detailed observational confrontation of the one-parameter dynamical DE equation-of-state parametrizations (9)-(13) presented in the previous section. We analyze several combinations of cosmological data, by considering the six cosmological parameters of the standard CDM paradigm: the baryon and the cold dark matter energy densities and , the ratio between the sound horizon and the angular diameter distance at decoupling , the reionization optical depth , and the spectral index and the amplitude of the scalar primordial power spectrum and . Moreover, for the various models we add the free parameter , which parametrizes the DE evolution. All these free parameters are explored within the range of the conservative flat priors listed in Table 1.
We derive the bounds on the cosmological parameters by analyzing the full range of the 2015 Planck temperature and polarization cosmic microwave background (CMB) angular power spectra, and we call this combination “CMB” Adam:2015rua ; Aghanim:2015xee . Additionally, we consider the Joint light-curve analysis sample from Supernovae Type Ia and we refer to this dataset as “JLA” Betoule:2014frx . Furthermore, we add the baryon acoustic oscillations (BAO) distance measurements, and we call them “BAO” Beutler:2011hx ; Ross:2014qpa ; Gil-Marin:2015nqa . Finally, we use the Hubble parameter measurements from the cosmic chronometers (CC) and we refer to them as “CC” Moresco:2016mzx .
In order to analyze statistically the several combinations of datasets, exploring the different dynamical DE scenarios, we use our modified version of the publicly available Monte-Carlo Markov Chain package Cosmomc Lewis:2002ah , including the support for the Planck data release 2015 Likelihood Code Aghanim:2015xee 111See http://cosmologist.info/cosmomc/. This has a convergence diagnostic based on the Gelman and Rubin statistic and implements an efficient sampling of the posterior distribution using the fast/slow parameter de-correlations Lewis:2013hha .
In this section we present the observational constraints and their implications for all the one-parameter DE parametrizations (9)-(13). All the models will be confronted initially with CMB data alone, and then with different combinations of cosmological data. In the following subsections we investigate the various models separately in detail, presenting their observational consequences.
iv.1 Model I
We start by investigating Model I of (9), namely . In Fig. 2 we can see the effect of different values on the temperature and matter power spectra. The results of the observational analysis of this model can be seen in Table 2, where we display the 68% and 95% confidence level (CL) constraints for various quantities, while the full contour plots are presented in Fig. 3.
As we observe, the CMB data alone allow for a very large value of the Hubble constant at present and moreover its error bars are significantly large: at 68% CL ( at 95% CL). The constraint on is actually very close to its local measurements Riess:2016jrr , recently confirmed by R18 and Birrer:2018vtm .
The present value of the DE equation-of-state parameter for CMB alone is found to prefer a phantom dark energy scenario, namely , at more than 95% CL. Consequently, the matter density parameter decreases and acquires a very low value ( at 68% CL). However, since these and are anti-correlated, while is positively correlated with (see Fig. 3) this does not correspond to the alleviation of the tension of Planck indirect estimation with the direct measurements from cosmic shear experiments like the Canada France Hawaii Lensing Survey (CFHTLenS) Heymans:2012gg ; Erben:2012zw , the Kilo Degree Survey of450 deg of imaging data (KiDS-450) Hildebrandt:2016iqg , and the Dark Energy Survey (DES) Abbott:2017wau .
When the BAO data are added to CMB ones, the constraints on the model parameters are significantly improved and the error bars on most of the parameters, in particular , , and , are decreased. The mean value of the Hubble constant slightly shifts towards a lower value, and the DE equation of state at present, , moves towards a smaller one ( at 68% CL) comparing to its estimation from CMB alone ( at 68% CL). As one can see, the CMB+BAO data also assure the validity of at more than 99% CL. The interesting output of this analysis is that the constraint on is again found to be very close to its estimation by local measurements Riess:2016jrr .
The addition of JLA to the former data set combination further improves the cosmological constraints, as one can clearly see from Table 2. In particular, we see that again shifts down and up, with decreasing error bars. An analogous improvement of the bounds can be seen for and . Although the estimation of from this analysis decreases in comparison to the previous results of CMB and CMB+BAO, within 95% CL it can still match the direct estimation Riess:2016jrr . Furthermore, the DE equation of state at present is again found to be in phantom regime.
