# The Inflaton that Could : Primordial Black Holes and Second Order
Gravitational Waves from Tachyonic Instability induced in Higgs-$R^{2}$
Inflation

Dhong Yeon Cheong  Kazunori Kohri  Seong Chan Park

###### Abstract

The running of the Higgs self coupling may lead to numerous phenomena in early
universe cosmology. In this paper we introduce a scenario where the Higgs
running induces turns in the trajectory passing a region with tachyonic mass,
leading to a temporal tachyonic growth in the curvature power spectrum. This
effect induced by the Higgs leaves phenomena in the form of primordial black
holes and stochastic gravitational waves, where proposed GW observatories will
be able to probe in the near future.

##  1 Introduction

We are now officially entering the era of gravitational wave (GW)
observatories. After the discovery of GWs from LIGO/VIRGO, the database
including gravitational wave signals of binary systems increased significantly
[1, 2]. Recently, NANOGrav and several pulsar timing arrays reported a
background which may represent a stochastic gravitational wave background [3,
4, 5, 6, 7, 8], where scenarios incorporate solar-mass primordial black holes
[9, 10, 11, 12]. Many proposed GW observatories (e.g. LISA [13, 14], DECIGO
[15, 16], Einstein Telescope [17], SKA [18], etc.) expect to cover a wide
range of frequencies, further unraveling physics occurring in the early
universe [19, 20].

Numerous scenarios in our early universe may produce stochastic GW backgrounds
(SGWB), which include cases that induce stochastic GWs at the second order (a
comprehensive review regarding this topic can be found in [21]). Processes
induced by inflation gained much interest with a localized enhancement in the
curvature power spectrum [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 10, 11, 64, 65, 66, 67, 68, 69, 70,
71, 72, 73, 74], in correlation with copious primordial black hole (PBH)
production [75, 76, 77, 78].111 For a recent review on PBHs see [79] and
references within.

Among many inflationary models, Higgs inflation [80] and its extensions [81,
82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100,
101, 102, 103, 104, 105, 106, 107, 108, 53, 109, 110, 63, 111, 112, 113, 114,
115, 116] gain immense interest as it incorporates the Standard Model scalar
with a nonminimal coupling to gravity, and it provides the best fit to current
cosmic microwave background (CMB) observations. The running behavior of the
Standard Model Higgs is also incorporated in the potential, which prospects
numerous phenomena in our cosmology [85, 88, 93, 97]. A general setup
incorporating this is the Higgs-$R^{2}$ inflation, where two scalars, namely
the scalaron and the SM Higgs generate a two field potential [91, 92, 117, 94,
118, 102, 101, 99, 105, 103, 108, 106, 53, 110, 119, 120, 121, 63, 112, 113,
116].

Intriguingly, the running of the Higgs self coupling running can induce much
richer phenonema. We discussed the parameters that induce an inflection point
in the model describable in the framework of an effective single field case in
our previous work [53]. There, we concluded that an enhanced curvature
perturbation can be produced by a near-inflection point induced by the Higgs
running, resulting in a tight correlation with the PBH mass and the CMB
spectral index. In this paper, we revisit the Higgs running and show cases
where the inflaton possesses turns in its trajectory and approaches the hill
in the potential at $h=0$, which exhibits a tachyonic mass.222Previous studies
on the tachyonic instability in Higgs-$R^{2}$ focused on the preheating era
[106, 119, 120].  Isocurvature perturbations grow exponentially, which induce
a rapid growth of the curvature perturbations (this mechanism, mainly
incorporating a rapid turn in the non-geodesic field space has taken interest
in the past several years [57, 58, 59, 62, 63, 122, 72, 123]). This in turn
displays a sharp bump in the curvature power spectrum, where this local
feature is probe-able in the form of PBHs and stochastic GWs in a wide range
of masses and frequencies. The mass and abundance of the PBHs, and
correspondingly the energy density and the frequency of the stochastic GWs
depend on the parameter choices $(\xi,\lambda)$, which allows one to connect
and probe low energy Standard Model measurements with proposed gravitational
wave measurements.

This paper is organized as follows, we introduce the Higgs-$R^{2}$ setup
including the running behavior of the Higgs. We analyze the background
dynamics of the inflaton and classify the trajectory in steps. We then compute
the curvature and isocurvature perturbations of the model and its
corresponding PBH abundance and GW spectrum. We conclude with the implications
of the results.

##  2 Inflation action

The action for the Higgs-$R^{2}$ inflation in the Jordan frame is given as

$S_{J}=\int
d^{4}x\sqrt{-g_{J}}\left[\frac{M_{P}^{2}}{2}\left(R_{J}+\frac{\xi(\mu)\,h^{2}}{M_{P}^{2}}R_{J}+\frac{R_{J}^{2}}{6M^{2}}\right)-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}h\nabla_{\nu}h-\frac{\lambda(\mu)}{4}h^{4}\right],$
(2.1)

with the Higgs, $h$, in the unitary gauge, the reduced Planck mass being
$M_{P}=1/\sqrt{8\pi G}\simeq 2.44\times 10^{18}~{}{\rm GeV}$, and the scalaron
mass $M\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}M_{P}/\xi$ introduced to match the dimensions. We take the
Higgs self coupling running $\lambda\left(\mu\right)$ at a scale $\mu$. The
scalaron, $s$, is defined via

$\sqrt{\frac{2}{3}}\frac{s}{M_{P}}=\ln\left(1+\frac{\xi
h^{2}}{M_{P}^{2}}+\frac{R_{J}}{3M^{2}}\right)\equiv\Omega(s).$ (2.2)

Weyl transformation yields the action in Einstein frame where
$g_{\mu\nu}=e^{\Omega(s)}g^{J}_{\mu\nu}$ with two scalar fields,
$(\phi^{a})=(s,h)$ appearing in the scalar potential $U(\phi^{a})$. As a
consequence, the kinetic terms involve a nontrivial field space metric
$G_{ab}$:

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R-\frac{1}{2}G_{ab}g^{\mu\nu}\nabla_{\mu}\phi^{a}\nabla_{\nu}\phi^{b}-U(\phi^{a})\right],$
(2.3) $\displaystyle U(\phi^{a})\equiv
e^{-2\Omega(s)}\left\\{\frac{3}{4}M_{P}^{2}M^{2}\left(e^{\Omega(s)}-1-\frac{\xi(\mu)h^{2}}{M_{P}^{2}}\right)^{2}+\frac{\lambda\left(\mu\right)}{4}h^{4}\right\\}.$
(2.4)

Explicitly, the field space metric is given for $(s,h)$ as

$G_{ab}=\begin{pmatrix}1&&0\\\ 0&&e^{-\Omega(s)}\end{pmatrix}.$ (2.5)

The parameters $\left.(M,\xi,\lambda)\right|_{\mu}$ running in scale $\mu$ by
the Standard Model and scalaron interactions follow 1-loop beta functions in
the form [124, 125, 126, 127, 128, 101, 108, 110]

