# Spectrum oscillations from features in the potential of single-field
inflation

I. Dalianis, G.P. Kodaxis, I.D. Stamou, N. Tetradis and A. Tsigkas-Kouvelis

###### Abstract

We study single-field inflationary models with steep step-like features in the
potential that lead to the temporary violation of the slow-roll conditions
during the evolution of the inflaton. These features enhance the power
spectrum of the curvature perturbations by several orders of magnitude at
certain scales and also produce prominent oscillatory patterns. We study
analytically and numerically the inflationary dynamics. We describe
quantitatively the size of the enhancement, as well as the profile of the
oscillations, which are shaped by the number and position of the features in
the potential. The induced tensor power spectrum inherits the distinctive
oscillatory profile of the curvature spectrum and is potentially detectable by
near-future space interferometers. The enhancement of the power specrtum by
step-like features, though significant, may be insufficient to trigger the
production of a sizeable number of primordial black holes if radiation
dominates the energy density of the early universe. However, it can result in
sufficient black hole production if the universe is dominated by non-
relativistic matter. For the latter scenario, we find that deviations from the
standard monochromatic profile of the mass spectrum of primordial black holes
are possible because of the multiple-peak structure of the curvature power
spectrum.

##  1 Introduction

###  1.1 Oscillations in the power spectrum

Inflationary models that predict deviations from scale invariance at small
scales have been attracting a lot of attention in recent years. During the
early evolution of the universe, a strong enhancement of the spectrum of
primordial scalar perturbations can trigger the gravitational collapse and the
formation of primordial black holes (PBHs), which may survive until today in
significant numbers in order to be detectable [1, 2, 3, 4]. This possibility
has been studied in great detail during the last years. (For reviews with
extensive lists of references, see [5, 6, 7, 8].) In addition, a potentially
observable stochastic background of gravitational waves (GWs) is generated
through the coupling of scalar and tensor modes at second order [9, 10, 11,
12, 13, 14, 15, 16, 17, 18]. The induced tensors are suppressed by the small
value of the scalar perturbations at the CMB scales [19], but may be sizeable
if the primordial density perturbations are enhanced at small scales. In this
way, the relic GW stochastic background may provide a direct probe of the very
early cosmic history. The detection prospects of induced GWs open a new window
to probe the inflationary dynamics at small scales, for which cosmic microwave
background (CMB) observables lack sensitivity.

A primordial scalar spectrum with a strong enhancement can be realized in
various setups, such as through inflationary potentials that contain a near-
inflection point [20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 27, 32, 31, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44], multi-field inflation [45, 46, 47,
48, 49, 50, 51, 52, 53, 54, 55, 56, 57], modified gravity [58, 59, 60, 61, 62,
63], curvaton models [64, 65, 66, 67], sound speed modulation and parametric
resonance [68, 69, 70, 71, 72, 73]. It can also be realized when the
inflationary potential features a step-like change [74, 75, 76, 77, 78], a
framework that was revisited recently in [79]. It is very interesting that the
enhancement profiles produced by these inflationary models may be
distinguishable. Different inflationary realizations yield power spectra with
a wide or narrow peak, oscillations or a multi-peak structure.

In this work we focus on power spectra with oscillations around the peak. The
oscillatory pattern is distinctive and possibly detectable, indicating a sharp
feature in the inflationary dynamics. It can be caused by the reentry of
$k$-modes in the horizon, a change in the sound speed [80], the backreaction
of the entropy modes on the adiabatic modes in multi-field inflation [81, 82],
or by a step in the inflaton potential [74, 83, 75, 76, 77, 84, 85, 78, 86,
87, 88, 89, 79]. The last example is the minimal realization of a sharp
feature that involves single-field inflation dynamics and a canonical kinetic
term. Motivated by the original proposal [74], where the effects of singular
points in the inflationary potential were studied, we study here smooth
variations of the basic setup, focusing on model-independent features. We
compute analytically and numerically the evolution of the curvature
perturbations and find a strong enhancement of the scalar spectrum. In
addition, we observe a burst of oscillations, generated solely by step-like
changes in the inflaton potential.

As we show in the following section, a sharp drop in the potential of the
inflaton field detunes the relative phase between the real and imaginary parts
of the curvature perturbation, so that oscillations in the amplitude of the
spectrum appear, while no reentry of modes takes place. The characteristic
period of the oscillations depends on the position of the feature, while
interference patterns are also apparent. We demonstrate that, even though the
strong features in the underlying inflaton evolution may not be simple and the
range of generated spectra extensive, an analytical understanding of their
form is feasible. This is achieved by approximating the time-dependent
inflaton background through a series of “pulses” that affect the evolution of
the fluctuations.

The amplitude of the peak of the spectrum of curvature perturbations ${\cal
P_{R}}(k)$ is determined by the characteristics of the features in the
inflaton potential [79], so that significant PBH production, along with a
detectable GW signal, can be generated. It is exciting that the shape
characteristics of the peak of the curvature spectrum are also imprinted on
the spectrum of GWs. In general, the ${\cal P_{R}}(k)$ characteristics are not
clearly visible in the mass spectrum of the fractional PBH abundance
$f_{\text{PBH}}$, which appears predominantly monochromatic, mostly sensitive
to the amplitude of the peak. Even though a universal behavior also appears in
the GW spectrum [90], especially for smooth scalar spectra, the tensor
perturbations are much more informative [91, 17, 92, 93] and can display more
clearly features originating in the scalar spectrum. In this way, the
detection of stochastic GWs is a portal to the primordial spectrum of scalar
perturbations at small scales, which can also be used to test the PBH dark
matter scenario. Moreover, it can provide details of the possible strong
features in the inflationary dynamics.

