# Early dark energy in the pre- and post-recombination epochs

Adrià Gómez-Valent1<EMAIL_ADDRESS>Ziyang Zheng1
<EMAIL_ADDRESS>Luca Amendola1<EMAIL_ADDRESS>heidelberg.de Valeria Pettorino2<EMAIL_ADDRESS>Christof Wetterich1
<EMAIL_ADDRESS>1 Institut für Theoretische Physik,
Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg,
Germany  2 AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot,
Sorbonne Paris Cité, F-91191 Gif-sur-Yvette, France

###### Abstract

Dark energy could play a role at redshifts $z\gg\mathcal{O}(1)$. Many
quintessence models possess scaling or attractor solutions where the fraction
of dark energy follows the dominant component in previous epochs of the
Universe’s expansion, or phase transitions may happen close to the time of
matter-radiation equality. A non-negligible early dark energy (EDE) fraction
around matter-radiation equality could contribute to alleviate the well-known
$H_{0}$ tension. In this work we constrain the fraction of EDE using two
approaches: first, we use a fluid parameterization that mimics the plateaux of
the dominant components in the past. An alternative tomographic approach
constrains the EDE density in binned redshift intervals. The latter allows us
to reconstruct the evolution of $\Omega_{de}(z)$ before and after the
decoupling of the Cosmic Microwave Background (CMB) photons. We have employed
Planck data 2018, the Pantheon compilation of supernovae of Type Ia (SNIa),
data on galaxy clustering, the prior on the absolute magnitude of SNIa by
SH0ES, and weak lensing data from KiDS+VIKING-450 and DES-Y1. When we use a
minimal parameterization mimicking the background plateaux, EDE has only a
small impact on current cosmological tensions. We show how the constraints on
the EDE fraction weaken considerably when its sound speed is allowed to vary.
By means of our binned analysis we put very tight constraints on the EDE
fraction around the CMB decoupling time, $\lesssim 0.4\%$ at $2\sigma$ c.l. We
confirm previous results that a significant EDE fraction in the radiation-
dominated epoch loosens the $H_{0}$ tension, but tends to worsen the tension
for $\sigma_{8}$. A subsequent presence of EDE in the matter-dominated era
helps to alleviate this issue. When both the SH0ES prior and weak lensing data
are considered in the fitting analysis in combination with data from CMB, SNIa
and baryon acoustic oscillations, the EDE fractions are constrained to be
$\lesssim 2.6\%$ in the radiation-dominated epoch and $\lesssim 1.5\%$ in the
redshift range $z\in(100,1000)$ at $2\sigma$ c.l. The two tensions remain with
a statistical significance of $\sim 2-3\sigma$ c.l.

Cosmology: observations – Cosmology: theory – cosmological parameters – dark
energy – dark matter

##  I Introduction

Various tensions in cosmological observations could indicate that the minimal
model for dark energy, namely the cosmological constant, may be insufficient.
The observed present value of the Hubble parameter $H_{0}$ seems to be higher
than the one inferred from the Cosmic Microwave Background (CMB) and other
observations measuring the early state of the Universe Aghanim  _et al._
(2020a); Riess  _et al._ (2021); Verde  _et al._ (2019); Di Valentino  _et
al._ (2021a). On the other hand, the amount of large-scale structure (LSS) in
the current Universe could be lower than the one inferred from early cosmology
Aghanim  _et al._ (2020a); Joudaki  _et al._ (2020); Wright  _et al._ (2020);
Heymans  _et al._ (2021); Di Valentino  _et al._ (2021b). If these tensions
grow with future data, and systematics are under control, a possibility is
that dark energy is dynamical Wetterich (1988); Peebles and Ratra (1988);
Ratra and Peebles (1988) in one form or another.

Dynamical dark energy can be associated with a scalar field whose potential
and kinetic energies evolve with time and are (almost) homogeneously
distributed in the Universe. Such models of quintessence could help to
understand why dark energy plays an important role in present cosmology
without an extreme tuning of model parameters. The early cosmological
evolution may obey a scaling behavior Wetterich (1988); Copeland  _et al._
(1998) for which the dark energy density is proportional to the dominant
energy density in radiation or matter. Due to its decrease with time the
present matter density is very small in Planck units, and the same would hold
for the dynamical dark energy density. Such models lead to a small fraction of
“early dark energy” (EDE) for high redshift $z$, in contrast to a negligible
value for the cosmological constant.

Some cosmological event or change of evolution properties has to occur in
order to end such a scaling solution and induce the much slower evolution of
the scalar field that can be inferred from the observed accelerated expansion
of the Universe and present dark energy domination (this is for example the
case of growing neutrino cosmologies, see Wetterich (2007); Amendola  _et al._
(2008); Pettorino  _et al._ (2010)). We treat here the dynamics of this end of
the scaling solution as unknown and gather information on the amount of EDE at
different cosmological epochs from observations in a model-independent way.
Early dark energy arises also in many other types of models, as for example in
ultra-light axion-like models Poulin  _et al._ (2018) or in those linked to
late phase transitions Niedermann and Sloth (2021); Gogoi  _et al._ (2021);
Niedermann and Sloth (2020). Our analysis is thought to cover a large class of
EDE-models.

Dynamical dark energy could be coupled to dark matter Wetterich (1995);
Amendola (2000). This modifies both the background evolution and the formation
of structure. Many models of modified gravity are actually equivalent to some
version of coupled dark energy Wetterich (2015); Pettorino and Baccigalupi
(2008). A nonzero coupling impacts indeed the tensions in cosmological
observations Pettorino (2013); Ade  _et al._ (2016) \- for a recent update see
Gómez-Valent  _et al._ (2020) and references therein. In the present paper we
investigate the case of uncoupled quintessence in order to find out if some
appropriate time evolution of EDE could remove the tensions, as advocated, for
example, in different versions of “new EDE” Poulin  _et al._ (2019); Agrawal
_et al._ (2019); Smith  _et al._ (2020); Niedermann and Sloth (2020); Gogoi
_et al._ (2021); Niedermann and Sloth (2021).

The amount of early dark energy is, however, severely constrained by the
combination of several cosmological observations Doran and Robbers (2006);
Calabrese  _et al._ (2011); Pettorino  _et al._ (2013); Hojjati  _et al._
(2013); Ade  _et al._ (2016); Karwal and Kamionkowski (2016); Hill  _et al._
(2020); Ivanov  _et al._ (2020); Chudaykin  _et al._ (2020); Seto and Toda
(2021); D’Amico  _et al._ (2021). These constraints depend on the detailed
time evolution of EDE, see e.g. Pettorino  _et al._ (2013) for an estimate
within different parameterizations and different epochs at which EDE
starts/ends to be relevant. The aim of the present paper is to continue this
study with a more detailed understanding of the constraints on EDE for various
different redshift ranges, and assessment of the possible role of EDE for the
observed tensions in the light of these constraints. For this purpose we
propose a new, more general parameterization that encompasses most models
encountered so far, as well as a tomographic approach. We further investigate
the impact of assumptions on the sound speed on constraining EDE. In our
analysis we include the CMB temperature, polarization and lensing data from
Planck 2018, the Pantheon compilation of supernovae of Type Ia (SNIa), data on
galaxy clustering from several surveys, the prior on the absolute magnitude of
SNIa obtained from the first steps of the cosmic distance ladder by SH0ES, and
weak lensing data from KiDS+VIKING-450 and DES-Y1. This data set constitutes a
significant improvement with respect to previous analyses, e.g. Pettorino  _et
al._ (2013); Ade  _et al._ (2016).

##  II Early dark energy

Dark energy models may have different amounts of dark energy at different
epochs, which distinguishes them from a $\Lambda$CDM model. Measuring the
amount of EDE at different epochs in the past can then be the key to
distinguish such scenarios and possibly falsify a cosmological constant. EDE
affects observables in various ways. First, the presence of EDE at decoupling
can change the position and height of the peaks in the CMB Doran  _et al._
(2001a); Pettorino  _et al._ (2013); Ade  _et al._ (2016): this is a
consequence of the fact that EDE affects the distance to last scattering
surface, as well as the relative amount of DE with respect to other components
(such as dark matter or neutrinos). EDE can also impact the CMB anisotropies
through the early integrated Sachs-Wolfe effect. Furthermore, EDE suppresses
the growth of structure after last scattering Ferreira and Joyce (1997); Doran
_et al._ (2001b); Caldwell  _et al._ (2003): a smaller number of clusters can
form with respect to a cosmological constant scenario Grossi and Springel
(2009); the lensing potential is also weaker, with an impact on both weak
lensing and the smoothing of CMB peaks at large multipoles due to CMB lensing.

It is then clear that a sufficient amount of EDE, if compatible with other
observations, can potentially have a direct impact on both $H_{0}$ and
$\sigma_{8}$ tensions. In the following we first design a general
parameterization able to mimic the background dominant component, and then
proceed with a tomographic analysis in different redshift bins.

###  II.1 Parametric EDE

Figure 1: Functions $\Omega_{de}(z)$ (left plot) and
$w_{de}(z)=p_{de}(z)/\rho_{de}(z)$ (right plot) obtained using the
parametrizations described in Sec. II.1, for some illustrative values of the
parameters.

We are interested in building a simple parameterization of the DE density that
allows us to reproduce the behavior of uncoupled quintessence models with
scaling or attractor solutions, while keeping a certain degree of generality
and flexibility. The presence of scaling or attractor solutions is one of the
most interesting features of DE models, as it allows to retrieve the current
acceleration phase starting from a wide range of initial conditions. They
represent a class of DE models that are promising to alleviate the fine tuning
issues that affect a cosmological constant. For this reason, we want our
parameterization to be able to generate two plateaux in
$\Omega_{de}(z)=\rho_{de}(z)/\rho_{c}(z)$, with $\rho_{de}(z)$ and
$\rho_{c}(z)=3H^{2}(z)/8\pi G$ the dark energy and critical energy densities
in the Universe, respectively. The first plateau occurs in the radiation-
dominated epoch (RDE), and the second one in the matter-dominated era (MDE).
These plateaux can possibly have different heights. This is what happens for
instance in quintessence models with a single exponential potential
$V(\phi)=V_{0}e^{-\sqrt{8\pi G}\lambda\phi}$ Wetterich (1988); Copeland  _et
al._ (1998); Barreiro  _et al._ (2000), where the fractions of EDE in the RDE
and MDE depend only on one parameter, $\lambda$ (cf. Appendix A).

