# Higgs-like spectator field as the origin of structure

Alexandros Karam<EMAIL_ADDRESS>Laboratory of High Energy and
Computational Physics, National Institute of Chemical Physics and Biophysics,
Rävala pst. 10, 10143 Tallinn, Estonia  Tommi Markkanen
<EMAIL_ADDRESS>Laboratory of High Energy and Computational Physics,
National Institute of Chemical Physics and Biophysics, Rävala pst. 10, 10143
Tallinn, Estonia  Helsinki Institute of Physics, P.O. Box 64, FIN-00014
University of Helsinki, Finland  Luca Marzola<EMAIL_ADDRESS>Laboratory
of High Energy and Computational Physics, National Institute of Chemical
Physics and Biophysics, Rävala pst. 10, 10143 Tallinn, Estonia  Sami Nurmi
<EMAIL_ADDRESS>Helsinki Institute of Physics, P.O. Box 64, FIN-00014
University of Helsinki, Finland  Department of Physics, University of
Jyväskylä, P.O. Box 35, FI-40014 University of Jyväskylä, Finland  Martti
Raidal<EMAIL_ADDRESS>Laboratory of High Energy and Computational
Physics, National Institute of Chemical Physics and Biophysics, Rävala pst.
10, 10143 Tallinn, Estonia  Arttu Rajantie<EMAIL_ADDRESS>Department of Physics, Imperial College London, London, SW7 2AZ, United
Kingdom

###### Abstract

We show that the observed primordial perturbations can be entirely sourced by
a light spectator scalar field with a quartic potential, akin to the Higgs
boson, provided that the field is sufficiently displaced from vacuum during
inflation. The framework relies on the indirect modulation of reheating, which
is implemented without any direct coupling between the spectator field and the
inflaton and does not require non-renormalisable interactions. The scenario
gives rise to local non-Gaussianity with $f_{\rm NL}\simeq 5$ as the typical
signal. As an example model where the indirect modulation mechanism is
realised for the Higgs boson, we study the Standard Model extended with right-
handed neutrinos. For the Standard Model running we find, however, that the
scenario analysed does not seem to produce the observed perturbation.

Introduction. The cosmological perturbations observed in the Cosmic Microwave
Background (CMB) are most commonly interpreted as a product of the inflaton
field quantum fluctuations, which are stretched to macroscopic scales during
the inflationary expansion in the early Universe.

Alternatively, the observed perturbation could be sourced by fluctuations of
spectator fields which have no dynamical effect during inflation but affect
the dynamics later on. Well-known examples include the curvaton model Lyth and
Wands (2002); Moroi and Takahashi (2001); Enqvist and Sloth (2002); Linde and
Mukhanov (1997); Mollerach (1990) and the modulated reheating scenario Kofman
(2003); Dvali  _et al._ (2004). In the latter, the inflaton field decay width
is modulated by spectator field fluctuations which sources the curvature
perturbation and typically gives rise to significant non-Gaussianities
Ichikawa  _et al._ (2008); Litsa  _et al._ (2020a). Modulated reheating has
been extensively studied in the past, most usually in setups which rely on the
direct coupling of the inflaton field to light spectator scalars – for
instance, the Higgs boson. We refer the reader to Ref. Ichikawa  _et al._
(2008) for the general formalism of modulated reheating, and to Refs.
Kobayashi  _et al._ (2012); Fujita and Harigaya (2016); Lu  _et al._ (2020);
Choi and Huang (2013); De Simone  _et al._ (2013); Cai  _et al._ (2013);
Freese  _et al._ (2017); Lu  _et al._ (2020); Litsa  _et al._ (2020b, a) for
models with a specific focus on the Higgs field. Further related work can be
found in Refs. Chambers and Rajantie (2008a, b); Chambers  _et al._ (2010);
Fujita  _et al._ (2013, 2014). Spectator field fluctuations can also
indirectly modulate the reheating, by introducing spatial dependence in the
inflaton decay width through kinematic blocking and thereby source the
primordial perturbation Fujita and Harigaya (2016); Lu  _et al._ (2020); Choi
and Huang (2013); Cai  _et al._ (2013); Freese  _et al._ (2017); Karam  _et
al._ (2020). In our previous work Karam  _et al._ (2020), we analysed the
consequences of the Standard Model (SM) Higgs field in this framework,
assuming that prior to the reheating process the field was in the vacuum state
determined by the equilibrium configuration on a de Sitter background
Starobinsky (1986); Starobinsky and Yokoyama (1994). Although the Higgs boson
does give rise to a significant scalar perturbation spectrum, such equilibrium
state generically leads to power spectra characterised by a blue tilt
Markkanen  _et al._ (2019); Herranen  _et al._ (2014), incompatible with the
observed red-tilted CMB spectrum. In spite of that, quite naturally, spectator
fields not in their vacuum state can generate the observed spectrum. Both in
the modulated reheating and curvaton models, primordial perturbations with a
red-tilted spectrum can be generated by spectator fields that possess non-
vanishing mean values and slowly roll towards their respective vacuum states.
In the curvaton setup, a crucial requirement is that after the end of
inflation the spectator energy density needs to grow comparable to the
dominant energy component in order for the mechanism to source significant
perturbations. In the case of the Higgs boson, or of any spectator field with
a quartic potential, this is problematic because the energy density of the
field never dilutes slower than the dominant radiation component during the
Hot Big Bang epoch Figueroa and Byrnes (2017); Choi and Huang (2013). On the
other hand, in the modulated reheating scenario, such constraint can be
avoided as the efficiency of the mechanism does not necessarily depend on the
spectator energy density.

