Bayesian comparison of non-standard cosmologies using type ... - arXiv

arXiv:1603.06563v2 [astro-ph.CO] 22 Mar 2016

Prepared for submission to JCAP

Bayesian comparison of non-standard cosmologies using type Ia supernovae and BAO data B. Santos, N. Chandrachani Devi and J. S. Alcaniz Departamento de Astronomia, Observatório Nacional, 20921-400, Rio de Janeiro – RJ, Brasil E-mail: [email protected], [email protected], [email protected]

Abstract. We use the most recent type Ia supernovae (SNe Ia) observations to perform a statistical comparison between the standard ΛCDM model and its extensions (wCDM and w(z)CDM) and some alternative cosmologies, namely: the Dvali–Gabadadze–Porrati (DGP) model, a power-law f (R) scenario in the metric formalism and an example of vacuum decay (Λ(t)CDM) cosmology in which the dilution of pressureless matter is attenuated with respect to the usual a−3 scaling due to the interaction of the dark matter and dark energy fields. We perform a Bayesian model selection analysis using the Affine-Invariant Monte-Carlo Ensemble sampler. In order to obtain the posterior distribution for the parameters of each model, we use the Joint Lightcurve Analysis (JLA) SNe Ia compilation containing 740 events in the interval 0.01 < z < 1.3. The data are analysed with the SALT-II light-curve fitter and the model selection is then performed by computing the Bayesian evidence of each model and the Bayes factor between them. The results indicate that the JLA data only cannot distinguish the standard ΛCDM from its alternatives but their combination with current measurements of baryon acoustic oscillations can. We provide a rank order for the models considered and also discuss the influence of the H0 priors on the results. Keywords: cosmology: observations – distance scale – cosmological parameters; cosmology: theory – dark energy; statistics: model selection ArXiv ePrint: 1603.06563

Contents 1

Introduction

1

2

Non-standard cosmological models 2.1 Dark energy models with constant equation-of-state 2.1.1 Dynamical dark energy models 2.2 Vacuum decay model 2.3 DGP model 2.4 f (R)-gravity models

2 2 2 3 4 4

3

Bayesian Model Selection

5

4

Data 4.1 Type Ia supernovae 4.2 Baryon acoustic oscillations

6 6 7

5

Methodology

8

6

Results

10

7

Conclusions

12

1

Introduction

Almost two decades ago, distance measurements of type Ia supernovae (SNe Ia) provided the first direct evidence for a late-time cosmic acceleration [1, 2]. Nowadays, this phenomenon is also confirmed from independent data, such as, for instance, the most recent measurements of the baryon acoustic oscillations (BAO) from galaxy surveys (see, e.g., [3] for a recent review on BAO measurements). From the theoretical side, however, the absence of a firm physical mechanism responsible for the present acceleration of the Universe has given rise to a number of alternative explanations. In general, mechanisms of cosmic acceleration are explored in two different ways: either introducing a new field in the framework of the Einstein’s general theory of relativity (GR), the so-called dark energy, or introducing modifications in GR at very large scales. In the general relativistic framework, the simplest explanation is to posit the existence of a cosmological constant Λ, a spatially homogeneous component whose pressure and energy density are related by pΛ = wρΛ , with the equation-of-state (EoS) parameter w = −1. However, as well known, the standard ΛCDM model (cosmological constant Λ plus cold dark matter) provides a good fit for a large number of observational datasets without addressing some important theoretical issues, such as the fine tuning of the Λ value and the cosmic coincidence problems [4–6]. If the cosmological term Λ is null or it is not decaying in the course of the expansion, as discussed in the so-called vacuum decay or Λ(t) cosmologies [7–11], an alternative possibility (which also does not address the above issues) is to assume the presence of an extra degree of freedom in the form of a minimally coupled scalar field φ (quintessence field). Among other things, what observationally may distinguish Λ or Λ(t) from φ is the time-dependency of the EoS parameter of quintessence fields, whose behaviour has been parameterised phenomenologically by several authors (see, e.g., [12–16] and references therein).

–1–

The observed cosmic acceleration can also be seen as the first evidence of a breakdown of GR on large scales rather than a manifestation of another ingredient in the cosmic budget. The most usual examples of cosmologies derived from modified or extended theories of gravity include f (R) models, in which terms proportional to powers of the Ricci scalar R are added to the EinsteinHilbert Lagrangian [17–23], and higher dimensional braneworld models, in which extra dimension effects drive the current cosmic acceleration by changing the energy balance in a modified Friedmann equation [24–30]. Since very little is known about the nature of the physical mechanism driving the cosmic acceleration, an important way to improve our understanding of this phenomenon is to use cosmological observations to constrain and select its many approaches. In this paper we use the most recent type Ia supernovae (SNe Ia) observations, the Joint Lightcurve Analysis (JLA) SNe Ia compilation containing 740 events in the interval 0.01 < z < 1.3, to perform Bayesian model selection analysis using the Affine-Invariant Monte-Carlo Ensemble sampler [31]. We consider in our analysis different classes of cosmological models and show that, while the SNe Ia data alone cannot distinguish between the standard cosmology and some of its alternatives, a joint analysis with current BAO data is able to rule out some scenarios. We also discuss the role of the priors on the present expansion rate H0 on the results. This paper is organised as follows. In section 2 we present the cosmological models considered in our analysis. The Bayesian framework of model selection is briefly discussed in section 3. The data sets and methodology used in the analysis are discussed in sections 4 and 5, respectively. We present and discuss the model comparison results in section 6. We summarise our main conclusions in section 7.

