vac}=\rho^{0}_{\Lambda}$ and $p_{\rm vac}=-\rho^{0}_{\Lambda}$ are constants
since no evolution in time is considered within the standard model of
cosmology. We denote the total neutrino contribution as
$\rho_{\nu}\equiv\rho^{\rm R}_{\nu}+\rho^{\rm NR}_{\nu}$ while for the
pressure we have $p_{\nu}\equiv p^{\rm R}_{\nu}+p^{\rm NR}_{\nu}$. We can
easily work out the expression for the normalized Hubble function in terms of
the relative energy densities $\Omega^{0}_{i}=\rho^{0}_{i}/\rho^{0}_{c}$

$E^{2}(a)\equiv\frac{H^{2}(a)}{H^{2}_{0}}=\left(\Omega^{0}_{b}+\Omega^{0}_{dm}\right)a^{-3}+\Omega^{0}_{\gamma}{a^{-4}}+\frac{\rho_{\nu}(a)}{\rho^{0}_{c}}+\Omega^{0}_{\Lambda}$
(1.115)

This function is fundamental to characterize a cosmological model at the
background level.

Now that we have the expression of the Hubble function for the $\Lambda$CDM,
as well as, the energy density for each component, we can compute two
important moments in the cosmic history: the equality time between matter and
radiation and the moment where the Universe transits from the decelerating
regime to the accelerated one. For the sake of simplicity we will consider
that all neutrinos are massless and of course relativistic, i.e. $N_{\rm
eff}=3.046$ in (1.109).
The first of the aforementioned points is easy to find since all we need to do
is equal the corresponding expressions for energy density of matter and
radiation $\rho_{r}(z_{\rm eq})=\rho_{m}(z_{\rm eq})$ and find the value
$z_{\rm eq}$ that meets the equality

$z_{\rm eq}=\frac{\Omega^{0}_{m}}{\Omega^{0}_{r}}-1\simeq 3405.$ (1.116)

Where we have used the values $\omega_{b}=0.02237$ and $\omega_{dm}=0.1200$
from the Planck 2018 TTTEEE+lowE+lensing [31] results. Beyond this point, the
non-relativistic matter becomes the dominant component of the Universe which
means that the structure formation process can start since the pressure from
the radiation component progressively decreases its preventive effect. At the
recombination time, when baryons were released from photons, the structure
formation process speeds up, and gives rise to the structure we can observe in
the present time.

Regarding the second point we need to employ the definition of the
deceleration parameter commonly denoted as

$q\equiv-\frac{\ddot{a}}{aH^{2}}=-1-\frac{a}{2H^{2}}\frac{dH^{2}}{da}.$
(1.117)

Obviously the transition point has to happen much later than the period where
the radiation was the dominant component, consequently we can neglect the
contribution of $\Omega_{r}(a)$ to simplify the expression. In the
$\Lambda$CDM, under the considerations that we mentioned, the deceleration
parameter takes the form

$q=\frac{1}{2}\frac{\Omega^{0}_{m}{a^{-3}}-2\Omega^{0}_{\Lambda}}{\Omega^{0}_{\Lambda}+\Omega^{0}_{m}{a^{-3}}}.$
(1.118)

To find the transition point $z_{\rm tr}$ we only need to impose the condition
$q=0$, which is fulfilled by the following value of the redshift

$z_{\rm tr}=\left(\frac{2\Omega^{0}_{\Lambda}}{\Omega^{0}_{m}}\right)^{1/3}-1$
(1.119)

At this point, where the CC term is the dominant component, the Universe
enters in a phase of accelerating expansion where galaxies are moving away
from each other.

####  Inflation

It is worthwhile to dedicate a few words to explain inflation, which is a
brief period of the cosmic history where an exponential expansion of the space
takes place during the early Universe [24, 25, 26, 27]. It is important to
make crystal-clear the fact that we have no direct experimental evidence for
inflation, however, as it happens with DE, the indirect evidences supporting
inflation are numerous and it turns out to be really complicate to explain the
current Universe without taking it into account. So, hereafter we assume
inflation as a part of the standard model of cosmology. This is the reason why
a short explanation of the general features of the mechanism and its main
consequences are required. See [62, 63, 64, 65, 66, 67, 68, 69, 70, 71] to
know more about different inflationary models.

As we have already mentioned, the inflationary phase consists in a very rapid
expansion of the space which happens after the Big Bang and before the
radiation dominated epoch. As we have seen previously, in order to have an
accelerating expanding Universe, the energy budget, must be dominated by a
component with negative pressure. As expected there are several ways to
implement inflation. Probably, one of the most studied types in the
literature, are the so-called inflaton models where inflation is driven by a
scalar field, called inflaton. Even if we cannot say that there is a standard
model for inflation, here we are going to present the most important features
of it within the context of inflaton models. To know more about alternatives
to inflaton models see [72]. The associated energy density and pressure can be
written in the usual way for a scalar field model:

$\displaystyle\rho_{\Phi}$ $\displaystyle=\frac{1}{2}\dot{\Phi}^{2}+U(\Phi)$
(1.120) $\displaystyle p_{\Phi}$
$\displaystyle=\frac{1}{2}\dot{\Phi}^{2}-U(\Phi)$ (1.121)

where $\Phi$ is a 1-dimensional field in natural units and $U(\Phi)$ is the
potential considered in the different inflationary models. The dots represent
derivative with respect to the cosmic time. As it can be appreciated from
(1.121) what it takes to get a negative pressure is a field configuration
where the potential energy density dominates over the kinetic term. This can
be realized if we consider the slow-roll approximation where the scalar field
slowly evolves from the false vacuum of the potential towards the absolute
minimum. Due to the slow evolution, the kinetic energy can be completely
neglected and $\rho_{\Phi}$ can be well approximated as a constant. In this
scenario, clearly the energy density associated to the scalar field quickly
becomes the dominant component of the Universe which means we can write the
Friedmann equation as:

$\frac{1}{a}\frac{da}{dt}=\sqrt{\frac{8\pi{G_{N}}\rho_{\Phi}}{3}}\equiv
H_{\Lambda}={\rm const}.$ (1.122)

The above differential equation can be easily solved to find the scale factor
for the inflationary period

$a(t)=a_{0}e^{H_{\Lambda}(t-t_{0})}\quad t>t_{0}$ (1.123)

where $a_{0}$ and $t_{0}$ are the scale factor and the cosmic time at the
beginning of inflation respectively. As the scalar field goes down the hill of
the potential, the potential energy density decreases whereas the kinetic
energy increases. The end of inflation takes place when $\dot{\Phi}^{2}\simeq
U(\Phi)$, i.e. the potential term is no longer the dominant one, and as a
consequence, the pressure is not negative anymore being the ultimate
consequence the reduction of the accelerated expansion of the Universe. Once
this has happened the scalar field falls into the ground state and starts to
oscillate around the minimum, giving rise to the process known as reheating.
Unfortunately our knowledge about this mechanism is very poor but the main
idea is to transfer the remaining energy, from the inflaton field, to the
creation of standard model particles which filled the Universe.

The natural question, having reached this point, is: why do we need an early
period of expansion ? So far, we have only explained what is inflation and the
main features of a toy model that allows us to implement it via a scalar
field. It is time then to list the problems that inflation solves:

  * •

Horizon Problem: As we do know now the CMB is highly isotropic. Regardless of
the direction we look at in the sky we observe an spectra coming from an
approximate black body with temperature $T_{\gamma}(a=1)=2.72548\pm 0.00057$K
[56] and tiny anisotropies of order
$\Delta{T_{\gamma}}/T_{\gamma}\sim\mathcal{O}(10^{-5})$ [31]. This fact points
out that early on the CMB photons were in thermal equilibrium. After some
moments of reflection a question pops up in our mind, how is this possible ?
Should not photons coming from different directions have different temperature
since they were never in causal contact ? What is more, we can extend these
type of question to the structure formation observed in the Universe, how is
possible that the Universe is isotropic and homogeneous ? This is known as the
horizon problem. Inflation provides a beautiful solution to this problem since
previously to the rapid expansion, caused by the inflation, the particles were
in causal contact, reaching the thermal equilibrium, and it was after the end
of inflation when the particles became disconnected from each other and evolve
independently.

  * •

Flatness Problem: From the Friedmann equation we can obtain the following
relation valid for any value of the scale factor $1-\Omega_{\rm
tot}(a)=\Omega_{k}(a)=-k/\dot{a}^{2}$, where $k$ is the curvature parameter
which determines the geometry of the Universe. If we derive the relation
w.r.t. the cosmic time we are lead to the relation

$\frac{d}{dt}\left(1-\Omega_{\rm
tot}(a)\right)=-2\frac{\ddot{a}}{\dot{a}}\left(1-\Omega_{\rm tot}(a)\right).$
(1.124)

For most of the cosmic history, until $z\sim 1$, the Universe has undergone a
phase of decelerating expansion, which means that
$\ddot{a}/\dot{a}=-|\ddot{a}/\dot{a}|$, consequently the quantity
$\left(1-\Omega_{\rm tot}(a)\right)$ grows with time. If now, we take the
value from Planck 2018 TTTEEE+lowE+lensing+BAO [31], $\Omega^{0}_{k}=0.0007\pm
0.0019$, we can infer that in order to get such a small value for
$\Omega^{0}_{k}$ the value of $\Omega_{k}(a)$, at the inflation time, had to
be really small. This is known as the flatness problem. Without inflation, we
need to invoke a very precise fine-tuning in the initial conditions. However
with inflation this situation is reached naturally due to the fact that, using
(1.123) we can get $1-\Omega_{\rm tot}(a)\sim e^{-2H_{\Lambda}(t-t_{0})}$,
which means that because of inflation $\Omega_{\rm tot}\simeq 1$ right after
the end of inflation.

  * •

Exotic-relics problem: Some exotic particles, such as the magnetic monopole,
are predicted within a given model of grand unified theory, however they have
never been observed. An accelerated expansion of the Universe, as the one
predicted by inflation, could have diluted their density, making today almost
impossible to observe them.

  * •

Origin of the structure formation: One of the most important advantages of
inflation is the obtaining of a mechanism capable to explain the origin of the
structure formation. The quantum fluctuations of the inflaton field, generated
during inflation, are stretched up to macroscopic scales due to the
exponential scale factor. It is generally claimed that these amplified
fluctuations later served as the seed of the structures that we currently
observe.

####  Problems and tensions of the $\Lambda$CDM model

So far, the cosmological standard model, with the CC as a building block,
turned out to be very successful in explaining the different phenomena
observed. The explanation of the well-known baryonic acoustic oscillations
(BAO) or the accurate prediction of the CMB spectra which contains
anisotropies in the temperature
$\Delta{T_{\gamma}}/T_{\gamma}\sim\mathcal{O}(10^{-5})$ are good examples yet
not the only ones.
However, as we shall see in this section, the $\Lambda$CDM suffers from some
problems and tensions, which may be interpreted either as systematic errors
affecting the measurements or as a signal of new physics. Since we have not
the ability to discern whether or not the observational data are affected by
some sort of systematic errors we are going to focus on the theoretical
problems and how they can be related with the possibility of having physics
beyond the standard model. The first one of them has already been treated in
extension, the cosmological constant problem, the tremendous mismatch between
the predicted value for the vacuum energy density in the context of QFT,
namely $\rho^{\rm QFT}_{\Lambda,{\rm ind}}$ and the observed value
$\rho^{0}_{\Lambda}$ which turns out to be $\rho^{\rm QFT}_{\Lambda,{\rm
ind}}/\rho^{0}_{\Lambda}\sim\mathcal{O}(10^{55})$ in natural units. Despite
the great efforts made so far we have to admit we are still far from solving
the problem. The available cosmological data can only be explained, in the
context of GR, if there is some form of repulsive energy which causes the
accelerated expansion of the Universe. The $\Lambda$CDM contemplates the
simplest of all possibilities, namely a constant vacuum energy density whose
value is equal to the observed one $\rho_{\rm vac}=\rho^{0}_{\Lambda}$ at any
time in cosmic history. Even if the match in the comparison among the
cosmological data and the theoretical predictions within the $\Lambda$CDM is
astonishing, we still have some questions pending to deal with: which is the
origin of the term $\rho^{0}_{\Lambda}$ ? Why precisely this tiny value ? Why
this term remains constant while the other components evolve as the Universe
expands ? The pressing need to answer these type of questions is one of the
main reasons that drives the seek for other cosmological models beyond the
$\Lambda$CDM.
Another theoretical problem related with $\rho^{0}_{\Lambda}$ is the cosmic
coincidence problem. It can be formulated simply with just one question: why,
precisely now, $\rho^{0}_{\Lambda}/\rho^{0}_{m}\sim\mathcal{O}(1)$ ? While it
is true that this fact can be just a coincidence, it is completely normal to
ask ourselves, how on earth, if $\rho_{m}\sim a^{-3}$ while
$\rho^{0}_{\Lambda}$ remains constant, they are of the same order in the
present time ? As stated in [37] in order to consider that this fact is just a
coincidence $\rho^{0}_{\Lambda}$ should not be linked with $\rho_{m}(a)$ and
the equality of both energy densities could have happened at any moment of the
cosmic history. It is not the main goal of this thesis to go into the details
of this controversial problem, to know more about the issue [73] and
references therein.
The cosmological constant problem, as well as, the cosmic coincidence problem
are theoretical problems that are rooted in our inability to explain the
origin and the possible evolution of the $\rho_{\rm vac}$ term. Nevertheless,
the $\Lambda$CDM is affected also by problems related with the apparent
discrepancy between the obtained values for some parameters when different
types of cosmological data are employed. Let us briefly summarize (they will
be widely discussed in the next chapters) the main two problems of this type.
The first of them is known as the $\sigma_{8}$-tension, and as the name
already indicates, affects the value of the parameter $\sigma_{8}(z)$, the
root-mean-square mass fluctuations on $R_{8}=8h^{-1}$Mpc scales

$\sigma^{2}_{8}(z)=\frac{1}{2\pi^{2}}\int^{\infty}_{0}dk{k^{2}}P_{m}(k,z)W^{2}(kR_{8}),$
(1.125)

where $P_{m}(k,z)$ is the matter power spectrum and $W(kR_{8})$ is the top hat
smoothing function. Recently, instead of quantifying the tension through the
parameter $\sigma_{8}(0)$, another parameter has been employed

$S_{8}=\sigma_{8}(0)\sqrt{\frac{\Omega^{0}_{m}}{0.3}}$ (1.126)

which as can be appreciated also involves the value of the relative matter
energy density at present time. The obtained value for the above parameter in
the $\Lambda$CDM model, considering the data Planck 2018 TTTEEE+lowE+lensing
[31] is $S_{8}=0.832\pm 0.013$. On the other hand, the values obtained by
different collaborations, using the weak gravitational lensing of galaxies,
are DES [74] $S_{8}=0.777^{+0.036}_{-0.038}$, KiDS-450 [75] $S_{8}=0.745\pm
0.039$ and HSC [76] $S_{8}=0.780^{+0.030}_{-0.033}$ respectively. In order to
properly quantify the tension among the obtained value by the Planck
collaboration and the weak gravitational lensing measurements we use the
following useful expression

$T_{X}=\frac{|X_{i}-X_{j}|}{\sqrt{\sigma^{2}_{X_{i}}+\sigma^{2}_{X_{j}}}}$
(1.127)

where $X$ is the observable we are considering to account for the tension and
$X_{i},X_{j}$ are the measurements that are under comparison. Therefore, with
respect to the Planck value, the tension for the before mentioned
collaborations takes the value, DES $T_{S_{8}}=1.40\sigma$, KiDS
$T_{S_{8}}=2.12\sigma$ and HSC $T_{S_{8}}=1.53\sigma$. Only in the KiDS case
the tension is above $2\sigma$ so we can say that for the moment the tension
is far from reaching a critical level. However it is important to keep an eye
on the future measurements obtained from the weak gravitational lensing to see
if the tension completely disappears or conversely, starts to increase. It is
worthwhile to mention some models able to relieve the tension, some examples
are: the running vacuum models [77, 78, 79], models that include the $A_{L}$
lensing phenomenological parameter [31, 80], modified gravity models [81, 82]
or a Brans-Dicke model with a cosmological constant [83].