The induced GWs may be detected in the near future by the current and planned
detectors. The LIGO collaboration has already produced upper limits in such
stochastic backgrounds [94]. The searches will be further extended by a
network of operating and designed gravitational wave detectors that will probe
a vast range of different frequency bands. Pulsar time array (PTA) GW
experiments [95] have a sensitivity to the nano-Hz frequency band, space-based
interferometers like LISA [96], Taiji [97], Tianqin [98], Decigo [99, 100] are
mostly sensitive to milli-Hz and deci-Hz frequency bands, and the LIGO/Virgo
and Einstein telescope [101] ground-based interferometers are sensitive to
larger frequencies.

###  1.2 The steps in the inflaton potential

In the following section we shall discuss in detail the oscillatory patterns
in the spectra of curvature perturbations and induced GWs that arise from
steep steps in the inflaton potential. The steps connect regions in which the
potential varies smoothly and the slow-roll conditions are satisfied. The
basic pattern corresponds to the vacuum energy having one or more transition
points at which it jumps from one constant value to another [79]. One can
speculate that these points may correspond to values of the inflaton field at
which certain modes, whose quantum fluctuations contribute to the vacuum
energy, decouple very quickly. Decoupling effects become visible when the
effective potential is regularized in a mass-sensitive scheme. Also, the
dependence of the potential on an energy scale, or a coarse-graining length,
can be analysed through the Wilsonian approach to the renormalization group,
see for example refs. [102, 103] for a particular implementation. The
resulting renormalization-group equation for the potential can capture the
decoupling behavior. However, our fundamental lack of understanding of the
nature of vacuum energy or the cosmological constant does not permit a
quantitative calculation of these effects.

Some intuition on this issue can be obtained by considering the role of
underlying symmetries. A specific framework, which we shall use as the basis
for the potentials that we shall consider, is provided by the models of
$\alpha$-attractors in supergravity [104, 105]. A toy model that demonstrates
the role of symmetries is described by the Lagrangian [106]

${\cal
L}=\sqrt{-g}\left[\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi+\frac{1}{12}\chi^{2}R(g)-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}\phi^{2}R(g)-\frac{1}{36}F^{2}(\phi/\chi)\left(\chi^{2}-\phi^{2}\right)^{2}\right],$
(1.1)

which is invariant under the conformal transformation

$g_{\mu\nu}\to e^{-2\sigma(x)}g_{\mu\nu},~{}~{}~{}~{}~{}~{}\phi\to
e^{\sigma(x)}\phi,~{}~{}~{}~{}~{}~{}\chi\to e^{\sigma(x)}\chi.$ (1.2)

For constant $F(\phi/\chi)$, there is a global $SO(1,1)$ symmetry that keeps
$\chi^{2}-\phi^{2}$ constant. The field $\chi$ does not have any physical
degrees of freedom and can be eliminated through the gauge-fixing condition
$\chi^{2}-\phi^{2}=6$. (All dimensionful quantities are expressed in units of
$M_{\rm Pl}$.) We parametrize the fields as
$\chi=\sqrt{6}\cosh(\varphi/\sqrt{6})$, $\phi=\sqrt{6}\sinh(\varphi/\sqrt{6})$
[106]. The Lagrangian becomes

${\cal
L}=\sqrt{-g}\left[\frac{1}{2}R(g)-\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-F^{2}\left(\tanh\frac{\varphi}{\sqrt{6}}\right)\right].$
(1.3)

A constant function $F(x)$, which preserves the $SO(1,1)$ symmetry, results in
a cosmological constant in this formulation. The value of the cosmological
constant is not constrained by the symmetry and is arbitrary.

We can introduce a minimal deformation of the symmetry by assuming that $F(x)$
takes two different values over two continuous ranges of $x$, with a rapid
transition in between. A stronger deformation that has been used extensively
in the literature assumes that $F(x)$ has a polynomial form. We shall employ a
combination of the above choices by assuming that $F(x)$ has the schematic
form

$F(x)=x^{n}+\sum_{i}A_{i}\,\Theta(x-x_{i}),$ (1.4)

allowing for more than one transition points. In order to avoid unphysical
features in the evolution of the inflaton, each step-function is replaced by a
continuous function with a sharp transition at $x_{i}$. A more general
framework is provided by the $\alpha$-attractors [104, 105, 106]. The
Lagrangian includes an additional free parameter $\alpha$ and takes the form

${\cal
L}=\sqrt{-g}\left[\frac{1}{2}R(g)-\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-F^{2}\left(\tanh\frac{\varphi}{\sqrt{6\alpha}}\right)\right].$
(1.5)

The potentials that result from our assumption for the function $F(x)$ with
positive $A_{i}$ are generalizations of the potential of the Starobinsky model
[107], with the addition of one or more steep steps. Allowing for negative
values of $A_{i}$ makes it possible to include inflection points in the
potential as well. As our analysis focuses on the phenomenological
consequences of general features in the potential, we consider parameters
$A_{i}$ that can take values over the whole real axis.

###  1.3 Plan of the paper

The plan of the paper is as follows: In the next section we present an
analytical discussion of the oscillatory patterns that can appear in the
spectrum of curvature perturbations when the inflaton potential contains step-
like features. We first establish our notation and identify the relevant
parameters for the analysis of oscillations. We next use a simple toy model,
neglecting the expansion of the background, in order to demonstrate how the
detuning of the relative phase between the real and imaginary parts of the
perturbation generates an oscillatory pattern in its amplitude. We then
present an analytical study of the oscillatory form of the curvature power
spectra that may result from strong features in the inflaton potential. This
is possible if the effect on the fluctuations is modelled by a series of
positive or negative “pulses” that correspond to the deviations from the slow-
roll regime. In section 3 we study explicit inflationary realizations with
step-like features in the framework of $\alpha$-attractors, paying particular
attention to the consistency with the CMB constraints. For these inflationary
models we examine the production of PBHs and their mass distribution, as well
as the spectrum of the induced GWs. We elaborate on the relations and
similarities between the patterns appearing in the curvature and tensor power
spectra. The last section 4 contains our conclusions. All dimensionful
quantities are given in units of $M_{\rm Pl}$ throughout the paper, unless the
units are explicitly stated.

