# Can neutrino-assisted early dark energy models ameliorate the $H_{0}$
tension in a natural way?

Diogo H. F. de Souza<EMAIL_ADDRESS>Rogerio Rosenfeld
<EMAIL_ADDRESS>Instituto de Física Teórica da Universidade
Estadual Paulista and ICTP South American Institute for Fundamental Research,
R. Dr. Bento Teobaldo Ferraz, 271, Bloco II, Barra-Funda - São Paulo/SP,
Brasil.

###### Abstract

The idea of neutrino-assisted early dark energy ($\nu$EDE), where a coupling
between neutrinos and the scalar field that models early dark energy (EDE) is
considered, was introduced with the aim of reducing some of the fine-tuning
and coincidence problems that appear in usual EDE models. In order to be
relevant in ameliorating the $H_{0}$ tension, the contribution of EDE to the
total energy density ($f_{\text{EDE}}$) should be around 10% near the redshift
of matter-radiation equality. We verify under which conditions $\nu$EDE models
can fulfill these requirements for a model with a quartic self-coupling of the
EDE field and an exponential coupling to neutrinos. We find that in the
situation where the EDE field is frozen initially, the contribution to
$f_{\text{EDE}}$ can be significant but it is not sensitive to the neutrino-
EDE coupling and does not address the EDE coincidence problem. On the other
hand, if the EDE field starts already dynamical at the minimum of the
effective potential, it tracks this time-dependent minimum that presents a
feature triggered by the neutrino transition from relativistic to
nonrelativistic particles. This feature generates $f_{\text{EDE}}$ in a
natural way at around this transition epoch, that roughly coincides with the
matter-radiation equality redshift. For the set of parameters that we
considered we did not find values that satisfy the requirements on the
background cosmological evolution to mitigate the Hubble tension in a natural
way in this particular $\nu$EDE model.

##  I Introduction

Despite of its great success in being able to describe a vast amount of
observables from different epochs of the evolution of the Universe, the
standard spatially flat $\Lambda$CDM model has recently being under stress
mainly from the so-called Hubble tension arising from measurements of the
Hubble constant from the cosmic microwave background (CMB) data in comparison
to local measurements (for recent reviews, see [1, 2, 3]). One of the most
promising solutions to the Hubble tension is a class of models called Early
Dark Energy (EDE) [4, 5, 6, 7, 8, 9]. The current status of EDE models
proposed to resolve the Hubble tension was recently reviewed in [10].

A non-negligible amount of a new dark energy component that becomes dynamical
at around the matter-radiation equality era and quickly dissipates afterwards
would decrease the sound horizon that enters the indirect measurement of the
Hubble constant from the CMB data, leading to an increase in the estimated
value of the Hubble parameter today; hence EDE can ameliorate the Hubble
tension [11]. In fact, the existence of EDE has already been proposed in the
context of late quintessence models with global attractor solutions and its
effect on the CMB and structure formation were studied in [12] and [13]. In
the more recent versions of EDE motivated by the Hubble tension, a scalar
field with an axion-like or a pure $\phi^{4}$ potential was proposed to model
the EDE component [5, 14, 6]. It has been shown that if EDE contributes with a
fraction $f_{\text{EDE}}\approx 10\%$ of the total energy density of the
Universe in an epoch with redshift around $\log_{10}(z)\sim 3.5$, its presence
is expected to decrease the statistical significance of the Hubble tension
(see, eg, [7]).

However, in the standard EDE scenario, some level of fine-tuning is necessary
in order to solve the Hubble tension: the scalar field mass and its initial
conditions need to have specific values in order for the EDE model to
contribute the required amount of energy density at around the matter-
radiation equality epoch. This is sometimes referred to as the EDE coincidence
problem.

Some models were proposed to address the EDE coincidence problem by
introducing a coupling of the EDE field to matter fields. For instance, a
coupling of EDE to dark matter as a trigger for its dynamics was recently
studied in [15, 16].