$\displaystyle\beta_{\alpha}$
$\displaystyle=-\frac{1}{16\pi^{2}}\frac{\left(1+6\xi\right)^{2}}{18},$ (2.6)
$\displaystyle\beta_{\xi}$
$\displaystyle=+\frac{1}{16\pi^{2}}\left(\xi+\frac{1}{6}\right)\left(12\lambda+6y_{t}^{2}-\frac{3}{2}g^{\prime
2}-\frac{9}{2}g^{2}\right),$ (2.7) $\displaystyle\beta_{\lambda}$
$\displaystyle=\beta_{\text{SM}}+\frac{1}{16\pi^{2}}\frac{2\xi^{2}\left(1+6\xi\right)^{2}M^{4}}{M_{P}^{4}},$
(2.8)

with $\alpha={M_{P}^{2}}/{12M^{2}}$ and $\beta_{\rm SM}$ being the Standard
Model contribution [129]. Choosing the renormalization prescription to be
$\mu\simeq\sqrt{h^{2}}$, we perform a standard parameterization of the
parameters $\lambda(\mu)\,,\,\,\xi(\mu)$ 333The running of $\alpha$ with the
parameters considered in this paper are characterized as
$|\beta_{\alpha}|\sim\mathcal{O}(1)$. Note that this is infinitesimal to
typical $\alpha\simeq\mathcal{O}(10^{10})$ values needed for successful
inflation. Therefore we safely neglect its running effects and take $\alpha$
as a constant. around the $\lambda$ minimum field value $h_{m}$

$\displaystyle\left.\lambda\left(\mu\right)\right|_{\mu=h}$
$\displaystyle=\lambda_{m}+\frac{\beta_{2}^{\text{SM}}}{\left(16\pi^{2}\right)^{2}}\ln^{2}\left(\sqrt{\frac{h^{2}}{h_{m}^{2}}}\right)=\lambda_{m}+b\ln^{2}\left(\sqrt{\frac{h^{2}}{h_{m}^{2}}}\right)$
(2.9) $\displaystyle\left.\xi\left(\mu\right)\right|_{\mu=h}$
$\displaystyle=\xi_{0}+2{\beta_{\xi}^{0}}\ln\left(\sqrt{\frac{h^{2}}{h_{m}^{2}}}\right)=\xi_{0}+b_{\xi}\ln\left(\sqrt{\frac{h^{2}}{h_{m}^{2}}}\right)$
(2.10)

with $\lambda_{m}\equiv\lambda(h_{m})\sim\mathcal{O}(10^{-6})$,
$\xi_{0}\equiv\xi(h_{m})\sim\mathcal{O}(1)$ , $\beta_{2}^{\text{SM}}\sim 0.5$,
$\beta_{\xi}^{0}\equiv\beta_{\xi}(h_{m})\sim-0.01$, $\mu_{m}=h_{m}\sim
10^{17}-10^{18}\,\,\text{GeV}$ as denoted in [130, 131]. 444We take
$\lambda_{m}>0$ to guarantee the stability of the Higgs potential during
inflation.  The 1-loop $\beta$-functions indicate that the running effects are
most significant in $\lambda$, with many orders changing while running from EW
scales to Planck scales, whereas the $\xi$ parameter running is insignificant
over this running range maintaining the same order. Throughout the paper we
take $M_{P}=1$ and focus on the $h>0$ region unless specified.

##  3 Background evolution

The action eq. (2.3) yields the equations of motion for the homogeneous
background fields and the Friedmann equation with the metric [99]

$\displaystyle ds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}$ (3.1)

gives, expressed incorporating the ‘curved field space metric’ effects,

$\displaystyle D_{t}\dot{\phi}^{a}+3H\dot{\phi}^{a}+G^{ab}D_{b}U$
$\displaystyle=0,$ (3.2) $\displaystyle 3H^{2}$
$\displaystyle=\frac{1}{2}\dot{\phi}_{0}^{2}+U$ (3.3)

with the covariant derivatives
$D_{a}\phi^{b}=\partial_{a}\phi^{b}+\Gamma^{b}_{ca}\phi^{c}$,
$\Gamma^{b}_{ca}=\frac{1}{2}G^{be}\left(\partial_{c}G_{ae}+\partial_{a}G_{ec}-\partial_{e}G_{ca}\right)$,
$D_{t}X^{a}=\dot{X}^{a}+\Gamma_{bc}^{a}\dot{\phi}^{b}X^{c}$, and
$\dot{\phi}_{0}^{2}=G_{ab}\dot{\phi}^{a}\dot{\phi}^{b}$.

The trajectory then takes a unique path in the field space $(s,h)$. The
parameterization of this curve can be described by constructing a set of
orthogonal unit vectors $T^{a}(t)$ and $N^{a}(t)$ where the former is tangent
to the path and the latter is normal to it, as depicted in figure 1.
Explicitly,

$T^{a}=\frac{\dot{\phi}^{a}}{\dot{\phi}_{0}}\,\,,\,\,\,\,N_{a}=\sqrt{\text{det}{\,G}}\,\epsilon_{ab}T^{b}$
(3.4)

with $\epsilon_{ab}$ being the 2 dimensional Levi-Civita symbol.

Figure 1: Tangent and normal vectors $\vec{T}$, $\vec{N}$ in field space
$(s,h)$.

Projection of the equations of motion to the tangent vector

$\ddot{\phi}_{0}+3H\dot{\phi}_{0}+U_{T}=0$ (3.5)

with $U_{T}\equiv T^{a}U_{a}$. Projections to the normal vector $N^{a}$ gives

$\frac{DT^{a}}{dt}=-\frac{U_{N}}{\dot{\phi}_{0}}N^{a}.$ (3.6)

We now define the slow-roll parameters by generalizing the setup to a
multifield scenario. They take the form

$\epsilon\equiv-\frac{\dot{H}}{H^{2}}=\frac{\dot{\phi}_{0}^{2}}{2H^{2}}\,\,,\,\,\,\,\eta^{a}\equiv-\frac{1}{H\dot{\phi}_{0}}D_{t}\dot{\phi}^{a}.$
(3.7)

Note that, the $\eta^{a}$ parameter is now a vector in the sense that there
are two degrees in the field space. The same decomposition to $T^{a}$ and
$N^{a}$ can be performed to this parameter as well, leading to

$\eta^{a}=\eta_{\parallel}T^{a}+\eta_{\perp}N^{a}$ (3.8)

with

$\displaystyle\eta_{\parallel}\equiv-\frac{\ddot{\phi}_{0}}{H\dot{\phi}_{0}}\,\,,\,\,\,\,\eta_{\perp}\equiv\frac{U_{N}}{\dot{\phi}_{0}H}.$
(3.9)

Given that $\eta_{\parallel}$ is along the tangential direction of the
trajectory, it can be regarded as the extension to the normal $\eta$ slow roll
parameter in single field inflation. $\eta_{\perp}$ on the other hand, can
also be inserted into eq. (3.6) in the sense that

$\frac{DT^{a}}{dt}=-H\eta_{\perp}N^{a}\equiv-\dot{\theta}N^{a}$ (3.10)

hence, the $\eta_{\perp}$ parameter precisely shows how quickly the tangential
direction $T^{a}$ is varying in time. The two parameters are related as
$\dot{\theta}\equiv H\eta_{\perp}$.

