February 2023 IPMU23-0005

Fitting power spectrum of scalar perturbations for primordial black hole
production during inflation

Daniel Frolovsky a and Sergei V. Ketov a,b,c,#

a Interdisciplinary Research Laboratory, Tomsk State University

36 Lenin Avenue, Tomsk 634050, Russia

b Department of Physics, Tokyo Metropolitan University

1-1 Minami-ohsawa, Hachioji-shi, Tokyo 192-0397, Japan

c Kavli Institute for the Physics and Mathematics of the Universe (WPI)

The University of Tokyo Institutes for Advanced Study,

Kashiwa 277-8583, Japan

#<EMAIL_ADDRESS>
Abstract

We propose a simple analytic fit for the power spectrum of scalar (curvature)
perturbations during inflation, in order to describe slow roll of inflaton and
formation of primordial black holes in the early universe, in the framework of
single-field models. Our fit is given by a sum of the power spectrum in the
slow-roll approximation, needed for a viable description of the cosmic
microwave background radiation in agreement with Planck/BICEP/Keck
measurements, and the log-normal (Gaussian) fit for the power spectrum
enhancement (peak) needed for efficient production of primordial black holes.
We use the T-type $\alpha$-attractor models in order to describe slow-roll
inflation. Demanding the location and height of the peak to yield the masses
of primordial black holes in the asteroid-size window allowed for the whole
(current) dark matter to be composed of the primordial black holes, we find
the restrictions on the remaining parameters and, most notably, on the width
of the peak.

##  1 Introduction

The inflationary paradigm was initially proposed as a possible solution to the
internal problems of the standard (Einstein-Friedmann) cosmology such as the
horizon problem, the flatness problem and the problem of initial conditions
[1, 2]. It was later recognized that inflation in the early universe may be a
solution to the structure formation problem also [3]. A major recognition of
the inflationary paradigm came with its success in explaining the
inhomogeneity and anisotropy of the cosmic microwave background (CMB)
radiation [4].

The underlying physics of inflation is still unknown but there is no shortage
of theoretical models of inflation. The simplest single-field models of
chaotic inflation are based on the quintessence (scalar-tensor gravity) or the
modified $F(R)$-gravity theories. More recently, the quintessence models were
further generalized by adding a near-inflection point to the inflaton
potential below the inflationary scale, leading to a peak in the power
spectrum of scalar perturbations that later collapse to primordial black holes
(PBH) [5, 6, 7, 8]. 111See also Refs. [9, 10, 11, 12] and the references
therein for observational constraints on PBH and their formation in single-
field inflationary models. PBH are also considered as a good (non-particle)
candidate for the present dark matter [13, 14].

Usually, one begins with a particular inflationary model having a specific
scalar potential, and then one numerically derives the power spectrum by using
the Mukhanov-Sasaki equation [15, 16]. In this paper, we begin with analytic
modeling of the power spectrum of scalar (curvature) perturbations for
possible PBH production in agreement with CMB measurements. We choose the
simplest fit as a sum of the CMB power spectrum in the slow-roll approximation
and the log-normal shape of the peak. This allows us to get analytic smooth
sewing of both spectra with the minimal number of parameters and a possibility
to analytically explore the whole parameter space, which is often difficult in
a numerical approach. As regards the CMB power spectrum, we describe it with
the help of the T-type $\alpha$-attractor models of inflation [17, 18] in
order to get the simplest form of the spectrum during slow roll. As a result,
we find new restrictions on the parameters of PBH production.

Our paper is organized as follows. In Sec. 2 we review the T-type
$\alpha$-attractor models of inflation, which are used as the baseline models
for CMB in the next Sections. The power spectrum of scalar perturbations
during inflation in the slow-roll approximation (relevant to CMB) is derived
in Sec. 3. Section 4 is devoted to our fit of the power spectrum of scalar
perturbations for both CMB and PBH production, and the related spectrum of
induced gravitational waves (GW). Our conclusion is Sec. 5.