Another similar mechanism was proposed in [17, 18], where a coupling of the
EDE field to neutrinos was introduced. There are two distinct consequences of
introducing such a coupling: (i) a modification in the original EDE field
potential leading to a new effective potential and (ii) a dynamical effect of
a “kick” to the EDE field. This kick is associated with the transition of
neutrinos from being relativistic to nonrelativistic particles when the
temperature of the universe falls below $T={\cal O}(m_{\nu})$, which causes a
jump in the trace of the neutrino energy-momentum tensor. This jump can
displace the value of the EDE field that then starts its oscillations and
dissipation. This mechanism can naturally trigger the EDE field at around the
matter-radiation equality since the temperature at this epoch is approximately
at the scale of neutrino masses.

In this paper, we explore this neutrino-assisted EDE model ($\nu$EDE) in more
detail. In order to do this, we developed an implementation of the $\nu$EDE
model in CAMB111https://github.com/cmbant/CAMB at the background level to
analyse the evolution of the EDE scalar field in the time-dependent effective
potential given some initial conditions and model parameters and the
computations are consistently performed in the Einstein frame. We study two
cases for the initial conditions: the field starting either at some arbitrary
value or starting at the the minimum of the effective potential (see, eg, Brax
et al. [19]). In the first case we consider the field initially frozen at an
arbitrary value by Hubble friction
($|V^{\prime\prime}_{\text{eff}}/H^{2}|<1$). In the second case, we chose
parameters of the model such that the field has already relaxed to the minimum
of the effective potential.

We show that in the first case the model can result in a cosmologically
significant amount of $f_{\text{EDE}}$ at the redshift required to ameliorate
the Hubble tension. However, using the approach proposed in [16], we point out
that this result is not due to the neutrino-EDE coupling and it does not
address the EDE coincidence problem. In the second case, the model provides a
natural value for the initial condition of the EDE field given by the minimum
of the effective potential [17, 18, 20]. Nevertheless, we find that the
contribution to $f_{\text{EDE}}$ is too small for addressing the Hubble
tension.

An important feature of the $\nu$EDE model is the neutrino kick [17, 18] which
may occur by a similar mechanism previously studied in [19, 21, 22, 23]. The
latter are chameleon models addressing the late dark energy with a different
class of self-interaction potentials. In the model studied here we do not see
the effects of the neutrino kick on the EDE field with a displacement from the
minimum of the effective potential by solving its field evolution equation.
Rather, we show that the jump in the trace of the neutrino energy-momentum
tensor is reflected in a change in the minimum of the effective potential that
the EDE is following adiabatically. It is this change that leads to an
increase in $f_{\text{EDE}}$.

This paper is organized as follows. In Section II we present a summary of the
model, and proceed to Section III for a description of the methods used in the
background analysis. Our results are presented in Section IV where we
differentiate the discussion in the two regimes described above: in Section
IV.1 the field is initially frozen by Hubble drag while in Section IV.2 the
field is initially dynamical. In both scenarios we compute the fraction of EDE
and assess the importance of the neutrino-EDE coupling. We present our
conclusions in Section V.

##  II Neutrino-assisted EDE

The idea of neutrinos coupled to a dark energy field was introduced in the so-
called “Mass-Variable Neutrinos” (MaVaN) models, since this coupling induces a
variation of the neutrino mass due to the cosmic evolution of the background
dark energy (also known as quintessence) field [24, 25]. The effects of mass-
varying neutrinos on the CMB anisotropies and large scale structures (LSS),
both at the background and perturbative levels, were studied in [26, 27, 28].
The issue of fine-tuning was discussed in the context of quintessence models
in [29] and more recently in MaVaN models in [30]. We now present a brief
summary of the model.

Consider a general action of the following form:

$\displaystyle S$ $\displaystyle=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}M_{pl}^{2}R-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi-V(\phi)\right]$
$\displaystyle+S_{dm}[\Psi_{dm},\tilde{g}^{dm}_{\mu\nu}]+S_{b}[\Psi_{b},\tilde{g}^{b}_{\mu\nu}]$
$\displaystyle+S_{\gamma}[\Psi_{\gamma},\tilde{g}^{\gamma}_{\mu\nu}]+S_{\nu}[\Psi_{\nu},\tilde{g}^{\nu}_{\mu\nu}],$
(1)

