lsection [5.8em] 2.3em

Hearing Higgs with Gravitational Wave Detectors

Alberto Salvio

Physics Department, University of Rome Tor Vergata,
via della Ricerca Scientifica, I-00133 Rome, Italy

I. N. F. N. - Rome Tor Vergata,
via della Ricerca Scientifica, I-00133 Rome, Italy

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Abstract

The relic gravitational wave background due to tensor linear perturbations
generated during Higgs inflation is computed. Both the Standard Model and a
well-motivated phenomenological completion (that accounts for all the
experimentally confirmed evidence of new physics) are considered. We focus on
critical Higgs inflation, which improves on the non-critical version and
features an amplification of the tensor fluctuations. The latter property
allows us to establish that future space-borne interferometers, such as
DECIGO, BBO and ALIA, may detect the corresponding primordial gravitational
waves.

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Email<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Relic background of gravitational waves from inflation
  3. 3 (Critical) Higgs inflation
    1. 3.1 Classical aspects
    2. 3.2 Quantum aspects and critical version
  4. 4 Relevant gravitational wave detectors
  5. 5 The Standard Model case
  6. 6 A well-motivated BSM scenario
    1. 6.1 Definition of the $a\nu$MSM
    2. 6.2 Inflationary gravitational waves in the $a\nu$MSM
  7. 7 Conclusions

##  1 Introduction

The observation in 2015 of a gravitational wave (GW) event (named GW150914)
from binary black hole mergers [1, 2] has opened the era of GW astronomy and
has naturally reinforced the interest in any source of gravitational
radiation.

From the point of view of particle physics there are several interesting
phenomena that can generate GWs within the reach of present or future
detectors. Examples are provided by strong first-order phase transitions. The
phase transitions in the Standard Model (SM), such as the electroweak (EW) one
or the confinement/deconfinement transition in quantum chromodynamics (QCD),
are not of this type. Therefore, the observation of a GW event that is
unambiguously generated by the EW or QCD transitions would be considered as a
clear evidence of new physics.

Other interesting sources of GWs are the quantum tensor fluctuations that are
generated during inflation. A detection of such GW background would provide
further evidence for inflation and, remarkably, a direct observational
confirmation of the quantum nature of gravity. Furthermore, since inflation is
often implemented via quantum scalar fields and typically involves extremely
high energies, detecting this type of GWs would give us valuable information
on the microscopic theory governing the fundamental interactions between
particles.

The experimental confirmation of the existence of the Higgs boson in 2012 [3,
4] has further increased the interest in the already very popular Higgs
inflation (HI) possibility, where the inflaton is identified with the Higgs
field [5]. In Refs. [5, 6, 7, 8] a viable HI was achieved by introducing a
very large non-minimal coupling between the Higgs and the Ricci scalar.

It was pointed out [9], however, that such a coupling would break perturbative
unitarity at a scale much below the Planck mass. While this does not directly
exclude the HI of [5, 6, 7, 8] as unitarity can still be satisfied non
perturbatively, it does prevent us from computing with current techniques the
implications of the theory in some sub-Planckian energy regimes and for some
background metrics [10]. Furthermore, when the mentioned coupling is very
large, it is necessary to tune the high energy values of some parameters in
order to preserve the quantum inflationary predictions: if this is not done
large higher derivative terms in the effective action are generated by quantum
corrections, changing the output of the model [11, 12, 13] (see also Ref [14]
for a related study).

These problems can be avoided [15, 16] by realizing HI very close to the
critical surface in the parameter space [17, 18, 19], which is located at the
border between the absolute stability and the metastability of the EW vacuum
[20]. Indeed, this version, known as critical Higgs inflation (CHI), features
a much smaller value of the non-minimal coupling, solving both the problems
mentioned above. At the same time, remarkably, CHI leads to larger quantum
tensor fluctuations [15, 16, 17, 18, 19] and one may hope that the
corresponding relic background of GWs could be detected, leading to the
exciting possibility that the Higgs manifests itself in GW astronomy.

The purpose of this paper is to compute the GW spectrum (as a function of
frequency) associated with the tensor linear perturbations in CHI and
investigate whether this gravitational radiation can be detected with future
GW interferometers.

HI (including the critical version) was originally proposed in the SM, which
is very successful in predicting the results of most of particle physics
experiments. However, one has to keep in mind that the SM certainly needs to
be extended. Indeed, the observed neutrino oscillations and dark matter are
enough to establish the existence of some physics beyond the SM (BSM). Due to
the very high energy scales during inflation, it is conceivable that the extra
degrees of freedom could affect both the inflationary observables and the
relic background of GWs. For this reason it is relevant to investigate the
production of inflationary GWs both in the SM and in a scenario that can
account for the experimentally confirmed evidence of BSM physics. We do so in
this work.

Here is an outline of the paper. In Sec. 2 we provide in a model-independent
way the gravitational wave spectral density generated by the inflationary
tensor fluctuations without committing ourselves to HI and without assuming a
specific model. We do make, however, some simplifying assumptions regarding
the properties of the extra light particles and the reheating phase, which are
satisfied in HI and CHI (and in the specific models we consider in the rest of
the paper). In Sec. 3 we give the main properties of CHI that are used in the
subsequent sections. A description of future GW detectors that would be able
to test CHI is given in Sec. 4. The SM case is then analyzed in Sec. 5, while
Sec. 6 is devoted to the corresponding analysis in a well-motivated
phenomenological completion of the SM below the Planck scale [21, 16, 22]. The
conclusions are given in Sec. 7.

##  2 Relic background of gravitational waves from inflation

GWs produced during inflation as quantum tensor fluctuations form today a
relic stochastic background of GWs (see Ref. [23] for a textbook introduction
to this topic). If sufficiently strong this background could be detected in
future GW experiments. Therefore, in this section we provide the key formulæ
for such background in a way that is as model independent as possible, hoping
that this could be relevant not only for the specific models of Secs. 5 and 6,
but also for some future activities.