We close the analysis by adding the CC dataset, nevertheless the results for the data combination CMB+BAO+JLA+CC do not exhibit significant differences from the previous case CMB+BAO+JLA.
In summary, the observational analysis for Model I shows that at more than 95% CL for CMB only, while the tension on seems to be alleviated. The addition of JLA shifts towards lower values, but still in agreement within 2 with Riess:2016jrr , while the addition of CC does not affect the results significantly. The contour plot in the plane can be seen in lower left graph of Fig. 3. The preference for a phantom DE equation of state is due to the better fit of the large scales of the temperature power spectrum, that prefers a lower quadrupole with respect to the CDM scenario, as can be clearly seen in Fig. 4.
Finally, comparing the results obtained for Model I with the constraints released by the Planck collaboration Aghanim:2018eyx for the CDM or CDM models, we can notice that for Model I using only CMB data the value is well constrained by the data and is close to its directly measured value Riess:2016jrr , while it has a slightly lower limit for the Planck extended scenarios. On the other hand, when adding the BAO data the value is still high and in agreement with Riess:2016jrr , while in the Planck case the Hubble constant decreases leading to the aforementioned tension.
iv.2 Model II
In this subsection we investigate Model II of (10), namely . In Fig. 5 we present the effect of on the temperature and matter power spectra. Additionally, in Table 3 we summarize the observational constraints arising from various data combinations, while the full contour plots are presented in Fig. 6.
Comparing the results with those of Model I above, we can see that for Model II and using the CMB data alone we find that acquires higher values ( at 68% CL) and indicates a strong evidence for a phantom equation of state which remains at more than 95% CL. Moreover, similarly to Model I, for Model II we also observe that the combinations CMB+BAO, CMB+BAO+JLA and CMB+BAO+JLA+CC significantly improve the constraints and reduce the error bars on the parameters. In particular, the mean value shifts towards lower values and we find at more than 95% CL for all data combinations. Note that from Table 3 we see that for the data set CMB+BAO the estimated value of is in agreement within 1 standard deviation with the local estimation of Riess:2016jrr and thus the -tension is alleviated ( is higher than the one estimated by Planck 2015 Ade:2015xua for the base CDM scenario, and it is in perfect agreement to Riess:2016jrr ). Additionally, for the last two combinations CMB+BAO+JLA and CMB+BAO+JLA+CC we observe that while the Hubble constant is always in agreement within with Riess:2016jrr , in contrast to Model I prefers a lower phantom mean value, but still with high significance. The contour plot in the plane can be seen in lower left graph of Fig. 6. Finally, similarly to the previous model, the tension is not reconciled.
iv.3 Model III
We proceed to the investigation of Model III of (11), namely . In Fig. 7 we depict the effect of on the temperature and matter power spectra. Moreover, in Table 4 we summarize the observational constraints arising from various data combinations, while the full contour plots are presented in Fig. 8.
As we can see, for the CMB data alone the Hubble parameter acquires an even larger mean value in comparison to the previous models, while the DE equation-of-state parameter at present obtains a smaller value, namely at 95% CL. Similarly to the previous model, we find that the inclusion of any external data set, namely BAO, JLA or CC, to CMB significantly improves the constraints, and is still valid up to 95% CL. For the combination of CMB+BAO data we see that the estimated value of (at 68% CL) is perfectly in agreement to its local estimation of Riess:2016jrr , alleviating the -tension. Moreover, concerning we can note that it is constrained to be (at 68% CL) which is phantom at more than . The addition of JLA to CMB+BAO decreases the error bars on , and , while within 95% CL this model seems to alleviate the tension on . The contour plot in the plane can be seen in lower left graph of Fig. 8. Finally, the combination CMB+BAO+JLA+CC does not offer any notable differences compared to the analysis with CMB+BAO+JLA, and thus similar conclusions are achieved.
iv.4 Model IV
In this subsection we investigate Model IV of (12), namely . In Fig. 9 we present the effect of on the temperature and matter power spectra. Additionally, in Table 5 we summarize the observational constraints arising from various data combinations, while the full contour plots are presented in Fig. 10.