2

Non-standard cosmological models

As mentioned earlier, the late-time cosmic acceleration is usually explored in two different ways: either including an extra component in the right-hand side of Einstein’s field equations or modifying gravity at large scales. In this work, we select models of both cases under the framework of a flat Friedmann-Robertson-Walker (FRW) metric. In what follows, we briefly discuss the scenarios considered in our analysis. 2.1

Dark energy models with constant equation-of-state

General relativistic scenarios with a constant dark energy equation-of-state (EoS) w generalises the standard ΛCDM model in which w = −1. In what follows, we refer to this model as wCDM model. The corresponding Friedmann equation for this cosmology is given by E(z)2 = Ωm,0 a−3 + Ωde,0 a−3(1+w) ,

(2.1)

where E(z) = H(z)/H0 is the normalised Hubble parameter and Ωm,0 and Ωde,0 correspond, respectively, to the current values of clustered matter (baryonic and dark) and dark energy density parameters, which obeys the normalisation condition Ωde,0 = 1 − Ωm,0 . 2.1.1

Dynamical dark energy models

A more general case can be studied by allowing the equation-of-state of the dark energy component to vary as a function of the cosmological scale factor a. In this case, the Friedmann equation takes the form " Z 1 # 1 + w(a0 ) 0 E(z)2 = Ωm,0 a−3 + Ωde,0 exp 3 da . (2.2) a0 a

–2–

In order to discriminate the dynamical dark energy (the time-varying nature of EoS) from that of Λ cosmological constant, we consider two kinds of w(a) parameterisations. First, the so-called Chevallier–Polarski–Linder (CPL) parameterisation [13, 14], given by: w(a) = w0 + wa (1 − a) ,

(2.3)

where w0 stands for the EoS’s value today whereas wa describes its time evolution. For this parameterisation, the last term of Eq. (2.2) is written as: Ωde,0 a−3(1+w0 +wa ) exp [−3wa (1 − a)] .

(2.4)

As discussed by Wang and Tegmark (2004), the above parameterisation cannot be extended to the entire history of the universe since it blows up exponentially in the future (a → ∞) for wa > 0. Therefore, we also consider a second dynamical dark energy parameterisation suggested by ref. [15]: w(z) = w0 + wa

z(1 + z) , 1 + z2

(2.5)

which is well-behaved over the entire cosmic evolution and mimics a linear-redshift evolution at low redshift. For this parameterisation (referred to it as BA parameterisation), the last term of Eq. (2.2) can be written as: 3wa Ωde,0 (1 + z)3(1+w0 ) (1 + z2 ) 2 . (2.6) Previous studies have shown that bounds on the w0 and wa parameters allow this dark energy component to remain sub-dominant at z >> 1. For details about the classification of different dark energy behaviours using parameterisation (2.5), we refer the reader to ref. [15]. 2.2

Vacuum decay model

An interesting attempt to account for the cosmological constant problems has also been discussed in the context of interacting dark matter and dark energy cosmologies. A number of ideas have been examined in these lines (see, e.g., [7–10, 32–36] and references therein). The model analysed in our study has a time-dependent cosmological term Λ(t) in which the vacuum energy density ρΛ decays with the expansion of the Universe as [10, 37] ρΛ = ρ˜ Λ,0 +

ρdm,0 −3+ a , 3−

(2.7)

where the  determines the diluting power of the dark matter density ρdm with respect to the usual a−3 as ρdm ∝ a−3+ . Depending upon the positive or negative values of , the energy is transferred either from dark energy to dark matter or vice-versa, respectively. In such scenarios, dark matter will no longer be independently conserved, such that a˙ ρ˙ dm + 3 ρdm = −ρ˙ Λ . a The Friedmann equation for this class of models is given by [38] " # 3Ωdm,0 −3+ ˜ E(z) = Ωb,0 a−3 + a + ΩΛ,0 , 3−

(2.8)

(2.9)

˜ Λ,0 = ΩΛ,0 − 3Ωdm,0 /(3 − ). There is an extra degree of freedom compared to the standard where Ω ΛCDM model due to such interaction (for more details on this class of models, see ref. [36]).

–3–

2.3

DGP model

The Dvali–Gabadadze–Porrati (DGP) model [24] is an example of alternative approach which governs cosmic acceleration via modification of the Einstein’s General Relativity, driven by higherdimensional theories. In this model, our 4-dimensional universe is confined to a 3D brane, embedded in a 5-dimensional bulk space-time with an infinite extra dimension. The energy-momentum tensor only resides on the brane surface whereas the gravitational field equations are driven by the 5-dimensional Einstein tensor and the 4-dimensional Einstein tensor of the induced metric on the brane. Only gravity is allowed to propagate off the 3-brane into the bulk and this induced effect on the brane leads to an accelerated expansion. A cross-over length scale, where the interaction between the effective 4D and 5D gravities takes 2 /2M 3 , and the Friedmann equation is modified as [26, 27]: place, is given by rc = MPl 5 s E(z) =

1 ρ 1 , + 2+ 2 2r 3MPl 4rc c

(2.10)

where ρ is the energy density of the cosmic fluid. Note that in the limit of H ∼ rc−1 , a self-accelerating solution is attained asymptotically which is the main feature of this model (see refs. [24, 39, 40] for details). The above equation can be rewritten as q p E(z) = Ωm,0 a−3 + Ωrc + Ωrc . (2.11) Here Ωrc represents the density parameter associated with the crossover scale, Ωrc = 1/(4rc2 H02 ). Under the flat FRW framework, the normalisation condition is given by Ωrc = [(1 − Ωm,0 )2 /4]. 2.4

f (R)-gravity models

The simplest extension of General Relativity can be obtained by considering additional terms proportional to powers of the Ricci scalar R in the Einstein–Hilbert Lagrangian, the so-called f (R) gravity. Differently from general relativistic scenarios, f (R) cosmology can naturally drive an accelerating cosmic expansion without introducing a dark energy field [17]. We consider the Einstein–Hilbert action in the Jordan frame including f (R) function of the Ricci scalar as Z √ f (R) −g 2 d4 x + S matter (gµν ) , (2.12) S = 2k where k2 = 8πG (G is a bare gravitational constant) and S matter represents the action of the matter minimally coupled to gravity. We assume the metric formalism, in which the connections are assumed to be the Christoffel symbols and the variation of the action is taken with respect to the metric. In a flat FRW spacetime, the field equations for the action (2.12) is given by ! k Rf0 − f 2 00 H = 0 ρ+ − 3H R˙ f , (2.13) 3f 2 " # k 1 2 2 000 00 00 0 ˙ ˙ ˙ ¨ 2H + 3H = − 0 p + R f + 2H R f + R f + ( f − R f ) , (2.14) f 2 where a prime denotes derivative with respect to R (for more on f (R) cosmologies, we refer the reader to refs. [21, 23]. In what follows, we will consider the power-law f (R) model f (R) = R − β/Rn ,

–4–

(2.15)

Model ΛCDM wCDM CPL BA Λ(t)CDM DGP f (R)

Equation (2.1) (w = −1) (2.1) (2.3) (2.5) (2.9) (2.11) (2.15)

Free parameters Ωm,0 , H0 Ωm,0 , H0 , w Ωm,0 , H0 , w0 , wa Ωm,0 , H0 , w0 , wa Ωdm,0 , Ωb,0 , H0 ,  Ωm,0 , H0 Ωm,0 , H0 , n

Table 1. Summary of models considered in the analysis along with the free parameters.

which satisfies all the viability conditions of f (R) models, as discussed by ref. [41], and reduces to the ΛCDM model for n = 0 and β = 4.38. For analysis involving BAO data we add a radiation term, Ωγ,0 = 2.469 × 10−5 h−2 [42], to all Friedmann equations above. A summary of the cosmological models considered in our analysis is given in table 1.

