Probing dynamics of dark energy with latest observations

TL;DR

This paper tests the standard cosmological model against recent observations, finds hints of dark energy evolution, and forecasts that future surveys could decisively detect such dynamics with high confidence.

Contribution

It introduces a combined analysis of observational data to probe dark energy dynamics and forecasts the potential of upcoming surveys to confirm these findings.

Findings

Current data mildly favor evolving dark energy.

Different observational datasets are in tension within the $ ext{Lambda}$CDM model.

Future surveys like DESI could detect dark energy evolution at 7σ confidence.

Abstract

We examine the validity of the $\Lambda$CDM model, and probe for the dynamics of dark energy using latest astronomical observations. Using the $Om(z)$ diagnosis, we find that different kinds of observational data are in tension within the $\Lambda$CDM framework. We then allow for dynamics of dark energy and investigate the constraint on dark energy parameters. We find that for two different kinds of parametrisations of the equation of state parameter $w$, a combination of current data mildly favours an evolving $w$, although the significance is not sufficient for it to be supported by the Bayesian evidence. A forecast of the DESI survey shows that the dynamics of dark energy could be detected at $7\sigma$ confidence level, and will be decisively supported by the Bayesian evidence, if the best fit model of $w$ derived from current data is the true model.

Tables

Table 1. Table 1: Constraints on dark energy parameters using current data and simulated data (numbers quoted in parenthesis) respectively.

Parametrisation I
$w_0$	$w_1$	$w_2$	$w_3$	$w_4$	$\sqrt{\left|\Delta {\chi }^2\right|}$	$\Delta \ln E$
$-1.02\pm 0.04\left(0.01\right)$	$0$	$0$	$0$	$0$	$0.4\left(0.8\right)$	$-2.3\left(-3.4\right)\pm 0.3$
$-1.08\pm 0.10\left(0.05\right)$	$0.26\pm 0.40\left(0.21\right)$	$0$	$0$	$0$	$0.7\left(5.2\right)$	$-3.9\left(6.1\right)\pm 0.3$
$-1.18\pm 0.17\left(0.08\right)$	$1.50\pm 1.75\left(0.67\right)$	$-2.34\pm 3.21\left(1.14\right)$	$0$	$0$	$1.1\left(5.4\right)$	$-7.1\left(3.1\right)\pm 0.3$
$-1.07\pm 0.17\left(0.10\right)$	$-1.42\pm 2.40\left(1.22\right)$	$12.1\pm 10.2\left(3.75\right)$	$-17.7\pm 12.6\left(3.32\right)$	$0$	$1.8\left(5.6\right)$	$-8.4\left(0.5\right)\pm 0.3$
$-1.00\pm 0.18\left(0.09\right)$	$0.38\pm 2.72\left(1.59\right)$	$-15.8\pm 21.2\left(9.29\right)$	$72.0\pm 62.3\left(20.0\right)$	$-79.6\pm 55.0\left(13.4\right)$	$2.2\left(6.0\right)$	$-8.8\left(0.0\right)\pm 0.3$
Parametrisation II
$w_0$	$w_1$	$w_2$	$w_3$	$w_4$	$\sqrt{\left|\Delta {\chi }^2\right|}$	$\Delta \ln E$
$-1.03\pm 0.04\left(0.03\right)$	$4.98\pm 2.87\left(0.61\right)$	$5.38\pm 2.43\left(0.39\right)$	$13.3\pm 6.42\left(0.40\right)$	$0$	$2.6\left(7.4\right)$	$-2.2\left(14.0\right)\pm 0.3$
$-1.03\pm 0.05\left(0.03\right)$	$4.77\pm 2.86\left(0.64\right)$	$5.61\pm 2.46\left(0.41\right)$	$13.8\pm 7.57\left(0.84\right)$	$4.90\pm 2.84\left(1.82\right)$	$2.6\left(7.5\right)$	$-2.0\left(14.2\right)\pm 0.3$

