# Preheating in Einstein-Cartan Higgs Inflation: Oscillon formation

Matteo Piani  and Javier Rubio

###### Abstract

We make use of classical lattice simulations in $3+1$ dimensions to study the
preheating stage of Higgs Inflation in Einstein-Cartan gravity. Focusing for
concreteness on a simplified scenario involving the seminal Nieh-Yan term, we
demonstrate the formation of dense and spatially localized oscillon
configurations constituting up to 70% of the total energy density. The
emergence of these meta-stable objects may lead to a prolonged period of
matter domination, effectively modifying the post-inflationary history of the
Universe as compared to the metric and Palatini counterparts. Notably, the
creation of oscillons comes together with a significant gravitational wave
signal, whose typical frequency lies, however, beyond the range accessible by
existing and planned gravitational wave experiments. The impact of the
Standard Model gauge bosons and fermions and the potential extension of our
results to more general Einstein-Cartan settings is also discussed.

##  1 Introduction

The constant improvement in the precision of measurements of the Cosmic
Microwave Background (CMB) [1, 2] has led to the establishment of the
inflationary paradigm [3, 4, 5] as the best candidate to tackle the conceptual
problems of the hot Big Bang while generating the primordial density
perturbations seeding structure formation. Despite this success, the nature of
the inflaton field remains unknown and its role could be played by any
candidate able to mimic a scalar field in the slow-roll regime. Indeed, while
the Planck and BICEP2 collaborations have set strong constraints on the scale
dependence of the density power spectrum and the tensor-to-scalar ratio
encoding the amount of primordial gravitational waves [2], large classes of
inflationary models remain still compatible with observations [6]. On top of
that, knowing the shape of the inflationary potential is a priori not
sufficient to determine the whole post-inflationary evolution. In particular,
any inflationary model must come together with a graceful exit mechanism able
to transfer the energy stored in the homogeneous inflaton condensate to the
Standard Model (SM) degrees of freedom, leading with it to the onset of the
usual hot Big Bang theory [7, 8, 9]. This poses significant limitations for
many particle physics scenarios in which the inflaton-matter couplings are not
experimentally known.

Among the many possible particle physics embeddings of the early Universe
accelerated expansion, Higgs Inflation (HI) [10, 11] stands out as an ideal
candidate for studying the post-inflationary dynamics. In fact, assuming no
Beyond SM physics between the electroweak and the Planck scale, it is a priori
possible to compute the running of the experimentally known SM couplings,
allowing, in principle, for a comprehensive study of the preheating stage.
This scenario has been studied in different formulations of gravity, ranging
from the standard metric case [10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]
to the Palatini [23, 24, 25, 26, 27, 28, 29, 30], teleparallel [31], Einstein-
Cartan [32, 33, 34, 35] and metric-affine formulations [36, 37, 38], including
also several scale-invariant extensions [39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 54, 55, 56]. Conversely, the post-inflationary
evolution has been studied both analytically and numerically only for the
metric [57, 58, 59, 60, 61, 62, 63, 64, 65] and Palatini [66, 67] cases.

When matter is disregarded it is always possible to map all these different
formulations back to the metric one. However, this ceases to be true in the
presence of fermions and non-minimal couplings with scalar fields (see [37,
68] for a detailed explanation). Such feature becomes explicit when turning to
the Einstein-frame, where the Lagrangian reduces to the one of metric gravity,
plus a modified potential and a non-canonical kinetic term for the Higgs
field. Although one could argue that such scenario could be also attained by
directly modifying the minimally coupled Standard Model Higgs Lagrangian in
the metric formulation, the choice of operators in that case would remain
arbitrary. On the contrary, the shape of the theory defining functions in the
framework under consideration is completely determined by gravitational
interactions, naturally providing a set of selection rules for the higher-
order operators in the Einstein-frame. In this context, the specific
formulation one uses provides a selection criterium which leads to different
outcomes, therefore breaking the degeneracy between all the possible
gravitational theories.

In this work, we make use of fully-fledged numerical lattice simulations in
$3+1$ dimensions to study the preheating stage of HI in Einstein-Cartan (EC)
gravity. Restricting ourselves to the radial degree of freedom of the Higgs
field, we find that, contrary to what happens in the metric and Palatini
formulations of the theory, this scenario allows for the formation of dense
and spatially-localized oscillon configurations constituting up to $70\%$ of
the total energy density for suitable model parameters. In spite of lacking a
conserved charge associated with them, these pseudo-solitonic objects can be
rather long-lived, leading to a prolonged matter-domination era that
effectively alters the minimum duration of the inflationary phase needed to
solve the usual hot Big Bang problems. Additionally, we find that such
structures can source a sizeable gravitational wave signal, thus providing an
alternative observational channel for this appealing inflationary model. The
emergence of oscillons is expected to be robust against the inclusion of gauge
fields, since the accumulation of these species turns out to be severely
suppressed by their perturbative decay into fermions for a large number of
oscillations, significantly exceeding the oscillon formation time.

The paper is structured as follows. In Section 2 we review the embedding of HI
in the EC gravity, highlighting on the differences with the metric and
Palatini formulations. Section 3 is devoted to the study of the preheating
stage in the scalar sector of theory, starting from a simple linear analysis
to subsequently perform a fully non-linear characterization of oscillon
formation via 3+1 classical lattice simulations. Aspects such as the abundance
and energy distribution of oscillons or the associated production of
gravitational waves at formation are addressed in detail, comparing the latter
with the sensitivity and frequency range of current and future GWs
experiments. The potential impact of the SM gauge boson and fermions is
considered in Section 4. Finally, our conclusions are presented in Section 5,
where we argue on possible extensions and improvements of our treatment.

##  2 Higgs Inflation in Einstein-Cartan Gravity

The EC formulation of gravity assumes the tetrads $e^{A}_{\mu}$ and spin
connection $\omega_{\mu}^{AB}$ ${(\omega_{\mu}^{AB}=-\omega_{\mu}^{BA}})$ as
fundamental gravitational variables [69, 70, 71, 72], with the Greek letters
related to the usual spacetime coordinate basis and the Latin ones to an
orthonormal non-coordinate basis displaying covariance under local Lorentz
transformations. The associated metrics and connections entering the covariant
derivatives in the two bases,

$D_{\mu}V^{\alpha}=\partial_{\mu}V^{\alpha}+\Gamma^{\alpha}_{\sigma\mu}V^{\sigma}\,,\hskip
28.45274ptD_{\mu}V^{A}=\partial_{\mu}V^{A}+\omega_{\mu}^{AB}V_{B}\,,$ (2.1)

are related through

$g_{\mu\nu}=e^{A}_{\mu}e^{B}_{\nu}\eta_{AB}\,,\hskip
28.45274pt\Gamma^{\mu}_{\;\;\nu\rho}=e^{\mu}_{A}\left(\partial_{\rho}e^{A}_{\nu}+\omega^{A}_{\rho
B}e^{B}_{\nu}\right)\,,$ (2.2)

with the second expression following directly from the condition
$D_{\mu}e_{\nu}^{A}=0$, without any symmetry assumption on the lower indices.
Beyond the curvature tensor, this theory accounts for a non-vanishing torsion
contribution

$T^{\mu}_{\;\;\nu\rho}=\Gamma^{\mu}_{\;\;\nu\rho}-\Gamma^{\mu}_{\;\;\rho\nu}\,,$
(2.3)

obtained by acting with the commutator of two covariant derivatives on a given
scalar (or a vector, upon subtracting curvature contributions).

As compared to General Relativity, this construction offers some conceptual
advantages. Firstly, EC gravity can be viewed as the gauge theory of the
Poincaré group, placing gravitational interactions on equal footing with other
fundamental forces in Nature [69, 70]. Secondly, the independent treatment of
the metric and the connection in EC gravity avoids the necessity of
introducing specific boundary terms in order to obtain the Einstein equations
of motion [73]. Finally, the spin connection can be easily coupled to
fermions, making the EC formulation particularly useful in describing the
behaviour of the SM in the presence of gravity.

In the context of scalar-tensor theories, the existence of torsion opens up
the possibility of including additional gravitational operators beyond those
usually considered in metric HI, namely the usual SM Higgs sector and the
Einstein-Hilbert term modified by the addition of a non-minimal coupling to
gravity, 111We adopt here a mostly plus signature and work in Planckian units
$M_{P}^{2}\equiv 8\pi G=1$.