##  2 Single-field models of slow-roll inflation for CMB

As the baseline models of large-single-field inflation, described by the
standard quintessence action

$S[g_{\mu\nu},\phi]=\int d^{4}x\,\sqrt{-g}\left\\{{{M^{2}_{\rm
Pl}}\over{2}}R-{{1}\over{2}}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-V(\varphi)\right\\}~{}~{},$
(1)

we choose the T-type $\alpha$-attractors [17, 18] with the canonical inflaton
potential

$V(\varphi)=V_{0}\tanh^{2}\left({{\varphi/M_{\rm
Pl}}\over{\sqrt{6\alpha}}}\right)\equiv V_{0}r^{2}~{},\quad
r=\tanh{{\varphi/M_{\rm Pl}}\over{\sqrt{6\alpha}}}~{},$ (2)

where the constant $V_{0}$ specifies the scale of inflation, and the
$\alpha>0$ is the free parameter of the order one.

This model is a viable model of large-field slow-roll inflation with a nearly
flat potential, whose inflationary solution is an attractor describing chaotic
inflation, being very close to the Starobinsky model [19] in the case of
$\alpha=1$. The CMB tilt of scalar perturbations, predicted by the T-model is
given by the simple formula [20]

$n_{s}=1-{{2}\over{N_{e}}}~{},$ (3)

in terms of e-folds $N_{e}$ as the running variable describing time evolution,
$N_{e}(k)=\ln(k_{\rm final}/k)$ as the function of scale $k$ [21]. The CMB
tensor-to-scalar ratio $r$ is approximately $(N_{e}\gg 1)$ given by [17, 18]

$r_{\rm\alpha}\approx{{12\alpha}\over{N_{e}^{2}}}~{},$ (4)

providing the comfortable theoretical prediction against future measurements
of $r$. Indeed, the current CMB measurements by Planck/BICEP/Keck
collaborations [22, 23, 24] give

$n_{s}=0.9649\pm 0.0042\quad(68\%~{}{\rm C.L.})~{},\quad
r<0.036\quad(95\%~{}{\rm C.L.})~{}~{},$ (5)

while they are in good agreement with Eqs. (3) and (4) with the best fit close
to $N_{e}=55$.

The T-model potential (2) is symmetric under the sign change
$\varphi\to-\varphi$ but is not periodic. Periodicity of inflaton potential is
assumed in the models of natural inflation and PBH production, where inflaton
is identified with an axion, see e.g., Ref. [25]. The periodicity in the
$\alpha$-attractor models can be achieved via changing the global shape of the
potential without affecting slow roll inflation by replacing the function
$\tanh\phi$ in Eq. (2) by the periodic (Jacobi) elliptic function
$sn(\phi\left|\right.k)$ with the elliptic modulus $0<k^{2}<1$, due to the
known approximation

$sn(\phi\left|\right.k)\approx\tanh\phi\quad{\rm for}\quad k^{2}\to
1~{}~{},\quad\phi={{\varphi/M_{\rm Pl}}\over{\sqrt{6\alpha}}}~{},$ (6)

thus combining the theoretically attractive features of chaotic inflation and
natural inflation.

The generalization of the simplest T-model potential (2) to the form [17, 18]

$V_{\rm gen.}(\varphi)=f^{2}\left(\tanh{{\varphi/M_{\rm
Pl}}\over{\sqrt{6\alpha}}}\right)$ (7)

with a monotonically increasing (during slow roll) function $f(r)$,
$r=\tanh{{\varphi/M_{\rm Pl}}\over{\sqrt{6\alpha}}}$, can be used for
engineering a near-inflection point in the potential, leading to a peak
(enhancement) in the power spectrum of scalar perturbations, needed for PBH
formation [26, 27]. 222The generalizations of the Starobinsky model and the
E-type $\alpha$-attractors, accommodating a near-inflection point for PBH
production, were proposed in Refs. [28] and [29], respectively.

