# Emergent Dark Energy, neutrinos and cosmological tensions

Weiqiang Yang<EMAIL_ADDRESS>Department of Physics, Liaoning Normal
University, Dalian, 116029, People’s Republic of China  Eleonora Di Valentino
<EMAIL_ADDRESS>Jodrell Bank Center for Astrophysics,
School of Physics and Astronomy, University of Manchester, Oxford Road,
Manchester, M13 9PL, UK.  Supriya Pan<EMAIL_ADDRESS>Department of Mathematics, Presidency University, 86/1 College Street, Kolkata
700073, India  Olga Mena<EMAIL_ADDRESS>IFIC, Universidad de Valencia-
CSIC, 46071, Valencia, Spain

###### Abstract

The Phenomenologically Emergent Dark Energy model, a dark energy model with
the same number of free parameters as the flat $\Lambda$CDM, has been proposed
as a working example of a minimal model which can avoid the current
cosmological tensions. A straightforward question is whether or not the
inclusion of massive neutrinos and extra relativistic species may spoil such
an appealing phenomenological alternative. We present the bounds on $M_{\nu}$
and $N_{\rm eff}$ and comment on the long standing $H_{0}$ and $\sigma_{8}$
tensions within this cosmological framework with a wealth of cosmological
observations. Interestingly, we find, at $95\%$ confidence level, and with the
most complete set of cosmological observations, $M_{\nu}\sim
0.21^{+0.15}_{-0.14}$ eV and $N_{\rm eff}=3.03\pm 0.32$ i.e. an indication for
a non-zero neutrino mass with a significance above $2\sigma$. The well known
Hubble constant tension is considerably easened, with a significance always
below the $2\sigma$ level.

###### pacs:

98.80.-k, 95.36.+x, 95.35.+d, 98.80.Es

##  I Introduction

Despite the success of the standard cosmological $\Lambda$CDM paradigm to
explain the current observations, our picture of the universe, and more
concretely, of its current accelerated expansion, is far to be complete. There
are a number of unsolved questions and pending inconsistencies that need to be
addressed. For example, the simplest cosmological scenario based on General
Relativity together with a positive cosmological constant as the dark energy
component faces the well-known $H_{0}$ and $S_{8}$ tensions111The parameter
$S_{8}$ is proportional to $\sigma_{8}\sqrt{\Omega_{m}}$, with $\sigma_{8}$
indicating the amplitude of the small-scale density fluctuations., which have
motivated plenty of searches for new physics scenarios, ranging from the
nature of dark matter and/or dark radiation Dvorkin  _et al._ (2014);
Berezhiani  _et al._ (2015); Bernal  _et al._ (2016); Ko and Tang (2016);
Kumar and Nunes (2016); Buen-Abad  _et al._ (2018); Ko and Tang (2017); Chacko
_et al._ (2016); Gariazzo  _et al._ (2017); Feng  _et al._ (2017); Zhao  _et
al._ (2017a); D’Eramo  _et al._ (2018); Alexander and McDonough (2019);
Escudero and Witte (2020); Vattis  _et al._ (2019); Alcaniz  _et al._ (2019);
Blinov and Marques-Tavares (2020); Clark  _et al._ (2020); Archidiacono  _et
al._ (2020); Hryczuk and Jodł owski (2020) dark energy Di Valentino  _et al._
(2015); Poulin  _et al._ (2019); Vagnozzi (2019); Di Valentino  _et al._
(2016); Huang and Wang (2016); Karwal and Kamionkowski (2016); Zhao  _et al._
(2017b); Di Valentino  _et al._ (2017a); Raveri  _et al._ (2017); Di Valentino
_et al._ (2018a, b); Solá  _et al._ (2017); Mörtsell and Dhawan (2018);
Banihashemi  _et al._ (2019); Yang  _et al._ (2019a); Poulin  _et al._ (2018);
Sakstein and Trodden (2019); Niedermann and Sloth (2019); Yang  _et al._
(2019b); Shafieloo  _et al._ (2018); Li  _et al._ (2019); Yang  _et al._
(2020); Li and Shafieloo (2020); Di Valentino  _et al._ (2020a, b); Chudaykin
_et al._ (2020); Alestas  _et al._ (2020); Niedermann and Sloth (2020), exotic
dark matter-dark energy interactions Xia and Wang (2016); Di Valentino  _et
al._ (2017b); Kumar and Nunes (2017); Yang  _et al._ (2018a, 2019c); Kumar
_et al._ (2019); Di Valentino  _et al._ (2019a); Pan  _et al._ (2019a);
Martinelli  _et al._ (2019); Di Valentino  _et al._ (2019b, 2020c); Yang  _et
al._ (2018b); Pan  _et al._ (2019b); Yang  _et al._ (2017a); Pan  _et al._
(2020a), modified gravity Dirian (2017); Renk  _et al._ (2017); Khosravi  _et
al._ (2019); Peirone  _et al._ (2018); Nunes (2018); Yan  _et al._ (2019);
Raveri (2019); Cai  _et al._ (2019); Frusciante  _et al._ (2019); Wang and
Mota (2020), and many others Kreisch  _et al._ (2019); Banihashemi  _et al._
(2018); Ó Colgáin  _et al._ (2019); Guo  _et al._ (2019); Agrawal  _et al._
(2019); Arendse  _et al._ (2019); Garcia-Quintero  _et al._ (2019); Hart and
Chluba (2019); Liu  _et al._ (2020); Di Valentino  _et al._ (2019c); Yang  _et
al._ (2019d); Desmond  _et al._ (2019); Ivanov  _et al._ (2020); Colgáin and
Yavartanoo (2019); Visinelli  _et al._ (2019); Berghaus and Karwal (2019); Di
Valentino  _et al._ (2020d); Jedamzik and Pogosian (2020); Pan  _et al._
(2020b); Wu  _et al._ (2020); Ye and Piao (2020); Ballesteros  _et al._
(2020); Braglia  _et al._ (2020); Ballardini  _et al._ (2020); Keeley  _et
al._ (2020); Capozziello  _et al._ (2020); Gonzalez  _et al._ (2020), most of
them involving extensions of the parameter space compared to the $\Lambda$CDM
model.