$S_{\rm SM+EH}=\int d^{4}x\sqrt{-g}\left[\frac{1+\xi
h^{2}}{2}g^{\mu\nu}R_{\mu\nu}(\Gamma)-\frac{1}{2}g^{\mu\nu}\partial_{\mu}h\partial_{\nu}h-\frac{\lambda}{4}(h^{2}-v^{2})^{2}\right]\,,$
(2.4)

with $h$ the Higgs field in the unitary gauge, $\xi$ a dimensionless coupling
constant to be determined from observations, $\lambda$ the Higgs self-
coupling, and $v\simeq 250$ GeV its vacuum expectation value. In the context
of EC gravity, building the most general theory containing only a massless
spin-2 gravitational degree of freedom requires the introduction of a plethora
of operators. As shown explicitly in Ref. [35] (see also Refs. [32, 33] for
specific choices), the most general graviscalar sector beyond (2.4) including
at most quadratic operators in the derivatives of all fields takes the form

$\displaystyle S_{\rm T}$ $\displaystyle=\int
d^{4}x\,\sqrt{-g}\Bigg{[}v^{\mu}\partial_{\mu}Z^{v}+a^{\mu}\partial_{\mu}Z^{a}$
(2.5)
$\displaystyle+\frac{1}{2}\Big{(}G_{vv}v_{\mu}v^{\mu}+2G_{va}v_{\mu}a^{\mu}+G_{aa}a_{\mu}a^{\mu}+G_{\tau\tau}\tau_{\alpha\beta\gamma}\tau^{\alpha\beta\gamma}+\tilde{G}_{\tau\tau}\epsilon^{\mu\nu\rho\sigma}\tau_{\lambda\mu\nu}\tau^{\lambda}_{~{}\rho\sigma}\Big{)}\Bigg{]}\,,$

with

$Z^{v/a}=\zeta^{v/a}_{h}h^{2}\,,\hskip
42.67912ptG_{ij}=c_{ij}\left(1+\xi_{ij}h^{2}\right)\ ,$ (2.6)

no summation repeated $i,j$ indices and $\xi^{v/a}_{h}$, $c_{ij}$ and
$\xi_{ij}$ constants. Here

$v_{\mu}=T^{\nu}_{~{}\mu\nu}\,,\hskip
28.45274pta_{\mu}=\epsilon_{\mu\nu\rho\sigma}T^{\nu\rho\sigma}\,,\hskip
28.45274pt\tau_{\mu\nu\rho}=\frac{2}{3}\left(T_{\mu\nu\rho}-v_{[\nu}g_{\rho]\mu}-T_{[\nu\rho]\mu}\right)\
,$ (2.7)

stand respectively for the vector, pseudo-vector and reduced tensor
irreducible components of the torsion tensor (2.3),

$T_{\mu\nu\rho}=e_{\mu
A}T^{A}_{\nu\rho}=\frac{2}{3}v_{[\nu}g_{\rho]\mu}-\frac{1}{6}a^{\sigma}\epsilon_{\mu\nu\rho\sigma}+\tau_{\mu\nu\rho}\,,$
(2.8)

with the square brackets standing for anti-symmetrization in the corresponding
indices.

In order to get a more intuitive picture of the inflationary and post-
inflationary dynamics in this rather complicated theory, it is convenient to
move to the Einstein-frame, where the Higgs field is minimally coupled to
gravity and no explicit torsion operator is present. This can be achieved in
two steps. First, we perform a Weyl rescaling of the metric with
$g_{\mu\nu}\rightarrow(1+\xi h^{2})^{-1}g_{\mu\nu}$. Second, we solve the
equation of motion for the anti-symmetric part of the connection, i.e. the one
encoding torsion, and plug back the solution into the action. The explicit
computation has already been developed in Refs. [35, 32, 52], to which we
refer to the interested reader for details. The final action $S=S_{\rm
SM+EH}+S_{\rm T}$ takes the compact form

$S=\int d^{4}x\sqrt{-g}\Bigg{[}\frac{R}{2}-\frac{1}{2}K(h)(\partial
h)^{2}-V(h)\Bigg{]}\,,$ (2.9)

with

$K(h)=\frac{1+c\,h^{2}}{(1+\xi h^{2})^{2}}\,,\hskip
42.67912ptV(h)=\frac{\lambda}{4}\frac{h^{4}}{(1+\xi h^{2})^{2}}\,,$ (2.10)

and

$c(h)=\xi+6\xi^{2}+4f(h)\frac{G_{aa}(\zeta^{v}_{h})^{2}+G_{vv}({\zeta^{a}_{h}})^{2}-G_{va}{\zeta^{v}_{h}}{\zeta^{a}_{h}}}{G_{vv}G_{aa}-G_{va}^{2}}\;.$
(2.11)

Here we have explicitly neglected the Higgs vacuum expectation value, since
this will not be relevant for our analysis of inflation and preheating. For
the purposes of this paper, we will mainly be interested in EC scenarios
leading to field-independent $c$ values. For illustration purposes only, in
what follows we will consider a simple scenario with

$\displaystyle c_{vv}=-\frac{2}{3}\,,\hskip 28.45274ptc_{va}=0\,,\hskip
28.45274ptc_{aa}=\frac{1}{24}\,,$ (2.12)
$\displaystyle\xi_{vv}=\xi_{aa}=-\zeta^{v}_{h}=\xi\,,\hskip
28.45274pt\xi_{va}=0\,,\hskip
28.45274pt\zeta^{a}_{h}=\frac{1}{4}\xi_{\eta}\;,$

leading effectively to a parameter

$c=\xi+6\xi^{2}_{\eta}\,.$ (2.13)

In this limit, the general equation (2.5) reduces to a simple interaction term
involving the seminal Nieh-Yan topological invariant [74, 75]

$S_{\rm T}=-\frac{1}{4}\int
d^{4}x\>\xi_{\eta}h^{2}\partial_{\mu}\left(\sqrt{-g}\epsilon^{\mu\nu\rho\sigma}T_{\nu\rho\sigma}\right)~{},$
(2.14)

with $\epsilon^{\mu\nu\rho\sigma}$ the totally anti-symmetric tensor
($\epsilon_{0123}=1$) and $\xi_{\eta}$ a non-minimal coupling to be determined
from observations. As shown in Refs. [32, 33, 55], this specific choice
provides a smooth parametric interpolation between the metric
($\xi_{\eta}=\xi$) and Palatini formulations of HI ($\xi_{\eta}=0$), a
property that, as we will show in Section 3, will turn out to be critical for
oscillon formation. Note, however, that most of our results can be easily
extended to more general settings within the EC multiverse (2.5), provided
that they effectively lead to a constant value of $c$ in the field range of
interest.

For constant $c$, the non-canonical kinetic term in Eq. (2.9) can be made
canonical by performing a field redefinition

$\frac{d\chi}{dh}=\sqrt{K(h)}\,.$ (2.15)

Solving this differential equation and inverting the result in the small
($c\,h^{2}\ll 1$) and large field ($c\,h^{2}\gg 1$) regimes, we obtain the
effective Einstein-frame potential

$V\simeq\begin{cases}\dfrac{\lambda}{4}\chi^{4}\,,&\hskip
14.22636pt\textrm{for}\hskip 14.22636pt\chi<\chi_{c}~{}\,,\\\
\dfrac{\lambda}{4\xi^{2}}\left[1-\exp\left(-\dfrac{2\xi|\chi|}{\sqrt{c}}\right)\right]^{2}\,,&\hskip
14.22636pt\textrm{for}\hskip 14.22636pt\chi\gg\chi_{c}\,,\end{cases}$ (2.16)

with $\chi_{c}=1/\sqrt{c}$ a crossover scale depending only on the parameter
$c$. This expression flattens out exponentially at large field values,
allowing for inflation to take place with the usual slow-roll initial
conditions. In this regime, our scenario coincides essentially with that of
$\alpha$-attractor E-models [76], upon the formal replacement $c\to
6\xi^{2}\alpha$, but in this case with a discrete $\mathds{Z}_{2}$ symmetry.
Although such a reflection invariance does not affect at all the inflationary
dynamics (other than allowing, of course, for inflation with negative and
positive field values), it will have important consequences for the preheating
phase, as we will show explicitly in Section 3.