3

Bayesian Model Selection

Bayesian inference is a way to describe the relationship between the model (or hypotheses), the data and the prior information about the model parameters. In a parameter estimation problem, the starting point for Bayesian data analysis is to compute the joint posterior for a set Θ of free parameters given the data, D, through the Bayes’ Theorem [43]: P(Θ|D, M) = L(D|Θ, M) P(Θ|M)/E(D|M), where P, L, P and E are the shorthands for the posterior, the likelihood, the prior and the Evidence1 , respectively. In short, the Bayes’ Theorem updates our previous knowledge about some model parameters in the light of a given data set. It is important to note that the Evidence E, the denominator of the Bayes’ Theorem, is just a normalisation constant and is uninteresting for parameter estimation, since it does not depend upon the model parameters. However, in a model comparison problem, the Evidence is used to evaluate the model’s performance in the light of the data by integrating the product L P over the full parametric space of the model: Z E(D|M) =

L(D|Θ, M) P(Θ|M) dΘ .

(3.1)

M

Therefore, the Evidence is the average value of the likelihood L over the entire model parameter space that is allowed before we observe the data. The most important characteristic of the Evidence is its application of the Occam’s razor to the model selection problem. It rewards the models that fit well the data and are also predictive, moving the average of the likelihood in Eq. (3.1) towards higher values than in the case of a model which fits poorly or is not very predictive (either is too complex or have a large number of parameters) [44]. This concept has been widely applied in cosmology (see, e.g, [45–49]). It is used to discriminate two competing models by taking the ratio Ei Bi j ≡ , (3.2) Ej which is also known as the Bayes’ factor of the model i relative to the model j. If each model is assigned equal prior probability, the Bayes factor gives the posterior odds of the two models. 1

Also called Bayesian evidence, marginal likelihood or model likelihood.

–5–

ln Bi j 0−1 1 − 2.5 2.5 − 5 >5

Strength of the evidence Inconclusive Weak Moderate Strong

Table 2. A new version of the Jeffreys’ scale for the Bayes factor values discussed by ref. [50] and adopted in our analysis. The second column shows the strength of the evidence of the model Mi relative to the model M j . Negative values of ln Bi j means support in favour of the model M j .

To rank the models of interest, we adopted the scale showed in table 2 to interpret the values of ln Bi j = ln (Ei /E j ) in terms of the strength of the evidence of a chosen reference model. This scale, suggested by ref. [50], is a revised and more conservative version of the so-called Jeffreys’ scale [51]. Note that the labels attached to the Jeffreys’ scale are empirical: it depends on the problem being investigated. Thus, for an experiment for which | ln Bi j | < 1, the evidence in favour of the model Mi is usually interpreted as inconclusive (see ref. [50] for a more complete discussion about this scale).

4 4.1

Data Type Ia supernovae

In this work, we focus primarily on current distance measurements of SNe Ia to perform an observational comparison of the cosmologies discussed in the previous section. We use the so-called Joint Light-Curve Analysis (JLA) sample which is an extension of the compilation provided by ref. [52] (referred to C11 compilation), containing a set of 740 spectroscopically confirmed SNe Ia. JLA is a compilation of several low-redshift (z < 0.1) samples, the full three-year SDSS-II supernova survey [53] sample within redshift 0.05 < z < 0.4, the first three years data of the SNLS survey [52, 54] up to redshift z < 1 and a few high redshift Hubble Space Telescope (HST) SNe [55] in the interval 0.216 < z < 1.755. The photometry of SDSS and SNLS was re-calibrated and the SALT2 model is retrained using the joint data set. Theoretically, the distance modulus predicted by the homogeneous and isotropic, flat FRW universe is given by dL (z, Θ) µ(z, Θ) = 5 log , (4.1) 10 pc with the luminosity distance dL defined as dL (z) = (1 + z)

z

Z 0

dz0 , E(z0 )

(4.2)

where E(z) = H(z)/H0 is the normalised Hubble parameter. However, from the observational point of view, the distance modulus of a SNe Ia is obtained by a linear relation from its light curve: µ = mB − (MB − α × x1 + β × c) ,

(4.3)

where mB represents the observed peak magnitude in rest-frame B band, x1 is the time stretching of the light curve, and c is the supernova color at maximum brightness. These three light-curve parameters mB , x1 and c have a different values for each supernova and is derived directly from the light curves. The nuisance parameters α and β are assumed to be constants for all the supernovae, but different for different cosmological models. Following directly ref. [56], we also assume a step

–6–

function relation for the absolute magnitude MB with the host stellar mass (Mstellar ) to cooperate the effect of host galaxy properties on MB . Using Eqs. (4.1)–(4.3), one can obtain the predicted magnitude, mB (z, Θ), for each one of the cosmological model discussed in the previous section. The free parameters of our analysis corresponding to the JLA measurements are α, β, MB and ∆ M . Using the observed magnitude measurements mB (z) of the JLA sample (table F.3 of ref. [56]) and the predicted ones from Eqs. (4.1) and (4.3), the MCMC simulations for the JLA SNe Ia sample were performed by assuming a multivariate Gaussian likelihood of the type LJLA (D|Θ) = exp[−χ2JLA (D|Θ)/2] ,

(4.4)