If we go beyond the simplest $\Lambda$CDM scenario, some of the problems
associated to the standard cosmology may be solved. Unfortunately, some new
complications naturally arise. On the other hand, (assuming the spatial
flatness of the Friedmann-Lemaître-Robertson-Walker (FLRW) geometry), the
number of free parameters in the $\Lambda$-cosmology is only six. Any
extension of the $\Lambda$-cosmology in terms of new degrees freedom naturally
exacerbates the $\chi^{2}$ of the corresponding observational analysis
compared to the $\Lambda$-cosmology. Therefore, the construction of a
_minimal_ cosmological model is very important. The modifications could also
rely on the gravitational sector of the theory Brans and Dicke (1961);
Banerjee and Pavón (2001a, b); Capozziello (2002); Das  _et al._ (2006);
Nojiri and Odintsov (2006); Sotiriou and Faraoni (2010); De Felice and
Tsujikawa (2010); Nojiri and Odintsov (2011); Capozziello and De Laurentis
(2011); de Haro and Amorós (2013); Cai  _et al._ (2016); Paliathanasis (2016);
Paliathanasis  _et al._ (2016); Nunes  _et al._ (2016, 2017); Paliathanasis
(2017); Dimakis  _et al._ (2017); Nojiri  _et al._ (2017); Nunes  _et al._
(2018); Paliathanasis (2019); Das  _et al._ (2019), rather than in the stress-
energy tensor Copeland  _et al._ (2006); Basilakos  _et al._ (2011);
Paliathanasis and Tsamparlis (2014); Dimakis  _et al._ (2016); Basilakos  _et
al._ (2018); Dimakis  _et al._ (2019); Papagiannopoulos  _et al._ (2020);
Banerjee and Das (2005); Banerjee and Ghosh (2006); Banerjee  _et al._ (2007);
de Haro  _et al._ (2016a); de Haro and Elizalde (2016); de Haro (2017); de
Haro  _et al._ (2016b); Pan and Sharov (2017); Sharov  _et al._ (2017); Pan
(2018); Pan  _et al._ (2018a); Yang  _et al._ (2019e, 2017b); Pan  _et al._
(2018b); Yang  _et al._ (2019f); Pan  _et al._ (2020c, d).

With such a wide model building perspective in mind, the authors of Ref. Li
and Shafieloo (2019) proposed a possible alternative framework, the
Phenomenologically Emergent Dark Energy (PEDE) model222Perhaps the readers
might be interested in a recent work on the Emergent Dark Energy having field
theoretic origin Barker  _et al._ (2020). However, in this work we shall focus
on the PEDE model proposed in Li and Shafieloo (2019) and recently extended
with one extra parameter in the dark sector in Ref. Li and Shafieloo (2020)..
This scenario has exactly the same number of free parameters as in the
spatially flat $\Lambda$-cosmology and it has been shown to provide a solution
to the $H_{0}$ tension. The very same model was investigated further in Ref.
Pan  _et al._ (2020e) (see also Ref. Rezaei  _et al._ (2020)) considering its
complete evolution including both the cosmological background plus
perturbations and analyzed using the most recent cosmological observations,
obtaining similar results concerning the $H_{0}$ tension. Of course the
success of the PEDE model in principle may rely on the different ingredients
included in our Universe’s description, and the nice features of this model
may disappear if one starts to include extra parameters such as the dark
matter or dark radiation properties. Thus, the motivation of this article is
to check how the extra degrees of freedom in terms of the neutrino properties
could affect the constraints on the Hubble constant within this model
scenario. Actually, it is well known that these additional parameters can
potentially solve the sound horizon problem Knox and Millea (2020); Arendse
_et al._ (2019), improving the agreement with the BAO data, contrarily to
other parameters as, for example, the spatial curvature. According to the
observational data, neutrino oscillations have robustly established the
existence of neutrino masses and their impact in cosmology is crucial (see
e.g. Refs. Lesgourgues and Pastor (2006); Lattanzi and Gerbino (2018); Aker
_et al._ (2019)).

In order to assess the viability of the PEDE model, we include the total
neutrino mass ($M_{\nu}$) as well as the effective number of neutrino species
($N_{\rm eff}$) in the minimal PEDE scenario explored throughout Ref. Liu and
Miao (2020) and consider three extended cosmological schemes, namely, PEDE
$+M_{\nu}$, PEDE $+N_{\rm eff}$ and PEDE $+M_{\nu}+N_{\rm eff}$.

The manuscript is organized as follows. Section II introduces the PEDE model.
In Sec. III we describe the cosmological probes used to examine the models.
After that, in Sec. IV, we discuss the results for all the analyzed models to
conclude in Sec. V with our main findings.

##  II Phenomenologically Emergent Dark Energy

Starting from a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) line
element, $ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2})$, where $a(t)$ is the
expansion scale factor of the universe and $(t,x,y,z)$ are the co-moving
coordinates, and assuming that the matter sector is minimally coupled to
gravity (described by the Einstein gravity), the gravitational field equations
can be written as

$\displaystyle H^{2}=\frac{8\pi
G}{3}\left(\rho_{\nu}+\rho_{r}+\rho_{m}+\rho_{DE}\right);$ (1) $\displaystyle
2\dot{H}+3H^{2}=-8\pi G\left(p_{\nu}+p_{r}+p_{m}+p_{DE}\right),$ (2)

where $G$ is the Newton’s gravitational constant, the dot represents the time
derivative, $H\equiv\dot{a}/a$ is the Hubble rate of the FLRW universe;
$(\rho_{i},p_{i})$ [$i=\nu,r,m,{\rm DE}$] are respectively the energy density
and pressure of massive neutrinos, radiation, pressure-less matter
(baryons+pressure-less dark matter) and the dark energy fluid. All the
components of the stress energy tensor are assumed to be of barotropic nature
and show no interactions among themselves. The conservation equation for each
fluid is therefore $\dot{\rho}_{i}+3H(p_{i}+\rho_{i})=0$. If the dark energy
fluid has a time-dependent cosmological constant, then using the conservation
equation for DE, one can easily show that the energy density of DE (with
respect to the critical energy density) defined as $\Omega_{DE}\equiv\rho_{\rm
DE}/\rho_{crit,0}$ evolves with the cosmic time (or alternatively with respect
to the redshift $z$) as:

$\displaystyle\tilde{\Omega}_{DE}(z)=\Omega_{DE,0}\;\exp\left[3\;\int_{0}^{z}\frac{1+w_{DE}(z^{\prime})}{1+z^{\prime}}dz^{\prime}\right]~{},$
(3)

where $\Omega_{\rm DE,0}$ is the present value of $\tilde{\Omega}_{\rm DE}$
and $w_{\rm DE}(z)=\frac{p_{\rm DE}}{\rho_{\rm DE}}$ is the equation-of-state
of the dark energy fluid. Usually, by prescribing the equation-of-state for
DE, one can determine the evolution of the DE density. Alternatively, one may
apply the reverse engineering method by prescribing the parametric form for
$\tilde{\Omega}_{\rm DE}$ and consequently determine other cosmological
parameters. The Phenomenological Dark Energy Emergent model follows this
reverse mechanism Li and Shafieloo (2019) and use as cosmic time the number of
_ten-foldings_ of the scale factor333The scale factor and the redshift are
related as $a\equiv\frac{1}{1+z}$., proposing a phenomenological description
of the dark energy fluid as follows:

$\displaystyle\tilde{\Omega}_{DE}(z)=\Omega_{DE,0}\Bigl{[}1-\tanh(\log_{10}(1+z))\Bigr{]}~{}.$
(4)

It is interesting to note that the PEDE model defined above has no extra free
parameters, i.e. it has exactly the same number of free parameters as the flat
$\Lambda$CDM model and also only one describing the dark energy fluid
properties: $\Omega_{DE,0}$, the current dark energy energy density. Using the
conservation equation for the DE and using Eq. (3), the dark energy equation
of state can be derived as:

$\displaystyle
w_{DE}(z)=\frac{1}{3}\frac{d\ln\tilde{\Omega}_{DE}}{dz}(1+z)-1~{}.$ (5)

The authors of Ref. Li and Shafieloo (2019) have shown that the dark energy
equation of state in the PEDE model at early times would be $-1-2/(3\ln 10)$
to evolve asymptotically to $-1$ in the far future. The current value is
$w_{DE}(z=0)=-1-1/(3\ln 10)$.