The main quantity we would like to compute is the following dimensionless
function of the frequency $f$ (the spectral density)

$\Omega_{\rm GW}(f)\equiv\frac{f}{\rho_{\rm cr}}\frac{d\rho_{\rm GW}}{df},$
(2.1)

where $\rho_{\rm cr}\equiv 3H_{0}^{2}\bar{M}_{\rm Pl}^{2}$ is the critical
energy density, $\bar{M}_{\rm Pl}$ is the reduced Planck mass, $H_{0}$ is the
present value of the Hubble rate and $\rho_{\rm GW}$ is the energy density
carried by the stochastic background. Indeed, this is the quantity normally
constrained in the context of interferometers and pulsar timing array
experiments. $\Omega_{\rm GW}(f)$ is related to the power spectrum of tensor
fluctuations at the present time ${\cal P}_{t,0}(f)$ by

$\Omega_{\rm GW}(f)=\frac{\pi^{2}}{3H_{0}^{2}}f^{2}{\cal P}_{t,0}(f).$ (2.2)

In turn ${\cal P}_{t,0}(f)$ can be expressed in terms of the power spectrum of
tensor fluctuations ${\cal P}_{t,{\rm in}}(f)$ (at the time when the mode with
frequency $f$ re-entered the horizon after inflation) through the transfer
function ${\cal T}_{\rm GW}$:

${\cal P}_{t,0}(f)=|{\cal T}_{\rm GW}(f)|^{2}{\cal P}_{t,{\rm in}}(f).$ (2.3)

Let us recall now the definition of ${\cal T}_{\rm GW}$. As we will write
explicitly below, this quantity is related to the tensor fluctuations of the
metric $g_{\mu\nu}$, that appear in111As usual $\mu,\nu,...$ are spacetime
indices, while $i,j,...$ are space indices. $\delta g_{\mu\nu}$ is the
fluctuation of the metric $g_{\mu\nu}$, $\tau$ is the conformal time and the
background metric has the spatially-flat Friedmann-Robertson-Walker (FRW)
form:  $ds^{2}=a(\tau)^{2}\left(d\tau^{2}-\delta_{ij}dx^{i}dx^{j}\right).$
(2.4) $\delta g_{ij}(\tau,\vec{x})=a^{2}(\tau)h_{ij}(\tau,\vec{x})$. Using the
conformal Newtonian gauge, the $h_{ij}$, besides being symmetric
($h_{ij}=h_{ji}$), also satisfy

$h_{ii}=0,\qquad\partial_{i}h_{ij}=0.$ (2.5)

It is convenient to perform a Fourier transform

$\displaystyle
h_{ij}(\tau,\vec{x})=\int\frac{d^{3}q}{(2\pi)^{3}}e^{i\vec{q}\cdot\vec{x}}\sum_{\lambda=\pm
2}h_{\lambda}(\tau,\vec{q})e^{\lambda}_{ij}(\hat{q}),$ (2.6)

where $e^{\lambda}_{ij}(\hat{q})$ are the usual polarization tensors for
helicity $\lambda=\pm 2$. We recall that for $\hat{q}$ along the third axis
the polarization tensors that satisfy (2.5) are given by

$e^{+2}_{11}=-e^{+2}_{22}=1/\sqrt{2},\quad
e^{+2}_{12}=e^{+2}_{21}=i/\sqrt{2},\quad e^{+2}_{3i}=e^{+2}_{i3}=0,\quad
e^{-2}_{ij}=(e^{+2}_{ij})^{*}$ (2.7)

and for a generic momentum direction $\hat{q}$ we can obtain
$e^{\lambda}_{ij}(\hat{q})$ by applying to (2.7) a rotation that connects the
third axis with $\hat{q}$. The polarization tensors defined in this way obey
$e^{\lambda}_{ij}(\hat{q})(e^{\lambda^{\prime}}_{ij}(\hat{q}))^{*}=2\delta^{\lambda\lambda^{\prime}}.$
The EOM of $h_{\lambda}$ in the vacuum is

$h_{\lambda}^{\prime\prime}+2{\cal
H}h_{\lambda}^{\prime}+q^{2}h_{\lambda}^{\prime}=0,$ (2.8)

where ${\cal H}\equiv a^{\prime}/a$, $q\equiv|\vec{q}|$ and a prime denotes
the derivative with respect to $\tau$. Eq. (2.8) has to be solved with the
initial condition that at the time $\tau_{\rm in}$ the function $h_{\lambda}$
reduces to the constant value at horizon exit after inflation. Both this
condition and Eq. (2.8) depend on $\vec{q}$ only through $q$ so we can write
that $h_{\lambda}$ is a function of $\tau$ and $q$ only, i.e.
$h_{\lambda}(\tau,q)$. The transfer function is defined by

$h_{\lambda}(\tau_{0},q)\equiv{\cal T}_{\rm GW}(f)h_{\lambda}(\tau_{\rm
in},q),$ (2.9)

where a subscript $0$ denotes the present time, and $q$ is related to the
frequency $f$ via $f\equiv q/(2\pi)$.

The free-streaming of the active neutrinos, the photon and other light
particles (if any) could generate a non-vanishing right-hand side of Eq.
(2.8). However, for the frequency range relevant for ground-based and space-
borne interferometers and pulsar timing arrays (i.e. $10^{-9}$ Hz $\lesssim
f\lesssim 10^{3}$ Hz) the neutrino and the photon free-streaming does not
affect the amplitude of GWs [24]. Should the specific model at hand contain
extra light particles, one would have to take into account a possible sizeable
effect of these extra species on Eq. (2.8).

The GWs that are within the sensitivities of ground-based and space-borne
interferometers as well as pulsar timing arrays all correspond to tensor modes
that re-entered the horizon during the radiation dominated era much before the
time of radiation-matter equality. Solving Eq. (2.8) for these modes and
performing a time average, the transfer function can be approximated by

${\cal T}_{\rm
GW}(f)\simeq\frac{a(\tau_{q})}{\sqrt{2}a_{0}}=\frac{a(\tau_{q})}{\sqrt{2}}$
(2.10)

where $\tau_{q}$ is the time when the mode with momentum $q$ re-entered the
horizon, i.e. ${\cal H}(\tau_{q})=q$, and, as usual, we have conventionally
set $a_{0}=1$.