χ2JLA (Θ) = [mB − mB (Θ)]T C −1 [mB − mB (Θ)] ,

(4.5)

with where C corresponds to the covariance matrix of the distance modulus µ, estimated accounting various statistical and systematic uncertainties. The light-curve fit statistical uncertainties, the systematic uncertainties associated with the calibration, the light-curve model, the bias correction and the mass step uncertainty are described in details in the section 5 of ref. [56], whereas the systematic uncertainties related to the peculiar velocity corrections and the contamination of the Hubble diagram by non-Ia are described briefly in ref. [52]. The uncertainty in redshift due to peculiar velocities, the uncertainty in magnitudes due to gravitational lensing, and the intrinsic deviation in SN magnitude are also taken into account while calibrating it. Using the JLA sample, [56] claimed to have provided the most restrictive constraints so far, i.e., w = −1.027 ± 0.055 (assuming w = constant) and Ωm,0 = 0.295 ± 0.034 (for a flat ΛCDM model). Therefore, it is interesting to perform a similar analysis for non-standard cosmological models, calibrating the data to each cosmology and checking their constraining power on the model parameters. 4.2

Baryon acoustic oscillations

Besides the JLA supernovae dataset, we also consider in our analysis the measurements of BAO in the galaxy distribution. The BAO in the primordial plasma have striking effects on the anisotropies of CMB and the large scale structure of matter. The measurements of the characteristic scale of the BAO in the correlation function of matter distribution provides a powerful standard ruler to probe the angular-diameter distance versus redshift relation and the Hubble parameter evolution. This distance-redshift relation can be obtained from the matter power spectrum and calibrated by the CMB anisotropy data. Usually, the BAO distance constraints are reported as a combination of the angular scale and the redshift separation. This combination is obtained by performing a spherical average of the BAO scale measurement and is given by r s (zdrag ) dz = , (4.6) DV (z) where " #1/3 cz 2 DV (z) = DC (z) (4.7) H(z) Rz is the volume-averaged distance [57] and DC (z) = 0 dz0 /H(z0 ) is the comoving angular diameter distance. In Eq. (4.6), r s (zdrag ) is the radius of the comoving sound horizon at the drag epoch zdrag when photons and baryons decouple [58], Z ∞ c s (z) r s (z) = dz , (4.8) zdrag H(z)

–7–

Survey 6dFGS MGS BOSS LOWZ SDSS(R) BOSS CMASS

z 0.106 0.15 0.32 0.35 0.57

dz (z) 0.3360 ± 0.0150 0.2239 ± 0.0084 0.1181 ± 0.0024 0.1126 ± 0.0022 0.0726 ± 0.0007

Reference [60] [61] [62] [63] [62]

Table 3. BAO distance measurements considered in this work.

q h i where c s (z) = c/ 3 1 + (3Ωb,0 /4Ωγ,0 )(1 + z)−1 is the sound speed in the photon-baryon fluid, and Ωb,0 and Ωγ,0 are the present values of baryon and photon density parameters, respectively. Table 3 shows the BAO distance measurements employed in this work. In addition to this data, we also include three correlated measurements of dz (z = 0.44) = 0.073, dz (z = 0.6) = 0.0726 and dz (z = 0.73) = 0.0592 from the WiggleZ survey [59], with the following inverse covariance matrix: C −1

   1040.3 −807.5 336.8    = −807.5 3720.3 −1551.9 .   336.8 −1551.9 2914.9

(4.9)

Using the same methodology applied to the JLA SNe Ia compilation, we also consider a multivariate Gaussian likelihood for the BAO data set. For each survey listed in the first column of the table 3, the chi-square is given by dz,survey − dz (zsurvey , Θ) = σi "

χ2survey (D|Θ)

#2 ,

(4.10)

where dz,survey and dz (zsurvey , Θ) are the observed and theoretical dz , respectively, and σsurvey is the error associated with each observed value. However, for the WiggleZ data the chi-square is of the form     χ2WiggleZ (D|Θ) = dz,i − dz (Θ) T C −1 dz,i − dz (Θ) . (4.11) Then, the BAO likelihood is directly obtained by the product of the individual likelihoods as LBAO = L6dFGS × LMGS × LLOWZ × LSDSS(R) × LCMASS × LWiggleZ . Similarly, the joint likelihood for the JLA SNe Ia compilation and the BAO data is given by Ljoint = LJLA × LBAO .

5

Methodology

While the idea of the Bayes’ Theorem is simple to understand, the computation of the posterior and the Evidence can be difficult both analytically, since the necessary integrals cannot be evaluated in closed form, and numerically, meaning that the integrations can be very time consuming when the dimension of the parametric space is large. In order to solve this problem, a widely used practice is to sample from the posterior by applying Markov Chain Monte Carlo (MCMC) techniques (we refer the reader to refs. [64–67] for some MCMC algorithms and to refs. [68, 69] for applications of some of those algorithms in cosmology). In this work, we applied an algorithm relying on the Python2 package emcee [70], an implementation of the Affine-Invariant MCMC Ensemble sampler by ref. [31]. This method received its name because the algorithm’s performance is invariant under linear transformations of the parameter 2

https://www.python.org.

–8–

Parameter H0 Ωm,0 Ωdm,0 Ωb,0  w w0 wa n α β MB ∆M

Prior Normal(73.8, 2.4) Uniform(10, 150) Uniform(0, 1) Uniform(0, 1) Uniform(0, 1) Uniform(-5, 5) Uniform(-2, 1) Uniform(-10, 10) Uniform(-10, 10) Uniform(-10, 10) Uniform(0, 0.5) Uniform(0, 5) Uniform(-30, -10) Uniform(-0.5, 0.5)

Model or data associated All models All models All models except Λ(t)CDM Λ(t)CDM Λ(t)CDM Λ(t)CDM wCDM CPL and BA CPL and BA f (R) JLA data JLA data JLA data JLA data

Table 4. Priors on the free parameters.