Similarly to the work carried out in Ref. Pan  _et al._ (2020e), we have also
taken into account the evolution at the level of perturbations. In the next
section we describe the cosmological datasets and the statistical methods to
constrain the different cosmological scenarios.

##  III Cosmological data and Methodology

In order to analyze the PEDE framework in the presence of massive neutrinos
and extra relativistic species, we have used a wealth of cosmological
observations that we shall detail in what follows.

  1. 1.

Cosmic Microwave Background (CMB): We use the latest CMB data from the Planck
2018 legacy release Aghanim  _et al._ (2018a, 2019)444Concretely, we use the
CMB temperature and polarization angular power spectra plikTTTEEE+lowl+lowE of
Aghanim  _et al._ (2018a, 2019).

  2. 2.

CMB Lensing: We include the Planck 2018 CMB lensing reconstruction power
spectrum data Aghanim  _et al._ (2018b).

  3. 3.

Baryon acoustic oscillation (BAO) data: We use BAO data from various galaxy
surveys that include 6dFGS Beutler  _et al._ (2011), SDSS-MGS Ross  _et al._
(2015), and BOSS DR12 Alam  _et al._ (2017), as considered by the Planck 2018
data analyses Aghanim  _et al._ (2018a).

  4. 4.

Hubble constant (R19): The measurement of the Hubble constant yielding
$H_{0}=74.03\pm 1.42$ km/s/Mpc at $68\%$ CL by Riess et al. Riess  _et al._
(2019) has been incorporated into our analysis. This measurement shows a
$4.4\sigma$ tension with that extracted from the CMB within the minimal
$\Lambda$CDM model.

  5. 5.

Supernovae Type Ia: we exploit measurements of SNIa luminosity distances from
the Pantheon sample Scolnic  _et al._ (2018), comprising 1048 data points
spanned over the redshift interval $z\in[0.01,2.3]$.

  6. 6.

Dark energy survey (DES): We use the galaxy clustering and cosmic shear
measurements from DES combined-probe Year 1 results Troxel  _et al._ (2018);
Abbott  _et al._ (2018); Krause  _et al._ (2017), as adopted by the Planck
collaboration in their 2018 final data analyses Aghanim  _et al._ (2018a).

For the analyses, we shall use a modified version of the publicly available
Markov Chain Monte Carlo (MCMC) code CosmoMC Lewis and Bridle (2002); Lewis
_et al._ (2000) package (see http://cosmologist.info/cosmomc/). This code
supports the new 2018 Planck likelihood Aghanim  _et al._ (2019), implements
an efficient sampling of the posterior distribution using the fast/slow
parameter decorrelations Lewis (2013), and has a convergence diagnostic based
on the Gelman-Rubin statistics Gelman and Rubin (1992). Finally, we have
computed the Bayesian Evidence (BE) for all the scenarios considered in this
work. We have actually followed the MCEvidence code Heavens  _et al._ (2017a,
b), where it has been shown that the BE can directly be computed using only
the MCMC chains which are used to constrain the parameter space. A concise
discussion on BE can be found in Yang  _et al._ (2018b). So, using the
MCEvidence code we have computed the logarithm of the Bayes factor, $B_{ij}$
of the cosmological scenarios with respect to the base $\Lambda$CDM model.
That means we calculate $\ln B_{ij}$ (where $i=$ PEDE, or PEDE + $M_{\nu}$,
PEDE + $N_{\rm eff}$, or PEDE + $M_{\nu}$ \+ $N_{\rm eff}$ and
$j=\Lambda$CDM), and after that we use the revised Jeffreys scale shown in
Table 1 in order to quantify the strength of evidence of the cosmological
scenarios.

$\ln B_{ij}$ | Strength of evidence for model $M_{i}$
---|---
$0\leq\ln B_{ij}<1$ | Weak
$1\leq\ln B_{ij}<3$ | Definite/Positive
$3\leq\ln B_{ij}<5$ | Strong
$\ln B_{ij}\geq 5$ | Very strong
Table 1: Revised Jeffreys scale which quantifies the strength of evidence of
the proposed cosmological model (identified through the index $i$) with
respect to the base cosmological model $\Lambda$CDM (identified through the
symbol $j$).

##  IV Results

In this Section we first discuss the results obtained within the minimal PEDE
scenario. Then, we enlarge the model to account for the neutrino parameters.
We shall consider three different cases: a first one where we allow for the
total neutrino mass $M_{\nu}$ to freely vary, a second one where we introduce
the possibility of having an extra dark radiation component parameterized by
$N_{\rm eff}$, and finally when we consider the previous two neutrino
parameters to freely vary simultaneously.

###  IV.1 PEDE

We begin the analysis with the simple and minimal PEDE model, updating the
earlier work of Ref. Pan  _et al._ (2020e) by means of Planck 2018 data. The
results are summarized in Tab. 2.

Comparing the new constraints with those one in Table II of Ref. Pan  _et al._
(2020e), we notice that there is no such significant shifts on the
cosmological parameters (with the exception of $\tau$ and $ln(10^{10}A_{s})$
due to the new low-$\ell$ polarization data), and the conclusions about the
PEDE model solving the Hubble constant tension still hold. Indeed, having the
same degrees of freedom of the $\Lambda$CDM scenario, the PEDE model can
strongly relieve the $H_{0}$ tension, leading it to negligible statistically
significant levels: for instance, the tension is reduced to the $1\sigma$
level for Planck CMB data alone. Also
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{m}/0.3}$ shifts now towards smaller
values, showing a better agreement with cosmic shear measurements. The fact
that the $H_{0}$ tension is strongly alleviated in the PEDE scenario is due to
the fact that the dark energy equation of state within this phenomenological
picture is negative, $w_{DE}(z=0)=-1/(3\ln 10)-1\simeq-1.145$, as it is well
known that a phantom dark energy equation of state can raise the value of
$H_{0}$ Di Valentino  _et al._ (2016, 2020c), especially when dealing with CMB
data only. When $w$ is allowed to lie in the phantom region, the parameter
$H_{0}$ must be increased to leave unchanged the location of the CMB acoustic
peaks. The addition of BAO measurements mildly softens this degeneracy.