where $\Psi_{dm}$, $\Psi_{b}$, $\Psi_{\gamma}$ and $\Psi_{\nu}$ are the dark
matter, baryons, photon, and neutrino fields, respectively, $M_{Pl}\equiv(8\pi
G)^{-1/2}$ is the reduced Planck mass and $g$ is the determinant of the metric
$g_{\mu\nu}$ in the Einstein-frame. The Jordan-frame metric,
$\tilde{g}_{\mu\nu}$, for each species are related with $g_{\mu\nu}$ by
$\tilde{g}^{i}_{\mu\nu}=A^{2}_{i}(\phi)g_{\mu\nu}$ where $i$ represents all
the components cited above. It is interesting to note that in the case
$A_{dm}=A_{b}=A_{\gamma}=A_{\nu}=1$ and
$V(\phi)=\mu^{4}[1-\text{cos}(\phi/f)]^{n}$, where $f$ is the energy scale
where a shift symmetry is spontaneously broken and $\mu$ is a small explicit
shift symmetry-breaking breaking energy scale that generates an axion-like
potential, one obtains the canonical uncoupled axion-like EDE model [4, 5].
When $A_{b}=A_{\gamma}=A_{\nu}=1$, $A_{dm}\equiv
A=\text{exp}(\beta\phi/M_{pl})$ and $V(\phi)=\lambda\phi^{4}$ we obtain the
chameleon EDE model, recently studied in [15]. Finally, when
$A_{dm}=A_{b}=A_{\gamma}=1$, $A_{\nu}\equiv A=\text{exp}(\beta\phi/M_{pl})$
and $V(\phi)=\lambda\phi^{4}/4$ we obtain the neutrino-assisted EDE ($\nu$EDE)
model studied in [17, 18]. This is the model considered in the following.
Notice that $\nu$EDE has the same framework and is described by the same
equations as for MaVaNs model, the main difference being that for the latter
case, the quintessence potential is $V(\phi)=V_{0}\exp(-\sigma\phi)$ which is
required to achieve the late time cosmic acceleration.

The coupling function that changes the metric between the Einstein and Jordan
frames can be related to the time-varying neutrino mass as
$m_{\nu}(\phi)=m_{\nu 0}A(\phi)$ where $m_{\nu 0}$ is the neutrino mass today.
The average neutrino energy density ($\rho_{\nu}$) and pressure ($p_{\nu}$)
are given in terms of an integral over the comoving momentum ($q$) of the
usual unperturbed Fermi-Dirac distribution $f_{0}(q)$:

$\displaystyle\rho_{\nu}(\phi)$ $\displaystyle=\frac{1}{a^{4}}\int
q^{2}dqd\Omega\epsilon f_{0}(q),$ (2) $\displaystyle p_{\nu}(\phi)$
$\displaystyle=\frac{1}{3a^{4}}\int
q^{2}dqd\Omega\frac{q^{2}}{\epsilon}f_{0}(q),$ (3) $\displaystyle f_{0}(q)$
$\displaystyle=\frac{g_{\nu}}{h^{3}_{P}}\frac{1}{e^{q/{k_{B}T_{\nu 0}}}+1},$
(4) $\displaystyle\epsilon^{2}$ $\displaystyle=q^{2}+a^{2}m_{\nu}^{2}(\phi),$
(5)

where $g_{\nu}$, $T_{\nu 0}$, $h_{P}$, $k_{B}$ are the number of spin degrees
of freedom, the neutrino temperature today, the Planck and Boltzmann constant,
respectively. In equation (4) we have assumed that neutrinos decoupled from
the primordial plasma when they are still relativistic, so the phase-space
distribution preserve the Fermi-Dirac form with a dependence only on the
comoving momentum.

Taking the time derivative of Eq. (2) and using that the energy conservation
is determined by the coupled system of neutrinos and the scalar field, we
obtain the modified Klein-Gordon equation

$\displaystyle\ddot{\phi}+3H\dot{\phi}+\frac{dV(\phi)}{d\phi}$
$\displaystyle=-\frac{d\,\text{ln}\,m_{\nu}(\phi)}{d\phi}\left(\rho_{\nu}-3p_{\nu}\right),$
(6)

where dot is the derivative with respect to time. The self-interaction EDE
potential assumed in this work is

$V(\phi)=\lambda\phi^{4}/4.$ (7)

This is the potential used in the so-called “rock ’n’ roll” models of EDE
[31].