In the generalized T-models (7) slow-roll inflation occurs for large positive
values of the inflation field $\varphi$ with an approximate scalar potential
of the E-type [30] as $(M_{\rm Pl}=1)$

$V(\varphi)=f_{\infty}^{2}-4f_{\infty}{f^{\prime}}_{\infty}\,e^{-\sqrt{{{2}\over{3\alpha}}}\varphi}+{\cal
O}\left(e^{-2\sqrt{{{2}\over{3\alpha}}}\varphi}\right)~{},$ (8)

where we have introduced the parameters
$f_{\infty}=\left.f\right|_{\varphi\to\infty}$ and
${f^{\prime}}_{\infty}=\left.\partial_{\varphi}f\right|_{\varphi\to\infty}$.
The constant in front of the second term in Eq. (8) can be chosen at will by a
constant shift of the inflaton field $\varphi$, so that the potential (8) can
be simplified to

$V(\varphi)=V_{0}\left(1-e^{-\sqrt{{{2}\over{3\alpha}}}\varphi}\right)+{\cal
O}\left(e^{-2\sqrt{{{2}\over{3\alpha}}}\varphi}\right)~{},$ (9)

which implies Eqs. (3) and (4).

The $\alpha$-attractors with $\alpha\neq 1$ do not have a simple description
on the dual $F(R)$-gravity side, see e.g., Ref. [31] for details of the
correspondence. The Starobinsky function $F(R)={{M^{2}_{\rm
Pl}}\over{2}}(R+{{R^{2}}\over{6m_{\rm inf.}^{2}}})$ on the modified gravity
side arises in the case of $\alpha=1$ and $f(r)=\sqrt{3}m_{\rm inf.}M_{\rm
Pl}r/(r+1)$, where $m_{\rm inf.}$ is the inflaton (scalaron) mass. In general,
the exact dual $F(R)$ gravity function associated with any inflaton potential
$V$ in the model (1) is only known in the parametric (implicit) form, see Eqs.
(2.7) and (2.8) in Ref. [31], as

$\displaystyle R$ $\displaystyle=$ $\displaystyle\left[{{\sqrt{6}}\over{M_{\rm
Pl}}}V_{,\varphi}+{{4V}\over{M^{2}_{Pl}}}\right]\exp\left(\sqrt{{{2}\over{3}}}{{\varphi}\over{M_{Pl}}}\right)~{}~{},$
(10) $\displaystyle F$ $\displaystyle=$ $\displaystyle{{M^{2}_{\rm
Pl}}\over{2}}\left[{{\sqrt{6}}\over{M_{\rm
Pl}}}V_{,\varphi}+{{2V}\over{M^{2}_{\rm
Pl}}}\right]\exp\left(2\sqrt{{{2}\over{3}}}{{\varphi}\over{M_{\rm
Pl}}}\right)~{}.$ (11)

When $\alpha\neq 1$, or $A(R)\neq const.$ in the slow-roll approximation with
the potential (2), we find that the $F$-function can be approximated in the
form

$F(R)={{M^{2}_{\rm Pl}}\over{2}}\left[R+A(R){{R^{2}}\over{6m^{2}}}\right]$
(12)

with the function

$A(R)\approx 1-{{3}\over{4}}\varepsilon~{},$ (13)

where $\varepsilon$ is the standard slow-roll parameter

$\varepsilon={{M^{2}_{\rm
Pl}}\over{2}}\left({{V_{,\varphi}}\over{V}}\right)^{2}~{}~{},$ (14)

and $R\approx 12H^{2}\approx 4V/M^{2}_{\rm Pl}$ in terms of the Hubble
function $H$ and the potential $V$.

The particular examples of the generalized T-models, suitable for inflation
and PBH production, can be obtained by expanding the $f(r)$-function in Taylor
series and tuning the expansion coefficients [30, 26].