##  4 Cosmological perturbations

Having the background evolutions, we now perturb the action and describe the
scalar perturbations of the model. The notations used are based on [132, 133]
(see also [134, 135, 136, 137, 138]).

The fields $\phi^{a}$ and the metric can be perturbed as

$\displaystyle\phi^{a}(t,\vec{x})$
$\displaystyle=\phi_{0}^{a}(t)+\delta\phi^{a}(t,\vec{x}),$ (4.1)
$\displaystyle ds^{2}$
$\displaystyle=-(1+2\psi)dt^{2}+a(t)^{2}(1-2\psi)\delta_{ij}dx^{i}dx^{j}.$
(4.2)

Implementing the basis $T^{a}$ and $N^{a}$ to the perturbations allows the
following gauge invariant fields

$\displaystyle v_{T}$
$\displaystyle=aT_{a}\delta\phi^{a}+a\frac{\dot{\phi}_{0}}{H}\psi\equiv
aT_{a}Q^{a}$ (4.3) $\displaystyle v_{N}$
$\displaystyle=aN_{a}\delta\phi^{a}\equiv aN_{a}Q^{a}$ (4.4)

with $Q^{a}\equiv\delta\phi^{a}+\frac{\dot{\phi}^{a}}{H}\psi$ being the
Mukhanov-Sasaki variable. In terms of these variables we also define the
comoving curvature/isocurvature perturbation

$\displaystyle\mathcal{R}=\frac{H}{a\dot{\phi}_{0}}v_{T}\equiv\frac{H}{\dot{\phi}_{0}}Q_{T}$
(4.5)
$\displaystyle\mathcal{S}=\frac{H}{a\dot{\phi}_{0}}v_{N}\equiv\frac{H}{\dot{\phi}_{0}}Q_{N}.$
(4.6)

The perturbed action up to second order is then

$S^{(2)}=\frac{1}{2}\int
d^{4}xa^{3}\left[\frac{\dot{\phi}_{0}^{2}}{H^{2}}\dot{\mathcal{R}}^{2}-\frac{\dot{\phi}_{0}^{2}}{H^{2}}\frac{(\nabla\mathcal{R})^{2}}{a^{2}}+{\dot{Q}_{N}}^{2}-\frac{(\nabla
Q_{N})^{2}}{a^{2}}+4\dot{\phi}_{0}\eta_{\perp}\dot{\mathcal{R}}Q_{N}-M_{\text{eff}}^{2}Q_{N}^{2}\right]$
(4.7)

where $M_{\text{eff}}^{2}$ is

$M_{\text{eff}}^{2}=U_{NN}+H^{2}\epsilon\mathbb{R}-\dot{\theta}^{2}.$ (4.8)

The equations of motion are

$\displaystyle\ddot{\mathcal{R}}+\left(3+2\epsilon-2\eta_{\parallel}\right)H\dot{\mathcal{R}}+\frac{k^{2}}{a^{2}}\mathcal{R}=-2\frac{H^{2}}{\dot{\phi}_{0}}\eta_{\perp}\left[\dot{Q}_{N}+\left(3-\eta_{\parallel}+\frac{\dot{\eta}_{\perp}}{H\eta_{\perp}}\right)HQ_{N}\right]$
(4.9)
$\displaystyle\ddot{Q}_{N}+3H\dot{Q}_{N}+\left(\frac{k^{2}}{a^{2}}+M_{\text{eff}}^{2}\right)Q_{N}=2\dot{\phi}_{0}\eta_{\perp}\dot{\mathcal{R}}.$
(4.10)

Both eq. (4.9) and eq. (4.10) incorporate mixing between $\mathcal{R}$ and
$Q_{N}$, with the mixing proportional to $\eta_{\perp}$. A naive estimation
yields when $\eta_{\perp}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}1\rightarrow\dot{\theta}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}H$,
mixing between the two perturbations become significant and $Q_{N}$ can source
$\mathcal{R}$.

In addition to the mixing, eq. (4.8) incorporates the essentials that
determine the dynamics of $Q_{N}$. $M_{\text{eff}}^{2}$ can take a negative
value either through 1) $U_{NN}<0$, 2) $\mathbb{R}<0$, corresponding to a
hyperbolic geometry in the field space, 3) $\dot{\theta}^{2}>U_{NN}$. In any
case, a tachyonic isocurvature mass then modifies the equations of motion for
$k^{2}/a^{2}<|M_{\text{eff}}^{2}|$ to be

$\ddot{Q}_{N}+3H\dot{Q}_{N}-\left(\left|M_{\text{eff}}^{2}\right|\right)Q_{N}\simeq
0.$ (4.11)

Hence $Q_{N}$ can exhibit an exponential growth due to the tachyonic mass.
This growth can be more rapid than cases implementing a USR phase.

##  5 Inflaton and perturbation evolution

Figure 2: Potential and field trajectory (black) of the setup. Note that the
inflaton follows a well defined valley with a positive $M_{\text{eff}}^{2}$,
allowing the trajectory to be independent to the initial conditions (top-
left). The potential then exhibits a region where the valley disappears,
inducing a turn in the trajectory and approaches the hill (top-right). After
evolving along this tachyonic hill, the inflaton falls back down into a
valley.

We now turn our interest to the dynamics in the critical Higgs-$R^{2}$ setup,
starting with the trajectory, which is schematically depicted in figure 2.

  * •

Stage 1: Initially the inflaton starts rolling down a well defined valley,
satisfying the slow-roll conditions. The large and positive
$M_{\text{eff}}^{2}$ ensures the isocurvature perturbation $Q_{N}$ be
suppressed. This initial valley allows the large scale predictions (e.g. CMB)
to be insensitive to the initial values of the inflaton, simply speaking it
exhibits an attractor prediction.

  * •

Stage 2: Then the inflaton rolls down in the Higgs direction, approaching the
hill at $h=0$.

  * •

Stage 3: Once the inflaton climbs up the hill where $h\approx 0$, the inflaton
exhibits a turn, and now evolves in the $s$ direction. The isocurvature mass
$M_{\text{eff}}^{2}$ becomes large and negative, with its value mainly
determined by $\xi$. Its precise value takes the form

$\displaystyle
M_{\text{eff}}^{2}\simeq\frac{1}{\dot{s}^{2}+e^{-\sqrt{\frac{2}{3}}{s}}\dot{h}^{2}}\left(e^{\sqrt{\frac{2}{3}}{s}}\dot{s}^{2}\frac{\partial^{2}U}{\partial
h^{2}}\right)\simeq-3M^{2}\left[\xi_{0}+b_{\xi}\ln\left(\frac{h}{h_{m}}\right)+\frac{9}{2}b_{\xi}\right]\left(1-e^{-\sqrt{\frac{2}{3}}{s}}\right).$
(5.1)

The inflaton then exhibits another turn back into the valley.