Parameters | Planck 2018 | Planck 2018+BAO | Planck 2018+R19 | Planck 2018+BAO+R19 | Full
---|---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.1200_{-0.0015-0.0025}^{+0.0013+0.0028}$ | $0.1215_{-0.0010-0.0019}^{+0.0010+0.0020}$ | $0.1193_{-0.0012-0.0023}^{+0.0012+0.0023}$ | $0.12093_{-0.00092-0.0019}^{+0.00096+0.0019}$ | $0.12032_{-0.00079-0.0015}^{+0.00077+0.0016}$
$\Omega_{b}h^{2}$ | $0.02238_{-0.00015-0.00030}^{+0.00015+0.00029}$ | $0.02227_{-0.00013-0.00025}^{+0.00013+0.00026}$ | $0.02243_{-0.00014-0.00027}^{+0.00014+0.00029}$ | $0.02232_{-0.00013-0.00025}^{+0.00013+0.00026}$ | $0.02236_{-0.00013-0.00024}^{+0.00012+0.00025}$
$100\theta_{MC}$ | $1.04092_{-0.00031-0.00062}^{+0.00032+0.00062}$ | $1.04076_{-0.00030-0.00056}^{+0.00030+0.00057}$ | $1.04102_{-0.00031-0.00062}^{+0.00030+0.00063}$ | $1.04082_{-0.00031-0.00056}^{+0.00028+0.00060}$ | $1.04085_{-0.00031-0.00055}^{+0.00028+0.00059}$
$\tau$ | $0.0544_{-0.0080-0.016}^{+0.0080+0.016}$ | $0.0527_{-0.0078-0.015}^{+0.0073+0.016}$ | $0.0555_{-0.0077-0.016}^{+0.0080+0.016}$ | $0.05279_{-0.0070-0.014}^{+0.0073+0.015}$ | $0.04756_{-0.0070-0.015}^{+0.0077+0.014}$
$n_{s}$ | $0.9651_{-0.0044-0.0084}^{+0.0044+0.0081}$ | $0.9617_{-0.0038-0.0075}^{+0.0038+0.0074}$ | $0.9670_{-0.0040-0.0081}^{+0.0040+0.0081}$ | $0.9632_{-0.0041-0.0073}^{+0.0037+0.0076}$ | $0.9634_{-0.0037-0.0070}^{+0.0036+0.0072}$
${\rm{ln}}(10^{10}A_{s})$ | $3.044_{-0.018-0.033}^{+0.016+0.034}$ | $3.044_{-0.017-0.030}^{+0.015+0.032}$ | $3.045_{-0.016-0.034}^{+0.016+0.032}$ | $3.043_{-0.015-0.029}^{+0.015+0.029}$ | $3.030_{-0.014-0.030}^{+0.014+0.030}$
$\Omega_{m0}$ | $0.2734_{-0.0092-0.016}^{+0.0078+0.017}$ | $0.2822_{-0.0060-0.012}^{+0.0061+0.012}$ | $0.2691_{-0.0068-0.014}^{+0.0068+0.013}$ | $0.2788_{-0.0055-0.011}^{+0.0056+0.011}$ | $0.2753_{-0.0049-0.0088}^{+0.0046+0.0093}$
$\sigma_{8}$ | $0.8579_{-0.0078-0.015}^{+0.0079+0.016}$ | $0.8612_{-0.0076-0.014}^{+0.0070+0.014}$ | $0.8566_{-0.0076-0.016}^{+0.0076+0.015}$ | $0.8597_{-0.0068-0.013}^{+0.0068+0.014}$ | $0.8522_{-0.0059-0.013}^{+0.0061+0.013}$
$H_{0}$ | $72.35_{-0.79-1.5}^{+0.78+1.5}$ | $71.54_{-0.55-1.1}^{+0.53+1.1}$ | $72.76_{-0.67-1.3}^{+0.65+1.3}$ | $71.84_{-0.52-1.0}^{+0.51+1.1}$ | $72.16_{-0.44-0.87}^{+0.44+0.86}$
$S_{8}$ | $0.819_{-0.018-0.030}^{+0.016+0.032}$ | $0.835_{-0.013-0.025}^{+0.013+0.023}$ | $0.811_{-0.015-0.029}^{+0.014+0.028}$ | $0.829_{-0.012-0.024}^{+0.012+0.023}$ | $0.8163_{-0.0089-0.018}^{+0.0091+0.017}$
$r_{\rm{drag}}$ | $147.084_{-0.285-0.608}^{+0.327+0.557}$ | $146.816_{-0.235-0.473}^{+0.260+0.457}$ | $147.206_{-0.269-0.534}^{+0.271+0.540}$ | $146.903_{-0.241-0.449}^{+0.241+0.471}$ | $147.022_{-0.203-0.396}^{+0.208+0.419}$

Table 2: Observational constraints on the simple PEDE scenario at 68% and 95%
CL using various cosmological datasets. The full combination refers to the
datasets Planck 2018+BAO+R19+Pantheon+DES+Lensing.

###  IV.2 PEDE $+M_{\nu}$

We report the constraints at 68% and 95% CL on the cosmological parameters for
the PEDE $+M_{\nu}$ model in Tab. 3 and we show the 1D posterior distributions
and 2D correlation plots in Fig. 1.

Comparing Tab. 3 with Tab. 2 it is evident that including a total neutrino
mass free to vary does not change the constraints on the six free cosmological
parameters of the model, showing the robustness of the model. However, both
$H_{0}$ and $S_{8}$ are affected by the introduction of $M_{\nu}$, because of
their important negative correlations, as can be noticed from Fig. 1. A freely
varying $M_{\nu}$ will imply a slightly lower value for the Hubble constant
and $S_{8}$. This correlation is also very well known within the $\Lambda$CDM
model, preventing the possibility of combining the Hubble constant direct
measurements with the CMB because shifting down the $H_{0}$ mean value
exacerbates further their tension. In the PEDE model, however, even if there
is a slight shift of the mean value of $H_{0}$ when introducing $M_{\nu}$ as a
free parameter, there is an increase of the corresponding error bars.
Consequently, the Hubble constant value reported by Planck 2018 in the PEDE
scenario is in agreement within 1$\sigma$ with $H_{0}$ measurements from R19
also when massive neutrinos are present, allowing us to combine the two
measurements, with the aim of breaking the $M_{\nu}$ versus $H_{0}$
correlation and obtain more stringent constraints. Therefore, while we have at
95% CL for Planck 2018 alone $M_{\nu}<0.26$ eV, we find for Planck 2018+R19
$M_{\nu}<0.15$ eV. Contrarily to the $\Lambda$CDM scenario, the addition of
BAO data does not improve the upper limit at 95% CL on the total neutrino mass
($M_{\nu}<0.27$ eV for Planck 2018+BAO), and this is due to the appearance of
a peak for $M_{\nu}$ different from zero in the 1D posterior distribution (see
the red curve in Fig. 1). We find, at 68% CL,
$M_{\nu}=0.139^{+0.065}_{-0.093}$ eV.

When considering all the data combined, i.e. Planck
2018+BAO+Pantheon+R19+DES+lensing, the Hubble constant tension persists albeit
at the (mild) 1.8$\sigma$ level, and the $S_{8}$ parameter is shifted towards
a lower value. Finally, we have an indication at more than $2\sigma$ for a
total neutrino mass different from zero, i.e $M_{\nu}=0.19^{+0.14}_{-0.16}$
eV. The fact that the neutrino mass bound is larger in the PEDE scenario is
due, again, to the strong degeneracy between the dark energy equation of state
and $M_{\nu}$. It is well-known that the neutrino mass bounds are less
constraining within the phantom region Vagnozzi  _et al._ (2018); Roy
Choudhury and Choubey (2018), because in this region the normalized expansion
rate is smaller than within the $\Lambda$CDM picture and therefore one needs
to increase $M_{\nu}$ and/or $H_{0}$ to compensate for such an effect.
Concerning the cosmic shear tension, it is also alleviated, as a larger
neutrino mass implies a lower value of $S_{8}$, as can be noticed from Fig. 1.