The slow-roll evolution of inflaton with e-folds $N$ as the running (time)
variable is described by the (non-linear) equation of motion, obtained from
the standard (Klein-Gordon) equation minimally coupled to gravity in the
(spatially flat) universe, when the acceleration term is ignored,

${{1}\over{M^{2}_{\rm Pl}}}\left({{d\varphi}\over{dN}}\right)^{2}={{d\ln
V}\over{dN}}~{}.$ (15)

This equation has an exact solution in the case of the T-model potential (2),
with

$\varphi/M_{\rm Pl}=\sqrt{2N_{0}}~{}{\rm
arcosh}\left({{N}\over{N_{0}}}\right)~{},\quad N\gg N_{0}>0~{},$ (16)

where the (implicit) integration constant is associated with constant shifts
of the field $\varphi$. The solution implies

${{N-N_{0}}\over{N+N_{0}}}=\tanh^{2}\left({{\varphi/M_{\rm
Pl}}\over{\sqrt{6\alpha}}}\right)~{},\quad N_{0}={{3\alpha}\over{4}}~{}~{},$
(17)

and gives a very simple potential $V(N)$ of the T-model in the slow-roll
approximation,

$V(N)=V_{0}\left({{N-N_{0}}\over{N+N_{0}}}\right)~{}~{}.$ (18)

The Hubble function $H(N)$ is also simply related to the potential $V(H)$ is
the slow-roll approximation via the Friedmann equation

$H^{2}(N)={{V(N)}\over{3M^{2}_{\rm Pl}}}~{}.$ (19)

The relations between the potential $V(N)$, the running tensor-to-scalar ratio
$r(N)$ and the slow-roll parameter $\varepsilon(N)$ in the slow-roll
approximation are very simple too,

$r(N)=16\varepsilon(N)=8{{d\ln
V}\over{dN}}={{12\alpha}\over{N^{2}-N_{0}^{2}}}={{12\alpha}\over{N^{2}-(3\alpha/4)^{2}}}~{},$
(20)

leading to a bit more precise formula than Eq. (4).

The very simple form (18) of the T-potential $V(N)$ in the slow-roll
approximation is one of the reasons why we choose the T-models as our baseline
models in this paper.

##  3 Power spectrum of scalar perturbations in slow-roll approximation

Primordial scalar perturbations ($\zeta$) and primordial tensor perturbations
(primordial gravitational waves $g$) are defined by a perturbed Friedmann-
Lemaitre-Robertson-Walker (FLRW) metric,

$ds^{2}=dt^{2}-a^{2}(t)\left(\delta_{ij}+h_{ij}(\vec{r})\right)dx^{i}dx^{j}~{}~{},\qquad
i,j=1,2,3~{},$ (21)

where

$h_{ij}(\vec{r})=2\zeta(\vec{r})\delta_{ij}+\sum_{a=1,2}g^{(a)}(\vec{r})e^{(a)}_{ij}(\vec{r})~{}~{},\quad
H={{da/dt}\over{a}}~{},$ (22)

in terms of the local basis $e^{(a)}$ obeying the relations $e^{i(a)}_{i}=0$,
$g^{(a)}_{,j}e^{j(a)}_{i}=0$ and $e^{(a)}_{ij}e^{ij(a)}=1$.

The primordial spectrum $P_{\zeta}(k)$ of scalar (density) perturbations is
defined by the 2-point correlator of scalar perturbations,

$\left\langle\zeta^{2}(\vec{r})\right\rangle=\int
dk{{P_{\zeta}(k)}\over{k}}~{}.$ (23)

The CMB power spectrum can be described by the Harrison-Zeldovich fit

$P^{\rm HZ}_{\zeta}(k)\approx 2.21^{+0.07}_{-0.08}\times
10^{-9}\left({{k}\over{k_{*}}}\right)^{n_{s}-1}~{}~{},$ (24)

near the pivot scale $k_{*}=0.05~{}{\rm Mpc}^{-1}$, or in the slow-roll (SR)
approximation by

$P^{\rm SR}_{\zeta}(k)\approx
P_{0}\ln^{2}{\left({{k}\over{k_{\text{final}}}}\right)}~{},\quad P_{0}=const.$
(25)

The power spectrum $P_{\zeta}(N)$ is simply related to the potential $V(N)$ in
the slow-roll approximation via the standard relation, see e.g., Refs. [5,
29],

$P_{\zeta}(N)={{V^{2}}\over{12\pi^{2}M^{4}_{\rm
Pl}}}\left({{dV}\over{dN}}\right)^{-1}~{}~{}.$ (26)

It also implies

$1-n_{s}={{d\ln P_{\zeta}(N)}\over{dN}}~{}.$ (27)

In the case of the potential (18), we find very simple equations,

$P^{\rm SR}_{\zeta}(N)={{V_{0}}\over{18\pi^{2}M^{4}_{\rm
Pl}\alpha}}(N-N_{0})^{2}\equiv P_{0}(N-N_{0})^{2}~{},$ (28)

and

$n_{s}=1-{{2}\over{N-N_{0}}}~{}~{},$ (29)

where the last equation reproduces Eq. (3).