With this aim in mind, we generalize the parameterizations proposed earlier in
Pettorino  _et al._ (2013); Ade  _et al._ (2016) and we consider here a DE
density with the following form,

$\rho_{de}(z)=\rho_{1}(1+z)^{4}+\rho_{2}(1+z)^{3}+\rho_{3}(1+z)^{3(1+w)}\,,$
(1)

parameterized by the constant energy densities $\rho_{1}$, $\rho_{2}$ and
$\rho_{3}$ and by a constant equation of state parameter (EoS) $w$. We call
this parameterization EDEp, where the "p" reminds us of the ‘plateaux’ that
characterise it. The last term of (1) is able to mimic the behavior of a late-
time dynamical DE with the $w$CDM form Turner and White (1997), whereas the
first two terms produce the plateaux in the RDE and MDE. It is useful to write
the constants $\rho_{1}$ and $\rho_{2}$ in terms of dimensionless parameters,
as follows

$\rho_{1}=\chi_{1}\Omega^{(0)}_{r,*}\rho^{(0)}_{c}\qquad;\qquad\rho_{2}=\chi_{2}\Omega^{(0)}_{m,*}\rho^{(0)}_{c}\,,$
(2)

where $\Omega^{(0)}_{r,*}$ and $\Omega^{(0)}_{m,*}$ are the current density
parameters of radiation and matter, respectively, computed considering three
massless neutrinos111Notice that $\Omega^{(0)}_{r,*}$ and $\Omega^{(0)}_{m,*}$
are not equal to $\Omega^{(0)}_{r}$ and $\Omega^{(0)}_{m}$. The latter are the
true radiation and matter density parameters, since they take into account the
non-relativistic behavior of the massive neutrino of $0.06$ eV in the late-
time universe that we have considered in all our numerical analyses.
$\Omega^{(0)}_{r,*}$ and $\Omega^{(0)}_{m,*}$ have been defined for practical
purposes, to allow an easier definition of the EDE fractions in the RDE and
MDE in terms of the dimensionless parameters $\chi_{1}$ and $\chi_{2}$,
respectively, cf. (3). This simplifies the implementation of our EDEp
parameterization in the Einstein-Boltzmann code, see Sec. III. The relative
difference between $\Omega^{(0)}_{m}$ and $\Omega^{(0)}_{m,*}$ is $\sim 0.4\%$
for a total neutrino mass of $0.06$ eV, whereas the one between
$\Omega^{(0)}_{r}$ and $\Omega^{(0)}_{r,*}$ is $\sim 13\%$., and
$\rho_{c}^{(0)}\equiv\rho_{c}(z=0)$ is the current critical energy density.
The superscripts $(0)$ denote quantities evaluated at present. In this work we
consider standard General Relativity (GR) and a flat Friedmann-Lemaître-
Robertson-Walker universe. If we want the DE density to be positive during the
cosmic expansion we have to demand $\chi_{1},\chi_{2},\rho_{3}>0$. Notice that
deep in the RDE and MDE the fractions of energy in the Universe contained in
the form of (early) DE are constant and given by

$\Omega^{\rm RD}_{ede}=\frac{\chi_{1}}{1+\chi_{1}}\qquad;\qquad\Omega^{\rm
MD}_{ede}=\frac{\chi_{2}}{1+\chi_{2}}\,.$ (3)

The present dark energy density, $\rho^{(0)}_{de}$, can be directly computed
from the Hubble parameter, $H_{0}=100h$ km/s/Mpc, and the reduced CDM and
baryon density parameters, $\omega_{cdm}=\Omega_{cdm}^{(0)}h^{2}$ and
$\omega_{b}=\Omega_{b}^{(0)}h^{2}$. Thus, one of the three $\rho_{i}$’s
appearing in (1) can be expressed in terms of the other two, e.g.
$\rho_{3}=\rho^{(0)}_{de}-\rho_{1}-\rho_{2}$. It is clear then that in this
EDE parametrization we deal with three additional parameters with respect to
the $\Lambda$CDM, so we have nine cosmological parameters in total, namely the
spectral index $n_{s}$ and amplitude $A_{s}$ of the primordial power spectrum,
$H_{0}$, $\omega_{b}$, $\omega_{cdm}$, the optical depth to reionization,
$\tau$, together with $w$, $\chi_{1}$ and $\chi_{2}$ (or, equivalently,
$\Omega^{\rm RD}_{ede}$ and $\Omega^{\rm MD}_{ede}$). We consider the minimal
configuration with a massive neutrino of $0.06$ eV and two massless neutrinos.

For typical (low) values of $\Omega^{\rm RD}_{ede}$ and $\Omega^{\rm
MD}_{ede}$ much smaller than one the DE fraction in EDEp can be approximated
by

$\Omega_{de}(z)=\frac{\Omega^{\rm MD}_{ede}+\Omega^{\rm
RD}_{ede}\left(\frac{1+z}{1+z_{eq}}\right)+\left(\frac{\Omega_{de}^{(0)}}{1-\Omega_{de}^{(0)}}-\Omega^{\rm
MD}_{ede}\right)(1+z)^{3w}}{1+\frac{1+z}{1+z_{eq}}+\frac{\Omega_{de}^{(0)}}{1-\Omega_{de}^{(0)}}(1+z)^{3w}}\,,$
(4)

with $z_{eq}$ the redshift at the matter-radiation equality time.

As the DE fluid under consideration is covariantly conserved, i.e. it does not
interact with the other species, we can make use of the following equation to
compute its associated pressure,

$\displaystyle\dot{\rho}_{de}+$ $\displaystyle 3H(\rho_{de}+p_{de})=0$ (5)
$\displaystyle\longrightarrow
p_{de}(z)=\frac{\rho_{1}}{3}(1+z)^{4}+w\rho_{3}(1+z)^{3(1+w)}\,,$

where the dot refers to a derivative with respect to the cosmic time. The
corresponding equation of state parameter can be obtained from the ratio of
(5) and (1), i.e. $w_{de}(z)=p_{de}(z)/\rho_{de}(z)$. It is clear that
$w_{de}=1/3$ and $w_{de}=0$ in the RDE and MDE, respectively, and
$w_{de}\approx w$ at present for low values of $\chi_{2}$. For the
perturbations, in our main analyses we take the sound speed of the DE fluid to
be equal to the speed of light in the DE rest frame Ballesteros and
Lesgourgues (2010), i.e. $\hat{c}_{s}=1$, so in our model the DE does not
cluster efficiently. We will study also what happens when we allow for values
of $\hat{c}_{s}<1$. Our simple parametrization is able to reproduce the
phenomenology of a quintessence model with a potential energy density that
reduces to an exponential function of the scalar field in the RDE and MDE.
These exponential potentials are controlled by two different constant
parameters, which allow for the two independent plateaux of $\Omega_{ede}$.
See Appendix A for details.

It is useful to compare our parametrization (1) or just (4) with those
previously studied in Doran and Robbers (2006); Calabrese  _et al._ (2011);
Pettorino  _et al._ (2013); Ade  _et al._ (2016). If we set $\Omega^{\rm
RD}_{ede}=\Omega^{\rm MD}_{ede}$, the phenomenology of (1) is almost identical
to the one found in the EDE1 parametrization of Pettorino  _et al._ (2013)
(originally proposed by Doran and Robbers (2006) and also employed by
Calabrese  _et al._ (2011); Ade  _et al._ (2016)), and when $w$ is set to $-1$
also to the EDE2 parametrization of Pettorino  _et al._ (2013). If we set
$\Omega^{\rm RD}_{ede}=0$ in (1) EDE is completely negligible during the RDE,

$\rho_{de}(z)=\rho_{2}(1+z)^{3}+\rho_{3}(1+z)^{3(1+w)}\,.$ (6)

We denote this particular case of EDEp as EDEpMD, to remind us that the
plateau is in this case following the matter component. The EDE3
parametrization of Pettorino  _et al._ (2013); Ade  _et al._ (2016) can be
retrieved to a good approximation if we set $\Omega^{\rm RD}_{ede}=0$ and
$w=-1$ and turn the second term on in (1) at a particular ‘threshold’ redshift
$z_{\rm thr}$, in the MDE. This is what we call EDEpMD,thr:

$\rho_{de}(z)=\rho_{2}(1+z)^{3}\theta(z_{{\rm
thr}}-z)+\rho_{3}(1+z)^{3(1+w)}\,,$ (7)

with $\theta$ the Heaviside step function. For $z_{\rm thr}\to\infty$ we
recover the pure EDEpMD parametrization (6). We show typical shapes of the
functions $\Omega_{de}(z)$ and $w_{de}(z)$ obtained with EDEp, EDEpMD and
EDEpMD,thr in Fig. 1. Finally, the behavior of the EDE4 parametrization of
Pettorino  _et al._ (2013) can be also reproduced if we set $\Omega^{\rm
MD}_{ede}=0$ and $w=-1$ and deactivate the first term in (1) at a specific
value of the scale factor in the RDE. Thus, with (1) we can study the same
scenarios as in Pettorino  _et al._ (2013); Ade  _et al._ (2016), and also
analyze what happens when we do not only have one single plateau, but two, one
in the RDE and another one in the MDE. Hence, we will be able to compare the
fitting results obtained with our parameterization with those reported in
previous works in the literature, and also appreciate the evolution of the
constraints on the EDE fraction with the data in the last years. In addition,
by allowing departures of $w$ from $-1$ we can also see what is the effect of
the late-time dynamics of DE on the fitting results. On top of this, we also
explore different values of the EDE sound speed $\hat{c}_{s}$. This shows that
our parameterization includes previous ones but is more general.

Coupled quintessence models with an interaction between dark energy and dark
matter can possess scaling solutions too, see Amendola (2000), and Gómez-
Valent  _et al._ (2020) for recent constraints. In some running vacuum models
(RVMs) with a vacuum energy density $\rho_{\rm vac}=\frac{3}{8\pi G}(c_{0}+\nu
H^{2})$ there is a plateau with $\Omega_{\rm vac}^{\rm RD}=\Omega_{\rm
vac}^{\rm MD}\simeq\nu$ Solà Peracaula  _et al._ (2018), and in the so-called
Ricci RVMs, in which $\rho_{\rm vac}=\frac{3}{8\pi G}(c_{0}+\nu R/12)$ with
$R$ the Ricci scalar, the vacuum energy density follows the one of non-
relativistic matter during the RDE and MDE, with $\Omega_{\rm vac}^{\rm
MD}\simeq\nu/4$ during the MDE Solà Peracaula  _et al._ (2021). In these
models, though, the pressure of the dark energy/vacuum component is different
from the one of the dominant species in the radiation- and matter-dominated
eras, in contrast to what happens in the uncoupled quintessence models
described before, and there are also important differences at the linear
perturbations level. Differences in perturbations are also found in Brans-
Dicke cosmology, but in this case both, the energy density and pressure of the
effective DE fluid, evolve as for relativistic and non-relativistic matter in
the RDE and MDE, respectively, for reasonable values of the Brans-Dicke
parameter $\omega_{\rm BD}$ Solà Peracaula  _et al._ (2019, 2020). Some of
these models are able to loosen the cosmological tensions, see the
aforementioned references for further details. Here we focus in the non-
interacting scenarios with scaling behavior discussed above, assuming GR. We
present the constraints on EDEp (1), EDEpMD (6) and EDEpMD,thr (7) in Sec.
V.1.

