Gauging the cosmic acceleration with recent type Ia supernovae data sets

Distance measurements to type Ia supernovae (SNe Ia) at high redshifts led to the astonishing discovery of the cosmic acceleration in 1998 Perlmutter:1998np ; Riess:1998cb . Within a general relativistic based description of gravity, with the background expansion equipped with an FLRW (Friedmann-Lemaitre-Robertson-Walker) metric it is necessary the inclusion of some unknown form of energy density (dark energy) responsible for driving such dynamics. The simplest explanation relies on the Einstein’s cosmological constant $\Lambda$, which is equivalent at the background level to a fluid with an equation of state parameter $w_{de}=p_{de}/\rho_{de}=-1$. However, there are also alternative descriptions for the accelerated expansion of the universe as for example modifications of Einstein’s gravity, backreaction mechanisms or viscous effects among many others. In all the above approaches (including the standard one) the evidence for acceleration appears from fitting data and realizing that the parameter space of such models leading to the accelerated expansion is the statistically favored one.

Another way to probe the acceleration is to perform kinematical tests with data where no assumptions about the gravitational sector or the material content of the universe are made. Within this class of tests one can cite parameterizations of the deceleration parameter $q(z)$ Elgaroy:2006tp ; Shafieloo:2009ti , the scale factor $a(t)$ Wang:2005yaa or the expansion rate $H(z)$ John:2005bz ; Nair:2012bs , as well as cosmographical tests employing a series expansion in the redshift $z$ review_cosmography ; Luongo:2015zgq ; cosm16 , however see busti2015 for limitations of such an approach.

In this work we revisit another model-independent estimator for accelerated expansion described in great detail by Schwarz and Seikel Seikel:2007pk ; Seikel:2008ms . The idea is based on falsifying the null hypothesis that the universe never experienced an accelerated expansion. Although such estimator does not provide the moment at which the transition to acceleration takes place (such criticism has been discussed in Ref. Mortsell:2008yu ) the analyses with the GOLD Riess:2006fw , ESSENCEWoodVasey:2007jb and UNION Kowalski:2008ez SNe Ia data sets have provided strong evidence in favor of acceleration Seikel:2007pk ; Seikel:2008ms .

Our aim in this work is twofold: First, we check the robustness of the estimator by testing it against mock catalogues of known cosmological expansions (e.g., $\Lambda$CDM, Einstein-de Sitter, Milne model and 3 other models which are not accelerated today but were accelerated in the past). This analysis allows us to understand the estimator and to gauge the level of accuracy expected for the statistical evidence (given in $\sigma$ levels) obtained with actual data sets. Second, we update the results of such estimator for recent SNe Ia data sets like the UNION2.1 Suzuki:2011hu and the Joint Light-Curve analysis (JLA) Betoule:2014frx samples.

Concerning the confrontation of a model independent estimator for acceleration with the recent JLA sample, it is worth noting that recently Ref. Dam:2017xqs has argued that the timescape model (with insignificant acceleration rate) fits the JLA sample with a likelihood that is statistically indistinguishable from the standard cosmological model. Even more intriguing, Ref.Nielsen:2015pga has pointed out that the JLA sample provides only marginal evidence for acceleration. However, Ref. Rubin:2016iqe has criticized the former result by arguing that the statistical model used in Ref.Nielsen:2015pga is deficient to account for changes in the observed SN light-curve parameter distributions with redshift. According to Ref. Rubin:2016iqe evidence for acceleration using SNe Ia only is $11.2\sigma$ in a flat universe. Thus, in our work we are also able to revisit this discussion by analyzing the evidence for acceleration in the JLA sample from a different perspective.

In the next section we review the estimator using it in section 3 with the UNION2.1 Suzuki:2011hu and in section 4 with the Joint-Lightcurve-Analysis (JLA) Betoule:2014frx samples. We conclude in the final section.

We review in this section the estimator developed in Ref. Seikel:2007pk by Seikel & Schwarz. The main idea here is to provide a quantitative measure of the accelerated dynamics of the universe in a model-independent way. The construction of such estimator is based on the definition of the deceleration parameter

$$q(z)=\frac{H^{\prime}(z)}{H(z)}(1+z)-1$$

(1)

where the prime denotes derivative with respect to the redshift. An accelerated background expansion at a certain redshift is indicated if $q(z)<0$.

In the the standard cosmology one expects that the dynamical evolution of the universe underwent a transition from the decelerated phase to the accelerated one at some transition redshift $z_{t}$. Deep in the matter dominated epoch the deceleration parameter assumes the value $\sim 0.5$ (similarly to the Einstein-de Sitter universe). As the effect of dark energy (or modified gravity) becomes relevant for the expansion in comparison to the matter density, then $q(z)$ turns to negative values. For the standard cosmology $q(z)$ is seen in the black line of Fig. 1.

