Strong evidence for an accelerating universe
Balakrishna S. Haridasu Vladimir V. Luković Rocco D'Agostino Nicola Vittorio
 

  
  
  
  Gran Sasso Science Institute (INFN), Viale Francesco Crispi 7, I-67100 L’Aquila, Italy Dipartimento di Fisica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1, I-00133, Roma, ItalySezione INFN, Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1, I-00133, Roma, Italy

  sandeep.haridasu@gssi.infn.it

  

  A recent analysis of the Supernova Ia data claims a 'marginal' (∼3σ) evidence for a cosmic acceleration. This result has been complemented with a non-accelerating R_h=ct cosmology, which was presented as a valid alternative to the ΛCDM model. In this paper we use the same analysis to show that a non-marginal evidence for acceleration is actually found. We compare the standard Friedmann models to the R_h=ct cosmology by complementing SN Ia data with the Baryon Acoustic Oscillations, Gamma Ray Bursts and Observational Hubble datasets. We also study the power-law model which is a functional generalisation of R_h=ct. We find that the evidence for late-time acceleration is beyond refutable at a 4.56σ confidence level from SN Ia data alone, and at an even stronger confidence level (5.38σ) from our joint analysis. Also, the non-accelerating R_h=ct model fails to statistically compare with the ΛCDM having a Δ(AIC)∼30.
 
 
Strong evidence for an accelerating universe
    
  	Balakrishna S. Haridasu<ref>
    Vladimir V. Luković<ref>,<ref>
    Rocco D'Agostino<ref>,<ref>
    Nicola Vittorio<ref>,<ref>
  
    Received / Accepted 
=========================================================================================================================================






§ INTRODUCTION

The very first evidence for an accelerated expansion of the universe was obtained using the SN Ia observations in <cit.> and <cit.>, which has been further confirmed with the most recent supernova data <cit.>. Other low redshift data such as the Observational Hubble parameter-OHD <cit.>, Baryon Acoustic Oscillations-BAO <cit.>, also support an accelerating universe. As an independent observation, the Cosmic Microwave Background-CMB radiation has been in excellent concert with these results and has provided with the most stringent constraints on the cosmological models <cit.>. These observations have established the ΛCDM as the concordance model of cosmology and the late-time acceleration has been a well accepted phenomenon. 

However, <cit.> have used a modified statistical model for the analysis of the supernova JLA dataset <cit.> and claimed that the evidence for the acceleration is marginal (≲ 3σ). The modification has been done 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, <cit.> 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 ∼ 4.2 σ. In this paper we want to show that even with the less flexible modelling of <cit.> the evidence for acceleration is very strong. We also extend the joint analysis done in <cit.> with the inclusion of Gamma Ray Bursts-GRB dataset <cit.>. 

The marginal evidence for an accelerating universe quoted in <cit.> 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 R_h=ct cosmology <cit.>, 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 <cit.>. Also, the R_h=ct model has often been regarded as a Milne universe <cit.> and several physical issues against this interpretation have been raised <cit.>. In fact, this model preserves the linear expansion by imposing the constraint on the total equation of state (EoS) (ρ_tot + 3p_tot=0), without requiring ρ_tot = p_tot = 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 <cit.>. In a recent work, <cit.> presented a linear coasting model in f(R) gravity, which is functionally equivalent to the R_h=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 <cit.>. In addition, it has been shown in <cit.> 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 R_h=ct model has been tested time and again and contradictory conclusions have been presented. <cit.> have pointed out some of the problems in this model using the cosmic chronometers data from <cit.> and radial BAO measurements, along with several model-independent diagnostics, to show that R_h=ct is not a viable model. In <cit.> several claims against R_h=ct were refuted, and a list of works favouring R_h=ct over ΛCDM was compiled in <cit.>. Power-law cosmologies with n ≥ 1 have been explored against data in several works such as <cit.>, finding n ∼ 1.5 consistently. In a more recent work <cit.>, power-law and R_h=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, we believe 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 present paper is structured as follows. A brief introduction to the models is given in <Ref>. We describe the data and method used for our joint analysis in <Ref>. Our results and discussion are given in <Ref>. 