The observed CMB window into inflation does not allow us to reconstruct the
full inflaton scalar potential from the power spectrum beyond the slow-roll
region. The well-known reconstruction formula, proposed by Hodges and
Blumentahl [21] in the form

${{1}\over{V(N)}}=-{{1}\over{12\pi^{2}M^{4}_{\rm
Pl}}}\int{{dN}\over{P_{\zeta}(N)}}~{},$ (30)

requires knowing the full power spectrum at different scales and the limits of
integration. Moreover, the reconstruction procedure should be based on getting
exact solutions to the Mukhanov-Sasaki equation instead of the slow-roll
solution in Eq. (28), see e.g., Ref. [32] for some examples. It is not our
purpose in this paper to reconstruct the inflaton potential beyond its
qualitative features. Nevertheless, it may be possible for the CMB region
under some additional assumptions, e.g., when assuming a very low value of the
tensor-to-scalar ratio $r$, which implies a small correction $\delta V$ to the
constant $V_{0}$ defining the inflationary scale, i.e. the inflaton potential
in the form $V=V_{0}+\delta V$ with $\left|\delta V\right|\ll V_{0}$. Then Eq.
(30) is simplified to

$\delta V(N)={{V_{0}}\over{12\pi^{2}M^{4}_{\rm
Pl}}}\int{{dN}\over{P_{\zeta}(N)}}~{},$ (31)

while the integration constants merely rescale $V_{0}$ and shift $N$. Equation
(15) also gets simplified to

${{V_{0}}\over{M^{2}_{\rm
Pl}}}\left({{d\varphi}\over{dN}}\right)^{2}={{d(\delta V)}\over{dN}}~{}~{}.$
(32)

Then a partial reconstruction of the scalar potential becomes possible from
the CMB power spectrum $P_{\zeta}(N)$ of scalar perturbations without knowing
the power spectrum of tensor perturbations, which is also true for the
$\alpha$-attractors when the parameter $\alpha$ is small enough, $\alpha\leq
1$, with

$V(\varphi)\approx
V_{0}\left(1-e^{-\sqrt{{{2}\over{3\alpha}}}\varphi}\right)~{}~{},$ (33)

as in Eq. (9). This is yet another reason for us to take the T-models of
$\alpha$-attractors as our baseline models of inflation and generalize their
power spectrum of scalar perturbations by adding a peak at higher values of
$k$.

##  4 Log-normal fit for a peak and GW spectrum

The log-normal fit is the simplest (Gaussian) description of a peak in the
power spectrum, see e.g., Ref. [33]. A power-law ansatz for the peak was
considered in Ref. [34]. In this paper we propose another ansatz for the power
spectrum, combining the CMB spectrum in the slow-roll approximation with the
log-normal fit for the enhancement (peak) of the power spectrum needed for PBH
formation at a lower scale,

$\displaystyle
P_{\zeta}(k)=P_{0}\ln^{2}{\left({{k}\over{k_{\text{final}}}}\right)}+A{{\exp{\displaystyle\Big{[}-{{\ln^{2}{\left({\textstyle{k\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{k_{\rm
peak}}$}}}}\right)}}\over{2\sigma^{2}}}}\Big{]}}\over{\sqrt{2\pi}\sigma}}~{}~{},$
(34)

where $k_{\rm peak}$ is a position of the peak, $\sigma>0$ is the width of the
peak and $A$ is the normalization of the peak amplitude,
$A\approx(\sqrt{2\pi}\sigma)0.01$, needed for efficient PBH production (about
$10^{7}$ higher than the CMB amplitude). The normalization factor $P_{0}$ is
given by Eq. (28),