Without doubt, the most worrisome and consequently the most studied of the
tensions is the $H_{0}$-tension, namely, the great discrepancy between the
value from Planck 2018 TTTEEE+lowE+lensing [31] $H_{0}=67.36\pm 0.54$ km/s/Mpc
and the local measurements from the SH0ES collaboration [84] $H_{0}=73.5\pm
1.4$ km/s/Mpc obtained with the cosmic distance ladder method using and
improved calibration of the Cepheid period-luminosity relation. As can be
easily checked the tension between the Planck value and the SH0ES one reaches
the disturbing level of $T_{H_{0}}=4.09\sigma$. The persistence in the
discrepancies over the last few years turns out to be really puzzling since we
are still far to conclude if the tension is rooted either in some sort of
still unknown systematic or perhaps is the clearest sign we have right now of
new physics. There are numerous attempts in the literature trying to solve
this tension, here we just mention some of them: invoking an extra
relativistic neutrino d.o.f. [31, 85, 86, 87], a coupled DE with an scalar
field potential [88] and early dark energy models (EDE) [89, 90]. Another
interesting possibility is the modification of the standard model at the
recombination period since it has been proved that the value of $H_{0}$ is
closely related with the comoving sound horizon $r_{d}$ [91, 92, 93, 94]. The
aforementioned options are based on the idea to solve the tension within the
GR paradigm, however the consideration of the Brans-Dicke theory opens a wide
range of interesting possibilities to alleviate the tension [83, 95].
Other cosmological data can shed some light on the issue. The SPTPol survey
[96], which also deals with CMB data, obtains a higher value for the Hubble
constant $H_{0}=71.29\pm 2.12$ km/s/Mpc. The data extracted, for the H0LICOW
team [97], from six gravitational lensed quasar point out also to a high value
$H_{0}=73.3^{+1.8}_{-1.7}$km/s/Mpc, which is in tension with the value
obtained, also from strong gravitational data, by the TDCOSMO+SLACS team [98]
$H_{0}=67.4^{+4.1}_{-3.2}$km/s/Mpc which clearly tells us that there are still
a lot of stuff to understand. Finally the Gravitational Waves field turns out
to be really promising and it is claimed that in the next decade it will
provide accurate data [99, 100, 101].
A final comment about the aforementioned $\sigma_{8}$-tension and
$H_{0}$-tension is in order. We have to bear in mind that, while one tries to
solve one of the tensions, should not make the other one worse. The $H_{0}$
and $\sigma_{8}$ are correlated so we have to be careful to not accommodate to
much one of the values while we lost sight the other one. This is the only
acceptable way to deal with these intriguing tensions.

###  Alternatives to the $\Lambda$CDM

In this section we present some of the models beyond the standard one that
have been proposed over the years. What all of them have in common, of course,
is the fact that they deviate in one way or another from the theoretical
framework where the $\Lambda$CDM is placed. As it is well-known the lack of a
convincing explanation for the accelerated expansion of the Universe has
motivated cosmologists to look for alternatives that do. Not only we are
chasing for models able to provide a more physical mechanism for the expansion
of the Universe but also we are interested in those models that can better
handle the aforementioned tensions that affect the concordance model. No need
to say that owing to the tremendous success of the $\Lambda$CDM, in explaining
most of the cosmological observations, the alternative models should not
depart to much from its theoretical predictions. For instance, as we shall see
by considering a mild evolution of the rigid cosmological constant term we can
fit the CMB data as well as the $\Lambda$CDM does but we can also improve the
description of the large scale structure data. In regards to the cosmological
data, the increasing number of new and very precise measurements will allow us
to put tight constraints over the models making possible to rule out those
that do not pass the observational tests. In the meantime we should remind
open-minded and consider all the possibilities.

####  Dark energy parameterizations

The simplest possible extension of the $\Lambda$CDM model consists in endowing
the DE density, $\rho_{\rm DE}$ with some sort of evolution over time and
consider, at the same time, that it is self-conserved, i.e. $\dot{\rho}_{\rm
DE}+3H(1+w_{\rm DE}(a))\rho_{\rm DE}=0$. While it is possible to build models
just by considering complicate functional forms for the EoS parameter, $w_{\rm
DE}(a)$, here we want to keep things as simple as possible, this is why we
only study two well-known different parameterizations, namely the XCDM [102]
and the CPL [103, 104, 105]. The first of them consists in considering $w_{\rm
DE}(a)=w_{0}={\rm const}.$, nevertheless, this model differs from the standard
one because $w_{0}\neq-1$, consequently the corresponding dark energy density
can be written as $\rho_{\rm DE}=\rho^{0}_{\Lambda}{a^{-3(1+w_{0})}}$. Being
$\rho^{0}_{\Lambda}$ the present measured value. The normalized Hubble
function can be written as:

$E^{2}(a)=(\Omega^{0}_{b}+\Omega^{0}_{dm})a^{-3}+\Omega^{0}_{\gamma}{a^{-4}}+\frac{\rho_{\nu}(a)}{\rho^{0}_{c}}+\Omega^{0}_{\Lambda}{a^{-3(1+w_{0})}}.$
(1.128)

For $w_{0}=-1$ it boils down to that of the $\Lambda$CDM with rigid CC, as
expected. On the other hand for $w_{0}\gtrapprox-1$ the XCDM mimics
quintessence, whereas for $w_{0}\lessapprox-1$ it mimics phantom DE.
A slightly more sophisticated DE parameterization is furnished by the CPL, in
which one assumes that the generic DE entity has a slowly varying EoS of the
form

$w_{\rm DE}(a)=w_{0}+w_{1}(1-a)=w_{0}+w_{1}\frac{z}{1+z}.$ (1.129)

The CPL parameterization, in contrast to the XCDM one, gives room for a time
evolution of the dark energy EoS owing to the presence of the additional
parameter $w_{1}$, which satisfies $0<|w_{1}|\ll|w_{0}|$ with
$w_{0}\gtrapprox-1$ or $w_{0}\lessapprox-1$. The corresponding normalized
Hubble function for the CPL can be easily computed:

$E^{2}(a)=(\Omega^{0}_{b}+\Omega^{0}_{dm})a^{-3}+\Omega^{0}_{\gamma}{a^{-4}}+\frac{\rho_{\nu}(a)}{\rho^{0}_{c}}+\Omega^{0}_{\Lambda}{a^{-3(1+w_{0}+w_{1})}}e^{-3w_{1}(1-a)}.$
(1.130)

Both the XCDM and the CPL parameterizations can be thought of as a kind of
baseline frameworks to be refereed to in the study of dynamical DE. They can
be used as a fiducial models to which we can compare other, more
sophisticated, models for the dynamical DE.

####  Running vacuum models

This kind of models is build upon the idea that the vacuum energy density
should be a time depending quantity in cosmology. It is difficult to conceive
an expanding Universe with a vacuum energy density that has remained immutable
since the origin of time. Rather, a smoothly evolving energy density that
inherits its time-dependence from cosmological variables, such as the Hubble
rate $H(t)$, or the scale factor $a(t)$ is not only a qualitatively more
plausible and intuitive idea, but it is also suggested by fundamental physics,
in particular by QFT in curved space-time.

Remember that the renormalized ZPE contribution coming from the quantum
fluctuations of a non-minimally coupled scalar field, in the FLRW metric,
gives rise to the expression (1.78), which is useful to explore the running of
the $\rho_{\rm vac}$, when we move from one scale to another. Once we insert
the observational obtained value, at some particular scale, we are able to
compute the value at another scale. Having said that, let us first relate the
different values of the vacuum energy density at the present energy scale
characterized by $M=H_{0}$ with the typical energy scale of most of the grand
unified theories (GUT) $M_{0}=M_{X}\sim\mathcal{O}(10^{16})$ GeV. We are
interested in the description of the current Universe, consequently, those
terms of order $\sim\mathcal{O}(H^{4})$ (which include the terms
$\dot{H}^{2}$, $H\ddot{H}$ and $\dot{H}H^{2}$) can be neglected. Considering
what we just said, (1.78), takes the following form:

$\rho_{\rm vac}(M=H_{0})\equiv\rho^{0}_{\Lambda}=\rho_{\rm
vac}(M_{0}=M_{X})+\frac{3}{16\pi^{2}}\left(\xi-\frac{1}{6}\right)H^{2}_{0}\left[M^{2}_{X}+m^{2}\ln\left(\frac{H_{0}}{M_{X}}\right)^{2}\right].$
(1.131)

Where $\rho^{0}_{\Lambda}$ is the observational measurement at present time.
The above expression can be simplified by defining the running parameter
$\nu_{\rm eff}$:

$\rho^{0}_{\Lambda}=\rho_{\rm vac}(M_{0}=M_{X})+\frac{3\nu_{\rm
eff}}{8\pi}H^{2}_{0}{M^{2}_{\rm pl}}$ (1.132)

where

$\nu_{\rm
eff}=\frac{1}{2\pi}\left(\frac{1}{6}-\xi\right)\frac{M^{2}_{X}}{M^{2}_{\rm
pl}}\left(1+\frac{m^{2}}{M^{2}_{X}}\ln\left(\frac{H_{0}}{M_{X}}\right)^{2}\right).$
(1.133)

This dimensionless parameter is different from zero as long as
$\xi\neq\frac{1}{6}$. Due to the presence of the ratio $M^{2}_{X}/M^{2}_{\rm
pl}\ll 1$ we can always ensure that its value will be small. Once again we
need to make use of the expression (1.78), but this time aiming to relate the
value of the vacuum energy density at $M=H$ with the value already calculated
(1.132) at $M_{0}=M_{X}$. The results can be cast as follows:

$\rho_{\rm vac}(H)=\rho^{0}_{\Lambda}-\frac{3\nu_{\rm
eff}}{8\pi}H^{2}_{0}{M^{2}_{\rm pl}}+\frac{3\nu_{\rm
eff}(H)}{8\pi}H^{2}{M^{2}_{\rm pl}}$ (1.134)

where we have employed the definition

$\nu_{\rm
eff}(H)=\frac{1}{2\pi}\left(\frac{1}{6}-\xi\right)\frac{M^{2}_{X}}{M^{2}_{\rm
pl}}\left(1+\frac{m^{2}}{M^{2}_{X}}\ln\left(\frac{H}{M_{X}}\right)^{2}\right).$
(1.135)

As stated we are interested in the post-inflationary Universe, therefore the
values that we are going to consider for $H(a)$ will not be much different
from the present value $H_{0}$. This fact allows us to make the approximation
$\nu_{\rm eff}(H)\simeq\nu_{\rm eff}$. All in all, the final expression can be
written as:

$\rho_{\rm vac}(H)\simeq\rho^{0}_{\Lambda}+\frac{3\nu_{\rm
eff}}{8\pi}(H^{2}-H^{2}_{0})M^{2}_{\rm pl}=\rho^{0}_{\Lambda}+\frac{3\nu_{\rm
eff}}{8\pi{G_{N}}}(H^{2}-H^{2}_{0}).$ (1.136)

For $\nu_{\rm eff}>0$, the above expression can be conceived as if the vacuum
decays into matter, since it value is larger in the past. Whereas for
$\nu_{\rm eff}<0$ is the other way around. See [106, 107, 108, 109, 88, 110]
to know more about interacting models. While it is true that the expression
displayed in (1.136) contains only the contribution of the ZPE coming from a
scalar field, can serve as an ansatz to study in a phenomenological way the
running vacuum models.
Actually, (1.136) can be understood, as a particular case of a more general
expression of the vacuum energy density considered in analogy to the
renormalization group equation, that can be treated as an ansatz, but well
motivated in the computations within a QFT [111, 33]:

$\frac{d\rho_{\rm
vac}}{d\ln\mu^{2}}=\frac{1}{4\pi^{2}}\sum_{i}\left[a_{i}{M^{2}_{i}}\mu^{2}+b_{i}\mu^{4}+c_{i}\frac{\mu^{6}}{M^{2}_{i}}+\dots\right].$
(1.137)

This equality describes in a very general way the possible running of the
vacuum energy density triggered by the quantum effects coming from the
different matter fields, either bosons or fermions. The coefficients
$a_{i},b_{i},c_{i}\dots$ are dimensionless and $M_{i}$ represents the masses
of the particles that contribute to the running of the vacuum. The energy
scale $\mu$ is typically identified with $\sim aH^{2}+b\dot{H}$. Notice that
$\rho_{\rm vac}(H)$ can only involve even powers of the Hubble rate $H$ or
powers of the derivative $\dot{H}$, because of the covariance of the effective
action.

Integrating (1.137), we can get an expression valid for the current Universe

$\rho_{\rm
vac}(H)=\frac{3}{8\pi{G_{N}}}\left(c_{0}+\nu{H^{2}}+\alpha{\dot{H}}\right)+\mathcal{O}(H^{4})$
(1.138)

being $\alpha$ and $\nu$ dimensionless parameters and $c_{0}$ is a
2-dimensional constant in natural units. We emphasize that we always keep
$c_{0}\neq 0$ so as to ensure a smooth $\Lambda$CDM limit when the
dimensionless coefficients are set to zero. These dimensionless coefficients
can be computed in the context of QFT from the ratio squared of the different
masses to the Planck mass and they can be interpreted as the $\beta$-function
of the running of the vacuum energy density. The theoretical predictions
establish $\alpha,\nu\sim 10^{-6}-10^{-3}$ [111], however their values, at
this moment, must be computed in a phenomenological way by constraining the
models with the cosmological data and a proper statistical analysis.

So far, we have justified the functional form of the vacuum energy density
within the context of QFT, nevertheless, in order to allow the dynamics of
$\rho_{\rm vac}(H)$, we are forced to introduce an extra ingredient with
respect to the standard model of cosmology. In this thesis we are going to
meanly consider three different scenarios derived from the local energy
conservation equation:

$\frac{d}{dt}\left[G(t)\sum_{N}\rho_{N}\right]+3G(t)H\sum_{N}\left(\rho_{N}+p_{N}\right)=0$
(1.139)

where, as usually, $N$ runs for the different components considered, e.g.
baryons, dark matter, neutrinos, photons and vacuum. Let us generically denote
by $w_{\rm DE}$ the equation of state parameter for the vacuum energy density,
since as we are about to see in some cases we consider that it can mildly
evolve with time. The different possibilities that we study in different
chapters of this thesis are:

Scenario I: $\dot{G}=0$, $w_{\rm DE}=-1$ and
$\sum_{N}\dot{\rho}_{N}+3H(\rho_{N}+p_{N})=0$

In this first scenario the Newton coupling is treated as if it were a
constant, $G=G_{N}$ and we consider dark energy as if it were vacuum, which
means $\rho_{\rm vac}+p_{\rm vac}=0$ in the local conservation equation. So,
here we consider an exchange of energy between the ordinary matter and the
vacuum component, which of course is not present in the $\Lambda$CDM model.

Scenario II: $\dot{G}\neq 0$, $w_{\rm DE}=-1$ and
$\frac{d}{dt}(\rho_{m}+\rho_{r})+3H(\rho_{m}+p_{m}+\rho_{r}+p_{r})=0$

Here the matter components $\rho_{m}=\rho_{b}+\rho_{dm}+\rho^{\rm NR}_{\nu}$
and $\rho_{r}=\rho_{\gamma}+\rho^{\rm R}_{\nu}$ follow exactly the same
evolution laws than in the concordance model. In this case the vacuum does not
interact with the ordinary matter but with the Newton coupling, which leaves
us with the following equation:

$\frac{\dot{\rho}_{\rm vac}}{\rho_{\rm
vac}+\rho_{m}+\rho_{r}}=-\frac{\dot{G}}{G}.$ (1.140)

In this scenario not only the vacuum energy density inherits some dynamics,
due to the expansion of the Universe, but also $G=G(t)$ does too. As cannot be
otherwise, within that type of models, the predicted evolution for the Newton
coupling is very slow being actually logarithmic $G(a)\simeq
G_{N}\left(1+\epsilon\ln(a)\right)$ with $|\epsilon|\ll 1$.

Scenario III: $\dot{G}=0$, $w_{\rm DE}\neq-1$ and
$\sum_{N}\dot{\rho}_{N}+3H(\rho_{N}+p_{N})=0$

On this last case we allow a tiny departure of the pure vacuum behaviour by
considering a non-trivial expression for $w_{\rm DE}(t)$, while $\rho_{\rm
vac}$ still has the functional form displayed in (1.138) and interacts, like
in Scenario I, with ordinary matter.

Something that should not be omitted is the fact that it is also possible to
accommodate the inflation mechanism within the RVM’s. In order to produce the
desired effects $\rho_{\rm vac}$ should acquire the following form:

$\rho_{\rm vac}(H)=\frac{3}{8\pi
G_{N}}\left(\bar{c}_{0}+\bar{\nu}{H^{2}}+\bar{\alpha}\frac{H^{4}}{H^{2}_{I}}\right)$
(1.141)

where $\bar{\nu}$ and $\bar{\alpha}$ are dimensionless parameters whereas the
constants $H_{I}$ and $\bar{c}_{0}$ have dimension 1 and 2 respectively. Upon
considering not only terms of order $\sim\mathcal{O}(H^{2})$ but also those of
order $\sim\mathcal{O}(H^{4})$ it is possible to produce inflation in the
period right before the radiation dominated era. Quite remarkable is the fact
that terms like $\sim H^{4}$ can be generated in string-inspired mechanism,
establishing a very intriguing connection with the RVM’s [112, 113]. In order
to know more details see [72] and references therein. Within this model we
also find a scale factor of the form (1.123), as could not be otherwise, since
we are interested in an expanding Universe completely dominated by a component
with a negative pressure.

In this thesis the RVM’s are widely tested, putting them under the light of a
large string of observational data, like the CMB spectrum, the BAO
measurements or the important LSS observations (see [114] for an updated
analysis). As expected the results depends of the dataset employed, but
interestingly, as long as the triad CMB+BAO+LSS is included in the analysis an
important conclusion can be extracted: the RVM’s are favoured over the
$\Lambda$CDM.

