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A&A
Volume 600, April 2017
Article Number L1
Number of page(s) 5
Section Letters
DOI https://doi.org/10.1051/0004-6361/201730469
Published online 17 March 2017

© ESO, 2017

1. Introduction

The very first evidence for an accelerated expansion of the Universe was obtained using the supernova Ia (SN Ia) observations in Riess et al. (1998) and Perlmutter et al. (1999), which has been further confirmed with the most recent supernova data (Betoule et al. 2014). Other low-redshift data such as the observational Hubble parameter (OHD, Jimenez & Loeb 2002), baryon acoustic oscillations (BAO, Eisenstein et al. 2005) also support an accelerating Universe. As an independent observation, the cosmic microwave background (CMB) radiation has agreed excellently with these results and has provided the most stringent constraints on the cosmological models (Ade et al. 2016). These observations have established the ΛCDM as the concordance model of cosmology, and the late-time acceleration has been a widely accepted phenomenon.

However, Nielsen et al. (2016) have used a modified statistical model for the analysis of the supernova JLA dataset (Betoule et al. 2014) and claimed that the evidence for the acceleration is marginal (≲ 3σ). The modification was made by assuming an intrinsic variation in the SN absolute magnitude and in the light curve (colour and stretch) corrections, which were modelled as Gaussian. More recently, Rubin & Hayden (2016) have strongly criticised this approach as incomplete and suggested using redshift dependent ad hoc functions for these corrections, presenting the evidence for acceleration to be ~ 4.2σ. In this paper we want to show that even with the less flexible modelling of Nielsen et al. (2016) the evidence for acceleration is very strong. We also extend the joint analysis done in Luković et al. (2016) with the inclusion of gamma-ray bursts (GRB) dataset (Wei 2010).

The marginal evidence for an accelerating universe quoted in Nielsen et al. (2016) implies a scenario with very low dark matter and dark energy densities. As this scenario converges towards the Milne model, it has been complemented with the Rh = ct cosmology (Melia & Shevchuk 2012; Melia 2012b,a), which features a non-accelerating linear expansion of the universe. This model essentially advocates that the Hubble sphere is same as the particle horizon of the universe (Bikwa et al. 2012). Also, the Rh = ct model has often been regarded as a Milne universe (Mitra 2014; Bilicki & Seikel 2012) and several physical issues against this interpretation have been raised (Lewis 2013; Lewis & van Oirschot 2012). In fact, this model preserves the linear expansion by imposing the constraint on the total equation of state (EoS) (ρtot + 3ptot = 0), without requiring ρtot = ptot = 0. It is interesting to note that this model also coincides with the linear coasting models that are discussed in the context of modified gravity (Gehlaut et al. 2003; Dev et al. 2001, 2002). In a recent work, Kumar (2016) presented a linear coasting model in f(R) gravity, which is functionally equivalent to the Rh = ct model. A linear expansion model can be generalised to a power-law model with an exponent n, which was brought up as an alternative to the standard model as it does not have the flatness and horizon problems (Sethi et al. 2005). In addition, it has been shown in Dolgov (1997) that classical fields coupling to spacetime curvature can give rise to a back-reaction from singularities, which can change the nature of expansion from exponential to power-law.

The Rh = ct model has been tested time and again, and contradictory conclusions have been presented. Bilicki & Seikel (2012) have pointed out some of the problems in this model using the cosmic chronometer data from Moresco et al. (2012b,a) and radial BAO measurements, along with several model-independent diagnostics, to show that Rh = ct is not a viable model. In Melia (2015) several claims against Rh = ct were refuted, and a list of works favouring Rh = ct over ΛCDM was compiled in Melia (2017). Power-law cosmologies with n ≥ 1 have been explored against data in several works such as Gehlaut et al. (2003), Dev et al. (2008), Sethi et al. (2005), Zhu et al. (2008), Shafer (2015), Rani et al. (2015), Dolgov et al. (2014), finding n ~ 1.5 consistently. In a more recent work (Shafer 2015), power-law and Rh = ct models were tested with SN Ia and BAO datasets and were found to be highly disfavoured against ΛCDM. As most of these works have used older data for their analyses, this is a good occasion to revise the constraints using more recent data and hence statistically verify the viability of these models against ΛCDM.

The paper is structured as follows. A brief introduction to the models is given in Sect. 2. We describe the data and method used for our joint analysis in Sect. 3. Our results and discussion are given in Sect. 4.