Parameters | Planck 2018 | Planck 2018+BAO | Planck 2018+R19 | Planck 2018+BAO+R19 | Full
---|---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.1201_{-0.0014-0.0027}^{+0.0014+0.0028}$ | $0.1210_{-0.0011-0.0023}^{+0.0012+0.0021}$ | $0.1194_{-0.0012-0.0023}^{+0.0012+0.0023}$ | $0.1207_{-0.0011-0.0022}^{+0.0011+0.0021}$ | $0.11969_{-0.00086-0.0017}^{+0.00084+0.0017}$
$\Omega_{b}h^{2}$ | $0.02237_{-0.00015-0.00030}^{+0.00015+0.00030}$ | $0.02229_{-0.00014-0.00026}^{+0.00013+0.00026}$ | $0.02244_{-0.00014-0.00027}^{+0.00014+0.00028}$ | $0.02233_{-0.00013-0.00027}^{+0.00013+0.00026}$ | $0.02240_{-0.00013-0.00025}^{+0.00013+0.00025}$
$100\theta_{MC}$ | $1.04090_{-0.00032-0.00062}^{+0.00031+0.00063}$ | $1.04075_{-0.00029-0.00060}^{+0.00030+0.00058}$ | $1.04102_{-0.00029-0.00059}^{+0.00029+0.00060}$ | $1.04085_{-0.00029-0.00058}^{+0.00029+0.00057}$ | $1.04087_{-0.00028-0.00058}^{+0.00028+0.00055}$
$\tau$ | $0.0543_{-0.0082-0.015}^{+0.0074+0.016}$ | $0.0534_{-0.0082-0.015}^{+0.0073+0.016}$ | $0.0544_{-0.0079-0.014}^{+0.0072+0.015}$ | $0.0532_{-0.0074-0.015}^{+0.0072+0.016}$ | $0.0535_{-0.0090-0.015}^{+0.0076+0.017}$
$n_{s}$ | $0.9651_{-0.0044-0.0087}^{+0.0045+0.0089}$ | $0.9628_{-0.0038-0.0073}^{+0.0039+0.0078}$ | $0.9667_{-0.0041-0.0081}^{+0.0041+0.0083}$ | $0.9636_{-0.0039-0.0075}^{+0.0039+0.0074}$ | $0.9649_{-0.0035-0.0073}^{+0.0039+0.0071}$
${\rm{ln}}(10^{10}A_{s})$ | $3.044_{-0.016-0.030}^{+0.015+0.032}$ | $3.044_{-0.017-0.031}^{+0.015+0.033}$ | $3.043_{-0.015-0.030}^{+0.015+0.031}$ | $3.044_{-0.015-0.030}^{+0.016+0.032}$ | $3.042_{-0.016-0.031}^{+0.016+0.033}$
$\Omega_{m0}$ | $0.277_{-0.015-0.025}^{+0.009+0.028}$ | $0.2875_{-0.0088-0.016}^{+0.0076+0.017}$ | $0.2684_{-0.0090-0.016}^{+0.0077+0.017}$ | $0.2804_{-0.0071-0.013}^{+0.0061+0.013}$ | $0.2846_{-0.0079-0.015}^{+0.0078+0.015}$
$\sigma_{8}$ | $0.852_{-0.010-0.045}^{+0.024+0.034}$ | $0.842_{-0.017-0.042}^{+0.026+0.039}$ | $0.859_{-0.010-0.029}^{+0.016+0.025}$ | $0.853_{-0.013-0.037}^{+0.022+0.033}$ | $0.826_{-0.021-0.034}^{+0.018+0.036}$
$H_{0}{\rm[km/s/Mpc]}$ | $72.1_{-0.9-2.8}^{+1.5+2.5}$ | $70.98_{-0.75-1.6}^{+0.85+1.5}$ | $72.85_{-0.77-1.7}^{+0.89+1.6}$ | $71.68_{-0.60-1.3}^{+0.70+1.2}$ | $71.17_{-0.86-1.5}^{+0.76+1.6}$
$M_{\nu}{\rm[eV]}$ | $<0.103\,,<0.26$ | $0.139_{-0.093}^{+0.065}<0.27$ | $<0.067\,,<0.15$ | $0.094_{-0.088}^{+0.029}<0.21$ | $0.187_{-0.073-0.16}^{+0.090+0.14}$
$S_{8}$ | $0.818_{-0.016-0.033}^{+0.017+0.033}$ | $0.824_{-0.017-0.035}^{+0.019+0.034}$ | $0.812_{-0.015-0.029}^{+0.015+0.030}$ | $0.824_{-0.015-0.034}^{+0.018+0.033}$ | $0.804_{-0.012-0.023}^{+0.012+0.023}$
$r_{\rm{drag}}$ | $147.068_{-0.304-0.621}^{+0.305+0.601}$ | $146.917_{-0.260-0.503}^{+0.261+0.521}$ | $147.190_{-0.272-0.548}^{+0.276+0.527}$ | $146.955_{-0.256-0.507}^{+0.255+0.515}$ | $147.124_{-0.225-0.432}^{+0.219+0.432}$

Table 3: Observational constraints at 68% and 95% CL using various
cosmological datasets within the extended PEDE $+M_{\nu}$ scenario. Figure 1:
One-dimensional marginalized posterior distributions for some selected
parameters and two dimensional $68\%$ and $95\%$ CL joint contours for several
combinations of the parameters within the extended scenario PEDE+$M_{\nu}$
using various cosmological datasets.

###  IV.3 PEDE $+N_{\rm eff}$

We report the constraints at 68% and 95% CL on the cosmological parameters for
PEDE $+N_{\rm eff}$ model in Tab. 4 and we show the 1D posterior distributions
and 2D correlation plots in Fig. 2.

Comparing Tab. 4 with Tab. 2 it is evident that including a number of
relativistic species at recombination $N_{\rm eff}$ free to vary introduces
the already known correlation between the cosmological parameters present in
the $\Lambda$CDM scenario. Therefore, we have an increase of the error bars,
and, since the $N_{\rm eff}$ mean value is, for all the combination of data,
with the exception of Planck 2018+R19, lower than the expected value $3.045$
Mangano  _et al._ (2005); de Salas and Pastor (2016); Akita and Yamaguchi
(2020), we have a shift down in the value of the parameters positively
correlated with $N_{\rm eff}$, as $\Omega_{c}h^{2}$, $n_{s}$ and $H_{0}$, and
a shift up on the value of those negatively correlated, such as $\theta_{MC}$.

Contrarily to what happens within the $\Lambda$CDM $+N_{\rm eff}$ scenario, in
the PEDE $+N_{\rm eff}$ model we have Planck 2018 in agreement with R19 at
1.2$\sigma$, justifying their combination. The value of $N_{\rm eff}$ will be
shifted from $N_{\rm eff}=2.93^{+0.18}_{-0.19}$ at 68% CL for Planck 2018
(slightly low but perfectly consistent with the expected value) to $N_{\rm
eff}=3.08\pm 0.14$ at 68% CL for Planck 2018+R19.

