determined from the theory of inflation. Thanks to (E.29) we can compute
$f(a)$:

$f(a)=\frac{a}{\delta_{m}(a,\vec{k})}\frac{d\delta_{m}(a,\vec{k})}{da}=\frac{a}{D(a)}\frac{dD(a)}{da}=\frac{a}{2P_{m}(a,\vec{k})}\frac{dP_{m}(a,\vec{k})}{da}.$
(E.30)

We have used $D(a)\sim\sqrt{P_{m}(a,\vec{k})}$. The observable
$f(a)\sigma_{8}(a)$ is computed at subhorizon scales and for most of the
cosmological models this leads to a second order differential equation for the
density contrast, which can be generally expressed as:

$\delta^{\prime\prime}_{m}+F(a,\vec{p})\delta^{\prime}_{m}+G(a,\vec{p})\delta_{m}=0$
(E.31)

where the dependence on $\vec{k}$ has been removed and the primes represent
derivatives with respect to the scale factor. The functions $F(a,\vec{p})$ and
$G(a,\vec{p})$ can vary from one model to another and they depend on the free
parameters embodied in the vector $\vec{p}$.

In order to compute $\sigma_{8}(a)$, which is the root mean square of matter
fluctuations on spheres of radius $R_{8}=8{h^{-1}}$ Mpc at the given value of
the scale factor, we need to employ the following expression:

$\sigma^{2}_{8}(a)=\frac{1}{2\pi^{2}}\int^{\infty}_{0}d{k}{k}^{2}P_{m}(a,\vec{k})W^{2}({k}R_{8})$
(E.32)

where ${k}\equiv|\vec{k}|$ and $W({k}R)$ is the top hat smoothing function:

$W({k}R)=\frac{3\left({\rm sin}({k}R)-{k}R{\rm cos}({k}R)\right)}{({k}R)^{2}}$
(E.33)

Usually the data is given in terms of the redshift instead of the scale factor
and without depending on $\vec{k}$, namely: $f(z)\sigma_{8}(z)$.

We want to stress the fact that the LSS data turn out to be crucial to
differentiate between models that have very similar behaviour at the
background level but not at the perturbation level. It is important to employ
a set of $f(z)\sigma_{8}(z)$ points totally free from correlations or double
counting in order to test the models in the fairest way possible. In Chapters
2-7 this idea has been followed carefully and when the correlations between
the data points have been provided, by the experimental teams, they have been
duly taken into account. To conclude this part we want to note that part of
the information, embodied in the data points, comes from such small scales
that non-linear effects come into play and therefore they need to be modeled.
Great improvements have been made in this area, turning the redshift-space
distortions data in a robust and reliable source of data.

#### Gravitational Lensing

General Relativity tells us that the presence of matter creates a local
curvature in space-time. As a consequence when a ray of light is affected by
this curvature its path is bent. This phenomenon causes a process known as
Gravitational Lensing. The light coming from distant galaxies is bent by the
gravitational effect exerted by the massive bodies present in the path of the
photon towards us. Therefore the real image of the distant galaxy, which is
behind the massive bodies but in the same line of sight, appears to be
distorted. In the recent years the improvements in the techniques to measure
this effect, carried out by the experimental teams [245, 75, 97, 98], have
turned this field into a very promising one and the information extracted can
be used to constraint cosmological models.

Depending on the amount of bending produced by the intervening masses we
consider two different regimes: Weak Lensing (WL) or Strong Lensing (SL).

Weak Lensing

The experimental teams like KiDS-450, DES or the HSC are able to measure the
Weak Lensing statistical distortion of angles and shapes of galaxy images
caused by the presence of inhomogeneities in the low-redshift Universe [75,
245, 352, 353, 76, 74]. The two-point correlation functions of these angle
distortions are related to the power spectrum of matter density fluctuations,
and from it, it is possible to obtain constraints on the parameter combination
$S_{8}=\sigma_{8}\sqrt{\Omega_{m}/0.3}$. However, one have to be careful when
including this data in the analysis, because of the following reason: at the
computational level, non-linear effects for small angular scales are
calculated with the Halofit module [383] (in case CLASS is used) which only
works fine for the $\Lambda$CDM and minimal extensions of it, as XCDM [102] or
CPL parametrization of the dark energy EoS parameter [103, 104]. Thus, it is
may not be able to model accurately the potential low-scale particularities of
the model under study. Indeed it is possible to remove these low angular
scales from the analysis, in order to avoid the operation of Halofit, but in
that case the loss of information is very big, and $S_{8}$ remains basically
unconstrained. 343434See http://kids.strw.leidenuniv.nl/sciencedata.php for
more details.