###  II.2 Tomographic EDE

We further consider the possibility of binning the amount of $\Omega_{de}(z)$,
similarly to what previously explored in Pettorino  _et al._ (2013); Ade  _et
al._ (2016), and also in our EDEpMD,thr parametrization. While in such cases,
though, only the fraction of dark energy below the threshold redshift at which
it would become non negligible was let free to vary, here we constrain the EDE
density in different bins; we then perform a tomographic analysis using the
data sets described in Sec. IV. For convenience we fix 11 bins, as indicated
in Table 1: the right column identifies the amplitude within each of them. At
low redshifts, $z<5$, we assume a $w$CDM behavior Turner and White (1997).
Here, again, $\rho_{de}^{(0)}$ can be directly determined from $H_{0}$,
$\omega_{b}$ and $\omega_{cdm}$; the other constant densities $\rho_{i}$, with
$i=A,...,J$ (cf. Table 1), are left free in our Monte Carlo runs, together
with $w$ and the six usual $\Lambda$CDM parameters. We keep $\hat{c}_{s}=1$.
Our main aim is to see how much EDE we can have in each bin, and therefore
which shape of $\Omega_{de}(z)$ is preferred by the data, regardless of its
complexity. Our binned method is similar (but not equal) to the one employed
in Hojjati  _et al._ (2013). The corresponding fitting results and
reconstructed shapes of $\Omega_{de}(z)$ obtained with different data set
combinations are shown and discussed in Sec. V.2.

Redshift bin | $\rho_{de}(z)$
---|---
$z\leq 5$ | $\rho_{de}^{(0)}(1+z)^{3(1+w)}$
$5<z\leq 10$ | $\rho_{A}(1+z)^{3}$
$10<z\leq 50$ | $\rho_{B}(1+z)^{3}$
$50<z\leq 200$ | $\rho_{C}(1+z)^{3}$
$200<z\leq 500$ | $\rho_{D}(1+z)^{3}$
$500<z\leq 1000$ | $\rho_{E}(1+z)^{3}$
$1000<z\leq 2000$ | $\rho_{F}(1+z)^{4}$
$2000<z\leq 3000$ | $\rho_{G}(1+z)^{4}$
$3000<z\leq 5000$ | $\rho_{H}(1+z)^{4}$
$5000<z\leq 10^{4}$ | $\rho_{I}(1+z)^{4}$
$z>10^{4}$ | $\rho_{J}(1+z)^{4}$
Table 1: Binning of the EDE density employed in the tomographic analysis
described in Sec II.2.

##  III Methodology

We have implemented the various parametrizations of Sec. II.1 and also the
binned $\rho_{de}(z)$ described in Sec. II.2 in our own modified version of
the Einstein-Boltzmann code CLASS Blas  _et al._ (2011). We have used it to
compute the background history and solve the linear perturbations equations,
which is of course crucial to obtain all the theoretical quantities that we
need to confront our models to the cosmological data, see Sec. IV. We have put
constraints on the parameters of our models through the usual Bayesian
exploration of the parameter space. Its high dimensionality is not only due to
the pure cosmological parameters entering the models, but also to the nuisance
parameters that are required to model the theory and experiment systematics,
as it happens e.g. with the CMB foregrounds (cf. the references in Sec.
IV.1.1). Such large parameter spaces must be explored with Monte Carlo
techniques as the Metropolis-Hastings algorithm Metropolis  _et al._ (1953);
Hastings (1970). For this purpose we have employed the Monte Carlo sampler
MontePython Audren  _et al._ (2013). We have used flat priors for the
cosmological parameters in common with the $\Lambda$CDM model, with widths
that fully respect the Planck 2018 uncertainties Aghanim  _et al._ (2020a).
Regarding the priors on the EDE fractions, previous studies in the literature
showed that they are always lower than $\sim 10\%$ Pettorino  _et al._ (2013);
Ade  _et al._ (2016), so we have used the following flat prior on $\chi_{i}$
(with $i=1,2$) (2) in our Monte Carlo analyses of the parametrizations of Sec.
II.1: $0<\chi_{i}<0.12$; and, of course, similar priors have been imposed in
each bin of the tomographic analysis of Sec. II.2.

##  IV Data

###  IV.1 Description of the individual data sets

In this section we list the various data sets used in this study. We explain
the most important features of each of them and provide the corresponding
references.

####  IV.1.1 Cosmic microwave background

We consider the full Planck 2018 TTTEEE+lowE likelihood Aghanim  _et al._
(2020a). It includes the data on the CMB temperature (TT) and polarization
(EE) anisotropies, and also their cross-correlations (TE) at low and high
multipoles. We call this data set, in short, CMBpol. In some of our fits we
also include the data on CMB lensing, i.e. we consider the Planck 2018
TTTEEE+lowE+lensing likelihood Aghanim  _et al._ (2020a, b). We denote the
resulting CMB data set as CMBpolens.

####  IV.1.2 Supernovae of Type Ia

We use the observed apparent magnitude and redshifts from the standardized
1048 SNIa of the Pantheon compilation Scolnic  _et al._ (2018). We have duly
taken into account the existing correlations among them through the
corresponding covariance matrix. The absolute magnitude of these SNIa, $M$, is
left free in the fitting analysis, and we impose a prior on it, see Sec.
IV.1.3 for details.

####  IV.1.3 Prior on $M$

There exist a $\sim 4.1\sigma$ tension between the values of the Hubble
parameter measured by SH0ES Riess  _et al._ (2021),

$H_{0,\rm SH0ES}=(73.2\pm 1.3)\,{\rm km/s/Mpc}\,,$ (8)

and the one inferred from the CMB anisotropy maps by Planck 2018 under the
assumption of the flat $\Lambda$CDM model Aghanim  _et al._ (2020a),
$H_{0}=(67.36\pm 0.54)$ km/s/Mpc, Verde  _et al._ (2019); Di Valentino  _et
al._ (2021a). It is still unclear whether this tension is due to some sort of
new physics beyond the standard model Di Valentino  _et al._ (2021c) or to the
presence of systematic errors in the data, see e.g. Aylor  _et al._ (2017);
Efstathiou (2020); Mortsell  _et al._ (2021); Freedman (2021), but in this
work we want to study its status in the context of the EDE models described in
Sec. II, assuming that the tension has a physical origin. We do so by using in
some of our fitting analyses the SH0ES effective calibration prior on the
absolute magnitude of the SNIa as provided in Camarena and Marra (2020),

$M_{\rm SH0ES}=-19.2191\pm 0.0405\,.$ (9)

The latter is obtained from the calibration of nearby SNIa (at $z\lesssim
0.01$) with Cepheids Riess  _et al._ (2019). As explained in Camarena and
Marra (2020), it is better to use this prior rather than the one on $H_{0}$
Reid  _et al._ (2019); Riess  _et al._ (2021), especially when it is combined
with data from supernovae compilations that include the same SNIa in the
Hubble flow considered by the SH0ES team. This is the case e.g. of the
Pantheon compilation, which we employ in all our fits, see Sec. IV.1.2 and
IV.2. Hence, using the prior on $M$ (9) instead of $H_{0}$ (8) we avoid
unwanted double-counting issues. In any case, the results obtained from the
two priors is almost indistinguishable for models with no abrupt features at
very low redshift ($z\lesssim 0.1$) Benevento  _et al._ (2020); Camarena and
Marra (2021), as the ones we analyze in this work or those of Gómez-Valent
_et al._ (2020); Solà Peracaula  _et al._ (2020). In some of our Tables we
also provide the best-fit values of $M$. This allows us to quantity the ‘$M$
tension’, i.e. the tension between the latter and (9). We show in Sec. V.2
that the statistical level of the SH0ES-Planck tension can be in some cases
quite different when formulated in terms of the Hubble parameter and the
absolute magnitude of SNIa.

####  IV.1.4 Baryon acoustic oscillations

The interactions between photons and baryons before the decoupling of the
former at $z_{dec}\sim 1100$ caused the so-called baryon acoustic oscillations
(BAO). Their imprint can be observed as a peak in the two-point correlation
function of matter fluctuations at a scale of about $147$ Mpc, which can be
used as a standard ruler to constrain cosmological models if one assumes that
it is not partially caused e.g. by any peculiar feature in the primordial
power spectrum Cole  _et al._ (2005); Eisenstein  _et al._ (2005). Several
galaxy surveys have been able to provide precise data on BAO, either in terms
of the dilation scale $D_{V}$,

$\frac{D_{V}(z)}{r_{d}}=\frac{1}{r_{d}}\left[D_{M}^{2}(z)\frac{cz}{H(z)}\right]^{1/3}\,,$
(10)

with $D_{M}=(1+z)D_{A}(z)$ being the comoving angular diameter distance and
$r_{d}$ the sound horizon at the baryon drag epoch, or even by splitting (when
possible) the transverse and line-of-sight BAO information and hence being
able to provide data on $D_{A}(z)/r_{d}$ and $H(z)r_{d}$ separately, with some
degree of correlation. The surveys provide the values of the measurements at
some effective redshift(s). We employ the following BAO data points in this
work:

  * •

$D_{V}/r_{d}$ at $z=0.122$ provided in Carter  _et al._ (2018), which combines
the dilation scales previously reported by the 6dF Galaxy Survey (6dFGS)
Beutler  _et al._ (2011) at $z=0.106$ and the one obtained from the Sloan
Digital Sky Survey (SDSS) Main Galaxy Sample at $z=0.15$ Ross  _et al._
(2015).

  * •

The anisotropic BAO data measured by BOSS using the LOWZ ($z=0.32$) and CMASS
($z=0.57$) galaxy samples Gil-Marín  _et al._ (2017).

  * •

The dilation scale measurements by WiggleZ at $z=0.44,0.60,0.73$ Kazin  _et
al._ (2014). The galaxies contained in the WiggleZ catalog are located in a
patch of the sky that partially overlaps with those present in the CMASS
sample by BOSS. Nevertheless, the two surveys are independent, work under
different seeing conditions, instrumental noise, etc. and target different
types of galaxies. The correlation between the CMASS and WiggleZ data has been
quantified in Beutler  _et al._ (2016), where the authors estimated the
correlation coefficient to be $\lesssim 2\%$. This justifies the inclusion of
the WiggleZ data in our analysis, although their statistical weight is much
lower than those from BOSS and in practice their use does not have any
important impact on our results.

  * •

$D_{A}/r_{d}$ at $z=0.81$ measured by the Dark Energy Survey (DES) Abbott  _et
al._ (2019).