As we see, the estimations of the Hubble parameter for all the dataset combinations are shifted towards higher values than CDM. For the CMB data only, acquires values comparable with Model III, i.e. at 68% CL. As before, the inclusion of any external data set significantly improves the constraints on the cosmological parameters, decreasing the error bars. A common feature for all the analysis is that remains in the phantom regime at more than 95% CL. Furthermore, for this model the CMB+BAO+JLA and CMB+BAO+JLA+CC data combinations favor a phantom DE equation of state at many standard deviations. Additionally, it is clearly seen that the -tension is alleviated for all the combinations considered, apart from the CMB data alone which predict a quite high value. Finally, the contour plot in the plane can be seen in lower left graph of Fig. 10.
iv.5 Model V
We close our analysis with the investigation of Model V of (13), namely . In Fig. 11 we show the effect of on the temperature and matter power spectra. Additionally, in Table 6 we summarize the observational constraints arising from various data combinations, while the full contour plots are presented in Fig. 12.
As we observe, we can clearly notice that this model maintains a similar trend compared to the previous four dynamical DE models. The present value of the DE equation-of-state parameter is constrained in the phantom regime up to 99% CL. The Hubble parameter acquires a very large value for the CMB data only ( at 68% CL) with large error bars, however for the other data combinations and its errors bars decrease, and it becomes clear that the -tension can be alleviated. Finally, the contour plot in the plane can be seen in lower left graph of Fig. 12.
V Model Comparison and the Bayesian Evidence
In the previous section we performed the observational analysis and we extracted the constraints on the various cosmological parameters of the five examined models. Concerning the free parameter of the models, in Fig. 13 we summarize the results, and we present them in a nutshell in a Whisker graph showing the 68% and 95% CL. As we mentioned above, we observe that in all models a phantom DE equation-of-state parameter at current time is favored. Moreover, from Fig. 13 we may also note that the extracted for Model II and III using the common datasets, namely, CMB+BAO, CMB+BAO+JLA and CMB+BAO+JLA+CC, is relatively close to the cosmological constant bundary , compared to the other three models.
Additionally, in order to present in a more transparent way the alleviation of the -tension, in Fig. 14 we summarize the contour plots in the plane for all the examined models. From the figure one can notice that the parameters and are correlated to each other.
The question that arises naturally is which of the five models exhibits a better behavior, and moreover how efficient are they comparing to standard CDM cosmology. Hence, we close our work with examining the Bayesian evidence of each of the five models analyzed above, compared to the reference CDM cosmological scenario. The Bayesian evidence plays a crucial role in determining the observational support of any cosmological model. The involved calculation is performed through the publicly available code MCEvidence Heavens:2017hkr ; Heavens:2017afc 222See github.com/yabebalFantaye/MCEvidence .. We mention that MCEvidence needs only the MCMC chains that are used to extract the parameters of the models.
In Bayesian analysis one needs to evaluate the posterior probability of the model parameters , given a particular observational dataset with any prior information for a model . Using the Bayes theorem one can write
where the quantity refers to the likelihood as a function of with the prior information. The quantity that appears in the denominator of (14) is known as the Bayesian evidence used for the model comparison. Let us note that this Bayesian evidence is the integral over the non-normalized posterior , given by
Now, for any cosmological model and the reference model (the reference model is the one with respect to which we compare the observational viability), the posterior probability is given by the following law:
where the quantity is the Bayes factor of the model with respect to the reference model . Depending on different values of (or equivalently ) we quantify the observational support of the model over the model . Here we use the widely accepted Jeffreys scales Kass:1995loi shown in Table 7, which imply that for the observational data support model more strongly than model . The negative values of reverse the conclusion, that is the reference model is preferred over .