§ MODELS


In this section we briefly describe the standard ΛCDM model, power-law and R_h=ct cosmologies, which we test 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,
 
    H(z)^2 = H_0^2( Ω_m(1+z)^3  + Ω_k(1+z)^2 +Ω_Λ(1+z)^3(1+w)),

where H_0 is the present expansion rate, while Ω_m, Ω_Λ and Ω_k  are the dimensionless density parameters. 
For the flat ΛCDM model, Ω_m =1-Ω_Λ. We test the extensions of ΛCDM model, namely the kΛCDM model with the constraint Ω_m =1-Ω_Λ-Ω_k, and the flat wCDM model with w as a free parameter.
 The second Friedmann equation, ä/a=- 4π/3 G∑_iρ_i(1+3w_i), gives us insight into the necessary conditions to be satisfied for assessing the dynamics of expansion rate. The criteria for acceleration can be derived as: Ω_m≤Ω_Λ/2 for kΛCDM and w ≤ -1/(3 Ω_Λ) 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) ∝ t^n, with the Hubble equation H(z) = H_0(1+z)^1/n. Here, n>1 implies an accelerated scenario. Although motivated physically with a total EoS parameter w_tot=-1/3, the flat R_h=ct model coincides with the power-law model for n=1. 
It is worthwhile noting that <Ref> reduces to the functional form of a power-law model for selected parameter values:   Ω_m = 0, Ω_Λ=1 and  

    w = 2-3n/3n.


The luminosity distance for all these models with their corresponding H(z) is written as,
 
    D_L(z) ≡{    (1+z) c ( ∫_0^zξ/H(ξ))        for  Ω_k=0 
       (1+z) c/H_0√(-Ω_k)sin( √(-Ω_k) H_0∫_0^zξ/H(ξ))        for  Ω_k≠0  
    .
 
 The theoretical distance modulus is defined as μ_th =  5log[D_L(Mpc)] + 25. The angular diameter distance is D_A(z)=D_L(z)/(1+z)^2, which is used in the modelling of BAO data.
 


§ DATA AND METHOD


We test the models described in the <Ref> against data in the redshift range 0<z≲ 8. We use the observables SN Ia, BAO, OHD and GRB that are uncorrelated. We perform a joint analysis using all the datasets together by defining a combined likelihood function. We keep the description of the data to a minimum as we refer to <cit.> for most of it. 

We use the JLA dataset <cit.> consisting of 740 SN, which already provides an empirical correction to the absolute magnitude,

    M^corr_B = M_B -α s + β c,

In the statistical method we implement <cit.>, the stretch s, colour c corrections and the absolute magnitude M_B are all considered random Gaussian variables without any redshift dependence. Such an 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 <cit.>, which is why the correction in 's' and 'c' have to be explicitly included when they are modelled as distributions. As anticipated in the introduction, we use the less flexible modelling of <cit.> 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 short comings have been discussed in <cit.>. It is of high importance to study these effects, which we shall address in a forthcoming paper. The SN Ia likelihood L_ SN used here is described in <cit.>. 

The BAO data is available for the compound observable D_V defined in <cit.>,

    D_V(z) = [(1+z)^2 D_A^2(z)c z/H(z)]^1/3 ,

It is important to note that the observable D_V is usually presented as a ratio with r_d, the sound horizon at the drag epoch. For the purpose of model selection we fit r_d as a free parameter instead of using the standard fit function for the drag epoch z_d <cit.>, as it is not very straight forward to use it for the power law cosmologies. A similar approach was also implemented in <cit.>. Hence, the parameters H_0 and r_d are now degenerate, and BAO data by itself is only able to constraint the combination r_d× H_0 and Ω_m.  
To avoid correlations among different BAO data points, we use only six measurements taken from <cit.> also summarised in Table 1 of <cit.>. 
A simple likelihood for the uncorrelated data is then implemented as,

    L_ BAO∝exp[-12∑_i=1^6( r_d/D_V^i-r_d/D_V(z_i)/σ_r_d/D_V^i)^2].


The measurements of the expansion rate have been estimated using the differential age (DA) method suggested in <cit.>, which considers pairs of passively evolving red galaxies at similar redshifts to obtain z/ t. 
We use a compilation of 30 uncorrelated DA points taken from <cit.> obtained using BC03 models.
We implement a simple likelihood function assuming all the data are uncorrelated. 

    L_ OHD∝exp[-12∑_i=1^30( H_i-H(z_i)/σ_H_i)^2].


Finally, we use the GRB dataset comprising of 109 observations compiled with the well known Amati relation <cit.>. 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 I and II of <cit.>. The distance modulus μ_GRB and the corresponding standard deviation can be defined as,

    μ_GRB   =   5/2(log_10[ (1+z)/4π(E_p,i/300 )^bS_bolo^-1/100 ^2] + λ) 
    σ_μ_GRB   =   (5/2log(10))^2[(b σ_E_p,i/E_p,i)^2 + (σ_S_bolo/S_bolo)^2+σ_sys^2].