  * •

Stage 4: The inflaton once again rolls into the well defined valley, with a
large and positive $M_{\text{eff}}^{2}$.

Let’s look into Stage 1, 2, 3 in more detail.

###  5.1 Stage 1.

In this region, where $M_{\text{eff}}^{2}>H^{2}>0$ and the turn rate
$\dot{\theta}\ll H$, the equations of motion simply resemble standard
effective single field, slow-roll results. The equations take the form, with
the transformation of the time variable to efolds using
$N_{e}=\int^{t}dt^{\prime}H(t^{\prime})$

$\displaystyle\frac{d^{2}\mathcal{R}_{k}}{dN_{e}^{2}}+\left(3+\epsilon-2\eta_{\parallel}\right)\frac{d{\mathcal{R}_{k}}}{dN_{e}}+\frac{k^{2}}{a^{2}H^{2}}\mathcal{R}$
$\displaystyle=0$ (5.2)
$\displaystyle\frac{d^{2}{Q}_{N,k}}{dN_{e}^{2}}+3\frac{d{Q}_{N,k}}{dN_{e}}+\left(\frac{k^{2}}{a^{2}H^{2}}+\frac{M_{\text{eff}}^{2}}{H^{2}}\right)Q_{N,k}$
$\displaystyle=0$ (5.3)

which, neglecting the slow-roll parameters as they are suppressed, give the
generally known solutions with ${\epsilon_{k}^{2}\equiv k^{2}/a^{2}H^{2}}$
[122, 123]

$\displaystyle\mathcal{R}_{k}(N_{e})$
$\displaystyle=e^{-\frac{3}{2}N_{e}}\left[c_{1}\,e^{-\frac{N_{e}}{2}\sqrt{9-4\epsilon_{k}^{2}}}+c_{2}\,e^{\frac{N_{e}}{2}\sqrt{9-4\epsilon_{k}^{2}}}\right]$
(5.4) $\displaystyle{Q}_{N,k}(N_{e})$
$\displaystyle=e^{-\frac{3}{2}N_{e}}\left[c_{3}\,e^{-\frac{N_{e}}{2}\sqrt{9-4\frac{M_{\text{eff}}^{2}}{H^{2}}-4\epsilon_{k}^{2}}}+c_{4}\,e^{\frac{N_{e}}{2}\sqrt{9-4\frac{M_{\text{eff}}^{2}}{H^{2}}-4\epsilon_{k}^{2}}}\right]$
(5.5)

where $c_{1},\,c_{2},\,c_{3},\,c_{4}$ are determined by the conditions at the
in-horizon state. We can see that for the case when $\epsilon_{k}^{2}\ll 1$,
i.e. out of the horizon,
$\mathcal{R}_{k}(N_{e})\propto\mathcal{R}_{0}+\mathcal{R}_{1}e^{-3N_{e}}$ and
has a mode freezing out, being constant deep outside the horizon. The
isocurvature mode $Q_{N,k}$, in contrast, always is suppressed as
$Q_{N,k}(N_{e})\propto e^{-\frac{3}{2}N_{e}}e^{\pm
i\frac{M_{\text{eff}}}{H}N_{e}}$ due to $M_{\text{eff}}\gg H$ in this regime,
and therefore is exponentially suppressed in the deep out of horizon region,
giving negligible effects on cosmological observables.

We now obtain standard effective single field slow-roll observables. As the
inflaton evolution in this era remains in the same $h$ order, we approximate
$\xi(h)\simeq\xi_{0}$. In this period, the fields resemble the approximate
relation in the large-scale observable region

$s_{v}\approx\sqrt{\frac{3}{2}}\ln\left[1+\frac{4(\lambda_{m}+3M^{2}\xi_{0}^{2})h^{2}+(7h-5h_{m})(h-h_{m})b}{12M^{2}\xi_{0}}\right]$
(5.6)

allowing us to combine the fields to be parameterized with the scalaron only,
leading to the slow roll parameters

$\epsilon_{H}\simeq\epsilon_{V}\equiv\frac{1}{2}\left.\left[\frac{U_{s}(s,h(s))}{U(s,h(s))}\right]^{2}\right|_{s=s_{*}}\,\,,\,\,\,\,\,\eta_{\parallel}\simeq\eta_{V}\equiv\left.\frac{U_{ss}(s,h(s))}{U(s,h(s))}\right|_{s=s_{*}}$
(5.7)

with $s_{*}$ being the scalaron field value at the CMB pivot scale and

$n_{s}\equiv 1+\frac{d\ln{\mathcal{P}_{\mathcal{R}}(k)}}{d\ln k}\simeq
1-6\epsilon_{V}+2\eta_{V}\,\,,\,\,\,\,\,r\simeq 16\epsilon_{V}.$ (5.8)

The inflationary epoch exhibits slow-roll with $\eta_{H}\ll 1$, and the
$\frac{\lambda(h)}{4}h^{4}$ term in the potential eq. 2.4 is orders smaller
than other terms. This lets us reasonally take the inflationary efolds
$N_{\text{inf}}=N_{\text{end}}-N_{\text{pivot}}\approx\frac{3}{4}e^{\sqrt{\frac{2}{3}}s_{*}}$,
which resembles an $R^{2}$ inflation-like form. Therefore, the above
expressions can be approximately expressed to

$\displaystyle n_{s}\approx$
$\displaystyle\,\,1-\frac{2}{N_{\text{inf}}}-\frac{9}{2N_{\text{inf}}^{2}}$
(5.9)
$\displaystyle+\frac{M^{2}\xi_{0}^{2}\,b}{\lambda_{m}\left(\lambda_{m}+3M^{2}\xi_{0}^{2}\right)}\left[2+\frac{3\left(2+3\ln\left(\frac{(\lambda_{m}+3M^{2}\xi_{0}^{2})h_{m}^{2}}{4M^{2}\xi_{0}N_{\text{inf}}}\right)\right)}{2N_{\text{inf}}}+\frac{27\left(1+\ln\left(\frac{(\lambda_{m}+3M^{2}\xi_{0}^{2})h_{m}^{2}}{4M^{2}\xi_{0}N_{\text{inf}}}\right)\right)}{4N_{\text{inf}}^{2}}\right]$

and

$r\approx\frac{12}{N_{\text{inf}}^{2}}+\frac{2M^{2}\xi_{0}^{2}\,b}{\lambda_{m}(\lambda_{m}+3M^{2}\xi_{0}^{2})N_{\text{inf}}}\left[12\ln\left(\frac{4M^{2}\xi_{0}N_{\text{inf}}}{\left(\lambda_{m}+3M^{2}\xi_{0}^{2}\right)h_{m}^{2}}\right)+\frac{9\left(2\ln\left(\frac{4M^{2}\xi_{0}N_{\text{inf}}}{\left(\lambda_{m}+3M^{2}\xi_{0}^{2}\right)h_{m}^{2}}\right)-1\right)}{N_{\text{inf}}}\right]$

(5.10)

where $N_{\text{end}}$ describes the efolds at the end of inflation, and
$N_{\text{pivot}}$ represents the efolds at the CMB pivot scale. Therefore the
additional logarithmic running of the Higgs self coupling shifts the spectral
index of the curvature power spectrum to larger values compared to the pure
Higgs-$R^{2}$ case with a constant Higgs self coupling.