|Strength of evidence for model|
In Table 8 we present the values of calculated for the five one-parameter DE models (9)-(13) analyzed in the previous section, for various observational datasets, compared to the reference CDM scenario. From the values of we can see that Model II and Model III present a better behavior than the other three analyzed models. However, comparing to all models the reference CDM scenario is favored. Nevertheless, we mention here that, interestingly enough, the one-parameter DE parametrizations considered in the present work seem to behave similarly or be less disfavored with respect to CDM scenario comparing with two-parameter DE parametrizations Ma:2011nc ; Feng:2011zzo ; Yang:2017alx ; Pan:2017zoh . This is an indication that one-parameter DE models can indeed be efficient in describing the universe evolution.
|Dataset||Model||Strength of evidence for reference CDM scenario|
|CMB+BAO+JLA||Model I||Very Strong|
|CMB+BAO+JLA+CC||Model I||Very Strong|
|CMB+BAO+JLA||Model IV||Very Strong|
|CMB+BAO+JLA+CC||Model IV||Very Strong|
|CMB+BAO+JLA||Model V||Very Strong|
|CMB+BAO+JLA+CC||Model V||Very Strong|
Vi Summary and Conclusions
The phenomenological parametrizations of DE equation of state can be very helpful for the investigation of DE features, since they are of general validity and can describe the DE sector independently of whether it is an extra peculiar fluid in the framework of general relativity or it is effectively of gravitational origin. However, although in the literature there has been a large amount of research on DE parametrizations which involve two or more free parameters, the one-parameter parametrizations seem to be underestimated.
In this work we performed a detailed observational confrontation of several one-parameter DE parametrizations, with various combination datasets. In particular, we used data from cosmic microwave background (CMB) observations, from Joint light-curve analysis sample from Supernovae Type Ia observations (JLA), from baryon acoustic oscillations (BAO) distance measurements, as well as from cosmic chronometers Hubble parameter measurements (CC), and we additionally performed various combined analyses.
Our analyses revealed that all the examined one-parameter dynamical DE models favor a phantom DE equation-of-state at present time , and this remains valid at more than 95% CL, confirming the result obtained in various other works in different contexts DiValentino:2017iww ; DiValentino:2015ola ; DiValentino:2016hlg ; DiValentino:2017zyq ; Yang:2017ccc ; Yang:2017zjs ; Mortsell:2018mfj ; Yang:2018euj ; Yang:2018uae . The inclusion of any external dataset to CMB improves the fitting and decreases the errors significantly without any change in the conclusion. Concerning the present value of the Hubble parameter , we found that the CMB data alone leads to large error bars, however the inclusion of other datasets decreases them significantly, with the favored value being in perfect agreement with its direct measurements. Hence, we deduce that one-parameter DE models can provide a solution to the known -tension between local measurements and Planck indirect ones. This is one of the main results of the present work. Nevertheless, the possible -tension does not seem to be reconciled, since in all models the favored value is similar to the Planck’s estimated one.
Finally, in order to examine which of the five models is better fitted to the data, as well as in order to compare it with CDM cosmological scenario, we performed a Bayesian analysis. As we saw Model II and Model III are relatively close to CDM (this can also be viewed from the Whisker graph in Fig. 13 where for Model II and Model III are relatively close to compared to other models). However, the reference CDM scenario is still favored comparing to all models. Nevertheless, these one-parameter DE models have similar or better efficiency in fitting the data comparing with the two-parameter DE parametrizations analyzed in the literature, taking into account their advantage that they are more economical and have one free parameter less. This is an indication that one-parameter DE models can indeed be efficient in describing the universe evolution, and thus they deserve a thorough investigation.
Acknowledgements.WY was supported by the National Natural Science Foundation of China under Grants No. 11705079 and No. 11647153. EDV acknowledges support from the European Research Council in the form of a Consolidator Grant with number 681431. SC acknowledges the Mathematical Research Impact Centric Support (MATRICS), project reference no. MTR/2017/000407, by the Science and Engineering Research Board, Government of India. This article is based upon work from CANTATA COST (European Cooperation in Science and Technology) action CA15117, EU Framework Programme Horizon 2020.