$P_{0}={{V_{0}}\over{18\pi^{2}\alpha M_{\text{Pl}}^{4}}}~{}~{},$ (35)

see also Eq. (25). When $V_{0}\sim m_{\rm inf.}^{2}M_{\text{Pl}}^{2}$, $m_{\rm
inf.}\sim 10^{-5}M_{\text{Pl}}$ and $\alpha\sim 1$, we get
$P_{0}\sim\mathcal{O}(10^{-12})$ . The power spectrum (34) can be rewritten,
using the e-foldings variable $N$ via the relation

$d\ln{k}=-dN~{},$ (36)

to the simple form

$\displaystyle
P_{\zeta}(N)=P_{0}\left(N-N_{0}\right)^{2}+A{{\exp{\displaystyle\Big{[}-{{\left(N-N_{\rm
peak}\right)^{2}}\over{2\sigma^{2}}}}\Big{]}}\over{\sqrt{2\pi}\sigma}}~{}~{}.$
(37)

We choose $k_{\text{\rm final}}=e^{N_{e}}$ $\text{Mpc}^{-1}$ $\approx 7.7\cdot
10^{23}~{}\text{Mpc}^{-1}$.

The PBH masses can be estimated by the relation [9, 10, 11, 12]

${{M_{\rm PBH}(k)}\over{M_{\odot}}}\simeq
10^{-16}\left({{k}\over{10^{14}\mathrm{Mpc}^{-1}}}\right)^{-2}~{}.$ (38)

We choose $k_{\rm peak}$ or $N_{\rm peak}$ to get $M_{\rm PBH}$ within the
current observational window for PBH as the whole dark matter [9, 10, 11],
i.e. between $10^{17}$ g and $10^{21}$ g. For example, when $k_{\rm
peak}\approx 10^{13}~{}\text{Mpc}^{-1}$, we get $M_{\rm PBH}\approx 2\cdot
10^{19}$ g. The profile of the power spectrum in given on Fig. 1 for some
values of $\sigma$.

Figure 1: The power spectrum with the parameters $P_{0}=6.57\cdot 10^{-13}$,
$k_{\rm peak}=10^{13}~{}\text{Mpc}^{-1}$, $k_{\text{\rm final}}=7.7\cdot
10^{23}$ $\text{Mpc}^{-1}$ for $\sigma>1$.

Equations (27) and (37) imply the spectral tilt

$n_{s}=1-{{2\left(N-N_{0}\right)-{{A\left(N-N_{\rm
peak}\right)e^{-{{\left(N-N_{\rm
peak}\right){}^{2}}\over{2\sigma^{2}}}}}\over{P_{0}\sqrt{2\pi}\sigma}}}\over{{{Ae^{-{{\left(N-N_{\rm
peak}\right){}^{2}}\over{2\sigma^{2}}}}}\over{P_{0}\sqrt{2\pi}\sigma}}+\left(N-N_{0}\right){}^{2}}}~{}~{}.$
(39)

It follows from Eq. (39) versus Eq. (3) that the tilt $n_{s}$ gets the
exponentially small corrections (back reaction) from the peak. To
quantitatively evaluate an impact of the back reaction, we introduce the
dimensionless parameters for the relative scales,

$\mu_{L}={{k_{\rm left}-k_{*}}\over{k_{\rm left}}}\quad{\rm
and}\quad\mu_{R}={{k_{\rm final}-k_{\rm right}}\over{k_{\rm final}}}~{}~{},$
(40)

characterizing the separation between the CMB pivot scale $k_{*}$ and the left
end $k_{\rm left}$ of the peak, and the separation between the end of
inflation $k_{\rm final}$ and the right end $k_{\rm right}$ of the peak,
respectively. Since the CMB pivot scale and the PBH scales have to be
separated, it implies $k_{*}<k_{\rm left}$ or $\mu_{L}>0$. The exponential
corrections in Eq. (39) are negligible when $k_{\rm left}\geq 10^{3}~{}{\rm
Mpc}^{-1}$. On the other hand, the right end of the peak must be within
inflation, so that $\mu_{R}>0$. We expect $k_{\rm right}$ to be close to the
end of inflation. 333Particle production is also more efficient toward the end
of inflation [35].