Knowing the potential in canonical field variables, we can proceed now with
the standard inflationary analysis, computing explicitly the slow-roll
parameters,

$\epsilon\equiv\dfrac{1}{2}\left(\dfrac{V_{,\chi}}{V}\right)^{2}\simeq\dfrac{8\xi^{2}}{c}\exp\left(-\dfrac{4\xi|\chi|}{\sqrt{c}}\right)~{},\hskip
28.45274pt\eta\equiv\dfrac{V_{,\chi\chi}}{V}\simeq-\frac{8\xi^{2}}{c}\exp\left(-\dfrac{2\xi|\chi|}{\sqrt{c}}\right)\,,$
(2.17)

and the relation between the inflaton field value and the number of $e$-folds
of inflation,

${\cal
N}(\chi)=\int_{\chi_{\text{end}}}^{\chi}\frac{d\chi^{\prime}}{\sqrt{2\epsilon(\chi^{\prime})}}\simeq\frac{c}{8\xi^{2}}\exp\left(\dfrac{2\xi|\chi|}{\sqrt{c}}\right)\hskip
14.22636pt\longrightarrow\hskip
14.22636pt|\chi(\mathcal{N})|=\frac{\sqrt{c}}{2\xi}\log\left(\frac{8\xi^{2}\mathcal{N}}{c}\right)~{},$
(2.18)

with

$\chi_{\text{end}}=\frac{\sqrt{c}}{2\xi}\log\left(1+\frac{2\sqrt{2}\xi}{\sqrt{c}}\right)$
(2.19)

the field value at which inflation ends [$\epsilon(\chi_{\text{end}})\equiv
1$]. This allows us to determine the amplitude of the primordial spectrum of
density perturbations $A_{s}=V/(24\pi^{2}M_{P}^{4}V)$, its tilt
$n_{s}=1+2\eta-6\epsilon$ and the tensor-to-scalar ratio $r=16\epsilon$ at the
time at which the pivot scale $k_{*}=0.05\,{\rm Mpc}^{-1}$ exits the horizon
(see figure 1),

$A_{s}=\frac{\lambda\mathcal{N}_{*}^{2}}{12\pi^{2}c}\,,\hskip
34.1433ptn_{s}=1-\frac{2}{\mathcal{N}_{*}}~{},\hskip
34.1433ptr=\frac{2c}{\xi^{2}\mathcal{N}_{*}^{2}}\,.$ (2.20)

The precise value of the number of $e$-folds ${\cal N}_{*}$ to be inserted in
these expressions depends on the whole post-inflationary evolution, and in
particular, on how long it takes after inflation to enter the phase of
radiation domination needed for successful Big Bang Nucleosynthesis. Note also
that the observational constraint on the CMB-normalization, $A_{s}=2.1\cdot
10^{9}$ [1], restricts only the ratio of couplings determining the amplitude
of the inflationary potential, namely

$\frac{c}{\lambda}=\frac{2}{5}\cdot 10^{7}\,\mathcal{N}_{*}^{2}\,.$ (2.21)

Figure 1: Numerical results (blue curve) in the $n_{s}-r$ plane for the action
(2.9) with different values of the non-minimal coupling $\xi$ (following the
procedure in [33, 55]), together with the recently updated constraints (green
shaded regions) from the BICEP/Keck collaboration [77], at 68% and 95% C.L.

Therefore, as far as inflation is concerned, a change in the Higgs self-
coupling $\lambda$ can be always compensated by a change of the parameter $c$,
meaning that a strict relation between cosmological observables and SM
parameters can only be established once the running of $\lambda$ is known. In
the context of Einstein-Cartan gravity, this is expected to depend not only on
the usual SM couplings but also on the actual strength of five- and six-
dimensional Higgs-fermion and fermion-fermion interactions appearing when
torsion is integrated out [70], potentially extended with additional non-
minimal couplings [78, 79, 80, 81, 34, 35]. In the lack of a proper
computation of the SM running in this context, we will consider in what
follows a fiducial value $\lambda=10^{-3}$. This choice is commensurate with
the value obtained using the pure SM running for the top quark pole masses in
Refs. [82, 83], while coinciding also with the favoured one in the Palatini
formulation of HI [29]. 222For a discussion of HI for negative values of
$\lambda$, we refer the reader to Refs. [59, 84].

A simple inspection of the inflationary observables (2.20) reveals several
interesting regimes. For $\xi=\xi_{\eta}$, one recovers the metric HI
predictions, where both the spectral tilt and the tensor-to-scalar ratio are
almost insensitive to the specific value of the non-minimal couplings to
gravity. On the contrary, for $\xi\gtrsim\xi_{\eta}^{2}$, we rather have
$c\simeq\xi$, recovering effectively the predictions of the Palatini HI
scenario, and in particular the strong $1/\xi$ suppression in the tensor-to-
scalar ratio. The intermediate regime, $\xi_{\eta}<\xi<\xi_{\eta}^{2}$, leads
to a tensor-to-scalar ratio proportional to $\xi_{\eta}^{2}/\xi^{2}$, with
$\xi_{\eta}$ fixed to a value $\xi_{\eta}\simeq\sqrt{2\lambda/3}\cdot
10^{3}\,\mathcal{N}_{*}$ by Eq. (2.21) and $\xi$ almost independent of the CMB
normalization. Since preheating has already been extensively studied both in
the metric [57, 58, 59, 60, 61, 62, 63, 64, 65] and Palatini [66, 67] regimes,
we will focus in what follows on the uncharted range
$\xi_{\eta}<\xi<\xi_{\eta}^{2}$.

##  3 Preheating in the scalar sector

Immediately after the end of inflation, the vast majority of the energy
density of the Universe is stored in the zero mode of the Higgs field, which
will start oscillating around the minimum of the potential (2.16), leading
with it to periodically time-varying masses for all the SM particles coupled
to it and potentially inducing particle creation. The direct creation of SM
fermions out of the Higgs condensate is, however, severely restricted by Pauli
blocking effects [85, 86], leaving the production of Higgs excitations and
electroweak gauge bosons as main depletion channels.

As a first step to study the complicated preheating dynamics of EC HI, let us
concentrate on the pure Higgs sector, neglecting momentarily its coupling to
the SM gauge bosons and fermions. In this regime, the evolution of the
inflaton field in a Friedmann-Lemaître-Robertson-Walker background obeys a
single-field Klein-Gordon equation

$\ddot{\chi}+3H\dot{\chi}-a^{-2}\nabla^{2}\chi+V_{,\chi}=0~{},$ (3.1)

with $a$ the scale factor, the dots denoting derivatives with respect to the
cosmic time $t$ and the Hubble rate $H=\dot{a}/a$ evolving according to the
Friedmann equations

$3H^{2}=\frac{1}{2}\dot{\chi}^{2}+\frac{1}{2a^{2}}(\nabla\chi)^{2}+V(\chi)\,,\hskip
28.45274pt\quad\dot{H}=-\frac{1}{2}\dot{\chi}^{2}-\frac{1}{6a^{2}}(\nabla\chi)^{2}\,.$
(3.2)

The initial stages of preheating can be understood by splitting the inflaton
field into a homogeneous component $\bar{\chi}(t)$ satisfying the zero-
gradient limit of Eqs. (3.1) and (3.2) and a small perturbation
$\delta\chi({\bf x},t)$. The combination of the background equations of motion
leads to the rather enlightening expression [66]

$\frac{dH}{d\bar{h}}=-\frac{\dot{\bar{\chi}}^{2}}{2\dot{\bar{h}}}=-\frac{1}{2}\sqrt{6H^{2}-2V[\bar{\chi}(\bar{h})]}\frac{d\bar{\chi}}{d\bar{h}}\,,$
(3.3)

with $V[\bar{\chi}(\bar{h})]$ the $h$-dependent potential in Eq. (2.10) and
the field redefinition $d\bar{\chi}/d\bar{h}$ completely encoding the specific
dependence on the EC formulation of gravity under consideration, cf. Eq.
(2.15). A simple inspection of the associated kinetic function (2.10) at the
intermediate field values $\bar{h}_{c}\ll\bar{h}\ll\bar{h}_{end}$ reveals
different behaviours for different values of $c$. In particular, for
$c\rightarrow\xi+6\xi^{2}$, corresponding to the metric HI limit
$\xi_{\eta}=\xi$, the kinetic function remains parametrically large,
$d\bar{\chi}/d\bar{h}\approx\sqrt{6}\xi\bar{h}/M_{P}\gg 1$, leading to a quick
damping of the oscillations as $\bar{h}$ approaches zero. On the other hand,
for $c\rightarrow\xi$, corresponding to the Palatini HI limit $\xi_{\eta}=0$,
we have approximately $d\bar{\chi}/d\bar{h}\approx 1$, making the damping of
the oscillations significantly less efficient and allowing for the inflaton
field to return closer to the inflationary plateau for a given number of
oscillations. These recursive field incursions have a dramatic effect on the
production of Higgs excitations, as becomes evident when considering the mode
equations for the linear perturbations $\delta\chi$ in Fourier space,

$\ddot{\delta\chi_{\textbf{k}}}+3H\dot{\delta\chi_{\textbf{k}}}+\left(\textbf{k}^{2}/a^{2}+V_{,\chi\chi}\right)\delta\chi_{\textbf{k}}=0\,,$
(3.4)

with $\bf k$ the associated wave-vector. In particular, the effective square
frequency of this damped harmonic oscillator becomes non-positive definite
whenever the Higgs field accelerates towards the minimum of the potential from
beyond the inflection point $\chi_{i}=\sqrt{c}/(2\xi)\ln 2$. This translates
into a tachyonic amplification of long-wave fluctuations
($\textbf{k}^{2}/a^{2}<|V_{,\chi\chi}|$) that can significantly enhance the
efficiency of other Bose stimulation effects. The maximum momentum that will
grow due to tachyonic instability is given by the minimum of the second
derivative $V_{,\chi\chi}$, which turns out to be located at a parametrically
fixed value

$\chi_{\rm min}=\frac{\sqrt{c}\log(2)}{\xi}\,,$ (3.5)

such that

$\frac{|\mathbf{k}_{\rm max}|}{a}=\sqrt{|V_{,\chi,\chi}(\chi_{\rm
min})|}=2^{-3/2}M\simeq 0.35M\,,$ (3.6)

with

$M^{2}\equiv V_{,\chi\chi}|_{\chi_{c}\rightarrow 0}=\frac{2\lambda}{c}$ (3.7)

the oscillation frequency of the condensate for $\chi_{c}\rightarrow 0$.