  * •

The anisotropic BAO data from the extended BOSS Data Release 16 (DR16) quasar
sample at $z=1.48$ Neveux  _et al._ (2020).

  * •

The combined measurement of the anisotropic BAO information obtained from the
Ly$\alpha$ absorption and quasars of the final data release (SDSS DR16) of
eBOSS, at $z=2.334$ du Mas des Bourboux  _et al._ (2020).

####  IV.1.5 Weak lensing

In some of our fitting analyses we employ a prior on the composite quantity
$S_{8}=\sigma_{8}(\Omega_{m}^{(0)}/0.3)^{0.5}$, with $\sigma_{8}$ the root-
mean-square (rms) of mass fluctuations at scales of $8h^{-1}$ Mpc. More
concretely, we use the prior

$S_{8}=0.762^{+0.025}_{-0.024}\,,$ (11)

obtained from the combined tomographic weak gravitational lensing (WL)
analysis of the Kilo Degree Survey (KiDS +VIKING-450) and the Dark Energy
Survey (DES-Y1) Joudaki  _et al._ (2020). This value is in tension with the
Planck 2018 TT,TE,EE+lowE +lensing $\Lambda$CDM best-fit value
($S_{8}=0.832\pm 0.013$) at the level of $\sim 2.5\sigma$. Other weak lensing
studies find similar results, see e.g. Joudaki  _et al._ (2018); Wright  _et
al._ (2020); Heymans  _et al._ (2021). They point to a lower amount of LSS
formation in the Universe than the one that can be accommodated by the
concordance model. This tension is usually formulated directly in terms of the
$\sigma_{8}$ parameter when information from galaxy clustering measurements is
considered Macaulay  _et al._ (2013); Solà Peracaula  _et al._ (2018);
Nesseris  _et al._ (2017); Heymans  _et al._ (2021), e.g. using data from
redshift-space distortions (RSD), see Sec. IV.1.6. The prior (11) has been
obtained under the strong model assumption of the flat $\Lambda$CDM, so
strictly speaking it can only be used in fitting analyses of the concordance
model. Nevertheless, we use it in some of our fitting runs too in order to
study the ability of our EDE models to loosen the $S_{8}$/$\sigma_{8}$
tension, and the impact of the WL data on our constraints on the EDE fraction.

CMBpol+SNIa
---
Parameter |  $\Lambda$CDM |  $w$CDM | EDEp | EDEpMD
$\omega_{b}$ | $0.02239^{+0.00014}_{-0.00015}$ | $0.02237^{+0.00016}_{-0.00015}$ | $0.02238^{+0.00017}_{-0.00016}$ | $0.02234^{+0.00015}_{-0.00016}$
$\omega_{cdm}$ | $0.1199^{+0.0014}_{-0.0013}$ | $0.1204\pm 0.0014$ | $0.1218^{+0.0016}_{-0.0015}$ | $0.1208^{+0.0014}_{-0.0015}$
$\tau$ | $0.055^{+0.007}_{-0.008}$ | $0.054\pm 0.008$ | $0.054\pm 0.008$ | $0.053^{+0.007}_{-0.008}$
$n_{s}$ | $0.9659\pm 0.0044$ | $0.9646^{+0.0044}_{-0.0045}$ | $0.9642^{+0.0044}_{-0.0045}$ | $0.9642^{+0.0043}_{-0.0046}$
$H_{0}$ [km/s/Mpc] | $67.60^{+0.59}_{-0.61}$ | $68.55^{+1.12}_{-1.10}$ | $68.71\pm 1.16$ | $68.61^{+1.09}_{-1.16}$
$\sigma_{8}$ | $0.811^{+0.007}_{-0.008}$ | $0.823\pm 0.014$ | $0.817^{+0.014}_{-0.015}$ | $0.818^{+0.015}_{-0.014}$
$r_{d}$ [Mpc] | $147.02^{+0.28}_{-0.32}$ | $146.92\pm 0.30$ | $146.18^{+0.66}_{-0.43}$ | $146.75^{+0.36}_{-0.30}$
$w$ | $-1$ | $-1.039^{+0.035}_{-0.039}$ | $-1.050^{+0.041}_{-0.040}$ | $-1.053^{+0.038}_{-0.042}$
$\Omega^{\rm RD}_{ede}\,(\%)$ | $0$ | $0$ | $<0.91\,(<2.08)$ | $0$
$\Omega^{\rm MD}_{ede}\,(\%)$ | $0$ | $0$ | $<0.27\,(<0.69)$ | $<0.29\,(<0.69)$
CMBpolens+SNIa+$M$
Parameter |  $\Lambda$CDM |  $w$CDM | EDEp | EDEpMD
$\omega_{b}$ | $0.02257^{+0.00015}_{-0.00014}$ | $0.02241^{+0.00014}_{-0.00016}$ | $0.02245^{+0.00015}_{-0.00019}$ | $0.02239\pm 0.00015$
$\omega_{cdm}$ | $0.1179\pm 0.0012$ | $0.1200\pm 0.0015$ | $0.1212\pm 0.0016$ | $0.1206\pm 0.0012$
$\tau$ | $0.057^{+0.007}_{-0.008}$ | $0.054\pm 0.008$ | $0.055\pm 0.008$ | $0.056^{+0.007}_{-0.008}$
$n_{s}$ | $0.9709\pm 0.0044$ | $0.9658^{+0.0044}_{-0.0047}$ | $0.9659^{+0.0045}_{-0.0044}$ | $0.9656^{+0.0041}_{-0.0042}$
$H_{0}$ [km/s/Mpc] | $68.56^{+0.56}_{-0.54}$ | $70.55^{+0.86}_{-0.88}$ | $70.63^{+0.86}_{-0.82}$ | $70.20^{+0.52}_{-0.69}$
$\sigma_{8}$ | $0.811^{+0.007}_{-0.008}$ | $0.838\pm 0.014$ | $0.830^{+0.014}_{-0.013}$ | $0.835\pm 0.010$
$r_{d}$ [Mpc] | $147.36^{+0.28}_{-0.29}$ | $146.98^{+0.33}_{-0.30}$ | $146.21^{+0.73}_{-0.44}$ | $146.79\pm 0.29$
$w$ | $-1$ | $-1.098^{+0.035}_{-0.032}$ | $-1.099^{+0.034}_{-0.032}$ | $-1.099^{+0.030}_{-0.025}$
$\Omega^{\rm RD}_{ede}\,(\%)$ | $0$ | $0$ | $<1.14\,(2.44)$ | 0
$\Omega^{\rm MD}_{ede}\,(\%)$ | $0$ | $0$ | $<0.22\,(0.52)$ | $<0.22\,(0.54)$
Table 2: The mean fit values and $68.3\%$ confidence limits for the
$\Lambda$CDM, the $w$CDM, EDEp (1) and EDEpMD (6), using the CMBpol+SNIa and
CMBpolens+SNIa+$M$ data sets (cf. Sec. IV). For $\Omega^{\rm RD}_{ede}$ and
$\Omega^{\rm MD}_{ede}$ we also show the $2\sigma$ limits inside the
parentheses. See the comments in Sec. V.1.

Some works in the literature Poulin  _et al._ (2019); Smith  _et al._ (2020);
Murgia  _et al._ (2021); Smith  _et al._ (2021) have reported significant
hints in favor of an ultra-light axion-like EDE model in which the Hubble
function is larger than in the $\Lambda$CDM during the pre-recombination epoch
due to the presence of a scalar field in the Universe with an associated
constant and very large potential energy density that at $z\sim 3000-5000$
decays as fast as (or even faster than) radiation. This excess of energy,
which can be of about $\sim 7-10\%$ at the maximum of $\Omega_{ede}$, produces
a decrease of the sound horizon $r_{d}$ that has to be compensated by an
increase of $H_{0}$ to keep the location of the first peak of the CMB
temperature power spectrum intact. These models lead, though, to much larger
values of $\omega_{cdm}$, and this generates also a larger growth of matter
perturbations in the late-time universe. Hence, the $H_{0}$ tension in this
model is alleviated, but at the expense of worsening the $S_{8}/\sigma_{8}$
one Hill  _et al._ (2020); Ivanov  _et al._ (2020); D’Amico  _et al._ (2021).
Something similar happens in the New EDE model proposed in Niedermann and
Sloth (2021, 2020) or the early modified gravity model studied in Braglia  _et
al._ (2021). Although the $S_{8}$/$\sigma_{8}$ tension is not as statistically
significant as the $H_{0}$ one (cf. Sec. IV.1.3), it is interesting to check
whether it is possible for our EDE models to alleviate both tensions at the
same time. This could happen, in principle, since our models have some
features that are not present in Poulin  _et al._ (2019); Niedermann and Sloth
(2020, 2021); Smith  _et al._ (2020); Hill  _et al._ (2020); Ivanov  _et al._
(2020); Murgia  _et al._ (2021); Smith  _et al._ (2021); D’Amico  _et al._
(2021). In particular, our models also allow for: (i) a non-negligible
fraction of EDE during the MDE that could help to slow down the formation of
structures in the Universe due to the inability of EDE to cluster; and (ii) a
late-time dynamical dark energy component, which can also help in this
direction.

Sanchez (2020) has raised some concerns about the use of $\sigma_{8}$ and
derived quantities as $S_{8}$. He suggests the use of $\sigma_{12}$, defined
as the rms linear theory variance at the fixed scale of $12$ Mpc, and
$S_{12}=\sigma_{12}(\omega_{m}/0.14)^{0.4}$. We provide the values of these
parameters in some of our Tables, together with the usual $\sigma_{8}$ and
$S_{8}$. We will show in Sec. V.2 that the enhancement in the LSS formation
processes observed in previous works Hill  _et al._ (2020); Ivanov  _et al._
(2020); D’Amico  _et al._ (2021) for different EDE models is also found for
those described in Sec. II when $\sigma_{12}$ and $S_{12}$ are considered,
instead of the usual $\sigma_{8}$ and $S_{8}$. We will see, though, that this
issue can be mitigated by allowing the presence of EDE in the MDE, after the
decoupling of the CMB photons.

####  IV.1.6 Redshift-space distortions and direct peculiar velocity
measurements

LSS Information can be also extracted from statistical measurements of the
anisotropic clustering of galaxies in redshift space, and from the direct
measurement of peculiar velocities when redshift-independent distance
indicators are available. Galaxy redshift surveys obtain constraints on the
product of the growth rate of structure, $f(z)=\frac{d\ln\delta_{m}(a)}{d\ln
a}$, and $\sigma_{8}(z)$. These are the measurements that we include in our
data set:

  * •

The data point at $z=0.035$ obtained from the joint analysis of 6dFGS and SDSS
peculiar velocities Said  _et al._ (2020).