Strong Lensing

In Chapter 7 we use the data extracted from the six gravitational lensed
quasars of variable luminosity reported by the H0LICOW team. They measure the
time delay produced by the deflection of the light rays due to the presence of
an intervening lensing mass. After modeling the gravitational lens it is
possible to compute the so-called time delay distance $D_{\Delta{t}}$ (cf.
[97] and references therein). The fact of being absolute distances (instead of
relative, as in the SNIa and BAO datasets) allows them to directly constrain
the Hubble parameter in the context of the $\Lambda$CDM as follows:
$H_{0}=73.3^{+1.7}_{-1.8}$ km/s/Mpc . It turns out that for the three sources
B1608+656, RX51131-1231 and HE0435-1223, the posterior distribution of
$D_{\Delta t}$ can be well approximated by the analytical expression of the
skewed log-normal distribution,

$\mathcal{L}(D_{\Delta t})=\frac{1}{\sqrt{2\pi}(D_{\Delta
t}-\lambda_{D})\sigma_{D}}\,{\rm exp}\left[-\frac{(\ln(D_{\Delta
t}-\lambda_{D})-\mu_{D})^{2}}{2\sigma^{2}_{D}}\right]\,,$ (E.34)

where the corresponding values for the fitted parameters $\mu_{D}$,
$\sigma_{D}$ and $\lambda_{D}$ are reported in Table 3 of [97]. On the other
hand, the former procedure cannot be applied to the three remaining lenses,
i.e. SDSS 1206+4332, WFI 2033-4723 and PG 1115+080. From the corresponding
Markov chains provided by H0LICOW353535http://shsuyu.github.io/H0LiCOW/site/
we have instead constructed the associated analytical posterior distributions
of the time delay angular diameter distances for each of them. Taking
advantage of the fact that the number of points in each bin is proportional to
$\mathcal{L}(D_{\Delta t})$ evaluated at the average $D_{\Delta t}$ for each
bin, we can get the values for $-\ln(\mathcal{L}/\mathcal{L}_{max})$ and fit
the output to obtain its analytical expression as a function of $D_{\Delta
t}$. The fitting function can be as accurate as wanted, e.g. using a
polynomial of order as high as needed. The outcome of this procedure is used
instead of (E.34) for the three aforesaid lenses. Interestingly, the non-
detection of Strong Lensing time delay variations can be used to put an upper
bound on $\dot{G}/G$ at the redshift and location of the lens [384].

#### Prior on $H_{0}$

In Chapters 5 and 7 we include, as a prior, the value of the Hubble parameter
measured by the SH0ES collaboration, $H_{0}=(73.24\pm 1.74)$ km/s/Mpc [193]
and $H_{0}=(73.5\pm 1.4)$ km/s/Mpc [84] respectively. Both measures have been
obtained by applying the cosmic distance ladder method, based on chaining
different techniques, to measure distance to celestial objects, to finally
obtain the desired final distance. The second measurement includes some
refinements in the method which includes an improved calibration of the
Cepheid period-luminosity relation. It is based on distances obtained from
detached eclipsing binaries located in the Large Magellanic Cloud, masers in
the galaxy NGC $4258$ and Milky Way parallaxes. This measurement is in
$4.1\sigma$ tension with the value obtained by the Planck 2018 team under the
TTTEEE+lowE+lensing dataset, and using the $\Lambda$CDM model,
$H_{0}=(67.36\pm 0.54)$ km/s/Mpc. [31]. This represents the most acute tension
affecting the standard model of cosmology and as cannot be otherwise has
triggered a search for models able to accommodate the SH0ES measurement while
the CMB data is included in the analysis.

#### CMB

Without doubt the cosmic microwave background is one of the most important
tools at our disposal to test cosmological models. It provides us with a
unique opportunity to obtain information from the early Universe, namely at
redshifts of order $z\sim\mathcal{O}(1000)$, and then if it is combined with
the late-time cosmological data we obtain a robust data set to tighten the
constrains on free parameters. In order to obtain the perturbation equations,
that contain the physics behind the CMB, one must deal with the coupled
differential equations expressing the contribution of each component which
includes, among other things, the interaction between photons and baryons. See
[55] and [61] for a detailed description.