- (1) S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys. 61, 1 (1989).
- (2) E. J. Copeland, M. Sami and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D 15, 1753 (2006).
- (3) Y. F. Cai, E. N. Saridakis, M. R. Setare and J. Q. Xia, Quintom Cosmology: Theoretical implications and observations, Phys. Rept. 493, 1 (2010).
- (4) S. Nojiri and S. D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy, eConf C 0602061, 06 (2006) [Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)].
- (5) S. Capozziello and M. De Laurentis, Extended Theories of Gravity, Phys. Rept. 509, 167 (2011).
- (6) Y. F. Cai, S. Capozziello, M. De Laurentis and E. N. Saridakis, “f(T) teleparallel gravity and cosmology, Rept. Prog. Phys. 79, no. 10, 106901 (2016).
- (7) M. Chevallier and D. Polarski, Accelerating universes with scaling dark matter, Int. J. Mod. Phys. D 10, 213 (2001).
- (8) E. V. Linder, Exploring the expansion history of the universe, Phys. Rev. Lett. 90, 091301 (2003).
- (9) A. R. Cooray and D. Huterer, Gravitational lensing as a probe of quintessence, Astrophys. J. 513, L95 (1999).
- (10) G. Efstathiou, Constraining the equation of state of the universe from distant type Ia supernovae and cosmic microwave background anisotropies, Mon. Not. Roy. Astron. Soc. 310, 842 (1999).
- (11) P. Astier, Can luminosity distance measurements probe the equation of state of dark energy, Phys. Lett. B 500, 8 (2001).
- (12) J. Weller and A. Albrecht, Future supernovae observations as a probe of dark energy, Phys. Rev. D 65, 103512 (2002).
- (13) H. K. Jassal, J. S. Bagla and T. Padmanabhan, Observational constraints on low redshift evolution of dark energy: How consistent are different observations?, Phys. Rev. D 72, 103503 (2005).
- (14) E. V. Linder and D. Huterer, How many dark energy parameters?, Phys. Rev. D 72, 043509 (2005).
- (15) Y. g. Gong and Y. Z. Zhang, Probing the curvature and dark energy, Phys. Rev. D 72, 043518 (2005).
- (16) S. Nesseris and L. Perivolaropoulos, Comparison of the legacy and gold snia dataset constraints on dark energy models, Phys. Rev. D 72, 123519 (2005).
- (17) B. Feng, M. Li, Y. S. Piao and X. Zhang, Oscillating quintom and the recurrent universe, Phys. Lett. B 634, 101 (2006).
- (18) J. Q. Xia, G. B. Zhao, H. Li, B. Feng and X. Zhang, Features in Dark Energy Equation of State and Modulations in the Hubble Diagram, Phys. Rev. D 74, 083521 (2006).
- (19) S. Basilakos and N. Voglis, Virialization of cosmological structures in models with time varying equation of state, Mon. Not. Roy. Astron. Soc. 374, 269 (2007).
- (20) S. Nojiri and S. D. Odintsov, The Oscillating dark energy: Future singularity and coincidence problem, Phys. Lett. B 637, 139 (2006).
- (21) E. N. Saridakis, Theoretical Limits on the Equation-of-State Parameter of Phantom Cosmology, Phys. Lett. B 676, 7 (2009).
- (22) E. M. Barboza, Jr. and J. S. Alcaniz, A parametric model for dark energy, Phys. Lett. B 666, 415 (2008).
- (23) E. N. Saridakis, Phantom evolution in power-law potentials, Nucl. Phys. B 819, 116 (2009).
- (24) S. Dutta, E. N. Saridakis and R. J. Scherrer, Dark energy from a quintessence (phantom) field rolling near potential minimum (maximum), Phys. Rev. D 79, 103005 (2009).
- (25) E. N. Saridakis, Quintom evolution in power-law potentials, Nucl. Phys. B 830, 374 (2010).
- (26) J. Z. Ma and X. Zhang, Probing the dynamics of dark energy with novel parametrizations, Phys. Lett. B 699, 233 (2011).