Figure 3: (Left) Contour of $\frac{dU}{dh}=0$, with the trajectory of the
inflaton colored in red. Once $h$ reaches $h_{local\,\,min}$,
$\frac{dU}{dh}>0$, making the inflaton fall down towards the hill. (Right)
Trajectory for several parameter sets, where the dashed black line expresses
the field value for $N_{\text{inf}}=50$. The position where hill-climbing
occurs depends on $\xi$ values.

###  5.2 Stage 2.

This stage contains the initial deviation from the valley. Recall that for the
Higgs-$R^{2}$ potential, the inflaton initially follows a valley well defined
by $\partial_{h}U=0$, in which the scalaron field at the valley $s_{v}$ takes
the expression

$s_{v}\simeq\sqrt{\frac{3}{2}}\ln\left[\frac{6M^{2}\xi_{0}+2h^{2}\left(\lambda_{m}+3M^{2}\xi_{0}^{2}\right)+b\,h^{2}\ln\left(\frac{h}{h_{m}}\right)\left(2\ln\left(\frac{h}{h_{m}}\right)+1\right)}{6M^{2}\xi_{0}}\right].$
(5.11)

This trajectory in general, can have critical points in the $(h,s)$ plane,
being

$\frac{h_{local\,\,max}}{h_{m}}\simeq
e^{-\frac{3}{4}-\frac{\sqrt{5b^{2}-16b\lambda_{m}-48bM^{2}\xi_{0}^{2}}}{4b}}\,\,,\,\,\,\frac{h_{local\,\,min}}{h_{m}}\simeq
e^{-\frac{3}{4}+\frac{\sqrt{5b^{2}-16b\lambda_{m}-48bM^{2}\xi_{0}^{2}}}{4b}}.$
(5.12)

Note that these extremal points in the field space exist when the following
conditions are satisfied

$0\leq\xi_{0}\lesssim\frac{1}{4\sqrt{3}M}\sqrt{5b-16\lambda_{m}}\,\,,\,\,\,\lambda_{m}<\frac{5b}{16}.$
(5.13)

We focus on the trajectory point $h_{local\,\,min}$. Once the inflaton hits
this point, $\partial_{h}U>0$ in the order

$\left.\frac{\partial U}{\partial
h}\right|_{s_{v}(h_{local\,\,min}),\,h_{local\,\,min}\pm\delta}=A\,\delta^{2}+\mathcal{O}(\delta^{3})$
(5.14)

for both $h_{local\,\,min}+\delta$ and $h_{local\,\,min}-\delta$ with
$\delta>0$, with $A$ being a positive constant. Therefore, the inflaton starts
rolling down towards the $h$ direction, approaching the hill at $h=0$. This is
depicted in figure 3. where the figure focuses on the transition region, and
it shows that the potential exhibits a region where $dU/dh>0$ giving a roll-
down towards the hill.

###  5.3 Stage 3.

This stage is precisely where the large and negative isocurvature mass induces
an exponential growth in the isocurvature perturbation $Q_{N}$. The evolution
of the perturbations are depicted in figure 4.

Figure 4: (Top) Perturbation evolution for the $k$ mode that exits the horizon
at $N_{\text{exit}}=44$. The efolds when the inflaton starts and stops rolling
down the hill are denoted as $N_{1}$ and $N_{2}$ respectively. The black-
dashed line represents the growth
$\mathcal{P}_{\mathcal{S}}\propto\exp\left[\left(\frac{2|M_{\text{eff}}|}{H}-3\right)N_{e}\right]$.
(Bottom) $M_{\text{eff}}^{2}/H^{2}$ and $\dot{\theta}^{2}/H^{2}$ evolution.
The tachyonic $M_{\text{eff}}$ induces an exponential growth in the
$\mathcal{P}_{\mathcal{S}}$, and due to the mixing between curvature and
isocurvature perturbations this enhancement is translated over to
$\mathcal{P}_{\mathcal{R}}$.

Figure 5: Compilation of the perturbations for $k_{\text{exit}}$ leaving at
certain scales, which corresponds to a definite $N_{\text{exit}}$. We observe
that after the tachyonic increase of the isocurvature, and consequentially the
curvature perturbation, the isocurvature part exponentially decays, and the
perturbations become adiabatic at scales evaluated at the end of inflation
($N_{\text{end}}$ in the plot, shown at the star).

From the first turn, due to the mixing term in the equations of motion, the
perturbations mix and as $\dot{\theta}^{2}/H^{2}>1$, the $\mathcal{R}$
experiences a slight bump in its evolution, however its effect is negligible.

The period when the inflaton rolls down the hill, where the starting efolds at
$N_{1}$ and the end efolds at $N_{2}$, the isocurvature mass takes
$M_{\text{eff}}^{2}/H^{2}\ll 0$, therefore the isocurvature perturbation
$Q_{N}$ is dominated by the exponential growth from the negative isocurvature
mass. Recalling the isocurvature perturbations equations eq. (4.10), eq. (5.3)
while neglecting the source terms ,as this is precisely the case when the
inflaton rolls down the $h=0$ hill,

$\frac{d^{2}{Q}_{N,k}}{dN_{e}^{2}}+3\frac{d{Q}_{N,k}}{dN_{e}}+\left(\frac{k^{2}}{a^{2}H^{2}}-\frac{|M_{\text{eff}}^{2}|}{H^{2}}\right)Q_{N,k}=0$
(5.15)

with the solutions

${Q}_{N,k}(N_{e})=e^{-\frac{3}{2}N_{e}}\left[d_{3}\,e^{-\frac{N_{e}}{2}\sqrt{9-4\frac{M_{\text{eff}}^{2}}{H^{2}}-4\epsilon_{k}^{2}}}+d_{4}\,e^{\frac{N_{e}}{2}\sqrt{9-4\frac{M_{\text{eff}}^{2}}{H^{2}}-4\epsilon_{k}^{2}}}\right]\xrightarrow[|M_{\text{eff}}^{2}|\gg
H^{2}]{\epsilon_{k}^{2}\ll
1}d_{4}~{}e^{\left(\frac{|M_{\text{eff}}|}{H}-\frac{3}{2}\right)N_{e}}$ (5.16)

with the $M_{\text{eff}}^{2}$ in this period taking the form of eq. (5.1).
Consequentially,