The question now is, how is this temperature field described today ? The
deviations with respect to the mean value of the CMB temperature can be
expressed as $\delta{T}=\delta{T}(z,\vec{x},\hat{n})$, where $\hat{n}$ is a
unitary vector that indicates the direction of the incoming photons. The CMB
can only be observed here in the Earth (actually near to the Earth thanks to
the satellites) and at present time, $z=0$. Taking this into account, while
the temperature field depends on $z$, $\vec{x}$ and $\hat{n}$, in practise, we
are only able to measure the dependence on the direction, or in other words
the anisotropies. It turns out to be convenient expand $\delta{T}$ in terms of
spherical harmonics, which can be understood as a kind of Fourier
transformation in 2-dimensions:

$\frac{\delta{T}(z,\vec{x},\hat{n})}{T(z)}=\sum^{\infty}_{\ell=1}\sum^{\ell}_{m=-\ell}a_{{\ell}m}(z,\vec{x})Y(\hat{n})$
(E.35)

being $Y(\hat{n})$ the spherical harmonics containing the angular dependence.
Upon inverting (E.35) we can get the expression for the $a_{\ell{m}}$
coefficients

$a_{\ell{m}}=\int d\Omega
Y^{*}(\hat{n})\frac{\delta{T}(z,\vec{x},\hat{n})}{T(z)}.$ (E.36)

They are characterized by having zero mean value but non-null variance, whose
expression can be obtained from:

$C_{\ell}=\frac{1}{2\ell+1}\sum^{\ell}_{m=-\ell}|a_{{\ell}m}|^{2}.$ (E.37)

This is the observable used to compare the theoretical predictions with the
observed power spectrum of the CMB. $\ell$ is known as the multipole moment
and is related to the angle $\theta$ through $\theta=\pi/\ell$. This last
relation means that the low-$\ell$ multipoles contain the information from the
large scale whereas the high-$\ell$ multipoles the small scale one. It is not
possible to have an analytical expression for $C_{\ell}$ since they involve
quantities that can only be obtained by getting the numerical solutions of a
coupled system of differential equations. So, they must be obtained
numerically. In Chapters 6-7 we have used the Einstein-Boltzmann system solver
CLASS [265] to do so.

It is possible to compress, in a very efficient way, the CMB information in
what is called as CMB distance priors, namely $R$ (shift parameter),
$\ell_{a}$ (acoustic length) and their correlations with the reduced baryon
mass parameter and the spectral index $(\omega_{b},n_{s})$. It has been shown
in [385] that although this compression loses information compared to the full
CMB likelihood, such information loss becomes negligible when more data is
added. The theoretical expressions for $R$ and $\ell_{a}$ are:

$\displaystyle R(\vec{p})$
$\displaystyle=\sqrt{\Omega_{m}}\int_{0}^{z_{*}}\frac{dz}{E(z)}$ (E.38)
$\displaystyle\ell_{a}(\vec{p})$
$\displaystyle=\pi(1+z_{*})\frac{D_{A}(z_{*})}{r_{s}(z_{*})}$ (E.39)

where $z_{*}$ is the redshift at decoupling. Its precise value depends weakly
on the parameters, and it is obtained from the fitting formula [386]:

$z_{*}=1048\,\left(1+0.00124\,\omega_{b}^{-0.738}\right)\left(1+g_{1}\omega_{m}^{g_{2}}\right)\,,$
(E.40)

with

$\displaystyle
g_{1}=\frac{0.0783\,\omega_{b}^{-0.238}}{1+39.5\,\omega_{b}^{0.763}}$
$\displaystyle g_{2}=\frac{0.560}{1+21.1\omega_{b}^{1.81}}\,.$ (E.41)

### Statistical analysis

As we have stated, statistics is a fundamental part of the analysis of a given
cosmological model. Here we are going to present the fundamental concepts that
we must consider if we want to constraint the parameter space of the models
under study. Furthermore, we will provide the statistical tools that allow us
to compare the performance of the different models, even if they have
different number of degrees of freedom.