In this work we study a modulated reheating mechanism that allows a Higgs-like
spectator field $h$ – that is, driven by a quartic potential – to source the
observed primordial perturbation spectrum despite having a subdominant energy
density throughout the cosmic evolution. The setup is similar to that of Ref.
Markkanen  _et al._ (2019), but in this work we concentrate on the limit of
large spectator field values which allows to generate the observed red
spectral tilt. We discuss how the mechanism could be implemented in extensions
of the SM, with the purpose of identifying the spectator field with the Higgs
boson. The main conclusion of our work is that a set-up with a quartic
spectator with no coupling to the inflaton and without non-renormalizable
operators can solely generate a curvature perturbation that is consistent with
current observational bounds.

The model. Following Ref. Karam  _et al._ (2020), we consider a Lagrangian
given by

$\displaystyle{\cal L}$
$\displaystyle=\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}m^{2}_{\phi}\phi^{2}-\frac{\lambda_{\phi}}{4}\phi^{4}+\frac{1}{2}(\partial
h)^{2}-\frac{\lambda_{h}}{4}h^{4}$ (1)
$\displaystyle+i\bar{\Psi}{\partial}\\!\\!\\!/\Psi-y_{\phi}\bar{\Psi}\Psi\phi-
y_{h}\bar{\Psi}\Psi h+\text{ H.c. },$

where $h$ is the spectator scalar field, the scalar singlet $\phi$ is the
inflaton and $\Psi$ is a Dirac fermion. The form of the inflaton potential
considered in Eq. (1) is not a prerequisite of our setup and is only used to
provide a simple and concrete template.

It is crucial to our purposes that the decay rate of the inflaton is subject
to the indirect modulation from a spectator field,

$\Gamma(h)=\frac{y_{\phi}^{2}m_{\phi}}{8\pi}\left[1-\frac{(2y_{h}h)^{2}}{m_{\phi}^{2}}\right]^{3/2}\,,$
(2)

such that the decay channel is blocked by kinematics when the spectator field
has the value

$h\geqslant\frac{{m_{\phi}}}{{2y_{h}}}\equiv h_{\rm kin}\,.$ (3)

Since the Lagrangian also allows for decay of the spectator field, for
simplicity we restrict here to parameters such that the effective mass of the
spectator field, $\sqrt{3\lambda}h$, is much smaller than that of the fermion,
$y_{h}h$, and thus the process can be safely ignored. It is worth pointing out
that if we identify $h$ with the SM Higgs boson, the remaining couplings to SM
fields, such as gauge bosons, can be safely neglected as they do not lead to
any significant depletion of the Higgs field value over time scales relevant
for the modulation effect, corresponding to ${\cal O}(1)$ Higgs oscillations
after the inflationary expansion Enqvist  _et al._ (2016). Note also that
while the tree-level action (1) does not directly couple the inflaton to the
Higgs boson, such a coupling is generated by fermions at the loop level.
However, because of the small values of the Yukawa coupling $y_{h}$ that we
are considering in this work, we neglect these radiative effects.

Denoting by $h_{*}$ the field value at the horizon crossing of a mode
$k_{*}=a_{*}H_{*}$ during inflation, we now focus on the case $h_{*}>h_{\rm
kin}$, indicating that the inflaton decay is kinematically blocked by the
spectator field via the indirect modulation mechanism. We assume the $h$-field
to be light if compared to the inflation scale, hence it fluctuates locally
around its VEV and the kinematic blocking of the inflaton decay is thus lifted
at different times in different locations in the Universe. This implements the
modulated reheating mechanism Dvali  _et al._ (2004) in our scheme and allows
for the production of significant curvature perturbations Karam  _et al._
(2020); Ichikawa  _et al._ (2008); Choi and Huang (2013). Specifically, we
require that the spectator field effective mass satisfies
$V^{\prime\prime}(h_{*})/(3H_{*}^{2})<0.01$, which for our model translates to
$\lambda_{h}h_{*}^{2}<0.01H_{*}^{2}$. For $h_{*}<M_{\rm P}$, the condition
implies $\Omega_{h*}\lesssim 10^{-3}$, so the spectator field energy density
is necessarily subdominant with respect to the inflaton contribution.

