Modelindependent cosmic acceleration and type Ia supernovae intrinsic luminosity redshift dependence
Key Words.:
cosmology: observations – cosmological parameters – supernovae: individual: SNIa luminosity evolutionAbstract
Context:The CDM model is the current standard model in cosmology thanks to its ability to reproduce the observations. The first observational evidence for this model appeared roughly 20 years ago from the type Ia supernovae (SNIa) Hubble diagram from two different groups. However, there has been some debate in the literature concerning the statistical treatment of SNIa, and their ability to prove the cosmic acceleration.
Aims:In this paper we relax the standard assumption that SNIa intrinsic luminosity is independent of the redshift, and we examine whether it may have an impact on our cosmological knowledge; more precisely, on the accelerated nature of the expansion of the Universe.
Methods:In order to be as general as possible, we do not specify a given cosmological model, but we reconstruct the expansion rate of the Universe through a cubic spline interpolation fitting the observations of the different cosmological probes: SNIa, baryon acoustic oscillations (BAO), and the highredshift information from the cosmic microwave background (CMB).
Results:We show that when SNIa intrinsic luminosity is not allowed to vary as a function of the redshift, cosmic acceleration is definitely proven in a modelindependent approach. However, allowing for a redshift dependence, a nonaccelerated reconstruction of the expansion rate is able to fit, as well as CDM, the combination of SNIa and BAO data, both treating the BAO standard ruler as a free parameter (not entering on the physics governing the BAO), or adding the recently published prior from CMB observations. We further extend the analysis by including the CMB data. In this case we also consider a third way to combine the different probes by explicitly computing from the early Universe physics, and we show that a nonaccelerated reconstruction is able to nicely fit this combination of low and highredshift data. We also check that this reconstruction is compatible with the latest measurements of the growth rate of matter perturbations. We finally show that the value of the Hubble constant () predicted by this reconstruction is in tension with modelindependent measurements.
Conclusions:We present a modelindependent reconstruction of a nonaccelerated expansion rate of the Universe that is able to nicely fit all the main background cosmological probes. However, the predicted value of is in tension with recent direct measurements. Our analysis points out that a final, reliable, and consensual value for would be critical to definitively prove the cosmic acceleration in a modelindependent way.
1 Introduction
The cosmological concordance model (CDM), mainly composed of cold dark matter and dark energy, provides an extremely precise description of the properties of our Universe with very few parameters. However, recent observations [Planck Collaboration et al. (2016a); Betoule et al. (2014); Beutler et al. (2011)] show that these components form about 95% of the energy content of the Universe, and their true nature remain still unknown. The evidence for an accelerated expansion, coming from the type Ia supernovae (SNIa) Hubble diagram [Riess et al. (1998); Perlmutter et al. (1999)], was key to consider the CDM as the concordance model. But there has recently been a debate in the literature wondering whether SNIa data alone, or combined with other lowredshift cosmological probes, can prove the accelerated expansion of the Universe [Nielsen et al. (2016); Shariff et al. (2016); Rubin & Hayden (2016); Ringermacher & Mead (2016); Tutusaus et al. (2017); Dam et al. (2017); Lonappan et al. (2017); Haridasu et al. (2017); Lin et al. (2017); Luković et al. (2018)]. For instance, the authors in Nielsen et al. (2016) claim that, allowing for the varying shape of the light curve and extinction by dust, they find that SNIa data are still quite consistent with a constant rate of expansion, while the authors in Rubin & Hayden (2016) claim, redoing this analysis, a 11.2 confidence level for acceleration with SNIa data alone in a flat universe.
In SNIa analyses it is usually assumed that two different SNIa in two different galaxies with the same colour, stretch of the lightcurve, and host stellar mass, have on average the same intrinsic luminosity, independently of the redshift. In this work we follow the approach of our previous analysis [Tutusaus et al. (2017)], and we relax this assumption by allowing these SNIa to have different intrinsic luminosities as a function of the redshift. Relaxing this redshift independence assumption has also been considered in other analyses [Wright (2002); Drell et al. (2000); Linden et al. (2009); Nordin et al. (2008); Ferramacho et al. (2009)]. In Tutusaus et al. (2017) it was shown that a nonaccelerated power law cosmology was able to fit the main lowredshift cosmological probes: SNIa, the baryon acoustic oscillations (BAO), the Hubble parameter as a function of the redshift (), and measurements of the growth of structures (), when some intrinsic luminosity redshift dependence is allowed. Nevertheless, this specific powerlaw model is excluded when considering cosmic microwave background (CMB) information (as it was shown in [Tutusaus et al. (2016)]), and recently confirmed by the authors of Riess et al. (2018), who showed that such a model cannot fit the latest SNIa observations at , even when accounting for some luminosity evolution. In this paper we extend our previous study with a modelindependent analysis, and we include the latest BAO observations as well as the complementary highredshift CMB data. In order to be as general as possible, we follow the approach of Bernal et al. (2016) and reconstruct the expansion rate at late times through a cubic spline interpolation.
2 Cosmological probes
In this section we present the different cosmological probes considered in the analysis, as well as the specific data sets used.
2.1 Type Ia supernovae
Type Ia supernovae are considered standardizable candles and they are useful to measure cosmological distances and break some degeneracies present in other cosmological probes. The standard observable used in SNIa measurements is the socalled distance modulus,
(1) 