space, making it a good tool to sample from a highly degenerate Gaussian (or even a non-Gaussian) distribution as well as from an uncorrelated isotropic one. This method works by moving several chains (or walkers) in parallel through the parameter space. The proposal distribution for a given walker depends on the positions of all the other walkers in the ensemble, since they carry information about the underlying distribution. This can dramatically improve the convergence in comparison to the classical Metropolis–Hastings algorithm (see ref. [71] for a recent benchmark of the Metropolis– Hastings, Nested sampling and Affine-Invariant methods using CMB measurements). For all models discussed in section 2, we ran 250 walkers for 400 iterations (steps) in order to get a sample with N = 105 points. However, before these 400 iterations, we also ran a burn-in phase for each model by computing the exponential autocorrelation time τexp at each iteration. The stopping criterion was chosen such that the burn-in phase stops if i > 20 × max(τexp ), where i is the iteration at the burn-in phase and max(τexp,i ) is the maximum τexp,i among all the dimensions of the parameter space at the i-th burn-in iteration. This is a very conservative stopping criterion and is more than enough to ensure that initialisation was forgotten [71]. Since it is difficult to integrate (3.1) numerically, emcee uses an alternative way to draw the posterior distribution, the so-called parallel-tempering MCMC [72, 73]. This method works by sampling from a modified posterior P ∝ L1/T P, where T (≥ 1) is called the “temperature” of the chain. All the statistics are obtained from the chain with T = 1 only, from which the usual posterior of the Bayes’ Theorem is recovered. By running multiple MCMC’s at different temperatures, the procedure explores well very different parts of the parameter space: the hot walkers (large T ) can easily sample from regions of low probability while the cold ones (low T ) are better to sample from the peaks of the likelihood, improving the convergence in the case of a likelihood with more than one mode [70]. The Evidence is estimated by this method after applying the thermodynamic integration technique 3 [74]. In this work, we ran our simulations using 20 temperatures, which is the emcee default value. It is also worth mentioning that Bayesian inference (both parameter estimation and model selection) strongly depends on the priors P(Θ|M) chosen for the free parameters. In our analysis, we applied flat (uniform) priors for almost all free parameters of the models analysed. The only excep3

More information about how emcee estimates the Evidence can be found at http://dan.iel.fm/emcee/current/ user/pt.

–9–

0.165

α

0.150 0.135 0.120 3.4

β

3.2 3.0 2.8

∆M

0.00 0.05 0.10

α

5

0 0.0 5 0.0 0

0.1

0.1

β

3.4

3.2

3.0

2.8

20 0.1 35 0.1 50 0.1 65

0.1

Ωm0

0.4

0.3

0.2

0.15

∆M

Figure 1. Credible Intervals (68% and 95%) for the ΛCDM model using the JLA SNe Ia compilation. The diagonal plots show the posterior distribution for each parameter marginalised with respect to all the other parameters.

tion is H0 , for which we assumed both a flat and a Gaussian prior of H0 = 73.8 ± 2.4 km s−1 Mpc−1 , as given by ref. [75]. All the priors used are shown in table 4. In the next section, we also discuss the influence of the H0 prior on our results by considering a uniform prior in the interval [10, 150].

6

Results

In figure 1 we show the parametric space of Ωm,0 and the nuisance parameters α, β, MB and ∆ M for the standard ΛCDM model. These results are obtained using the JLA SNe Ia sample considering the Gaussian prior on H0 displayed in table 4. As shown in the figure, our results are in excellent agreement with those of ref. [56] (see figure 9 and table 10 of that reference for comparison). Similar plots for the other cosmological models considered in this analysis are not shown for brevity. Our main results are summarised in table 5 where the first, second and third sub-tables correspond to the results obtained using the JLA SNe sample alone, BAO measurements alone and a

– 10 –

Model Λ(t)CDM DGP ΛCDM f (R) wCDM CPL BA ΛCDM f (R) wCDM Λ(t)CDM BA CPL DGP Λ(t)CDM ΛCDM wCDM f (R) BA CPL DGP

ln E

ln B0i Strength of the evidence JLA sample −359.835 −0.363 Inconclusive −360.005 −0.194 Inconclusive −360.198 0 — −360.695 0.497 Inconclusive −361.105 0.906 Inconclusive −363.774 3.576 Moderate −363.919 3.721 Moderate BAO data −5.442 0 — −5.668 0.226 Inconclusive −6.156 0.714 Inconclusive −6.419 0.977 Inconclusive −6.920 1.478 Weak −7.142 1.700 Weak −12.127 6.685 Strong Joint JLA + BAO analysis −366.109 −0.607 Inconclusive −366.716 0 — −366.980 0.263 Inconclusive −367.075 0.358 Inconclusive −368.570 1.853 Weak −368.677 1.960 Weak −393.566 26.850 Strong

Table 5. Bayesian Evidence for the different cosmologies considered in the analysis. The results were obtained using the Gaussian prior on H0 , as shown in table 4.

joint analysis of SNe and BAO, respectively. These results consider Gaussian prior on H0 shown in table 4. We first observe that the current SNe Ia data alone cannot rule out any of the cosmological models studied in this analysis while the joint analysis with BAO data is much more effective to this end. This is clearly seen in the last sub-table of table 5, where three models (CPL, BA and DGP) are at least weakly disfavoured with respect to the standard model. We also note that among all, the most dramatic change in the rank of the models happens for the DGP model. Although providing a good fit for the SNe Ia data (even better than the standard cosmology), the joint analysis with BAO measurements reveals that this scenario is strongly disfavoured with respect to the ΛCDM model. The above results should be compared with the ranking order provided by ref. [76]. These authors used 103 SNe Ia from the Sloan Digital Sky Survey-II Supernova Survey along with two data points of the CMB/BAO ratio to rank a number of alternative cosmologies, some of which being also considered in the current study. Their analysis was performed using two different SNe Ia light-curve fitting, i.e., MLCS and SALT-II, and the model ranking is done using the Bayesian (BIC) and Akaike (AIC) information criteria. Comparing our table 5 with table 3 (SALT-II) of ref. [76], we found a good agreement between the rank order of both analyses. On the other hand, a comparison with the ranking order presented in their table 3 (MLCS) shows a complete disagreement. This is fully consistent with the overall conclusions of ref. [76] and shows the influence of the SNe Ia light-curve fitting on the parameter estimation and model selection.

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Model DGP ΛCDM wCDM f (R) Λ(t)CDM CPL BA ΛCDM DGP f (R) Λ(t)CDM wCDM BA CPL ΛCDM DGP f (R) Λ(t)CDM wCDM CPL BA

ln E

ln B0i Strength of the evidence JLA sample −360.215 −0.278 Inconclusive −360.493 0 – −360.746 0.253 Inconclusive −360.806 0.313 Inconclusive −360.973 0.480 Inconclusive −363.998 3.505 Moderate −364.324 3.830 Moderate BAO data −5.541 0 – −5.815 0.274 Inconclusive −5.889 0.349 Inconclusive −6.473 0.932 Inconclusive −6.478 0.937 Inconclusive −7.545 2.004 Weak −7.596 2.055 Weak Joint JLA + BAO analysis −365.603 0 – −365.664 0.062 Inconclusive −365.853 0.251 Inconclusive −366.902 1.299 Weak −367.032 1.429 Weak −368.963 3.360 Moderate −369.151 3.549 Moderate

Table 6. Bayesian Evidence for the different cosmologies considered in the analysis. The results were obtained using the flat prior on H0 , as shown in table 4.