In addition, it is interesting to notice that taking the derivative of Eq. (2)
with respect to the scalar field results in

$d\rho_{\nu}(\phi)/d\phi=(d\,\text{ln}\,m_{\nu}/d\phi)(\rho_{\nu}-3p_{\nu}).$
(8)

For definiteness we are going to adopt a coupling of the form:

$A(\phi)=\exp(\beta\phi/M_{pl}),$ (9)

where $\beta$ is a dimensionless constant factor giving the coupling strength
between neutrinos and the EDE field. The fact that in EDE models the energy
density must dissipate quickly after recombination ensures that
$m_{\nu}(\phi)\rightarrow m_{\nu 0}$ today. With this choice of interaction
the Klein-Gordon equation takes the familiar form:

$\ddot{\phi}+3H\dot{\phi}+\frac{dV(\phi)}{d\phi}=-\frac{\beta}{M_{pl}}\Theta_{\nu},$
(10)

where $\Theta_{\nu}\equiv-\rho_{\nu}+3p_{\nu}$ is the trace of the energy-
momentum tensor for the neutrinos. Therefore, one can define an effective
potential where

$V^{\prime}_{\text{eff}}=V^{\prime}(\phi)-\frac{\beta}{M_{pl}}\Theta_{\nu}.$
(11)

and using equation (8) we write the redshift-dependent effective potential as

$V_{\text{eff}}(\phi)=V(\phi)+\rho_{\nu}(\phi,z).$ (12)

An important quantity throughout this work is the redshift-dependent minimum
of the effective potential given by [18]222Notice that since we are working in
the Einstein frame there is an implicit dependence of $\Theta_{\nu}$ on the
field and the coupling constant $\beta$. This dependence can be neglected only
if $|\beta\phi/M_{Pl}|\ll 1$. We take this dependence into account in our
numerical code.:

$\displaystyle\phi_{\text{min}}$
$\displaystyle=-\left(\frac{\beta\Theta_{\nu}}{\lambda M_{Pl}}\right)^{1/3},$
(13)

The free parameters of the model are the scalar field self-coupling constant
$\lambda$, a parameter $\beta$ that characterizes the scalar field coupling to
neutrinos, and the initial conditions for the field $\phi_{i}$ and
$\dot{\phi}_{i}$. In the following we will adopt three degenerate neutrinos
with $\sum m_{\nu 0}=0.3$ eV which is below the upper bound found in [32] with
Planck data only.

##  III A background analysis of the neutrino-assisted EDE

As previously stated, the free parameters of the model are $\lambda$, $\beta$,
and $\phi_{i}$ (we will assume $\dot{\phi}_{i}=0$, i.e. the field is frozen
initially). In order to gain some intuition on the parameter $\lambda$, notice
that in the axion-like EDE potential with
$V(\phi)=\mu^{4}[1-\text{cos}(\phi/f)]^{n}$, an expansion around the minimum
for $n=1$ gives the axion mass $m_{a}=\mu^{2}/f$ and the quartic coupling
$\lambda=m_{a}^{2}/f^{2}$. In scalar field models, it is well-known that the
field becomes dynamical (i.e. it “thaws”) when its mass is of the order of the
Hubble parameter. Therefore, if we want the EDE field to become dynamical
before recombination ($H\approx(\text{1 eV})^{2}/M_{Pl}\approx{\cal
O}(10^{-27})$ eV) and assuming a typical symmetry breaking scale of
$f=10^{15}$ GeV one expects very small values of the self-coupling of
$\lambda={\cal O}(10^{-100})$. Even in anharmonic potentials with $n>1$, this
behavior is present [5], and in our case the value of $\lambda$ controls the
epoch when EDE becomes dynamical, with a redshift usually denoted by $z_{c}$.
More generally, the field becomes dynamical when [19]

$\frac{V^{\prime\prime}_{\text{eff}}}{H^{2}}=\mathcal{O}(1).$ (14)

Another important quantity in EDE models is the maximum contribution of the
EDE component to the total energy budget of the universe at $z_{c}$, denoted
by $f_{\text{EDE}}$:

$f_{\text{EDE}}(z_{c})=\frac{\rho_{\text{EDE}}(z_{c})}{\rho_{\text{tot}}(z_{c})},$
(15)

and it is determined by the model parameters. In fact, in an EDE model there
is a simple relation between $\lambda$, $\phi_{i}$ and $z_{c}$:

$\lambda\approx\frac{T_{0}^{4}}{9M_{Pl}^{2}\phi_{i}^{2}}(1+z_{c})^{4},$ (16)

where $T_{0}$ is the CMB temperature today.