####  Quintessence scalar field models $\phi$CDM

The literature is full of various attempts to explain, the CC and
consequently, the current acceleration of the Universe employing scalar
fields, see [115, 116, 117, 118, 119, 120, 121, 122, 106, 123] and references
therein. The main idea behind this type of models is to describe DE in terms
of some sort of scalar $\phi$ field with its corresponding potential
$V(\phi)$, which can be the tail of a more complex effective potential in
which inflation is also incorporated. We want to remark that, as the tittle of
the section suggests, we are only interested in the quintessence scalar field
models. The associated energy density $\rho_{\phi}$ evolves slowly with time
until the scalar field becomes the dominant component, thus causing the
accelerated expansion of the Universe we can see at the present time. The
scalar field models possess a well-defined local Lagrangian description. The
total action, including the EH term and the matter part, can be written, for a
minimally coupled scalar field, in the following way:

$S_{\rm tot}=S_{\rm EH}+S_{\phi}+S_{m}=\frac{M^{2}_{\rm pl}}{16\pi}\int
d^{4}x\sqrt{-g}\left[R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right]+S_{m}.$
(1.142)

Here $M_{\rm pl}=1/\sqrt{G_{N}}=1.22\times 10^{19}{\rm GeV}$ is the Planck
mass in natural units. As it can be appreciated in this convention the scalar
field is taken to be dimensionless. Once we have the analytical expression for
the scalar field action we can easily obtain the corresponding energy-momentum
tensor thanks to:

$T^{\phi}_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta{S_{\phi}}}{\delta{g^{\mu\nu}}}=\frac{M^{2}_{\rm
pl}}{16\pi}\left[\partial_{\mu}\phi\partial_{\nu}\phi-
g_{\mu\nu}\left(\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+V(\phi)\right)\right].$
(1.143)

At this point we want to remark that we consider that the scalar field is an
homogeneous quantity which means that at the background level is just a
function of time. To know the details about the perturbation equations for the
scalar field models see Appendix B. We can easily obtain the energy density
and the pressure from the expression of the energy-momentum tensor:

$\rho_{\phi}=T^{\phi}_{00}=\frac{M^{2}_{\rm
pl}}{16\pi}\left(\frac{\dot{\phi}^{2}}{2}+V(\phi)\right)\quad
p_{\phi}=T^{\phi}_{ii}=\frac{M^{2}_{\rm
pl}}{16\pi}\left(\frac{\dot{\phi}^{2}}{2}-V(\phi)\right)$ (1.144)

The dots represent derivatives w.r.t. the cosmic time. Unlike the $\Lambda$CDM
model the EoS parameter for the DE component is $w_{\phi}\neq-1$, this can be
seen from

$w_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}=\frac{\dot{\phi}^{2}-2V(\phi)}{\dot{\phi}^{2}+2V(\phi)}.$
(1.145)

In order to produce the late-time acceleration, the condition $w_{\phi}<-1/3$
must be fulfilled, which implies $\dot{\phi}/2\ll V(\phi)$. Taking this into
account the EoS parameter can be approximated by
$w_{\phi}\simeq-1+\dot{\phi}^{2}/V(\phi)$. Due to the extra degree of freedom,
embodied in the scalar field, it is possible to obtain the Klein-Gordon
equation for a non-coupled model, upon functional derivative of $S_{\phi}$
with respect to the scalar field and considering the FLRW metric:

$\ddot{\phi}+3H\dot{\phi}+\frac{\partial{V(\phi)}}{\partial\phi}=0.$ (1.146)

We must not fall into the error of thinking that the scalar field models are
free from the old CC constant problem or equivalently free from some sort of
fine-tuning problem. In order to have the proper shape the considered
potential must adjust at least one free parameter to be able to produce the
late-time accelerated expansion. To see a comprehensive list of possibilities
for the potential check Table 1.1 of [37]. Also read Section 1.2.4 (Scalar
fields in the late-time Cosmology) to get a summary of the different attempts
based on the idea of scalar fields throughout the modern history of cosmology.

####  Brans-Dicke model with cosmological constant

The Brans-Dicke theory [124] contains an additional gravitational degree of
freedom as compared to GR, and therefore it genuinely departs from GR in a
fundamental way. The new d.o.f. is represented by the scalar BD-field $\psi$,
which is non-minimally coupled to curvature, $R$. The BD-action reads as
follows 777We use natural units, therefore $\hbar=c=1$ and $G_{N}=1/M_{\rm
pl}^{2}$, where $M_{\rm pl}\simeq 1.22\times 10^{19}$ GeV is the Planck mass.
Regarding the sign convention we employ the $(+,+,+)$. See Appendix A for the
details.

$\displaystyle S_{\rm BD}=\int
d^{4}x\sqrt{-g}\left[\frac{1}{16\pi}\left(R\psi-\frac{\omega_{\rm
BD}}{\psi}g^{\mu\nu}\partial_{\nu}\psi\partial_{\mu}\psi\right)-\rho_{\Lambda}\right]+\int
d^{4}x\sqrt{-g}\,{\cal L}_{m}(\Phi_{i},g_{\mu\nu})\,.$ (1.147)

The (dimensionless) factor in front of the kinetic term of $\psi$, i.e.
$\omega_{\rm BD}$, will be referred to as the BD-parameter. The last term of
(1.147) stands for the matter action $S_{m}$, which is constructed from the
Lagrangian density of the matter fields, collectively denoted as $\Phi_{i}$.
There is no potential for the BD-field $\psi$ in the original BD-theory, but
we admit the presence of a CC term associated to $\rho_{\Lambda}$. The scalar
field, $\psi$ has dimension $2$ in natural units (i.e. mass dimension
squared), in contrast to the dimension $1$ of ordinary scalar fields. The
effective value of $G$ at any time is thus given by $1/\psi$, and of course
$\psi$ must be evolving very slowly with time. The field equations of motion
ensue after performing variation with respect to both the metric and the
scalar field $\psi$. While the first variation yields

$\psi\,G_{\mu\nu}+\left(\Box\psi+\frac{\omega_{\rm
BD}}{2\psi}\left(\nabla\psi\right)^{2}\right)\,g_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}\psi-\frac{\omega_{\rm
BD}}{\psi}\nabla_{\mu}\psi\nabla_{\nu}\psi=8\pi\left(\,T_{\mu\nu}-g_{\mu\nu}\rho_{\Lambda}\right)\,,$
(1.148)

the second variation gives the wave equation for $\psi$, which depends on the
curvature scalar $R$. The latter can be eliminated upon tracing over the
previous equation, what leads to a most compact result:

$\Box\psi=\frac{8\pi}{2\omega_{\rm BD}+3}\,\left(T-4\rho_{\Lambda}\right)\,.$
(1.149)

Here we have assumed that both $\omega_{\rm BD}$ and $\rho_{\Lambda}$ are
constants. To simplify the notation, we have written $(\nabla\psi)^{2}\equiv
g^{\mu\nu}\nabla_{\mu}\psi\nabla_{\nu}\psi$. In the first field equation,
$G_{\mu\nu}=R_{\mu\nu}-(1/2)Rg_{\mu\nu}$ is the Einstein tensor, and on its
r.h.s. $T_{\mu\nu}=-(2/\sqrt{-g})\delta S_{m}/\delta g^{\mu\nu}$ is the
energy-momentum tensor from matter. In the field equation for $\psi$, $T\equiv
T^{\mu}_{\mu}$ is the trace of the energy-momentum tensor for the matter part
(relativistic and non-relativistic). The total energy-momentum tensor as
written on the r.h.s. of (1.148) is the sum of the matter and vacuum parts and
adopts the perfect fluid form:

$\tilde{T}_{\mu\nu}=T_{\mu\nu}-\rho_{\Lambda}g_{\mu\nu}=p\,g_{\mu\nu}+(\rho+p)U_{\mu}{U_{\nu}}\,,$
(1.150)

with $\rho\equiv\rho_{m}+\rho_{r}+\rho_{\Lambda}$ and $p\equiv
p_{m}+p_{r}+p_{\Lambda}=p_{r}+p_{\Lambda}$, where $p_{m}=0$,
$p_{r}=(1/3)\rho_{r}$ and $p_{\Lambda}=-\rho_{\Lambda}$ stand for the
pressures of dust matter (which includes baryons and DM), radiation and
vacuum, respectively. As in GR, we consider a possible CC term or constant
vacuum energy density, $\rho_{\Lambda}$, in the BD-action (1.147). The quantum
matter fields usually induce an additional, and very large, contribution to
$\rho_{\Lambda}$, this is of course the origin of the CC Problem. Let us write
down the field equations in the flat FLRW metric,
$ds^{2}=-dt^{2}+a^{2}\delta_{ij}dx^{i}dx^{j}$. Using the total density $\rho$
and pressure $p$ as indicated above, Eq. (1.148) renders the two independent
equations

$3H^{2}+3H\frac{\dot{\psi}}{\psi}-\frac{\omega_{\rm
BD}}{2}\left(\frac{\dot{\psi}}{\psi}\right)^{2}=\frac{8\pi}{\psi}\rho$ (1.151)
$2\dot{H}+3H^{2}+\frac{\ddot{\psi}}{\psi}+2H\frac{\dot{\psi}}{\psi}+\frac{\omega_{\rm
BD}}{2}\left(\frac{\dot{\psi}}{\psi}\right)^{2}=-\frac{8\pi}{\psi}p$ (1.152)

whereas (1.149) yields

$\ddot{\psi}+3H\dot{\psi}=-\frac{8\pi}{2\omega_{\rm BD}+3}(3p-\rho)\,.$
(1.153)

Here dots indicate derivatives with respect to the cosmic time and
$H=\dot{a}/a$ is the Hubble rate. For constant $\psi=1/G_{N}$, the first two
equations reduce to the Friedmann and pressure equations of GR, and the third
requires $\omega_{\rm BD}\to\infty$ for consistency. By combining the above
equations we expect to find a local covariant conservation law, similar to GR.
This is because there is no interaction between matter and the BD field. It
can indeed be checked by explicit calculation. Although the calculation is
more involved than in GR, the final result turns out to be the same:

$\dot{\rho}+3H(\rho+p)=\sum_{N}\dot{\rho}_{N}+3H(\rho_{N}+p_{N})=0\,,$ (1.154)

where the sum is over all components, i.e. baryons, dark matter, radiation,
neutrinos and vacuum. Here we take the point of view that all components are
separately conserved in the main periods of the cosmic evolution. In
particular, the vacuum component obviously does not contribute in the sum
since it is assumed to be constant and $\rho_{\Lambda}+p_{\Lambda}=0$ . The
expression (1.154) can also be obtained upon lengthy but straightforward
computation of the covariant derivative on both sides of Eq. (1.148) and using
the Bianchi identity satisfied by $G_{\mu\nu}$ and the field equation of
motion for $\psi$.

####  Modified gravity models

So far we have considered models that assume the presence of dark energy
(being a constant or a time-evolving quantity), however it is convenient to be
open-minded and consider different alternatives present in the literature. See
[55] in order to know about modified gravity models. Of course, we cannot
simply omit the experimental evidence we have nowadays of how galaxies move
away from each other in an accelerated way. Actually, in the previous section
we have considered one of the first attempts of cosmological model not based
on GR, namely the BD model, where gravity is not only mediated by the tensor
$G_{\mu\nu}$ but also by the BD-field which is usually denoted as $\psi$. One
can replace there the $\rho_{\Lambda}$ by a potential $U(\psi)$ and then
remove completely the presence of a pure cosmological constant. We shall
comment more about that in Chapter 7.

Of course it is possible to introduce a modification in the field equations
without using a scalar field, and a classical example of that are the $f(R)$
models [125, 126, 127, 128, 129], whose action take the general form:

$S=\frac{1}{16\pi{G_{N}}}\int d^{4}x\sqrt{-g}f(R)+S_{m}(\Phi_{i},g_{\mu\nu}).$
(1.155)

As in other cases here $S_{m}$ denotes the matter action and $\Phi_{i}$ refer
to the matter fields in general. Note that the EH action is just a particular
case of the above action with $f(R)=R-2\Lambda$. It goes without saying that
not all the forms for the function $f(R)$ are valid, since the considered
model has to pass not only the cosmological test, but also the astrophysical
ones in case no screening mechanism is considered. A brief comment on the
frame where the models are considered turns out to be necessary. We say that
we work in the Einstein frame when the Ricci scalar, appearing in the action,
is not multiplied by a constant or replaced by a function of $R$, like in the
case of $f(R)$ models, and it is also not multiplied by some sort of function
of the scalar field, like in the BD model. If any of the above conditions are
not fulfilled we say we are in the Jordan frame. Both frames are linked
through a conformal transformation of the metric

$\hat{g}_{\mu\nu}=\Omega^{2}{g_{\mu\nu}}$ (1.156)

where the function $\Omega$ can vary depending on the model considered. There
is no general agreement about which of the frames is the physical one, in case
any of them are. In this thesis we adopt the convention to consider the
physical frame as that one where the model is originally formulated.

If we go back to the $f(R)$ models, in order to obtain the field equations,
one has to choose between two different approaches. The first one is the
metric formalism, where the connection between the Christoffel symbols
$\Gamma^{\mu}_{\nu\lambda}$ and the metric $g_{\mu\nu}$ is the usual one. The
other option is known as the Palatini formalism, which considers that both
functions (the Christoffel symbols and the metric) must be considered as
independent. Since for all the models studied in this thesis we assume the
Levi-Civita connection, we stick to the former case. By considering the same
process employed to obtain Einstein’s field equations, vary the action with
respect to $g_{\mu\nu}$ it is possible to obtain the following equations:

$F(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}{F(R)}+g_{\mu\nu}\Box{F(R)}=8\pi{G_{N}}T_{\mu\nu}.$
(1.157)

In the above expression appear $R_{\mu\nu}$ and $T_{\mu\nu}$, which are the
usual Ricci tensor and energy-momentum tensor respectively, whereas the term
$F(R)\equiv\partial{f}/\partial{R}$ has been introduced to simplify the
notation. If we take the trace of (1.157) we end up with the following
relation:

$3\Box{F(R)}+F(R)R-2f(R)=8\pi{G_{N}}T$ (1.158)

being $T$ the trace of the energy-momentum tensor. What (1.158) means is that
there is a propagating scalar degree of freedom due to the presence of the
term $\Box{F(R)}$.

If the FLRW metric, $ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}{dx^{j}}$ is
assumed, we can write the cosmological equations as:

$\displaystyle 3H^{2}$
$\displaystyle=8\pi{G_{N}}\left(\rho_{m}+\rho_{r}+\rho_{\rm DE}\right)$
(1.159) $\displaystyle 2\dot{H}+3H^{2}$
$\displaystyle=-8\pi{G_{N}}\left(p_{m}+p_{r}+p_{\rm DE}\right)$ (1.160)

where the terms $\rho_{\rm DE}$ and $p_{\rm DE}$ are defined as:

$\displaystyle\rho_{\rm DE}$
$\displaystyle\equiv\frac{1}{8\pi{G_{N}}}\left(3H^{2}(1-F)+\frac{(FR-f)}{2}-3H\dot{F}\right)$
(1.161) $\displaystyle p_{\rm DE}$
$\displaystyle\equiv\frac{1}{8\pi{G_{N}}}\left((2\dot{H}+3H^{2})(F-1)+2H\dot{F}+\ddot{F}+\frac{(f-FR)}{2}\right).$
(1.162)

Upon considering the effective expressions for the energy density and the
pressure it is possible to get the following local conservation equation
$\dot{\rho}_{\rm DE}+3H\left(\rho_{\rm DE}+p_{\rm DE}\right)=0$. If we
continue with the analogy of dark energy we are allowed to define an EoS
parameter given by the expression:

$w_{\rm DE}=\frac{p_{\rm DE}}{\rho_{\rm
DE}}=-1+\frac{2\ddot{F}-2H\dot{F}-4\dot{H}F}{6H^{2}(1-F)+FR-f-6H\dot{F}}.$
(1.163)

The above expression is useful to detect deviations from a pure GR model.

##  First evidence of running cosmic vacuum: challenging the concordance model

We start our study of different models, with the capacity to provide a more
general framework that the $\Lambda$CDM does, in order to accommodate a
possible time evolution of the dark energy (DE) component, by studying the
Running Vacuum Models (RVM’s). These models were introduced in the previous
chapter and are mainly characterized by having a vacuum energy density that
evolves as the Universe expands. The particular functional form of
$\rho_{\Lambda}=\rho_{\Lambda}(H)$ is motivated in the context of quantum
field theory (QFT) in curved space-times.
We would like to remark that our aim here is neither to solve the cosmological
constant (CC) problem nor to solve the coincidence problem, but to study the
aforementioned models from a phenomenological perspective.
To do so, we will consider a large set of cosmological data
SNIa+BAO+$H(z)$+LSS+BBN+CMB in order to test the possibility that the
$\Lambda$-term and the associated vacuum energy density
$\rho_{\Lambda}=\Lambda/(8\pi{G})$ could actually be a dynamical (“running”)
quantity whose rhythm of variation might be linked to the Universe expansion
rate, $H(a)$. The idea is to check if this possibility helps to improve the
description of the overall cosmological data as compared to the rigid
assumption $\Lambda=$const. inherent to the concordance $\Lambda$CDM model.
The plan of the chapter is as follows. In Section 2.1 we describe the
different types of running vacuum models (RVM’s) that will be considered in
this study. In Section 2.2 we fit these models to a large set of cosmological
data on distant type Ia supernovae (SNIa), baryonic acoustic oscillations
(BAO), the known values of the Hubble parameter at different redshift points,
the large scale structure (LSS) formation data, the BBN bound on the Hubble
rate, and, finally, the CMB distance priors from WMAP and Planck. We include
also a fit of the data with the standard XCDM parameterization, which serves
as a baseline for comparison. In Section 2.3 we present a detailed discussion
of our results, and finally in Section 2.4 we deliver our conclusions.