#### Model fitting

It is time to properly introduce the methods that have been used in this
thesis in order to get the results displayed in the corresponding tables and
figures that can be found in Chapters 2-7. First of all, we should bear in
mind, the ultimate goal of the model fitting procedure, namely: extracting
information on the cosmological parameters, that characterize the studied
model, from the considered observational data. The more precise data, covering
a wide range of redshift we have, the better constrained the free parameters
of a given model will be. As a clear example of this statement one can take
the evolution of the constrains on the $\Omega_{\Lambda}$ parameter over the
years, since as mentioned in the introductory chapter, $\Omega_{\Lambda}$ was
found to be different from zero at $\sim 5\sigma$ c.l. employing only distant
supernovae, while by considering the recent CMB data from the Planck team this
evidence rises until the astonishing level of $\sim 94\sigma$ c.l. As it is
well-known, the $\Lambda$CDM model, has proven to be very successful in
passing the cosmological observational tests over the last years. What we want
to know now is whether models that allow small deviations with respect to
$\Lambda$CDM’s predictions are favoured by the data.

Two approaches can be taken in order to extract the desired information from
the cosmological data, the frequentist and the bayesian. Determining the
conceptual differences among them is beyond the scope of this thesis, so, we
are just going to focus on explaining the bayesian approach since it has been
proven to be the most convenient one to study cosmological models. It is based
on the famous Bayes theorem, which can be expressed as:

$\displaystyle p(\theta|x,M)=\frac{p(x|\theta,M)p(\theta|M)}{p(x|M)}\,.$
(E.42)

In words, it says that the posterior probability of $\theta$ (which represents
the free parameters), namely $p(\theta|x,M)$, given the model $M$ and the data
$x$ is equal to the probability of the data given the parameters of the model
(i.e. the likelihood $p(x|\theta,M)$) times the prior probability of $\theta$,
$p(\theta|M)$, divided by the probability of the data $x$ (usually in the form
of a probability distribution function (PDF)).

What we are looking for, is the expression of the posterior probability
function, which contains the information on how the characterizing parameters
of a given model are distributed once the data $x$ have been considered in
addition to some previous knowledge on the parameters encoded in the priors.
The function $p(x|\theta,M)$ is usually referred as the likelihood and is
related to the $\chi^{2}(\theta)$ (as long as the data can be described by a
Gaussian function) function defined in (E.1) in the following way:

$\mathcal{L}=\mathcal{L}_{0}e^{-\frac{\chi^{2}(\theta)}{2}}.$ (E.43)

Where $\mathcal{L}_{0}$ is just a normalization constant. Note that while in
the previous section we have employed $\vec{p}$ to denote the vector of free
parameters, here for convenience we use $\theta$. The likelihood function
gives us information about the parameters, $\theta$, given the data. In
particular, it says how likely a given data would be for a specific set of
values of the parameters $\theta$. It is important to remark that it is not a
probability distribution function of the parameters. In the next section we
will provide more information about the function $p(x|M)$ due to the fact that
it plays an important role in the model selection procedure. For the purpose
of this section what is important is the relation

$p(\theta|x,M)\propto p(x|\theta,M)p(\theta|M)$ (E.44)

since it allows us to sample the posterior distribution function once we have
established the shape of the priors and the form of the likelihood function.
This is tantamount to say that we will be working with the kernel of the
posterior instead of with the full probability distribution function. The
kernel is the part of the probability distribution function that remains when
all the parts that do not depend on the parameters are removed. This concept
can be easily understood with a simple example. Consider a random variable $X$
which is distributed according to the expression of a Gaussian function
$p(X|\sigma,\mu)=e^{-(X-\mu)^{2}/2\sigma^{2}}/\sqrt{2\pi\sigma^{2}}$, then the
kernel would be in this case $e^{-(X-\mu)^{2}/2\sigma^{2}}$.

Let us briefly discuss the general features of some of the most common model
fitting procedures. The first one is the method known as the grid method,
which is a quite straight-forward approach. Basically it consists in building
a grid for the involved parameters in the parameter space and compute the
value of the likelihood function at each point of the grid. Once all the
calculations are done and thanks to the relation (E.44) it is possible to
estimate not only those values of the parameters that maximizes the likelihood
function (or minimizes the $\chi^{2}$ function) but also the errors associated
to the free parameters that represent the d.o.f. of the model under
consideration. This method is very robust as it does not make any assumption
about the shape of the posterior distribution, only relies on the fact that
the probability distribution functions of the experimental measurements are
correct. However it has an important drawback, even if we consider that we are
able to make a reasonable guess about where is the maximum of the likelihood
function in order to be sure that it is into the grid that we build. The main
problem is, as could not be otherwise, the computational time. When our model
contains more than 3-4 parameters, and note that this is perfectly reasonable,
this method requires an enormous amount of time to compute the value of the
likelihood function at all the desired point.