$\displaystyle\mathcal{P}_{\mathcal{S}}(k_{\text{exit}},N_{e})=\frac{k_{\text{exit}}^{3}}{2\pi^{2}}\frac{H^{2}}{\dot{\phi}_{0}^{2}}\langle
Q_{N,k}\,Q_{N,k}\rangle=\mathcal{P}_{\mathcal{S}}(k_{\text{exit}},N_{1})~{}e^{\left(\frac{2|M_{\text{eff}}|}{H}-{3}\right)(N_{e}-N_{1})}$
(5.17)

therefore $Q_{N,k}$, and consequentially $\mathcal{P}_{\mathcal{S}}$ grows
exponentially during this period. $\mathcal{R}$, however, does not grow
instantaneously, precisely due to the fact that $\dot{\theta}^{2}/H^{2}\ll 1$
in this period. Then, as the second turn with $\dot{\theta}^{2}/H^{2}\gg 1$
occurs, the enhanced $Q_{N}$ is then sourced to $\mathcal{R}$, now also
exponentially growing and decaying according to the mixing, will then stop
evolving in the superhorizon limit when $\dot{\theta}^{2}/H^{2}\ll 1$ occurs
again.

One may raise the following question:is eq. (2.9) a sufficient approximation
for this scenario, judging by the fact that $h$ approaches 0. The answer is
yes. The turn in the trajectory occurs around eq. (5.12) where
$h_{local\,\,min}\sim h_{m}$. The region where $h\ll h_{m}$, the potential
term $U\supset\frac{\lambda(h)}{4}h^{4}e^{-2\sqrt{\frac{2}{3}}s}$ is
subdominant.

##  6 Power spectrum and PBH abundance

We numerically compute the cosmological perturbations in this scenario using
PyTransport [139]. Here we present several parameter sets exhibiting this
large local feature in the power spectrum, enlisted in Table 1. The Planck CMB
pivot scale is set as $k_{*}=0.05~{}\text{Mpc}^{-1}$.

Figure 6: The observational curvature power spectrum
$\mathcal{P}_{\mathcal{R}}(k,N_{\text{end}})$ for the benchmark parameter sets
in Table 1.

Set | $M(M_{P})$ | $\xi_{0}$ | $\lambda_{m}(\times 10^{-6})$ | $\beta_{2}$ | $\beta_{\xi}^{0}$ | $h_{m}\,(M_{P})$ | $k_{max}\,(\text{Mpc}^{-1})$ | $\mathcal{P}_{\mathcal{R},max}$ | $n_{s}$ | $r$
---|---|---|---|---|---|---|---|---|---|---
$1$ | $1.3\times 10^{-5}$ | $4.0$ | $4.1743336$ | $0.5$ | $-0.01$ | $0.21$ | $3.9\times 10^{12}$ | $0.032$ | $0.967$ | $0.004$
$2$ | $1.3\times 10^{-5}$ | $3.5$ | $4.1003376$ | $0.5$ | $-0.01$ | $0.21$ | $1.2\times 10^{11}$ | $0.027$ | $0.967$ | $0.004$
$3$ | $1.3\times 10^{-5}$ | $3.0$ | $4.0109148$ | $0.5$ | $-0.01$ | $0.21$ | $1.6\times 10^{9}$ | $0.034$ | $0.967$ | $0.004$
$4$ | $1.3\times 10^{-5}$ | $2.5$ | $3.8998765$ | $0.5$ | $-0.01$ | $0.21$ | $6.5\times 10^{6}$ | $0.02$ | $0.967$ | $0.004$

Table 1:  Several benchmark parameter sets ($M$, $\xi_{0}$, $\lambda_{m}$,
$\beta_{2}=(4\pi)^{4}\,b$, $\beta_{\xi}^{0}=b_{\xi}/2$, $h_{m}$) and their
corresponding small-scale observables.

The corresponding curvature power spectrum $\mathcal{P}_{\mathcal{R}}(k)$ for
these parameter sets are depicted in figure 6. Each exhibit a near-scale
invariant power spectrum with $n_{s}$ giving perfect consistency with current
Planck CMB observations [140, 141]. Note that the CMB predictions also are
consistent among parameters, regardless of the position of the localized peak.
This is precisely due to the fact that the tachyonic enhancement of the
curvature/isocurvature perturbations induces an exponential increase,
requiring a much shorter period on how long this instability sustains. This
amplified $\mathcal{P}_{\mathcal{R}}$ at small scales can also lead to copious
PBH production, which depending on the mass of the PBH can account for the
majority of the dark matter in our universe. Taking a peaks theory approach
[142, 143, 144, 145, 146, 147, 148, 149], where the details are in Appendix
A., we depict the PBH abundance
$f_{\text{PBH}}(M_{\text{PBH}})=\Omega_{\text{PBH}}/\Omega_{\text{DM}}$ in
figure 7., taking a common critical density contrast $\delta_{c}=0.41$ [150,
151] and a Gaussian window function
$W(k,R)=\exp\left(-\frac{k^{2}R^{2}}{2}\right)$. One major feature the
tachyonic instability-induced $\mathcal{P}_{\mathcal{R}}$ enhancement composes
is that it can cover all viable mass ranges of PBHs, which is in contrast with
our previous USR induced PBH studies [53].

Figure 7: The corresponding PBH abundances for the benchmark parameter sets in
Table 1. The observational constraints are taken from [79, 152]. Depending on
the parameters the tachyonic instability-induced perturbations one can induce
scenarios where PBHs can consist a significant amount of dark matter within
the constraint bounds. Set-4 is not depicted in the figure range due to a
smaller peak value $\mathcal{P}_{\mathcal{R},max}$.  Figure 8: Parameter
sensitivity on the inflaton trajectory. For demonstration, we take the
parameter $\lambda_{m}$ tuning for Set. 1 in Table 1. The percentage level
corresponds to the parameter $\delta\lambda_{m}/\lambda_{m}$.

Figure 9: (Left )The degree of parameter $\lambda_{m}$ tuning for Set. 1 in
Table 1. The percentage level corresponds to the parameter
$\delta\lambda_{m}/\lambda_{m}$. (Right) The degree of parameter $\xi_{0}$
tuning for Set. 1 in Table 1. The percentage level corresponds to the
parameter $\delta\xi_{0}/\xi_{0}$.

###  6.1 Degree of parameter tuning

Obviously, the tachyonic enhancement is subject to parameter tuning. The
evolution of the inflaton along the hill determines the amount of enhancement,
and as this period itself is unstable the parameters will need some degree of
tuning to have a noticeable perturbation growth.