where is the luminosity distance, and the comoving angular diameter distance.
The standardization of SNIa is based on empirical observation that they form a homogeneous class of objects, whose variability can be characterized by two parameters [Tripp (1998)]: the time stretching of the light curve () and the SNIa color at maximum brightness (). If we assume that different SNIa with identical colour, shape, and galactic environment have on average the same intrinsic luminosity for all redshifts, the distance modulus can be expressed as
(2) 
where corresponds to the observed peak magnitude in the band restframe, while and are nuisance parameters. Although the mechanism is not fully understood, it has been shown [Sullivan et al. (2011); Johansson et al. (2013)] that both and depend on properties of the host galaxy. In this work we use the joint lightcurve analysis from Betoule et al. (2014), where the authors approximately correct for these effects assuming that the absolute magnitude is related to the stellar mass of the host galaxy, , by a simple step function
(3) 
where and are two extra nuisance parameters. The authors also discard the additional dependency of on the host stellar mass because it does not have a significant impact on the cosmology.
Concerning the errors and the correlations of the measurements, we use the full covariance matrix provided in Betoule et al. (2014), where the authors have considered several statistical and systematic uncertainties, such as the error propagation of the lightcurve fit uncertainties, calibration, lightcurve model, bias correction, mass step, dust extinction, peculiar velocities, and contamination of nontype Ia supernovae. This covariance matrix depends on the and nuisance parameters, so when we sample the parameter space we recompute the covariance matrix at each step.
Allowing for some redshift dependence on the SNIa intrinsic luminosity, the distance modulus can be expressed as
(4) 
where stands for a nuisance term that accounts for the intrinsic luminosity dependence as a function of the redshift.
Although the mechanism of SNIa detonation is well understood, the difficulty of observing the system before becoming a SNIa leaves enough uncertainty to wonder whether considering a luminosity dependence with the redshift may have an effect on the cosmological conclusions. A varying gravitational constant, or a fine structure constant variation, could generate a luminosity dependence on the redshift, but our approach here is just to consider a phenomenological model to explore the degeneracy of SN distancedependent effects and the cosmological information. Different phenomenological models have been proposed for (see Tutusaus et al. (2017) and references therein). In this work we just consider Model B from Tutusaus et al. (2017), that has also been illustrated in Riess et al. (2018), where . A lower power contribution models a luminosity evolution dominant at lowredshift, while a higher power contribution lead to a luminosity evolution dominating at highredshift. It is important to notice that must be greater than , in order not to be degenerate with . When sampling the parameter space we limit .
2.2 Baryon acoustic oscillations
The baryon acoustic oscillations are the characteristic patterns observed in the galaxy distribution of the largescale structure of the Universe. They are characterized by the length of a standard ruler, , and, in the standard cosmological model, they are originated from sound waves propagating in the early Universe. The BAO scale corresponds then to the comoving sound horizon at the redshift of the baryon drag epoch,
(5) 
where and is the sound velocity as a function of the redshift,
(6) 
In this expression stands for the baryon density while corresponds to the photon density. Their ratio can be approximated [Eisenstein & Hu (1998)] by , with and the baryon energy density parameter. In this work we fix the temperature of the CMB to K [Fixsen (2009)].
However, it is known that models differing from the standard CDM framework may have a value for that is not compatible with [Verde et al. (2017b)], and it has also been shown that the computation of may have an effect on the trouble with the Hubble constant [Bernal et al. (2016)]. Moreover, there has recently been some analyses computing without any dependence on latetime Universe assumptions [Verde et al. (2017a)]. Because of all this, in this work we consider three different methods to include BAO data: either we compute with equation (5), or we let it free, or we include the prior Mpc from Verde et al. (2017a).
We consider both isotropic and anisotropic measurements of the BAO. The distance scale used for isotropic measurements is given by
(7) 
while for the radial and transverse measurements the distance scales are and , respectively.
We use the isotropic measurements provided by 6dFGS at [Beutler et al. (2011)] and by SDSS  MGS at [Ross et al. (2015)], as well as the results from WiggleZ at [Kazin et al. (2014)]. We also consider the anisotropic final results of BOSS DR12 at [Alam et al. (2017)], and the new anisotropic measurements from the eBOSS DR14 quasar sample [GilMarín et al. (2018)] at . These results have been obtained by measuring the redshift space distortions using the power spectrum monopole, quadrupole and hexadecapole. The authors in GilMarín et al. (2018) have shown that their results are completely consistent with different methods used for analyzing the same data [Hou et al. (2018); Zarrouk et al. (2018)]. We finally consider the latest results from the combination of the Ly forest autocorrelation function [Bautista, J. E. et al. (2017)] and the Lyquasar crosscorrelation function [du Mas des Bourboux, H. et al. (2017)] from the complete BOSS survey at . We take into account the covariance matrix provided for the measurements of WiggleZ, BOSS DR12, eBOSS DR14, we consider a correlation coefficient of 0.38 for the Ly forest measurements, and we assume measurements of different surveys to be uncorrelated. In order to take into account the nonGaussianity of the BAO observable likelihoods far from the peak, we follow Bassett & Afshordi (2010) by replacing the usual for a Gaussian likelihood observable by
(8) 
where the ratio stands for the detection significance, in units of , of the BAO feature. We consider a detection significance of for 6dFGS, for SDSSMGS and WiggleZ, for BOSS DR12, for eBOSS DR14, and for the Ly forest.