On the other hand, using entropy conservation and again the fact that the
tensor modes re-entered the horizon during the radiation dominated era, one
obtains

$a(\tau_{q})\simeq\left(\frac{g_{*}(T_{q})}{\bar{g}_{*}}\right)^{1/2}\left(\frac{\bar{g}^{S}_{*}}{g^{S}_{*}(T_{q})}\right)^{2/3}\frac{H_{0}\Omega_{R}^{1/2}}{q},$
(2.11)

where $T_{q}$ is the value of the (photon) temperature $T$ when the mode with
momentum $q$ enters the horizon and $g^{(S)}_{*}$ is the (entropy) effective
number of relativistic species. These functions of $T$ are defined as usual by

$\displaystyle g_{*}(T)$ $\displaystyle\equiv$
$\displaystyle\sum_{b}g_{b}\left(\frac{T_{b}}{T}\right)^{4}+\frac{7}{8}\sum_{f}g_{f}\left(\frac{T_{f}}{T}\right)^{4}$
(2.12) $\displaystyle g^{S}_{*}(T)$ $\displaystyle\equiv$
$\displaystyle\sum_{b}g_{b}\left(\frac{T_{b}}{T}\right)^{3}+\frac{7}{8}\sum_{f}g_{f}\left(\frac{T_{f}}{T}\right)^{3},$
(2.13)

where the sum over $b$ runs over all relativistic bosons, that over $f$ runs
over all relativistic fermions (with number of spin or helicity states $g_{b}$
and $g_{f}$ and temperatures $T_{b}$ and $T_{f}$, respectively). Moreover, in
Eq. (2.11) $\bar{g}^{(S)}_{*}$ is the value of $g^{(S)}_{*}$ at a reference
temperature $T_{r}$ below that of $e^{\pm}$ annihilation, but such that the
three active neutrinos are still relativistic. The quantity $\Omega_{R}$
denotes as usual the ratio of (today’s) energy density in radiation to
$\rho_{\rm cr}$, which can be conventionally expressed in terms of the
effective number of neutrino species $N_{\rm eff}^{(\nu)}$ and the ratio
$\Omega_{\gamma}$ of the photon energy density
$\rho_{\gamma}=\pi^{2}T_{0}^{4}/15$ to $\rho_{\rm cr}$:

$h^{2}\Omega_{R}=h^{2}\Omega_{\gamma}\left[1+N_{\rm
eff}^{(\nu)}\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\right],$ (2.14)

where as usual $T_{0}$ is today’s temperature and $h\equiv
H_{0}/(100\leavevmode\nobreak\ {\rm km\,s^{-1}\,Mpc^{-1}})$.

Note that $h^{2}\Omega_{R}$ can be computed once $N_{\rm eff}^{(\nu)}$ is
known because $h^{2}\Omega_{\gamma}$ is determined by the known value of
$T_{0}$ [25]:

$h^{2}\Omega_{\gamma}=h^{2}\frac{\rho_{\gamma}}{\rho_{\rm
cr}}=\frac{\pi^{2}T_{0}^{4}}{45\bar{M}_{\rm
Pl}^{2}}\left(\frac{h}{H_{0}}\right)^{2},$ (2.15)

which is independent of the value of $H_{0}$ because this quantity only
appears in the ratio $h/H_{0}$. At the temperature $T_{r}$, by definition, all
active neutrinos are relativistic and would contribute to $N_{\rm
eff}^{(\nu)}$, but it is important to keep in mind that near the present epoch
at least two active neutrinos are non relativistic (see Refs. [26, 27] for
recent bounds on their masses). The value of $\Omega_{R}$ depends on whether
today the lightest neutrino is relativistic or not. In order to compute the
active neutrino contribution to $N_{\rm eff}^{(\nu)}$ and to the parameters
$\bar{g}_{*}$ and $\bar{g}^{S}_{*}$, which also enter Eq. (2.11), one should
recall that a relativistic neutrino species after $e^{\pm}$ annihilation
features a temperature $T_{\nu}=(4/11)^{1/3}T$. Moreover, generically $N_{\rm
eff}^{(\nu)}$ and the parameters $\bar{g}_{*}$ and $\bar{g}^{S}_{*}$ can also
receive a contribution from extra species that are relativistic at the
temperature $T_{0}$ and $T_{r}$, respectively (if any).

In order to use (2.11) we need to estimate the dependence of $T_{q}$ on $q$.
This can be obtained by applying the entropy conservation in the form

$g_{*}^{S}(T_{q})T_{q}^{3}a^{3}(\tau_{q})=g_{*0}^{S}T_{0}^{3}.$ (2.16)

The quantity $g_{*0}^{S}\equiv g_{*}^{S}(T_{0})$ certainly receives a
contribution from photons, but can also receive a contribution from one active
neutrino (depending on its mass) and other possible extra species that are
relativistic at the present time (if any). By using (2.11) in (2.16) one finds

$T_{q}=\left(\frac{g_{*0}^{S}}{\bar{g}_{*}^{S}}\right)^{1/3}\frac{c(T_{q})T_{0}q}{H_{0}\Omega_{R}^{1/2}},$
(2.17)

where $c(T)$ is a slowly-varying function of $T$ given by

$c(T)=\left(\frac{g_{*}(T)}{\bar{g}_{*}}\right)^{-1/2}\left(\frac{\bar{g}^{S}_{*}}{g^{S}_{*}(T)}\right)^{-1/3}.$
(2.18)

Eqs. (2.17) and (2.18) can be used to estimate $T_{q}$ as a function of $q$:
we can first approximate $c(T_{q})$ with its value at the $q$-independent
temperature $T_{q0}\equiv T_{r}$, i.e. $c(T_{q})\simeq c(T_{q0})=1$, which
gives in the right-hand side of Eq. (2.17)

$T_{q1}\equiv\left(\frac{g_{*0}^{S}}{\bar{g}_{*}^{S}}\right)^{1/3}\frac{T_{0}q}{H_{0}\Omega_{R}^{1/2}}.$
(2.19)

Next we approximate $c(T_{q})\simeq c(T_{q1})$, which gives through (2.17)

$T_{q2}\equiv\left(\frac{g_{*0}^{S}}{\bar{g}_{*}^{S}}\right)^{1/3}\frac{c(T_{q1})T_{0}q}{H_{0}\Omega_{R}^{1/2}},$
(2.20)

Reiterating this procedure a few times one obtains a good estimate of $T_{q}$.
In practise, $T_{q}\simeq T_{q2}$ is already good enough for our purposes.

One can obtain a more convenient expression for $\Omega_{\rm GW}(f)$ by
writing

${\cal P}_{t,{\rm in}}(f)=r(q_{*})A_{\cal
R}(q_{*})\left(\frac{f}{f_{*}}\right)^{n_{t}},$ (2.21)

where $q_{*}$ is a pivot scale used in cosmic microwave background (CMB)
observations, $A_{\cal R}$ is the amplitude of the (scalar) curvature
perturbation, $r$ is the tensor-to-scalar ratio, $f_{*}=q_{*}/(2\pi)$ and
$n_{t}$ is the spectral index of tensor perturbations. In the specific case of
single-field inflation (which is the relevant one in CHI) $n_{t}=-r/8$, but
Eq. (2.21) holds even in multi-field inflationary scenarios. By inserting
(2.21) in (2.2)-(2.3) one obtains

$\Omega_{\rm GW}(f)=\frac{\pi^{2}}{3H_{0}^{2}}f^{2}|{\cal T}_{\rm
GW}(f)|^{2}r(q_{*})A_{\cal R}(q_{*})\left(\frac{f}{f_{*}}\right)^{n_{t}}.$
(2.22)

This formula might be changed by the reheating era, but for the frequency
range relevant for ground-based and space-borne interferometers as well as
pulsar timing arrays, we checked that these corrections are negligible for a
reheating temperature $T_{\rm RH}$ above the $10^{10}$ GeV scale. As we will
see in Sec. 3, this is the case in HI, which we focus on in this work.