To show this, we choose the Set 1. parameters and vary $\lambda_{m}$ and
$\xi_{0}$ and explicitly allowing the $h<0$ region.555We take $\lambda_{m}$
and $\xi_{0}$ as the tuning parameter as it encapsulates the role on
sustaining the instability period.  As depicted in figure 8., the field
evolution along the $h=0$ hill is sensitive to the parameter choices. This
difference to the evolution period on the hill leads to a difference in the
curvature power spectrum peak, depicted in figure 9. Notice that in order to
sustain an enhancement in the curvature power spectrum to be
$\mathcal{P}_{\mathcal{R}}\sim 10^{-2}$, the parameter
$\delta\lambda_{m}/\lambda_{m}\equiv(\lambda_{m}^{dev}-\lambda_{m})/\lambda_{m}\sim\mathcal{O}(10^{-4})\,\%$,
with $\lambda_{m}^{dev}$ being the parameter that deviates from the benchmark
set parameter, which is in similar orders with fine-tuning degrees in single-
field polynomial inflation models as well [39]. The $\xi_{0}$ parameter, on
the other hand, is an order less sensitive to achieve the same order of
enhancement, with
$\delta\xi_{0}/\xi_{0}\equiv(\xi_{0}^{dev}-\xi_{0})/\xi_{0}\sim\mathcal{O}(10^{-3})\,\%$,
Noticeably, if one requires the detectability of the SGWB, the required
enhancement softens to $\mathcal{P}_{\mathcal{R}}\sim 10^{-4}\,(10^{-5})$,
hence the tuning of the parameters also decrease to an order less to be
$\delta\lambda_{m}/\lambda_{m}\sim\mathcal{O}(10^{-3})\,\%$,
$\delta\xi_{0}/\xi_{0}\sim\mathcal{O}(10^{-2})\,\%$.

##  7 Stochastic gravitational wave background (SGWB) at the second order

Figure 10: The corresponding GW abundances for the benchmark parameter sets in
Table 1, overlayed over various gravitational wave observatory’s sensitivity
curves [153, 154]. The white region at nano-Hertz frequency corresponds to the
NANOGrav 12.5 year results [3].

The amplified curvature perturbations also lead to a copious amount of
stochastic GWs. The current energy density fraction per logarithmic wavelength
of the GWs from second order is, following [155, 156]

$\displaystyle\Omega_{\text{GW}}(\eta_{0},k)=c_{g}\frac{\Omega_{r,0}}{6}\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du\left(\frac{4v^{2}-(1+v^{2}-u^{2})^{2}}{4uv}\right)^{2}\overline{\mathcal{I}^{2}(v,u)}\mathcal{P}_{\mathcal{R}}(kv)\mathcal{P}_{\mathcal{R}}(ku)$
(7.1)

where $\Omega_{r,0}\approx 5.38\times 10^{-5}$ is the current radiation energy
density fraction [140], $x\equiv k\eta$ with $\eta$ being the conformal time,
$c_{g}\equiv\frac{a_{f}^{4}\rho_{r}(\eta_{f})}{\rho_{r}(\eta_{0})}=\frac{g_{*}}{g_{*}^{0}}\left(\frac{{g_{*}^{0}}_{S}}{{g_{*}}_{S}}\right)^{4/3}\approx
0.4$ by taking the current universe effective energy and entropy degree of
freedom as $g_{*}^{0}=3.36$ and ${g_{*}^{0}}_{S}=3.91$ respectively. The
degree of freedom at the evaluation of perturbations take the value
${g_{*}}={g_{*}}_{S}=106.75$. $\overline{\mathcal{I}^{2}(v,u)}$ is expressed
analytically in radiation-domination as [155, 156, 157].

$\displaystyle\overline{\mathcal{I}^{2}(v,u)}=\frac{1}{2}\left[\frac{3(u^{2}+v^{2}-3)}{4u^{3}v^{3}}\right]^{2}$
$\displaystyle\left[\left(-4uv+(u^{2}+v^{2}-3)\ln\left|\frac{3-(u+v)^{2}}{3-(u-v)^{2}}\right|\right)^{2}\right.$
$\displaystyle+\left.\pi^{2}(u^{2}+v^{2}-3)^{2}\Theta(v+u-\sqrt{3})\right].$
(7.2)

The current universe SGWB energy fraction $\Omega_{\text{GW}}\,h^{2}$ for our
benchmark parameters are depicted in figure 10. Note that due to the nature of
the tachyonic instability, the GW spectrum can span over all frequencies from
aLIGO frequencies all the way up to PTA frequencies. Utilizing current and
future GW observatories, we can obtain useful information on the $\xi$ and
$\lambda(h)$ parameters, ultimately gaining information on the running
behavior of the SM Higgs self coupling for high energy scales. Noticeably, our
Set. 4 parameters directly correspond to the recently reported NANOGrav
results. Therefore, these running parameters and nonminimal coupling values,
if the results are confirmed to be indeed a SGWB, will be able to be a
possible source of the observed SGWB.

##  8 Conclusion and Discussions

In this work we demonstrated that the $\lambda$ running in Higgs-$R^{2}$
inflation can be implemented in the effective production of enhanced curvature
perturbations, consequentially leading to amplified second order stochastic
gravitational wave productions and possibly primordial black holes. The two
field potential characterized by the scalaron and Higgs $(s,h)$ exhibits a
temporary valley structure breakdown induced by the running of $\lambda(h)$.
The inflaton then rolls down towards the hill at $h=0$, where there is a
tachyonic instability that depends on the scalaron mass $M$ and the nonminimal
coupling $\xi$. Isocurvature perturbations are exponentially enhanced, which
are transferred to the curvature perturbation as the inflaton rolls back down
the hill and settles at the lower field valley. Compared to our previous work
that implemented a ultra-slow-roll phase [53], the tachyonic enhancement
presented in this work occurs in a much shorter duration, hence allowing for
effectively all mass ranges of primordial black hole dark matter without
conflicting Planck CMB observables. It also allows parameters that coincide
with the recently reported stochastic process by NANOGrav and other PTA
observatories.

There are several aspects to foresee from here. In this work we effectively
parameterized the $\lambda$ running with the parameters
$\lambda_{m},~{}b,~{}h_{m}$. In principle these will correspond to the
physical running parameters $m_{top},~{}\alpha_{s},~{}m_{W}$ at low energies,
therefore providing a connection between the low energy SM parameters and the
inflationary features. A full parameter scan of these SM parameters that
incorporate a sizable $\mathcal{P}_{\mathcal{R}}$ at a certain scale will
provide information on the running parameters of our Standard Model Higgs,
which we seek to pursue in a future work [158].

Also, in this work we only considered Gaussian fluctuations in the
cosmological perturbations. It is well known that multi-field inflation,
especially those incorporating a tachyonic instability may exhibit high levels
on non-Gaussianity, which will alter the predictions of SGWB, and PBH
abundances [159, 160, 161, 162, 163]. We leave the computation of non-
Gaussianity and its impact on small scale observables for a future work.