A much more efficient approach is the so-called Fisher matrix method, which is
based on the central limit theorem that claims that any well behaved
distribution function can be approximated by a Gaussian distribution near its
maximum. In other words, even if we do not know the shape of the posterior
distribution, we can obtain at least an approximation, considering that the
likelihood is well-described by a multivariate Gaussian near the maximum, and
once again, the relation (E.44). Let us write down the approximation taken
into account:

$\mathcal{L}(\theta)\simeq\mathcal{L}(\hat{\theta})e^{-\frac{1}{2}(\theta_{i}-\hat{\theta}_{i})F_{ij}(\theta_{j}-\hat{\theta}_{j})}$
(E.45)

where by $\hat{\theta}$ we mean the vector of best fit values, which maximizes
the likelihood. The term $F_{ij}$ is called the Fisher matrix and its
expression will be clarified in a moment. Note that the above expression
implies that the likelihood is not only a Gaussian function of the data but
also of the parameters. Depending on the shape of the priors the posterior
distribution is also a multivariate-Gaussian distribution. Taking the
logarithm on both sides of (E.45) and expanding at second order near
$\hat{\theta}$ we can obtain the expression:

$\ln(\mathcal{L}(\theta))\simeq\ln(\mathcal{L}(\hat{\theta}))+\frac{1}{2}\frac{\partial^{2}\ln{\mathcal{L}(\theta)}}{\partial\theta_{i}\partial\theta_{j}}\Bigg{|}_{\theta=\hat{\theta}}(\theta_{i}-\hat{\theta}_{i})(\theta_{j}-\hat{\theta}_{j}).$
(E.46)

We have not included the first derivative of the logarithm of the likelihood
function because the expression is evaluated at $\hat{\theta}$ which by
definition is the vector of parameters that extremizes (in this case
maximizes) the likelihood. Now we are able to identify the Fisher matrix with
its expression

$F_{ij}\equiv\frac{\partial^{2}\ln{\mathcal{L}(\theta)}}{\partial\theta_{i}\partial\theta_{j}}\Bigg{|}_{\theta=\hat{\theta}}.$
(E.47)

Even if it is not possible to compute analytically the Fisher matrix, very
often we can easily compute it numerically due to the smooth behaviour of the
function around the maximum. What makes this method extremely useful is the
fact that once the Fisher matrix is computed it is straight-forward to get the
errors of the parameters and the two-dimensional contour plots which are going
to be ellipses. In order to marginalize over the parameters we are not
interested in, we only need to remove the corresponding rows and columns from
the inverse of the Fisher matrix. On the other hand, to obtain the length of
the semiaxes of the ellipsoidal contours we need to take the square root of
the eigenvalues of the inverse of the Fisher matrix. Some numerical problems
could arise from the procedure to get the inverse of the Fisher matrix, so we
must be careful with that. As it can be understood this method implies a
considerable reduction in the computational time in comparison to the grid
method even if we study models with a large number of free parameters.
Nevertheless, it is important to remark that this is only accurate, as long
as, the likelihood function and consequently the posterior distribution can be
well-approximated by a multivariate Gaussian function, otherwise the
deviations from the normal distribution can infer biased estimation for the
errors. In order to obtain the results presented in Chapters 2-5 the Fisher
matrix method has been employed. However, we have carefully studied possible
deviations from the Gaussian behaviour. See the before mentioned chapters to
know all the details.

We will finish this brief summary by talking about the well-known Monte Carlo
Markov Chains (MCMC) method. We remember that the main disadvantage of the
grid method is the huge computationally time that is required to explore the
desired piece of the parameter space. The solution provided to this issue by
the MCMC method is to explore the landscape of the space parameter under study
with random jumps. In other words, using the likelihood function and the
selected priors, according to the relation (E.44) the MCMC techniques allow us
to explore the parameter space and keep those points that are accepted by the
selection algorithm employed (see for instance [195, 196] ). Once this is done
we end up with a collection of n-dimensional points labeled by the values of
the free parameters. We call this the Markov Chains, and from them we can get
the corresponding confidence level for the involved parameters, the
correlations among them and the contour plots. As the reader may have noticed,
we have not mentioned that the likelihood function must have a Gaussian shape.
This can be considered the main advantage of the MCMC method over the Fisher
matrix approach. In other words, we do not need to make any assumption about
the form of the likelihood function which allows us to study cases where the
non-gaussianities may have an important role. On the contrary, clearly the
MCMC method is going to take more time than the Fisher matrix method. A
natural question at this point could be, how long should the Markov Chains be
? A convergence criterion must be applied at this point, see for instance
[387].