The solution to the Hubble tension in EDE models should fulfill the basic
requirements [9, 16]

$\displaystyle f_{\text{EDE}}\sim 0.1~{}\text{and}~{}\log_{10}z_{c}\sim 3.5$
(17)

and that the EDE field is dynamical just prior to matter-radiation equality.
However, usual EDE models suffer from a coincidence problem due to the
arbitrary initial conditions. Extensions of EDE models including interactions
with other components such as dark matter [15] or neutrinos [17, 18] may
address the EDE coincidence problem via some triggering mechanism [16]. In
order to investigate the impact of the new coupling $\beta$ to the fraction of
EDE produced in the extended model we introduce the quantity

$\displaystyle\Delta\equiv\frac{f_{\text{EDE}}}{f_{\text{EDE}}^{\beta=0}},$
(18)

which is defined by the ratio between the fraction of EDE for the coupled
($\beta\neq 0$) and uncoupled ($\beta=0$) scenarios. In the limit that
$\Delta\rightarrow 1$ the $f_{\text{EDE}}$ is mainly sourced by the bare
potential [16].

In the top panel of Fig. 1 we show an example of the background evolution for
cold dark matter (CDM), photons ($\gamma$), neutrinos $\nu$ and the EDE scalar
field plus a cosmological constant ($\phi+\Lambda$) where the conditions in
Eq. (17) are satisfied. In the bottom panel one can see that in this case the
quantity $\Delta$ is close to 1 at the EDE peak ($z\sim 2\times 10^{3}$),
indicating that the coupling to neutrinos is not relevant to generating
$f_{\text{EDE}}$. We will show below cases with the opposite situation.

Figure 1: The background evolution for cold dark matter (CDM), photons
($\gamma$), neutrinos $\nu$ and the scalar field plus a cosmological constant
($\phi+\Lambda$). The parameters used are $\beta=10$,
$\phi_{i}=-0.6\,{M}_{Pl}$ and $\log_{10}\lambda=-109$. The surge of EDE is
around at $z\sim 2\times 10^{3}$. The bottom panel shows the quantity defined
in Eq. (18) which is the ratio between the coupled and uncoupled cases.

##  IV Initial value of the EDE field

Previously we assumed an arbitrary initial value for the EDE field. We now
want to address this issue in more detail. We identify two situations: a) the
field is still frozen initially and starts to follow the minimum of the
effective potential after its thawing and b) the field is already dynamical
initially and therefore it follows the minimum of the effective potential.
This separation is somewhat arbitrary since at large enough redshifts the
field is frozen. However, in practical terms, we need to choose an initial
redshift from which we start evolving the field numerically. The separation
between these two regimes at a given redshift $z_{i}$ can be estimated by the
condition

$\frac{V^{\prime\prime}_{\text{eff}}(\phi_{\text{min}})}{H^{2}}=3\lambda^{1/3}\beta^{2/3}\left(\frac{\Theta_{\nu}}{H^{3}M_{Pl}}\right)^{2/3}\Bigg{|}_{z_{i}}={\cal
O}(1).$ (19)

In Fig. 2 we show the contours in the parameter space $(\lambda,\beta)$ where
the field is already dynamical at some given redshifts. The figure provides
estimates of the values of these coupling separating the two cases.

Figure 2: Values of $\lambda$ and $\beta$ for which
$|V^{\prime\prime}_{\text{eff}}/H^{2}-1|=0.1$ for $z_{i}=10^{3},10^{4}$ and
$10^{5}$. For regions above the curves the field is dynamical at the
corresponding redshift.

The motivation for studying these two cases separately below is that in the
first case one could argue that the initial value of the frozen field does not
have a natural value whereas in the second case the value of the field is
naturally close to the minimum of the effective potential. Indeed we find
different dynamical behaviours in these two cases.