We illustrate those considerations by our numerical calculations with the
results displayed on Figs. 2 and 3 for various values of $k_{\rm peak}$ and
$\sigma$ against the observed values of the tilt $n_{s}$ in Eq. (5). The black
curves in the ($k_{\rm peak},\sigma)$-plane correspond to the condition
$\mu_{R}=0$. The area above the black curve and the white area are forbidden.

Figure 2: The impact of Eq. (39) on the parameters of our model for
$10^{11}~{}{\rm Mpc}^{-1}k_{\rm peak}\leq 10^{12}~{}{\rm Mpc}^{-1}$ (on the
left) and for $10^{12}~{}{\rm Mpc}^{-1}\leq k_{\rm peak}\leq 10^{13}~{}{\rm
Mpc}^{-1}$ (on the right) in the $(\sigma,k_{\rm peak})$-plane. The (excluded)
area above the black curve leads to the right end of the peak after the end of
inflation. The other parameters are $P_{0}=6.57\cdot 10^{-13}$ and
$k_{\text{\rm final}}=7.7\cdot 10^{23}~{}\text{Mpc}^{-1}$.
Figure 3: The impact of Eq. (39) on the parameters for $10^{14}~{}{\rm
Mpc}^{-1}\leq k_{\rm peak}\leq 10^{15}~{}{\rm Mpc}^{-1}$ in the
$(\sigma,k_{\rm peak})$-plane. The (excluded) area above the black curve leads
to the right end of the peak after the end of inflation. The other parameters
are $P_{0}=6.57\cdot 10^{-13}$ and $k_{\text{\rm final}}=7.7\cdot
10^{23}~{}\text{Mpc}^{-1}$.

For example, when $N_{e}=55$ and $N_{0}=3/4$ (or $\alpha=1$), we get
$n_{s}\approx 0.9631$ from Eq. (29), whereas we get $n_{s}\approx 0.9649$
after taking into account the exponential terms in Eq. (39) with the
parameters $k_{\rm peak}=6\cdot 10^{11}~{}{\rm Mpc}^{-1}$ and $\sigma=3.945$.

Our analysis allows us to restrict (from above) the possible peak width values
$\sigma$ at fixed $k_{\rm peak}$ and duration of inflation $N_{e}$ (or
$n_{s}$). We summarize those restrictions in Table 1. 444When $k_{\rm
peak}>10^{15}~{}\text{Mpc}^{-1}$, the PBH masses are lower than the Hawking
evaporation limit of $10^{15}g$ for black holes.

@$M_{\rm PBH},\text{g}$@ |  @$k_{\rm peak},\text{Mpc}^{-1}$ @ | $\sigma$
---|---|---
$10^{21}$ | $1.41\cdot 10^{12}$ | $\leq 3.89$
$10^{20}$ | $4.46\cdot 10^{12}$ | $\leq 3.73$
$10^{19}$ | $1.41\cdot 10^{13}$ | $\leq 3.56$
$10^{18}$ | $4.46\cdot 10^{13}$ | $\leq 3.40$
$10^{17}$ | $1.41\cdot 10^{14}$ | $\leq 3.23$
Table 1: The PBH masses $M_{\rm PBH}$, the scales $k_{\rm peak}$ and the upper
bounds on $\sigma$.

The spectrum of the induced GW can be derived by using the standard formula
obtained in the second order with respect to perturbations [36],

$\begin{array}[]{c}\displaystyle\Omega_{GW}(k)={{\Omega_{r,0}}\over{32}}\int\limits_{0}^{\infty}dv\int\limits_{|1-v|}^{1+v}du{{\mathcal{T}(u,v)}\over{u^{2}v^{2}}}P_{\zeta}(vk)P_{\zeta}(uk),\\\\[17.07164pt]
\displaystyle\mathcal{T}(u,v)={\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{4}$}}}}\Big{[}{{4v^{2}-(1+v^{2}-u^{2})^{2}}\over{4uv}}\Big{]}^{2}\Big{(}{{u^{2}+v^{2}-3}\over{2uv}}\Big{)}^{4}\Big{[}\Big{(}\ln\Big{|}{{3-(u+v)^{2}}\over{3-(u-v)^{2}}}\Big{|}\\\\[8.53581pt]
\displaystyle-\,{{4uv}\over{u^{2}+v^{2}-3}}\Big{)}^{2}+\pi^{2}\Theta\big{(}u+v-\sqrt{3}\big{)}\Big{]}~{}~{},\end{array}$
(41)