Model | $h$ | $\omega_{b}=\Omega_{b}h^{2}$ | $n_{s}$ | $\Omega_{m}$ | $\nu_{\rm eff}$ | $\omega$ | $\chi^{2}_{\rm min}/dof$ | $\Delta{\rm AIC}$ |  $\Delta{\rm BIC}$
---|---|---|---|---|---|---|---|---|---
$\Lambda$CDM | $0.693\pm 0.003$ | $0.02255\pm 0.00013$ | $0.976\pm 0.003$ | $0.294\pm 0.004$ | - | $-1$ | 90.44/85 | - | -
XCDM | $0.670\pm 0.007$ | $0.02264\pm 0.00014$ | $0.977\pm 0.004$ | $0.312\pm 0.007$ | - | $-0.916\pm 0.021$ | 74.91/84 | 13.23 | 11.03
A1 | $0.670\pm 0.006$ | $0.02237\pm 0.00014$ | $0.967\pm 0.004$ | $0.302\pm 0.005$ | $0.00110\pm 0.00026$ | $-1$ | 71.22/84 | 16.92 | 14.72
A2 | $0.674\pm 0.005$ | $0.02232\pm 0.00014$ | $0.965\pm 0.004$ | $0.303\pm 0.005$ | $0.00150\pm 0.00035$ | $-1$ | 70.27/84 | 17.87 | 15.67
G1 | $0.670\pm 0.006$ | $0.02236\pm 0.00014$ | $0.967\pm 0.004$ | $0.302\pm 0.005$ | $0.00114\pm 0.00027$ | $-1$ | 71.19/84 | 16.95 | 14.75
G2 | $0.670\pm 0.006$ | $0.02234\pm 0.00014$ | $0.966\pm 0.004$ | $0.303\pm 0.005$ | $0.00136\pm 0.00032$ | $-1$ | 70.68/84 | 17.46 | 15.26

Table 1: The best-fit values for the $\Lambda$CDM, XCDM and the RVM’s,
including their statistical significance ($\chi^{2}$-test and Akaike and
Bayesian information criteria, AIC and BIC, see the text). The large and
positive values of $\Delta$AIC and $\Delta$BIC strongly favor the dynamical DE
options (RVM’s and XCDM) against the $\Lambda$CDM (see text). We use $90$ data
points in our fit, to wit: $31$ points from the JLA sample of SNIa, $11$ from
BAO, $30$ from $H(z)$, $13$ from linear growth, $1$ from BBN, and $4$ from CMB
(see S1-S7 in the text for references). In the XCDM model the EoS parameter
$\omega$ is left free, whereas for the RVM’s and $\Lambda$CDM is fixed at
$-1$. The specific RVM fitting parameter is $\nu_{\rm eff}$, see Eq.(2.6) and
the text. For G1 and A1 models, $\nu_{\rm eff}=\nu$. The remaining parameters
are the standard ones ($h,\omega_{b},n_{s},\Omega_{m}$). The quoted number of
degrees of freedom ($dof$) is equal to the number of data points minus the
number of independent fitting parameters ($5$ for the $\Lambda$CDM, $6$ for
the RVM’s and the XCDM. The normalization parameter M introduced in the SNIa
sector of the analysis is also left free in the fit, cf. [130], but it is not
listed in the table). For the CMB data we have used the marginalized mean
values and standard deviation for the parameters of the compressed likelihood
for Planck 2015 TT,TE,EE + lowP data from [131], which provide tighter
constraints to the CMB distance priors than those presented in [81]

Model | $h$ | $\omega_{b}=\Omega_{b}h^{2}$ | $n_{s}$ | $\Omega_{m}$ | $\nu_{\rm eff}$ | $\omega$ | $\chi^{2}_{\rm min}/dof$ | $\Delta{\rm AIC}$ |  $\Delta{\rm BIC}$
---|---|---|---|---|---|---|---|---|---
$\Lambda$CDM | $0.692\pm 0.004$ | $0.02254\pm 0.00013$ | $0.975\pm 0.004$ | $0.295\pm 0.004$ | - | $-1$ | 86.11/78 | - | -
XCDM | $0.671\pm 0.007$ | $0.02263\pm 0.00014$ | $0.976\pm 0.004$ | $0.312\pm 0.007$ | - | $-0.920\pm 0.022$ | 73.01/77 | 10.78 | 8.67
A1 | $0.670\pm 0.007$ | $0.02238\pm 0.00014$ | $0.967\pm 0.004$ | $0.302\pm 0.005$ | $0.00110\pm 0.00028$ | $-1$ | 69.40/77 | 14.39 | 12.27
A2 | $0.674\pm 0.005$ | $0.02233\pm 0.00014$ | $0.966\pm 0.004$ | $0.302\pm 0.005$ | $0.00152\pm 0.00037$ | $-1$ | 68.38/77 | 15.41 | 13.29
G1 | $0.671\pm 0.006$ | $0.02237\pm 0.00014$ | $0.967\pm 0.004$ | $0.302\pm 0.005$ | $0.00115\pm 0.00029$ | $-1$ | 69.37/77 | 14.42 | 12.30
G2 | $0.670\pm 0.006$ | $0.02235\pm 0.00014$ | $0.966\pm 0.004$ | $0.302\pm 0.005$ | $0.00138\pm 0.00034$ | $-1$ | 68.82/77 | 14.97 | 12.85

Table 2: Same as in Table 1, but excluding from our analysis the BAO and LSS
data from WiggleZ, see point S5) in the text.

###  Two basic types of RVM’s

In an expanding Universe we may expect that the vacuum energy density and the
gravitational coupling are functions of the cosmic time through the Hubble
rate, thence $\rho_{\Lambda}=\rho_{\Lambda}(H(t))$ and $G=G(H(t))$. Adopting
the canonical equation of state $p_{\Lambda}=-\rho_{\Lambda}(H)$ also for the
dynamical vacuum, the corresponding field equations in the Friedmann-Lemaître-
Robertson-Walker (FLRW) metric in flat space become formally identical to
those with strictly constant $G$ and $\Lambda$:

$\displaystyle 3H^{2}=8\pi\,G(H)\,(\rho_{m}+\rho_{r}+\rho_{\Lambda}(H))$ (2.1)
$\displaystyle 3H^{2}+2\dot{H}=-8\pi\,G(H)\,(p_{r}-\rho_{\Lambda}(H))\,.\,$
(2.2)

The equations of state for the densities of relativistic ($\rho_{r}$) and dust
matter ($\rho_{m}$) read $p_{r}=(1/3)\rho_{r}$ and $p_{m}=0$, respectively.
Consider now the characteristic RVM structure of the dynamical vacuum energy:

$\displaystyle\rho_{\Lambda}(H;\nu,\alpha)=\frac{3}{8\pi G}\left(c_{0}+\nu
H^{2}+\frac{2}{3}\alpha\,\dot{H}\right)+{\cal O}(H^{4})\,,$ (2.3)

where $G$ can be constant or a function $G=G(H;\nu,\alpha)$ depending on the
particular model. The above expression is the form that has been suggested in
the literature from the quantum corrections of QFT in curved space-time (cf.
[33] and references therein). The terms with higher powers of the Hubble rate
have recently been used to describe inflation, see e.g. [132, 133] and [134],
but these terms play no role at present and will be hereafter omitted. The
coefficients $\nu$ and $\alpha$ have been defined dimensionless. They are
responsible for the running of $\rho_{\Lambda}(H)$ and $G(H)$, and so for
$\nu=\alpha=0$ we recover the $\Lambda$CDM, with $\rho_{\Lambda}$ and $G$
constants. The values of $\nu$ and $\alpha$ are naturally small in this
context since they can be related to the $\beta$-functions of the running. An
estimate in QFT indicates that they are of order $10^{-3}$ at most [111], but
here we will treat them as free parameters of the RVM and hence we shall
determine them phenomenologically by fitting the model to observations. As
previously indicated, a simple Lagrangian language for these models that is
comparable to the scalar field DE description may not be possible, as
suggested by attempts involving the anomaly-induced action [111, 33].

Two types of RVM will be considered here: i) type-G models, when matter is
conserved and the running of $\rho_{\Lambda}(H)$ is compatible with the
Bianchi identity at the expense of a (calculable) running of $G$; ii) type-$A$
models, in contrast, denote those with $G=$const. in which the running of
$\rho_{\Lambda}$ must be accompanied with a (calculable) anomalous
conservation law of matter. Both situations are described by the generalized
local conservation equation
$\nabla^{\mu}\left(G\,\tilde{T}_{\mu\nu}\right)=0$, where
$\tilde{T}_{\mu\nu}=T_{\mu\nu}+\rho_{\Lambda}\,g_{\mu\nu}$ is the total
energy-momentum tensor involving both matter and vacuum energy. In the FLRW
metric, and summing over all energy components, we find

$\frac{d}{dt}\,\left[G(\rho_{m}+\rho_{r}+\rho_{\Lambda})\right]+3\,G\,H\,\sum_{i=m,r}(\rho_{i}+p_{i})=0\,.$
(2.4)

If $G$ and $\rho_{\Lambda}$ are both constants, we recover the canonical
conservation law $\dot{\rho}_{m}+\dot{\rho}_{r}+3H\rho_{m}+4H\rho_{r}=0$ for
the combined system of matter and radiation. For type-G models Eq.(2.4) boils
down to $\dot{G}(\rho_{m}+\rho_{r}+\rho_{\Lambda})+G\dot{\rho}_{\Lambda}=0$
since $\dot{\rho}_{m}+3H\rho_{m}=0$ and $\dot{\rho}_{r}+4H\rho_{r}=0$ for
separated conservation of matter and radiation, as usually assumed. Mixed type
of RVM scenarios are possible, but will not be considered here.
We can solve analytically the type-G and type-A models by inserting equation
(2.3) into (2.1) and (2.2), or using one of the latter two and the
corresponding conservation law (2.4). It is convenient to perform the
integration using the scale factor $a(t)$ rather than the cosmic time. For
type-G models the full expression for the Hubble function normalized to its
current value, $E(a)=H(a)/H_{0}$, can be found to be

$\displaystyle\left.E^{2}(a)\right|_{\rm
type-G}=1+\left(\frac{\Omega_{m}}{\xi}+\frac{\Omega_{r}}{\xi^{\prime}}\right)$
$\displaystyle\times\left[-1+a^{-4\xi^{\prime}}\left(\frac{a\xi^{\prime}+\xi\Omega_{r}/\Omega_{m}}{\xi^{\prime}+\xi\Omega_{r}/\Omega_{m}}\right)^{\frac{\xi^{\prime}}{1-\alpha}}\right]\,,$
(2.5)

where $\Omega_{i}=\rho_{i0}/\rho_{c0}$ are the current cosmological parameters
for matter and radiation, and we have defined

$\xi=\frac{1-\nu}{1-\alpha}\equiv 1-\nu_{\rm eff}\,,\ \ \
\xi^{\prime}=\frac{1-\nu}{1-\frac{4}{3}\alpha}\equiv 1-\nu_{\rm
eff}^{\prime}\,.$ (2.6)

Figure 1: Likelihood contours in the $(\Omega_{m},\nu_{\rm eff})$ plane for
the values $-2\ln\mathcal{L}/\mathcal{L}_{max}=2.30$, $6.18,11.81$, $19.33$,
$27.65$ (corresponding to 1$\sigma$, 2$\sigma$, 3$\sigma$, 4$\sigma$ and
5$\sigma$ c.l.) after marginalizing over the rest of the fitting parameters
indicated in Table 1. We display the progression of the contour plots obtained
for model G2 using the 90 data points on SNIa+BAO+$H(z)$+LSS+BBN+CMB, as we
evolve from the high precision CMB data from WMAP9, Planck 2013 and Planck
2015 – see text, point S7). In the sequence, the prediction of the concordance
model ($\nu_{\rm eff}=0$) appears increasingly more disfavored, at an
exclusion c.l. that ranges from $\sim 2\sigma$ (for WMAP9), $\sim 3.5\sigma$
(for Planck 2013) and up to $4\sigma$ (for Planck 2015). Subsequent
marginalization over $\Omega_{m}$ increases slightly the c.l. and renders the
fitting values indicated in Table 1, which reach a statistical significance of
$4.2\sigma$ for all the RVM’s. Using numerical integration we can estimate
that $\sim 99.81\%$ of the area of the $4\sigma$ contour for Planck 2015
satisfies $\nu_{\rm eff}>0$. We also estimate that $\sim 95.47\%$ of the
$5\sigma$ region also satisfies $\nu_{\rm eff}>0$. The corresponding AIC and
BIC criteria (cf. Table 1) consistently imply a very strong support to the
RVM’s against the $\Lambda$CDM.

Note that $E(1)=1$, as it should. Moreover, for $\xi,\xi^{\prime}\to 1$ (i.e.
$|\nu,\alpha|\ll 1$) $\nu_{\rm eff}\simeq\nu-\alpha$ and $\nu_{\rm
eff}^{\prime}\simeq\nu-(4/3)\alpha$. In the radiation-dominated epoch, the
leading behaviour of Eq.(2.1) is $\sim\Omega_{r}\,a^{-4\xi^{\prime}}$, while
in the matter-dominated epoch is $\sim\Omega_{m}\,a^{-3\xi}$. Furthermore, for
$\nu,\alpha\to 0$, $E^{2}(a)\to
1+\Omega_{m}\,(a^{-3}-1)+\Omega_{r}(a^{-4}-1)$. This is the $\Lambda$CDM form,
as expected in that limit. Note that the following constraint applies among
the parameters:
$c_{0}=H_{0}^{2}\left[\Omega_{\Lambda}-\nu+\alpha\left(\Omega_{m}+\frac{4}{3}\,\Omega_{r}\right)\right]$,
as the vacuum energy density $\rho_{\Lambda}(H)$ must reproduce the current
value $\rho_{\Lambda}$ for $H=H_{0}$, using
$\Omega_{m}+\Omega_{r}+\Omega_{\Lambda}=1$. The explicit scale factor
dependence of the vacuum energy density, i.e.
$\rho_{\Lambda}=\rho_{\Lambda}(a)$, ensues upon inserting (2.1) into (2.3). In
addition, since the matter is conserved for type-G models, we can use the
obtained expression for $\rho_{\Lambda}(a)$ to also infer the explicit form
for $G=G(a)$ from (2.1). We refrain from writing out these cumbersome
expressions and we limit ourselves to quote some simplified forms. For
instance, the expression for $\rho_{\Lambda}(a)$ when we can neglect the
radiation contribution is simple enough:

$\rho_{\Lambda}(a)=\rho_{c0}\,a^{-3}\left[a^{3\xi}+\frac{\Omega_{m}}{\xi}(1-\xi-a^{3\xi})\right]\,,$
(2.7)

where $\rho_{c0}=3H_{0}^{2}/8\pi\,G_{0}$ is the current critical density and
$G_{0}\equiv G(a=1)$ is the current value of the gravitational coupling. Quite
obviously for $\xi=1$ we recover the $\Lambda$CDM form:
$\rho_{\Lambda}=\rho_{c0}(1-\Omega_{m})=\rho_{c0}\Omega_{\Lambda}=$const. As
for the gravitational coupling, it evolves logarithmically with the scale
factor and hence changes very slowly888This is a welcome feature already
expected in particular realizations of type-G models in QFT in curved space-
time [111, 33]. See also [135] . It suffices to say that it behaves as

$G(a)=G_{0}\,a^{4(1-\xi^{\prime})}\,f(a)\simeq G_{0}(1+4\nu_{\rm
eff}^{\prime}\,\ln\,a)\,f(a)\,,$ (2.8)

where $f(a)=f(a;\Omega_{m},\Omega_{r};\nu,\alpha)$ is a smooth function of the
scale factor. We can dispense with the full expression here, but let us
mention that $f(a)$ tends to one at present irrespective of the values of the
various parameters $\Omega_{m},\Omega_{r},\nu,\alpha$ involved in it; and
$f(a)\to 1$ in the remote past ($a\to 0$) for $\nu,\alpha\to 0$ (i.e.
$\xi,\xi^{\prime}\to 1$). As expected, $G(a)\to G_{0}$ for $a\to 1$, and
$G(a)$ has a logarithmic evolution for $\nu_{\rm eff}^{\prime}\neq 0$. Notice
that the limit $a\to 0$ is relevant for the BBN (Big Bang Nucleosynthesis)
epoch and therefore $G(a)$ should not depart too much from $G_{0}$ according
to the usual bounds on BBN. We shall carefully incorporate this restriction in
our analysis of the RVM, see later on.