2.3 Cosmic microwave background
The cosmic microwave background is an extremely powerful source of information due to the high precision of modern data. Furthermore it represents highredshift data, complementing lowredshift probes. As it was shown in Wang & Mukherjee (2007), a significant part of the information coming from the CMB can be compacted into a few numbers, the socalled reduced parameters: the scaled distance to recombination , the angular scale of the sound horizon at recombination , and the reduced density parameter of baryons . For a flat universe their expressions are given by
R&≡Ω_m H_0^2∫_0^z_CMB
\textdzH(z) ,
ℓ_a&≡πcrs(zCMB)∫_0^z_CMB\textdzH(z) ,
ω_b&≡Ω_b h^2 ,
where stands for the redshift of the last scattering
epoch. In this work we consider the data obtained from the Planck 2015
data release [Planck Collaboration et al. (2016b)], where the compressed likelihood
parameters are obtained from the Planck temperaturetemperature plus
the low Planck temperaturepolarization likelihoods. We
specifically consider the reduced parameters obtained when
marginalizing over the amplitude of the lensing power spectrum for the
lower values, since it leads to a more conservative approach, together
with their covariance matrix.
3 Methodology
In this section we first remind the standard CDM model and we then present the reconstruction method used to obtain the expansion rate as a function of the redshift. We give a detailed explanation of how we introduce each cosmological probe in the analysis, and we finally describe the method used to sample the parameter space.
3.1 The CDM model
The flat CDM model assumes a flat RobertsonWalker metric together with FriedmannLemaître dynamics, leading to the comoving angular diameter distance,
(9) 
and the FriedmannLemaître equation,
(10) 
where () stands for the matter (radiation) energy density parameter. We follow Planck Collaboration et al. (2016a) in computing the radiation contribution as
(11) 
where corresponds to the photon contribution
(12) 
In this work we fix the effective number of neutrinolike relativistic degrees of freedom to . When we consider only SNIa data, or SNIa combined with BAO data letting free, we fix the value of for the radiation contribution on CDM [see equations (11, 12)] to kmsMpc, since there is no sensitivity to in these cases. However, is left free for all the other cases and reconstructions in the rest of the work. The remaining parameters when fitting CDM to the data are and the corresponding nuisance parameters of the cosmological probes considered (see Table 1).
3.2 Expansion rate reconstruction method
We want our reconstruction to be as modelindependent as possible, and we impose a smooth and continuous expansion rate. Several modelindependent reconstruction methods have been used in the literature to reconstruct the dark energy equation of state parameter, or even the Hubble parameter. Among them let us mention the principal component analysis [Huterer & Starkman (2003); Crittenden et al. (2009); Liu et al. (2016); Said et al. (2013); Qin et al. (2015)], the Gaussian processes [Clarkson & Zunckel (2010); Holsclaw et al. (2010); Seikel et al. (2012); Yu et al. (2017); Busti et al. (2014); Wang & Meng (2017)], or, very recently, the Weighted Polynomial Regression method [GómezValent & Amendola (2018)]. In this work we follow the approach from Bernal et al. (2016), reconstructing the latetime expansion history by expressing in piecewise natural cubic splines. When we consider SNIa data alone, is specified by its values at 5 different “knots” in redshift: . Therefore, our reconstruction when analyzing SNIa data considers the following set of parameters with for being the 5 knots in redshift, the standard SNIa nuisance parameters, and the SNIa intrinsic luminosity evolution parameters.
When BAO data is added into the analysis we consider an extra knot in our reconstruction at . We follow two different approaches to include the BAO measurements: first we consider the product as a free parameter, and secondly we add information coming from the early Universe through the prior on from Verde et al. (2017a), Mpc. In the first case, the set of parameters considered in our reconstruction of is with for being the 6 knots in redshift, while in the second case we consider and separately .
When we finally add the reduced parameters for the CMB we need to specify up to earlytimes. In order to do this we add the seventh knot at computed according to a matter dominated model (with flat RobertsonWalker metric and FriedmannLemaître dynamics) with free and parameters [see equation (10)], and we extend the model up to very highredshift. The main idea in this reconstruction is to start at earlytimes following a matter dominated model (plus radiation and a negligible contribution of dark energy through a cosmological constant) and, when we start to have lowredshift data and a cosmological constant is still negligible with respect to the quantity of matter present in the Universe, we reconstruct through a cubic spline interpolation; in this way we give our reconstruction the freedom to choose the preferred expansion without specifying a particular model for dark energy. When analyzing the data we consider three different cases, depending on the way of introducing the BAO measurements. First, we consider as a free parameter, while, in a second place, we add the prior on from Verde et al. (2017a). In both cases, the set of parameters that enters into the reconstruction is given by , and we add the prior on [Planck Collaboration et al. (2016a)]. As a last case we compute the value of using equation (5). In this case the set of parameters is given by , and we add the prior on [Planck Collaboration et al. (2016a)].
In order to test the degeneracy between a SNIa intrinsic luminosity redshift dependence and cosmic acceleration, we consider different cases with and without evolution, so and can be removed from the analysis, and we also consider coasting reconstructions, in which the universe has a latetime constant expansion rate. More specifically, we fix the first 4 knots (3 for SNIa alone) such that is equal to at these points. See Table 1 for a summary of the different cases considered and the cosmological and nuisance parameters present in them.
Case  Cosmological probes  Cosmological parameters  Nuisance parameters 