For the Planck 2018+BAO dataset combination we have instead $N_{\rm
eff}=2.82\pm 0.17$ at 68% CL, i.e. 1$\sigma$ lower than the expected value,
and this is reflected on the shift of the Hubble constant parameter, raising
the $H_{0}$ tension to 2.1$\sigma$. For the Planck 2018+BAO+R19 combination we
again obtain a slightly larger $N_{\rm eff}=3.05\pm 0.14$ at 68% CL due to the
larger value of the Hubble constant when the R19 prior is considered in the
data analyses.

Finally, in the full dataset combination case, i.e. Planck
2018+BAO+Pantheon+R19+DES+lensing, we find that the Hubble constant tension
appears with a modest significance (1.7$\sigma$). We also find a very robust
constraint on the effective number of relativistic species, $N_{\rm
eff}=2.87^{+0.13}_{-0.14}$ at 68% CL, which is in perfect agreement with the
expectation of the Standard Model for three neutrino families. Therefore,
contrarily to what happens in the $\Lambda$CDM $+N_{\rm eff}$ scenario, in the
PEDE $+N_{\rm eff}$ model a larger value of the Hubble constant in agreement
with R19 does not necessarily imply an extra amount of dark radiation at
recombination. This is again due to the fact that the larger value of $H_{0}$
in this model is strongly related to the fact that currently $w_{\rm DE}<-1$,
becoming the dark energy equation-of-state more negative ($w_{\rm
DE}\simeq-1.34$) as we move back to early times.

Parameters | Planck 2018 | Planck 2018+BAO | Planck 2018+R19 | Planck 2018+BAO+R19 | Full
---|---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.1183_{-0.0030-0.0059}^{+0.0030+0.0061}$ | $0.1178_{-0.0030-0.0058}^{+0.0029+0.0059}$ | $0.2000_{-0.0027-0.0053}^{+0.0027+0.0053}$ | $0.1210_{-0.0026-0.0053}^{+0.0027+0.0055}$ | $0.1174_{-0.0025-0.0047}^{+0.0023+0.0048}$
$\Omega_{b}h^{2}$ | $0.02227_{-0.00022-0.00043}^{+0.00022+0.00044}$ | $0.02211_{-0.00018-0.00036}^{+0.00018+0.00036}$ | $0.02245_{-0.00017-0.00033}^{+0.00017+0.00032}$ | $0.02233_{-0.00016-0.00031}^{+0.00016+0.00031}$ | $0.02225_{-0.00015-0.00030}^{+0.00015+0.00030}$
$100\theta_{MC}$ | $1.04112_{-0.00043-0.00084}^{+0.00044+0.00085}$ | $1.04118_{-0.00045-0.00086}^{+0.00044+0.00090}$ | $1.04095_{-0.00041-0.00080}^{+0.00041+0.00080}$ | $1.04085_{-0.00041-0.00079}^{+0.00042+0.00081}$ | $1.04120_{-0.00040-0.00076}^{+0.00039+0.00078}$
$\tau$ | $0.0535_{-0.0073-0.015}^{+0.0074+0.015}$ | $0.0515_{-0.0072-0.015}^{+0.0073+0.015}$ | $0.0552_{-0.0076-0.016}^{+0.0078+0.016}$ | $0.0528_{-0.0075-0.015}^{+0.0074+0.016}$ | $0.0483_{-0.0067-0.014}^{+0.0068+0.014}$
$n_{s}$ | $0.9605_{-0.0085-0.016}^{+0.0084+0.017}$ | $0.9541_{-0.0068-0.014}^{+0.0068+0.014}$ | $0.9679_{-0.0061-0.012}^{+0.0059+0.012}$ | $0.9630_{-0.0059-0.011}^{+0.0054+0.011}$ | $0.9581_{-0.0054-0.011}^{+0.0054+0.011}$
${\rm{ln}}(10^{10}A_{s})$ | $3.038_{-0.018-0.036}^{+0.018+0.036}$ | $3.032_{-0.018-0.034}^{+0.018+0.036}$ | $3.046_{-0.017-0.034}^{+0.017+0.035}$ | $3.044_{-0.017-0.034}^{+0.017+0.033}$ | $3.024_{-0.014-0.028}^{+0.014+0.029}$
$\Omega_{m0}$ | $0.2772_{-0.0098-0.019}^{+0.0097+0.020}$ | $0.2865_{-0.0073-0.013}^{+0.0065+0.014}$ | $0.2690_{-0.0070-0.014}^{+0.0070+0.015}$ | $0.2789_{-0.0057-0.011}^{+0.0058+0.012}$ | $0.2775_{-0.0051-0.0091}^{+0.0047+0.010}$
$\sigma_{8}$ | $0.852_{-0.012-0.024}^{+0.012+0.024}$ | $0.848_{-0.012-0.023}^{+0.012+0.024}$ | $0.859_{-0.011-0.022}^{+0.011+0.022}$ | $0.860_{-0.011-0.021}^{+0.011+0.022}$ | $0.8435_{-0.0089-0.017}^{+0.0088+0.017}$
$H_{0}{\rm[km/s/Mpc]}$ | $71.4_{-1.6-3.2}^{+1.6+3.3}$ | $70.1_{-1.3-2.4}^{+1.2+2.5}$ | $73.0_{-1.1-2.1}^{+1.1+2.1}$ | $71.85_{-0.98-1.9}^{+0.97+1.9}$ | $71.11_{-0.89-1.8}^{+0.88+1.8}$
$N_{\rm eff}$ | $2.93_{-0.19-0.37}^{+0.18+0.38}$ | $2.82_{-0.17-0.34}^{+0.17+0.34}$ | $3.08_{-0.14-0.28}^{+0.14+0.28}$ | $3.05_{-0.14-0.28}^{+0.14+0.29}$ | $2.87_{-0.14-0.26}^{+0.13+0.27}$
$S_{8}$ | $0.818_{-0.016-0.031}^{+0.016+0.032}$ | $0.829_{-0.013-0.026}^{+0.013+0.027}$ | $0.813_{-0.015-0.030}^{+0.015+0.030}$ | $0.829_{-0.013-0.027}^{+0.014+0.027}$ | $0.8111_{-0.0095-0.019}^{+0.0095+0.018}$
$r_{\rm{drag}}$ | $148.283_{-1.867-3.749}^{+1.888+3.865}$ | $149.158_{-1.785-3.528}^{+1.806+3.652}$ | $146.832_{-1.472-2.808}^{+1.464+2.922}$ | $146.906_{-1.482-2.942}^{+1.466+2.974}$ | $148.831_{-1.406-2.804}^{+1.352+2.845}$

Table 4: Observational constraints at 68% and 95% CL using various
cosmological datasets within the extended PEDE+$N_{\rm eff}$ scenario.  Figure
2: One-dimensional marginalized posterior distributions for some selected
parameters and two dimensional $68\%$ and $95\%$ CL joint contours for several
combinations of the parameters within the extended scenario PEDE+$N_{\rm eff}$
using various cosmological datasets.