As for the spectrum, at the equilibrium, spectator field fluctuations in a de
Sitter background result only in a blue tilted spectra Markkanen  _et al._
(2019); Herranen  _et al._ (2014) ($n_{s}>1$) and therefore cannot source the
observed primordial perturbation. The situation is different for field
configurations far from equilibrium, _i.e._ , in the mean field limit
$h_{*}>\sqrt{\langle h^{2}\rangle}_{\rm eq},$ when fluctuations around the
mean field value $h_{*}$ yield either red or blue tilts. In the setup
specified by Eq. (1), the out-of-equilibrium configuration
$h_{*}>\sqrt{\langle h^{2}\rangle}_{\rm eq}$ corresponds to an atypical field
configuration, or initial condition, which we suppose to be accidentally
realised in our observable patch. This is similar to what is often assumed in
the context of the curvaton scenario, for example see Lyth and Wands (2002).
Alternatively, one might think of modifying the spectator field potential such
that the initial condition could be dynamically realised. Further study of
this question is however beyond the scope of this work. Here we simply assume
the mean field limit $h_{*}>\sqrt{\langle h^{2}\rangle}_{\rm eq}$ due to the
phenomenological reason that it is required to get the observed spectral tilt.
Using the de Sitter equilibrium result $\sqrt{\langle h^{2}\rangle}_{\rm
eq}\simeq 0.36H_{*}/\lambda_{h}^{1/4}$ Starobinsky and Yokoyama (1994), the
validity of the mean field limit requires $h_{*}>0.36H_{*}/\lambda_{h}^{1/4}$.
Combining this with the mass bound $\lambda_{h}h_{*}^{2}<0.01H_{*}^{2}$, our
constraints are specified by

$\frac{0.36}{\lambda_{h}^{1/4}}<\frac{h_{*}}{H_{*}}<\frac{0.1}{\lambda_{h}^{1/2}}~{},$
(4)

which further implies the bound $\lambda_{h}<0.006$.

Analytical estimates. Simple analytical estimates of the spectrum and non-
Gaussianity of curvature perturbations produced by the indirect modulation
mechanism can be obtained in the limit where the inflaton decays rapidly after
the kinematic blocking is lifted. This occurs when $h(t_{\rm kin})=h_{\rm
kin}$ for the first time in the evolution of the spectator field value. In the
discussion, we neglect the slow evolution of $h$ during the inflationary
expansion and approximate its value at the end of this process by $h_{\rm
end}\simeq h_{*}$. The spectator field starts to oscillate when $H_{\rm
osc}\approx\sqrt{3\lambda_{h}}h_{*},$ and the field behaviour prior to the
first zero crossing can be approximated quite accurately by Karam  _et al._
(2020)

$h\approx h_{*}\left(1-\frac{3}{2}{\rm
e}^{-\frac{27}{4}\frac{H}{\sqrt{3\lambda_{h}}h_{*}}}\right)\,,$ (5)

where $H\propto a^{-3/2}$ for a Universe dominated by inflaton oscillations in
a quadratic potential.

In order to ensure the prompt decay of the inflaton field during the first
available window allowed by kinematics, we impose that $\rho_{\phi}/\rho_{\rm
tot}$ falls below $10^{-5}$ as $|h|<h_{\rm kin}$ during the first half of the
first $h$ oscillation – i.e. the first period when $h$ moves from $h_{\rm
kin}$ through zero to $-h_{\rm kin}$. For $\rho_{\phi}/\rho_{\rm
tot}<10^{-5}$, the inflaton contribution to the number of $e$-folds from later
times, ${\cal O}(\rho_{\phi}/\rho_{\rm tot})\int Hdt$, can be neglected when
studying perturbations of order $\Delta N\sim 10^{-5}$. As a first
approximation, we can then approximate that the Universe changes from matter
to radiation domination at $t=t_{\rm kin}$, and from Eq. (5) it follows that

$H_{\rm
kin}(h_{*})=\frac{4}{27}\sqrt{3\lambda_{h}}h_{*}\,\ln\left(\frac{3h_{*}}{2(h_{*}-h_{\rm
kin})}\right)\,.$ (6)

By using $\rho_{\phi}\propto{\rm exp}(-\int\Gamma dt)$, and estimating the
integral over the first window allowed by kinematics as $\int\Gamma dt\sim
2\Gamma_{0}\int_{t_{0}}^{t_{k}}dt$, where $t_{0}$ and $t_{k}$ are determined
by solving Eq. (5) for $h(t_{k})=h_{\rm kin}$ and $h(t_{0})=0$, the fast decay
condition $-\int\Gamma dt>{\rm ln}10^{-5}$ translates into the bound

$h_{*}\lesssim 0.21m_{\phi}y_{\phi}y_{h}^{-1/2}\lambda_{h}^{-1/4}~{}.$ (7)

Here we have also assumed the limit $h_{\rm kin}\ll h_{*}$. The numerical
analysis presented below indeed confirms that this inequality well determines
the regime where our analytical approximations are fully applicable.