From Eq. (1) one can obtain the expansion rate $H(z)$ as a function of the deceleration parameter according to the integral equation

$${\rm ln}\,\frac{H(z)}{H_{0}}=\int^{z}_{0}\frac{1+q(\tilde{z})}{1+\tilde{z}}d\tilde{z},$$

(2)

where $H_{0}$ is the Hubble constant.

The null hypothesis proposed in Seikel:2007pk is that the universe has never expanded in an accelerated way i.e., $q(z)>0\,\forall\,z$. Hence, the direct consequence of applying this inequality to the above equation is

$${\rm ln}\,\frac{H(z)}{H_{0}}\geq\int^{z}_{0}\frac{1}{1+\tilde{z}}d\tilde{z}={\rm ln}\,(1+z),$$

(3)

which is equivalent to

$$H(z)\geq H_{0}(1+z).$$

(4)

Therefore, from the above result one can infer whether or not the universe experienced any event of accelerated expansion from direct measurements of the Hubble rate. This can be achieved for example using the so called cosmic chronometers, which are galaxies supposed to passively evolve in a certain redshift range $\Delta z$. Then, estimation of the stellar ages ($\Delta t$) in such objects lead to an estimation of $H(z)=-1/(1+z)\,dz/dt\sim-1/(1+z)\,\Delta z/\Delta t$. However the available number of $H(z)$ data is limited to a few dozens and the quality (in terms of the associated errors) is low.

In order to assess information on the background expansion of the universe the SNe Ia data is the most reliable observational tool. The quantity and quality of SNe Ia data have substantially increased in the last years and ongoing surveys will drastically improve SNe Ia samples in the near future. The crucial definition in supernovae cosmology is the luminosity distance. In a flat FLRW universe it reads

$$d_{L}(z)=(1+z)\int^{z}_{0}\frac{d\tilde{z}}{H(\tilde{z})}.$$

(5)

Now, in order to apply the inequality (4) in the context of supernovae data the definition $d_{L}$ turns into

$$d_{L}\leq(1+z)\frac{1}{H_{0}}\int^{z}_{0}\frac{d\tilde{z}}{1+\tilde{z}}=(1+z)\frac{1}{H_{0}}\ln(1+z).$$

(6)

The luminosity distance calculated in some theoretical model under investigation is compared to the observed quantities via the definition of the observed distance modulus

$$\mu=m-M=5\log(d_{L}/{\rm Mpc})+25,$$

(7)

where $M$ and $m$ are the absolute and apparent magnitudes, respectively.

For each supernova $i$ in the sample we can define the quantity

	$\displaystyle\Delta\mu_{obs}(z_{i})$	$\displaystyle=$	$\displaystyle\mu_{obs}(z_{i})-\mu(q=0)$
		$\displaystyle=$	$\displaystyle\mu_{obs}(z_{i})-5{\rm log}\left[\frac{1}{H_{0}}(1+z_{i})\ln(1+z_{i})\right]-25,$

which is the difference between its observed distance modulus $\mu_{obs}(z_{i})$ and the distance modulus of a universe with constant deceleration parameter $q=0$ at the redshift $z_{i}$.

The null hypothesis behind the estimator of Ref. Seikel:2007pk is that the universe never expanded in an accelerated way which corresponds to $\Delta\mu_{obs}{\leq}0$ for each supernova. Oppositely, positive $\Delta\mu_{\rm obs}$ values indicate acceleration.

The face value of $\Delta\mu_{obs}$ is of limited interest if its associated error $\sigma_{i}$ is not included. For each sample used in this work (UNION2.1 and JLA) we will obtain the error in the distance moduli of each supernovae $i$ ($\sigma_{i}$) from the available covariance matrix $C$ of the data.

One way to apply the estimator is via the the so called “single SNe Ia analysis” which corresponds to compute the quantity $\Delta\mu_{obs}$ for each SN individually. Although the single SN analysis presents some interesting results, it is however of limited statistical interest. A more reliable analysis of the estimator $\Delta\mu_{obs}$ is obtained with the so called “averaged SNe Ia analysis”. In this analysis we group a number N of SNe defining the mean value

$$\overline{\Delta\mu}=\frac{\sum_{i=1}^{N}g_{i}\,\Delta\mu_{obs}(z_{i})}{\sum_{i=1}^{N}g_{i}},$$

(9)

where the factor $g_{i}=1/\sigma^{2}_{i}$ enables data points with smaller errors contribute more to the average.