SNIa  SNIa  
SNIa+BAO free  SNIa+BAO  
SNIa+ev+BAO free  SNIa+BAO  
SNIa+BAO prior  SNIa+BAO  
SNIa+ev+BAO prior  SNIa+BAO  
SNIa+BAO free +CMB  SNIa+BAO+CMB  
SNIa+ev+BAO free +CMB  SNIa+BAO+CMB  
SNIa+BAO prior +CMB  SNIa+BAO+CMB  
SNIa+ev+BAO prior +CMB  SNIa+BAO+CMB  
SNIa+BAO compute +CMB  SNIa+BAO+CMB  
SNIa+ev+BAO compute +CMB  SNIa+BAO+CMB 
3.3 Fitting the data
In order to reconstruct the expansion rate as a function of the redshift, we fit the data minimizing the common function,
(13) 
where u stands for the model prediction, while
and hold for the observables and their
covariance matrix, respectively. We sample the parameter space to
minimize this function using the MIGRAD application from the
iminuit Python
package
We also compute the probability that a higher value for the occurs for a fit with degrees of freedom, where is the number of data points and is the number of parameters,
(14) 
where is the upper incomplete gamma function and the complete gamma function. This value provides us with a goodness of fit statistic. A probability close to 1 indicates that it is likely to obtain higher values than the minimum found, pointing to a good fit by the model. When we combine different probes, we minimize the sum of the individual functions for each probe, i.e., we assume the probes to be uncorrelated.
4 Results
In this section we present the results of the reconstruction of the expansion rate of the Universe as a function of the redshift for different sets of cosmological probes: SNIa, SNIa combined with BAO, and SNIa combined with both BAO and CMB data. We also comment on the linear growth of structures measurements, and the importance of the value of the Hubble constant, , to draw conclusions on the accelerated expansion of the Universe.
4.1 Case 1: SNIa
{sidewaystable}
Case  Model  d.o.f  