Figure 2: (Left) Field values for the end of inflation $\chi_{\rm end}$
(orange), the location of the inflection point $\chi_{i}$ (green) and the
field value associated to the minimum of $V_{,\chi\chi}$ (or the maximum
momentum growing due to tachyonic instability) as a function of the non-
minimal coupling $\xi$. The red line indicates the critical field value
$\chi_{c}$ below which the potential becomes quartic. As expected, in the
metric limit $\xi\simeq 1500$ inflation ends too close to the inflection
point, which combined with a higher Hubble friction prevents the field to re-
enter the tachyonic region of the potential in successive oscillations.
(Right) Schematic form of the Einstein-frame HI potential for different values
of the non-minimal coupling $\xi$, with $\xi_{1}<\xi_{2}<\xi_{3}$.

If the resulting fluctuations are not efficiently diluted by the expansion of
the Universe, they will eventually backreact on the homogeneous dynamics,
leading to the fragmentation of the condensate into a highly inhomogeneous
state with sizable overdensities. These overdensities can potentially collapse
under their own self-interactions, leading to the formation of periodically-
varying pseudo-solitonic objects or oscillons [87, 88, 89, 90, 91, 92, 93].
Indeed, such localized quasi-spherical configurations are known to appear in
many inflationary models involving a shallower-than-quadratic potential [92,
94, 95, 96, 97],333This feature allows for the partial cancellation of the
dispersion terms in Eq. (3.1), leaving effectively a free harmonic oscillator
with a frequency of the order of inflaton mass at the minimum. as the one
under consideration.

Having a potential that can support oscillon solutions is of course a
necessary, although not sufficient, condition for their formation. In order to
properly assess the formation of oscillons, we will resort in what follows to
non-perturbative $3+1$ classical lattice simulations. We will make use of the
recently developed $\mathcal{C}osmo\mathcal{L}attice$ package [98, 99]. This
public code allows to solve the preheating dynamics in a self-consistent way,
accounting not only for particle production but also for its backreaction
effects on the homogeneous Friedmann equations (3.2) and therefore on the
scale factor evolution. To this end, it makes use of a cubic box of comoving
size $L$, $N^{3}$ points uniformly distributed and periodic boundary
conditions. The values of $L$ and $N$ are chosen in such a way all the
relevant momenta involved in our simulations are always well within the
infrared (IR) and ultraviolet (UV) resolution in momentum space,

$k_{\rm IR}=\frac{2\pi}{L}\,,\hskip 42.67912ptk_{\rm
UV}=\frac{\sqrt{3}}{2}Nk_{\rm IR}\,,$ (3.8)

associated respectively with the size of the simulation box and the lattice
spacing.

Following the standard procedure, the initial conditions for the Higgs field
are determined by evolving the homogeneous version of Eq. (3.1) from the deep
slow-roll regime (${\cal N}_{*}=55$) till the end of inflation, which we
identify with the onset of our simulations ($t=0$). Fluctuations over this
homogeneous background are then added as Gaussian random fields, as done
usually for systems with short classicalization times as the one under
consideration. The subsequent temporal evolution of the system is performed
using a 2nd-order Velocity Verlet algorithm, which turns out to be sufficient
to keep the violation of energy conservation below $\mathcal{O}(10^{-3})$
during the whole simulation time. More precisely, the scale factor is evolved
using the second Friedmann equation

$\frac{\ddot{a}}{a}=\frac{1}{3}\langle-2E_{K}+E_{V}\rangle\,,$ (3.9)

while the first one,

$H^{2}=\frac{1}{3}\langle E_{\rm K}+E_{\rm G}+E_{\rm V}\rangle\,,$ (3.10)

is rather used as a check of energy conservation. In these expressions, we
have conveniently split the total energy density into its kinetic (K),
gradient (G) and potential (V) counterparts,

$E_{K}\equiv\frac{1}{2}\dot{\chi}^{2}\,,\hskip
28.45274ptE_{G}\equiv\frac{1}{2a^{2}}(\nabla\chi)^{2}\,,\hskip
28.45274ptE_{V}\equiv V\,,$ (3.11)

with $\langle\ldots\rangle$ denoting the spatial averaging of the
corresponding quantity over the simulation volume. Note also that the lattice
potential accounts for both the large and small field regimes in Eq. (2.16),
suitably interpolated in our lattice implementation. Whenever needed, we will
make also use of temporal averages of specific quantities over a given number
of oscillations, denoting them as $\langle\ldots\rangle_{T}$.

###  3.1 Oscillon formation

Changing the value of the non-minimal coupling $\xi$ is expected to
substantially affect the background dynamics and the associated growth of
perturbations, leading to different fragmentation processes. Indeed, as shown
in Fig. 2, the ratio $\chi_{\rm end}/\chi_{i}$, between the field value
$\chi_{\rm end}$ at the end of inflation and the location of the inflection
point $\chi_{i}$, increases with $\xi$, allowing the background field to
explore the tachyonic region of the potential for a higher number of
oscillations. At the same time, the Hubble rate immediately after the end of
inflation scales inversely proportional to $\xi$ ($H\sim V^{1/2}_{\rm
inf}\sim\sqrt{\lambda}/\xi$), so that higher values of this quantity translate
also into a less efficient expansion rate.

To characterize the preheating stage in EC HI and the potential formation of
oscillons444It is important to remark that, contrary to the vast literature on
the subject, our localized configurations arise from a potential involving a
quartic part, instead of a mere quadratic one, around the minimum. As this
does not affect the formation of localized objects, we will refer to them as
oscillons. configurations in this setting, we will consider two benchmark
values of $\xi$ leading to different fragmentation modalities. These values
are summarized in Table 1, where we present also the grid parameters, the
infrared and ultraviolet resolution of the corresponding reciprocal lattice in
units of the oscillation frequency (3.7) and the final number of $e$-folds of
preheating considered in our simulations. In both cases, the associated
tensor-to-scalar ratio lies substantially below the expected sensitivity of
CMB Stage-IV experiments [100], making the stochastic GWs background generated
during inflation completely undetectable in the near future.

$\xi$ | $L\left[M^{-1}\right]$ | N | $k_{\rm IR}\left[M\right]$ | $k_{\rm UV}\left[M\right]$ | $\Delta\mathcal{N}$
---|---|---|---|---|---
$5.0\cdot 10^{4}$ | 30 | 512 | 0.2 | 89 | 2
$2.5\cdot 10^{5}$ | 60 | 512 | 0.1 | 44 | 1
Table 1: Lattice parameters and resolution for the two benchmark scenarios
considered in our simulations. For each of these cases, we present also the
final number of $e$-folds of preheating considered in our simulations,
$\Delta\mathcal{N}$.

$\xi=5\cdot 10^{4}$

$\xi=2.5\cdot 10^{5}$

Figure 3: Oscillons configurations for our two benchmark scenarios at time
$Mt=500$, corresponding respectively to $\Delta{\cal N}=2~{}({\rm left})$ and
$\Delta{\cal N}=0.9~{}({\rm right})$. The contours in the plots represent
regions with overdensities $5$ times larger than the average. Once formed,
these objects remain located at almost fixed spatial positions for a large
number of oscillations. Due to the expanding box, their fixed physical size
shrinks, however, with time.

The typical outputs of our simulations are presented in Fig. 3 and in this
URL, where we display several snapshots of the 3-dimensional lattice volume as
well as computer-generated movies explicitly demonstrating the formation of
oscillons in EC HI. The emergence of these pseudo-solitonic objects manifests
itself also through several observables which we now proceed to describe:

Figure 4:  Lattice results for our two benchmark scenarios $\xi=5\cdot 10^{4}$
(left), $2.5\cdot 10^{5}$ (right). First row: Time evolution of the lattice-
averaged field value $\langle\chi\rangle$ (blue) and the homogeneous solution
(red dashed), as compared to the location of the inflection points
$\pm\chi_{i}$ (green). Second row: Time evolution of the root mean squared
(3.12) characterizing the growth of perturbation. When this quantity becomes
comparable with the amplitude of the background oscillations, fragmentation
takes place, and $\langle\chi\rangle$ starts deviating from the homogeneous
solution.

  * •

Field average and time-dependent dispersion: As shown in the upper panels of
Fig. 4, the coherent nature of the inflaton field at sufficiently early times
is well-captured by the field average $\langle\chi\rangle$, which undergoes
damped oscillations driven essentially by the homogeneous version of Eq.
(3.1). However, the cumulative tachyonic production of Higgs excitations
translates eventually into a sharp deviation from this coherent behaviour,
with the average field value drastically dropping down to zero at that time.