The curvature perturbation in Fourier space can be computed in terms of the
$\delta N$ expression Wands  _et al._ (2000) as

$\zeta({\bf k_{*}})=N^{\prime}\delta h_{*}({\bf
k_{*}})+\frac{1}{2}N^{\prime\prime}\delta h_{*}({\bf k_{*}})^{2}+\ldots,$ (8)

where $N$ is the number of $e$-folds from the horizon crossing
$k_{*}=a_{*}H_{*}$ to a final constant energy surface after the inflaton
decay, a prime denotes differentiation with respect to the field value $h_{*}$
at horizon crossing, and $\delta h_{*}^{n}$ are convolutions. Throughout this
work we concentrate on the limit where the inflaton contribution to the
curvature is negligible,
$\zeta_{\phi}\approx(1/\sqrt{2\epsilon_{*}})H_{*}/(2\pi M_{\rm P})\ll
10^{-5}$, and therefore omit it in all expressions.

When the condition in Eq. (7) is satisfied, we can well approximate the
evolution of the Universe using a sudden transition from matter to radiation
domination at $t_{\rm kin}$, which yields Karam  _et al._ (2020)
$N^{\prime}\approx-(1/6){H_{\rm kin}^{\prime}}/{H_{\rm kin}}$, with $H_{\rm
kin}$ given by Eq. (6). Substituting this into Eq. (8), the power spectrum of
curvature perturbation becomes

$\displaystyle{\cal P}_{\zeta}(k_{*})=\frac{{H^{\prime}_{\rm
kin}{}^{2}}}{{36H_{\rm
kin}^{2}}}\left(\frac{H_{*}}{2\pi}\right)^{2}\approx\frac{{1}}{{36h_{*}^{2}}}\left(\frac{{H_{*}}}{{2\pi}}\right)^{2}+{\cal
O}\left(\frac{h_{\rm kin}}{h_{*}}\right)^{2},$ (9)

where in the last step we expanded Eq. (6) in the limit $h_{*}\gg h_{\rm
kin}$. The spectral index is given by Wands  _et al._ (2002)

$n_{s}-1=-2\epsilon_{*}+2\frac{{\lambda_{h}h_{*}^{2}}}{{H_{*}^{2}}}~{},$ (10)

where $\epsilon\equiv-\dot{H}/{H^{2}}$ is computed at the horizon crossing.

Comparing our result with the observations, ${\cal P}_{\zeta}(k_{*})=(2.100\pm
0.030)\times 10^{-9}$ and $n_{\rm s}(k_{*})=0.965\pm 0.004$ at the pivot scale
$k_{*}=0.05\ {\rm Mpc}^{-1}$ Aghanim  _et al._ (2018), we find that the
analytical estimates agree with the measurements for $h_{*}/H_{*}\simeq 580$
and $\epsilon_{*}\simeq 0.018$ (assuming $\lambda_{h}\lesssim 10^{-8}$ so that
the positive contribution from the second term in Eq. (10) is negligible). For
$h_{*}\simeq 580H_{*}$, the assumed mean field condition (4) implies
$\lambda_{h}\gtrsim 1.5\times 10^{-13}$. Note also that the derivation of the
estimate (9) assumes the fast decay condition in Eq. (7), which constrains
from above the range of $\lambda_{h}$ values for which the analytical estimate
can be used. For instance, taking
$y_{\phi}=1,y_{h}=10^{-3},m_{\phi}/H_{*}=0.1$ and setting $h_{*}=580H_{*}$,
Eq. (7) yields $\lambda_{h}\lesssim 1.7\times 10^{-12}$ and, for this
particular example, cases with larger values of $\lambda_{h}$ need to be
studied numerically – see the results further below.