We adopt σ_sys = 0.7571, following the model independent calibration done in <cit.>. 
The likelihood for the GRB is defined as, 

    L_ GRB∝exp[-12∑_i=1^109(μ_GRB^i - μ_th^i/σ_μ_GRB^i)^2].


The joint likelihood for these four independent observables is given as L_ tot=L_ SNL_ OHDL_ BAOL_ GRB. We use the two most common criteria for model comparison in cosmology, namely the Akaike Information criteria (AIC) <cit.> and the Bayesian Information criteria (BIC) <cit.>.
The AIC and BIC values for large number of measurements are defined as,

    AIC   = -2log L^max + 2 N_p , 
    BIC   = -2log L^max + N_plog(N_data),
 
where, N_p and N_data are the number of parameters and data points, respectively. Δ(AIC) = AIC-AIC_ref 
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) 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. 



§ 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 <cit.> as the analysis in kΛCDM model evades non accelerating criterion by only ≲ 3σ. We reproduce this result and agree with this statement (see top panel of <Ref>). However, there is a strong prejudice for a flat universe from the CMB data (<cit.>), 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 Ω_m - w plane is much more significant (≥ 4.56σ) compared to the marginal (≥ 2.88σ) found in the Ω_m - Ω_Λ plane (see <Ref>). 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 Ω_m are considered. In <Ref> contours for the best-fit regions of 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 Ω_m - w allows us to explore points that correspond to the functional forms of R_h=ct and power-law models for specific values of the parameters in wCDM. The point (Ω_m,w)=(0,-1/3), shown in red in the bottom panel of <Ref>, phenomenologically reproduces the expansion law of the R_h=ct model (cf. <Ref>). Similarly, the point (Ω_m,w)=(0,-0.38) corresponds to the best-fit value of n=1.08 in power-law model from our joint analysis (cf. <Ref> and <Ref>).  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. <Ref>). In the Ω_m - w plane this point lies within the 2σ SN confidence region, in contrast to the R_h=ct at 4.56σ.  However, when the joint analysis is performed, these points are at 5σ and 5.5σ for the power-law and R_h=ct models, respectively.

The best-fit parameter values for the models considered in this work are presented in <Ref>. The values for the fitted r_d parameter are consistent with the estimates (see <cit.>) using the fit function for the drag epoch from <cit.>. 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 <cit.>, 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 <cit.> (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 <cit.> using X-ray cluster data.

It is clear that the H_0 estimates for the power-law and R_h=ct models are highly in tension with the direct estimate in <cit.>, which is already a well established problem for ΛCDM <cit.>. In this work, the joint analysis provides H_0=66.4±1.8 km s^-1/Mpc for the ΛCDM scenario. We note that this value is consistent with our previous estimate <cit.> but with a higher error due to the difference in the BAO analysis (see <Ref>). In any case, this value still remains in tension with the direct estimate at 2.7σ.



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



The analysis of SN Union 2.1 data done by <cit.> calls for a non accelerating scenario, as the R_h=ct model was claimed to perform on par with ΛCDM. This point was also taken by <cit.> 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 R_h=ct performs poorly when compared to ΛCDM with Δ(AIC)_SN∼ 22, while a power-law model is performing as good as ΛCDM with n∼1.28 from SN data. Our results are consistent with the previous work by <cit.>. The values of Δ(AIC)_SN obtained from the SN data alone are shown in <Ref>. 

Note that the Milne model was claimed to perform marginally worse in comparison to ΛCDM using the SN data alone <cit.>. In our work, we find for this model Δ(AIC)_SN= 9.8 high enough to reject a model. In any case, 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 <Ref>). The AIC statistics disfavours the power law models by Δ((AIC)_Joint)∼ 28 and the R_h=ct model by Δ((AIC)_Joint)∼ 30 in comparison to the ΛCDM model. In any case, our joint analysis shows that all the three models (Milne, power-law and R_h=ct) are strongly disfavoured with respect to ΛCDM (<Ref>). 



§ CONCLUSIONS



Contrary to the claim by <cit.>, we find that the SN data alone indicates an accelerating universe at more than 4.56 σ 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 R_h=ct model fails to explain at once these data resulting in Δ(AIC)_Joint∼ 30 with respect to ΛCDM. Although, the power-law model performs slightly better that the R_h=ct model, similarly fails with a Δ(AIC)_Joint∼ 28. Our analysis shows that the possibility of having models with an uniform rate of expansion 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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