As anticipated at the beginning of this section, a change in the non-minimal
coupling $\xi$ impacts both the number of times the field undergoes tachyonic
oscillations and the overall backreaction dynamics. In particular, for
$\xi=5\cdot 10^{4}$ and thanks to the $Z_{2}$ symmetry of the potential, the
homogeneous condensate is able to re-enter the instability region for about
consecutive 6-7 semi-oscillations before the cosmic expansion dampens out its
amplitude. Afterwards, the field continues oscillating in the quadratic part
of the potential, with the condensate remaining intact for about 8-9
additional semi-oscillations to finally fragment in less than one oscillation.
On the other hand, for $\xi=2.5\cdot 10^{5}$, the Higgs field spends most of
the time beyond the inflection point, leading to fragmentation in just 6 semi-
oscillations. This situation is reminiscent of what happens in the Palatini
formulation of HI, where the field incursions all the way back the
inflationary plateau lead to fragmentation in a single oscillation [66, 67].
Note, however, that in that case, the potential is essentially quartic all the
way up to the plateau, not being able therefore to support the formation of
oscillons.

A similar picture follows also from considering the root mean squared (rms)
characterizing the typical growth of perturbations,

${\rm rms}(\chi)=\sqrt{\langle\chi^{2}\rangle-\langle\chi\rangle^{2}}\,.$
(3.12)

As shown in the lower panels of Fig. 4, backreaction takes place precisely
when this quantity reaches the amplitude of the background component and the
average energy density stored in perturbations becomes comparable with that of
the inflaton condensate.

Figure 5: Lattice results for our two benchmark scenarios $\xi=5\cdot 10^{4}$
(left), $2.5\cdot 10^{5}$ (right). First row: Evolution of the volume-averaged
(semi-transparent) kinetic (blue), gradient (red) and potential (green)
contributions to the total energy, together with their time averages (opaque).
Second row: Time evolution of the equation of state parameter (blue) and its
time average (orange).

  * •

Energy budget and average equation of state: As previously mentioned, one of
the key ingredients allowing the inflaton condensate to re-enter the tachyonic
region of the potential is a relatively large ratio between the field value at
the end of inflation and the location of the inflection point. However, an
important contribution to the dynamics comes also from the expansion rate
regulating the damping of oscillations. In the presence of particle creation,
this is effectively determined by the relative contribution of kinetic,
gradient and potential counterparts to the total energy density, or,
equivalently, by the associated equation-of-state parameter,

$w\equiv\frac{\langle p\rangle}{\langle\rho\rangle}=\frac{\langle
E_{K}-E_{G}/3-E_{V}\rangle}{\langle E_{K}+E_{G}+E_{V}\rangle}\,.$ (3.13)

The evolution of the different energy components for our two benchmark
scenarios is displayed in the upper panels of Fig. 5. During the initial
tachyonic oscillations, the energy budget is dominated by the potential energy
contribution, reflecting the fact that the inflaton field spends more time
close to the turning points than to the minimum of the potential. This is also
apparent in the evolution of the equation-of-state parameter, displayed in the
lower panels of the same figure. In particular, while this quantity oscillates
wildly between $w=-1$ (potential energy domination) and $w=1$ (kinetic energy
domination), its temporal average lies always within the interval $-1<\langle
w\rangle_{T}<0$, accounting both for the early-time field excursions in the
plateau domain and the light tendency to equipartition between kinetic and
potential energy contributions once the oscillation amplitude enters the
quadratic part of the potential. This situation holds until the onset of
backreaction, where the growth of perturbations leads to the appearance of a
non-negligible gradient component that becomes commensurable or even equal to
the potential counterpart. At this fragmentation stage, the equation of state
approaches rapidly zero, in accordance with Eq. (3.13) and the energy budget
at that time. The created oscillons behave then collectively as pressureless
dust.

At this point, it is important to remark that, in the parameter range of our
simulations, backreaction happens always before the average field amplitude
drops below the threshold value $\chi_{c}$. Therefore, when fragmentation
happens and oscillons form, the Universe enters a sustained period of matter
domination regardless of the shape of the potential. This contrasts with the
naive expectation following from an incomplete homogeneous treatment, where a
radiation-dominated stage is expected to develop at field values
$\bar{\chi}<\chi_{c}$, where the EC HI potential (2.16) becomes approximately
quartic [101].

  * •

Power spectra:

Figure 6: Time evolution of the energy power spectra for our two benchmark
scenarios, with each spectrum taken at intervals of $\Delta(Mt)=10$. The
initial growth of perturbations at low momenta is clearly distinguishable, as
well as the broad peak appearing due to oscillon formation. Once oscillons
have formed, we observe a residual growth of momenta in the UV. This is due to
the presence of the quartic region at the bottom of the EC HI potential and
does not happen therefore in other purely quadratic models.

The unstable tachyonic band in Eq. (3.6) can be quantitatively observed in the
Higgs power spectrum $\mathcal{P}_{k}$ or two-point correlation function in
Fourier space, customarily defined through

$\langle\chi^{2}\rangle=\int\dfrac{k^{3}}{2\pi^{2}}d\ln
k\,\mathcal{P}_{k}~{}.$ (3.14)

As shown in Fig. 6, the associated infrared fluctuations experience a rapid
growth at the very early stages of preheating, where the average field value
exceeds the location of the inflection point $\chi_{i}$. As expected, this
growth is stronger for higher values of $\xi$, in agreement with the higher
efficiency and longer duration of the tachyonic regime in this case.

Due to self-resonance, the enhancement of perturbations continues even for
$\langle\chi\rangle<\chi_{i}$, albeit at a significantly slower rate.
Eventually, the growth of fluctuations is halted by backreaction effects and a
broad peak appears at a physical momentum $\rm k_{p}$, corresponding to a
typical oscillon size $R\sim(2\pi)/{\rm k_{p}}$. 555Since the momenta on the
lattice are co-moving and the oscillons have a fixed physical size, this scale
shift towards the UV due to the expansion of the Universe. It is interesting
to notice that increasing $\xi$ affects also slightly the typical excited band
and the final size of the oscillons. As we will see in the forthcoming
section, this has a very mild impact on the peak frequency of the produced GWs
signal.

  * •

Energy density histograms: The smoking gun that signals the actual existence
of quasi-spherical structures of a fixed physical size, rather than other
kinds of anisotropies, is the energy density distribution [93]. As shown in
Fig. 7, the histograms containing this quantity develop a flattening region at
high densities, representing the points on the lattice that lie within the
overdense oscillon configurations. This plateau-like feature encodes the local
conservation of energy in oscillons as compared to the decreasing energy
density of the simulation box $\langle\rho\rangle\propto a^{-3}$, so that the
plateau in the distribution becomes more and more pronounced at later times.

Note that while the oscillons remain highly subdominant in volume, they
dominate the total energy budget. In particular, the fraction of regions where
the energy density exceeds the average energy density $\langle\rho\rangle$ by
a threshold factor 5,

$f=\frac{\int_{\rho>5\langle 5\rangle}\rho\,dV}{\int\,\rho\,dV}\,,$ (3.15)

accounts for $\sim 70\%$ of the total energy.

Figure 7: Energy density histograms for the benchmark case $\xi=5\cdot
10^{4}$, with the heights of the bins normalized to set the area of the
histograms to $1$. One can clearly distinguish the transition from the
configuration before fragmentation (left), to the one slightly after oscillon
formation (centre), where the distribution starts flattening, signalling the
appearance of highly overdense regions with density contrasts
$\rho/\langle\rho\rangle>5$. As expected, the flatter part becomes more
pronounced at later times (left), as the energy in the oscillons remains
constant and the one in the background dilutes as matter.

###  3.2 Gravitational waves signal

In the previous section, we have seen how oscillons form in EC HI for a wide
range of the parameter space compatible with observations, with different
values of the non-minimal coupling $\xi$ leading to different fragmentation
dynamics. In this section, we explore whether fragmentation and oscillon
formation leads to distinctive GWs signatures.

The anisotropies in the inflaton configuration are expected to source linear
metric perturbation $h_{ij}$ through the wave equation

$\ddot{h}_{ij}+3H\dot{h}_{ij}-a^{-2}\nabla^{2}h_{ij}=2a^{-2}\Pi^{\rm
TT}_{ij}~{},$ (3.16)

with

$\Pi^{\rm TT}_{ij}=(\partial_{i}\chi\partial_{j}\chi)^{\rm TT}$ (3.17)

the transverse-traceless (TT) part of the associated anisotropic energy-
momentum tensor. Employing the GW module implemented in
$\mathcal{C}osmo\mathcal{L}attice$ [102], we can easily extract the energy
density power spectrum of the GWs signal, namely [98]

$\Omega_{\rm GW}=\frac{1}{3H^{2}}\frac{{\rm d}\rho_{\rm GW}}{{\rm
d}\log\,k}~{},\hskip 28.45274pt{\rm with}\hskip 28.45274pt\rho_{\rm
GW}=\frac{1}{4}\langle\dot{h}_{ij}\dot{h}_{ij}\rangle~{}.$ (3.18)

The results of the simulations for the two benchmark scenarios considered in
the previous section are shown in Fig. 8. We observe that the GWs spectrum
grows initially with time, developing a main peak at low momenta. When
fragmentation starts, the spectrum quickly rises, and, shortly after
fragmentation ends, there appears a subdominant secondary peak at a higher
frequency. Once the oscillons have formed the spectrum stops growing,
reflecting their individual character and roughly spherical symmetry.