As a final remark for this section, note that $h^{2}\Omega_{\rm GW}$ as well
as $T_{q}$ and ${\cal T}_{\rm GW}$ are independent of $H_{0}$ and thus their
determination is not affected by the problem of the Hubble tension (see Ref.
[28] for a recent review).

##  3 (Critical) Higgs inflation

In this work we focus on the case in which the role of the inflaton is played
by the Higgs field. However, we would like to provide a model-independent
treatment: in this section we do not assume to work within the SM.

A viable HI means in particular that, in the presence of other scalar fields,
the Higgs direction in the potential should be relatively flat compared to the
other ones. Moreover, there should be a non-minimal coupling between the Higgs
doublet $H$ and the Ricci scalar222For the signature of the metric
$g_{\mu\nu}$ we use the mostly minus convention, $g$ is the metric
determinant, $R\equiv g^{\mu\nu}R_{\mu\nu}$, where $R_{\mu\nu}\equiv
R_{\rho\mu\,\,\,\nu}^{\quad\rho}$,
$R_{\mu\nu\,\,\,\sigma}^{\quad\rho}\equiv\partial_{\mu}\Gamma_{\nu\,\sigma}^{\,\rho}-\partial_{\nu}\Gamma_{\mu\,\sigma}^{\,\rho}+\Gamma_{\mu\,\tau}^{\,\rho}\Gamma_{\nu\,\sigma}^{\,\tau}-\Gamma_{\nu\,\tau}^{\,\rho}\Gamma_{\mu\,\sigma}^{\,\tau}$
and $\Gamma_{\mu\,\nu}^{\,\sigma}$ is the Levi-Civita connection.  $R$, which
appears in the following term in the action

$-\int d^{4}x\sqrt{-g}\xi_{H}|H|^{2}R.$ (3.1)

This term is present together with the standard Einstein-Hilbert term in what
is known as the Jordan-frame action. By performing a field redefinition one
can trade the non-minimal coupling with a modification of the other terms
(minimally coupled to gravity) where the matter fields appear, such as the
scalar potential. This set of redefined fields is known as the Einstein frame
(see Refs. [29, 30] for a general discussion in the presence of an arbitrary
number of scalar, vector and fermion fields). As well-known, and as will
become clear shortly, the $\xi_{H}$ term in (3.1) is needed to have a
sufficiently flat Higgs potential in the Einstein frame.

Also, reheating after inflation occurs very efficiently in HI (both in the
critical and non-critical versions) [31, 32] because the Higgs has sizable
couplings to other SM particles. Even restricting to the SM, this leads to a
high reheating temperature, $T_{\rm RH}\gtrsim 10^{13}$ GeV. The presence of
extra particles in a BSM scenario can further increase $T_{\rm RH}$ because of
possible extra Higgs decay channels.

###  3.1 Classical aspects

Studying Higgs inflation in the unitary gauge, the Einstein-frame potential of
the canonically normalized Higgs field $\phi^{\prime}$ is given by (see e.g.
[15])

$U_{H}\equiv\frac{V_{H}}{\Omega_{H}^{4}}=\frac{\lambda_{H}\phi(\phi^{\prime})^{4}}{4(1+\xi_{H}\phi(\phi^{\prime})^{2}/\bar{M}_{\rm
Pl}^{2})^{2}},$ (3.2)

where $\phi$ is the Higgs field non-minimally coupled to gravity that is
related to $\phi^{\prime}$ through

$\frac{d\phi^{\prime}}{d\phi}=\Omega_{H}^{-2}\sqrt{\Omega_{H}^{2}+\frac{3\bar{M}_{\rm
Pl}^{2}}{2}\left(\frac{d\Omega_{H}^{2}}{d\phi}\right)^{2}},$ (3.3)

$\Omega_{H}^{2}$ is defined by

$\Omega_{H}^{2}\equiv 1+\frac{2\xi_{H}|H|^{2}}{\bar{M}_{\rm Pl}^{2}}$ (3.4)

and $V_{H}=\lambda_{H}\phi^{4}/4$ corresponds to the Higgs potential in the
Jordan frame, which involves the Higgs quartic coupling $\lambda_{H}$. Eq.
(3.2) tells us that $U_{H}$ becomes flat for a sufficiently large $\phi$ (and
thus $\phi^{\prime}$, as this is a monotonically increasing function of
$\phi$) at least for $\xi_{H}>0$. We impose this positivity condition in the
present work.

In a spatially flat FRW geometry the equations for the spatially homogeneous
field $\phi^{\prime}(t)$ and the cosmological scale factor $a(t)$ are

$\ddot{\phi}^{\prime}+\frac{\sqrt{3\dot{\phi}^{\prime
2}+6U_{H}}}{\sqrt{2}\bar{M}_{\rm
Pl}}\dot{\phi}^{\prime}+\frac{dU_{H}}{d\phi^{\prime}}=0$ (3.5)

and

$H_{I}^{2}=\frac{\dot{\phi}^{\prime 2}+2U_{H}}{6\bar{M}_{\rm Pl}^{2}},$ (3.6)

where a dot represents the derivative with respect to cosmic time $t$ and
$H_{I}\equiv\dot{a}/a$.

In the slow-roll approximation the parameters

$\epsilon_{H}\equiv\frac{\bar{M}_{\rm
Pl}^{2}}{2}\left(\frac{1}{U_{H}}\frac{dU_{H}}{d\phi^{\prime}}\right)^{2},\quad\eta_{H}\equiv\frac{\bar{M}_{\rm
Pl}^{2}}{U_{H}}\frac{d^{2}U_{H}}{d\phi^{\prime 2}}$ (3.7)

are small and one can simplify considerably the calculation of the functions
$\phi^{\prime}(t)$ and $a(t)$ and the inflationary observables.