where $\Omega_{r,0}=8.6\cdot 10^{-5}$. Our numerical results for a wide peak
with $\sigma>1$ are displayed on Fig. 4.

Figure 4: The induced GW spectrum for selected values $\sigma>1$ of a wide
peak in the power spectrum, with the parameters $P_{0}=6.57\cdot 10^{-13}$,
$k_{\rm peak}=10^{13}~{}\text{Mpc}^{-1}$ and $k_{\text{\rm final}}=7.7\cdot
10^{23}~{}\text{Mpc}^{-1}$.

The peak in the GW-spectrum associated with a wide ($\sigma>1$) peak in the
power spectrum can be analytically approximated as

$\Omega^{{\rm(peak)}}_{{\rm GW},r}\approx
0.125{{A^{2}}\over{\sigma^{2}}}\exp{\left[{{-\ln^{2}\left({{k}\over{k_{\rm
peak}}}\right)}\over{\sigma^{2}}}\right]}\sim 10^{-6}P^{2}_{\zeta}(k)~{}~{}.$
(42)

For comparison, in Fig. 5 we give our numerical results for the induced GW
spectrum with selected values $\sigma<1$ of a sharp peak in the power
spectrum. Then the GW spectrum is not given by a sum of contributions from the
peak and the slow-roll, while the simple relation to the power spectrum in Eq.
(42) is also not valid. Instead, the cross terms in Eq. (41) become
significant and the shape of the GW spectrum changes, see Fig. 5. This
phenomenon was also observed in Ref. [37].

Figure 5: The induced GW spectrum for selected values $\sigma<1$ of a sharp
peak in the power spectrum, with the parameters $P_{0}=6.57\cdot 10^{-13}$,
$k_{\rm peak}=10^{13}~{}\text{Mpc}^{-1}$ and $k_{\text{\rm final}}=7.7\cdot
10^{23}~{}\text{Mpc}^{-1}$.

##  5 Conclusion

Our investigation in this paper is based on the ansatz (34) for the power
spectrum of scalar (curvature) perturbations during inflation. The ansatz is
given by a sum of the CMB power spectrum in the slow-roll approximation and
the log-normal fit for the power spectrum enhancement (peak) needed for
efficient PBH production. The ansatz (34) is very simple, while we use the
slow-roll approximation and the T-type $\alpha$-attractor models of inflation
in order to justify the first term in Eq. (34). The second term in Eq. (34)
requires the scalar potential in those models to be generalized, e.g., via
engineering a near-inflection point and an ultra-slow-roll phase during
inflation, see e.g., Refs. [5, 6, 12, 26, 27, 28, 29, 30, 38, 39] for explicit
examples. We are aware that the slow-roll approximation is violated during the
ultra-slow-roll phase needed for a peak generation, and do not expect that Eq.
(34) is suitable for a full reconstruction of the inflaton scalar potential.
Instead, we take the power spectrum (34) for granted and study its
consequences, both analytically and numerically, in the context of CMB and PBH
as DM.

Our main results are given in Sec. 4 including our Figures and Table 1 that
summarizes the restrictions on the peak width $\sigma$ from above.

## Acknowledgements

This work was partially supported by Tomsk State University under the
development program Priority-2030. SVK was also supported by Tokyo
Metropolitan University, the Japanese Society for Promotion of Science under
the grant No. 22K03624, and the World Premier International Research Center
Initiative, MEXT, Japan.

One of the authors (SVK) is grateful to Shyam Balaji, Guillem Domenech,
Noriaki Kitazawa, Laura Iacconi, Misao Sasaki and Alexei Starobinsky for
discussions and correspondence.

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