- (27) L. Feng and T. Lu, A new equation of state for dark energy model, JCAP 1111, 034 (2011).
- (28) C. J. Feng, X. Y. Shen, P. Li and X. Z. Li, A New Class of Parametrization for Dark Energy without Divergence, JCAP 1209, 023 (2012).
- (29) A. De Felice, S. Nesseris and S. Tsujikawa, Observational constraints on dark energy with a fast varying equation of state, JCAP 1205, 029 (2012).
- (30) X. m. Chen, Y. Gong, E. N. Saridakis and Y. Gong, Time-dependent interacting dark energy and transient acceleration, Int. J. Theor. Phys. 53, 469 (2014).
- (31) S. Basilakos and J. Solá, Effective equation of state for running vacuum: ‘mirage’ quintessence and phantom dark energy, Mon. Not. Roy. Astron. Soc. 437, no. 4, 3331 (2014).
- (32) E. Di Valentino, A. Melchiorri and J. Silk, Reconciling Planck with the local value of in extended parameter space, Phys. Lett. B 761, 242 (2016).
- (33) R. Chávez, M. Plionis, S. Basilakos, R. Terlevich, E. Terlevich, J. Melnick, F. Bresolin and A. L. González-Morán, Constraining the dark energy equation of state with H galaxies, Mon. Not. Roy. Astron. Soc. 462, no. 3, 2431 (2016).
- (34) E. Di Valentino, A. Melchiorri, E. V. Linder and J. Silk, Constraining Dark Energy Dynamics in Extended Parameter Space, Phys. Rev. D 96, 023523 (2017).
- (35) E. Di Valentino, Crack in the cosmological paradigm, Nat. Astron. 1, 569 (2017).
- (36) G. B. Zhao et al., Dynamical dark energy in light of the latest observations, Nat. Astron. 1, 627 (2017).
- (37) W. Yang, R. C. Nunes, S. Pan and D. F. Mota, Effects of neutrino mass hierarchies on dynamical dark energy models, Phys. Rev. D 95, 103522 (2017).
- (38) R. J. F. Marcondes and S. Pan, Cosmic chronometers constraints on some fast-varying dark energy equations of state [arXiv:1711.06157 [astro-ph.CO]].
- (39) W. Yang, S. Pan and A. Paliathanasis, Latest astronomical constraints on some nonlinear parametric dark energy models, Mon. Not. Roy. Astron. Soc. 475, 2605 (2018).
- (40) S. Pan, E. N. Saridakis and W. Yang, Observational Constraints on Oscillating Dark-Energy Parametrizations, Phys. Rev. D 98, no. 6, 063510 (2018).
- (41) S. Vagnozzi, S. Dhawan, M. Gerbino, K. Freese, A. Goobar and O. Mena, Constraints on the sum of the neutrino masses in dynamical dark energy models with are tighter than those obtained in CDM, Phys. Rev. D 98, no. 8, 083501 (2018).
- (42) V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rept. 215, 203 (1992).
- (43) C. P. Ma and E. Bertschinger, Cosmological perturbation theory in the synchronous and conformal Newtonian gauges, Astrophys. J. 455, 7 (1995).
- (44) K. A. Malik and D. Wands, Cosmological perturbations, Phys. Rept. 475, 1 (2009).
- (45) R. Adam et al. [Planck Collaboration], Planck 2015 results. I. Overview of products and scientific results, Astron. Astrophys. 594, A1 (2016).
- (46) N. Aghanim et al. [Planck Collaboration], Planck 2015 results. XI. CMB power spectra, likelihoods, and robustness of parameters, Astron. Astrophys. 594, A11 (2016).
- (47) M. Betoule et al. [SDSS Collaboration], Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples, Astron. Astrophys. 568, A22 (2014).
- (48) F. Beutler et al., The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant, Mon. Not. Roy. Astron. Soc. 416, 3017 (2011).
- (49) A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden and M. Manera, The clustering of the SDSS DR7 main Galaxy sample I. A 4 per cent distance measure at , Mon. Not. Roy. Astron. Soc. 449, no. 1, 835 (2015).