  * •

The two data points provided by the Galaxy and Mass Assembly survey (GAMA) at
$z=0.18$ Simpson  _et al._ (2016) and $z=0.38$ Blake  _et al._ (2013).

  * •

The four points at $z=0.22,0.41,0.60,0.78$ measured by WiggleZ Blake  _et al._
(2011).

  * •

The RSD measurements by BOSS from the power spectrum and bispectrum of the
DR12 galaxies contained in the LOWZ ($z=0.32$) and CMASS ($0.57$) samples Gil-
Marín  _et al._ (2017).

  * •

The two points at $z=0.60,0.86$ reported by the VIMOS Public Extragalactic
Redshift Survey (VIPERS) Mohammad  _et al._ (2018).

  * •

The point at $z=0.77$ by VIMOS VLT Deep Survey (VVDS) Guzzo  _et al._ (2008);
Song and Percival (2009).

  * •

The Subaru FMOS galaxy redshift survey (FastSound) measurement at $z=1.36$
Okumura  _et al._ (2016).

  * •

The measurement by eBOSS DR16 at $z=1.48$ Neveux  _et al._ (2020).

We call this data set RSD in short because most of these points are obtained
from the analysis of redshift-space distortions. Whenever this data set is
combined with the BAO data we take into account the correlations reported in
Gil-Marín  _et al._ (2017); Neveux  _et al._ (2020).

###  IV.2 Combined data sets

We list here the data set combinations that we have used in this work. We have
built these combinations from the individual data sets of Sec. IV.1. The
corresponding fitting results are presented and discussed in Sec. V.

In view of the significance of the $H_{0}$ tension, we deem it is of utmost
importance to study and compare the constraints on our EDE models obtained:
(i) with a minimal data set composed by CMBpol+SNIa; and (ii) adding on top of
the latter the SH0ES prior on the absolute magnitude of SNIa, i.e. using
CMBpol+SNIa+$M$. The SNIa data help to break the strong degeneracies found in
the $w$-$H_{0}$ plane when only CMB data are used in the analysis (see e.g.
Alestas  _et al._ (2020)), and this is why we add the Pantheon compilation to
the Planck 2018 data in these minimal data configurations. Their comparison
allows us to quantify the impact of the SH0ES prior on the fitting results and
on the ability of the models to loosen the $H_{0}$/$M$ tension.

The properties and limitations of the EDEp and EDEpMD parametrizations are
already grasped with the aforementioned minimal data sets (cf. Table 2). For
EDEpMD,thr we study also the effect of the CMB lensing and BAO+RSD data when
combined with CMBpol+SNIa. We provide the corresponding constraints in figures
2 and 3. For the analyses of the binned $\rho_{de}(z)$ described in Sec. II.2
we report our results in Table 3, where we explicitly test the impact of BAO
and the weak lensing data, by considering not only the minimal data sets
described in the previous paragraph, but also adding the information on BAO
and BAO+WL. In addition, we redo the fitting analyses considering also the CMB
lensing in order to quantify its impact. Tables 4 and 5 complement the
discussion of our tomographic study.

##  V Results and discussion

We present and discuss now the results obtained from the fitting analyses of
the EDE parametrizations and the tomographic EDE described in Secs. II.1 and
II.2, respectively, using the data sets listed in the previous section.

###  V.1 Results for the parametric analysis

The mean fit values and corresponding uncertainties for the various
cosmological parameters in the EDEp and EDEpMD parametrizations obtained with
the baseline CMBpol+SNIa dataset are reported in Table 2. The constraints on
the fraction of early dark energy in the EDEp parametrization in the
radiation- and matter-dominated epochs, $\Omega^{\rm RD}_{ede}$ and
$\Omega^{\rm MD}_{ede}$, are very strong. They lie below $\sim 2\%$ and $\sim
0.7\%$, respectively, at the $2\sigma$ c.l. It is interesting to observe that
the upper limit of $\Omega^{\rm MD}_{ede}$ in the EDEpMD parametrization
coincides with the one obtained in the more general EDEp. The constraints on
$\Omega^{\rm MD}_{ede}$ and $\Omega^{\rm RD}_{ede}$ in EDEp are quite
independent. Actually, we have checked that the correlation coefficient
between these two parameters is pretty small, $\sim 5.6\%$. As already noticed
in Pettorino  _et al._ (2013), the low upper limits on $\Omega^{\rm RD}_{ede}$
and $\Omega^{\rm MD}_{ede}$ are due to the very tight constraint on the
fraction of EDE around the CMB decoupling time. The latter acts as an anchor
for $\Omega^{\rm RD}_{ede}$ and even more for $\Omega^{\rm MD}_{ede}$, since
in the last scattering surface the matter energy density is already $\sim 3$
times larger than the radiation one.

Another result from Table 2 to remark is that EDEp cannot alleviate
significantly the $H_{0}$ and $\sigma_{8}$ tensions. The shape of the early
dark energy density seems to be too restricted in these parametrization. There
is a slight increase of $H_{0}$ in EDEp and EDEpMD with respect to the
$\Lambda$CDM, but it is mainly due to the dynamics of the late-time DE, and
this is why the major part of the effect is already found with the $w$CDM
parametrization Turner and White (1997).

Figure 2: Constraints at $2\sigma$ c.l. on the fraction of EDE in the MDE as a
function of its sound speed $\hat{c}_{s}$ and under the CMBpol+SNIa data set.
We show these constraints through the ratio $\Omega^{\rm MD}_{ede}/\Omega^{\rm
MD}_{cdm}$, with $\Omega^{\rm MD}_{cdm}$ the fraction of CDM during the MD,
given by
$\Omega^{MD}_{cdm}=\omega_{cdm}/[(1+\chi_{2})(\omega_{cdm}+\omega_{b})]$ up to
a very small correction due to massive neutrinos. We have employed here the
EDEpMD parametrization (6) with $w=-1$. When $\hat{c}_{s}\to 0$ we find that
the constraints on $\Omega^{\rm MD}_{ede}$ loosen and approach to those
obtained for cold dark matter, as expected, since in this limit the DE behaves
exactly as CDM during the MDE and there exists an obvious degeneracy between
these two parameters.

When we include the SH0ES prior in our fitting analysis we increase, of
course, the value of the Hubble parameter, see Table 2. The tension with the
distance ladder determination is now only of $\sim 1.7\sigma$, but this is
again mainly thanks to the lowering of $w$222The upper bound of $\Omega^{\rm
RD}_{ede}$ in the EDEp parametrization increases by $0.36\%$ with respect to
when we do not include the SH0ES prior, whereas the value of $\Omega^{\rm
MD}_{ede}$ decreases by $0.17\%$ (cf. Table 2). The increase of $\Omega^{\rm
RD}_{ede}$ helps to alleviate the $H_{0}$ tension, although this effect is
only marginal. It forces the decrease of $\Omega^{\rm MD}_{ede}$ to fulfill
the strong constraint on the fraction of EDE around the CMB decoupling time.,
which now lies more in the phantom region ($3\sigma$ away below $w=-1$)
Banerjee  _et al._ (2021). The values of $w$ and $H_{0}$ are almost identical
to those found in the $w$CDM parametrization, and EDE does not have any
important impact on the $H_{0}$ tension in the context of the EDEp
parametrization. The loosening of the $H_{0}$ tension is accompanied by a
slight worsening of the $\sigma_{8}$ one due to the positive correlation
between $H_{0}$ and $\sigma_{8}$. Phantom dark energy leads to lower values of
the DE density in the past and this produces, in turn, an increase of the
structure formation processes in the Universe. It seems that the EDE density
has a too restricted form in the parameterizations under study here. They
allow for plateaux and generalize previous studies, but still seem to be quite
constrained and unable to resolve the tensions. In the next section we then
investigate the binned tomographic approach described in Sec. II.2. Before
moving on, however, we discuss briefly two options to weaken the constraints
on the fraction of EDE.

Figure 3: Constraints on $\Omega^{\rm MD}_{ede}$ at $95\%$ c.l. obtained from
the fitting analyses of the EDEpMD,thr parametrization (7), and under the data
sets shown in the legend. Figure 4: Constraints for $H_{0}$, $\sigma_{8}$ and
$w$ at $1\sigma$ and $2\sigma$ c.l. obtained with the EDEpMD,thr
parametrization (7) for different values of the threshold redshift $z_{thr}$
and using the same combined data sets as in Fig. 3.

Much weaker constraints on the fraction of EDE can be obtained by allowing for
values of $\hat{c}^{2}_{s}\ll 1$. The only difference between our EDE and cold
dark matter during the MDE is found at the perturbations level. Both are
pressureless fluids at the background level, but have different $\hat{c}_{s}$.
The latter is equal to 1 for EDE and 0 for the dark matter. Hence, we expect
the constraints on $\Omega^{\rm MD}_{ede}$ to loosen if we decrease the value
of $\hat{c}_{s}$ of EDE, and even obtain a full degeneracy between the
fraction of EDE and CDM during the MDE in the limit $\hat{c}_{s}\to 0$. This
is actually what happens, as we explicitly show in Fig. 2. We have to say,
though, that the change in the sound speed does not help to alleviate the
cosmological tensions, neither. For instance, under the CMBPol+SNIa data set
the Hubble parameter remains close to $\sim 67.5$ km/s/Mpc, and the $1\sigma$
uncertainty is lower than $\sim 0.8$ km/s/Mpc regardless of the value of
$\hat{c}_{s}$ under consideration when the late-time DE dynamics is switched
off.

One can also get weaker constraints on EDE in the MDE by activating EDE at
lower redshifts. In order to study this effect we explore the EDEpMD,thr
parametrization (7). In Fig. 3 we provide the $2\sigma$ c.l. bounds on
$\Omega^{\rm MD}_{ede}$ obtained with the baseline data set CMBpol+SNIa with
and without late-time dark energy dynamics, and also adding the CMB lensing
and the BAO+RSD data sets. When $z_{\rm thr}\to\infty$ we recover the
constraints obtained in the EDEpMD model, of course, but when we allow for
lower values of the threshold redshift (below the CMB decoupling one) we get
larger upper bounds on $\Omega^{\rm MD}_{ede}$, which depend on the concrete
data set under consideration and also on $z_{\rm thr}$. We report the results
obtained with $z_{\rm thr}=10,50,200,500,\infty$. The addition of the Planck
2018 CMB lensing to the CMBpol+SNIa baseline data set leads to stronger
constraints on $\Omega^{\rm MD}_{ede}$. Its value decreases by a $\sim 25\%$
$\forall{z_{\rm thr}}$. When we also include the BAO+RSD data the decrease is
even bigger, $\sim 75\%$. If we turn off the late-time dynamics of DE we also
obtain tighter bounds on $\Omega^{\rm MD}_{ede}$, just because in this case we
remove the degeneracy between this parameter and $w$. Our Fig. 3 can be
compared with Figs. 6-7 of Pettorino  _et al._ (2013) and Fig. 11 of Ade  _et
al._ (2016), which were obviously obtained with older data sets. In Pettorino
_et al._ (2013) the authors employed CMB data from WMAP9 combined with small
scale measurements from the South Pole Telescope (SPT), whereas in Ade  _et
al._ (2016) the authors employed the Planck 2015 CMB likelihood and studied
the impact of some other background and weak lensing data sets, as described
in Sec. 4 in that reference. The results presented in this section constitute
a significant update obtained with the Planck 2018 likelihood and also other
more recent background and LSS data (cf. Sec. IV for details).