The standard deviation of the mean value is defined by

$$\sigma_{\overline{\Delta\mu}}=\left[\frac{\sum^{N}_{i=1}g_{i}\left[\Delta\mu_{obs}\left(z_{i}\right)-\overline{\Delta\mu}\right]^{2}}{(N-1)\sum^{N}_{i=1}g_{i}}\right]^{1/2}.$$

(10)

For example, using (10) , Ref. Seikel:2008ms points out averaged statistical evidences for acceleration such that $4.3\sigma$ for the GOLD sample and $7.2\sigma$ using the 2008 UNION sample, both assuming a flat expansion.

The Joint Light-Curve analysis (JLA) Betoule:2014frx totals 740 SNe Ia including several low-redshift samples $(z<0.1)$, the SDSS-II data $(0.05<z<0.4)$, three years data from SNLS $(0.2<z<1)$ and a few high redshift from the Hubble Space Telescope (HST).

The free parameters fitted in the observed distance modulus for the catalog we use are $\alpha=0.141$, $\beta=3.101$, $M_{B}=-19.05$ and $\delta=-0.070$ which have been fitted with the $\Lambda$CDM cosmology.

With such JLA sample we perform the single SNe Ia analysis as one can see the results in Fig. 4 and Table 6. Clearly, this sample presents a smaller dispersion than the UNION2.1 but still with a similar signal-to-noise ratio.

The averaged analysis per bin in the JLA sample is presented in Table (7). In order to study the effects of nearby SNe Ia we both exclude again all SNe Ia at redshifts $z<0.2$ as well as the SNe belonging to the $low-z$ sub-catalog.

Rather than using the traditional approach of fitting SNe Ia data with the $\Lambda$CDM model to assess the best-fit values of the cosmological parameters we have studied the late-time cosmic acceleration with a model-independent estimator $\Delta\mu_{obs}$.

The essence of this estimator is to falsify the null hypothesis that the universe never expanded in an accelerated way. From our analysis with mock catalogs in section 3.1 we have found however that the estimator actually provides an averaged balance between the accelerated and decelerated periods. Although the models based on the equation of state parameter (12) can be seen as unrealistic expansions, they serve as counterexamples to show that if data has such untypical distribution the estimator would fail in providing evidence for acceleration in cosmologies that experienced an accelerated epoch. This is in fact related to the fact that $\Delta\mu$ can not be mapped into the deceleration parameter $q(z)$. Therefore, the message here is that one should carefully use this estimator.

Following the spirit of Ref. Seikel:2007pk ; Seikel:2008ms and assuming that the estimator can be used for a $\Lambda$CDM-like distribution of data as provided by the UNION2.1 and JLA samples, we have also updated the evidence for acceleration obtained from recent catalogs. For the JLA data set we have found robust evidence (see Table 7) favoring acceleration in a flat FLRW expansion.

It is also evident the strong dependence of the estimator on $H_{0}$. The larger the $H_{0}$ value adopted for the estimator, stronger is the evidence favoring acceleration. All reasonable values for $H_{0}$ lead to positive statistical confidence favoring acceleration.

However, it is worth noting that the UNION2.1 and JLA sample used here with fixed light curve parameters $\alpha,\beta$, $M_{B}$ and $\delta$ are not actually model-independent since the $\Lambda$CDM model has been adopted in the data fitting. Then, unless the light curve parameters are obtained in a pure astrophysical manner (in the sense their values do not depend on the cosmology adopted) this analysis also can not be regarded as a model-independent one. Attempts to do that by using $H(z)$ data from cosmic chronometers are possible javier2016 , although systematics regarding the stellar population synthesis model are not negligible busti2014 . The estimator can be adapted to the $H(z)$ data via the inequality 4. We also checked with the 36 data points for $H(z)$ compiled in Ref. Yu:2017iju that the evidence favoring acceleration becomes $12.1\,\sigma\,(H_{0}=68.31),17.8\,\sigma\,(H_{0}=70.00)$ and $23.3\,\sigma\,(H_{0}=73.24)$. Again, though this result clearly shows how the estimator has a strong dependence on $H_{0}$, there is no doubt about the cosmic acceleration even for lower $H_{0}$ values.

A more reliable test would verify acceleration independently on the light curve parameters or properly taken into account them. In fact, the light-curve parameters could be even non constant for all SNe. Recent investigations suggest a non trivial dependence of the of the stretch-luminosity parameter $\alpha$ and the color-luminosity parameter $\beta$ on the redshift Li:2016dqg or with respect to the host galaxy morphology Henne:2016mkt .

A future perspective for our work is the development of a new estimator for assuring cosmic acceleration in a full model-independent way.

HV and SG thank CNPq and FAPES for partial support. VCB is supported by FAPESP/CAPES agreement under grant number 2014/21098-1 and FAPESP under grant number 2016/17271-5.