The second order term in Eq. (8) contributes to the local bispectrum with a
non-Gaussianity amplitude $f_{\rm
NL}=(5/6)N^{\prime\prime}/(N^{\prime}{}^{2})$ given by

$f_{\rm NL}=5\left(1-\frac{H_{\rm kin}^{\prime\prime}H_{\rm kin}}{H_{\rm
kin}^{\prime 2}}\right)\approx 5+{\cal O}\left(\frac{h_{\rm
kin}}{h_{*}}\right)^{2}~{},$ (11)

where, in the last step, we have again expanded Eq. (6) in the limit $h_{*}\gg
h_{\rm kin}$. The prediction $f_{\rm NL}\sim 5$ is a characteristic signal of
our mechanism in the limit of fast decay (i.e. when Eq. (7) holds), and when
$h_{*}\gg h_{\rm kin}$ so that ${\cal O}(h_{\rm kin}/{h_{*}})$ terms in Eq.
(6) can be neglected. This analytical result is confirmed by the numerical
analysis presented below. We remark that this level of non-Gaussianity is
still compatible with the current bound $f_{\rm NL}^{\rm local}=-0.9\pm 5.1$
Akrami  _et al._ (2019) at the $1\sigma$ confidence level. Importantly, the
next generation surveys will reach the sensitivity required to either detect
the signal or rule out the mechanism as the source of the primordial
perturbation in the aforementioned limits. In particular, upcoming large-scale
structure probes such as DESI and Euclid are expected to constrain the
primordial non-Gaussianity with uncertainties of $\mathcal{O}(1)$ Font-Ribera
_et al._ (2014); Amendola  _et al._ (2018); Mueller  _et al._ (2019).

Numerical results. The analytical approximations we proposed apply to a regime
where the inflaton decays efficiently in the kinematically allowed window
prior to the first zero crossing of the spectator field value. To go beyond
this limit, we numerically solve the following system of equations

$\displaystyle\dot{\rho}_{\phi}+3H\rho_{\phi}$ $\displaystyle=$
$\displaystyle-\Gamma(h)\rho_{\phi},$ (12)
$\displaystyle\ddot{h}+3H\dot{h}+\lambda_{h}h^{3}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\dot{\rho_{\rm r}}+4H\rho_{\rm r}$
$\displaystyle=$ $\displaystyle\Gamma(h)\rho_{\phi},$ $\displaystyle 3M_{\rm
P}^{2}H^{2}$ $\displaystyle=$ $\displaystyle\rho_{\phi}+\rho_{\rm
r}+\frac{1}{2}\dot{h}^{2}+\frac{\lambda_{h}}{4}h^{4}~{},$

where $\Gamma$ denotes the real part of Eq. (2). The initial conditions are
determined by matching the solution of the system composed by Eq. (12) with
the first line replaced by $\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=0$, at a
matching time chosen well after the end of inflation (so that $\langle
w_{\phi}\rangle=0$) and well before $t_{\rm kin}$, so that $\Gamma=0$. The
initial conditions for this second system of equations are set by the
inflationary slow roll solution, together with $\dot{h}_{*}=0,\rho_{r*}=0$ and
the initial spectator field value $h_{*}$. Note that modelling the inflaton
decay by inserting the diffusion term $\Gamma\dot{\phi}$ in its equation of
motion would not be justified as $\dot{\phi}/\Gamma\gtrsim\phi$ when the decay
takes place. We have checked that our results do not depend on the choice of
the matching time within the given boundaries.

In our computation we set the inflaton parameters in Eq. (1) to
$m_{\phi}=1.55\times 10^{9}$ GeV, $\lambda_{\phi}=3.75\times 10^{-21}$ and
$y_{\phi}=1$, which yields $H_{*}\approx 1.6\times 10^{10}$ GeV and
$\epsilon_{*}\approx 0.017\dots 0.018$ at the horizon crossing of the pivot
scale $k_{*}=0.05$ ${\rm Mpc}^{-1}$. The precise values vary depending on the
reheating process controlled by the spectator field couplings. For this choice
of inflaton parameters, the inflaton contribution to the curvature
perturbation is negligible, $\zeta_{\phi}^{2}\sim 10^{-17}$, and will
therefore be omitted henceforth. Note also that our results are not limited to
the precise form of the inflaton potential used in Eq. (1). Any other
potential leading to same $H_{*}$ and $\epsilon_{*}$, and reducing to a
quadratic form at the end of inflation, would give essentially the same
results for the curvature perturbation induced by the modulation mechanism.

To compute the curvature perturbation, we construct a grid of initial values
$h_{*}$ (the results shown in the figures are computed using a grid of $100$
points and a step size $\Delta\chi_{*}=0.78H_{*}$). For each grid point we
numerically evolve the system Eq. (12) up to a common final $\rho$ chosen such
that $\rho_{\phi}/\rho<10^{-10}$ and determine the corresponding number of
$e$-folds $N(h_{*})$. From this data we numerically evaluate the first and
second derivatives of $N(h_{*})$ with respect to the initial field value
$h_{*}$, corresponding to the coefficients of the $\delta h_{*}(k_{*})$ powers
in Eq. (8). For the spectator field perturbations at the horizon crossing
$\delta h_{*}(k_{*})$ in Eq. (8), we use the linear perturbation theory result
for the two point function amplitude, ${\cal P}_{\delta
h_{*}}(k_{*})=(H_{*}/(2\pi))^{2}$, and neglect any non-Gaussianity in $\delta
h_{*}(k_{*})$. The spectrum of the curvature perturbation and the non-
Gaussianity parameter $f_{\rm NL}$ are then given by