###  IV.1 Field initially frozen

In Figure 3 we plot the effective potential given in Eq. (12) as a function of
the field for different values of the redshift starting at $z_{i}=10^{5}$ for
$\beta=10$ and $\lambda=10^{-109}$, where $\rho_{\nu}(\phi,z)$ is given by Eq.
(2). At high redshifts, the neutrino contribution dominates and it is a smooth
function of $\phi$. At lower redshifts, the contribution from $V(\phi)$ to the
effective potential starts to become more relevant. For these parameters, we
plot the evolution of the field for three different values of $\phi_{i}$: one
starting at the minimum of the effective potential and the other two are
chosen arbitrarily for comparison. We also show the minimum of the effective
potential given by Eq. (13) that, due to the coupling, evolves with redshift.

In all the cases the field is frozen at $z_{i}$, corresponding to the first
line in the plot. We find numerically that for the initial values of the field
that we chose it remains frozen until approximately the redshift that
satisfies Eq. (16) is reached. Notice that as the neutrino contribution
becomes small at lower redshifts, the minimum of the effective potential
approaches the minimum of the bare potential $V(\phi)$ at $\phi=0$.

After thawing the field starts to roll in the effective potential tracking and
oscillating around its minimum. This behaviour of the field is shown both by
the insert zoom in Figure 3 and at the top panel of Figure 4, where one can
see the field relaxing to $\phi=0$, dissipating the energy density contained
in it.

Figure 3: The orange lines are snapshots for the effective potential for
decreasing values of redshift (the first line from top to bottom starts at
$z_{i}=10^{5}$ and the redshift interval between two subsequent lines is
$\log_{10}(z_{n}/z_{n+1})\approx 0.41$). The black stars indicate the minimum
of the effective potential and the blue, purple and red dots are the scalar
field evolution starting at different initial conditions as indicated in the
label. Notice that in the case represented by the purple dots the field starts
at the minimum of the effective potential but does not follow it until
thawing. Also is shown a zoom of the region where the field is dynamical and
oscillates around the minimum of the effective potential. In this figure we
used $\beta=10$ and $\lambda=10^{-109}$.

In the bottom panel of Figure 4 we show the contribution to $f_{\text{EDE}}$
in this example.

Figure 4: The scalar field evolution and the fraction of energy density for
the same three initial conditions as in Fig. 3: $\phi_{i}/M_{Pl}=-0.6,-0.24,0$
for the blue, purple and red lines, respectively. The dashed lines in the
bottom panel shows the uncoupled case with all other parameters kept fixed at
the same values.

One can see that there could be a significant contribution depending on the
initial value of the field. However, the coupling to neutrinos does not seem
to play a large role in this case. We investigate this issue by computing the
evolution of the field with the same initial conditions but without a coupling
to neutrinos ($\beta=0$). This is shown as the dashed lines in the bottom
panel of Figure 4, where one can see that although the coupling introduces
changes in the behaviour of the field the modification to the maximum of
$f_{\text{EDE}}$ is negligible. In the case of the largest contribution, we
find $|f_{\text{EDE}}/f_{\text{EDE}}^{\beta=0}-1|=5\times 10^{-4}$. Therefore,
in this case one needs the usual fine-tuning of initial conditions found in
regular EDE models in order to have a significant contribution to
$f_{\text{EDE}}$.

###  IV.2 Field initially dynamical

In this subsection we are interested to study the case where the field is
already dynamical at the initial redshift considered. The motivation is that
in this case the field is already tracking the minimum of the effective
potential and hence its value is not arbitrary. We will show below that in
order for the field to become dynamical at an early redshift, a larger value
of the self-coupling constant is necessary but this leads to a small fraction
of EDE.

We begin by showing some examples of this behaviour in Figure 5. We choose
parameters in such a way that the field becomes dynamical and starts to
oscillate around the minimum of the effective potential at some $z={\cal
O}(10^{5})$. More specifically, we start the field evolution at a high
redshift $z=10^{7}$ at $\phi_{i}/M_{Pl}=-10^{-4}$, where it is still frozen
and show how it approaches the minimum of the effective potential at $z={\cal
O}(10^{5})$ for different values of $\beta$ and $\lambda$. We also show that
the difference is minimal when compared to the evolution starting at
$z=10^{5}$ with the field at the minimum of the effective potential.