##  2 Analytical calculation of the spectrum of curvature perturbations

###  2.1 General considerations

In this section we discuss an approximate analytical treatment of the spectrum
of curvature perturbations in cases that the slow-roll approximation is
strongly violated. We assume that the inflaton potential displays the standard
plateau that can lead to an almost scale-invariant spectrum. In addition, it
contains a strong feature within a finite range of field values, which can
lead to the violation of the slow-roll conditions or even cause inflation to
cease momentarily. In order to be as model independent as possible, we do not
focus on specific potentials with these properties.

We consider the most general scalar metric perturbation around the Friedmann-
Robertson-Walker (FRW) background [108]

$ds^{2}=a^{2}(\tau)\left\\{(1+2\phi)d\tau^{2}-2B_{,i}\,dx^{i}d\tau-\left((1-2\psi)\delta_{ij}+2E_{,ij}\right)dx^{i}dx^{j}\right\\},$
(2.1)

with $B_{,i}=\partial_{i}B$, $E_{,ij}=\partial_{i}\partial_{j}E$. The inflaton
field can be split into a background and a perturbation:
$\varphi(\tau)+\delta\varphi(\tau,x)$. A gauge-invariant field perturbation
can be defined as $v=a\left(\delta\varphi+(\varphi^{\prime}/{\cal
H})\psi\right)$, satisfying the Mukhanov-Sasaki equation [109, 110]

$v^{\prime\prime}-\nabla^{2}v-\frac{z^{\prime\prime}}{z}\,v=0,$ (2.2)

with $z=a\varphi^{\prime}/{\cal H}$. The primes and the Hubble parameter refer
to derivatives with respect to conformal time. The gauge-invariant comoving
curvature perturbation $R=-v/z$ satisfies

$R_{k}^{\prime\prime}+2\frac{z^{\prime}}{z}R_{k}^{\prime}+k^{2}R_{k}=0$ (2.3)

in Fourier space.

We shall use the number of efoldings $N$ as the independent variable for the
evolution of the perturbations. The Hamilton-Jacobi slow-roll parameters are
defined through the relations

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{V(\varphi)}{3M_{\rm
Pl}^{2}-\frac{1}{2}\varphi_{,N}^{2}}$ (2.4) $\displaystyle\varepsilon_{H}$
$\displaystyle=$ $\displaystyle-\frac{d\ln
H}{dN}=\frac{\varphi^{2}_{,N}}{2M_{\rm Pl}^{2}}$ (2.5) $\displaystyle\eta_{H}$
$\displaystyle=$
$\displaystyle\varepsilon_{H}-\frac{1}{2}\frac{d\ln\varepsilon_{H}}{dN}=\frac{\varphi^{2}_{,N}}{2M_{\rm
Pl}^{2}}-\frac{\varphi_{,NN}}{\varphi_{,N}},$ (2.6)

where $H=e^{-N}{\cal H}$ is the Hubble parameter defined through cosmic time,
and subscipts denote derivatives with respect to $N$. The parameter $z$ is
defined as $z=e^{N}\,\varphi_{,N}.$ The effective equation of state for the
background is $w=-1+2\varepsilon_{H}/3$. The equation for the curvature
perturbation takes the form

$R_{k,NN}+f(N)\,R_{k,N}+\frac{k^{2}}{e^{2N}H^{2}}R_{k}=0,$ (2.7)

with the quantity

$f(N)=3+\frac{2\varphi_{,NN}}{\varphi_{,N}}-\frac{\varphi_{,N}^{2}}{2M_{\rm
Pl}^{2}}=3+\varepsilon_{H}-2\eta_{H}$ (2.8)

playing a crucial role in determining the qualitative behaviour of the
solutions. In the slow-roll regime it acts as a generalized friction term.
However, if $\eta_{H}$ becomes positive and large it can lead to a dramatic
enhancement of the perturbations.

In the approximation that the slow-roll parameters are neglected and $H$ is
assumed to remain constant, the solution of eq. (2.7) can be expressed in
terms of the Bessel functions $J_{\pm 3/2}$ as

$R_{k}(N;C_{p},C_{m},3)=A\,e^{-\frac{3}{2}\,N}\left(C_{p}J_{3/2}\left(e^{-N}\frac{k}{H}\right)+C_{m}\,J_{-3/2}\left(e^{-N}\frac{k}{H}\right)\right),$
(2.9)

where we take $A$ to be real without loss of generality. For the values
$C_{p}=1$, $C_{m}=i$ the two Bessel functions combine into the Hankel function
of the first kind $H^{(1)}_{3/2}$. The curvature pertubation is
$R_{k}(N;1,i,3)\propto\left(e^{-ik\tau}/\sqrt{k}\right)\,(1-i/(k\tau))/a(\tau)$,
where the conformal time is $H\tau=-e^{-N}=-1/a$. For $\tau\to-\infty$ this is
the standard expression for the Bunch-Davies vacuum in the slow-roll regime,
which is taken as the initial condition for the evolution of the fluctuations.
For $\tau\to 0^{-}$ the curvature perturbation approaches a constant value
$\propto k^{-3/2}$ as the mode with wavenumber $k$ moves out of the horizon
and freezes. The power spectrum of curvature perturbations
$\Delta_{R}^{2}=({k^{3}}/{2\pi^{2}})|R_{k}|^{2}$ is scale invariant. It is
important to notice that the value of the curvature perturbation at late
times, or $N\to\infty$, comes from the second term in eq. (2.9), as the first
one vanishes. In this sense, it is the absolute value of $C_{m}$ that
determines the power spectrum.