Figure 8: Gravitational waves spectra for our two benchmark scenarios. The
first scales to grow are those related to the inhomogeneities arising during
the initial growth of perturbations and the fragmentation process. In a second
moment, we observe a secondary peak at higher momenta, due to the oscillon
formation and relaxation to a spherical shape. Afterwards, the growth stops,
as there are no more strong dynamical inhomogeneities to produce GWs.

The proper connection of the GW spectrum at the time of emission to that at
present times requires the knowledge of the full cosmological history upon
fragmentation, and in particular, of the average lifetime of the created
oscillon configurations. Unfortunately, observing the fate of these very
stable objects within a realistic scenario like the one under consideration
requires not only extensive computational power but also a detailed lattice
implementation of the Higgs couplings to other Standard Model degrees of
freedom, a task that significantly exceeds the proof-of-principle scope of
this paper (cf., however, Section 4). In order to estimate the current
amplitude $\Omega_{0,{\rm GW}}$ and frequency $f_{0}$ of the spectrum, we
follow therefore a poor-man’s parametric approach based on two working
approximations. First, we will presume the Universe to evolve adiabatically
until the present day, ensuring entropy conservation. Second, we will consider
that, once oscillons have formed, the associated equation-of-state parameter
evolves gradually to that of radiation domination, the moment at which we
assume the SM matter content to be completely thermalized. 666Note that,
although most likely true in a SM setting like the one under consideration
[59, 103], the onset of radiation domination does not necessarily coincide
with that of thermalization. In particular, the former requires only the
effective energy dominance of relativistic particles, which might have,
however, a non-thermal spectrum, see e.g. Refs. [104, 105]. With these
assumptions, we get [106]

$\displaystyle f_{0}=\frac{k_{p}}{2\pi a_{0}}=\dfrac{k_{p}}{a_{\rm e}\rho_{\rm
e}^{1/4}}e^{{-\frac{1}{4}(1-3\bar{w})\Delta{\cal N}_{\rm RD}}}\,4\cdot
10^{10}Hz~{},$ (3.19) $\displaystyle\Omega_{0,{\rm GW}}~{}h^{2}\simeq
1.6\times 10^{-5}e^{-(1-3\bar{w})\Delta{\cal N}_{RD}}\,\Omega_{\rm e,GW}\,,$
(3.20)

with $a_{0}$ the scale factor today, the quantities $a_{\rm e}$, $\rho_{\rm
e}$ and $\Omega_{\rm e,GW}$ denoting respectively the scale factor, the total
energy density and the gravitational waves’ energy fraction at the time of
emission and $\bar{w}$ the mean equation-of-state parameter in the $e$-folds
interval ${\cal N}_{\rm RD}$ between GW emission and the onset of radiation
domination. As explicit in Eq. (3.19), there are two main factors contributing
to the redshift of the maximum available signal. On the one hand, the likely
existence of a prolonged intermediate stage with an equation of state
$0<\bar{w}<1/3$. On the other hand, $f_{0}$ will be smaller for higher values
of the energy density at the time of formation. Nonetheless, a higher energy
density during preheating implies also a smaller horizon and hence higher
momenta in the spectrum, so that opposite contributions from $k_{p}$ and
$\rho_{e}$ tend to cancel out. As usually happens in preheating scenarios, the
resulting amplitude of the signal is commensurable with the sensitivity of
current GWs experiments, but the peak frequency turns out to be ${\cal O}(\rm
GHz)$, lying therefore far away from the available observational window777See
[107] for experimental proposals in the $\rm MHz-GhZ$ window.. As a
consequence, the main distinctive feature of the EC formulation, compared to
the Palatini and metric one, is the possibility of a prolonged stage of matter
domination, with consequences on the inflationary observables.

##  4 Towards preheating in the Standard Model

Now that we have established the presence of oscillons in the scalar sector of
EC-HI, let us turn our attention to the other SM components, namely the
massive electroweak bosons and fermions. As shown in Refs. [57, 58, 59, 60,
66], the generation of these two species is intrinsically intertwined, with
the Bose enhancement of the former strongly depending on their perturbative
decay into fermions.

The so-called Combined preheating formalism developed in Ref. [58] (cf. also
Ref. [59]) allows for a semi-analytic treatment of the gauge boson production
in Higgs inflation, under several simplifying assumptions. First, it replaces
the usual Higgs-gauge boson interactions following from the SM covariant
derivatives with global couplings between the Higgs field and three scalar
degrees of freedom playing the role of gauge bosons. This approximation is
well-supported by several numerical studies, explicitly demonstrating the
qualitative equivalence of global and Abelian interactions during the initial
stages of preheating, where the non-linearities associated with gauge boson
self-interactions can be still safely neglected [108, 109, 110]. Second, the
formalism makes use of a WKB approximation in order to determine the
production of gauge boson excitations at every semi-oscillation of the
inflaton field. This well-established procedure takes into account the
spontaneous and stimulated emission of particles at the bottom of the
potential, as well as potential interference effects among semi-oscillations
[111]. Third, Combined preheating accounts for the potential decay of the
produced gauge bosons into the SM quarks and leptons by considering effective
decay widths proportional to the gauge boson masses, as done, for instance, in
instant preheating scenarios [112]. This approximation explicitly neglects a
plethora of annihilations processes, which are, however, subdominant at early
times or low gauge boson number densities [59]. All together the Combined
preheating assumptions give rise to a master, phase-averaged, equation
relating the number densities $n_{k}(j)$ of each gauge boson field
polarization at two consecutive zero crossings [59]

$\left[\frac{1}{2}+n_{k}((j+1)^{+})\right]=(1+2C_{k}(j))\left[\frac{1}{2}+n_{k}(j^{+})\,e^{-\Theta(j)}\right]\,,$
(4.1)

with $C_{k}(j)$ a $k$-dependent IR window function depending on the small-
amplitude behaviour of the corresponding gauge boson mass,

$\Theta(j)=\int_{t_{j}}^{t_{j}+1}\Gamma(t)dt$ (4.2)

the mean decay rate into fermions among two consecutive crossings $t_{j-1}$
and $t_{j}$ and $\Gamma(t)$ the instantaneous rate evaluated on the background
equations of motion. The explicit form of these building functions can be
obtained by transforming the usual (scalarized) Higgs-gauge boson
interactions, $\sim g_{i}^{2}/8\,h^{2}A^{2}$, into the Einstein frame, with
$g_{i}=g,g/\cos\,\theta_{w}$ for the $A=W^{\pm}$ and $Z$ bosons, $g$ the
$U(1)_{Y}$ coupling and $\theta_{w}$ the weak mixing angle. Upon performing an
additional field redefinition $A\to\Omega A$ for canonically normalizing the
resulting kinetic sector, we obtain an interaction term888For simplicity we
will always refer to quantities related to the $W^{\pm}$ bosons, Although the
same formulas, with the appropriate couplings, apply to the $Z$ boson.

$\frac{{\cal L}_{int}}{\sqrt{-g}}=\frac{1}{2}m^{2}A^{2}\,,$ (4.3)

with

$m^{2}=\dfrac{g^{2}}{4}\dfrac{h^{2}}{1+\xi
h^{2}}=\begin{cases}\dfrac{g^{2}}{4}\dfrac{\chi^{2}}{1+\xi\chi^{2}}~{},&\hskip
28.45274pt\chi<\chi_{c}~{},\\\
\dfrac{g^{2}}{4\xi}\left[1-\exp\left(-\dfrac{2\xi|\chi|}{\sqrt{c}}\right)\right]~{},&\hskip
28.45274pt\chi\gg\chi_{c}\,,\end{cases}$ (4.4)

an effective mass approaching the constant value $g^{2}/(4\xi)$ in the large
field regime and dropping to zero at each minimum-crossing of the inflaton
field, where the adiabaticity condition ${\dot{m}/m^{2}<1}$ is maximally
violated and gauge boson production takes place. In this latter regime, the
equation of motion for the $A$-field perturbations takes the form

$\ddot{\delta A_{\textbf{k}}}+3H\dot{\delta
A_{\textbf{k}}}+\left(\textbf{k}^{2}/a^{2}+m^{2}(\chi)\right)\delta
A_{\textbf{k}}=0~{}.$ (4.5)

with

$m^{2}(\chi)\simeq\begin{cases}\dfrac{g^{2}}{4}\chi^{2}~{},&\hskip
14.22636pt\textrm{for}\hskip 14.22636pt\chi<\chi_{c}~{},\\\
\dfrac{g^{2}}{2\sqrt{c}}|\chi|~{},&\hskip 14.22636pt\textrm{for}\hskip
14.22636pt\chi\gg\chi_{c}~{},\end{cases}$ (4.6)

depending only on the non-minimal coupling $\xi$ through the constant $c$,
which, for a given value of $\lambda$, is almost completely determined by the
CMB normalization, cf. Eq. (2.20).