$\displaystyle{\cal P}_{\zeta}(k_{*})$ $\displaystyle=$ $\displaystyle
N^{\prime}(h_{*})^{2}\left(\frac{H_{*}}{2\pi}\right)^{2}$ (13) $\displaystyle
f_{\rm NL}(k_{*})$ $\displaystyle=$
$\displaystyle\frac{5}{6}\frac{N^{\prime\prime}(h_{*})}{N^{\prime}(h_{*})^{2}}~{},$
(14)

and the spectral index by Eq. (10). All these quantities are readily evaluated
from the numerical data. The results for ${\cal P}_{\zeta}$, $n_{\rm s}$ and
$f_{\rm NL}$ at the pivot scale $k_{*}=0.05\;{\rm Mpc}^{-1}$ are shown in
Figs. 1, 2 and 3, respectively. We have checked that decreasing the final
value of $\rho$ does not affect the results, confirming thereby that the
curvature perturbation has relaxed to a constant well within the time span of
the numerical computation.

Figure 1: The amplitude of the power spectrum sourced by the modulation
mechanism computed at the pivot scale $k_{*}=0.05\;{\rm Mpc}^{-1}$. Here
$H_{*}$ denotes the Hubble rate at the horizon exit of $k_{*}$. The horizontal
line indicates the central value of the Planck data Aghanim  _et al._ (2018).
The circles mark the smallest field value which satisfies the mean-field
condition assumed in our analysis.  Figure 2: The spectral index at the pivot
scale $k_{*}=0.05\;{\rm Mpc}^{-1}$. The horizontal line indicates the central
value of the Planck data Aghanim  _et al._ (2018).  Figure 3: The local non-
Gaussianity parameter $f_{\rm NL}$ at the pivot scale $k_{*}=0.05\;{\rm
Mpc}^{-1}$. The green and orange horizontal dashed lines indicate the
$1\sigma$ and $2\sigma$ confidence intervals of the Planck data Akrami  _et
al._ (2019), respectively.

The starting points of the lines in Figs. 1, 2 and 3, marked by circles,
correspond to the smallest field value for which the mean field condition in
Eq. (4) holds for each value of $\lambda_{h}$. The fast decay region, where
the inflaton decay essentially completes during the first zero crossing of the
spectator field, is approximately determined by the condition Eq. (7), which
for the parameters chosen as in the figures reads $h_{*}/H_{*}\lesssim
0.6\lambda_{h}^{-1/4}$. Increasing $\lambda_{h}$ makes the inflaton decay less
efficient as $\Gamma(0)/H_{\rm kin}\propto(\sqrt{\lambda_{h}}/h_{*})^{-1}$
and, correspondingly, the upper limit of the fast decay region decreases with
larger values of $\lambda_{h}$. As expected, in the fast decay regime the
numerical results agree relatively well with the analytical approximations in
Eqs. (9), (10) and (11). For the cases shown in Figs. 1 and 3, the analytical
estimates for ${\cal P}_{\zeta}$ and $f_{\rm NL}$ (obtained by using the full
form of Eq. (6) in Eqs. (9) and (11)) respectively deviate less than $10$% and
$15$% from the corresponding numerical results in the fast decay region
$h_{*}/H_{*}\lesssim 0.6\lambda_{h}^{-1/4}$.

For $h_{*}$ values not in the fast decay region (7), i.e. for
$h_{*}/H_{*}\gtrsim 0.6\lambda_{h}^{-1/4}$ when parameters are chosen as in
Figs. 1, 2 and 3, the decay is no longer completed during the first window
allowed by kinematics and the analytical estimates cease to be applicable. For
each value of $\lambda_{h}$ in Fig. 3, the end of the fast decay region
$h_{*}/H_{*}\sim 0.6\lambda_{h}^{-1/4}$ coincides with the regime where
$f_{\rm NL}$ starts to grow towards large positive values. Increasing $h_{*}$
further delays the inflaton decay successively to the second, third, or
following zero crossing of the spectator field value, leading to a drastic
amplification of the non-Gaussianity and increasingly complicated non-
monotonous forms for both ${\cal P}_{\zeta}(h_{*})$ and $f_{\rm NL}(h_{*})$,
as seen in Figs. 1 and 3, respectively. This appears to be a generic feature
of the setup, indicating that configurations not in the fast decay regime (7)
tend to generate unacceptably large non-Gaussianity, possibly apart from tuned
configurations around points where $f_{\rm NL}$ accidentally crosses zero.