2. Models

In this section we briefly describe the standard ΛCDM model, power-law and Rh = ct cosmologies that we tested to asses the late-time acceleration. The dominant components of the ΛCDM model at late times are cold dark matter (CDM), treated as dust, and dark energy (DE) fluid with an EoS parameter w = −1. The corresponding Friedmann equation is given by (1)where H0 is the current expansion rate, while Ωm, ΩΛ, and Ωk are the dimensionless density parameters. For the flat ΛCDM model, Ωm = 1−ΩΛ. We tested the extensions of ΛCDM model, namely the kΛCDM model, with the constraint Ωm = 1−ΩΛ−Ωk, and the flat wCDM model with w as a free parameter. The second Friedmann equation, , gives us insight into the conditions that need to be satisfied to assess the dynamics of the expansion rate. The criteria for acceleration can be derived as Ωm ≤ ΩΛ/ 2 for kΛCDM and w ≤ −1/(3ΩΛ) for wCDM. We can asses the evidence for acceleration by estimating the confidence level with which the criteria are satisfied.

In a flat power-law cosmological model the scale factor evolves in time as a(t) ∝ tn, with the Hubble equation H(z) = H0(1 + z)1 /n. Here, n> 1 implies an accelerated scenario. Although motivated physically with a total EoS parameter wtot = −1/3, the flat Rh = ct model coincides with the power-law model for n = 1. It is worthwhile noting that Eq. (1) reduces to the functional form of a power-law model for selected parameter values: Ωm = 0, ΩΛ = 1 and (2)The luminosity distance for all these models with their corresponding H(z) is written as (3)The theoretical distance modulus is defined as μth = 5log  [DL(Mpc)] + 25. The angular diameter distance is DA(z) = DL(z)/(1 + z)2, which is used in the modelling of BAO data.

3. Data and method

We tested the models described in the Sect. 2 against data in the redshift range 0 <z ≲ 8. We used the observables SN Ia, BAO, OHD, and GRB, which are uncorrelated. We performed a joint analysis using all the datasets together by defining a combined likelihood function. We kept the description of the data to a minimum as we refer to Luković et al. (2016) for most of it.

We used the JLA dataset (Betoule et al. 2014) consisting of 740 SN, which provides an empirical correction to the absolute magnitude, (4)In the statistical method we implemented (Nielsen et al. 2016), the stretch s, colour c corrections, and the absolute magnitude MB are all considered as random Gaussian variables without any redshift dependence. This assumption does not account for the selection effects in s and c corrections. The JLA dataset has been corrected for the selection bias only in the apparent magnitude (Betoule et al. 2014), which is why the correction in s and c have to be explicitly included when they are modelled as distributions. As described in the introduction, we used the less flexible modelling of Nielsen et al. (2016) to show that even in this case the evidence for acceleration is very strong. Different methods for treating the selection bias in the SN data and their shortcomings have been discussed in Kessler & Scolnic (2017). It is of high importance to study these effects, which we will address in a forthcoming paper. The SN Ia likelihood ℒSN used here is described in Nielsen et al. (2016), Shariff et al. (2016), Luković et al. (2016).

The BAO data are available for the compound observable DV defined in Eisenstein et al. (2005), (5)It is important to note that the observable DV is usually presented as a ratio with rd, the sound horizon at the drag epoch. For the purpose of the model selection, we fit rd as a free parameter instead of using the standard fit function for the drag epoch zd (Eisenstein et al. 1998), as it is not very straightforward to use it for the power-law cosmologies. A similar approach was also implemented in Shafer (2015). Hence, the parameters H0 and rd are now degenerate, and BAO data by itself are only able to constrain the combination rd × H0 and Ωm. To avoid correlations among different BAO data points, we used only six measurements taken from Beutler et al. (2011), Anderson et al. (2014), Ross et al. (2015), Delubac et al. (2015), Font-Ribera et al. (2014), also summarised in Table 1 of Luković et al. (2016). A simple likelihood for the uncorrelated data was then implemented as (6)The measurements of the expansion rate were estimated using the differential age (DA) method suggested in Jimenez & Loeb (2002), which considers pairs of passively evolving red galaxies at similar redshifts to obtain dz/ dt. We used a compilation of 30 uncorrelated DA points taken from Simon et al. (2005), Stern et al. (2010), Moresco et al. (2012a), Moresco et al. (2016), Moresco (2015), Zhang et al. (2014) obtained using BC03 models. We implemented a simple likelihood function assuming all the data are uncorrelated. (7)Finally, we used the GRB dataset comprising of 109 observations compiled with the well-known Amati relation (Amati et al. 2002, 2008, 2009). The dataset has 50 GRBs at z< 1.4 and 59 GRBs at z> 1.4, in a total range of 0.1 <z< 8.1. The dataset is given in Tables 1 and 2 of Wei (2010). The distance modulus μGRB and the corresponding standard deviation can be defined as We adopted σsys = 0.7571, following the model-independent calibration in Feng & Li (2016). The likelihood for the GRB is defined as (10)The joint likelihood for these four independent observables is given as ℒtot = ℒSNℒOHDℒBAOℒGRB. We used the two most common criteria for a model comparison in cosmology, namely the Akaike information criteria (AIC; Akaike 1974) and the Bayesian information criteria (BIC; Schwarz et al. 1978). The AIC and BIC values for large number of measurements are defined as where Np and Ndata are the number of parameters and data points. The Δ(AIC) = AIC−AICref criterion takes into account the number of parameters to estimate the amount of information lost in one model when compared to a reference model, in our case, ΛCDM. We define the Δ(BIC) as similar to Δ(AIC). A negative value of the Δ(AIC) or Δ(BIC) indicates that the model in comparison performs better than the reference model.