The above picture is modified when the function $f(N)$ of eq. (2.8) deviates
from a constant value equal to 3. For small values of $\varepsilon_{H}$,
$\eta_{H}$ the deviations from scale invariance can be computed analytically
through the standard slow-roll analysis. However, our interest lies with
strong modifications of $\varepsilon_{H}$, $\eta_{H}$ that result in the
enhancement of the spectrum by several orders of magnitude.

The typical forms of the effective-friction function $f(N)$ that we would like
to analyse are depicted in fig. 1. These examples result from an inflaton
potential used in ref. [79],

$V(\varphi)=V_{0}\left(1+\frac{1}{2}C\left(1+\tanh(c\varphi)\right)+B\varphi\right),$
(2.10)

for specific choices of its parameters. The function $f(N)$ remains close to
3, apart from a range of efoldings in which it deviates strongly from this
value. Similar features can be obtained with other types of potentials in
single- or multi-field inflation, such as potentials with inflection points,
or multiple inflationary stages. The pattern can be repeated several times.
When $f(N)$ becomes negative it induces a strong enhancement of the spectrum.
The modifications to the spectrum appear for wavenumbers of density
perturbations deep in the nonlinear regime today.

Figure 1:  The inflaton potential $V(\varphi)$ of eq. (2.10), the evolution of
the inflaton $\varphi$ (dashed line) and its derivative $\varphi_{,N}$ (solid
line), and the function $f(N)$ defined in eq. (2.8), for two choices of the
parameters of the potential: First row: $C=-0.3$, $c=100$, $B=-0.03$. Second
row: $C=0.00058$, $c=100$, $B=-0.03$.

In order to obtain an analytical solution, we model $f(N)$ through a sequence
of square “pulses”, each with constant $f(N)=\kappa_{i}\not=3$. At early and
late times we assume that the inflaton is in a slow-roll regime, with
negligible slow-roll parameters, so that $f(N)=3$ and the curvature
perturbation is given by eq. (2.9). We approximate the Hubble parameter $H$ as
almost constant. This is a good approximation, as our focus is on
modifications of the spectrum by several orders of magnitude. In comparison,
the change in the Hubble parameter for an inflection point in the potential is
less than 1%, while for a step in the potential it is of order 10%. We use an
arbitrary normalization for the number of efoldings by absorbing a factor of
$\exp(N_{0})$ in $k$, where $N_{0}$ corresponds to the actual number of
efoldings since the beginning of inflation until the moment in time that we
denote by $N=0$. In practice this means that the physical value of the
wavenumber is $\exp(N_{0})\,k$.

Our starting point is the solution (2.9), which defines the initial condition
for $N\to-\infty$. For $C_{p}=1$, $C_{m}=i$, this expression corresponds to
the Bunch-Davies vacuum. We neglect slow-roll corrections and approximate the
evolution through eq. (2.9) until the value of $N$ at which the first
nontrivial “pulse” appears in $f(N)$. In the following subsection we analyse
the modification of the curvature perturbation induced by this and the
following “pulses”, until the system returns to a slow-roll regime. For
$N\to\infty$ the solution becomes constant. We are interested in the relative
increase of the asymptotic value of $|C_{m}|$ in comparison to the value
$|C_{m}|=1$ corresponding to a scale-invariant spectrum. In this sense, the
value of the $k$-independent parameter $A$ in eq. (2.9) is not of interest to
us. This parameter would determine the amplitude of the spectrum in the CMB
region, and needs to be adjusted to a phenomenologically correct value.

An important point concerns the form of $f(N)$. Negative values of this
function result only from $\eta_{H}$ taking large positive values, as can be
seen through eqs. (2.5), (2.6), (2.8). In general, large deviations from 3 can
result from the term $2\varphi_{,NN}/\varphi_{,N}$ being the dominant one in
eq. (2.8). The integral of $f(N)-3$ over $N$, from an early to a late slow-
roll regime separated by nontrivial evolution, is

$\int_{N_{e}}^{N_{l}}dN(f(N)-3)=\int_{N_{e}}^{N_{l}}dN(\varepsilon_{H}-2\,\eta_{H})=2\log\frac{\left(\varphi_{,N}\right)_{l}}{\left(\varphi_{,N}\right)_{e}}+\log\frac{H_{l}}{H_{e}}=\log\frac{\left(dH/dN\right)_{l}}{\left(dH/dN\right)_{e}},$
(2.11)

where we have used the definitions (2.5), (2.6). This quantity is
approximately zero for inflaton potentials with a strong feature localized
within a region supporting slow-roll inflation and with similar values of
$dH/dN$ before and after the feature. In this work we neglect the slow-roll
corrections and analyse only the very large enhancement resulting from such a
strong feature, by imposing the constraint that positive and negative “pulses”
have integrated areas that cancel.

###  2.2 Toy-model analysis

Several features that appear in the spectra that we study in the following
sections can be understood in a much simpler context. We are interested in the
effect of a “pulse” on the evolution of a mode with a free-wave initial
condition. It is instructive to ignore the background expansion and consider
the toy-model equation

$R_{k,tt}+\kappa R_{k,t}+k^{2}R_{k}=0.$ (2.12)

The solutions are oscillatory with an amplitude that gets suppressed or
enhanced, depending on the sign of the friction parameter $\kappa$. It is
straightforward to derive the solution for a friction term that vanishes at
all times apart from the interval $0<t<t_{p}$, by requiring the continuity of
the solution and its first derivative at $t=0$ and $t_{p}$.

Figure 2:  The real part (dashed curve), imaginary part (dot-dashed curve) and
amplitude (solid curve) of the solution of eq. (2.12) with $k=2$, for a
“pulse” in the interval $0\leq t\leq 0.5$. We also display the “pulse”, with a
rescaled maximum $\kappa/5$. Left plot: $\kappa=5$. Right plot: $\kappa=-5$.