Next we quote the solution for type-A models. As indicated, in this case we
have an anomalous matter conservation law. Integrating (2.4) for $G=$const.
and using (2.3) in it one finds
$\rho_{t}(a)\equiv\rho_{m}(a)+\rho_{r}(a)=\rho_{m0}a^{-3\xi}+\rho_{r0}a^{-4\xi^{\prime}}$.
We have assumed, as usual, that there is no exchange of energy between the
relativistic and non-relativistic components. The standard expressions for
matter and radiation energy densities are recovered for $\xi,\xi^{\prime}\to
1$. The normalized Hubble function for type-A models is simpler than for
type-G ones. The full expression including both matter and radiation reads:

$\left.E^{2}(a)\right|_{\rm
type-A}=1+\frac{\Omega_{m}}{\xi}\left(a^{-3\xi}-1\right)+\frac{\Omega_{r}}{\xi^{\prime}}\left(a^{-4\xi^{\prime}}-1\right)\,.$
(2.9)

From it and the found expression for $\rho_{t}(a)$ we can immediately derive
the corresponding $\rho_{\Lambda}(a)$:

$\rho_{\Lambda}(a)=\rho_{\Lambda
0}+\rho_{m0}(\xi^{-1}-1)(a^{-3\xi}-1)+\rho_{r0}(\xi^{\prime-1}-1)(a^{-4\xi^{\prime}}-1)\,.$
(2.10)

Once more for $\nu,\alpha\to 0$ (i.e. $\xi,\xi^{\prime}\to 1$) we recover the
$\Lambda$CDM case, as easily checked. In particular one finds
$\rho_{\Lambda}\to\rho_{\Lambda{0}}=$const. in this limit.

Figure 2: As in Fig. 1, but for model A2. Again we see that the contours tend
to migrate to the $\nu_{\rm eff}>0$ half plane as we evolve from WMAP9 to
Planck 2013 and Planck 2015 data. Using the same method as in Fig. 1, we find
that $\sim 99.82\%$ of the area of the $4\sigma$ contour for Planck 2015 (and
$\sim 95.49\%$ of the corresponding $5\sigma$ region) satisfies $\nu_{\rm
eff}>0$. The $\Lambda$CDM becomes once more excluded at $\sim 4\sigma$ c.l.
(cf. Table 1 for Planck 2015).  Figure 3: As in Fig. 1 and 2, but for model
XCDM and using Planck 2015 data. The $\Lambda$CDM is excluded at $\sim
4\sigma$ c.l. (cf. Table 1).

###  Fitting the vacuum models to the data

In order to better handle the possibilities offered by the type-G and type-A
models as to their dependence on the two specific vacuum parameters
$\nu,\alpha$, we shall refer to model G1 (resp. A1) when we address type-G
(resp. type-A) models with $\alpha=0$ in Eq.(2.3). In these cases $\nu_{\rm
eff}=\nu$. When, instead, $\alpha\neq 0$ we shall indicate them by G2 and A2,
respectively. This classification scheme is used in Tables 1-2 and 5-7, and in
Figs. 1-6. In the tables we are including also the XCDM (cf. Section 2.3) and
the $\Lambda$CDM.
To this end, we fit the various models to the wealth of cosmological data
compiled from distant type Ia supernovae (SNIa), baryonic acoustic
oscillations (BAO), the known values of the Hubble parameter at different
redshift points, $H(z_{i})$, the large scale structure (LSS) formation data
encoded in $f(z_{i})\sigma_{8}(z_{i})$, the BBN bound on the Hubble rate, and,
finally, the CMB distance priors from WMAP and Planck, with the corresponding
correlation matrices in all the indicated cases. Specifically, we have used
$90$ data points (in some cases involving compressed data) from $7$ different
sources S1-S7, to wit:

$z$ | $H(z)$ | References
---|---|---
$0.07$ | $69.0\pm 19.6$ | [136]
$0.09$ | $69.0\pm 12.0$ | [137]
$0.12$ | $68.6\pm 26.2$ | [136]
$0.17$ | $83.0\pm 8.0$ | [138]
$0.1791$ | $75.0\pm 4.0$ | [139]
$0.1993$ | $75.0\pm 5.0$ | [139]
$0.2$ | $72.9\pm 29.6$ | [136]
$0.27$ | $77.0\pm 14.0$ | [138]
$0.28$ | $88.8\pm 36.6$ | [136]
$0.3519$ | $83.0\pm 14.0$ | [139]
$0.3802$ | $83.0\pm 13.5$ | [140]
$0.4$ | $95.0\pm 17.0$ | [138]
$0.4004$ | $77.0\pm 10.2$ | [140]
$0.4247$ | $87.1\pm 11.2$ | [140]
$0.4497$ | $92.8\pm 12.9$ | [140]
$0.4783$ | $80.9\pm 9.0$ | [140]
$0.48$ | $97.0\pm 62.0$ | [141]
$0.5929$ | $104.0\pm 13.0$ | [139]
$0.6797$ | $92.0\pm 8.0$ | [139]
$0.7812$ | $105.0\pm 12.0$ | [139]
$0.8754$ | $125.0\pm 17.0$ | [139]
$0.88$ | $90.0\pm 40.0$ | [141]
$0.9$ | $117.0\pm 23.0$ | [138]
$1.037$ | $154.0\pm 20.0$ | [139]
$1.3$ | $168.0\pm 17.0$ | [138]
$1.363$ | $160.0\pm 33.6$ | [142]
$1.43$ | $177.0\pm 18.0$ | [138]
$1.53$ | $140.0\pm 14.0$ | [138]
$1.75$ | $202.0\pm 40.0$ | [138]
$1.965$ | $186.5\pm 50.4$ | [142]
Table 3: Current published values of $H(z)$ in units [km/s/Mpc] obtained using the differential-age technique (see the quoted references and point S4 in the text). Survey | $z$ | $f(z)\sigma_{8}(z)$ | References
---|---|---|---
6dFGS | $0.067$ | $0.423\pm 0.055$ | [143]
SDSS-DR7 | $0.10$ | $0.37\pm 0.13$ | [144]
GAMA | $0.18$ | $0.29\pm 0.10$ | [145]
| $0.38$ | $0.44\pm 0.06$ | [146]
DR12 BOSS | $0.32$ | $0.427\pm 0.052$ | [147]
| $0.57$ | $0.426\pm 0.023$ |
WiggleZ | $0.22$ | $0.42\pm 0.07$ | [148]
| $0.41$ | $0.45\pm 0.04$ |
| $0.60$ | $0.43\pm 0.04$ |
| $0.78$ | $0.38\pm 0.04$ |
2MTF | $0.02$ | $0.34\pm 0.04$ | [149]
VIPERS | $0.7$ | $0.380\pm 0.065$ | [150]
VVDS | $0.77$ | $0.49\pm 0.18$ | [151]
|  |  | [152]
Table 4: Current published values of $f(z)\sigma_{8}(z)$. See the text, S5).

S1) The SNIa data points from the SDSS-II/SNLS3 Joint Light-curve Analysis
(JLA) [130]. We have used the $31$ binned distance modulus fitted to the JLA
sample and the compressed form of the likelihood with the corresponding
covariance matrix.

S2) 5 points on the isotropic BAO estimator $r_{s}(z_{d})/D_{V}(z_{i})$:
$z=0.106$ [153], $z=0.15$ [154], $z_{i}=0.44,0.6,0.73$ [155], with the
correlations between the last 3 points.

S3) 6 data points on anisotropic BAO estimators: 4 of them on
$D_{A}(z_{i})/r_{s}(z_{d})$ and $H(z_{i})r_{s}(z_{d})$ at $z_{i}=0.32,0.57$,
for the LOWZ and CMASS samples, respectively. These data are taken from [147],
based on the Redshift-Space Distortions (RSD) measurements of the power
spectrum combined with the bispectrum, and the BAO post-reconstruction
analysis of the power spectrum (cf. Table 5 of that reference), including the
correlations among these data encoded in the provided covariance matrices. We
also use 2 data points based on $D_{A}(z_{i})/r_{s}(z_{d})$ and
$D_{H}(z_{i})/r_{s}(z_{d})$ at $z=2.34$, from the combined LyaF analysis
[156]. The correlation coefficient among these 2 points are taken from [157]
(cf. Table II of that reference). We also take into account the correlations
among the BAO data and the corresponding $f\sigma_{8}$ data of [147] – see S5)
below and Table 4.

S4) $30$ data points on $H(z_{i})$ at different redshifts, listed in Table 3.
We use only $H(z_{i})$ values obtained by the so-called differential-age
techniques applied to passively evolving galaxies. These values are
uncorrelated with the BAO data points. See also [158, 159, 160] and [161],
where the authors make only use of Hubble parameter data in their analyses. We
find, however, indispensable to take into account the remaining data sets to
derive our conclusions on dynamical vacuum, specially the BAO, LSS and CMB
observations. This fact can also be verified quite evidently in Figures 5-6,
to which we shall turn our attention in Section 2.3.

S5) $f(z)\sigma_{8}(z)$: 13 points. These are referred to in the text as LSS
(large scale structure formation). The actual fitting results shown in Table 1
make use of the LSS data listed in Table 4, in which we have carefully avoided
possible correlations among them (see below). Let us mention that although we
are aware of the existence of other LSS data points in the literature
concerning some of the used redshift values in our Table 4 – cf. e.g. [162,
163, 164, 165], we have explicitly checked that their inclusion or not in our
numerical fits has no significant impact on our main result, that is to say,
it does not affect the attained $\gtrsim 4\sigma$ level of evidence in favor
of the RVM’s. This result is definitely secured in both cases, but we have
naturally presented our final results sticking to the most updated data.

The following observation is also in order. We have included both the WiggleZ
and the CMASS data sets in our analysis. We are aware that there exists some
overlap region between the CMASS and WiggleZ galaxy samples. But the two
surveys have been produced independently and the studies on the existing
correlations among these observational results [166, 167] show that the
correlation is small. The overlap region of the CMASS and WiggleZ galaxy
samples is actually not among the galaxies that the two surveys pick up, but
between the region of the sky they explore. Moreover, despite almost all the
WiggleZ region (5/6 parts of it) is inside the CMASS one, it only takes a very
small fraction of the whole sky region covered by CMASS, since the latter is
much larger than the WiggleZ one (see, e.g. Figure 1 in [166]). In this paper,
the authors are able to quantify the correlation degree among the BAO
constraints in CMASS and WiggleZ, and they conclude that it is less than 4%.
Therefore, we find it justified to include the WiggleZ data in the main table
of results of our analysis (Table 1), but we provide also the fitting results
that are obtained when we remove the WiggleZ data points from the BAO and
$f(z)\sigma_{8}(z)$ data sets (see Table 2). The difference is small and the
central values of the fitting parameters and their uncertainties remain
intact. Thus the statistical significance of Tables 1 and 2 is the same.

S6) BBN: we have imposed the average bound on the possible variation of the
BBN speed-up factor, defined as the ratio of the expansion rate predicted in a
given model versus that of the $\Lambda$CDM model at the BBN epoch ($z\sim
10^{9}$). This amounts to the limit $|\Delta H^{2}/H_{\Lambda}^{2}|<10\%$
[168].

S7) CMB distance priors: $R$ (shift parameter) and $\ell_{a}$ (acoustic
length) and their correlations with $(\omega_{b},n_{s})$. For WMAP9 and Planck
2013 data we used the covariance matrix from the analysis of [169], while for
Planck 2015 data those of [131]. Our fitting results for the last case are
recorded in all our tables (except in Table 5 where we test our fit in the
absence of CMB distance priors $R$ and $\ell_{a}$). We display the final
contour plots for all the cases, see Figs. 1-2. Let us point out that in the
case of the Planck 2015 data we have checked that very similar results ensue
for all models if we use the alternative CMB covariance matrix from [81]. We
have, however, chosen to explicitly present the case based on [131] since it
uses the more complete compressed likelihood analysis for Planck 2015 TT,TE,EE
+ lowP data whereas [81] uses Planck 2015 TT+lowP data only.
Notice that G1 and A1 have one single vacuum parameter ($\nu$) whereas G2 and
A2 have two ($\nu,\alpha$). There is nonetheless a natural alignment between
$\nu$ and $\alpha$ for general type-G and A models, namely $\alpha=3\nu/4$, as
this entails $\xi^{\prime}=1$ (i.e. $\nu_{\rm eff}^{\prime}=0$) in Eq.(2.6).
Recall that for G2 models we have $G(a)\sim G_{0}\,a^{4(1-\xi^{\prime})}$ deep
in the radiation epoch, cf. Eq.(2.8), and therefore the condition
$\xi^{\prime}=1$ warrants $G$ to take the same value as the current one,
$G=G_{0}$, at BBN. For model G1 this is not possible (for $\nu\neq 0$) and we
adopt the aforementioned $|\Delta H^{2}/H_{\Lambda}^{2}|<10\%$ bound. We apply
the same BBN restrictions to the A1 and A2 models, which have constant $G$.
With this setting all the vacuum models contribute only with one single
additional parameter as compared to the $\Lambda$CDM: $\nu$, for G1 and A1;
and $\nu_{\rm eff}=\nu-\alpha=\nu/4$, for G2 and A2.

For the statistical analysis, we define the joint likelihood function as the
product of the likelihoods for all the data sets. Correspondingly, for
Gaussian errors the total $\chi^{2}$ to be minimized reads:

$\chi^{2}_{\rm tot}=\chi^{2}_{\rm SNIa}+\chi^{2}_{\rm
BAO}+\chi^{2}_{H}+\chi^{2}_{f\sigma_{8}}+\chi^{2}_{\rm BBN}+\chi^{2}_{\rm
CMB}\,.$ (2.11)

Each one of these terms is defined in the standard way, for some more details
see e.g. [170], although we should emphasize that here the correlation
matrices have been included. The BAO part was split as indicated in S2) and
S3) above. Also, in contrast to the previous analysis of [171], we did not use
here the correlated $Omh^{2}(z_{i},z_{j})$ diagnostic for $H(z_{i})$ data.
Instead, we use

$\chi^{2}_{\rm H}({\bf p})=\sum_{i=1}^{30}\left[\frac{H(z_{i},{\bf p})-H_{\rm
obs}(z_{i})}{\sigma_{H,i}}\right]^{2}\,.$ (2.12)

As for the linear structure formation data we have computed the density
contrast $\delta_{m}=\delta\rho_{m}/\rho_{m}$ for each vacuum model by
adapting the cosmic perturbations formalism for type-G and type-A vacuum
models. The matter perturbation, $\delta_{m}$, obeys a generalized equation
which depends on the RVM type. For type-A models it reads (as a differential
equation with respect to the cosmic time)

$\ddot{\delta}_{m}+\left(2H+\Psi\right)\,\dot{\delta}_{m}-\left(4\pi
G\rho_{m}-2H\Psi-\dot{\Psi}\right)\,\delta_{m}=0\,,$ (2.13)

where $\Psi\equiv-\frac{\dot{\rho}_{\Lambda}}{\rho_{m}}$. For
$\rho_{\Lambda}=$const. we have $\Psi=0$ and Eq.(2.13) reduces to the
$\Lambda$CDM form 999For details on these equations, confer the comprehensive
works [170], [172] and [173]. For type-G models the matter perturbation
equation is explicitly given in [171]. From here we can derive the weighted
linear growth $f(z)\sigma_{8}(z)$, where $f(z)=d\ln{\delta_{m}}/d\ln{a}$ is
the growth factor and $\sigma_{8}(z)$ is the rms mass fluctuation amplitude on
scales of $R_{8}=8\,h^{-1}$ Mpc at redshift $z$. It is computed from

$\small\sigma_{8}^{2}(z)=\frac{\delta_{m}^{2}(z)}{2\pi^{2}}\int_{0}^{\infty}k^{2}\,P(k,\vec{p})\,W^{2}(kR_{8})\,dk\,,$
(2.14)

with $W$ a top-hat smoothing function (see e.g. [170] for details). The linear
matter power spectrum reads $P(k,\vec{p})=P_{0}k^{n_{s}}T^{2}(k,\vec{p})$,
where $\vec{p}=(h,\omega_{b},n_{s},\Omega_{m},\nu_{\rm eff})$ is the fitting
vector for the vacuum models we are analyzing (including the $\Lambda$CDM, for
which $\nu_{\rm eff}=0$ of course), and $T(k,\vec{p})$ is the transfer
function, which we take from [174], upon introducing the baryon density
effects through the modified shape parameter $\Gamma$ [175, 176]. We have also
explicitly checked that the use of the effective shape of the transfer
function provided in [177] does not produce any change in our results.
The expression (2.16) at $z=0$ allows us to write $\sigma_{8}(0)$ in terms of
the power spectrum normalization factor $P_{0}$ and the primary parameters
that enter our fit for each model (cf. Table 1). We fix $P_{0}$ from

$\small
P_{0}=2\pi^{2}\frac{\sigma_{8,\Lambda}^{2}}{\delta^{2}_{m,\Lambda}(0)}\left[\int_{0}^{\infty}k^{2+n_{s,\Lambda}}T^{2}(k,\vec{p}_{\Lambda})W^{2}(kR_{8,\Lambda})dk\right]^{-1}\,,$
(2.15)

in which we have introduced the vector of fiducial parameters
$\vec{p}_{\Lambda}=(h_{\Lambda},\omega_{b,\Lambda},n_{s,\Lambda},\Omega_{m,\Lambda},0)$.
This vector is defined in analogy with the fitting vector introduced before,
but all its parameters are fixed and taken to be equal to those from the
Planck 2015 TT,TE,EE+lowP+lensing analysis [178] with $\nu_{\rm eff}=0$. The
fiducial parameter $\sigma_{8,\Lambda}$ is also taken from the aforementioned
Planck 2015 data. However, $\delta_{m,\Lambda}(0)$ in (2.15) is computable: it
is the value of $\delta_{m}(z=0)$ obtained from solving the perturbation
equation of the $\Lambda$CDM using the mentioned fiducial values of the other
parameters. Finally, from
$\sigma_{8}(z)=\sigma_{8}(0)\delta_{m}(z)/\delta_{m}(0)$ and plugging (2.15)
in (2.16) one finds:

$\small\sigma_{\rm
8}(z)=\sigma_{8,\Lambda}\frac{\delta_{m}(z)}{\delta_{m,\Lambda}(0)}\left[\frac{\int_{0}^{\infty}k^{2+n_{s}}T^{2}(k,\vec{p})W^{2}(kR_{8})dk}{\int_{0}^{\infty}k^{2+n_{s,\Lambda}}T^{2}(k,\vec{p}_{\Lambda})W^{2}(kR_{8,\Lambda})dk}\right]^{1/2}\,.$
(2.16)

Computing next the weighted linear growth rate $f(z)\sigma_{8}(z)$ for each
model under consideration, including the $\Lambda$CDM, all models become
normalized to the same fiducial model defined above. The results for
$f(z)\sigma_{8}(z)$ in the various cases are displayed in Fig. 4 together with
the LSS data measurements (cf. Table 4). We will further comment on these
results in the next section.