Finally, as mentioned earlier, the Bayesian approach depends on the prior information chosen for the free parameters. In order to quantify such dependence we also display in table 6 the rank of models when a flat prior on the value of H0 is assumed. Comparing the joint JLA plus BAO results of both analysis, one clearly sees that the evidence for the ΛCDM model is strengthened with respect to the wCDM, CPL and BA scenarios. Another interesting aspect is concerned to the flat DGP model. For the flat prior case, this model provides a good fit to both JLA SNe Ia and BAO data, which changes significantly its ranking position when compared with the results of table 5. Such differences emphasizes the importance of the value of the current expansion rate to this kind of analysis. Although not shown in the tables, we also verified that for the range of H0 values discussed by refs. [75, 77, 78], the ranking order of table 5 remains unaltered.

7

Conclusions

Given the current state of uncertainty that remains over the physical mechanism behind the observed acceleration of the Universe, an important way to improve our understanding of this phenomenon is to use cosmological observations to constrain its different approaches. In this paper, we have performed a Bayesian model selection statistics to rank some non-standard cosmological models in the light of the most recent SNe Ia (JLA compilation) and BAO data. Our analyses have showed

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that the JLA data alone are unable to distinguish between the standard ΛCDM scenario and some specific examples of coupled quintessence cosmologies (Λ(t)CDM), modified gravity models ( f (R) and DGP) and simple parameterisations of the dark energy component. In particular, we have shown that models like Λ(t)CDM and DGP provide the best-fit to JLA data, with their Bayesian evidence being inconclusive with respect to the standard ΛCDM model (see table 5). We have also shown that some of these results are significantly modified when a joint analysis of SNe Ia and BAO data is performed. In this case, the evidence for the ΛCDM model is strengthened with respect to the two time-dependent dark energy parameterisations (BA and CPL) considered in the analysis and also to the DGP model. For this latter model, for instance, the joint analysis shows that it becomes strongly disfavoured with respect to the standard cosmology, with ln B0i = 26.85, whereas the analysis using the JLA data alone provides ln B0i = −0.194. These results are in full agreement with previous ones using different data sets (see, e.g., [76]). An important aspect worth emphasising concerns the ranking position of the decaying vacuum cosmology considered in the analysis. As discussed earlier (see section 2.2), in this kind of models the dark energy field interacts with the pressureless component of dark matter in a process that violates adiabaticity and that constitutes a phenomenological attempt at alleviating the so-called coincidence problem [10, 37]. We have found that this scenario provides an excellent fit to both SNe Ia observations and SNe Ia plus BAO data when the currently accepted interval for H0 is considered. By assuming a flat prior on the present expansion rate, we have found a weak evidence for this model with respect to the ΛCDM. Finally, we have also explored the dependency of the Bayesian ranking order with two different priors on H0 and shown the importance of the value of the current expansion rate to the analysis (for a broad discussion on the importance of more precise measurements of H0 for cosmology and fundamental physics, see ref. [79]). For consistency, we have also verified in our analysis that the ranking order of table 5 remains unaltered for the range of H0 values currently more accepted in the literature, 67 ≤ H0 ≤ 75 (1σ) [75, 77, 78].

Acknowledgments B. Santos and N. Chandrachani Devi are supported by the National Observatory DTI-PCI program of the Brazilian Ministry of Science, Technology and Innovation (MCTI). J.S. Alcaniz thanks CNPq, FAPERJ and INEspaço/CAPES for the financial support.

References [1] Supernova Search Team collaboration, A. G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009–1038, [astro-ph/9805201]. [2] Supernova Cosmology Project collaboration, S. Perlmutter et al., Measurements of Ω and Λ from 42 high redshift supernovae, Astrophys. J. 517 (1999) 565–586, [astro-ph/9812133]. [3] E. Aubourg et al., Cosmological implications of baryon acoustic oscillation measurements, Phys. Rev. D 92 (2015) 123516, [1411.1074]. [4] S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61 (Jan, 1989) 1–23. [5] V. Sahni and A. A. Starobinsky, The case for a positive cosmological Λ-term, Int. J. Mod. Phys. D 9 (2000) 373–444, [astro-ph/9904398]. [6] T. Padmanabhan, Cosmological constant: The weight of the vacuum, Phys. Rept. 380 (jul, 2003) 235–320, [hep-th/0212290].