For an early-time solution $R_{k}(t)=e^{-ikt}$, the evolution is depicted in
fig. 2. We observe the suppression of the amplitude for positive $\kappa$ and
the enhancement for negative $\kappa$. However, the most striking feature is
the appearance of oscillations in the amplitude. Their origin lies in the
modification by the “pulse” of the relative phase between the real and
imaginary parts. For sufficiently large $|\kappa|$ the relative phase in the
late stage of the evolution almost vanishes (as in the plot), so that the
amplitude approaches zero at certain instances. In the cosmological context,
the oscillatory form of the evolution as a function of time can be transferred
to the spectrum of perturbations. At late times, each mode $k$ exits the
horizon and eventually freezes. This can occur at any point of the oscillatory
cycle, depending on the value of $k$. As a result, the asymptotic values of
the perturbations depend strongly on the freezing time, and the spectrum
displays oscillations as a function of $k$.

It is known that it is possible to obtain an oscillatory pattern in the
spectrum if inflation stops for a certain time interval, so that modes that
had exited the horizon reenter and start oscillating again until their next
exit. Our toy example implies a more general pattern: Any feature during the
evolution of the perturbations that detunes the relative phase between the
real and imaginary parts of the solution results in an oscillatory spectrum,
even if inflation is not halted.

Another interesting property of the late-time evolution is displayed in fig.
3: The relative suppression of the amplitude of a mode for a given positive
friction parametr $\kappa$ is larger for higher wavenumber $k$. This is
counter-intuitive at first sight, as one would expect the last term of eq.
(2.12) to become more dominant for larger $k$ and limit the suppression
induced by the second term. However, the opposite happens. For small $k$, the
strong friction tends to freeze the evolution during the “pulse”, so that the
real and imaginary parts resume their oscillations after the “pulse” with
amplitudes comparable to the initial ones. As a general rule of thumb, for a
duration of the “pulse” of order 1, a strong suppression of the solution
occurs for $k\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}\kappa$.

Figure 3:  The real part (dashed curve), imaginary part (dot-dashed curve) and
amplitude (solid curve) of the solution of eq. (2.12) for a “pulse” with
$\kappa=10$ in the interval $0\leq t\leq 0.5$. We also display the “pulse”,
with a rescaled maximum $\kappa/10$. Left plot: $k=10$; right plot: $k=3$, in
arbitrary units.

###  2.3 Analytical expressions

We turn next to the analysis of eq. (2.7). For constant $f(N)=\kappa$ the
solution involves a linear combination of the Bessel functions
$J_{\pm\kappa/2}$ and has the form

$R_{k}(N;C_{p},C_{m},\kappa)=Ae^{-\frac{1}{2}\kappa\,N}\left(C_{p}J_{\kappa/2}\left(e^{-N}\frac{k}{H}\right)+C_{m}\,J_{-\kappa/2}\left(e^{-N}\frac{k}{H}\right)\right).$
(2.13)

Let us suppose that the coefficients of the solution $C_{p_{i}},C_{m_{i}}$ are
known for a range of efoldings for which $\kappa$ takes a specific value
$\kappa_{i}$. If this range is followed by a transition at $N=N_{fi}$ to a
second range in which $\kappa$ takes a differrent value $\kappa_{f}$, we would
like to compute the corresponding values of the constants
$C_{p_{f}},C_{m_{f}}$, see fig. 4. This can be achieved by requiring the
continuity of the solution and its first derivative at $N=N_{fi}.$ The new
coefficients are given through the relation

$\begin{pmatrix}C_{p_{f}}\\\
C_{m_{f}}\end{pmatrix}=M(N_{fi},\kappa_{i},\kappa_{f},k)\begin{pmatrix}C_{p_{i}}\\\
C_{m_{i}}\end{pmatrix},$ (2.14)

where the matrix $M(N_{fi},\kappa_{i},\kappa_{f},k)$ has components

$\displaystyle M_{11}$ $\displaystyle=$ $\displaystyle
C\left(J_{-{\kappa_{f}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)J_{-1+\kappa_{i}/2}\left(e^{-N_{fi}}\frac{k}{H}\right)+J_{1-{\kappa_{f}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)J_{{\kappa_{i}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)\right)$
$\displaystyle M_{12}$ $\displaystyle=$ $\displaystyle
C\left(-J_{-{\kappa_{f}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)J_{1-{\kappa_{i}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)+J_{1-{\kappa_{f}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)J_{-{\kappa_{i}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)\right)$
$\displaystyle M_{21}$ $\displaystyle=$ $\displaystyle
C\left(-J_{{\kappa_{f}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)J_{-1+\kappa_{i}/2}\left(e^{-N_{fi}}\frac{k}{H}\right)+J_{-1+\kappa_{f}/2}\left(e^{-N_{fi}}\frac{k}{H}\right)J_{{\kappa_{i}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)\right)$
$\displaystyle M_{22}$ $\displaystyle=$ $\displaystyle
C\left(J_{{\kappa_{f}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)J_{1-{\kappa_{i}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)+J_{-1+\kappa_{f}/2}\left(e^{-N_{fi}}\frac{k}{H}\right)J_{-{\kappa_{i}}/{2}}\left(e^{-N_{fi}}\frac{k}{H}\right)\right),$

with

$C=\frac{\pi}{2}e^{\frac{1}{2}N_{fi}(-2+\kappa_{f}-\kappa_{i})}\frac{k}{H}\,\csc\left(\frac{\pi\kappa_{f}}{2}\right).$
(2.16)

The matrix has the property $M(N_{fi},\kappa_{m},\kappa_{f},k)\cdot
M(N_{fi},\kappa_{i},\kappa_{m},k)=M(N_{fi},\kappa_{i},\kappa_{f},k)$. This
implies that we can select the value $\kappa=3$ as a reference point for all
transitions between different values of $\kappa$.