To be successful, inflation must last long enough, which leads to a lower
bound on the number of e-folds

$N\equiv\int_{t_{b}}^{t_{e}}dt\,H_{I}(t),$ (3.8)

where $t_{e}$ is the time when inflation ends and $t_{b}$ is the time when the
various inflationary observables such as $A_{\cal R}$, the corresponding
spectral index $n_{s}$ and $r$ are determined through observations. In the
slow-roll approximation $N$ is expressed as a function of the field
$\phi_{b}^{\prime}$ (at $t_{b}$) rather than as a function of time,

$N=\int_{\phi^{\prime}_{e}}^{\phi^{\prime}_{\rm b}}\frac{U_{H}}{\bar{M}_{\rm
Pl}^{2}}\left(\frac{dU_{H}}{d\phi^{\prime}}\right)^{-1}d\phi^{\prime},$ (3.9)

where $\phi^{\prime}_{e}$ is the field value at the end of inflation.

###  3.2 Quantum aspects and critical version

At quantum level the formulæ of Sec. 3.1 remain approximately valid except
that one must consider $\lambda_{H}$ and $\xi_{H}$ as functions of
$\phi^{\prime}$. Since the definition of these functions can be ambiguous [6,
7, 18, 33, 34] here we adopt the quantization used in [15, 16], which can be
embedded in a UV completion of gravity [35, 36, 37, 38] (see Refs. [29, 30]
for reviews). In this approach the $\phi^{\prime}$-dependence of $\lambda_{H}$
and $\xi_{H}$ is obtained by solving the renormalization group equations
(RGEs), which depends on the specific model one considers. This procedure,
however, applies both to the critical and non-critical version of Higgs
inflation.

With this approach $n_{s}(q_{*}),A_{\cal R}(q_{*})$ and $r(q_{*})$ can be
computed through

$n_{s}(q_{*})=1-6\epsilon_{H}+2\eta_{H},\qquad r(q_{*})=16\epsilon_{H},\qquad
A_{\cal R}(q_{*})=\frac{U_{H}/\epsilon_{H}}{24\pi^{2}\bar{M}_{\rm Pl}^{4}},$
(3.10)

evaluated at $\phi_{b}^{\prime}$.

These theoretical predictions must satisfy the observational limits, which put
constraints on specific models realizing CHI. In 2018 Planck released its last
cosmological observations related to inflation [39]:

$n_{s}(q_{*})=0.9649\pm 0.0042\,\,(68\%{\rm CL}),\quad
r(q_{*})<0.076\,\,(95\%{\rm CL}),$ (3.11)

while for the curvature power spetrum

$A_{\cal R}(q_{*})=(2.10\pm 0.03)10^{-9},$ (3.12)

where the pivot scale $q_{*}=0.05\leavevmode\nobreak\ {\rm Mpc}^{-1}$ is used.

The main features of CHI are a bigger value of $r$ and a smaller value of
$\xi_{H}$ compared to the non-critical HI. The former property gives us the
hope to detect primordial gravitational waves in the near future, as we will
see in the next sections. The latter one pushes the scale of the perturbative
breaking of unitarity $\sim\bar{M}_{\rm Pl}/\xi_{H}$ (pointed out in Refs.
[9]) very close to the Planck scale, where anyhow new degrees of freedom are
needed to UV complete Einstein gravity. As shown in [10, 16] CHI is a viable
effective field theory: the cutoff (i.e. the maximal energy below which
consistency is possible) is much above the inflationary energy $U_{H}^{1/4}$.
In the rest of the paper we will assume that the new degrees of freedom
required by the UV completion do not already appear at or just above
$U_{H}^{1/4}$. The viability of such assumption has been established in [10,
16]. This guarantees that the predictions for the inflationary observables are
not significantly altered by the above-mentioned degrees of freedom.

##  4 Relevant gravitational wave detectors

How sensitive should a GW detector be in order to observe the spectral density
$h^{2}\Omega_{\rm GW}$ in (2.22)? This depends essentially on the four
quantities ${\cal T}_{\rm GW}$, $r(q_{*})$, $A_{\cal R}(q_{*})$ and $n_{t}$.
To make a rough estimate we can consider the SM, which will be analyzed in
more detail in Sec. 5. Using (2.10) and (2.11), this case corresponds to
${\cal T}_{\rm GW}\sim 10^{-21}$ Hz$/f$ and inserting in (2.22) one obtains
the order-of-magnitude estimate

$h^{2}\Omega_{\rm GW}(f)\sim
10^{-16}\left(\frac{r(q_{*})}{0.076}\right)\left(\frac{A_{\cal
R}(q_{*})}{2.10\times 10^{-9}}\right)\left(\frac{f}{f_{*}}\right)^{n_{t}}.$
(4.1)

Planck data [39] put some constraints on multi-field inflationary scenarios so
we can approximate $n_{t}\simeq-r/8$, which means that $h^{2}\Omega_{\rm
GW}(f)$ has a weak dependence on $f$ (cf. the bound on $r$ in (3.11)). So
using the observational information in (3.11) and (3.12), the estimate in
(4.1) tells us that a GW detector should reach a sensitivity corresponding to
$h^{2}\Omega_{\rm GW}\sim 10^{-16}$. This is some orders of magnitude below
the sensitivity of ground-based interferometers and the Laser Interferometer
Space Antenna (LISA) [40]. However, some future space-borne experiments may
reach it.

One of these is the proposed Japanese mission DECi-hertz Interferometer
Gravitational wave Observatory (DECIGO) [41], which would reach the
sensitivity to $h^{2}\Omega_{\rm GW}\sim 10^{-17}$ around the scale $f\sim
0.1$ Hz. Indeed, the primary objective of DECIGO is to directly observe the
beginning of the universe.

Another space-based gravitational-wave detector that can reach a similar
sensitivity is the proposed Big Bang Observer (BBO) [42, 43], which is
intended as a follow on mission to LISA. Although LISA’s sensitivity is
several orders of magnitude away from the required $10^{-16}$, BBO can reach
this goal. The Advanced Laser Interferometer Antenna (ALIA) [44, 42] is yet
another proposed space-based interferometer with similar characteristics,
although with a slightly smaller frequency scale where the maximal sensitivity
is reached.

##  5 The Standard Model case

Let us first apply these results to the SM or, more generically, to any model
where all extra fields are decoupled.