###  IV.4 PEDE $+M_{\nu}$+$N_{\rm eff}$

We report the constraints at 68% and 95% CL on the cosmological parameters for
PEDE $+M_{\nu}$+$N_{\rm eff}$ model in Tab. 5 and we show the 1D posterior
distributions and 2D correlation plots in Fig. 3. Comparing Tab. 5 with the
previous cases, we find a combination of all the characteristic features we
have already discussed in the previous sections, albeit with a larger increase
of the error bars.

For the Planck 2018 alone case we find in fact that $H_{0}=70.9^{+2.4}_{-1.8}$
m/s/Mpc at 68% CL, in agreement with R19 at 1.1$\sigma$, $M_{\nu}<0.30$ eV at
95% CL (slightly larger than the previous cases), and $N_{\rm eff}=2.91\pm
0.19$ at 68% CL, slightly lower but perfectly consistent with its standard
value, $3.045$. For Planck 2018 + BAO, and Planck 2018 + R19 instead, we have
exactly the same constraints on the neutrino parameters as we found within the
PEDE + $M_{\nu}$ and PEDE + $N_{\rm eff}$ scenarios.

Finally, for the full dataset combination, i.e. Planck
2018+BAO+Pantheon+R19+DES+lensing, we find that the Hubble constant tension is
lowered down to 1.8$\sigma$, a value of the total neutrino mass different from
zero well above $3\sigma$ significance $M_{\nu}=0.21^{+0.20}_{-0.18}$ eV, and
$N_{\rm eff}=3.03\pm 0.16$ at 68% CL, perfectly in agreement with the expected
canonical value. The value of $S_{8}$ is also considerably lower than what one
finds within the $\Lambda$CDM + $M_{\nu}$ \+ $N_{\rm eff}$ scenario.

Parameters | Planck 2018 | Planck 2018+BAO | Planck 2018+R19 | Planck 2018+BAO+R19 | Full
---|---|---|---|---|---
$\Omega_{c}h^{2}$ | $0.1181_{-0.0032-0.0057}^{+0.0029+0.0060}$ | $0.1180_{-0.0032-0.0057}^{+0.0028+0.0061}$ | $0.1198_{-0.0028-0.0055}^{+0.0028+0.0055}$ | $0.1212_{-0.0027-0.0053}^{+0.0027+0.0052}$ | $0.1193_{-0.0026-0.0053}^{+0.0026+0.0051}$
$\Omega_{b}h^{2}$ | $0.02225_{-0.00023-0.00046}^{+0.00022+0.00043}$ | $0.02216_{-0.00019-0.00037}^{+0.00019+0.00037}$ | $0.02245_{-0.00017-0.00033}^{+0.00017+0.00033}$ | $0.02235_{-0.00016-0.00031}^{+0.00016+0.00032}$ | $0.02239_{-0.00016-0.00033}^{+0.00016+0.00032}$
$100\theta_{MC}$ | $1.04112_{-0.00044-0.00087}^{+0.00045+0.00088}$ | $1.04112_{-0.00044-0.00087}^{+0.00045+0.00090}$ | $1.04096_{-0.00042-0.00081}^{+0.00042+0.00082}$ | $1.04080_{-0.00041-0.00080}^{+0.00041+0.00082}$ | $1.04094_{-0.00044-0.00078}^{+0.00039+0.00086}$
$\tau$ | $0.0534_{-0.0081-0.015}^{+0.0075+0.016}$ | $0.0525_{-0.0075-0.015}^{+0.0075+0.015}$ | $0.0550_{-0.0075-0.015}^{+0.0076+0.015}$ | $0.0533_{-0.0076-0.015}^{+0.0075+0.016}$ | $0.0550_{-0.0076-0.015}^{+0.0075+0.016}$
$n_{s}$ | $0.9598_{-0.0088-0.018}^{+0.0087+0.017}$ | $0.9562_{-0.0072-0.014}^{+0.0073+0.014}$ | $0.9674_{-0.0062-0.012}^{+0.0061+0.012}$ | $0.9643_{-0.0061-0.02}^{+0.0061+0.013}$ | $0.9648_{-0.0064-0.0131}^{+0.0064+0.0126}$
${\rm{ln}}(10^{10}A_{s})$ | $3.037_{-0.018-0.036}^{+0.018+0.037}$ | $3.035_{-0.018-0.036}^{+0.018+0.035}$ | $3.045_{-0.017-0.033}^{+0.017+0.033}$ | $3.045_{-0.017-0.035}^{+0.017+0.036}$ | $3.043_{-0.018-0.034}^{+0.017+0.035}$
$\Omega_{m0}$ | $0.282_{-0.020-0.031}^{+0.011+0.036}$ | $0.2896_{-0.0081-0.016}^{+0.0081+0.017}$ | $0.2682_{-0.0087-0.015}^{+0.0075+0.017}$ | $0.2807_{-0.0070-0.012}^{+0.0063+0.013}$ | $0.2861_{-0.0069-0.013}^{+0.0069+0.014}$
$\sigma_{8}$ | $0.842_{-0.015-0.057}^{+0.030+0.045}$ | $0.836_{-0.018-0.042}^{+0.023+0.039}$ | $0.861_{-0.012-0.030}^{+0.016+0.028}$ | $0.853_{-0.015-0.037}^{+0.021+0.034}$ | $0.821_{-0.015-0.030}^{+0.016+0.030}$
$H_{0}{\rm[km/s/Mpc]}$ | $70.9_{-1.8-4.4}^{+2.4+4.2}$ | $69.9_{-1.3-2.5}^{+1.2+2.5}$ | $73.0_{-1.1-2.2}^{+1.1+2.2}$ | $71.77_{-0.97-1.9}^{+0.96+1.9}$ | $70.92_{-0.92-1.8}^{+0.90+1.8}$
$M_{\nu}{\rm[eV]}$ | $<0.121\,,<0.30$ | $0.13_{-0.10}^{+0.05}<0.26$ | $<0.064\,,<0.15$ | $<0.127<0.23$ | $0.209_{-0.075-0.14}^{+0.074+0.15}$
$N_{\rm eff}$ | $2.91_{-0.19-0.36}^{+0.19+0.38}$ | $2.86_{-0.19-0.34}^{+0.17+0.35}$ | $3.07_{-0.15-0.28}^{+0.15+0.30}$ | $3.07_{-0.15-0.30}^{+0.15+0.30}$ | $3.03_{-0.16-0.32}^{+0.16+0.32}$
$S_{8}$ | $0.816_{-0.017-0.033}^{+0.017+0.033}$ | $0.821_{-0.016-0.035}^{+0.018+0.032}$ | $0.814_{-0.016-0.033}^{+0.017+0.031}$ | $0.825_{-0.015-0.033}^{+0.017+0.032}$ | $0.801_{-0.011-0.022}^{+0.011+0.023}$
$r_{\rm{drag}}$ | $148.456_{-1.897-3.692}^{+1.909+3.741}$ | $148.842_{-1.804-3.606}^{+1.940+3.600}$ | $146.951_{-1.532-2.999}^{+1.520+3.039}$ | $146.689_{-1.630-2.907}^{+1.522+3.045}$ | $147.352_{-1.707-3.092}^{+1.559+3.285}$