As it can be appreciated all the methods have advantages and disadvantages
which makes recommendable to bear in mind all of them in order to use the most
suitable one in each case. As stated at the beginning of this appendix the
model fitting is unavoidable if one wants to carry out a proper study of a
given model. No matter how well-looking are the theoretical predictions of
some particular model, if at the end of the day, the model is not able to pass
the experimental tests it must be ruled out, at least until new experimental
data come out.

#### Model selection

So far, we have been dealing with the problem of comparing the theoretical
predictions for a given model, with the observational data considered and how
we can narrowing down the parameter space through different methods like the
Fisher matrix or like the Monte Carlo Markov Chains techniques. When all this
is done is time for the next natural question: does, the model studied in the
first place, reproduce the data better than any other one ? Answering this
question is known as the model selection problem.

Among the different options, in this appendix, we focus on the selection
procedure called the Bayesian evidence or marginal likelihood. The
significance of the results obtained can be further appraised by computing the
Bayesian evidence, which is based on evaluating the posterior probability of a
model $M$ given the data $x$ and the priors, see e.g. [55]. The relevant
quantity we are looking for is the so-called Bayes factor, which does not
depend on the arbitrarily assigned prior probability to the given model $M$.
This factor can be obtained by making use of the Bayes theorem previously
mentioned (E.42). As can be appreciated by taking a look of that expression,
the quantity $p(x|M)$ does not depend on the values of $\theta$. Therefore, if
we normalize the posterior probability to one and integrate over all the
parameters $\theta$ on both sides of Eq. (E.42), the corresponding integral in
the numerator on the r.h.s. must be equal to $p(x|M)$:

$\displaystyle p(x|M)=\int d\theta\,p(x|\theta,M)\,p(\theta|M)\,.$ (E.48)

This likelihood integral over all the values that can take the parameters
$\theta$ is called the marginal likelihood or evidence [55].

An analogous formula to (E.42) can also be applied to estimate the posterior
probability that a model $M$ is true given a measured data set $x$. Thus,
following the same scheme, the posterior probability of the model given the
data, $p(M|x)$, must be equal to the probability of the data given the model
(irrespective of the values of $\theta$) – i.e. the marginal likelihood (E.48)
— times the prior probability of the model, divided by the PDF of the data.
Writing this relation for two cosmological models $M_{i}$ and $M_{j}$ that are
being compared, we find that the ratio of posterior probabilities of these
models is equal to the ratio of model priors times a factor:

$\frac{p(M_{i}|x)}{p(M_{j}|x)}=\frac{p(M_{i})}{p(M_{j})}\,B_{ij}\,.$ (E.49)

Such factor, $B_{ij}=p(x|M_{i})/p(x|M_{j})$, is the so-called Bayes factor of
the model $M_{i}$ with respect to model $M_{j}$. It gives the ratio of
marginal likelihoods (i.e. of evidences) of the two models. Note that it
coincides with the ratio of posterior probabilities of the models only if the
prior probabilities of these models are the same. This is generally assumed
(“Principle of Insufficient Reason”) and hence the comparison of the two
models $M_{i}$ and $M_{j}$ is usually performed directly in terms of $B_{ij}$.

In the literature it has been customary to define the Bayes information
criterion (BIC) through the parameter ${\rm BIC}=\chi^{2}_{\rm min}+n\,\ln N$,
where $\chi^{2}_{\rm min}$ is the minimum of the $\chi^{2}$ function, $n$ is
the number of independent fitting parameters and $N$ is the total number of
data points. One can show that the Bayes factor can be estimated through
$B_{ij}=e^{\Delta{\rm BIC}/2}$ where $\Delta{\rm BIC}={\rm BIC}_{j}-{\rm
BIC}_{i}$ is the difference between the values of BIC for models $M_{j}$ and
$M_{i}$ [242, 187], i.e.