– 13 –

[7] M. Özer and M. O. Taha, A possible solution to the main cosmological problems, Phys. Lett. B 171 (1986) 363–365. [8] K. Freese, F. C. Adams, J. A. Frieman and E. Mottola, Cosmology with decaying vacuum energy, Nucl. Phys. B 287 (1987) 797–814. [9] J. C. Carvalho, J. A. S. Lima and I. Waga, Cosmological consequences of a time-dependent Λ term, Phys. Rev. D 46 (sep, 1992) 2404–2407. [10] J. S. Alcaniz and J. A. S. Lima, Interpreting cosmological vacuum decay, Phys. Rev. D 72 (sep, 2005) 063516, [astro-ph/0507372]. [11] S. Carneiro, C. Pigozzo, H. A. Borges and J. S. Alcaniz, Supernova constraints on decaying vacuum cosmology, Phys. Rev. D 74 (2006) 023532, [astro-ph/0605607]. [12] G. Efstathiou, Constraining the equation of state of the universe from distant type Ia supernovae and cosmic microwave background anisotropies, Mon. Not. Roy. Astron. Soc. 310 (1999) 842–850, [astro-ph/9904356]. [13] M. Chevallier and D. Polarski, Accelerating universes with scaling dark matter, Int. J. Mod. Phys. D 10 (2001) 213–223, [gr-qc/0009008]. [14] E. V. Linder, Exploring the expansion history of the universe, Phys. Rev. Lett. 90 (mar, 2003) 091301, [astro-ph/0208512]. [15] E. M. Barboza and J. S. Alcaniz, A parametric model for dark energy, Phys. Lett. B 666 (sep, 2008) 415–419, [0805.1713]. [16] E. M. Barboza, J. S. Alcaniz, Z. H. Zhu and R. Silva, A generalized equation of state for dark energy, Phys. Rev. D 80 (2009) 043521, [0905.4052]. [17] S. Capozziello, Curvature quintessence, Int. J. Mod. Phys. D 11 (2002) 483–492, [gr-qc/0201033]. [18] S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner, Is cosmic speed-up due to new gravitational physics?, Phys. Rev. D 70 (aug, 2004) 043528, [astro-ph/0306438]. [19] J. Santos, J. S. Alcaniz, M. J. Reboucas and F. C. Carvalho, Energy conditions in f (R)-gravity, Phys. Rev. D 76 (2007) 083513, [0708.0411]. [20] T. P. Sotiriou and V. Faraoni, f (R) theories of gravity, Rev. Mod. Phys. 82 (2010) 451–497, [0805.1726]. [21] A. De Felice and S. Tsujikawa, f (R) theories, Living Rev. Rel. 13 (2010) 3, [1002.4928]. [22] M. Campista, B. Santos, J. Santos and J. S. Alcaniz, Cosmological consequences of exponential gravity in Palatini formalism, Phys. Lett. B 699 (2011) 320–324, [1012.3943]. [23] S. Capozziello and M. De Laurentis, Extended theories of gravity, Phys. Rept. 509 (2011) 167–321, [1108.6266]. [24] G. Dvali, G. Gabadadze and M. Porrati, 4D gravity on a brane in 5D minkowski space, Phys. Lett. B 485 (jul, 2000) 208–214, [hep-th/0005016]. [25] C. Deffayet, G. R. Dvali and G. Gabadadze, Accelerated universe from gravity leaking to extra dimensions, Phys. Rev. D 65 (2002) 044023, [astro-ph/0105068]. [26] C. Deffayet, S. J. Landau, J. Raux, M. Zaldarriaga and P. Astier, Supernovae, CMB, and gravitational leakage into extra dimensions, Phys. Rev. D 66 (2002) 024019, [astro-ph/0201164]. [27] J. S. Alcaniz, Some observational consequences of brane world cosmologies, Phys. Rev. D 65 (2002) 123514, [astro-ph/0202492]. [28] K. Freese and M. Lewis, Cardassian expansion: A model in which the universe is flat, matter dominated, and accelerating, Phys. Lett. B 540 (2002) 1–8, [astro-ph/0201229].

– 14 –

[29] V. Sahni and Y. Shtanov, Brane world models of dark energy, J. Cosmol. Astropart. Phys. 0311 (2003) 014, [astro-ph/0202346]. [30] M. D. Maia, E. M. Monte, J. M. F. Maia and J. S. Alcaniz, On the geometry of dark energy, Class. Quant. Grav. 22 (2005) 1623–1636, [astro-ph/0403072]. [31] J. Goodman and J. Weare, Ensemble samplers with affine invariance, Comm. App. Math. Comp. Sci. 5 (2010) 65–80. [32] M. Özer and M. O. Taha, A model of the universe free of cosmological problems, Nucl. Phys. B 287 (1987) 776–796. [33] H. A. Borges and S. Carneiro, Friedmann cosmology with decaying vacuum density, Gen. Relativ. Gravit. 37 (aug, 2005) 1385–1394, [gr-qc/0503037]. [34] I. L. Shapiro and J. Solà, On the possible running of the cosmological “constant”, Phys. Lett. B 682 (nov, 2009) 105–113, [0910.4925]. [35] S. Carneiro, M. A. Dantas, C. Pigozzo and J. S. Alcaniz, Observational constraints on late-time Λ(t) cosmology, Phys. Rev. D 77 (apr, 2008) 083504, [0711.2686]. [36] F. E. M. Costa and J. S. Alcaniz, Cosmological consequences of a possible Λ-dark matter interaction, Phys. Rev. D 81 (feb, 2010) 043506, [0908.4251]. [37] P. Wang and X.-H. Meng, Can vacuum decay in our universe?, Class. Quant. Grav. 22 (2005) 283–294, [astro-ph/0408495]. [38] P. C. Ferreira, J. C. Carvalho and J. S. Alcaniz, Probing interaction in the dark sector, Phys. Rev. D 87 (2013) 087301, [1212.2492]. [39] J. S. Alcaniz and N. Pires, Cosmic acceleration in brane cosmology, Phys. Rev. D 70 (2004) 047303, [astro-ph/0404146]. [40] A. Lue, The phenomenology of Dvali–Gabadadze–Porrati cosmologies, Phys. Rept. 423 (jan, 2006) 1–48, [astro-ph/0510068]. [41] L. Pogosian and A. Silvestri, The pattern of growth in viable f (R) cosmologies, Phys. Rev. D 77 (2008) 023503, [0709.0296]. [42] WMAP collaboration, E. Komatsu et al., Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmological interpretation, Astrophys. J. Suppl. 192 (2011) 18, [1001.4538]. [43] R. T. Bayes, An essay toward solving a problem in the doctrine of chances, Phil. Trans. Roy. Soc. Lond. 53 (1764) 370. [44] A. R. Liddle, Information criteria for astrophysical model selection, Mon. Not. Roy. Astron. Soc. 377 (2007) L74, [astro-ph/0701113]. [45] A. H. Jaffe, H0 and odds on cosmology, Astrophys. J. 471 (1996) 24, [astro-ph/9501070]. [46] M. P. Hobson, S. L. Bridle and O. Lahav, Combining cosmological datasets: Hyperparameters and Bayesian evidence, Mon. Not. Roy. Astron. Soc. 335 (2002) 377, [astro-ph/0203259]. [47] T. D. Saini, J. Weller and S. L. Bridle, Revealing the nature of dark energy using Bayesian evidence, Mon. Not. Roy. Astron. Soc. 348 (2004) 603, [astro-ph/0305526]. [48] R. Trotta, Applications of Bayesian model selection to cosmological parameters, Mon. Not. Roy. Astron. Soc. 378 (2007) 72–82, [astro-ph/0504022]. [49] D. Parkinson, P. Mukherjee and A. R. Liddle, A Bayesian model selection analysis of WMAP3, Phys. Rev. D 73 (2006) 123523, [astro-ph/0605003]. [50] R. Trotta, Bayes in the sky: Bayesian inference and model selection in cosmology, Contemp. Phys. 49 (2008) 71, [0803.4089]. [51] H. Jeffreys, Theory of Probability. Oxford Univ. Press, Oxford, 1961.