Figure 4: The $f(z)\sigma_{8}(z)$ data (Table 4) and the predicted curves by
the RVM’s, XCDM and the $\Lambda$CDM, using the best-fit values in Table 1.
Shown are also the values of $\sigma_{8}(0)$ that we obtain for all the
models. The theoretical prediction of all the RVM’s are visually
indistinguishable and they have been plotted using the same (blue) dashed
curve.

###  Discussion

Table 1 and Figures 1-2 present in a nutshell our main results. We observe
that the effective vacuum parameter, $\nu_{\rm eff}$, is neatly projected non
null and positive for all the RVM’s. The presence of this effect can be traced
already in the old WMAP9 data (at $\sim 2\sigma$), but as we can see it
becomes strengthened at $\sim 3.5\sigma$ c.l. with the Planck 2013 data and at
$\sim 4\sigma$ c.l. with the Planck 2015 data – see Figs. 1 and 2. For Planck
2015 data it attains up to $\gtrsim 4.2\sigma$ c.l. for all the RVM’s after
marginalizing over the other fitting parameters.

Model | $h$ | $\omega_{b}=\Omega_{b}h^{2}$ | $n_{s}$ | $\Omega_{m}$ | $\nu_{\rm eff}$ | $\omega$ | $\chi^{2}_{\rm min}/dof$ | $\Delta{\rm AIC}$ |  $\Delta{\rm BIC}$
---|---|---|---|---|---|---|---|---|---
$\Lambda$CDM | $0.679\pm 0.005$ | $0.02241\pm 0.00017$ | $0.968\pm 0.005$ | $0.291\pm 0.005$ | - | $-1$ | 68.42/83 | - | -
$X$CDM | $0.673\pm 0.007$ | $0.02241\pm 0.00017$ | $0.968\pm 0.005$ | $0.299\pm 0.009$ | - | $-0.958\pm 0.038$ | 67.21/82 | -1.10 | -3.26
A1 | $0.679\pm 0.010$ | $0.02241\pm 0.00017$ | $0.968\pm 0.005$ | $0.291\pm 0.010$ | $-0.00001\pm 0.00079$ | $-1$ | 68.42/82 | -2.31 | -4.47
A2 | $0.676\pm 0.009$ | $0.02241\pm 0.00017$ | $0.968\pm 0.005$ | $0.295\pm 0.014$ | $0.00047\pm 0.00139$ | $-1$ | 68.31/82 | -2.20 | -4.36
G1 | $0.679\pm 0.009$ | $0.02241\pm 0.00017$ | $0.968\pm 0.005$ | $0.291\pm 0.010$ | $0.00002\pm 0.00080$ | $-1$ | 68.42/82 | -2.31 | -4.47
G2 | $0.678\pm 0.012$ | $0.02241\pm 0.00017$ | $0.968\pm 0.005$ | $0.291\pm 0.013$ | $0.00006\pm 0.00123$ | $-1$ | 68.42/82 | -2.31 | -4.47

Table 5: Same as in Table 1, but removing both the $R$-shift parameter and the
acoustic length $l_{a}$ from our fitting analysis.

Model | $h$ | $\omega_{b}=\Omega_{b}h^{2}$ | $n_{s}$ | $\Omega_{m}$ | $\nu_{\rm eff}$ | $\omega$ | $\chi^{2}_{\rm min}/dof$ | $\Delta{\rm AIC}$ |  $\Delta{\rm BIC}$
---|---|---|---|---|---|---|---|---|---
$\Lambda$CDM | $0.685\pm 0.004$ | $0.02243\pm 0.00014$ | $0.969\pm 0.004$ | $0.304\pm 0.005$ | - | $-1$ | 61.70/72 | - | -
XCDM | $0.683\pm 0.009$ | $0.02245\pm 0.00015$ | $0.969\pm 0.004$ | $0.306\pm 0.008$ | - | $-0.991\pm 0.040$ | 61.65/71 | -2.30 | -4.29
A1 | $0.685\pm 0.010$ | $0.02243\pm 0.00014$ | $0.969\pm 0.004$ | $0.304\pm 0.005$ | $0.00003\pm 0.00062$ | $-1$ | 61.70/71 | -2.36 | -4.34
A2 | $0.684\pm 0.009$ | $0.02242\pm 0.00016$ | $0.969\pm 0.005$ | $0.304\pm 0.005$ | $0.00010\pm 0.00095$ | $-1$ | 61.69/71 | -2.35 | -4.33
G1 | $0.685\pm 0.010$ | $0.02243\pm 0.00014$ | $0.969\pm 0.004$ | $0.304\pm 0.005$ | $0.00003\pm 0.00065$ | $-1$ | 61.70/71 | -2.36 | -4.34
G2 | $0.685\pm 0.010$ | $0.02242\pm 0.00015$ | $0.969\pm 0.004$ | $0.304\pm 0.005$ | $0.00006\pm 0.00082$ | $-1$ | 61.70/71 | -2.36 | -4.34

Table 6: Same as in Table 1, but removing the points from the LSS data set
from our analysis, i.e. all the 13 points on $f\sigma_{8}$ .

It is also interesting to gauge the dynamical character of the DE by
performing a fit to the overall data in terms of the well-known XCDM
parametrization, in which the DE is mimicked through the density
$\rho_{X}(a)=\rho_{X0}\,a^{-3(1+\omega)}$ associated to some generic entity X,
which acts as an ersatz for the $\Lambda$ term; $\rho_{X0}$ being the current
energy density value of X and therefore equivalent to $\rho_{\Lambda 0}$, and
$\omega$ is the (constant) equation of state (EoS) parameter for X. The XCDM
trivially boils down to the rigid $\Lambda$-term for $\omega=-1$, but by
leaving $\omega$ free it proves a useful approach to roughly mimic a (non-
interactive) DE scalar field with constant EoS. The corresponding fitting
results are included in all our tables along with those for the RVM’s and the
$\Lambda$CDM. In Table 1 (our main table) and in Fig. 3, we can see that the
best fit value for $\omega$ in the XCDM is $\omega=-0.916\pm 0.021$.
Remarkably, it departs from $-1$ by precisely $4\sigma$.
Obviously, given the significance of the above result it is highly convenient
to compare it with previous analyses of the XCDM reported by the Planck and
BOSS collaborations. The Planck 2015 value for the EoS parameter of the XCDM
reads $\omega=-1.019^{+0.075}_{-0.080}$ [178] and the BOSS one is
$\omega=-0.97\pm 0.05$ [157]. These results are perfectly compatible with our
own result for $\omega$ shown in Table 1 for the XCDM, but in stark contrast
to our result their errors are big enough as to be also fully compatible with
the $\Lambda$CDM value $\omega=-1$. This is, however, not too surprising if we
take into account that none of these analyses included LSS data in their fits,
as explicitly indicated in their papers 101010Furthermore, at the time these
analyses appeared they could not have used the important LSS and BAO results
from [147], i.e. those that we have incorporated as part of our current data
set, not even the previous ones from [147]. The latter also carry a
significant part of the dynamical DE signature we have found here, as we have
checked.. In the absence of LSS data we would find a similar situation. In
fact, as our Table 6 clearly shows, the removal of the LSS data set in our fit
induces a significant increase in the magnitude of the central value of the
EoS parameter, as well as the corresponding error. This happens because the
higher is $|\omega|$ the higher is the structure formation power predicted by
the XCDM, and therefore the closer is such prediction with that of the
$\Lambda$CDM (which is seen to predict too much power as compared to the data,
see Fig. 4). In these conditions our analysis renders $\omega=-0.991\pm
0.040$, which is definitely closer to (and therefore compatible with) the
central values obtained by Planck and BOSS teams. In addition, this result is
now fully compatible with the $\Lambda$CDM, as in the Planck 2015 and BOSS
cases, and all of them are unfavored by the LSS observations. This is
consistent with the fact that both information criteria, $\Delta$AIC and
$\Delta$BIC, become now slightly negative in Table 6, which reconfirms that if
the LSS data are not used the $\Lambda$CDM performance is comparable or even
better than the other models. So in order to fit the observed values of
$f\sigma_{8}$, which are generally lower than the predicted ones by the
$\Lambda$CDM, $|\omega|$ should decrease. This is exactly what happens for the
XCDM, as well as for the RVM’s, when the LSS data are included in our analysis
(in combination with the other data, particularly with BAO and CMB data). It
is apparent from Fig. 4 that the curves for these models are then shifted
below and hence adapt significantly better to the data points.
Correspondingly, the quality of the fits increases dramatically, and this is
also borne out by the large and positive values of $\Delta$AIC and
$\Delta$BIC, both above $10$ (cf. Table 1).
The above discussion explains why our analysis of the observations through the
XCDM is sensitive to the dynamical feature of the DE, whereas the previous
results in the literature are not. It also shows that the size of the effect
found with such a parametrization of the DE essentially concurs with the
strength of the dynamical vacuum signature found for the RVM’s using exactly
the same data. This is remarkable, and it was not obvious a priori since for
some of our RVM’s (specifically for A1 and A2) there is an interaction between
vacuum and matter that triggers an anomalous conservation law, whereas for
others (G1 and G2) we do not have such interaction (meaning that matter is
conserved in them, thereby following the standard decay laws for relativistic
and non-relativistic components). The interaction, when occurs, is however
proportional to $\nu_{\rm eff}$ and thus is small because the fitted value of
$\nu_{\rm eff}$ is small. This probably explains why the XCDM can succeed in
nailing down the dynamical nature of the DE with a comparable performance.
However not all dynamical vacuum models describe the data with the same
efficiency, see e.g. [108, 179, 180]. In the XCDM case the departure from the
$\Lambda$CDM takes the fashion of “effective quintessence”, whereas for the
RVM’s it appears as genuine vacuum dynamics. In all cases, however, we find
unmistakable signs of DE physics beyond the $\Lambda$CDM (cf. Table 1), and
this is a most important result of our work.
As we have discussed in Section 2.1, for models A1 and A2 there is an
interaction between vacuum and matter. Such interaction is, of course, small
because the fitted values of $\nu_{\rm eff}$ are small, see Table 1. The
obtained values are in the ballpark of $\nu_{\rm eff}\sim{\cal O}(10^{-3})$
and therefore this is also the order of magnitude associated to the anomalous
conservation law of matter. For example, for the non-relativistic component we
have

$\rho_{m}(a)=\rho_{m0}a^{-3\xi}=\rho_{m0}a^{-3(1-\nu_{\rm eff})}\,.$ (2.17)

This behaviour has been used in the works by [181, 182] as a possible
explanation for the hints on the time variation of the fundamental constants,
such as coupling constants and particle masses, frequently considered in the
literature. The current observational values for such time variation are
actually compatible with the fitted values we have found here. This is an
intriguing subject that is currently of high interest in the field, see e.g.
[168] and [183]. For models G1 and G2, instead, the role played by $\nu_{\rm
eff}$ and $\nu_{\rm eff}^{\prime}$ is different. It does not produce any
anomaly in the traditional matter conservation law (since matter and radiation
are conserved for type-G models), but now it impinges a small (logarithmic)
time evolution on $G$ in the fashion sketched in Eq.(2.8). Thus we find, once
more, a possible description for the potential variation of the fundamental
constants, in this case $G$, along the lines of the above cited works, see
also [184]. There are, therefore, different phenomenological possibilities to
test the RVM’s considered here from various points of view.

Model | $h$ | $\omega_{b}=\Omega_{b}h^{2}$ | $n_{s}$ | $\Omega_{m}$ | $\nu_{\rm eff}$ | $\omega$ | $\chi^{2}_{\rm min}/dof$ | $\Delta{\rm AIC}$ |  $\Delta{\rm BIC}$
---|---|---|---|---|---|---|---|---|---
$\Lambda$CDM | $0.693\pm 0.006$ | $0.02265\pm 0.00022$ | $0.976\pm 0.004$ | $0.293\pm 0.007$ | - | $-1$ | 39.35/38 | - | -
XCDM | $0.684\pm 0.010$ | $0.02272\pm 0.00023$ | $0.977\pm 0.005$ | $0.300\pm 0.009$ | - | $-0.960\pm 0.033$ | 37.89/37 | -1.25 | -2.30
A1 | $0.681\pm 0.011$ | $0.02254\pm 0.00023$ | $0.972\pm 0.005$ | $0.297\pm 0.008$ | $0.00057\pm 0.00043$ | $-1$ | 37.54/37 | -0.90 | -1.95
A2 | $0.684\pm 0.009$ | $0.02252\pm 0.00024$ | $0.971\pm 0.005$ | $0.297\pm 0.008$ | $0.00074\pm 0.00057$ | $-1$ | 37.59/37 | -0.95 | -2.00
G1 | $0.681\pm 0.011$ | $0.02254\pm 0.00023$ | $0.972\pm 0.005$ | $0.297\pm 0.008$ | $0.00059\pm 0.00045$ | $-1$ | 37.54/37 | -0.90 | -1.95
G2 | $0.682\pm 0.010$ | $0.02253\pm 0.00024$ | $0.971\pm 0.005$ | $0.297\pm 0.008$ | $0.00067\pm 0.00052$ | $-1$ | 37.61/37 | -0.97 | -2.02

Table 7: Fitting results using the same data as in [81]

We may reassess the quality fits obtained in this work from a different point
of view. While the $\chi^{2}_{\rm min}$ value of the overall fit for any RVM
and the XCDM is seen to be definitely smaller than the $\Lambda$CDM one, it
proves very useful to reconfirm our conclusions with the help of the time-
honored Akaike and Bayesian information criteria, AIC and BIC, see [185, 186,
187]. They read as follows:

${\rm AIC}=\chi^{2}_{\rm min}+\frac{2nN}{N-n-1}\,,\ \ \ \ \ {\rm
BIC}=\chi^{2}_{\rm min}+n\,\ln N\,.$ (2.18)