Equations

$$Om(z)\equiv \frac{[H(z)/H_0{]}^2-1}{(1+z{)}^3-1}.$$

$$w(a)=\sum _{i=0}^{N_{\mathrm{p}}}{w}_i(1-a{)}^i$$

$$w(a)=w_0+w_1(1-a{)}^{w_2}\cos \left(w_{3}a+w_4\right)$$

$$\mathbf{P}\equiv \{{\omega }_b,{\omega }_c,{\Theta }_s,\tau ,n_s,A_s,w_0,...,w_4,\mathcal{N})$$

$$Om(Planck2015)$$

$$Om(\mathrm{gBAO})$$

$$Om(\mathrm{OHD})$$

$$Om(\mathrm{Ly}\alpha \mathrm{FB})$$

$$\Delta \ln E\equiv \ln E_{\mathrm{DDE}}-\ln E_{\Lambda \mathrm{CDM}}$$

$$E\equiv \int d^n\theta P(\theta )$$

Full text

I Introduction

The accelerating expansion of the Universe revealed by supernovae type Ia (SNIa) is one of the most significant discoveries in modern cosmology RiessPerl . In the framework of general relativity, the cosmic acceleration in the late Universe is due to dark energy (DE), a yet unknown energy component contributing to about two thirds of the total energy budget of the Universe. From astronomical observations, measurements of the equation of state parameter (EoS) $w$, which is the ratio of pressure to energy density of DE, can shed light on the nature of DE as different DE models can be characterised by $w$. For example, the cosmological constant $\Lambda$, which is one of the most popular DE models, predicts that $w=-1$, while in dynamical dark energy (DDE) models including quintessence quintessence , phantom phantom , quintom quintom and so on, $w$ evolves with redshift $z$. Hence reconstructing the $w(z)$ function from observations including cosmic microwave background (CMB), SNIa and large scale structure (LSS) measurements, is an efficient way to test dark energy models.

Performing a consistency check for the $\Lambda$CDM model, which has least number of model parameters compared with DDE models in general, using observations is a common starting point for phenomenological studies of dark energy. Interestingly, recent studies show that different kinds of observational data are in tension within the framework of the $\Lambda$CDM model Zhao17 ; Font-Ribera:2013wce ; Sahni:2014ooa ; Battye:2014qga ; Aubourg:2014yra ; P15 ; Addison:2015wyg ; Bernal:2016gxb ; Sola1 ; Sola2 ; DMS ; DMLS . In particular, Zhao et al. (2017) Zhao17 quantifies the tension using the Kullback-Leibler divergence Kullback:1951va , and uses a nonparametric DDE model to successfully relieve the tension. Their analysis basically shows that the tension within $\Lambda$CDM can be interpreted as a signal of dynamics of dark energy at a $3.5\sigma$ confidence level (CL).

In this paper, we perform a complementary study to Zhao et al. (2017). We first reinvestigate the tension between different datasets using the $Om$ Om1 ; Om2 diagnosis, and then reconstruct $w(z)$ following a parametric approach. We quantify the significance of $w\neq-1$ and perform a model selection using the Bayesian evidence on current and simulated future observational data.

This paper is organised as follows. In the next section we present the method and datesets used, and in section III we present the result, followed by a section of conclusion and discussion.

II Method and Data

In this section, we present the methodology used for quantifying the tension among datasets, for performing dark energy model parameter inference and for model selection. We also describe datasets used in this work.

II.1 The $Om$ diagnosis

The quantity $Om$ is defined as follows Om1 ; Om2 ,

[TABLE]

where $H(z)$ and $H_{0}$ are the Hubble parameter measured at redshift $z$ and [math] respectively. It is a useful diagnosis of any deviation from the $\Lambda$CDM model simply because $Om(z)=\Omega_{m}$ in $\Lambda$CDM. Thus any non-constancy of $Om(z)$ signals that $w\neq-1$, if the flatness of the Universe is assumed.

Observationally, $H_{0}$ can be directly measured in the local Universe, and $H(z)$ can be estimated from CMB, baryonic acoustic oscillations (BAO) redshift surveys using either galaxies (gBAO), or Lyman-$\alpha$ forest (Ly$\alpha$FB), or from the relative age of old and passively evolving galaxies following a cosmic chronometer approach (OHD).

II.2 Parametrisations of the Universe

In this work, we consider two kinds of parametrisations of $w(a)$, where $a$ is the scale factor of the Universe111For more parametrisations of $w(a)$, see PNP ..