CDM  682.89/735  0.915  
SNIa  Splines  681.38/731  0.905  
CS (3 knots)  717.60/734  0.661 
We first start considering only SNIa data. We present this case as an illustration of the reconstruction method used. The bestfit values for the cosmological and nuisance parameters are presented in Table 4.1 together with the 1 error bars, and the reconstructions are shown in Fig. 1. We show three different models in this case: the reconstruction through cubic splines (red), the reconstruction for a coasting universe (labelled CS) at lowredshift (fixing the first 3 knots  green), and CDM as a reference (black). We do not consider any SNIa luminosity evolution for the moment. In Table 4.1 we also provide the reader with the ratio of the over the number of degrees of freedom, and the probability from equation (14). In order to obtain the bands for the reconstructions we generate 500 splines from an dimensional Gaussian centered at the bestfit values and with the covariance matrix obtained from the fit to the data. We further require each spline to have a value smaller or equal than 1 with respect to the bestfit reconstruction.
In Table 4.1 we can clearly see that all the SNIa nuisance parameters values are compatible for the three models, and that a coasting universe shows a lower expansion rate when we increase the redshift with respect to the standard spline reconstruction. This is confirmed from Fig. 1 where we see that the expansion rate drops above . We can also observe in this figure that the bands increase when we increase the redshift, as expected, since there are less data points in this region. Comparing the three models, we observe that the spline reconstruction provides a slightly smaller value (681.38) than CDM (682.89), but the former has many more parameters in the model, so the ability of these models to fit the data is roughly the same, being slightly better for CDM () than the spline reconstruction (). However, the value obtained for the coasting reconstruction (717.60) is much larger than the previous values, which also implies that this model is less able to perfectly fit the data (). A detailed model comparison is beyond the scope of this work, since we are interested in studying the accelerated expansion of the Universe and the relation it may have with SNIa luminosity, not in proposing a new cosmological model to confront against CDM. However, the coasting reconstruction has a relative probability of exp( %, showing that a coasting universe at lowredshift is totally excluded, even using SNIa data alone, when SNIa intrinsic luminosity is assumed to be redshift independent. Notice also that, even if we ask the reconstruction to be nonaccelerated at lowredshift, it prefers to add some acceleration at earlier times (above ) than just having a constant velocity expansion.
4.2 Case 2: SNIa+BAO
{sidewaystable}
Case  Model  d.o.f  
CDM  698.64/753  0.922  
SNIa+BAO free  Splines 