4. Results and discussion

In this section we present the results obtained from our joint analysis for the models and data given in the earlier sections. We first present our assessment for the current accelerated state of the Universe and then comment on the model comparison using the AIC and BIC statistics.

The SN Ia Hubble diagram was claimed to be consistent with a uniform rate of expansion in Nielsen et al. (2016) as the analysis in kΛCDM model differs from the non-accelerating criterion by only ≲ 3σ. We reproduce this result and agree with this statement (see top panel of Fig. A.1). However, there is a strong prejudice for a flat Universe from the CMB data (Ade et al. 2016), and hence it is important to analyse SN Ia and other cosmological data in the context of a flat wCDM model. We find that the evidence for acceleration in the Ωm−w plane is much more significant (≥ 4.56σ) than the marginal (≥ 2.88σ) found in the Ωm−ΩΛ plane (see Fig. A.1). The claimed marginal evidence for acceleration that corresponds to a very low matter density becomes more significant in both kΛCDM and wCDM models when more physical values of Ωm are considered. In Fig. A.1 contours for the best-fit regions of the supernova dataset and our joint analysis are shown. The joint analysis improves the evidence for acceleration in the kΛCDM model to 4.98σ and in wCDM model to 5.38σ.

The parameter space Ωm−w allows us to explore points that correspond to the functional forms of Rh = ct and power-law models for specific values of the parameters in wCDM. The point (Ωm,w) = (0,−1/3), shown in red in the bottom panel of Fig. A.1, phenomenologically reproduces the expansion law of the Rh = ct model (cf. Eq. (1)). Similarly, the point (Ωm,w) = (0,−0.38) corresponds to the best-fit value of n = 1.08 in the power-law model from our joint analysis (cf. Table 1 and Eq. (2)). Using SN data alone, the best-fit model for the power-law cosmology is found to be n = 1.28 (corresponding to w = −0.48), implying an accelerating Universe (cf. Sect. 2). In the Ωm−w plane this point lies within the 2σ SN confidence region, in contrast to the Rh = ct at 4.56σ. However, when the joint analysis is performed, these points are at 5σ and 5.5σ for the power-law and Rh = ct models, respectively.

The best-fit parameter values for the models considered in this work are presented in Table 1. The values for the fitted rd parameter are consistent with the estimates (see Ade et al. 2016) using the fit function for the drag epoch from Eisenstein et al. (1998). In our analysis, the best-fit value for the index n is driven towards unity. This result is quite different form the previous n estimates (Dolgov et al. 2014; Shafer 2015; Rani et al. 2015), which consistently suggest n ~ 1.5. The modification in the statistical method for the SN analysis enables this change. While the value we find for BAO data alone (n = 0.94) is consistent with the value given by Shafer (2015) (n = 0.93), for BAO+SN data we find n = 1.11 in contrast to his n = 1.52. Our best-fit value for the joint analysis (n = 1.08 ± 0.02) is now consistent with n = 1.14 ± 0.05 found by Zhu et al. (2008) using X-ray cluster data.

It is clear that the H0 estimates for the power-law and Rh = ct models are very different from the direct estimate in Riess et al. (2016), which is a well-established problem for ΛCDM (Luković et al. 2016; Bernal et al. 2016). In this work, the joint analysis provides H0 = 66.4 ± 1.8 km s-1 Mpc-1 for the ΛCDM scenario. We note that this value is consistent with our previous estimate (Luković et al. 2016), but with a higher error due to the difference in the BAO analysis (see Sect. 3). In any case, this value still remains different from the direct estimate at 2.7σ.

Table 1

Best-fit parameters for the joint analysis of SN+BAO+ DA+GRB datasets, with 1σ errors.