In this case the temperature-frequency relation in Eq. (2.17) is uniquely
determined up to a small uncertainty due to the fact that the lightest
neutrino can be relativistic or non relativistic today:

$T_{2\pi f}\simeq
10^{7}\,\text{GeV}\,\frac{f}{\text{Hz}}\times\left\\{\begin{array}[]{c}3.87\quad\mbox{(relativistic
lightest neutrino)}\\\ 3.91\quad\mbox{(non-relativistic lightest
neutrino)}\end{array}\right..$ (5.1)

Here we focus on the case $T\gtrsim 100$ GeV as this corresponds to
frequencies that space-borne as well (as ground-based) GW detectors are
sensitive to. The uncertainty in (5.1) can be neglected within the
approximations performed in Sec. 2. The two cases above, however, lead to a
sizeable difference when computing the transfer function because of its
dependence on $N_{\rm eff}^{(\nu)}$, cf. Eqs. (2.10), (2.11) and (2.14):

${\cal T}_{\rm GW}(f)\simeq
10^{-21}\,\frac{\text{Hz}}{f}\times\left\\{\begin{array}[]{c}1.25\quad\mbox{(relativistic
lightest neutrino)}\\\ 1.13\quad\mbox{(non-relativistic lightest
neutrino)}\end{array}\right..$ (5.2)

The next step is to determine the inflationary observables appearing in
(2.22). We realize CHI in the SM as explained in Ref. [15]. Regarding the
determination of the dependence $\lambda_{H}$ and $\xi_{H}$ on
$\phi^{\prime}$, we solve here the RGEs given in the appendix of [45] with the
procedure outlined in [15]. Since we are at criticality, the values of the
relevant SM couplings at the EW scale are actually chosen differently compared
to the current central values, which would correspond to living too far from
the border between the absolute stability and the metastability of the EW
vacuum. In particular, we set $M_{t}\simeq 171.04$ GeV, which, following Ref.
[15], corresponds to the critical point (see Ref. [46] for the experimental
value of $M_{t}$).

In Fig. 1 we show $h^{2}\Omega_{\rm GW}(f)$ in two distinct cases: when the
lightest (active) neutrino is still relativistic today and when it is not. As
shown in the left plot of that figure the result depends on the number of
e-folds $N$, which is taken there to be around 60, and, more importantly, on
whether today the lightest neutrino is relativistic or not. In computing $N$
we have taken into account the exact equation of motion of the Higgs field
without using the slow-roll approximation. This is significantly more precise
than using the slow-roll expression because of the presence of a (quasi)
inflection point in the CHI potential [15]. In the right plot of Fig. 1 it is
shown how the primordial GWs generated by CHI are within the sensitivity of
the future space-borne GW detectors BBO, DECIGO and ALIA. The experimental
curves are power-law integrated sensitivity curves, which are determined
following the method described in [47, 48]. In Fig. 1 we also show the
predictions of a BSM scenario that will be discussed in Sec. 6.

Figure 1: The GW spectral density as a function of the frequency $f$ in the SM
. The two distinct cases of relativistic and non-relativistic lightest
neutrino are considered and this corresponds to the width of the quasi-
horizontal black band in the right plot (computed with $\xi_{H}(\bar{M}_{\rm
Pl})\simeq 12.3$). In the left plot we also display the dependence on the
number of e-folds $N$ around the value $60$. The SM GW spectral density is
also compared to sensitivity curves of future GW detectors in the right plot.
In the right plot we also give the predictions in the $a\nu$MSM corresponding
to the parameter values quoted in the text (purple quasi-horizontal dashed
lines).

Note that $M_{t}\simeq 171.04$ GeV that we used in the SM is about $2\sigma$
away from the current central value [46]. CHI in the SM also predicts
inflationary observables that are in tension with the most recent Planck
bounds [39], namely [15]

$n_{s}(q_{*})\simeq 0.97,\qquad r(q_{*})\simeq 0.076,\qquad A_{\cal
R}(q_{*})\simeq 2.2\times 10^{-9},$ (5.3)

where we used the value of $N$ quoted in Fig. 1. Similar tensions were also
pointed out in Ref. [49]. However, we should keep in mind that the SM
certainly has to be extended because of well-established observations (e.g.
neutrino oscillations and DM) as we discussed in the introduction.

##  6 A well-motivated BSM scenario

We then move to consider the $a\nu$MSM case [21, 16]. This is an SM extension
(defined in Sec. 6.1) that can account for all the experimentally confirmed
signals of BSM physics (neutrino oscillations and dark matter) and can solve
other issues of the SM (baryon asymmetry, the strong CP problem333The other
two fine-tuning problems of the SM (the Higgs mass and cosmological constant
problems) could be addressed, unlike the strong CP one, with anthropic
arguments [50]., the metastability of the EW vacuum and inflation) at the same
time.

###  6.1 Definition of the $a\nu$MSM

We now give the details of the $a\nu$MSM that are needed for the following
discussion (see Refs. [21, 16] for further details). To the SM fields one adds
three sterile neutrinos $N_{i}$ and the fields of the Kim-Shifman-Vainshtein-
Zakharov (KSVZ) QCD axion model [51] (two Weyl quarks $q_{1}$, $q_{2}$ neutral
under ${\rm SU(2)_{\it L}\times U(1)_{\it Y}}$ and a complex scalar $A$). This
provides a simple but realistic implementation of the Peccei-Quinn mechanism
[52], which solves the strong CP problem.

The SM Lagrangian, $\mathscr{L}_{\rm SM}$, is extended by adding three terms,

$\mathscr{L}=\mathscr{L}_{\rm SM}+\mathscr{L}_{N}+\mathscr{L}_{\rm
axion}+\mathscr{L}_{\rm gravity},$ (6.1)

$\mathscr{L}_{N}$ represents the sterile-neutrino-dependent piece:

$i\overline{N}_{i}\hbox{\hbox{${\partial}$}}\kern-6.25958pt{\hbox{${/}$}}N_{i}+\left(\frac{1}{2}N_{i}M_{ij}N_{j}+Y_{ij}L_{i}HN_{j}+{\rm
h.c.}\right),$ (6.2)

where the Majorana mass matrix $M$ is taken diagonal and real,
$M=\mbox{diag}(M_{1},M_{2},M_{3}),$ without loss of generality.
$\mathscr{L}_{\rm axion}$ is the KSVZ piece:

$\mathscr{L}_{\rm
axion}=i\sum_{j=1}^{2}\overline{q}_{j}\hbox{\hbox{${D}$}}\kern-7.39772pt{\hbox{${/}$}}\,q_{j}+|\partial
A|^{2}-(yq_{2}Aq_{1}+h.c.)-\Delta V(H,A),$

where $\Delta V(H,A)$ is the $A$-dependent term in the classical potential

$\Delta
V(H,A)\equiv\lambda_{A}(|A|^{2}-f_{a}^{2}/2)^{2}+\lambda_{HA}(|H|^{2}-v^{2})(|A|^{2}-f_{a}^{2}/2),$

$f_{a}$ is the axion decay constant and $v\simeq 174$ GeV is the EW breaking
scale. Finally,