Parameter | CMBpol+SNIa | CMBpol+SNIa+$M$ | CMBpol+SNIa+$M$+BAO | CMBpol+SNIa+$M$+BAO+WL
---|---|---|---|---
$\omega_{b}$ | $0.02257^{+0.00022}_{-0.00023}$ | $0.02277^{+0.00023}_{-0.00025}$ | $0.02282^{+0.00024}_{-0.00025}$ | $0.02259\pm 0.00021$
$\omega_{cdm}$ | $0.1222^{+0.0020}_{-0.0021}$ | $0.1221\pm 0.0022$ | $0.1223^{+0.0020}_{-0.0021}$ | $0.1200^{+0.0013}_{-0.0014}$
$\tau$ | $0.055\pm 0.009$ | $0.056\pm 0.009$ | $0.057^{+0.008}_{-0.009}$ | $0.053\pm 0.008$
$n_{s}$ | $0.9727^{+0.0072}_{-0.0073}$ | $0.9752^{+0.0067}_{-0.0069}$ | $0.9760^{+0.0070}_{-0.0071}$ | $0.9740^{+0.0061}_{-0.0067}$
$H_{0}$ [km/s/Mpc] | $68.29^{+1.26}_{-1.36}$ | $70.86^{+1.00}_{-1.10}$ | $70.38^{+0.84}_{-0.89}$ | $69.85^{+0.76}_{-0.77}$
$M$ | $-19.405\pm 0.032$ | $-19.342^{+0.023}_{-0.024}$ | $-19.350^{+0.019}_{-0.020}$ | $-19.365^{+0.018}_{-0.017}$
$\sigma_{8}$ | $0.854^{+0.023}_{-0.025}$ | $0.880^{+0.025}_{-0.029}$ | $0.877^{+0.026}_{-0.029}$ | $0.833^{+0.016}_{-0.017}$
$S_{8}$ | $0.869^{+0.025}_{-0.029}$ | $0.863\pm 0.029$ | $0.866^{+0.026}_{-0.028}$ | $0.819^{+0.014}_{-0.016}$
$\sigma_{12}$ | $0.840^{+0.019}_{-0.022}$ | $0.843\pm 0.022$ | $0.843^{+0.020}_{-0.022}$ | $0.806^{+0.012}_{-0.013}$
$S_{12}$ | $0.851^{+0.022}_{-0.025}$ | $0.854^{+0.024}_{-0.027}$ | $0.855^{+0.023}_{-0.026}$ | $0.810^{+0.014}_{-0.016}$
$r_{d}$ [Mpc] | $145.66^{+0.90}_{-0.70}$ | $145.22^{+0.95}_{-0.90}$ | $145.06^{+0.97}_{-0.89}$ | $146.51^{+0.64}_{-0.51}$
$w$ | $-1.037^{+0.043}_{-0.041}$ | $-1.070\pm 0.038$ | $-1.048^{+0.037}_{-0.034}$ | $-1.037^{+0.032}_{-0.031}$
Parameter | CMBpolens+SNIa | CMBpolens+SNIa+$M$ | CMBpolens+SNIa+$M$+BAO | CMBpolens+SNIa+$M$+BAO+WL
$\omega_{b}$ | $0.02257^{+0.00021}_{-0.00023}$ | $0.02274^{+0.00023}_{-0.00025}$ | $0.02277^{+0.00022}_{-0.00024}$ | $0.02266^{+0.00021}_{-0.00020}$
$\omega_{cdm}$ | $0.1215\pm 0.0016$ | $0.1211\pm 0.0017$ | $0.1214\pm 0.0016$ | $0.1193^{+0.0013}_{-0.0014}$
$\tau$ | $0.054^{+0.008}_{-0.009}$ | $0.056^{+0.008}_{-0.009}$ | $0.056\pm 0.008$ | $0.055^{+0.008}_{-0.007}$
$n_{s}$ | $0.9721^{+0.0063}_{-0.0071}$ | $0.9748^{+0.0067}_{-0.0071}$ | $0.9746^{+0.0068}_{-0.0069}$ | $0.9727^{+0.0061}_{-0.0068}$
$H_{0}$ [km/s/Mpc] | $68.25^{+1.27}_{-1.26}$ | $70.84^{+1.04}_{-1.07}$ | $70.21^{+0.80}_{-0.84}$ | $70.00^{+0.76}_{-0.73}$
$M$ | $-19.407^{+0.031}_{-0.030}$ | $-19.343^{+0.024}_{-0.025}$ | $-19.355\pm 0.018$ | $-19.362\pm 0.016$
$\sigma_{8}$ | $0.845^{+0.017}_{-0.020}$ | $0.868^{+0.019}_{-0.021}$ | $0.864^{+0.019}_{-0.021}$ | $0.839^{+0.013}_{-0.015}$
$S_{8}$ | $0.858^{+0.019}_{-0.021}$ | $0.848^{+0.020}_{-0.022}$ | $0.853^{+0.019}_{-0.020}$ | $0.824^{+0.012}_{-0.013}$
$\sigma_{12}$ | $0.831^{+0.014}_{-0.015}$ | $0.831^{+0.015}_{-0.016}$ | $0.833^{+0.015}_{-0.016}$ | $0.810^{+0.010}_{-0.011}$
$S_{12}$ | $0.841^{+0.016}_{-0.017}$ | $0.840^{+0.017}_{-0.020}$ | $0.843\pm 0.018$ | $0.815^{+0.011}_{-0.013}$
$r_{d}$ [Mpc] | $145.94^{+0.69}_{-0.57}$ | $145.65^{+0.82}_{-0.71}$ | $145.51^{+0.83}_{-0.71}$ | $146.46^{+0.56}_{-0.44}$
$w$ | $-1.035^{+0.040}_{-0.039}$ | $-1.067^{+0.037}_{-0.036}$ | $-1.050^{+0.034}_{-0.035}$ | $-1.045\pm 0.032$

Table 3: The mean fit values and $68.3\%$ confidence limits for the most
important cosmological (main+derived) parameters, obtained under different
data sets (with and without CMB lensing) for the binned $\rho_{de}(z)$
described in Sec. II.2. See Fig. 5 for the constraints on $\Omega_{de}(z)$,
and Sec. V.2 for a thorough discussion on these results.

In Fig. 4 we provide the corresponding constraints on $H_{0}$, $\sigma_{8}$
and $w$ at $1\sigma$ and $2\sigma$ c.l. in the EDEpMD,thr parameterization for
the same scenarios explored in Fig. 3. They support some of the comments made
in the previous paragraphs of this section, e.g. (i) the values of
$\sigma_{8}$ and $H_{0}$ remain close to those found in the $\Lambda$CDM
model. In other words, the tensions are not significantly alleviated in this
class of scaling early dark energy models; (ii) phantom values of $w<-1$ allow
us to decrease the $H_{0}$ tension very slightly; and (iii) larger values of
$w$ lead to lower values of $\sigma_{8}$ due to the presence of a larger
fraction of dark energy at low redshifts. This is why we get $w\sim-1$ when we
include the BAO+RSD data set.

The dedicated analysis presented in this section updated and generalized
previous constraints on this class of early dark energy models, and motivated
the study of the next section, in which we will reconstruct the shape of
$\Omega_{de}(z)$ without sticking to a restricted family of parametrizations.

###  V.2 Results for tomographic Dark Energy

Now we provide the results obtained in the tomographic model described in Sec.
II.2 in order to see whether more general shapes of $\Omega_{de}(z)$ can
loosen the cosmological tensions. This is in fact suggested by previous
analyses in the literature, see e.g. Poulin  _et al._ (2019); Agrawal  _et
al._ (2019); Smith  _et al._ (2020); Chudaykin  _et al._ (2020); Niedermann
and Sloth (2020); Gogoi  _et al._ (2021); Niedermann and Sloth (2021). Our
results are presented in Tables 3-5, and also in Fig. 5.

Table 3 and Fig. 5 confirm that a significant (and non-constant) fraction of
EDE in the RDE can alleviate the $H_{0}$ tension if it can be kept below $\sim
0.6\%$ at $2\sigma$ c.l. around the CMB decoupling time, i.e. at $z\sim
1000-2000$. For instance, from the analysis of the CMBpolens+SNIa and
CMBpolens+SNIa+$M$+BAO data sets we obtain $H_{0}=(68.25^{+1.27}_{-1.26})$
km/s/Mpc and $H_{0}=(70.21^{+0.80}_{-0.84})$ km/s/Mpc, respectively. They are
$2.77\sigma$ and $1.95\sigma$ below the SH0ES measurement (8), and the central
value is significantly lower when the SH0ES prior is not considered. Hill  _et
al._ (2020) reported similar results in the context of the ultra-light axion
model. Nevertheless, if we compare the values of $M$ obtained from these data
sets with the SH0ES value (9) we still obtain a tension of $3.80\sigma$ and
$3.07\sigma$, respectively. This means that in terms of $M$, the tension is
bigger than when it is formulated in terms of $H_{0}$, and the capability of
EDE of alleviating the tension is much lower, at least in the EDE framework we
are considering here.