In both cases, $n$ is the number of independent fitting parameters and $N$ the
number of data points used in the analysis. To test the effectiveness of a
dynamical DE model (versus the $\Lambda$CDM) for describing the overall data,
we evaluate the pairwise differences $\Delta$AIC ($\Delta$BIC) with respect to
the model that carries smaller value of AIC (BIC) – in this case, the RVM’s or
the XCDM. The larger these differences the higher is the evidence against the
model with larger value of AIC (BIC) – the $\Lambda$CDM, in this case. For
$\Delta$AIC and/or $\Delta$BIC in the range $6-10$ one may claim “strong
evidence” against such model; and, above 10, one speaks of “very strong
evidence” [185, 187]. The evidence ratio associated to rejection of the
unfavored model is given by the ratio of Akaike weights, $e^{\Delta{\rm
AIC}/2}$. Similarly, $e^{\Delta{\rm BIC}/2}$ estimates the so-called Bayes
factor, which gives the ratio of marginal likelihoods between the two models
[55]. Table 1 reveals conspicuously that the $\Lambda$CDM appears very
strongly disfavored (according to the above statistical standards) as compared
to the running vacuum models. Specifically, $\Delta$AIC is in the range
$17-18$ and $\Delta$BIC around $15$ for all the RVM’s. These results are fully
consistent and since both $\Delta$AIC and $\Delta$BIC are well above $10$ the
verdict of the information criteria is conclusive. But there is another
remarkable feature to single out at this point, namely the fact that the
simple XCDM parametrization is now left behind as compared to the RVM’s. While
the corresponding XCDM values of $\Delta$AIC and $\Delta$BIC are also above
$10$ (reconfirming the ability of the XCDM to improve the $\Lambda$CDM fit)
they stay roughly $4$ points below the corresponding values for the RVM’s.
This is considered a significant difference from the point of view of the
information criteria. Therefore, we conclude that the RVM’s are significantly
better than the XCDM in their ability to fit the data. In other words, the
vacuum dynamics inherent to the RVM’s seems to describe better the overall
cosmological data than the effective quintessence behaviour suggested by the
XCDM parametrization. Being the ratio of Akaike weights and Bayes factor much
bigger for the RVM’s than for the $\Lambda$CDM, the former appear definitely
much more successful than the latter. The current analysis undoubtedly
reinforces the conclusions of our previous study [171], with the advantage
that the determination of the vacuum parameters is here much more precise and
therefore at a higher significance level. Let us stand out some of the most
important differences with respect to that work: 1) To start with, we have
used now a larger and fully updated set of cosmological data; 2) The selected
data set is uncorrelated and has been obtained from independent analysis in
the literature, see points S1-S7) above and references therein; 3) We have
taken into account all the known covariance matrices among the data; 4) In
this work, $h$, $\omega_{b}$ and $n_{s}$ are not fixed a priori (as we did in
the previous one), we have now allowed them to vary in the fitting process.
This is, of course, not only a more standard procedure, but also a most
advisable one in order to obtain unbiased results. The lack of consensus on
the experimental value of $h$ is the main reason why we have preferred to use
an uninformative flat prior – in the technical sense – for this parameter.
This should be more objective in these circumstances, rather than being
subjectively elicited – once more in the technical sense – by any of these
more or less fashionable camps for $h$ that one finds in the literature, [188,
189, 190, 191, 192, 157, 178, 193], whose ultimate fate is unknown at present
(compare, e.g. the value from [178] with the one from [193], which is $\sim
3\sigma$ larger than the former); 5) But the most salient feature perhaps, as
compared to our previous study, is that we have introduced here a much more
precise treatment of the CMB, in which we used not only the shift parameter,
$R$, (which was the only CMB ingredient in our previous study) but the full
data set indicated in S7) above, namely $R$ together with $\ell_{a}$ (acoustic
length) and their correlations with $(\omega_{b},n_{s})$.

Figure 5: Reconstruction of the contour lines for model A2, under Planck 2015
CMB data (rightmost plot in Fig. 2) from the partial contour plots of the
different SNIa+BAO+$H(z)$+LSS+BBN+CMB data sources. The $1\sigma$ and
$2\sigma$ contours are shown in all cases. For the reconstructed final contour
lines we also plot the $3\sigma$, $4\sigma$ and $5\sigma$ regions.

Altogether, this explains the substantially improved accuracy obtained in the
current fitted values of the $\nu_{\rm eff}$ parameter as compared to [171].
In particular, in what concerns points 1-3) above we should stress that for
the present analysis we are using a much more complete and restrictive BAO
data set. Thus, while in our previous work we only used 6 BAO data points
based on the $A(z)$ estimator (cf. Table 3 of [194]), here we are using a
total of 11 BAO points (none of them based on $A(z)$, see S2-S3). These
include the recent results from [147], which narrow down the allowed parameter
space in a more efficient way, not only because the BAO data set is larger but
also owing to the fact that each of the data points is individually more
precise and the known correlation matrices have been taken into account.
Altogether, we are able to significantly reduce the error bars with respect to
the ones we had obtained in our previous work. We have actually performed a
practical test to verify what would be the impact on the fitting quality of
our analysis if we would remove the acoustic length $l_{a}$ from the CMB part
of our data and replace the current BAO data points by those used in [171].
Notice that the CMB part is now left essentially with the $R$-shift parameter
only, which was indeed the old situation. The result is that we recover the
error bars’ size shown in the previous papers, which are $\sim 4-5$ times
larger than the current ones, i.e. of order $\mathcal{O}(10^{-3})$. We have
also checked what would be the effect on our fit if we would remove both the
data on the shift parameter and on the acoustic length; or if we would remove
only the data points on LSS. The results are presented in Tables 5 and 6,
respectively. We observe that the $\Delta$AIC and $\Delta$BIC values become
$2-4$ points negative. This means that the full CMB and LSS data are
individually very important for the quality of the fit and that without any of
them the evidence of dynamical DE would be lost. If we would restore part of
the CMB effect on the fit in Table 5 by including the $R$-shift parameter in
the fitting procedure we can recover, approximately, the situation of our
previous analysis, but not quite since the remaining data sources used now are
more powerful.
It is also interesting to explore what would have been the result of our fits
if we would not have used our rather complete SNIa+BAO+$H(z)$+LSS+BBN+CMB data
set and had restricted ourselves to the much more limited one used by the
Planck 2015 collaboration in the paper [81]. The outcome is presented in Table
7. In contrast to [178], where no LSS (RSD) data were used, the former
reference uses some BAO and LSS data, but their fit is rather limited in scope
since they use only 4 BAO data points, 1 AP (Alcock-Paczynski parameter) data
point, and one single LSS point, namely $f\sigma_{8}$ at $z=0.57$, see details
in that paper. In contradistinction to them, in our case we used 11 BAO and 13
LSS data points, some of them very recent and of high precision [147]. From
Table 7 it is seen that with only the data used in [81] the fitting results
for the RVM’s are poor enough and cannot still detect clear traces of the
vacuum dynamics. In particular, the $\Delta$AIC and $\Delta$BIC values in that
table are moderately negative, showing that the $\Lambda$CDM does better with
only these data. As stated before, not even the XCDM parametrization is able
to detect any trace of dynamical DE with that limited data set, as the
effective EoS is compatible with $\omega=-1$ at roughly $1\sigma$
($\omega=-0.960\pm 0.033$). This should explain why the features that we are
reporting here have been missed till now.
We complete our analysis by displaying in a graphical way the contributions
from the different data sets to our final contour plots in Figs. 1-3. We start
analyzing the RVM’s case. For definiteness we concentrate on the rightmost
plot for model A2 in Fig. 2, but we could do similarly for any other one in
Figs 1-2. The result for model A2 is depicted in Fig. 5, where we can assess
the detailed reconstruction of the final contours in terms of the partial
contours from the different SNIa+BAO+$H(z)$+LSS+BBN+CMB data sources. This
reconstruction is presented through a series of three plots made at different
magnifications. In the third plot of the sequence we can easily appraise that
the BAO+LSS+CMB data subset plays a fundamental role in narrowing down the
final physical region of the $(\Omega_{m},\nu_{\rm eff})$ parameter space, in
which all the remaining parameters have been marginalized over. This
reconstruction also explains in very obvious visual terms why the conclusions
that we are presenting here hinge to a large extent on considering the most
sensitive components of the data. While CMB obviously is a high precision
component in the fit, we demonstrate in our study (both numerically and
graphically) that the maximum power of the fit is achieved when it is combined
with the wealth of BAO and LSS data points currently available.

Figure 6: Upper-left plot: two-dimensional $\Omega_{m}-\omega$ contours at
$1\sigma$ and $2\sigma$ c.l. for the XCDM, obtained with only the LSS data
set. The dotted contours in blue and purple are the exact ones, whilst the red
and green ellipses have been obtained using the Fisher’s approximation. Upper-
right plot: Same, but for the combination BAO+LSS. Lower-left plot: As in the
upper plots, but for the CMB data. Lower-right plot: The Fisher’s generated
contours at $1\sigma$ and $2\sigma$ c.l. for all the data sets: SNIa (dotted
lines), H(z) (solid lines), BAO (dot-dashed lines), LSS (dotted, very thin
lines) and CMB (solid lines, tightly packed in a very small, segment-shaped,
region at such scale of the plot). The exact, final, combined contours (from
$1\sigma$ up to $5\sigma$) can be glimpsed in the small colored area around
the center. See the text for further explanations and Fig. 3 for a detailed
view.

In Fig. 6 we show the corresponding decomposition of the data contours for the
XCDM model as well. In the upper-left plot we display the two-dimensional
contours at $1\sigma$ and $2\sigma$ c.l. in the $(\Omega_{m},\omega)$ plane,
found using only the LSS data set. The elliptical shapes are obtained upon
applying the Fisher matrix formalism [55], i.e. assuming that the two-
dimensional distribution is normal (Gaussian) not only in the closer
neighborhood of the best-fit values, but in all the parameter space. In order
to obtain the dotted contours we have sampled the exact distribution making
use of the Metropolis-Hastings Markov chain Monte Carlo algorithm [195, 196].
We find a significant deviation from the ideal perfectly Gaussian case. In the
upper-right plot we do the same for the combination BAO+LSS. The continuous
and dotted contours are both elliptical, which remarkably demonstrates the
Gaussian behaviour of the combined BAO+LSS distribution. Needless to say the
correlations among BAO and LSS data (whose covariance matrices are known) are
responsible for that, i.e. they explain why the product of the non-normal
distribution obtained from the LSS data and the Gaussian BAO one produces
perfectly elliptical dotted contours for the exact BAO+LSS combination.
Similarly, in the lower-left plot we compare the exact (dotted) and Fisher’s
generated (continuous) lines for the CMB data. Again, it is apparent that the
distribution inferred from the CMB data in the $(\Omega_{m},\omega)$ plane is
a multivariate normal. Finally, in the lower-right plot we produce the
contours at $1\sigma$ and $2\sigma$ c.l. for all the data sets in order to
study the impact of each one of them. They have all been found using the
Fisher approximation, just to sketch the basic properties of the various data
sets, despite we know that the exact result deviates from this approximation
and therefore their intersection is not the final answer. The final contours
(up to $5\sigma$) obtained from the exact distributions can be seen in the
small colored area around the center of the lower-right plot. The reason to
plot it small at that scale is to give sufficient perspective to appreciate
the contour lines of all the participating data. The final plot coincides, of
course, with the one in Fig.3, where it can be appraised in full detail.
As it is clear from Fig. 6, the data on the $H(z)$ and SNIa observable are not
crucial for distilling the final dynamical DE effect, as they have a very low
constraining power. This was also so for the RVM case. Once more the final
contours are basically the result of the combination of the crucial triplet of
BAO+LSS+CMB data (upon taking due care in this case of the deviations from
normality of the LSS-inferred distribution). The main conclusion is
essentially the same as for the corresponding RVM analysis of combined
contours in Fig. 5, except that in the latter there are no significant
deviations from the normal distribution behaviour, as we have checked, and
therefore all the contours in Fig. 5 can be accurately computed using the
Fisher’s matrix method.
The net outcome is that using either the XCDM or the RVM’s the signal in favor
of the DE dynamics is clearly pinned down and in both cases it is the result
of the combination of all the data sets used in our detailed analysis,
although to a large extent it is generated from the crucial BAO+LSS+CMB
combination of data sets. In the absence of any of them the signal would get
weakened, but when the three data sets are taken together they have enough
power to capture the signal of dynamical DE at the remarkable level of $\sim
4\sigma$.

###  Conclusions

To conclude, the running vacuum models emerge as serious alternative
candidates for the description of the current state of the Universe in
accelerated expansion. These models have a close connection with the possible
quantum effects on the effective action of QFT in curved space-time, cf. [33]
and references therein. There were previous phenomenological studies that
hinted in different degrees at the possibility that the RVM’s could fit the
data similarly as the $\Lambda$CDM, see e.g. the earlier works by [197, 135,
198, 199], as well as the more recent ones by [170] and [172], including of
course the study that precedes this work, [171]. However, to our knowledge
there is no devoted work comparable in scope to the one presented here for the
running vacuum models under consideration. The significantly enhanced level of
dynamical DE evidence attained with them is unprecedented, to the best of our
knowledge, all the more if we take into account the diversified amount of data
used. Our study employed for the first time the largest updated
SNIa+BAO+$H(z)$+LSS+BBN+CMB data set of cosmological observations available in
the literature. Some of these data (specially the BAO+LSS+CMB part) play a
crucial role in the overall fit and are substantially responsible for the main
effects reported here. Furthermore, recently the BAO+LSS components have been
enriched by more accurate contributions, which have helped to further enhance
the signs of the vacuum dynamics. At the end of the day it has been possible
to improve the significance of the dynamical hints from a confidence level of
roughly $3\sigma$, as reported in our previous study [171], up to the
$4.2\sigma$ achieved here. Overall, the signature of dynamical vacuum energy
density seems to be rather firmly supported by the current cosmological
observations. Already in terms of the generic XCDM parametrization we are able
to exclude, for the first time, the absence of vacuum dynamics ($\Lambda$CDM)
at $4\sigma$ c.l., but such limit can be even surpassed at the level of the
RVM’s and other related dynamical vacuum models, see the review [200].
It may be quite appropriate to mention at this point of our analysis the very
recent study of us [201], in which we have considered the well-known Peebles &
Ratra scalar field model with an inverse power law potential
$V(\phi)\propto\phi^{-{\alpha}}$ [202], where the power ${\alpha}$ here
should, of course, not be confused with a previous use of $\alpha$ for model
A2 in Section 2.1. In that study we consider the response of the Peebles &
Ratra model when fitted with the same data sets as those used in the current
work. Even though there are other recent tests of that model, see e.g. the
works by [203, 158, 204, 205, 206, 207], none of them used a comparably rich
data set as the one we used here. This explains why the analysis of [201] was
able to show that a non-trivial scalar field model, such as the Peebles &
Ratra model, is able to fit the observations at a level comparable to the
models studied here. In fact, the central value of the ${\alpha}$ parameter of
the potential is found to be non-zero at $\sim 4\sigma$ c.l., and the
corresponding equation of state parameter $\omega$ deviates consistently from
$-1$ also at the $4\sigma$ level. These remarkable features are only at reach
when the crucial triplet of BAO+LSS+CMB data are at work in the fitting
analysis of the various cosmological models. The net outcome of these
investigations is that several models and parametrizations of the DE do
resonate with the conclusion that there is a significant ($\sim 4\sigma$)
effect sitting in the current wealth of cosmological data. The effect looks
robust enough and can be unveiled using a variety of independent frameworks.
Needless to say, compelling statistical evidence conventionally starts at
$5\sigma$ c.l. and so we will have to wait for updated observations to see if
such level of significance can eventually be attained. In the meanwhile the
possible dynamical character of the cosmic vacuum, as suggested by the present
study, is pretty high and gives hope for an eventual solution of the old
cosmological constant problem, perhaps the toughest problem of fundamental
physics.

##  Dynamical dark energy: scalar fields and running vacuum

In the previous chapter, a careful and complete analysis of the RVM’s has been
carried out. This models are built around the possible connection between the
$\Lambda$-term and the vacuum energy. This assumption automatically leads us
to the following equation of state $p_{\Lambda}=-\rho_{\Lambda}$. Nevertheless
this is not the only possibility that can be considered as we shall see in
this chapter.
Another very interesting option is the one in which dark energy (DE) is
described in terms of a scalar field with some standard form for its
potential. These kind of models, commonly denoted as $\phi$CDM models, have a
local Lagrangian description. While it is true that the scalar field does not
play an important role until the late time Universe, it can be part of a more
general and complex potential acting in the inflationary period.
Unlike for the RVM’s models it is not possible to obtain an analytical
expression for the Hubble function of the scalar field models. As a
consequence we will work out some approximate solutions that will be used as
an initial conditions to solve numerically the differential equations. For the
sake of comparison, we present also the results obtained for the RVM as well
as the results for two very well-known DE parameterizations, the XCDM and the
CPL. The main upshot of the study is that a high evidence in favour of models
with a dynamical DE is found.
This chapter is organized as follows: In Section 3.1 we present the
theoretical background of the $\phi$CDM model. We list the main cosmological
equations and we write down their particular form when the Peebles & Ratra
potential $V(\phi)\sim\phi^{-\alpha}$ is chosen. We also find analytical
solutions in the Matter Dominated Epoch (MDE) and in the Radiation Dominated
Epoch (RDE). In sections 3.2 and 3.3 for convenience we provide the background
cosmological equations for the DE parameterizations before mentioned and for
the RVM respectively. We extend the study from the background level to the
perturbative level in Section 3.4 whereas in Section 3.5 we finally deliver
our conclusions.