$\mathscr{L}_{\rm gravity}=-\left(\frac{\bar{M}_{\rm
Pl}^{2}}{2}+\xi_{H}(|H|^{2}-v^{2})+\xi_{A}(|A|^{2}-f_{a}^{2}/2)\right)R-\Lambda,$
(6.3)

where $R$ is the Ricci scalar, $\xi_{H}$ and $\xi_{A}$ represent the non-
minimal couplings of $H$ and $A$ to gravity and $\Lambda$ is the cosmological
constant. In the $a\nu$MSM the inflaton is identified with the Higgs; like in
the SM, it is possible to do so with $\xi_{H}\sim\mathcal{O}(10)$, as
discussed in Ref. [16], when we are close to the frontier between the
stability and the metastability of the EW vacuum (critical Higgs inflation)444
A variant of the $a\nu$MSM, where the $M_{ij}$ are generated through the
vacuum expectation value of $A$ (which we denote by $\langle A\rangle$) and
$|A|$ takes part in a multi-field inflationary dynamics, has been subsequently
proposed in [53, 54, 55]. See also Ref. [56] for a study of the corresponding
inflationary GW spectrum. .

The active-neutrino masses $m_{i}$ ($i=1,2,3$) are obtained by diagonalizing
the matrix

$m_{\nu}=\frac{m_{D1}m_{D1}^{T}}{M_{1}}+\frac{m_{D2}m_{D2}^{T}}{M_{2}}+\frac{m_{D3}m_{D3}^{T}}{M_{3}},$
(6.4)

where the arrays $m_{Di}$ ($i=1,2,3$) are the columns of the Dirac mass matrix
$m_{D}=vY$ i.e.
$m_{D}=\left(\begin{array}[]{ccc}m_{D1}\,,&m_{D2}\,,&m_{D3}\end{array}\right).$
We then express $Y$ in terms of the $M_{i}$ and $m_{i}$ as done in Refs. [21,
16].

On the other hand, the PQ symmetry breaking induced by $\langle
A\rangle=f_{a}/\sqrt{2}$ gives to the extra quarks a Dirac mass
$M_{q}=yf_{a}/\sqrt{2}$ and to the extra scalar a squared mass

$M_{A}^{2}=f_{a}^{2}\left(2\lambda_{A}+\mathcal{O}\left(\frac{v^{2}}{f_{a}^{2}}\right)\right).$
(6.5)

Since

$f_{a}\gtrsim 10^{8}\leavevmode\nobreak\ \mbox{GeV}$ (6.6)

(see Ref. [57] for a review), the $\mathcal{O}\left(v^{2}/f_{a}^{2}\right)$
term is very small and will be neglected.

One may wonder whether the PQ symmetry breaking generates a first-order phase
transition in this model, such that one may hope to detect the corresponding
GWs as well. The answer to this question is negative [58, 59] (see also Ref.
[48] for a previous related discussion), although one can build different BSM
scenarios where instead that happens [58, 59, 60]. Nevertheless, as we show in
Sec. 6.2, CHI in the $a\nu$MSM can lead to an inflationary background of GWs,
which can be observed by future detectors.

Let us now discuss the other generic observational bounds that are relevant
for our purposes (see Refs. [21, 16] for a discussion of the remaining
observational bounds).

Regarding the active-neutrinos, here we take the currently most precise values
reported in [26, 27] for normal ordering (which is currently preferred) of the
following quantities: $\Delta m^{2}_{21}$, $\Delta m^{2}_{31}$ (where $\Delta
m_{ij}^{2}\equiv m_{i}^{2}-m_{j}^{2}$), the active-neutrino mixing angles and
the CP phase in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. One must
also take into account the cosmological upper bound on the sum of the neutrino
masses [61]

$\sum_{i}m_{i}<0.12\leavevmode\nobreak\ \mbox{eV}.$ (6.7)

Regarding the SM sector, we also have to fix the values of the relevant SM
couplings at the EW scale, which we identify conventionally with the top
mass555In the $a\nu$MSM (unlike in the SM) we take the central value
$M_{t}\simeq 172.5$ GeV for the top mass. $M_{t}\simeq 172.5$ GeV [46]. We
take the values computed in [20], which expresses these quantities in terms of
$M_{t}$, the Higgs mass $M_{h}\simeq 125.1$ GeV [62], the strong fine-
structure constant renormalized at the $Z$ mass, $\alpha_{s}(M_{Z})\simeq
0.1184$ [63] and $M_{W}\simeq 80.379$ GeV [62] (see the quoted references for
the uncertainties on these quantities).

Regarding the axion, besides the lower bound in (6.6) we have an upper bound
on $f_{a}$ that is obtained by requiring that dark matter is not overproduced.
Such bound has a mild logarithmic dependence on $\lambda_{A}$ (for e.g.
$\lambda_{A}$ around the scale $10^{-1}$, it is about $5\times 10^{10}$ GeV
[22]). When $f_{a}$ saturates this bound the axion accounts for the whole dark
matter [64, 65], while for smaller values the rest of dark matter can be
explained with the lightest sterile neutrino [66, 67, 68, 69], forming a two-
component DM [22].

Finally, as far as the determination of the dependence $\lambda_{H}$ and
$\xi_{H}$ on $\phi^{\prime}$ is concerned, we solve here the RGEs given in the
appendix of [22] as explained in [16].

###  6.2 Inflationary gravitational waves in the $a\nu$MSM

We now apply the formalism of Sec. 2 to determine the primordial GW spectrum
generated by CHI in the $a\nu$MSM.

Here we have an extra very light particle, the axion. Note that the effect of
axion free-streaming is only at the % level (see e.g. Fig. 8 of Ref. [56]) and
can be neglected within the approximations that have been performed here. This
means that we can keep solving the vacuum equation (2.8) for the tensor
perturbations $h_{\lambda}$.

A crucial ingredient to compute the GW spectrum of interest is $a(\tau_{q})$,
which has been estimated in Eq. (2.11). Let us analyse in turn the various
ingredients that are needed to determine $a(\tau_{q})$ in the $a\nu$MSM.

A first quantity that we need is the reference temperature $T_{r}$, which
enters $a(\tau_{q})$ because $\bar{g}^{(S)}_{*}$ is the value of $g^{(S)}_{*}$
at $T_{r}$. In the $a\nu$MSM we take $T_{r}$ by definition to be below that of
$e^{\pm}$ annihilation, but such that the three active neutrinos and the axion
are still relativistic. Such a temperature exists thanks to the lower bound on
$f_{a}$ in (6.6) (which provides an upper bound on the axion mass $m_{a}$, see
again Ref. [57] for a review).