Parameter | CMBpol+SNIa | CMBpol+SNIa+$M$ | CMBpol+SNIa+$M$+BAO | CMBpol+SNIa+$M$+BAO+WL
---|---|---|---|---
$H_{0}$ [km/s/Mpc] | $2.66\sigma$ | $1.40\sigma$ | $1.81\sigma$ | $2.22\sigma$
$M$ | $3.60\sigma$ | $2.62\sigma$ | $2.91\sigma$ | $3.31\sigma$
$\sigma_{8}$ | $2.89\sigma$ | $3.45\sigma$ | $3.32\sigma$ | $2.65\sigma$
$S_{8}$ | $2.93\sigma$ | $2.66\sigma$ | $2.85\sigma$ | $1.98\sigma$
$\sigma_{12}$ | $2.66\sigma$ | $2.67\sigma$ | $2.73\sigma$ | $1.82\sigma$
Parameter | CMBpolens+SNIa | CMBpolens+SNIa+$M$ | CMBpolens+SNIa+$M$+BAO | CMBpolens+SNIa+$M$+BAO+WL
$H_{0}$ [km/s/Mpc] | $2.73\sigma$ | $1.41\sigma$ | $1.95\sigma$ | $2.13\sigma$
$M$ | $3.71\sigma$ | $2.62\sigma$ | $3.07\sigma$ | $3.28\sigma$
$\sigma_{8}$ | $2.96\sigma$ | $3.63\sigma$ | $3.50\sigma$ | $3.03\sigma$
$S_{8}$ | $3.04\sigma$ | $2.67\sigma$ | $2.91\sigma$ | $2.25\sigma$
$\sigma_{12}$ | $2.69\sigma$ | $2.64\sigma$ | $2.71\sigma$ | $2.05\sigma$

Table 4: Statistical significance of the cosmological tensions for the
relevant parameters $H_{0}$, $M$, $\sigma_{8}$, $S_{8}$ and $\sigma_{12}$,
obtained under different data sets (with and without CMB lensing) for the
tomographic DE described in Sec. II.2. The tensions are quantified by using
the fitting values reported in Table 3 and the following measurements: the
Hubble parameter (8) and the absolute magnitude of SNIa (9) by SH0ES; the
value of $\sigma_{8}$ = 0.760 $\pm$ 0.022 provided in Heymans  _et al._
(2021), which is obtained from the cosmic shear analysis by KiDS-1000 and the
galaxy clustering data from BOSS. We use this value also to quantify the
tension in $\sigma_{12}$, since $\sigma_{12}\simeq\sigma_{8}$ for the
$\Lambda$CDM due to the low values of $H_{0}$ encountered in this model
Sanchez (2020); and the WL measurement of $S_{8}$ (11).

We also see that the large fraction of $\Omega_{de}(z)$ required in the RDE to
loosen the $H_{0}$ tension, which can be of about $\sim 4-5\%$ at $2\sigma$
c.l. according to some data sets that include the SH0ES prior on $M$ but no
LSS information, leads to higher values of $\omega_{cdm}$ and, to a lesser
extent, also of $n_{s}$, which in turn exacerbates the $\sigma_{8}$/$S_{8}$
tension. The former is needed to reduce the early integrated Sachs-Wolfe
effect introduced by the EDE. This is aligned with previous works that also
consider EDE in the pre-recombination epoch, see e.g. Hill  _et al._ (2020);
Ivanov  _et al._ (2020); D’Amico  _et al._ (2021); Murgia  _et al._ (2021);
Smith  _et al._ (2021). Tables 3 and 4 show that the tension decreases $\sim
1\sigma$ when it is analyzed through the LSS estimators $\sigma_{12}$ and
$S_{12}$ Sanchez (2020). Nevertheless, it does not disappear (see also Table
5). Under the CMBpolens+SNIa+$M$ data set the $\sigma_{8}$ tension is of
$3.63\sigma$, whereas for the $\sigma_{12}$ parameter it is of $2.64\sigma$,
and a similar decrease is observed when other data combinations are employed
in the fitting analysis, see again Table 4.

When the SH0ES prior on $M$ (9) is taken into account in absence of LSS data,
we get a $\sim 2\sigma$ evidence for the presence of a non-null EDE fraction
during the RDE, as in Poulin  _et al._ (2019)333There are some differences,
though, between the ultra-light axion-like field model of Poulin  _et al._
(2018, 2019) and our reconstructed EDE. For instance, in the former the EDE
sound speed is not constant and equal to one, but has a more complicated
dependence on the scale factor and the wavemode $k$ Poulin  _et al._ (2018).
Moreover, in the model of Poulin  _et al._ (2018, 2019) $\Omega_{de}(z)$ has a
sizeable peak at $z\sim 3000-5000$, but the EDE fraction is negligible at much
higher redshifts, whereas in our binned analysis the possibility of having a
non-negligible fraction of EDE deep in the RDE is not excluded by the data.
This is an interesting feature of EDE that obviously cannot be grasped with
the model studied in Poulin  _et al._ (2019, 2018), just because in that case
$\rho_{de}(z)\approx const.$ by construction during radiation-domination, when
the energy density of the scalar field is basically given by a flat (and very
large) potential and EDE behaves as a cosmological constant. Thus, for large
enough redshifts the EDE density in that model is always overcome by the one
of radiation.. Nevertheless, the inclusion of the weak lensing data from
KiDS+VIKING-450 and DES-Y1 (cf. Sec. IV.1.5) forces the EDE fraction in the
RDE to be again compatible at $1\sigma$ with $0$ in order to allow
$\omega_{cdm}$ to take values closer to the $\Lambda$CDM ones and not to
worsen the $\sigma_{8}/S_{8}$ tension. Notice that, as expected, the upper
bound on the fraction of EDE in the MDE that we obtain in our binned analysis
is larger than the one found with EDEp and its variants, even when the prior
on $S_{8}$ is not included (see Table 2 and Fig. 5). Indeed, higher fractions
of EDE in the MDE (see again Fig. 5) also allow to keep the amount of LSS more
under control. With the CMBpolens+SNIa+$M$+BAO+WL we obtain
$H_{0}=(70.00^{+0.76}_{-0.73})$ km/s/Mpc and $M=-19.362\pm 0.016$. They are in
$2.13\sigma$ and $3.28\sigma$ tension with the SH0ES values, respectively.
Again, the tension in $M$ is larger than in $H_{0}$ by $\sim 1\sigma$.
Regarding the LSS estimators, we obtain $S_{8}=0.824^{+0.012}_{-0.013}$ and
$\sigma_{12}=0.810^{+0.010}_{-0.011}$. The former is in $2.25\sigma$ tension
with (11), whereas the latter is compatible at $1\sigma$ with the value
obtained in the $\Lambda$CDM. The tensions in $H_{0}$ and $S_{8}$ can be kept
at $\sim 2\sigma$ c.l. under this concrete data set, as advocated by Murgia
_et al._ (2021).

Figure 5: Left plots: Reconstructed shapes of $\Omega_{de}(z)$ obtained from
the fitting analyses without the Planck 2018 CMB lensing data (cf. Sec.
IV.1.1). In the first and second rows we show the constraints in the region
$z\in[0,2\cdot 10^{4}]$ at $1\sigma$ and $2\sigma$ c.l., respectively. In the
last row we zoom in the redshift range $z\in[0,3000]$ to better appreciate the
details in the MDE. The tightest upper bound on $\Omega_{de}(z)$ is obtained
around the CMB decoupling time, i.e. at $z\sim 1000-2000$, where the data
force $\Omega_{de}(z)\lesssim 0.6\%$ at $2\sigma$ c.l.; Right plots: The same,
but including the CMB lensing data. See the comments in Sec. V.2.

The effect of the CMB lensing from Planck 2018 on the EDE fraction can be
appreciated by direct comparison of the plots in the left and right columns of
Fig. 5. When the CMB lensing is included, the upper bound on $\Omega_{de}(z)$
is reduced by $\sim 1\%$ for $z\gtrsim 5000$, by a $\sim 0.5\%$ for
$3000\lesssim z\lesssim 5000$ and by a smaller fraction at lower redshifts.
Some differences are found, though, depending on the other data sets employed
in the fitting analyses. The preferred matter densities decrease, although
they are still compatible at $1\sigma$ with the ones inferred without
including the CMB lensing. This leads also to a slight decrease on the LSS
estimators when the weak lensing prior is not considered. When the latter is
included, the values of $\sigma_{12}$ and $S_{12}$ (and also $\sigma_{8}$ and
$S_{8}$) remain stable under the addition of the CMB lensing likelihood.

We would like to remark the very low upper bounds that we obtain for the EDE
fraction in the redshift range $z\in(1000,2000)$, cf. Fig. 5. The exact value
for these upper bounds depend, of course, on the specific data set, e.g. the
constraints are a little bit weaker when the WL prior (11) is used in the
fitting analysis, but the EDE fraction when $\hat{c}_{s}=1$ is, in any case,
very strongly constrained in that epoch and lies always below $0.6-0.7\%$
regardless of the data set under consideration. Under the full data set, the
EDE fractions are constrained to be below $2.6\%$ in the radiation-dominated
epoch and $\lesssim 1-1.5\%$ in the redshift range $z\in(100,1000)$ at
$2\sigma$ c.l. This limits strongly the possible impact of EDE on the value of
the Hubble parameter or the present amount of structure.

##  VI Conclusions

In this work we have studied the phenomenological performance of a family of
flexible parametrizations for the dark energy density that are able to mimic
the scaling behavior that is encountered in a wide variety of quintessence
models. DE has been treated as a perfect fluid with two plateaus in
$\Omega_{de}(z)$, one in the matter-dominated epoch and another one in the
radiation-dominated era. We have put very tight constraints on the fraction of
EDE in the radiation- and matter-dominated epochs in the context of these
models. The CMBpol+SNIa data set already forces these two quantities to lie
below $2.44\%$ and $0.52\%$ at $2\sigma$ c.l., respectively. These strong
constraints are necessary to respect the upper bound on the amount of EDE at
the last scattering surface, as we have explicitly checked in our tomographic
analysis. We have found that this class of scaling EDE models does not lead to
a significant alleviation of the $H_{0}$ and $\sigma_{8}$ tensions. Larger EDE
fractions are allowed by the data if: (i) the sound speed for DE is fixed at
values $\hat{c}_{s}^{2}\ll 1$, since in this limit dark energy behaves as dark
matter during the matter-dominated epoch at the background and perturbations
levels and, hence, there is a huge degeneracy between these two components. If
we would assume a vanishing sound speed for dark energy there would be no way
to distinguish it from dark matter. We find indeed no bounds on the dark
energy fraction in this case. This changes drastically already for a rather
small sound speed of EDE of the order $10^{-4}$; and (ii) if EDE is switched
on at later times, already in the matter-dominated era. Nevertheless, the
cosmological tensions remain also in these cases.