Parametrisation I

Polynomial expansion Ade:2015rim

[TABLE]

where $N_{\rm p}$ defines the order of the polynomial expansion. Note that $N_{\rm p}=0$ and $N_{\rm p}=1$ are the $w$CDM model, in which $w$ is a constant, and the Chevallier-Polarski-Linder (CPL) model CP ; L respectively, and including higher order terms allows more general behaviour of $w(a)$. In this work, we consider cases with $N_{\rm p}\leqslant 4$.

Parametrisation II

Oscillatory function

Although Parametrisation I allows for oscillatory behaviours of $w(a)$ in general, it requires a large number of terms in order to properly approximate a periodic oscillatory function, e.g., a cosine function. Therefore we consider another kind of parametrisation as,

[TABLE]

This is a general cosine function that allows its mean, amplitude, period and phase to be free parameters. It is similar to the functional form used in Feng:2004ff but is more general in that the $(1-a)^{w_{2}}$ term allows the amplitude to vary with the scale factor.

Our parametrisation of the Universe is thus,

[TABLE]

where $\omega_{b}$ and $\omega_{c}$ are the baryon and cold dark matter physical densities, $\Theta_{s}$ is the angular size of the sound horizon at decoupling, $\tau$ is the optical depth, $n_{s}$ and $A_{s}$ are the spectral index and the amplitude of the primordial power spectrum, and $w_{0},...,w_{4}$ denote the above-mentioned dark energy EoS parameters. We marginalize over nuisance parameters $\mathcal{N}$ such as the intrinsic SN luminosity, galaxy bias, etc.

II.3 Observational datasets used

The datasets we consider in this work include the gBAO measurements that utilize the BOSS DR12 sample at nine effective redshifts BAOZhao ; BAOWang , the Ly$\alpha$FB measurements Delubac14 , the 6dFRS 6df and SDSS main galaxy sample MGS BAO measurements, the WiggleZ galaxy power spectra wigglez , the recent estimate of the Hubble constant $H_{0}$ obtained from local measurements of Cepheids R16 ($H_{0}$), the recent OHD measurements of $H(z)$ hz16 , the JLA sample of SNIa Betoule:2014frx , the weak lensing shear angular power spectra from CFHTLenS WL and the Planck 2015 CMB temperature and polarisation angular power spectra P15 .

For the purpose of forecast, we simulate future gBAO data assuming a DESI 222http://desi.lbl.gov/ sensitivity following DESI , and also consider a future space-based supernova mission described in EuclidSN .

II.4 Parameter estimation and model selection

We use a modified version of CAMB CAMB to calculate observables, and include dark energy perturbations following the approach developed in DEP . We perform a Markov Chain Monte Carlo (MCMC) global fitting of parameters listed in Eq (4) to a combination of datasets described in Sec. II.3 using a modified version of CosmoMC Lewis:2002ah , and use the PolyChord polychord plug-in of CosmoMC to compute the Bayesian evidence for the model selection.

III Result

We present our results in Table 1 and in Figs 1-3.

The quantity $Om(z)$ is estimated using $H(z)$ measurements from Planck 2015, gBAO, OHD and Ly$\alpha$FB respectively, with the recent $H_{0}$ measurement presented in R16 . To check the constancy of $Om(z)$ using each individual kind of datasets, and the consistency between different kinds of data, we fit constants to the $Om(z)$ measurements from Planck 2015, gBAO and OHD separately, and show the 68% CL constraints in cyan, blue and grey horizontal bands respectively in Fig 1. Specifically, we obtain,

[TABLE]

It is true that neither the Planck 2015, gBAO nor OHD dataset shows a significant deviation from a constant $Om$ given the level of uncertainty, however, the derived $Om$’s from Planck 2015, gBAO and OHD are different at larger than $2\sigma$ CL. Furthermore, the $Om$ values derived here are all smaller than $\Omega_{m}$ derived from Planck 2015 alone in the $\Lambda$CDM model P15 , which is $\Omega_{m}=0.315\pm 0.013$. This to some extent is due to the fact that the $H_{0}$ value used here, which is $73.24\pm 1.74$ km ${\rm s}^{-1}{\rm Mpc}^{-1}$, is significantly larger than that derived from Planck 2015, which is $67.31\pm 0.96$ km ${\rm s}^{-1}{\rm Mpc}^{-1}$. All these discrepancy among datasets suggests that the $\Lambda$CDM model may need to be extended.