Figure 5: The full colored lines are the field evolution for
$\phi_{i}=-10^{-4}M_{Pl}$ at $z_{i}=10^{7}$ and considering different values
of the parameters $\beta$ and $\lambda$ as indicated in the label. Each field
track its minimum as described by Eq. (13) and it is indicated by the
different stylized black lines. The dotted colored lines are the field
evolution starting at $z_{i}=10^{5}$ in their respective minimum for the same
set of parameters $\beta$ and $\lambda$.

We now turn to the contribution of the EDE field $\phi$ to the energy density
of the Universe, showing in Figure 6 the behaviour of $f_{\text{EDE}}$ for the
cases described above. We compare the numerical computation with the
analytical estimate obtained by assuming that the EDE field is at the minimum
of the effective potential (Eq. (13)) and that the dominant contribution comes
from the bare potential $V(\phi)$[18]:

$f_{\text{EDE}}=\frac{1}{12}\frac{\beta^{4/3}}{\lambda^{1/3}}\left(\frac{\Theta_{\nu}(\phi)}{H^{3/2}M_{Pl}^{5/2}}\right)^{4/3},$
(20)

computed assuming that $\beta\phi/M_{Pl}<1$ in order to use the approximate
expression Eq. (13). However, in the equation above we use the value of $\phi$
from CAMB. One can see that the analytical estimate is a reasonable
approximation of the average of the oscillations in $f_{\text{EDE}}$. We also
notice that even for large values of the coupling ($\beta=1000$) a
cosmologically significant level of $f_{\text{EDE}}$ is not obtained333In
principle it is possible to obtain a significant amount of EDE for still
larger values of $\beta$. Indeed, we find that $\beta\geq{\cal O}(10^{6})$
could in principle lead to $f_{\text{EDE}}={\cal O}(10\%)$ but these large
values of $\beta$ can not be considered natural.. The same behaviour is also
seen if one decreases the field self-coupling $\lambda$, as expected from Eq.
(20) shown in the bottom panel of Fig. 6: an increase in $f_{\text{EDE}}$ is
obtained for smaller values of $\lambda$ at the cost of a decrease in the
relevant redshift $z_{c}$. A more thourough investigation including a
comparison with data using a Bayesian analysis is necessary to reach a more
definitive conclusion of whether this case can help to ease the $H_{0}$
tension.

Figure 6: Fraction of EDE for the same set of parameters as in Fig. 5. The
different stylized black lines are the analytic prediction for the
$f_{\text{EDE}}$ according to Eq. (20).

A prominent feature of $\nu$EDE models is the kick in the EDE field provided
by the coupling with neutrinos that occurs when neutrinos transition from the
relativistic to the nonrelativistic regimes. An intriguing possibility is that
this mechanism could explain the EDE coincidence problem since the neutrino
mass energy scale roughly coincides with the matter-radiation equality epoch.
We now want to assess the impact of the neutrino coupling in the contribution
of the EDE field to the total energy density.

In order to evaluate the impact of the coupling to neutrinos we will use the
criteria in Eq. (18) and compare the two cases, with and without coupling. In
both cases we start the field evolution at the minimum of the effective
potential at $z=10^{5}$ in the situation where the field is already dynamical
but in the first case we study the full model whereas in the second case we
set the neutrino coupling $\beta$ to zero, keeping all other parameters equal.
In Figure 7 we show a comparison in $f_{\text{EDE}}$ for both cases, and one
can see that the effect of the neutrino-EDE coupling is prominent. Although
the contribution is the same initially, in the uncoupled case the field tends
to zero (the minimum of the bare potential) while the coupled case follows Eq.
(13) which for larger $\beta$ can significantly differ from zero. This results
in a large difference between the two cases, which impacts $f_{\text{EDE}}$ by
several orders of magnitude.

Figure 7: A comparison of $f_{\text{EDE}}$ between the coupled and uncoupled
($\beta=0$) scenarios with the field starting at the minimum of the effective
potential at $z=10^{5}$ while keeping fixed all the other cosmological and
model parameters. The purple line is the case with largest $f_{\text{EDE}}$
shown in the bottom of Fig. 6.