Parameter |  $\Lambda$CDM |  $w$CDM | Binned EDE
---|---|---|---
$\omega_{b}$ | $0.02255\pm 0.00013$ | $0.02246^{+0.00014}_{-0.00013}$ | $0.02282^{+0.00024}_{-0.00025}$
$\omega_{cdm}$ | $0.1183^{+0.0009}_{-0.0008}$ | $0.1197\pm 0.0010$ | $0.1223^{+0.0020}_{-0.0021}$
$\tau$ | $0.057^{+0.008}_{-0.009}$ | $0.055^{+0.007}_{-0.008}$ | $0.057^{+0.008}_{-0.009}$
$n_{s}$ | $0.9699^{+0.0034}_{-0.0038}$ | $0.9665^{+0.0038}_{-0.0041}$ | $0.9760^{+0.0070}_{-0.0071}$
$H_{0}$ [km/s/Mpc] | $68.49^{+0.35}_{-0.40}$ | $68.78^{+0.66}_{-0.78}$ | $70.38^{+0.84}_{-0.89}$
$M$ | $-19.394^{+0.009}_{-0.011}$ | $-19.371^{+0.013}_{-0.017}$ | $-19.350^{+0.019}_{-0.020}$
$\sigma_{8}$ | $0.807\pm{0.008}$ | $0.828^{+0.012}_{-0.013}$ | $0.877^{+0.026}_{-0.029}$
$S_{8}$ | $0.807^{+0.012}_{-0.010}$ | $0.817\pm 0.012$ | $0.866^{+0.026}_{-0.028}$
$\sigma_{12}$ | $0.792\pm 0.009$ | $0.802^{+0.012}_{-0.011}$ | $0.843^{+0.020}_{-0.022}$
$S_{12}$ | $0.794^{+0.011}_{-0.009}$ | $0.809^{+0.012}_{-0.015}$ | $0.855^{+0.023}_{-0.026}$
$r_{d}$ [Mpc] | $147.26^{+0.20}_{-0.23}$ | $147.02^{+0.26}_{-0.24}$ | $145.06^{+0.97}_{-0.89}$
$w$ | $-1$ | $-1.063^{+0.033}_{-0.028}$ | $-1.048^{+0.037}_{-0.034}$
Table 5: Comparison of the fitting results obtained for the $\Lambda$CDM,
$w$CDM and the binned $\rho_{de}(z)$, using the CMBpol+SNIa+$M$+BAO data set.

EDE can only have a larger impact on the cosmological tensions if
$\Omega_{de}(z)$ takes more flexible shapes that allow to respect the very
strict constraints found around the CMB decoupling time
($\Omega_{de}(z_{dec})\lesssim 0.4\%$ at $2\sigma$ c.l.), while still leading
to a significant EDE fraction in other epochs of the cosmic expansion. The
strong bound on EDE for redshift $200<z<1000$ ($\Omega_{de}(z)\lesssim 1\%$ at
$2\sigma$ c.l.) is rather impressive since it is entirely based on the
different clustering properties of dark energy and dark matter.

In general, when the SH0ES prior is not included in the fitting analysis there
is no significant shift in the value of the Hubble parameter when compared to
the $\Lambda$CDM, although the uncertainties clearly grow by a factor $2-3$.
This allows to decrease the $H_{0}$ tension to the $2.66\sigma$ c.l. under the
minimal CMBpol+SNIa data set. The addition of the CMB lensing has a very mild
effect, increasing the tension up to the $2.73\sigma$ level.

When the BAO data and the SH0ES prior are also considered, the $H_{0}$ tension
is reduced to the $\sim 2\sigma$ level by increasing the EDE fraction at the
radiation-dominated epoch. A $2\sigma$ preference for a non-null EDE density
in that epoch is obtained, $\Omega_{de}(z)\lesssim 4\%$ at $95\%$ c.l. Our
reconstructed $\Omega_{de}(z)$ allows for non-peaked shapes, in contrast to
what one finds e.g. in models based on ultra-light axions Poulin  _et al._
(2018, 2019). The model needs, though, values of the current dark matter
density much larger than the ones typically encountered in the concordance
model. This is to lower the early integrated Sachs-Wolfe effect down, which is
enhanced by the presence of EDE in the pre-recombination epoch. This
automatically leads to an increase of $\sigma_{8}$ and $S_{8}$ that worsens
the LSS tension, which lies now at the $3-3.5\sigma$ c.l. In terms of
$\sigma_{12}$, the tension is somewhat lower, but still remains at the $\sim
2.7\sigma$ level, as when the $M$ prior is not used in the analysis. It is
also to be noted that the $H_{0}$ tension still stays at the $3\sigma$ level
when formulated in terms of the absolute magnitude of SNIa $M$.

Finally, when we use the most complete data set, taking also the weak lensing
data into account, we find that the model can keep the $H_{0}$ and $S_{8}$
tensions at the $\sim 2\sigma$ c.l., thanks also to a slight increase of the
EDE fraction in the matter-dominated epoch, although, again, the $M$ tension
remains at $\gtrsim 3\sigma$.

In view of our results, it seems unlikely that EDE alone can provide a
satisfactory resolution of the cosmological tensions. Whether the latter have
or not a physical origin, or whether their statistical significance is as high
as claimed by some sectors of the cosmological community, is still a matter of
discussion and is certainly not a closed subject. Here we conclude that, in
any case, if the data sets employed in this study do not suffer from any
important systematic errors, uncoupled EDE is not able to relieve completely
the tensions. Under our full data set CMBpolens+SNIa+$M$+BAO+WL they remain at
$2-3\sigma$ c.l.

Nothing prevents, though, the solution to the cosmological tensions to be
multi-sided, rather than due to a single new physics component. EDE could
still play a significant role in this story. Some interesting directions to
explore in the future are: (i) a possible coupling of EDE to dark matter, see
e.g. Karwal  _et al._ (2021); (ii) the impact of more complicated behavior of
the EDE sound speed, parameterizing its time dependence, or performing a
tomographic analysis similar to the one we have carried out in this work for
the EDE density; and (iii) the potential degeneracy between EDE and other
cosmological parameters, as the neutrino masses, which could in principle help
to soften the $S_{8}$ tension while keeping the needed amount of non-
relativistic matter at the CMB decoupling time. The latter would be needed,
together with a higher EDE fraction in the radiation-dominated epoch, in order
to increase the value of the Hubble parameter. We leave these investigations
for a future work.

Acknowledgements

AGV is funded by the Deutsche Forschungsgemeinschaft (DFG) - Project number
415335479.

##  Appendix A EDEp and quintessence

The EDEp parametrization described in Sec. II.1 and analyzed in Sec. V.1 is
based on the formula (1) for the DE density and the covariant self-
conservation of the dark energy component, which leads to the expression of
its pressure, (5). We have treated the DE as a perfect fluid. However, for all
the epochs of the cosmic expansion in which the effective EoS parameter of the
dark energy fluid is larger than -1 it is also possible to formulate the model
in terms of a quintessence scalar field $\phi$. In particular, during the RDE
and MDE. Let us see how. The energy density and pressure associated to $\phi$
read, respectively,

$\rho_{\phi}=\frac{\dot{\phi}^{2}}{2}+V\qquad;\qquad
p_{\phi}=\frac{\dot{\phi}^{2}}{2}-V\,.$ (12)

Hence, we can write $\dot{\phi}$ and the potential as follows,

$\dot{\phi}=\sqrt{\rho_{\phi}+p_{\phi}}\longrightarrow
d\phi=\frac{da}{a}\sqrt{\frac{3}{8\pi
G}}\sqrt{\frac{\rho_{\phi}(a)+p_{\phi}(a)}{\rho_{m}(a)+\rho_{r}(a)+\rho_{\phi}(a)}}\,,$
(13) $V(a)=\frac{1}{2}\left[\rho_{\phi}(a)-p_{\phi}(a)\right]\,,$ (14)

identifying the DE perfect fluid density and pressure with the scalar field
ones, i.e. doing $\rho_{\phi}=\rho_{de}$ and $p_{\phi}=p_{de}$. We can plug
now (1) and (5) into (13) and (14), find $\phi(a)$ from the former and invert
it to get $a(\phi)$. This result can be then introduced in (14) to finally
obtain the shape of the potential needed to reproduce the phenomenology of our
EDEp parametrization. For the RDE and MDE the following formula applies,

$\phi(a)=\phi(a_{ini})+\frac{2}{\kappa}\sqrt{\frac{\chi_{1}}{1+\chi_{1}}}\int_{a_{ini}}^{a}\frac{d\bar{a}}{\bar{a}}\left[\frac{1+\frac{3\chi_{2}}{4\chi_{1}}\frac{\bar{a}}{a_{eq}}}{1+\left(\frac{1+\chi_{2}}{1+\chi_{1}}\right)\frac{\bar{a}}{a_{eq}}}\right]^{1/2}\,,$
(15)

with $\kappa\equiv\sqrt{8\pi G}$, $a_{eq}=a(t_{eq})$ the scale factor
associated to the matter-radiation equality time, and $a_{ini}$ the scale
factor at some moment deep in the RDE. $\phi_{ini}$ can be safely set to zero,
since the quantity that is physically relevant here is $\phi-\phi_{ini}$.
Unfortunately, the integral appearing in (15) cannot be solved analytically,
so we cannot compute the analytical formula of $V(\phi)$. Nevertheless, it is
sufficient for our purposes to show that deep enough in the RDE and MDE the
potential takes a simple exponential form.

For instance, if $\chi_{1}\neq 0$ and for sufficiently early times, in the RDE
we obtain,

$\phi(a)=\frac{2}{\kappa}\sqrt{\frac{\chi_{1}}{1+\chi_{1}}}\ln(a/a_{ini})\,,$
(16)

The potential can be written as follows,

$V_{\rm
RD}(\phi)=\frac{\chi_{1}}{3}\rho_{r}(a_{ini})\exp{\left[-2\kappa\phi\sqrt{1+\frac{1}{\chi_{1}}}\right]}\,,$
(17)

whereas in the MDE, for values of the scale factor
$a\gg\chi_{1}a_{eq}/\chi_{2}$, we find

$V_{\rm
MD}(\phi)=\frac{\chi_{2}}{2}\rho_{m}(a_{*})\exp{\left[-\kappa\sqrt{3\left(1+\frac{1}{\chi_{2}}\right)}(\phi-\phi_{*})\right]}$
(18)

with $a_{*}$ the scale factor associated to some time in the MDE. Now we can
use the scaling formula $\Omega_{\phi}=\frac{3}{\lambda^{2}}(1+\bar{w})$
Copeland  _et al._ (1998) that applies for quintessence with an exponential
potential $V=V_{0}e^{-\sqrt{8\pi G}\lambda\phi}$, where $\bar{w}$ is the EoS
of the dominant component in the Universe. From (17) and (18) we obtain during
the RDE $\lambda^{2}_{\rm RD}=4(1+\chi_{1})/\chi_{1}=4/\Omega^{\rm RD}_{ede}$,
and in the MDE $\lambda^{2}_{\rm MD}=3(1+\chi_{2})/\chi_{2}=3/\Omega^{\rm
MD}_{ede}$. This automatically leads to the formulas (3) for the EDE
fractions. This is sufficient to show that there exists a quintessence
potential able to produce the same plateaus in the RDE and MDE that we
encounter in the context of the simple EDEp parameterization (1)-(5). This
potential reduces to (17) and (18) in the epochs of interest. The number of
additional parameters that we have in this scalar field formulation coincides
with the one in the EDEp. Obviously, a full quintessence formulation of the
EDEp parametrization in the last stages of the cosmic expansion, during the
DE-dominated period, is only possible if $w>-1$ in (1).

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