For more general DE models parametrised by Eqs (2) and (3), we derive constraints on model parameters, which are shown in Table 1. For the polynomial expansion case, we increasingly add higher order terms to the $w$CDM model in the global fitting. We find that the $\chi^{2}$ can be reduced by $4.8$ at most for the $N_{\rm p}=4$ model. For the purpose of model selection, we also evaluate the logarithmic Bayesian factor,

[TABLE]

where

[TABLE]

denotes the Bayesian evidence, which is an integral of the probability distribution function of $n$-dimensional parameters $\theta$. We find that $\Delta{\rm ln}E$ is negative for all cases, meaning that neither of these DDE models is favoured over the $\Lambda$CDM model. For the $N_{\rm p}=4$ case, in which $w(z)$ is parametrised with five free parameters, is found to be not equal to $-1$ at $2.2\sigma$ CL, and the Bayesian factor is as low as $\Delta{\rm ln}E=-8.8\pm 0.3$, which strongly indicates current data do not support extending $\Lambda$ in this parametrisation.

For parametrisation II, we show results with and without the phase $w_{4}$ fixed, and find that whether $w_{4}$ varies or not does not change the result: $\chi^{2}$ is reduced by $6.8$ (a $2.6\sigma$ signal of $w\neq-1$) by four additional parameters with a Bayesian factor $\Delta{\rm ln}E=-2.2\pm 0.3$. Admittedly, although this model is also not supported by the Bayesian evidence, it is much less disfavoured than the $N_{\rm p}=4$ model in parametrisation I, and it fits to the data better.

In Figs 2 and 3, we reconstruct $w(z)$ using constraints on DE parameters we obtained. As shown, the best fit $w(z)$ models with all five DE parameters varied, which are shown in the far right panel of Fig 2, and in the lower panel in Fig 3, crosses $-1$ during evolution, and exhibits certain level of oscillations with respect to redshift $z$, which is consistent with the prediction of the model of oscillating quintom Feng:2004ff . We compare this result to the nonparametric reconstruction presented in Zhao17 . As shown, our result is consistent with that in Zhao et al. (2017) within $1\sigma$ CL.

To reinvestigate the tension among various datasets in DDE models, we over-plot $Om$ for the best fit DDE model as parametrised by Eq (3) (black solid). As shown, it is consistent with all datasets, signalling a release of tension among datasets.

To assess whether the best fit $w$ model found in this work will be supported by future observations, we take the best fit $w$ model as a fiducial model, create mock BAO and supernovae data assuming a DESI DESI and a future space-based supernova mission EuclidSN combined with Planck 2015 data, and repeat our analysis. We find that for parametrisation I, models of $N_{\rm p}=1,2$ will be supported by Bayesian evidences, with a signal of $w\neq-1$ at $5\sigma$ CL. Although the $N_{\rm p}=3,4$ models fit data better, they are not much preferred to the $\Lambda$CDM model even for the future data. On the other hand, future data support the oscillation model much more significantly. Namely, those models will be detected at more than $7\sigma$ CL with a large Bayesian factor of $\Delta{\rm ln}E=14\pm 0.3$.

IV Conclusion and Discussions

We revisit the consistency among various kinds of recent observations using the $Om$ diagnosis, and confirm that the tension exists among Planck 2015, gBAO, OHD, Ly$\alpha$FB and the new $H_{0}$ measurement in the $\Lambda$CDM model.

We therefore allow the dynamics of dark energy and perform parametric reconstruction of $w(z)$ with two kinds of parametrisations using a combination of current datasets, and using the simulated future data. We find that an oscillatory $w(z)$ across $-1$ during the evolution is mildly favoured by a combination of current observations at a confidence level of $2.6\sigma$ based on the improvement in $\chi^{2}$. This model can well relieve the tension among datasets. It is true that this is not sufficient for it to be supported by the Bayesian evidence, however, for future galaxy surveys with a sensitivity similar to DESI and space-based supernova surveys, the best-fit model derived in this work will be detected at a confidence level of $7\sigma$, and will be decisively supported by the Bayesian evidence.

Acknowledgements.

The authors are supported by NSFC Grant No. 11673025, and by a Key International Collaboration Grant from Chinese Academy of Sciences. GBZ is also supported by a Royal Society Newton Advanced Fellowship.