$\Delta{\rm BIC}=\Delta\chi^{2}_{\rm min}-\Delta n\,\ln N\,,$ (E.50)

$\Delta{\rm BIC}=2\ln B_{ij}$ |  Bayesian evidence of model ${M}_{i}$ versus $M_{j}$
---|---
$0<\Delta{\rm BIC}<2$ | weak evidence (consistency between both models)
$2<\Delta{\rm BIC}<6$ | positive evidence
$6<\Delta{\rm BIC}<10$ | strong evidence
$\Delta{\rm BIC}\geq 10$ | very strong evidence
$\Delta{\rm BIC}<0$ | counter-evidence against model $M_{i}$
Table 1: Conventional ranges of values of $\Delta{\rm BIC}$ used to judge the
observational support for a given model $M_{i}$ with respect to the reference
one $M_{j}$.

where $\Delta\chi^{2}_{\rm min}=\left(\chi^{2}_{\rm
min}\right)_{j}-\left(\chi^{2}_{\rm min}\right)_{i}$ is the difference between
the minimum values of $\chi^{2}$ for each model, and $\Delta n=n_{i}-n_{j}$ is
the difference in the number of independent fitting parameters of $M_{i}$ and
$M_{j}$, both describing the same $N$ data points. The added term to
$\Delta\chi^{2}_{\rm min}$, i.e. $\Delta n\,\ln N$, represents the penalty
assigned to the model having the largest number of parameters ($\Delta n>0$).
Indeed, if $M_{j}$ has less parameters than $M_{i}$ (i.e. $n_{i}>n_{j}$) we
expect that $\left(\chi^{2}_{\rm min}\right)_{i}<\left(\chi^{2}_{\rm
min}\right)_{j}$ and hence $\Delta\chi^{2}_{\rm min}>0$. However, since
$\Delta n>0$ the last term of (E.50) indeed penalizes the model with more
parameters and compensates in part for its (presumably smaller) value of
$\chi^{2}_{\rm min}$. Defined in this way, a positive value of the expression
(E.50) denotes that the model $M_{i}$ is better than model $M_{j}$, and the
larger is $\Delta{\rm BIC}$ the better is $M_{i}$ as compared to $M_{j}$.
Clearly, the Bayesian criterion allows a quantitatively implementation of
Occam’s razor.

Even though the simple and very economic formula (E.50) is useful and has been
applied in many places of the literature, see e.g. [77, 79, 78, 251] and
references therein, it is only an approximate formula. The exact value of
$\Delta{\rm BIC}$ associated to the full Bayes factor requires to evaluate

$\Delta{\rm BIC}=2\,\ln B_{ij}=2\,\ln\frac{\int
d\theta_{i}\,p(x|\theta_{i},M_{i})\,p(\theta_{i}|M_{i})}{\int
d\theta_{j}\,p(x|\theta_{j},M_{j})\,p(\theta_{j}|M_{j})}\,,$ (E.51)

where $\theta_{i}$ and $\theta_{j}$ are the two sets of free parameters
integrated over for each model. We refer to this (exact) value of $\Delta{\rm
BIC}$ as the full “Bayesian evidence” of model $M_{i}$ versus model $M_{j}$.
It represents the optimal implementation of Occam’s razor. Formula (E.50)
provides an estimate (sometimes good, sometimes rough) of the cumbersome
expression (E.51). The evaluation of the latter is carried out in practice
using the MCMC’s for the statistical analysis of the model and it can be a
formidable numerical task. Fortunately it can be speeded up with the help of
the recent numerical code MCEvidence [388]. In Table 1 we collect the ranges
of values of $\Delta{\rm BIC}$ that are conventionally used in the literature
to quantify the observational support of a given model $M_{i}$ with respect to
some reference one $M_{j}$ [242, 187]. Consistently with the approximate
formula discussed above, positive values of $\Delta{\rm BIC}$ in the indicated
intervals favor the model $M_{i}$ over model $M_{j}$. Usually $M_{j}$ is the
$\Lambda$CDM and $M_{i}$ is any of the considered beyond the standard model
(BSM). Small positive values near one just denote that the two models under
competition are comparable (i.e. that there is no marked preference of $M_{i}$
over $M_{j}$), but larger and positive values of $\Delta{\rm BIC}$ increase
the support of $M_{i}$ (BSM) versus $M_{j}$ ($\Lambda$CDM). Negative values of
$\Delta{\rm BIC}$, in contrast, would lead to the opposite conclusion, i.e.
that the BSM is penalized as compared to the $\Lambda$CDM and hence that the
latter is preferred.

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