Moreover, generically $N_{\rm eff}^{(\nu)}$ can receive an axion contribution,
which, however, is very small. The value of $\Omega_{R}$ that enters
$a(\tau_{q})$, therefore, essentially only depends on whether today the
lightest neutrino is relativistic or not, like in the SM.

Other quantities that appear in $a(\tau_{q})$ are $\bar{g}_{*}$ and
$\bar{g}^{S}_{*}$; besides the photons and active neutrinos, the QCD axions
respecting the lower bound in (6.6) gives an extra contribution to these
parameters (which is also quite small).

Finally, in the $a\nu$MSM $g_{*0}^{S}\equiv g_{*}^{S}(T_{0})$, which is one of
the quantities needed to compute the dependence of $T_{q}$ on $q$, can also
receive a contribution from the axion (depending on its mass).

These axion contributions can be computed explicitly knowing the decoupling
temperature666A simple formula for $T^{a}_{\rm dec}$ was proposed by [70]
$T^{a}_{\rm dec}\simeq 9.6\times
10^{6}\mbox{GeV}\left(\frac{f_{a}}{10^{10}\leavevmode\nobreak\
\mbox{GeV}}\right)^{2.246}.$ (6.8) We observe that this formula gives a
$T^{a}_{\rm dec}$ always below $f_{a}$ as long as $f_{a}\lesssim 10^{12}$ GeV,
which is required in order not to overproduce dark matter [22]. $T^{a}_{\rm
dec}$ of axions [70, 56] and their temperature after decoupling [57]

$T_{a}=T_{\nu}\left\\{\frac{2}{g^{S}_{*}(T_{h})-1}\left[\left(\frac{T}{T_{\nu}}\right)^{3}+\frac{21}{8}\right]\right\\}^{1/3}=T_{\nu}\left(\frac{43}{4(g^{S}_{*}(T_{h})-1)}\right)^{1/3},$
(6.9)

where $T_{h}$ represents some high value of $T$ at a time when all particles
shared the same temperature. Thermal axion production has been studied in
detail in Ref. [71] (see also Refs. [72, 70] for previous studies).

Moreover, realizing CHI in the $a\nu$MSM can also provide values of the
inflationary parameters in agreement with the most recent Planck bounds [39],
in addition to accounting for neutrino oscillations, dark matter and baryon
asymmetry, solving the strong CP problem and stabilizing the EW vacuum (all at
the same time). This is the case, for example, setting the experimental inputs
as in Sec. 6.1 and the values of the other parameters as follows:
$M_{1}=10^{11}\,$GeV, $M_{2}=6.4\times 10^{13}\,$GeV, $M_{3}>\bar{M}_{\rm
Pl}$, $f_{a}\simeq 5\times 10^{10}\,$GeV, $\lambda_{HA}(M_{A})\simeq 0.016$,
$\lambda_{A}(M_{A})\simeq 0.10$, $y(M_{A})=0.1$, $\xi_{H}(M_{A})\simeq 14$ and
$\xi_{A}(M_{A})\simeq-2.6$. This gives the following values of the
inflationary observables

$n_{s}(q_{*})\simeq 0.965,\qquad r(q_{*})=0.048,\qquad A_{\cal
R}(q_{*})=2.1\times 10^{-9}$ (6.10)

and a number of e-folds around 60. These values are in perfect agreement with
the Planck constraints in (3.11) and (3.12). Moreover, in the $a\nu$MSM all
tensions between theoretical predictions and experimental values, which occur
in the SM (cf. the end of Sec. 5), are removed. The rather high value of $r$
predicted by CHI in the $a\nu$MSM may be tested with other CMB observations,
such as those of BICEP and the Keck Array [73]. Future planned CMB experiments
(e.g. CMB S4 [74] and LiteBIRD [75]) will be sensitive to smaller values of
$r$ and thus will be able to test further CHI, even in a larger class of
models.

The corresponding GW predictions are compared with the BBO, DECIGO and ALIA
sensitivities in Fig. 1 (right plot). As clear from that plot, these space-
borne interferometers have the potential to detect directly these inflationary
GWs.

The model does not predict other GW sources that could hide such signals.
Indeed, The QCD and the electroweak transitions are too weak to generate GWs
both in the SM (as well known) and in the $a\nu$MSM: the BSM degrees of
freedom are either too massive or they couple too weakly to the SM particles
to obtain first order phase transitions, which are necessary to generate
observable GWs. Another possible source of GWs is preheating after inflation,
which can occur through parametric resonance or tachyonic effects. However,
such GWs are always peaked at a much higher frequency, $\gtrsim 10^{7}$ Hz
[76, 77] and so should not overlap significantly with the signals discussed
here. Finally, the amplitude of gravitational waves produced by QCD axion
string-wall networks is too small to be observed even in future gravitational
wave detectors with high sensitivities [78].

##  7 Conclusions

In this work we have explored the possibility to test the Higgs sector through
GW interferometers. In particular, we have computed the inflationary GW
background due to the tensor linear perturbations coming from quantum
fluctuations during HI. We have focused on CHI because in this case one can
push the scale of the perturbative breaking of unitarity due to $\xi_{H}$ very
close to the Planck scale, where anyhow new degrees of freedom are needed to
UV complete Einstein gravity. Furthermore, in the CHI, unlike in the non-
critical variant, it is not necessary to fine tune the high energy values of
any parameters in order to preserve the quantum inflationary predictions, as
commented in the introduction. This is possible because CHI features a
moderate value of $\xi_{H}$, of order 10.

After a rather general calculation of the inflationary GW spectral density
$\Omega_{\rm GW}(f)$ due to the tensor linear perturbations, both the SM and a
well-motivated phenomenological completion below the Planck scale have been
studied in detail. The latter model, dubbed the $a\nu$MSM, features three
right-handed neutrinos and a KSVZ axion sector in addition to the SM fields.
Despite being very simple, the $a\nu$MSM accounts for neutrino oscillations,
dark matter, the baryon asymmetry in the universe, solves the strong CP
problem and stabilizes the EW vacuum at the same time. While in the SM CHI is
in tension with some particle physics data (in particular $M_{t}$) and
cosmological CMB observations, the $a\nu$MSM allows us to implement CHI
without any tension with our current experimental knowledge.

We have established that, both in the SM and in the above-mentioned BSM
scenario, CHI generates an inflationary GW spectrum due to the tensor linear
perturbations that is within the reach of some space-borne GW interferometers,
such as DECIGO, BBO and ALIA.

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