We stress that the Milne model with Ωm = ΩΛ = 0 and Ωk = 1 does not correspond to the flat Rh = ct model. In fact, these two models share the same Hubble equation and EoS (ρ + 3p = 0), but do not have the same DL as the negative curvature in the Milne model corresponds to DL ∝ (1 + z)sinh(log (1 + z)) with Ωk = 1, whereas in the Rh = ct model DL ∝ (1 + z)log (1 + z). In the SN Ia Hubble diagram it is difficult to see any significant difference among the ΛCDM, Milne, and Rh = ct model predictions, but their performance can be more effectively tested with the information criteria.

Table 2

Δ(AIC) and Δ(BIC) comparisons for models with ΛCDM as the reference.

The analysis of SN Union 2.1 data described by Melia (2012a) calls for a non-accelerating scenario, as the Rh = ct model was claimed to perform on par with ΛCDM. This point was also taken by Nielsen et al. (2016), who analysed the JLA dataset with their improved statistical method. In our work, using the same technique for analysing the JLA data, we find that Rh = ct performs poorly when compared to ΛCDM with Δ(AIC)SN ~ 22, while a power-law model is performing as well as ΛCDM with n ~ 1.28 from SN data. Our results are consistent with the previous work by Shafer (2015). The values of Δ(AIC)SN obtained from the SN data alone are shown in Table 2.

We note that the Milne model was claimed to perform marginally worse than ΛCDM using the SN data alone (Nielsen et al. 2016). In our work, we find for this model Δ(AIC)SN = 9.8 high enough to reject a model. The Milne model fails to keep up when the high redshift GRB (Δ(AIC)GRB ~ 20) and BAO (Δ(AIC)BAO ~ 38) data are used, yielding a total Δ(AIC)Joint ~ 66.4 (see Table 2). The AIC statistics disfavours the power-law models by Δ((AIC)Joint) ~ 28 and the Rh = ct model by Δ((AIC)Joint) ~ 30 in comparison to the ΛCDM model. Our joint analysis shows that all the three models (Milne, power-law and Rh = ct) are strongly disfavoured with respect to ΛCDM (Table 2).

5. Conclusions

Contrary to the claim by Nielsen et al. (2016), we find that the SN data alone indicate an accelerating Universe at more than 4.56σ confidence level. This evidence becomes even stronger (5.38σ) when we perform the joint analysis combining SN, BAO, OHD, and GRB data. The non-accelerating Rh = ct model at once fails to explain these data, resulting in Δ(AIC)Joint ~ 30 with respect to ΛCDM. Although the power-law model performs slightly better that the Rh = ct model, it similarly fails with a Δ(AIC)Joint ~ 28. Our analysis shows that the possibility of models with a uniform expansion rate is excluded given the current low-redshift data. In conclusion, on one hand we re-assert that the current expansion of our Universe is accelerated, and on the other hand, that ΛCDM still constitutes the base line for a concordance model in cosmology.

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Appendix A: Additional figure

thumbnail Fig. A.1

Grey and violet confidence regions are obtained from the supernova alone and the joint analysis, respectively. Top panel: 1, 2, and 3σ confidence regions in the Ωm−ΩΛ parameter space for kΛCDM. The solid line identifies the flat ΛCDM models. The dashed line gives the no-acceleration criterion: Ωm = ΩΛ/ 2. The black point is where the 4.98σ confidence contour of the joint likelihood touches the dashed line. Bottom panel: 1, 2, and 3σ confidence regions in the Ωm−w parameter space for flat wCDM. As in the top panel, the dashed curve identifies the no-acceleration criterion: w = −1/(3(1−Ωm)). Likewise, the black point is where the 5.38σ confidence contour of the joint likelihood touches the dashed line.

All Tables

Table 1

Best-fit parameters for the joint analysis of SN+BAO+ DA+GRB datasets, with 1σ errors.

In the text
Table 2

Δ(AIC) and Δ(BIC) comparisons for models with ΛCDM as the reference.

In the text

All Figures

thumbnail Fig. A.1

Grey and violet confidence regions are obtained from the supernova alone and the joint analysis, respectively. Top panel: 1, 2, and 3σ confidence regions in the Ωm−ΩΛ parameter space for kΛCDM. The solid line identifies the flat ΛCDM models. The dashed line gives the no-acceleration criterion: Ωm = ΩΛ/ 2. The black point is where the 4.98σ confidence contour of the joint likelihood touches the dashed line. Bottom panel: 1, 2, and 3σ confidence regions in the Ωm−w parameter space for flat wCDM. As in the top panel, the dashed curve identifies the no-acceleration criterion: w = −1/(3(1−Ωm)). Likewise, the black point is where the 5.38σ confidence contour of the joint likelihood touches the dashed line.
In the text

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