696.46/748  0.911  
CS (4 knots) 

739.91/752  0.616  

CDM 


698.64/751  0.914  
SNIa+ev+BAO free  Splines 

694.21/746  0.912  
CS (4 knots) 

698.95/750  0.909  

CDM 

698.64/753 
0.922  
SNIa+BAO prior  Splines 
696.46/748 
0.911  
CS (4 knots) 
739.91/752 
0.616  

CDM 


698.35/751  0.915  
SNIa+ev+BAO prior  Splines 

694.37/746  0.912  
CS (4 knots) 

698.95/750  0.909  

After having shown how the reconstruction method works, and having applied it to SNIa data alone, we focus on the combination of SNIa and BAO data. As it is shown in Table 1 we consider two different ways to combine these data sets: we either let the product free, or we add a prior on . Since we consider the models with and without SNIa intrinsic luminosity evolution, and we always add CDM as a reference, we finally have 4 different subcases with the corresponding three models per subcase. The bestfit values and errors for the parameters for all these cases are shown in Table 4.2.
Let us first focus on the case where is treated as a free parameter. As it was the case with SNIa data alone, all the SNIa nuisance parameters have compatible values for the different models considered. However, the coasting reconstruction now does not show a reduced expansion rate at highredshift (adding or not SNIa luminosity evolution), due to the addition of the BAO data points above . We can also see that the value of obtained from the spline reconstruction is more or less compatible with the one obtained with CDM, but it is lower for the coasting reconstruction, adding SNIa intrinsic luminosity or not. Concerning the ability of the models to fit the data, the of the spline reconstruction is always slightly smaller than the CDM one (696.46 against 698.64, and 694.21 against 698.64 when we allow the SNIa luminosity to vary). But as it was the case before, the probability of providing a good fit is roughly the same for both models, being slightly better for CDM (0.911 against 0.922, and 0.912 against 0.914 when we account for evolution). It is also what can be seen in the reconstruction plot on the left panel of Fig. 2. With respect to the coasting reconstruction, we can see in Table 4.2 that, when SNIa intrinsic luminosity is allowed to vary, we obtain a value very close to the CDM one, thus giving also a good probability to correctly fit the data (0.909 against 0.912, for the standard reconstruction, and 0.914, for CDM).
Let us now focus on the combination of SNIa and BAO data with a prior on (two last rows of Table 4.2 and the right panel of Fig. 2). It allows us to obtain a constraint on , so we represent in this case the expansion rate by . All the bestfit values for the parameters are very close to the previous case, with nearly the same values and the same probabilities, since we have only added one data point and one parameter in the analysis. As before, a coasting universe provides a good fit to the data with a probability of 0.909 against 0.912, for the standard spline reconstruction, and 0.915 for CDM, when SNIa luminosity is allowed to vary. The interesting result from these cases is that the value found for for the spline reconstruction is always smaller than the one obtained for CDM, but still compatible, while it is significantly smaller for the coasting reconstruction, as it can be seen in the right plot of Fig. 2. This is consistent with the lower value found for in the previous cases for the coasting reconstruction.
4.3 Case 3: SNIa+BAO+CMB
{sidewaystable}
Case  Model  d.o.f  