- (50) H. Gil-Marín et al., The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: BAO measurement from the LOS-dependent power spectrum of DR12 BOSS galaxies, Mon. Not. Roy. Astron. Soc. 460, no. 4, 4210 (2016).
- (51) M. Moresco et al., A 6% measurement of the Hubble parameter at : direct evidence of the epoch of cosmic re-acceleration, JCAP 1605, no. 05, 014 (2016).
- (52) A. Lewis and S. Bridle, Cosmological parameters from CMB and other data: A Monte Carlo approach, Phys. Rev. D 66, 103511 (2002).
- (53) A. Lewis, Efficient sampling of fast and slow cosmological parameters, Phys. Rev. D 87, no. 10, 103529 (2013).
- (54) A. G. Riess et al., “A 2.4% Determination of the Local Value of the Hubble Constant,” Astrophys. J. 826, 56 (2016).
- (55) A. G. Riess et al., New Parallaxes of Galactic Cepheids from Spatially Scanning the Hubble Space Telescope: Implications for the Hubble Constant, Astrophys. J. 855, 136 (2018).
- (56) S. Birrer et al., H0LiCOW - IX. Cosmographic analysis of the doubly imaged quasar SDSS 1206+4332 and a new measurement of the Hubble constant, arXiv:1809.01274 [astro-ph.CO].
- (57) C. Heymans et al., CFHTLenS: The Canada-France-Hawaii Telescope Lensing Survey, Mon. Not. Roy. Astron. Soc. 427, 146 (2012).
- (58) T. Erben et al., CFHTLenS: The Canada-France-Hawaii Telescope Lensing Survey - Imaging Data and Catalogue Products, Mon. Not. Roy. Astron. Soc. 433, 2545 (2013).
- (59) H. Hildebrandt et al., KiDS-450: Cosmological parameter constraints from tomographic weak gravitational lensing, Mon. Not. Roy. Astron. Soc. 465, 1454 (2017).
- (60) T. M. C. Abbott et al. [DES Collaboration], Dark Energy Survey Year 1 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing, Phys. Rev. D 98, no. 4, 043526 (2018).
- (61) N. Aghanim et al. [Planck Collaboration], Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 [astro-ph.CO].
- (62) P. A. R. Ade et al. [Planck Collaboration], Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594, A13 (2016).
- (63) A. Heavens, Y. Fantaye, A. Mootoovaloo, H. Eggers, Z. Hosenie, S. Kroon and E. Sellentin, Marginal Likelihoods from Monte Carlo Markov Chains, arXiv:1704.03472 [stat.CO].
- (64) A. Heavens, Y. Fantaye, E. Sellentin, H. Eggers, Z. Hosenie, S. Kroon and A. Mootoovaloo, No evidence for extensions to the standard cosmological model, Phys. Rev. Lett. 119, no. 10, 101301 (2017).
- (65) R. E. Kass and A. E. Raftery, Bayes Factors, J. Am. Statist. Assoc. 90, no.430, 773 (1995).
- (66) E. Di Valentino, A. Melchiorri and O. Mena, Can interacting dark energy solve the tension?, Phys. Rev. D 96, no. 4, 043503 (2017).
- (67) E. Di Valentino, A. Melchiorri and J. Silk, Beyond six parameters: extending CDM, Phys. Rev. D 92, no.12, 121302 (2015).
- (68) W. Yang, S. Pan and D. F. Mota, Novel approach toward the large-scale stable interacting dark-energy models and their astronomical bounds, Phys. Rev. D 96, no. 12, 123508 (2017)
- (69) W. Yang, S. Pan and J. D. Barrow, Large-scale Stability and Astronomical Constraints for Coupled Dark-Energy Models, Phys. Rev. D 97, no. 4, 043529 (2018).
- (70) E. Mörtsell and S. Dhawan, Does the Hubble constant tension call for new physics?, arXiv:1801.07260 [astro-ph.CO].
- (71) W. Yang, S. Pan, E. Di Valentino, R. C. Nunes, S. Vagnozzi and D. F. Mota,