The next step is to define a matrix corresponding to a “pulse” of height
$\kappa$ above the value corresponding to the scale-invariant case. This
matrix can be defined as

$M_{pulse}(N_{1},N_{2},\kappa,k)=M(N_{2},\kappa,3,k)\cdot
M(N_{1},3,\kappa,k).$ (2.17)

As we explained earlier, the increase of the power spectrum relative to the
scale invariant one is given by the value of $|C_{m}|^{2}$ after a mode of
given $k$ has evolved past the strong features in the background. A product of
several $M_{pulse}$ matrices can reproduce the final values of the
coefficients $(C_{p},C_{m})$ of the Bessel functions $J_{\pm 3/2}$ after the
fluctuations have evolved from an initial configuration corresponding to
$(C_{p},C_{m})=(1,i)$ through a period of strong features in the function
$f(N)$. Clearly, it is possible to reconstruct any smooth function $f(N)$ in
terms of short intervals of $N$ during which the function takes constant
values. Multiplying the corresponding $M_{pulse}$ matrices would provide a
solution to the problem of the evolution of perturbations. However, such an
approach is not very efficient for a numerical solution. We are mainly
interested in obtaining intuitive analytical expressions for forms of $f(N)$
such as those depicted in fig. 1 and 4, for which a product of a small number
of $M_{pulse}$ matrices suffices.

Figure 4:  An illustration of the approximate form of the function $f(N)$ that
we assume for the analytical study. Left panel: a double-“pulse” model, with a
positive-friction “pulse” followed by a negative-friction one. Right panel:
the double-“pulse” form assumed in the three examples resulting in the spectra
of figs. 6, 7, 8.

Simple analytical expressions can be obtained in the limits of large and small
$k$, using the corresponding expansions of the Bessel functions. For a large
real argument we have

$J_{a}(z)=\sqrt{\frac{2}{\pi
z}}\left[\cos\left(z-\frac{a\pi}{2}-\frac{\pi}{4}\right)-\frac{4a^{2}-1}{8z}\sin\left(z-\frac{a\pi}{2}-\frac{\pi}{4}\right)+\mathcal{O}\left(z^{-2}\right)\right].$
(2.18)

Using this expression we find for large $k$

$M_{pulse}^{(\infty)}(N_{1},N_{2},\kappa,k)=e^{-\frac{1}{2}(N_{2}-N_{1})(\kappa-3)}\left\\{\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\frac{1}{8}(\kappa-3)\frac{H}{k}\begin{pmatrix}S_{11}&S_{12}\\\
S_{21}&S_{22}\end{pmatrix}\right\\}$ (2.19)

where

$\displaystyle S_{11}$ $\displaystyle=$ $\displaystyle
2e^{N_{1}}\sin\left(2e^{-N_{1}}\frac{k}{H}\right)-2e^{N_{2}}\sin\left(2e^{-N_{2}}\frac{k}{H}\right),$
$\displaystyle S_{12}$ $\displaystyle=$
$\displaystyle-e^{N_{1}}\left(1+\kappa+2\cos\left(2e^{-N_{1}}\frac{k}{H}\right)\right)+e^{N_{2}}\left(1+\kappa+2\cos\left(2e^{-N_{2}}\frac{k}{H}\right)\right),$
$\displaystyle S_{21}$ $\displaystyle=$ $\displaystyle
e^{N_{1}}\left(1+\kappa-2\cos\left(2e^{-N_{1}}\frac{k}{H}\right)\right)-e^{N_{2}}\left(1+\kappa-2\cos\left(2e^{-N_{2}}\frac{k}{H}\right)\right),$
$\displaystyle S_{22}$ $\displaystyle=$
$\displaystyle-2e^{N_{1}}\sin\left(2e^{-N_{1}}\frac{k}{H}\right)+2e^{N_{2}}\sin\left(2e^{-N_{2}}\frac{k}{H}\right).$
(2.20)

Keeping the leading contribution, we find that the power spectrum is scale
invariant at late times (or $N\to\infty$) for $k\rightarrow\infty$, but has a
value multiplied by the factor

$\left[\delta\Delta^{(\infty)}_{R}\right]^{2}=|C_{m}|^{2}=e^{-(N_{2}-N_{1})(\kappa-3)},$
(2.21)

relative to its scale-invariant value for modes that have sufficiently small
$k$, so that they exit the horizon and decouple very early with $C_{m}=i$,
without being affected by the features in $f(N)$. The exponent in the above
expression is simply the area of the “pulse” exceeding the value 3. For
$\kappa>3$ the spectrum is suppressed, while for $\kappa<3$ it is enhanced. By
breaking a general function $f(N)$ in infinitesimal “pulses”, it is easy to
see that the enhancement is equal to the integral of $f(N)-3$ over $N$. The
corrections subleading in $H/k$ introduce oscillatory patterns in the
spectrum. The characteristic periods can be deduced from eqs. (2.20). The
spectrum is expected to vanish at intervals $\delta k/H=e^{N_{1}}\pi$ and
$\delta k/H=e^{N_{2}}\pi$. Moreover, when $N_{1}\simeq N_{2}$ we expect
interference patterns.