CDM 

698.67/754  0.926  
SNIa+BAO free +CMB  Splines 

697.01/748  0.909  
CS (4 knots) 

740.32/752  0.612  

CDM 

698.66/752  0.918  
SNIa+ev+BAO free +CMB  Splines 

694.93/746  0.909  
CS (4 knots) 

699.44/750  0.906  

CDM 

699.21/755  0.927  
SNIa+BAO prior +CMB  Splines 

697.07/749  0.913  
CS (4 knots) 

740.51/753  0.620  

CDM 

699.17/753  0.920  
SNIa+ev+BAO prior +CMB  Splines 

694.93/747  0.913  
CS (4 knots) 

699.46/751  0.911  

CDM 

699.31/755  0.927  
SNIa+BAO compute +CMB  Splines 

697.09/749  0.912  
CS (4 knots) 

740.55/753  0.620  

CDM 

699.24/753  0.920  
SNIa+ev+BAO compute +CMB  Splines 

694.93/747  0.913  
CS (4 knots) 

699.47/751  0.911  

Observed  CDM  Splines  CS (4 knots)+ev  

1.7414  1.7385  1.7382  
301.68  301.67  301.65  
2.254  2.261  2.262 
As a last case we consider the combination of the three main background expansion cosmological probes: SNIa, BAO, and CMB. We have already presented two different ways to combine SNIa and BAO data, so when we add CMB data we keep this approach and, since we now include the physics of the early Universe, we add a third way consisting on computing the explicit value of . The bestfit values, with the 1 errors, for the parameters for these three subcases are presented in Table 4.3, and the corresponding reconstruction in Fig. 3.
Let us start with the combination considering a free parameter. Both assuming the SNIa intrinsic luminosity to be redshift independent or allowing it to vary, the three models provide compatible values for all the parameters except , which is significantly smaller for the coasting reconstruction, as it was already shown in the combination of SNIa and BAO data, and which is compensated by a larger . When SNIa luminosity is allowed to vary, a coasting reconstruction provides roughly the same (699.44) as CDM (698.66) with a slightly smaller probability (0.906 against 0.918), showing that a nonaccelerated expanding universe can fit the three main background probes when SNIa intrinsic luminosity is allowed to vary.
In a second place we add a prior on . All the bestfit values are compatible between the different models as before, except for and , which are smaller and larger for a coasting reconstruction, respectively, accounting for SNIa luminosity evolution or not. The obtained values are very similar, leading to very similar probabilities to correctly fit the data, and they show that a coasting reconstruction can correctly fit the data when SNIa luminosity evolution is accounted for. In the last place we compute using equation (5). All the bestfit values are compatible with the previous results, and compatible between the different models, except for and . It is also the case for the values and the corresponding probabilities. We conclude that a nonaccelerated universe can correctly fit the three main background probes when we account for a redshift dependence in the intrinsic luminosity of SNIa.
For completeness, we present in Fig. 4 the residuals to SNIa and BAO observations for three different models: CDM (black), the reconstruction through cubic splines (red), and the nonaccelerated model using a coasting reconstruction (green) taking into account SNIa intrinsic luminosity evolution. We also provide the predictions for the CMB quantities , , and in Table 2. All these predictions have been computed using the bestfit values for the parameters obtained from the global fit to the combination of SNIa, BAO, and CMB data, computing explicitly the value of using equation (5). From these results we can graphically see that all three models are perfectly able to fit the data; including the coasting reconstruction with SNIa luminosity evolution. As it can be seen in Table 4.3, a different approach when combining SNIa, BAO, and CMB data gives nearly the same values for the parameters, which leads to nearly the same predictions.
4.4 Growth rate
The measurements of the growth rate of matter perturbations offer an additional constraint on cosmological models. Their value depend on the theory of gravity used and it is well known that there are identical background evolutions with different growth rates [Piazza et al. (2014)]. Defining the linear growth factor of matter perturbations as the ratio between the linear density perturbation and the energy density, , we can derive the standard second order differential equation for [Peebles (1993)]
(15) 
where the dot stands for differentiation over the cosmic time. Neglecting second order corrections, this differential equation can be rewritten with derivatives over the scale factor [Dodelson (2003)]
(16) 
which is valid under the assumption that dark energy cannot be perturbed and does not interact with dark matter. We can now define the growth factor as
(17) 
and then compute the observable weighted growth rate as
(18) 
where stands for the observed value of the root mean square mass fluctuation amplitude on scales of Mpc at redshift (fixed to 0.8159 in this work [Planck Collaboration et al. (2016a)]), and represents the CDM Planck growth rate [Planck Collaboration et al. (2016a)]. In this work we consider the measurements of the weighted growth rate from the 6dFGS survey [Beutler et al. (2012)], the WiggleZ survey [Blake et al. (2012)], and the VIPERS survey [de la Torre et al. (2013)], as well as the different SDSS projects: SDSSII MGS DR7 [Howlett et al. (2015)] (with the main galaxy sample of the seventh data release), SDSSIII BOSS DR12 [Alam et al. (2017)] (with the LRGs from the 12th BOSS data release), and SDSSIV DR14Q [GilMarín et al. (2018)] (with the latest quasar sample of eBOSS). We have not included this data set in our fitting analysis for simplicity, but we show in Fig. 5 that, using the bestfit values for the parameters from the SNIa+BAO+CMB fit, the prediction for the three models considered (CDM, spline reconstruction, and coasting reconstruction with SNIa luminosity evolution) is in very good agreement with the observations. Notice that the values for the parameters used in Fig. 5 have been obtained computing the value of , but the results are equivalent using the other approaches for the combination of our three main data sets.
4.5 The Hubble constant
The Hubble constant, , is one of the most important parameters in modern cosmology, since it is used to construct time and distance cosmological scales. It was first measured by Hubble to be roughly 500 km/s/Mpc [Hubble (1929)]. Current data supports a value for close to 70 km/s/Mpc. However, nearly 100 years later there is still no consensus on its value. Local measurements already show some tension on the results depending on the calibration of SNIa distances [Riess et al. (2018); Tammann, G. A. & Reindl, B. (2013)]. Moreover, there is also some tension between the direct measurement of and the value inferred from the CMB assuming a CDM model [Planck Collaboration et al. (2016a)]. There has been many attempts in the literature to solve this discrepancy both from an observational and a theoretical perspective [see Bernal et al. (2016); GómezValent & Amendola (2018) and references therein for a detailed discussion on the trouble with ]. In this work we consider two very recent model independent measurements of in order to check its effect on the conclusions we can draw concerning the cosmic acceleration. We first consider the value obtained from the Hubble Space Telescope observations in Riess et al. (2018) (R18 in the following), km/s/Mpc. We then consider the value obtained with Gaussian Processes using SNIa data, and constraints on from cosmic chronometers in GómezValent & Amendola (2018) (GVA18 in the following), km/s/Mpc. This last value is closer to the one derived with an “inverse distance ladder” approach in Aubourg et al. (2015), , where the measurement assumes standard prerecombination physics but is insensitive to dark energy or space curvature assumptions. It is also closer to the value derived from the CMB observations using a flat CDM model, [Planck Collaboration et al. (2016a)].