###  $\phi$CDM with Peebles & Ratra potential

Suppose that the dark energy is described in terms of some scalar field $\phi$
with a standard form for its potential $V(\phi)$, see below. We wish to
compare its ability to describe the data with that of the $\Lambda$CDM, and
also with other models of DE existing in the literature. The data used in our
analysis will be the same one used in our previous studies [171, 77, 78]
namely data on the distant supernovae (SNIa), the baryonic acoustic
oscillations (BAO), the Hubble parameter at different redshifts, $H(z_{i})$,
the large scale structure formation data (LSS), and the cosmic microwave
background (CMB). We denote this string of data as SNIa+BAO+$H(z)$+LSS+CMB.
Precise information on these data and corresponding observational references
are given in the aforementioned papers and are also summarized in the caption
of Table 1. The main results of our analysis are displayed in Tables 1-3 and
Figures 1-5, which we will account in detail throughout our exposition.

We start by explaining our theoretical treatment of the $\phi$CDM model in
order to optimally confront it with observations. The scalar field $\phi$ is
taken to be dimensionless, being its energy density and pressure given by

$\rho_{\phi}=\frac{M^{2}_{pl}}{16\pi}\left[\frac{\dot{\phi}^{2}}{2}+V(\phi)\right]\,,\
p_{\phi}=\frac{M^{2}_{pl}}{16\pi}\left[\frac{\dot{\phi}^{2}}{2}-V(\phi)\right]\,.$
(3.1)

Here $M_{pl}=1/\sqrt{G}=1.22\times 10^{19}$ GeV is the Planck mass, in natural
units. As a representative potential we adopt the original Peebles & Ratra
(PR) form [202]:

$V(\phi)=\frac{1}{2}\kappa M_{pl}^{2}\phi^{-\alpha}\,,$ (3.2)

in which $\kappa$ and $\alpha$ are dimensionless parameters. These are to be
determined in our fit to the overall cosmological data. The motivation for
such potential is well described in the original paper [202]. In a nutshell:
such potential stands for the power-law tail of a more complete effective
potential in which inflation is also comprised. We expect $\alpha$ to be
positive and sufficiently small such that $V(\phi)$ can mimic an approximate
CC that is decreasing slowly with time, in fact more slowly than the matter
density. Furthermore, we must have $0<\kappa\ll 1$ such that $V(\phi)$ can be
positive and of the order of the measured value $\rho^{0}_{\Lambda}\sim
10^{-47}$ GeV4. In the late Universe the tail of the mildly declining
potential finally surfaces over the matter density (not far away in our past)
and appears as an approximate CC which dominates since then. Recent studies
have considered the PR-potential in the light of the cosmological data, see
e.g. [205, 206, 158, 204, 207]. Here we show that the asset of current
observations indicates strong signs of dynamical DE which can be parametrized
with such potential. In this way, we corroborate the unambiguous signs
recently obtained with independent DE models [171, 77, 78] and with a similar
level of confidence.

Model | $\Omega_{m}$ | $\omega_{b}=\Omega_{b}h^{2}$ | $n_{s}$ | $h$ | $\chi^{2}_{\rm min}/dof$
---|---|---|---|---|---
$\Lambda$CDM | $0.294\pm 0.004$ | $0.02255\pm 0.00013$ | $0.976\pm 0.003$ | $0.693\pm 0.004$ | 90.44/85

Table 1: The best-fit values for the $\Lambda$CDM parameters
$(\Omega_{m},\omega_{b},n_{s},h)$. We use a total of $89$ data points from
SNIa+BAO+$H(z)$+LSS+CMB observables in our fit: namely $31$ points from the
JLA sample of SNIa [130], $11$ from BAO [153, 154, 155, 147, 156, 157], $30$
from $H(z)$ [136, 137, 138, 139, 140, 141, 142], $13$ from linear growth [147,
143, 144, 145, 146, 148, 149, 150, 151, 152], and $4$ from CMB [131]. For a
summarized description of these data, see [77]. The quoted number of degrees
of freedom ($dof$) is equal to the number of data points minus the number of
independent fitting parameters ($4$ for the $\Lambda$CDM). For the CMB data we
have used the marginalized mean values and standard deviation for the
parameters of the compressed likelihood for Planck 2015 TT,TE,EE + lowP data
from [131]. The parameter M in the SNIa sector [130] was dealt with as a
nuisance parameter and has been marginalized over analytically. The best-fit
values and the associated uncertainties for each parameter in the table have
been obtained by numerically marginalizing over the remaining parameters [55]

The scalar field of the $\phi$CDM models satisfies the Klein-Gordon equation
in the context of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric:
$\ddot{\phi}+3H\dot{\phi}+{dV}/{d\phi}=0$, where $H=\dot{a}/a$ is the Hubble
function. In some cases the corresponding solutions possess the property of
having an attractor-like behaviour, in which a large family of solutions are
drawn towards a common trajectory [208, 121, 209]. If there is a long period
of convergence of all the family members to that common trajectory, the latter
is called a “tracker solution” [121, 209]. When the tracking mechanism is at
work, it funnels a large range of initial conditions into a common final state
for a long time (or forever, if the convergence is strict). Not all potentials
$V$ admit tracking solutions, only those fulfilling the “tracker condition”
$\Gamma\equiv V\,V^{\prime\prime}/(V^{\prime})^{2}>1$ [121, 209], where
$V^{\prime}=\partial V/\partial\phi$. For the Peebles-Ratra potential (3.2),
one easily finds $\Gamma=1+1/\alpha$, so it satisfies such condition precisely
for $\alpha>0$.
It is frequently possible to seek power-law solutions, i.e.
$\phi(t)=A\,t^{p}$, for the periods when the energy density of the Universe is
dominated by some conserved matter component
$\rho(a)=\rho_{1}\left({a_{1}}/{a}\right)^{n}$ (we may call these periods
“nth-epochs”). For instance, $n=3$ for the matter-dominated epoch (MDE) and
$n=4$ for the radiation-dominated epoch (RDE), with $a_{1}$ the scale factor
at some cosmic time $t_{1}$ when the corresponding component dominates. We
define $a=1$ as the current value. Solving Friedmann’s equation in flat space,
$3H^{2}(a)=8\pi G\rho(a)$, we find $H(t)={2}/(nt)$ as a function of the cosmic
time in the nth-epoch. Substituting these relations in the Klein-Gordon
equation with the Peebles-Ratra potential (3.2) leads to

$p=\frac{2}{\alpha+2}\,,\qquad
A^{\alpha+2}=\frac{\alpha(\alpha+2)^{2}M_{pl}^{2}\kappa
n}{4(6\alpha+12-n\alpha)}\,.$ (3.3)

From the power-law form we find the evolution of the scalar field with the
cosmic time:

$\phi(t)=\left[\frac{\alpha(\alpha+2)^{2}M_{pl}^{2}\kappa
n}{4(6\alpha+12-n\alpha)}\right]^{1/(\alpha+2)}t^{2/(\alpha+2)}\,.$ (3.4)

In any of the nth-epochs the equation of state (EoS) of the scalar field
remains stationary. A straightforward calculation from (3.1), (3.2) and (3.4)
leads to a very compact form for the EoS:

$w_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}=-1+\frac{\alpha n}{3(2+\alpha)}\,.$
(3.5)

Since the matter EoS in the nth-epoch is given by $\omega_{n}=-1+n/3$, it is
clear that (3.5) can be rewritten also as
$w_{\phi}=(\alpha\omega_{n}-2)/(\alpha+2)$. This is precisely the form
predicted by the tracker solutions [121, 209], in which the condition
$w_{\phi}<\omega_{n}$ is also secured since $|\alpha|$ is expected small. In
addition, $w_{\phi}$ remains constant in the RDE and MDE, but its value does
not depend on $\kappa$, only on $n$ (or $\omega_{n}$) and $\alpha$. The
fitting analysis presented in Table 2 shows that $\alpha={\cal O}(0.1)>0$ and
therefore $w_{\phi}\gtrsim-1$. It means that the scalar field behaves as
quintessence in the pure RD and MD epochs (cf. the plateaus at constant values
$w_{\phi}\gtrsim-1$ in Fig. 1). Notice that the behaviour of $w_{\phi}$ in the
interpolating epochs, including the period near our time, is not constant (in
contrast to the XCDM, see next section) and requires numerical solution of the
field equations. See also [210] for related studies.
We can trade the cosmic time in (3.4) for the scale factor. This is possible
using $t^{2}=3/(2\pi Gn^{2}\rho)$ (which follows from Friedmann’s equation in
the nth-epoch) and $\rho(a)=\rho_{0}a^{-n}=\rho_{c0}\,\Omega\,a^{-n}$, where
$\Omega=\Omega_{m},\Omega_{r}$ are the present values of the cosmological
density parameters for matter ($n=3$) or radiation ($n=4$) respectively, with
$\rho_{c0}=3H_{0}^{2}/(8\pi\,G)$ the current critical energy density. Notice
that $\Omega_{m}=\Omega_{dm}+\Omega_{b}$ involves both dark matter and
baryons. In this way we can determine $\phi$ as a function of the scale factor
in the nth-epoch. For example, in the MDE we obtain

$\phi(a)=\left[\frac{\alpha(\alpha+2)^{2}\bar{\kappa}}{9\times
10^{4}\omega_{m}(\alpha+4)}\right]^{1/(\alpha+2)}a^{3/(\alpha+2)}\,.$ (3.6)

Here we have conventionally defined the reduced matter density parameter
$\omega_{m}\equiv\Omega_{m}\,h^{2}$, in which the reduced Hubble constant $h$
is defined as usual from $H_{0}\equiv 100h\,\varsigma$, with $\varsigma\equiv
1{\rm km/s/Mpc}=2.133\times 10^{-44}{\rm GeV}$ (in natural units). Finally,
for convenience we have introduced in (3.6) the dimensionless parameter
$\bar{\kappa}$ through $\kappa\,M_{pl}^{2}\equiv\bar{\kappa}\,\varsigma^{2}$.

Model | $\omega_{m}=\Omega_{m}h^{2}$ | $\omega_{b}=\Omega_{b}h^{2}$ | $n_{s}$ | $\alpha$ | $\bar{\kappa}$ | $\chi^{2}_{\rm min}/dof$ | $\Delta{\rm AIC}$ |  $\Delta{\rm BIC}$
---|---|---|---|---|---|---|---|---
$\phi$CDM | $0.1403\pm 0.0008$ | $0.02264\pm 0.00014$ | $0.977\pm 0.004$ | $0.219\pm 0.057$ | $(32.5\pm 1.1)\times 10^{3}$ | 74.85/84 | 13.34 | 11.10

Table 2: The best-fit values for the parameter fitting vector (3.7) of the
$\phi$CDM model with Peebles & Ratra potential (3.2), including their
statistical significance ($\chi^{2}$-test and Akaike and Bayesian information
criteria, AIC and BIC, see the text). We use the same cosmological data set as
in Table 1. The large and positive values of $\Delta$AIC and $\Delta$BIC
strongly favor the $\phi$CDM model against the $\Lambda$CDM. The specific
$\phi$CDM fitting parameters are $\bar{\kappa}$ and $\alpha$. The remaining
parameters $(\omega_{m},\omega_{b},n_{s})$ are standard (see text). The number
of independent fitting parameters is $5$, see Eq.(3.7)– one more than in the
$\Lambda$CDM. Using the best-fit values and the overall covariance matrix
derived from our fit, we obtain: $h=0.671\pm 0.006$ and $\Omega_{m}=0.311\pm
0.006$, which allows direct comparison with Table 1. We find $\sim 4\sigma$
evidence in favor of $\alpha>0$. Correspondingly the EoS of $\phi$ at present
appears quintessence-like at $4\sigma$ confidence level: $w_{\phi}=-0.931\pm
0.017$.

Equation (3.6) is convenient since it is expressed in terms of the independent
parameters that enter our fit, see below. Let us note that $\phi(a)$, together
with its derivative $\phi^{\prime}(a)=d\phi(a)/da$, allow us to fix the
initial conditions in the MDE (a similar expression can be obtained for the
RDE). Once these conditions are settled analytically we have to solve
numerically the Klein-Gordon equation, coupled to the cosmological equations,
to obtain the exact solution. Such solution must, of course, be in accordance
with (3.6) in the pure MDE. The exact EoS is also a function
$w_{\phi}=w_{\phi}(a)$, which coincides with the constant value (3.5) in the
corresponding nth-epoch, but interpolates nontrivially between them. At the
same time it also interpolates between the MDE and the DE-dominated epoch in
our recent past, in which the scalar field energy density surfaces above the
non-relativistic matter density, i.e. $\rho_{\phi}(a)\gtrsim\rho_{m}(a)$, at a
value of $a$ near the current one $a=1$. The plots for the deceleration
parameter, $q=-\ddot{a}/aH^{2}$, and the scalar field EoS, $w_{\phi}(a)$, for
the best fit parameters of Table 2 are shown in Fig. 1. The transition point
from deceleration to acceleration ($q=0$) is at $z_{t}=0.628$, which is in
good agreement with the values obtained in other works [211, 212], and is also
reasonably near the $\Lambda$CDM one ($z_{t}^{\Lambda{\rm CDM}}=0.687$) for
the best fit values in Tables 1 and 2. The plots for $\phi(a)$ and the energy
densities are displayed in Fig. 2. From equations (2) and (6) we can see that
in the early MDE the potential of the scalar field decays as $V\sim
a^{-3\alpha/(2+\alpha)}\sim a^{-3\alpha/2}$, where in the last step we used
the fact that $\alpha$ is small. Clearly the decaying behaviour of $V$ with
the expansion is much softer than that of the matter density, $\rho_{m}\sim
a^{-3}$, and for this reason the DE density associated to the scalar field
does not play any role until we approach the current time. This fact is
apparent in Fig. 2 (right), where we numerically plot the dimensionless
density parameters $\Omega_{i}(a)=\rho_{i}(a)/\rho_{c}(a)$ as a function of
the scale factor, where $\rho_{c}(a)=3H^{2}(a)/(8\pi G)$ is the evolving
critical density.

As indicated above, the current value of the EoS can only be known after
numerically solving the equations for the best fit parameters in Table 2, with
the result $w_{\phi}(z=0)=-0.931\pm 0.017$ (cf. Fig. 1). Such result lies
clearly in the quintessence regime and with a significance of $4\sigma$. It is
essentially consistent with the dynamical character of the DE derived from the
non-vanishing value of $\alpha$ in Table 2.
In regard to the value of $h$, there is a significant tension between non-
local measurements of $h$, e.g. [178, 157, 189, 192, 213, 91, 214], and local
ones, e.g. [193]. Some of these values can differ by $3\sigma$ or more. For
the $\Lambda$CDM model we find $h=0.693\pm 0.004$ (cf. Table 1), which is in
between the ones of [178] and [193] and is compatible with the value presented
in [191]. For the $\phi$CDM, our best-fit value is $h=0.671\pm 0.006$ (cf.
caption of Table 2), which differs by more than $3\sigma$ with respect to the
$\Lambda$CDM one in our Table 1. Still, both remain perfectly consistent with
the recent estimates of $h$ from Hubble parameter measurements at intermediate
redshifts [161]. At the moment it is not possible to distinguish models on the
sole basis of $H(z)$ measurements. Fortunately, the combined use of the
different sorts of SNIa+BAO+$H(z)$+LSS+CMB data offers nowadays a real
possibility to elucidate which models are phenomenologically preferred.

Model | $\Omega_{m}$ | $\omega_{b}=\Omega_{b}h^{2}$ | $n_{s}$ | $h$ | $\nu$ | $w_{0}$ | $w_{1}$ | $\chi^{2}_{\rm min}/dof$ | $\Delta{\rm AIC}$ |  $\Delta{\rm BIC}$
---|---|---|---|---|---|---|---|---|---|---
XCDM | $0.312\pm 0.007$ | $0.02264\pm 0.00014$ | $0.977\pm 0.004$ | $0.670\pm 0.007$ | - | $-0.916\pm 0.021$ | - | 74.91/84 | 13.28 | 11.04
CPL | $0.311\pm 0.009$ | $0.02265\pm 0.00014$ | $0.977\pm 0.004$ | $0.672\pm 0.009$ | - | $-0.937\pm 0.085$ | $0.064\pm 0.247$ | 74.85/83 | 11.04 | 6.61
RVM | $0.303\pm 0.005$ | $0.02231\pm 0.00015$ | $0.965\pm 0.004$ | $0.676\pm 0.005$ | $0.00165\pm 0.00038$ | -1 | - | 70.32/84 | 17.87 | 15.63

Table 3: The best-fit values for the running vacuum model (RVM), together with
the XCDM and CPL parametrizations, including also their statistical
significance ($\chi^{2}$-test and Akaike and Bayesian information criteria,
AIC and BIC) as compared to the $\Lambda$CDM (cf. Table 1). We use the same
string of cosmological SNIa+BAO+$H(z)$+LSS+CMB data as in Tables 1 and 2. The
specific fitting parameters for these models are $\nu,w_{0},$ and
$(w_{0},w_{1})$ for RVM, XCDM and CPL, respectively. The remaining parameters
are standard. For the models RVM and XCDM the number of independent fitting
parameters is $5$, exactly as in the $\phi$CDM. For the CPL parametrization