Figure 9: Field values at which adiabaticity condition ${\dot{m}/m^{2}<1}$ is
first violated in our two benchmark scenarios (orange), as compared to the
homogeneous solution for the background (blue) and the crossover value
$\chi_{c}$ below which the Einstein-frame potential becomes quartic (red).
Note that, although adiabaticity is always broken in the quadratic part of the
potential, the value $\chi_{ad}$ approaches $\chi_{c}$ at higher $\xi$.

As shown in Fig. 9, the violation of adiabaticity condition $\dot{m}/m^{2}<1$
following from the numerical solution of the background equations of motion
happens generically at field values $\chi_{a}>\chi_{c}$, even though these two
scales approach each other for increasing values of $\xi$. Regarding possible
perturbative decays into fermions and gauge bosons, there are two key
conditions that must be satisfied at the same time. On one hand, for the decay
to be kinematically allowed, the condition $M>2m(2m_{f})$ must be fulfilled.
On the other hand, the decay can take place efficiently only if the rate
$\Gamma$ is larger than the Hubble rate $H$. While for the $W^{\pm}$ and $Z$
bosons the first condition is never fulfilled, the Higgs can still decay into
all the fermions, except the top quark, with different situations. As the
decay rate is proportional to the fermion mass, the Higgs will be more likely
to decay into the heavier fermions. Nonetheless, these decay channels remain
kinematically blocked for timescales much longer than the one for oscillon
formation. On the contrary, the decay into light fermions will be
kinematically open at early times, but the smallness of their Yukawa
couplings, and therefore decay rates, will make the decay inefficient on
timescales much longer than the ones relevant to preheating. Taking this into
account, together with the fact that we only aim at providing a qualitative
picture of this otherwise complicated dynamics, we will neglect in what
follows the contribution of the upper term in the right-hand side of Eq.
(4.6). In this limit, the scenario under consideration reduces conceptually to
that studied in the metric formulation of the theory [57, 58, 59, 60]. By
performing the same type of computations, the IR window function in Eq. (4.1)
takes the form999For the sake of simplicity, we neglect the slight differences
among $W^{\pm}$ and $Z$ encoded in Lorentz invariant phase space factors, cf.
Ref. [58] for details on this issue.  [58]

$C(x)=\pi^{2}\left[{\rm Ai}(-x^{2}){\rm Ai}^{\prime}(-x^{2})+{\rm
Bi}(-x^{2}){\rm Bi}^{\prime}(-x^{2})\right]^{2}\,,$ (4.7)

with $\rm Ai$ and $\rm Bi$ the Airy functions of first and second kind,
$x=k/aM\left(j/q\right)^{1/3}$, $j$ the number of inflaton zero crossings and
the parameter

$q=\frac{\sqrt{c}g^{2}\chi_{\rm
end}}{8\pi\lambda}=\frac{cg^{2}}{16\pi\lambda\,\xi}\log\left(1+\frac{2\sqrt{2}\xi}{\sqrt{c}}\right)\,,$
(4.8)

decreasing for increasing values of $\xi$. Assuming that the same
approximation holds in our case, gauge bosons will be produced with a typical
scale $k_{\rm gb}\sim q^{\frac{1}{3}}M$, which for our benchmark scenarios
$\xi=5\cdot 10^{4}$ and $2.5\cdot 10^{5}$, corresponds roughly to $k_{\rm
gb}/M\sim 20,10$, respectively. The depletion of the bosons produced at a
crossing is efficient whenever the factor $\Theta$ in Eq. (4.1) is large
enough to compensate for the growth of $n_{k}$ due to parametric resonance.
Taking into account that the specific form of the decay rate into SM particles
is proportional to the mass of the bosons,

$\Gamma=\dfrac{3g^{2}m}{16\pi}\,,$ (4.9)

we can compare the two effects for different values of $\xi$. On the one hand,
being the maximum decay rate proportional to the gauge bosons masses, and the
latter to $1/\sqrt{\xi}$, we have that for larger values of the non-minimal
coupling, the maximum $\Gamma$ will have a lower value. On the other hand,
increasing $\xi$ substantially modifies the background dynamics, resulting in
a higher period for the first few oscillations. To get an estimate of the
actual timescales on which the decay rate becomes inefficient, we can consider
the maximum value of the Bose enhancement factor happening at $x=0$, and
compare it with the suppression factor $e^{-\Theta}$, by evaluating the decay
rate on the homogeneous solution for the background. As shown in Fig. 10, in
both our scenarios the decay rate is highly efficient for the first
$\mathcal{O}(10^{2}-10^{3})$ zero crossings, significantly exceeding the
typical number of semi-oscillations needed for oscillon formation,
$\mathcal{O}(10-20)$.

Figure 10: Interplay between the Bose enhancement factor for $k=0$ and the
suppression due to gauge boson decay into fermions for a fiducial gauge
coupling $g^{2}=0.3$, as a function of the number of zero-crossings $j$. The
factor $\Theta$ is computed on the numerical solution for the background.
Although the two effects cancel out after $\mathcal{O}(10^{3})$ zero-
crossings, the actual modes can start growing slightly before, once the proper
recursive formula in Eq. (4.1) is taken into account. Nonetheless, we expect
the gauge boson production to be heavily subdominant during the first tens of
oscillations where fragmentation and oscillon formation takes place.

This results in two main effects. Firstly, for a large number of oscillations,
almost all the energy stored in the gauge bosons at each zero crossing is
transferred to SM fermions, making this the dominant channel for energy
transfer to SM particles other than the Higgs. Secondly, the SM gauge bosons
cannot efficiently accumulate, so their parametric resonance is substantially
delayed, preventing them from backreacting on the inflaton condensate, already
at the early stages of its evolution.

Now that we have established that the decay into fermion efficiently
suppresses the production of gauge bosons, it is natural to wonder how the
interplay between the inflaton and other SM particles can affect the formation
of oscillons. Based on our simple linear analysis it is clear that the amount
of energy transferred to fermions through the gauge bosons decay needs several
oscillations to become important. For instance, in the metric case [60], this
can take up to hundreds of oscillations. Taking into account that increasing
the non-minimal coupling reduces the number of oscillations before
fragmentation, we can infer that such transfer becomes less efficient for
larger $\xi$, leaving the process of oscillon formation unaltered. However, it
is important to point out that our previous analysis works only at a linear
level, and does not take into account possible non-linear effects associated
with the quick tachyonic growth of inflaton perturbations, as well as the
highly inhomogeneous state between fragmentation and oscillon formation.
Although it would be interesting to perform a fully non-linear analysis,
including both fermions and gauge bosons in our lattice, such a task presents
some technical difficulties. First, although
$\mathcal{C}osmo\mathcal{L}attice$ can in principle support the implementation
of the full Standard Model gauge group, this can only be done in the Jordan-
Frame, where the covariant derivatives are linear in the fields. Second, even
if the gauge fields are introduced as mere scalar fields [60, 67], the created
particles will be produced with typical momenta much higher than those
relevant for oscillons, therefore requiring simulations with very high
resolution, far beyond the scope of this paper. In particular, the coverage of
all momenta involved in the picture, especially at later times, would require
at least $N>1024$ lattice sites for realistic gauge couplings $g^{2}\simeq
0.3$. Such simulations would be extremely time-consuming, especially if all
three gauge bosons are taken into account. 101010Using a $N=512$ lattice, we
are able to confirm, however, that the impact of gauge bosons in the tachyonic
amplification of Higgs excitations remains subdominant until oscillon
formation for $g^{2}\simeq{\cal O}(10^{-2})$, even if the decay into fermions
is completely neglected and the depletion mechanism discussed above is absent.
Third, when introducing spinors on the lattice one encounters the infamous
fermion-doubling problem [113], leading to the appearance of additional copies
of the latter. Although this limitation could be circumvented with an
effective treatment along the lines of Ref. [60] (see also Ref. [114]), the
resolution of the IR and UV momenta involved would still remain.

##  5 Conclusions and outlook

Among the plethora of inflationary models in the literature, Higgs inflation
stands out as one rather minimalistic scenario not requiring the introduction
of additional degrees of freedom beyond the Standard Model content while
potentially allowing for a complete analysis of the preheating stage following
the inflation. In spite of its simplicity, the inclusion of a non-minimal
coupling of the Higgs field to gravity breaks the usual degeneracy among
metric-affine representations, with different formulations of gravity giving
rise to different inflationary predictions and preheating dynamics.