Analytical expressions for $k\to 0$ are more difficult to obtain because the
(1,2)-component of the matrix $M_{pulse}$ scales as $1/k$ in this limit. As a
result, the effect of several “pulses”, which involves the product of several
such matrices, is not described by a simple analytical expression. However,
the components (2,1) and (2,2), which are relevant for the spectrum, are
simpler. The $(2,1)$-component becomes nonzero only at order $(k/H)^{3}$,
while the $(2,2)$-component is equal to $1+{\cal O}\left((k/H)^{2}\right)$.
So, up to order $(k/H)^{2}$, the (2,2)-component is sufficient for the
calculation of the spectrum. We give the result for the sequence of two
“pulses”:

$\begin{split}M_{pulse}^{(0)}(N_{1},N_{2},N_{3},\kappa_{1},&\kappa_{2},k)|_{2,2}\equiv[M(N_{3},\kappa_{2},3,k)\cdot
M(N_{2},\kappa_{1},\kappa_{2},k)\cdot M(N_{1},3,\kappa_{1},k)]_{2,2}\\\
=1+\frac{1}{6}\left(\frac{k}{H}\right)^{2}\times&\Biggl{[}\frac{\kappa_{1}-3}{\kappa_{1}(\kappa_{1}-2)}\left(-2(\kappa_{1}-3)e^{N_{1}(\kappa_{1}-2)-N_{2}\kappa_{1}}+3(\kappa_{1}-2)e^{-2N_{1}}-\kappa_{1}e^{-2N_{2}}\right)\Biggr{.}\\\
&-\frac{2(\kappa_{1}-3)(\kappa_{2}-3)}{\kappa_{2}(\kappa_{1}-2)}\left(e^{N_{3}\kappa_{2}}-e^{N_{2}\kappa_{2}}\right)\left(e^{-2N_{2}-N_{3}\kappa_{2}}-e^{N_{1}(\kappa_{1}-2)-N_{2}\kappa_{1}-N_{3}\kappa_{2}}\right)\\\
\Biggl{.}&+\frac{\kappa_{2}-3}{\kappa_{2}(\kappa_{2}-2)}\left(-2(\kappa_{2}-3)e^{N_{2}(\kappa_{2}-2)-N_{3}\kappa_{2}}+3(\kappa_{2}-2)e^{-2N_{2}}-\kappa_{2}e^{-2N_{3}}\right)\Biggr{]}.\end{split}$
(2.22)

For $\kappa_{2}=3$ the second “pulse” is eliminated and only the first term in
the bracket survives, while for $\kappa_{1}=3$ the first “pulse” is eliminated
and the last term survives. For $\kappa_{1},\kappa_{2}\not=3$ there is a
mixing term, which indicates that the effects of the various “pulses” are not
simply additive, even within this approximation.

Figure 5:  The (2,2)-component of the matrix $M_{pulse}$ defined in eq. (2.17)
(solid curve), along with the approximations for large $k/H$ (dashed curve)
and small $k/H$ (dot-dashed curve). Left plot: $N_{1}=0$, $N_{2}=0.5$,
$\kappa=-5$. Right plot: $N_{1}=0$, $N_{2}=0.5$, $\kappa=10$.

The oscillatory behaviour of the solutions can be observed in the components
of the matrix $M_{pulse}$ defined in eq. (2.17). In fig. 5 we depict the
(2,2)-component of this matrix (solid lines) for $N_{1}=0$, $N_{2}=0.5$. This
component gives the leading contribution to the power spectrum. The left plot
corresponds to a negative-friction “pulse” with $\kappa=-5$ that causes the
enhancement of the spectrum. The right plot is obtained for positive friction
$\kappa=10$ that leads to suppression. The asymptotic expansions of this
component for large $k/H$ (dashed curve), given by eq. (2.19), and small $k/H$
(dot-dashed curve), given by eq. (2.22), are also plotted. Oscillatory
behaviour is observed, associated with interference patterns from two almost
equal frequencies corresponding to $\delta k=e^{N_{1}}\pi$ and $\delta
k=e^{N_{2}}\pi$. It is interesting that the oscillatory frequencies are
correctly reproduced by the asymptotic expansion even for small $k/H$. Another
feature that can be observed is the strong decrease of $M_{pulse}$ for
$k/H\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}\kappa/2$ for
positive $\kappa$, in agreement with the discussion at the end of the previous
subsection.

###  2.4 The integral of $f(N)$

The form of $f(N)$ that we assume in our discussion should result from the
time evolution of the parameters $\varepsilon_{H}$ and $\eta_{H}$. We saw at
the end of subsection 2.1 that the integral of this function over $N$ is
constrained by eq. (2.11). In this subsection we discuss the type of field
evolution, as given by the function $\varphi_{,N}$, that is consistent with
our approximate treatment.

The Mukhanov-Sasaki equation (2.2) implies that the Wronskian of each Fourier
mode of its solution

$W[v_{k}]=-i(v_{k}\,v_{k}^{*\prime}-v_{k}^{*}v_{k}^{\prime})$ (2.23)

remains constant during the evolution. Here a prime denotes a derivative with
respect to conformal time $\tau=-e^{-N}/H$. The solution of eq. (2.2) plays
the role of the mode function in the canonical quantization of the field $v$.
For the Bunch-Davies vacuum, the initial condition at early times, when
$k^{2}\gg z^{\prime\prime}/z$, is chosen such that the mode function has the
standard form in Minkowski spacetime. Selecting positive-energy solutions
fixes the sign of the Wronskian to be positive, while the appropriate
normalization results in $W[v_{k}]=1$. This choice is automatically preserved
at later times if $v_{k}$ is a solution of eq. (2.2). This can be seen by
multiplying eq. (2.2) by $v_{k}^{*}$ and subtracting the conjugate of the same
equation multiplied by $v_{k}$.