– 15 –

[52] SNLS collaboration, A. Conley et al., Supernova constraints and systematic uncertainties from the first 3 years of the Supernova Legacy Survey, Astrophys. J. Suppl. 192 (2011) 1, [1104.1443]. [53] SDSS collaboration, M. Sako et al., The data release of the Sloan Digital Sky Survey-II supernova survey, ArXiv e-prints (2014) , [1401.3317]. [54] SNLS collaboration, J. Guy et al., The Supernova Legacy Survey 3-year sample: type Ia Supernovae photometric distances and cosmological constraints, Astron. Astrophys. 523 (2010) A7, [1010.4743]. [55] A. G. Riess et al., New Hubble Space Telescope discoveries of Type Ia supernovae at z ≥ 1: Narrowing constraints on the early behavior of dark energy, Astrophys. J. 659 (2007) 98, [astro-ph/0611572]. [56] SDSS collaboration, M. Betoule et al., Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples, Astron. Astrophys. 568 (2014) A22, [1401.4064]. [57] SDSS Collaboration collaboration, D. J. Eisenstein et al., Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies, Astrophys. J. 633 (2005) 560–574, [astro-ph/0501171]. [58] D. J. Eisenstein and W. Hu, Baryonic features in the matter transfer function, Astrophys. J. 496 (1998) 605, [astro-ph/9709112]. [59] C. Blake et al., The WiggleZ dark energy survey: Joint measurements of the expansion and growth history at z < 1, Mon. Not. Roy. Astron. Soc. 425 (2012) 405, [1204.3674]. [60] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell et al., The 6dF Galaxy Survey: Baryon acoustic oscillations and the local Hubble constant, Mon. Not. Roy. Astron. Soc. 416 (2011) 3017, [1106.3366]. [61] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden and M. Manera, The clustering of the SDSS DR7 Main galaxy Sample – i. A 4 per cent distance measure at z = 0.15, Mon. Not. Roy. Astron. Soc. 449 (2015) 835, [1409.3242]. [62] BOSS collaboration, L. Anderson et al., The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Baryon acoustic oscillations in the Data Releases 10 and 11 galaxy samples, Mon. Not. Roy. Astron. Soc. 441 (2014) 24, [1312.4877]. [63] N. Padmanabhan, X. Xu, D. J. Eisenstein, R. Scalzo, A. J. Cuesta, K. T. Mehta et al., A 2 per cent distance to z = 0.35 by reconstructing baryon acoustic oscillations - i. Methods and application to the Sloan Digital Sky Survey, Mon. Not. Roy. Astron. Soc. 427 (2012) 2132, [1202.0090]. [64] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21 (1953) 1087–1092. [65] D. J. C. MacKay, Information Theory, Inference & Learning Algorithms. Cambridge University Press, New York, NY, USA, 2003. [66] J. Skilling, Nested sampling, AIP Conf. Proc. 735 (2004) 395–405. [67] F. Feroz, M. P. Hobson, E. Cameron and A. N. Pettitt, Importance nested sampling and the MultiNest algorithm, ArXiv e-prints (2013) , [1306.2144]. [68] A. Lewis and S. Bridle, Cosmological parameters from CMB and other data: A Monte Carlo approach, Phys. Rev. D 66 (2002) 103511, [astro-ph/0205436]. [69] P. Mukherjee, D. Parkinson and A. R. Liddle, A nested sampling algorithm for cosmological model selection, Astrophys. J. 638 (2006) L51–L54, [astro-ph/0508461]. [70] D. Foreman-Mackey, D. W. Hogg, D. Lang and J. Goodman, emcee: The MCMC hammer, Publ. Astron. Soc. Pac. 125 (2013) 306–312, [1202.3665]. [71] R. Allison and J. Dunkley, Comparison of sampling techniques for Bayesian parameter estimation, Mon. Not. Roy. Astron. Soc. 437 (2014) 3918–3928, [1308.2675].

– 16 –

[72] R. H. Swendsen and J.-S. Wang, Replica Monte Carlo simulation of spin-glasses, Phys. Rev. Lett. 57 (Nov, 1986) 2607–2609. [73] D. J. Earl and M. W. Deem, Parallel tempering: Theory, applications, and new perspectives, Phys. Chem. Chem. Phys. 7 (2005) 3910–3916. [74] P. M. Goggans and Y. Chi, Using thermodynamic integration to calculate the posterior probability in Bayesian model selection problems, AIP Conf. Proc. 707 (2004) 59–66. [75] A. G. Riess, L. Macri, S. Casertano, H. Lampeitl, H. C. Ferguson, A. V. Filippenko et al., A 3% solution: Determination of the Hubble constant with the Hubble Space Telescope and Wide Field Camera 3, Astrophys. J. 730 (2011) 119, [1103.2976]. [76] J. Sollerman, E. Mörtsell, T. M. Davis, M. Blomqvist, B. Bassett and e. a. Becker, A. C., First-Year Sloan Digital Sky Survey-II (SDSS-II) supernova results: Constraints on nonstandard cosmological models, Astrophys. J. 703 (oct, 2009) 1374–1385, [0908.4276]. [77] HST collaboration, W. L. Freedman et al., Final results from the Hubble Space Telescope key project to measure the Hubble constant, Astrophys. J. 553 (2001) 47–72, [astro-ph/0012376]. [78] W. L. Freedman, B. F. Madore, V. Scowcroft, C. Burns, A. Monson, S. E. Persson et al., Carnegie Hubble Program: A mid-infrared calibration of the Hubble constant, Astrophys. J. 758 (oct, 2012) 24, [1208.3281]. [79] S. H. Suyu, T. Treu, R. D. Blandford, W. L. Freedman, S. Hilbert, C. Blake et al., The Hubble constant and new discoveries in cosmology, ArXiv e-prints (feb, 2012) , [1202.4459].

– 17 –