Table 5: Observational constraints at 68% and 95% CL using various cosmological datasets within the extended PEDE + $M_{\nu}$ \+ $N_{\rm eff}$ scenario. Figure 3: One-dimensional marginalized posterior distributions for some selected parameters and two dimensional $68\%$ and $95\%$ CL joint contours for several combinations of the parameters within the extended scenario PEDE + $M_{\nu}$ \+ $N_{\rm eff}$ using various cosmological datasets.  Model | Datasets | $\ln B_{ij}$ |
---|---|---|---
PEDE | CMB | $0.6$ |
PEDE | CMB+BAO | $-1.7$ |
PEDE | CMB+R19 | $7.9$ |
PEDE | CMB+BAO+R19 | $3.7$ |
PEDE | CMB+BAO+R19+Pantheon+DES+Lensing | $-3.1$ |
PEDE + $M_{\nu}$ | CMB | $0.1$ |
PEDE + $M_{\nu}$ | CMB+BAO | $-1.0$ |
PEDE + $M_{\nu}$ | CMB+R19 | $6.9$ |
PEDE + $M_{\nu}$ | CMB+BAO+R19 | $3.7$ |
PEDE + $M_{\nu}$ | CMB+BAO+R19+Pantheon+DES+Lensing | $-1.5$ |
PEDE + $N_{\rm eff}$ | CMB | $0.2$ |
PEDE + $N_{\rm eff}$ | CMB+BAO | $-2.5$ |
PEDE + $N_{\rm eff}$ | CMB+R19 | $4.1$ |
PEDE + $N_{\rm eff}$ | CMB+BAO+R19 | $3.2$ |
PEDE + $N_{\rm eff}$ | CMB+BAO+R19+Pantheon+DES+Lensing | $-3.1$ |
PEDE + $M_{\nu}$ \+ $N_{\rm eff}$ | CMB | $0.3$ |
PEDE + $M_{\nu}$ \+ $N_{\rm eff}$ | CMB+BAO | $-0.9$ |
PEDE + $M_{\nu}$ \+ $N_{\rm eff}$ | CMB+R19 | $4.5$ |
PEDE + $M_{\nu}$ \+ $N_{\rm eff}$ | CMB+BAO+R19 | $3.4$ |
PEDE + $M_{\nu}$ \+ $N_{\rm eff}$ | CMB+BAO+R19+Pantheon+DES+Lensing | $-1.2$ |
Table 6: Summary of the $\ln B_{ij}$ values quantifying the evidence of fit of
the cosmological scenarios with respect to a base cosmological model
$\Lambda$CDM considering the observational datasets employed in this work. We
note that the negative value of $\ln B_{ij}$ indicates that $\Lambda$CDM is
preferred over the model.

##  V Summary and Conclusions

Our work has been mostly motivated by the measurement of one of the key
cosmological parameters – the Hubble constant ($H_{0}$) – which has been a
serious issue at present time demanding alternatives to the well known
$\Lambda$CDM cosmology. A few years back, it was noticed that the estimations
of $H_{0}$ by two different observational missions did not really match at
all. This mismatch is currently highly significant: the recent observational
results on this issue report that the estimated value of the Hubble constant
by Planck Aghanim  _et al._ (2018a) within the minimal $\Lambda$CDM paradigm
is more than $\sim 4\sigma$ apart from the estimation by Riess et al (SH0ES
collaboration) Riess  _et al._ (2019). While a possible pure systematic origin
of the tension Efstathiou (2014) is still under debate, a number of other
possible avenues have been followed in the literature. An independent high
versus low redshift discrepancy is that related to the substantial
discordances among CMB measurements from Planck and those from cosmic shear
concerning the matter perturbations at small scales. The tension is usually
quantified in terms of the $S_{8}$ parameter. To alleviate these tensions,
especially the Hubble constant one, some approaches have been followed
(dynamical or interacting dark energy, early dark energy, modified gravity and
other non-standard scenarios). Despite the large differences among these
cosmologies, all of them share a common feature, and it is the fact that
mostly all include an extension of the six parameter-$\Lambda$CDM cosmology.

Here instead we focus on a phenomenological alternative which does not imply
any extra degrees of freedom: the Phenomenological Emergent Dark Energy model
(PEDE), a possible _minimal_ scenario which has been shown to provide an
excellent solution to the long-standing $H_{0}$ tension. Also the values of
the $S_{8}$ parameter are lower than in the canonical $\Lambda$CDM cosmology
and, therefore, the model is a promising avenue towards solving the current
cosmological tensions. However, before establishing the PEDE model as a
concrete and complete framework which could serve as a guidance for model
building, it is mandatory to ensure that the resolution of the discrepancies
is stable against obvious extensions of the PEDE model. We have therefore re-
analyzed the PEDE scenario with the inclusion of massive neutrino species and
possible extra relativistic degrees of freedom, because they can help with the
sound horizon problem. We find that the cosmological tensions are still
alleviated: the Hubble constant tension is always below the $2\sigma$ level
and the value of $S_{8}$ is in a better agreement with cosmic shear estimates.
With the most complete combination of datasets, we have obtained $M_{\nu}\sim
0.21^{+0.15}_{-0.14}$ eV and $N_{\rm eff}=3.03\pm 0.32$ (with $95\%$ CL
errors), i.e. an indication for a non-zero neutrino mass with a significance
above $2\sigma$. However, the sound horizon is still $\sim 147$ Mpc, failing
in recovering the lower value preferred by the BAO data. While the sound
horizon issue is not solved in this case but based on other outcomes in this
article, one may believe that the PEDE model provides a new alternative
minimal and solid framework to inspire other phenomenological possibilities.
This feature is strengthened through the Bayesian Evidence analyses (see Table
6 where we present the values of $\ln B_{ij}$ for PEDE and its several
extensions) which clearly show that for CMB, CMB+R19 and CMB+BAO+R19, PEDE and
its all extensions are preferred over $\Lambda$CDM whilst for the remaining
two datasets, namely, CMB+BAO and CMB+BAO+R19+Pantheon+DES+Lensing,
$\Lambda$CDM is favored. If future laboratory measurements will find an
indication for $M_{\nu}\sim 0.21$ eV, this would strongly motivate
phenomenological PEDE like-scenarios, as alternatives to the $\Lambda$CDM
canonical picture.

###### Acknowledgements.

The authors thank the referee for some clarifying points which completed the
article. WY was supported by the National Natural Science Foundation of China
under Grants No. 11705079 and No. 11647153. EDV was supported from the
European Research Council in the form of a Consolidator Grant with number
681431. SP has been supported by the Mathematical Research Impact-Centric
Support Scheme (MATRICS), File No. MTR/2018/000940, given by the Science and
Engineering Research Board (SERB), Govt. of India. O.M. is supported by the
Spanish grants FPA2017-85985-P, PROMETEO/2019/083 and by the European Union
Horizon 2020 research and innovation program (grant agreements No. 690575 and
67489).

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