## Acknowledgments

We thank Minxi He, Sung Mook Lee, Shi Pi, Misao Sasaki for precious
discussions. This work was supported by JSPS KAKENHI grant number JP17H01131
(K.K.), MEXT KAKENHI grant number JP20H04750 (K.K.), and the National Research
Foundation grants funded by the Korean government (MSIP)
(NRF-2019R1A2C1089334)&(NRF-2021R1A4A2001897) (SCP).

##  Appendix A Peaks theory approach on the PBH abundance calculation.

In this appendix we address the peaks theory approach to calculate the PBH
abundance. The density contrast power spectrum is related to the curvature
power spectrum through the following formula

$\displaystyle\mathcal{P}_{\delta}(k)=\frac{4(1+w)^{2}}{(5+3w)^{2}}\left(\frac{k}{aH}\right)^{4}\mathcal{P}_{\mathcal{R}}(k)\rightarrow\frac{16}{81}\left(\frac{k}{aH}\right)^{4}\mathcal{P}_{\mathcal{R}}(k)\,\,\,\text{
in RD : $w=\frac{1}{3}$ . }$ (A.1)

We then smooth the density contrast to a typical scale $R=\frac{1}{aH}$, with
a window function $W(k,R)$ in Fourier space. Hence, the spectral index of the
density contrast with a window function will take the form

$\displaystyle\sigma_{i}^{2}(R)=\int_{0}^{\infty}\frac{dk}{k}k^{2i}W^{2}(k,R)\mathcal{P}_{\mathcal{\delta}}=\frac{16}{81}\int_{0}^{\infty}\frac{dk}{k}k^{2i}W^{2}(k,R)\mathcal{P}_{\mathcal{R}}$
(A.2)

with the $i=0$-th moment corresponding to the variance
$\sigma_{0}^{2}(R)\equiv\langle\delta^{2}(\mathbf{x},R)\rangle$ . The PBH mass
can be expressed as a function to the associated horizon mass

$\displaystyle M_{\text{PBH}}=\mathcal{K}M_{H}=\frac{\mathcal{K}}{2GH}$ (A.3)

with $M_{H}$ being the horizon mass, $\mathcal{K}$ being the collapse
efficiency. We take $\mathcal{K}=0.2$. Associating the mass with the relevant
wave number, we get the following scaling

$\displaystyle M_{\text{PBH}}=4.64\times
10^{15}~{}{\gamma}\left(\frac{g_{*}}{106.75}\right)^{-\frac{1}{6}}\left(\frac{k_{\text{PBH}}}{k_{*}}\right)^{-2}M_{\odot}$
(A.4)

with $k_{*}=0.05~{}\text{Mpc}^{-1}$ denoting the CMB pivot scale and
$g_{*}=106.75$ the relativistic degrees of freedom at that epoch.

In peaks theory [142] ( see also [143, 144, 145, 146, 147, 148, 149]) , the
PBH mass fraction
$\beta_{\text{PBH}}(M)\equiv\left.\frac{\rho_{\text{PBH}}(M)}{\rho_{\text{tot}}}\right|_{\text{formation}}$
is related to the peak number density over a criteria $\nu>\nu_{c}$

$\displaystyle
n(\nu_{c})=\frac{1}{(2\pi)^{2}}\left(\frac{\sigma_{2}}{\sqrt{3}\sigma_{1}}\right)^{3}\int_{\nu_{c}}^{\infty}d\nu\int_{0}^{\infty}d\xi_{1}\frac{f(\xi_{1})}{\sqrt{2\pi(1-\gamma^{2})}}\exp{\left[-\frac{1}{2}\left(\nu^{2}+\frac{(\xi_{1}^{2}-\gamma\nu)^{2}}{1-\gamma^{2}}\right)\right]}$
(A.5)

leading to the relation

$\displaystyle\beta_{\text{PBH}}(R)$
$\displaystyle=n(\nu_{c})(2\pi)^{3/2}R^{3}$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\left(\frac{R\sigma_{2}}{\sqrt{3}\sigma_{1}}\right)^{3}\int_{\nu_{c}}^{\infty}d\nu\int_{0}^{\infty}d\xi_{1}\frac{f(\xi_{1})}{\sqrt{2\pi(1-\gamma^{2})}}\exp{\left[-\frac{1}{2}\left(\nu^{2}+\frac{(\xi_{1}^{2}-\gamma\nu)^{2}}{1-\gamma^{2}}\right)\right]}.$
(A.6)

The parameters stated here are $\nu=\frac{\delta}{\sigma_{0}}$,
$\nu_{c}=\frac{\delta_{c}}{\sigma_{0}}$,$\gamma=\frac{\sigma_{1}^{2}}{\sigma_{0}\sigma_{2}}$,
and the $f(\xi_{1})$ function being

$\displaystyle f(\xi_{1})=\frac{1}{2}(\xi_{1}^{3}-$ $\displaystyle
3\xi_{1})\left(\text{erf}\left[\sqrt{\frac{5}{2}}\xi_{1}\right]+\text{erf}\left[\sqrt{\frac{5}{8}}\xi_{1}\right]\right)$
$\displaystyle+\sqrt{\frac{2}{5\pi}}\left\\{\left(\frac{8}{5}+\frac{31}{4}\xi_{1}^{2}\right)\exp\left[-\frac{5}{8}\xi_{1}^{2}\right]+\left(-\frac{8}{5}+\frac{1}{2}\xi_{1}^{2}\right)\exp\left[-\frac{5}{2}\xi_{1}^{2}\right]\right\\}.$
(A.7)

Taking a high-peak approximation $\gamma\nu\gg 1$ and $\gamma\simeq 1$, one
can obtain an analytic form of the $\beta_{\text{PBH}}(M_{\text{PBH}})$
associated with the PBH mass [143]

$\displaystyle\beta_{\text{PBH}}(M_{\text{PBH}})=\frac{1}{\sqrt{2\pi}}\left(\frac{R~{}\sigma_{1}}{\sqrt{3}\sigma_{0}}\right)^{3}\left(\nu_{c}^{2}-1\right)\exp\left(-\frac{\nu_{c}^{2}}{2}\right).$
(A.8)

The current day PBH energy density fraction against the total dark matter
energy density $f_{\text{PBH}}(M_{\text{PBH}})$ is obtained from
$\beta_{\text{PBH}}(M_{\text{PBH}})$ through the following relation

$\displaystyle
f_{\text{PBH}}\left(M_{\text{PBH}}\right)\equiv\left.\frac{\Omega_{\rm
PBH}}{\Omega_{\rm DM}}\right|_{\text{today}}\simeq 2.7\times
10^{8}\left(\frac{\mathcal{K}}{0.2}\right)^{\frac{1}{2}}\left(\frac{10.75}{g_{*}}\right)^{\frac{1}{4}}\left(\frac{M_{\odot}}{M_{\text{PBH}}}\right)^{\frac{1}{2}}\beta_{\text{PBH}}(M_{\text{PBH}}).$
(A.9)

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