As for the Yukawa coupling $y_{h}$, larger values of this quantity decrease
$h_{\rm kin}$ in Eq. (3). This narrows the window where the inflaton decay is
kinematically allowed around the spectator zero crossings and makes the decay
process less efficient. Correspondingly, the region where the spectator field
value crosses zero more than once before the decay, i.e. the region where the
fast decay condition (7) does not hold, is pushed towards smaller values of
$h_{*}$ as $y_{h}$ grows. The dependence of ${\cal P}_{\zeta}$ and $f_{\rm
NL}$ as functions of $y_{h}$ is illustrated in Figs. 4 and 5, respectively,
for a sample of $y_{h}$ values.

Figure 4: The dependence of ${\cal P}_{\zeta}$ on the Yukawa coupling $y_{h}$.
The horizontal line indicates the central value of the Planck data Aghanim
_et al._ (2018) Figure 5: The dependence of $f_{\rm NL}$ on the Yukawa
coupling $y_{h}$. The green and orange horizontal dashed lines indicate the
$1\sigma$ and $2\sigma$ confidence intervals of the Planck data Akrami  _et
al._ (2019), respectively.

Implications for inflationary model building. In our analysis, the inflaton
potential affects directly only the value of the spectral index Eq. (10). This
is not specific to our setup and applies also to generic modulated reheating
and curvaton models when the curvature perturbation is dominated by the
spectator field. We remark that a generic prediction of such setups is a
negligible tensor-to-scalar ratio $r$, which follows from the low inflationary
scale required to keep the perturbations sourced by the inflaton subdominant.
In particular, this means that potentials leading to too large tensor
perturbations in the inflaton dominated limit can use the mechanism proposed
here, as well as other modulated reheating or curvaton models, to achieve
agreement with observations. An extensive analysis of inflaton potentials
compatible with the curvaton framework was presented in Encyclopaedia
Curvatonis Vennin  _et al._ (2015) and qualitatively similar conclusions apply
to our setup.

Embedding into the SM. The above analyses relied on the interplay between the
inflaton, a generic spectator field characterised by a quartic potential and a
fermion that interacts with both the scalar fields. By regarding the
parameters in the Lagrangian (1) as free and working at the level of classical
potential, the presented analytic and numerical computations have highlighted
the limits where the framework can match the CMB observations. In Ref. Karam
_et al._ (2020) we have shown that the required modulation mechanism is
already implemented in a well-known extension of the SM, where the particle
content considers a singlet inflaton and right-handed Majorana neutrinos. It
is therefore of interest to investigate whether the same setup allows for the
generation of the CMB through the dynamics discussed in the present work.

If the spectator field is to be identified with the Higgs boson, our analysis
applies provided that the effective Higgs potential can be approximated by the
tree level form: $V(h)=\lambda_{h}(\mu)h^{4}/4$. The renormalization scale is
set to $\mu\sim\sqrt{y_{t}}h$, with $y_{\rm t}$ being the top-quark Yukawa
coupling, corresponding to the largest effective mass scale when $h_{*}\gg
H_{*}$. The effective potential has a maximum where $\lambda_{h}(h_{\rm
max})=-\beta_{h}(h_{\rm max})/4$ and the presented analysis, which used
$V=\lambda_{h}h^{4}/4$ with a constant coupling, is applicable only for
$|h|\lesssim|h_{\rm max}|$. Using SM two-loop beta functions in the $R_{\xi}$
gauge and $\rm\overline{MS}$ scheme, for a (world average) top-quark mass of
$m_{t}=172.9$ GeV we find $\lambda_{h}(h_{\rm max})\gtrsim{\cal O}(10^{-5})$.
However, to produce the observed spectral tilt in the quartic setup, we
require $\lambda_{h}(h_{*})<10^{-8}$ – see the discussion below Eq. (10).
Moreover, even smaller values in the ballpark of $\lambda_{h}(h_{*})\lesssim
10^{-11}$ are necessary to avoid large non-Gaussianities, as shown in Fig. 3.
The specific points where $f_{\rm NL}(h_{*})$ in Fig. 3 accidentally crosses
zero might represent a possible caveat. However, although we have not studied
this in detail, it is expected that even such configurations lead to too large
non-Gaussianity in the form of the trispectrum amplitude $g_{\rm NL}\propto
N^{\prime\prime\prime}/N^{\prime 3}$. We thus conclude that, assuming the SM
running of couplings, the Higgs boson seemingly fails to source the observed
primordial perturbation through the mechanism studied here.