In this paper, we have studied the preheating phase for Higgs Inflation in
Einstein-Cartan gravity using fully-fledged numerical lattice simulations in
$3+1$ dimensions. We have found that this scenario allows for the formation of
dense and spatially localized oscillon configurations, constituting up to 70%
of the total energy density for suitable model parameters. These pseudo-
solitonic objects survive for the entire simulation time, leading to a
prolonged period of matter domination that modifies the post-inflationary
history and therefore the minimum duration of the inflationary phase needed to
solve the hot Big Bang problems. Furthermore, we have shown that such
structures can source a significant gravitational wave signal, providing an
alternative observational channel for this inflationary model, besides the
usual stochastic background of primordial tensor perturbations generated
during inflation. Unfortunately, as usually happens in preheating scenarios,
the associated peak frequency turns out to be significantly larger than the
one accessible by current and planned GWs experiments.

While the results of this paper have been explicitly derived assuming a simple
Nieh-Yan interaction (2.14), they can be easily extended to the more general
constant $c$ scenarios within the EC multiverse111111For instance, a choice
$\xi_{vv}=\xi_{aa}=\xi$, $c_{av}=0$ in Eq. (2.11) translates also into a
constant $c$ value,  $c=\xi+6\xi_{\eta,{\rm eff}}^{2}\hskip
28.45274pt\textrm{with}\hskip 28.45274pt\xi_{\eta,{\rm
eff}}^{2}\equiv\xi^{2}+\frac{2}{3}\left(\frac{\zeta_{ha}^{2}}{c_{aa}}+\frac{\zeta_{hv}^{2}}{c_{vv}}\right)\,,$
(5.1) allowing to recover the phenomenology described in this paper for any
set of parameters $\zeta_{ha}$, $\zeta_{hv}$, $c_{aa}$, $c_{vv}$ leading
numerically to $\xi_{\eta,{\rm eff}}^{2}\simeq\xi_{\eta}^{2}$ and
commensurable $\xi$ values. [35], including also scale-invariant extensions
[55] or even TDiff generalizations [47, 115]. Indeed, the main condition for
oscillon formation is effectively encoded in the field redefinition
$d\bar{\chi}/d\bar{h}$ entering Eq. (3.3) or, equivalently, in the kinetic
function $K(h)$ appearing in the Einstein frame-action (2.9). Provided this
quantity is sufficiently close to one (but still larger), the inflaton field
will be allowed to exceed the inflection of the potential for a given number
of oscillations, building up fluctuations through tachyonic instabilities and
eventually fragmenting out into oscillon configurations.

Our results are only a first step towards the full characterization of the
preheating process in Einstein-Cartan HI. In particular, it would be
interesting to address several aspects not properly taken into account in this
study:

  1. 1.

Oscillons lifetime: Understanding the lifetime of oscillons is indeed of
uttermost importance for determining the precise value of the inflationary
observables to be confronted with observations. On top of that, oscillons
could also generate gravitational waves when they decay, with longer lifetimes
translating into higher chances of observing GWs within the frequency windows
of current and future gravitational wave experiments. In all of our
simulations, it is clear that oscillons have enough time to form before the
quartic coupling becomes relevant. However, their long-time classical and
quantum stability [116, 117, 118, 119, 120], as well as their possible
formation for higher values of $\xi$, remains to be checked. Unfortunately,
performing simulations for these scenarios turns out to be rather problematic,
as the quartic potential affects mainly the momenta in the far UV regime,
while the typical scale of oscillons is located in the IR. In this regard, it
would be interesting to test whether the presence of quartic self-interactions
can lead to parametric resonance effects within oscillons, along the lines of
Ref. [118]. Future research could also explore the impact of the complete SM
structure on the inflaton lifetime, accounting for rescattering effects
induced by non-Abelian interactions [109, 110] or potential spike effects in
longitudinal gauge degrees of freedom [61, 62, 63, 64, 65]. Note, however,
that the Riemannian spikes advocated in the latter references are expected to
be subdominant in our context. In particular, the combination of the
uncertainty principle and the approximate conservation of energy during the
first few oscillations, translates roughly into a typical energy scale $\Delta
E_{\rm sp}\sim\Delta t_{\rm sp}^{-1}\sim\dot{\chi}_{\rm
0}/\chi_{c}\sim\sqrt{c\,V_{\rm end}}\sim\beta\sqrt{\lambda}$ for the particles
created by the spike, with $\Delta t_{\rm sp}$ the time needed to cross the
spike, $\chi_{0}$ the field velocity at the minimum of the potential, $V_{\rm
end}$ the potential energy at the end of inflation and

$\beta\equiv\frac{\sqrt{c}}{\sqrt{c}+2\sqrt{2}\xi}\,.$ (5.2)

While $\beta\sim{\cal O}(1)$ in the metric formulation [64], this quantity is
significantly smaller in the EC HI settings considered in this paper,
$\beta\sim{\cal O}(10^{-2}-10^{-3})$, leading to a transfer of energy through
this mechanism of order $\rho_{\rm sp}/V_{\rm end}\sim 1/(8\pi^{2})(\Delta
E_{\rm sp})^{4}/V_{\rm end}\sim\lambda\,c\,\beta^{2}/(16\pi^{2})\sim{\cal
O}(10^{-2}-10^{-3})$.

  2. 2.

Gravitational perturbations and primordial black holes: Since oscillons
represent well-localized objects with high overdensities, it is natural to
wonder whether these quasi-spherical structures could potentially collapse
into black holes, a possibility not accounted for in our simulations, where
metric perturbations are completely disregarded. Assuming spherical symmetry,
the strength of the Newtonian potential at the surface of a single oscillon of
radius $R=(2\pi/{\rm k}_{p})$, mass $\mathcal{M}$, and core energy density
$\rho_{c}$ can be estimated as121212This result comes from the fact that
oscillons are expected to have a field amplitude at the core proportional to
that of the oscillating homogeneous condensate at the time of backreaction,
$\chi_{c}\sim\chi_{\text{br}}$, which result in
$\rho_{c}\sim\langle\rho\rangle_{\text{br}}$. [93]

$|\Phi|\sim\dfrac{\mathcal{M}}{8\pi R}\sim\dfrac{\rho_{c}R^{2}}{6}\sim
H^{2}_{\text{br}}R^{2}~{},$ (5.3)

with $H_{\text{br}}$ the value of the Hubble parameter at the time of
backreaction (fragmentation). For all the scenarios considered in this paper,
this compactness turns out to be significantly smaller than one, $|\Phi_{\rm
EC}|\sim 10^{-3}-10^{-4}$, supporting our approximations. We point out,
however, that this estimate addresses only the typical oscillon sizes that we
expect to find in our simulation, being a priori possible, although unlikely
due to the magnitude of $\Phi$, that some of them meet the condition for
collapse. Moreover, if stable enough, oscillons could eventually cluster on
long time scales [121, 122].

  3. 3.

Fermions in EC gravity: As mentioned when introducing EC gravity, one of the
main perks of this formulation is the possibility to naturally account for the
coupling of fermions to gravity, thanks to the presence of a non-vanishing
spin connection. However, the use of a theory with torsion leads inevitably to
the appearance of higher-order four-fermion and scalar-fermion interactions
appearing when torsion is integrated out [70]. Interestingly enough, these
operators can be used, for instance, to produce singlet fermions that can play
the role of dark matter, as done in Ref. [34]. Unfortunately, even if the
universality of gravitational couplings to fermions is then assumed, one would
need to take into account potential non-canonical kinetic terms for fermions,
increasing the freedom of the model, and reducing, therefore, its predictive
power. As long as the scale introduced by these operators is higher than the
inflationary one, we can expect their effect to be subdominant both in the
running of $\lambda$, as well as in the gauge bosons’ perturbative decay.
Nonetheless, they might play an important role if non-perturbative effects are
taken into account.

We leave the in-depth study of these interesting aspects for future work.

## Acknowledgments

M. P. (ORCID ID 0000-0002-2387-5948) acknowledges the Fundação para a Ciência
e a Tecnologia (FCT), Portugal, for the financial support to the Center for
Astrophysics and Gravitation-CENTRA, Instituto Superior Técnico, Universidade
de Lisboa, through the Project No. UIDB/00099/2020. M. P. thanks also the
support of this agency through the Grant No. SFRH/BD/151003/2021 in the
framework of the Doctoral Program IDPASC-Portugal. JR (ORCID ID
0000-0001-7545-1533) is supported by a Ramón y Cajal contract of the Spanish
Ministry of Science and Innovation with Ref. RYC2020-028870-I. The authors
acknowledge the Department of Theoretical Physics of the Universidad
Complutense de Madrid for allowing access to the cluster facilities utilized
for the research reported in this paper.

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