##  V Conclusion

Neutrino-assisted early dark energy models were introduced with the aim of
reducing some of the fine-tuning required in usual EDE models with respect to
the parameters characterizing the scalar field self-coupling and its initial
displacement. The idea that a “neutrino kick” occurring around the matter-
radiation equality epoch could trigger the EDE field is interesting for
addressing the EDE coincidence problem.

In this work we developed an implementation of the $\nu$EDE model in CAMB at
the background level to study $f_{\text{EDE}}$ and $\Delta$, quantifying the
EDE scalar field contribution to the total energy density and the impact of
the neutrino-EDE coupling, respectively. We have performed this analysis
considering two cases depending whether or not the EDE field is frozen at the
initial redshift of its evolution. We find that if the field is initially
frozen, its value can be arbitrary and the model can help in ameliorating the
Hubble tension. However, since $\Delta\sim 1$ the neutrino-EDE coupling is not
relevant in this case and the EDE coincidence problem is not addressed.

In contrast, as pointed out in [18, 20], the opposite scenario is attractive
since there is a natural initial value for the field: the minimum of the
effective potential, $\phi_{\text{min}}$, which is time-dependent. We show
that the field follows $\phi_{\text{min}}$ without experiencing a significant
dynamical kick from its evolution equation, ie without a displacement from the
minimum. However, $\phi_{\text{min}}$ is affected by the change in the trace
of the neutrino energy-momentum tensor, which generates a bump in
$f_{\text{EDE}}$ since the field follows the minimum.

We find that on average $f_{\text{EDE}}$ is well reproduced by the analytical
prediction in Eq. (20). Furthermore, from Fig. 7 one can see that in this
regime the neutrino-EDE coupling plays a non-negligible role, leading to large
modifications in $f_{\text{EDE}}$. However, the attained value of
$f_{\text{EDE}}$ does not satisfy the criteria in Eq. (17) for easing the
Hubble tension. One might think that decreasing the value of the self-coupling
$\lambda$ would increase the value of $f_{\text{EDE}}$, as follows from Eq.
(20). This is correct but one should notice that this would also imply in a
field that is frozen until later times (since it results in a smaller
$V^{\prime\prime}(\phi)$), which changes this model to the first case
discussed above. In addition, we verified that couplings as large as
$\beta=1000$ does not produce a significant amount of EDE.

We conclude that $\nu$EDE models, with a self-interaction potential and a
coupling of the type $V(\phi)\propto\phi^{4}$ and
$A(\phi)=\exp(\beta\phi/M_{Pl})$, respectively, do not seem to naturally
provide a significant amount of early dark energy which is one of the criteria
usually required for the model to ease the Hubble tension. However, a more
thourough investigation is certainly needed. Different potentials and
couplings should be analysed, such as the ones proposed in Ref. [16] where a
model with EDE interacting with dark matter is studied using a monodromy-type
potential and a $\phi^{2}$ coupling. Additionally, a successful model must not
only be able to alleviate the Hubble tension at the background level but also
produce perturbations that agree with the current CMB and LSS data. Hence, a
comparison with data using Bayesian statistics at the perturbation level
analysis (eg [27, 26, 28]) is necessary in order to reach a more definitive
conclusion of whether this class of models is capable of alleviating the
$H_{0}$ tension and possibly the $S_{8}$ tension as well. It could be
interesting to further study $\nu$EDE models using data from large galaxy
surveys, such as the Dark Energy Survey and the imminent Rubin Telescope’s
Legacy Survey of Space and Time.

###### Acknowledgements.

We are very much in debt to Mark Trodden, Jeremy Sakstein, Mariana González
and Qiuyue Liang for their thoughtful criticism of the first version of this
paper and for their patience in sharing their views on this subject. We would
like to thank Vivian Miranda, David Mota, João Rebouças and especially Elisa
Ferreira for useful suggestions about the text. The work of DHFS is supported
by a CAPES fellowship. RR is partially supported by a CNPq fellowship, the
project INCT do e-Universo, LIneA and Fapesp.

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