In Fig. 6 we represent the profile likelihood (assuming Gaussian likelihoods) for both the observed values of , R18 (black) and GVA18 (blue), and the values derived from the nonaccelerated reconstruction for the combination SNIa+BAO+CMB taking into account the SNIa intrinsic luminosity evolution. We present the three values obtained for the three approaches followed when combining the data sets: consider a free parameter (green), add a prior on it (yellow), or compute it explicitly (purple). From the figure alone it is clear that the value for the nonaccelerated reconstruction is in tension with R18 at more than 5, independently of the approach used when combining the data sets. More precisely, a nonaccelerated reconstruction is ruled out if we consider the R18 measurement at 5.65 (free , ), 6.56 (prior , ), or 6.62 (compute , ), showing that, with the R18 measurement, cosmic acceleration is proven even if some astrophysical systematics evolving with the redshift modify the intrinsic luminosity of SNIa. However, we can also see from the figure that if we consider the measured value from the Gaussian Processes, a nonaccelerated reconstruction shows a bit less than a 3 tension. More precisely, there is a tension of 2.42 (free ), 2.86 (prior ), or 2.90 (compute ). In this case, the measured value of points towards ruling out these reconstructions, but the tension is still far from the 5 threshold.
5 Conclusions
In this paper we have adressed the question whether relaxing the standard assumption that SNIa intrinsic luminosity does not depend on the redshift may have an impact on the conclusions that can be drawn on the accelerated nature of the expansion of the Universe. Although there is no theoretically motivated model for this luminosity evolution, it has not been proven that two SNIa in two galaxies with the same lightcurve, colour, and host stellar mass have the same intrinsic luminosity independently of the redshift. Moreover, with this kind of analysis we can distinguish between the effect of unknown astrophysical systematics varying with the redshift and the cosmological information.
The impact of SNIa luminosity evolution on our cosmological knowledge has already been adressed before [Wright (2002); Drell et al. (2000); Linden et al. (2009); Nordin et al. (2008); Ferramacho et al. (2009); Tutusaus et al. (2016, 2017)], but in this work we have extended the analysis by including the physics of the early Universe (), and thus considering the main background cosmological probes: SNIa, BAO, and the CMB. In order to be as general as possible, we have not imposed a cosmological model, but we have reconstructed the expansion rate of the Universe using a cubic spline interpolation.
We have first applied, as an illustration of the method, the reconstruction to SNIa data alone with the standard SNIa luminosity independence assumption. We have shown that with this assumption cosmic acceleration is definitely preferred against a local nonaccelerated universe.
In a second step we have added the latest BAO data to our analysis. We have considered two different ways to combine it with SNIa data: either we have considered as a free parameter, or we have added a prior on coming from CMB observations, without any dependence on latetime Universe assumptions. In both cases we have seen that a nonaccelerated universe is able to fit the data nearly as nicely as CDM, when we allow the SNIa intrinsic luminosity to vary as a function of the redshift.
Next, we have extended the data sets in the analysis by adding the information coming from the CMB through the reduced parameters. In order to deal with this information we have been forced to specify the model up to very high redshifts. We have decided to follow a matter dominated model (plus radiation and a negligible dark energy contribution in the form of a cosmological constant) from the early Universe down to , where we start to have lowredshift data. We have then coupled the model to our spline reconstruction. In other words, we consider a matterradiation dominated model at the early Universe and, when we start to have lowredshift data and a cosmological constant is still negligible, we allow the expansion rate to vary freely without specifying any dark energy model. When adding the CMB data we follow three different approaches: treat as a free parameter, add a prior on it, or compute it assuming that the BAO and the CMB share the same physics. In all three cases we have seen that a nonaccelerated model is able to nicely fit the data, when SNIa intrinsic luminosity is allowed to vary, including the information on the early Universe coming from the CMB.
For simplicity we have not added the measurements for the growth rate of matter perturbations, but we have checked that using the bestfit values from the global fit SNIa+BAO+CMB we are able to correctly predict the latest measurements.
After having seen that if SNIa intrinsic luminosity does depend on the redshift, the main cosmological probes are not able to rule out a nonaccelerated model, we focus on the impact that the Hubble constant may have on this question. We have considered two different model independent recent measurements of : km/s/Mpc (R18) from Riess et al. (2018), and km/s/Mpc (GVA18) from GómezValent & Amendola (2018). We have shown that if we consider the R18 value, cosmic acceleration is proven at more than 5.65 for a general expansion rate reconstruction [for which we get (free ), (prior ), and (compute )], even if SNIa intrinsic luminosity varies as a function of the redshift due to any astrophysical unknown systematic. It is important to say, though, that if we consider the GVA18 value, a nonaccelerated reconstruction for the expansion rate is at a 3 tension with the measurement, but still below the 5 detection.
In conclusion, if SNIa intrinsic luminosity varies as a function of the redshift, a nonaccelerated universe is able to correctly fit all the main background probes. However, the value of turns out to be a key ingredient in the conclusions we can draw concerning the cosmic acceleration. If we take it into account we are close to claim an accelerated expansion of the Universe using an approach very independent of the cosmological model assumed, and even if SNIa intrinsic luminosity varies. A final consensus on a direct measurement of and its precision will be decisive to finally prove the cosmic acceleration independently of the cosmological model and any redshift dependent astrophysical systematic that may remain in the SNIa analysis.
Acknowledgements.
We thank Adam G. Riess and Daniel L. Shafer for very fruitful comments which helped to noticeably improve this work. This work has been carried out thanks to the support of the OCEVU Labex (ANR11LABX0060) and of the Excellence Initiative of AixMarseille University  A*MIDEX, part of the French âInvestissements dâAvenirâ programme.Footnotes
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