One may of course ask if it is possible to circumvent these problems in
extended setups with new physics coupled to the Higgs boson. Here we do not
address this important question in detail, but we briefly comment on possibly
viable phenomenological structures. First, a negative threshold correction
$\lambda_{h}(h_{*})\rightarrow\lambda_{h}(h_{*})+\delta\lambda_{h}\lesssim
10^{-10}$ could make the scenario work, at the cost of fine-tuning
$\delta\lambda_{h}$ and $h_{*}$ close to the threshold scale. Also new scalar
degrees coupled to the Higgs would act in favour of the scenario as they
contribute positively to $\beta_{h}$ and make it possible to decrease
$\lambda_{h}(h_{\rm max})=-\beta_{h}(h_{\rm max})/4$. An interacting fixed
point $\lambda_{h}(\mu)\lesssim 10^{-11}$, $\beta_{h}(\mu)=0$ at, or below,
the scale $h_{*}$ would also work, and it would be interesting to investigate
if such fixed point could arise, for instance, from anomalously broken scale
invariance. More generally, the tree-level Higgs potential could differ from
the quartic form, for example due to the non-minimal curvature coupling $\xi
Rh^{2}$ or couplings to other fields with non-vanishing VEVs. This would
change the reheating dynamics compared to our analysis, leading to potentially
different conclusions. One could also think of scenarios where $h_{*}$ is
located beyond the vacuum present at large field values, contrary to what we
have assumed. Suppose, for instance, that the corresponding minimum $h_{\rm
min}$ is brought to sub-Planckian values by the interaction with an additional
scalar field used to generate the right-handed neutrino Majorana masses via
symmetry breaking. One could then arrange $h_{*}>h_{\rm kin}>h_{\rm min}$, so
that the reheating would commence at $h\sim h_{\rm kin}$ and, if thermal
corrections rapidly lift the minimum $h_{\rm min}$, the Higgs could relax into
the electroweak vacuum. A more careful assessment of these possibilities,
however, requires dedicated analyses that go beyond the scope of this work.

Conclusions. We have shown that a spectator field with a quartic potential can
alone source the observed primordial perturbation through the modulated
reheating mechanism, realised without direct couplings between the spectator
and the inflaton field and by using only renormalisable operators. To obtain a
red spectrum , the spectator field needs to be sufficiently displaced from
vacuum during inflation, its quartic coupling $\lambda_{h}\lesssim 10^{-8}$
and, as in generic spectator models, the inflationary dynamics needs to yield
a suitable value for the slow roll parameter $\epsilon_{\rm H}$ at the horizon
crossing of observable modes. The setup gives rise to primordial non-
Gaussianities of the local type. We find that when the inflaton decay
completes sufficiently fast (during the first spectator oscillation that lifts
the kinematical blocking associated to the modulation channel), the bispectrum
amplitude is set by $f_{\rm NL}\sim 5$, well in agreement with the current
bounds and a testable signature by upcoming surveys. For $\lambda_{h}\lesssim
10^{-11}$, this is the case for a relatively broad range of initial spectator
field values $h_{*}$. For larger values of $\lambda_{h}$, the inflaton decay
is slower which gives rise to a growing tension between obtaining the observed
amplitude of perturbations and maintaining the non-Gaussianity within the
observational bounds.

Because the primordial perturbations are not directly related to the inflaton
potential, the proposed framework, as well as modulated reheating and curvaton
models in general, allows a wide range of inflationary models to come in
agreement with present data, including those built on quadratic and quartic
potentials. The predicted scalar-to-tensor ratio is generally negligible.

The scenario we discussed allows, in principle, for a straightforward
identification of the spectator field with the Higgs boson in a popular
extension of the Standard Model with Majorana right-handed neutrinos. Although
the required indirect modulation mechanism can be effectively implemented,
matching the observed perturbation needs a modification of the Standard Model
renormalisation group equations. In fact, using the Standard Model running and
approximating the Higgs effective potential with the tree-level term, we find
that the required value of the quartic coupling, $\lambda_{h}\lesssim
10^{-11}$, cannot be obtained in the regime where a quartic form well
approximates the full potential. Whereas more precise studies that use the
full form of the Higgs effective potential are needed in order to fully assess
the possibility, we have briefly discussed promising phenomenological
extensions of the Standard Model which could allow the Higgs boson to source
the observed perturbations in the considered scenario based on a quartic
potential. It would be of interest to study how these proposals could be
realised in concrete particle physics setups.

Acknowledgement. This work was supported by the EU Marie Curie grant 786564,
the European Regional Development Fund through the CoE program grant TK133,
the Mobilitas Pluss grants MOBTT5, MOBTT86, and the Estonian Research Council
grants PRG356, PRG1055 and PRG803. A.R. was supported by the U.K. Science and
Technology Facilities Council grants ST/P000762/1 and ST/T000791/1 and
Institute for Particle Physics Phenomenology Associateship.

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