We have based our analysis on the curvature perturbation $R_{k}$, related to
$v_{k}$ through $R_{k}=-v_{k}/z$, with $z=e^{N}\varphi_{,N}$. The consistency
of our approximation of describing $f(N)$ through a sequence of “pulses”
implies a specific form of $\varphi_{,N}$ during the evolution through the
strong features in the potential. We can deduce this form by considering the
Wronskian of $R_{k}$

$W[R_{k}]=-i(R_{k}\,R_{k}^{*\prime}-R_{k}^{*}R_{k}^{\prime})=\frac{W[v_{k}]}{z^{2}}=\frac{1}{z^{2}}.$
(2.24)

The solution (2.13) gives

$W[R_{k}]\propto
i\left(C_{p}C_{m}^{*}-C_{m}C_{p}^{*}\right)\exp((1-\kappa)N).$ (2.25)

Consistency with eq. (2.24) requires that
$\varphi_{,N}\propto\exp((\kappa-3)N/2)$. The inflaton “velocity” must grow
fast with $N$ for $\kappa>3$, and decay for $\kappa<3$. This is the behaviour
observed in fig. 1. We have already mentioned that any function $f(N)$ can be
reconstructed as a sequence of very short “pulses” of variable height
$\kappa$. For small $N$ the change of $\varphi_{,N}$, starting from some
initial value at $N=0$, is linear in $N$ with a slope proportional to
$\kappa$. Thus, by breaking $f(N)$ into many “pulses” one can obtain the
required evolution of $\varphi_{,N}$ as a function of $N$. In this sense our
analysis is very general. For consistency, of course, the deduced evolution
must result from an appropriate inflaton potential.

Our main aim is to obtain an intuitive understanding of the form of the
spectrum by focusing on the gross properties of $f(N)$. Let us consider a
feature in the evolution resulting from two successive “pulses” with heights
$\kappa_{1}$ and $\kappa_{2}$, between early and late slow-roll regimes with
$\kappa=3$. The solution after the feature is traversed is given by eq. (2.9)
with

$\begin{pmatrix}C_{p}\\\ C_{m}\end{pmatrix}=M(N_{3},\kappa_{2},3,k)\cdot
M(N_{2},\kappa_{1},\kappa_{2},k)\cdot
M(N_{1},3,\kappa_{1},k)\begin{pmatrix}1\\\ i\end{pmatrix},$ (2.26)

where the matrix $M$ is given by eq. (LABEL:Mab). Before the “pulse” we have
$i(C_{p}C_{m}^{*}-C_{m}C_{p}^{*})/2=1$, while after the “pulse” one finds

$\frac{i}{2}(C_{p}C_{m}^{*}-C_{m}C_{p}^{*})=e^{-(n_{2}-n_{1})(\kappa_{1}-3)-(n_{3}-n_{2})(\kappa_{2}-3)}.$
(2.27)

The exponent is exactly (minus) the integral of $f(N)-3$. By comparing the
Wronskian $W[R_{k}]$ at late and early times (before and after the “pulse”),
it becomes clear the the quantity (2.27) is equal to the ratio
$\left(\varphi^{2}_{,N}\right)_{e}/\left(\varphi^{2}_{,N}\right)_{l}$, with
both quantities being constant. In this way we reproduce the result of eq.
(2.11), under our assumption that $H_{l}/H_{e}\simeq 1$.

Let us summarize the basic points: According to our assumptions, the system is
in a slow-roll regime during an early and a late period, with values of the
Hubble parameter that we have approximated as equal. We can assume that the
values of $\varphi_{,N}$ are also approximately equal during these periods.
These assumptions isolate the effect of the strong feature in the intermediate
part of the evolution from the properties in the slow-roll regimes. During the
intermediate part the inflaton “velocity” $\varphi_{,N}$ changes fast, by
growing or decaying depending on the sign of $f(N)-3$, as observed in fig. 1.
The integral of $f(N)-3$ over $N$ must vanish for $\varphi_{,N}$ to have equal
values at early and late times. For realistic situations one must take into
account the breaking of scale invariance in the slow-roll regimes as well.
However, these are included in the standard slow-roll analysis and are not of
interest to us here.

Finally, it can be checked through the asymptotic form of the Bessel functions
that for both $k\to 0$ and $k\to\infty$, and for a vanishing integral of
$f(N)-3$, we have $(C_{p},C_{m})=(1,i)$ at all times during the evolution.
This indicates that the low- and high-$k$ modes are not affected by the
presence of the feature. As a result the scale-invariant form of the spectrum
is modified only for a finite range of wavenumbers $k$.

###  2.5 Form of the spectrum

In this subsection we consider three examples of spectra that display the
features we discussed in the previous subsection. The range of possible
spectra is large, as we do not focus on a particular underlying model, but
simply consider various forms of the function $f(N)$ defined in eq. (2.8). We
assume that the integral of $f(N)-3$ over $N$ vanishes, so that the spectrum
is scale invariant with the same amplitude for very low and very high
wavenumbers $k$. We focus only on the relative enhancement of the spectrum at
intermediate scales as a result of the presence of strong features in the
underlying inflaton evolution. As the absolute scale of the spectrum is not of
interest for our discussion, we set $A=1$ in eq. (2.9). We discuss next three
particular examples of the form of the friction function $f(N)$.

In our first example (Ex. 1) the spectrum results from a function $f(N)$ of
the qualitative form depicted in the first line of fig. 1 and displayed
explicitly in fig. 4. The feature consists of a positive-friction “pulse” with
$\kappa_{1}=80$ in the interval between $N_{1}=0$ and $N_{2}=0.2$, followed by
a negative-friction “pulse” with $\kappa_{2}=-3$ in the interval between
$N_{2}=0.2$ and $N_{3}=2.77$. The value of the spectrum for a given value of
$k/H$ is equal to $|C_{m}|^{2}$, where $(C_{p},C_{m})$ is given by eq. (2.26).
The result is depicted by the middle curve of the top plot in fig. 6, in the
$k$-range $k/H=10^{-2}-10^{3}$ that corresponds to $N\simeq-4.6$ up to $6.9$.
In the same figure we also display the spectra that would result from a single
“pulse”. These are computed from the expression

$\begin{pmatrix}C_{p}\\\ C_{m}\end{pmatrix}=M(N_{2},\kappa_{1},3,k)\cdot
M(N_{1},3,\kappa_{1},k)\begin{pmatrix}1\\\ i\end{pmatrix},$ (2.28)

for the positive-friction “pulse” (lower curve in fig. 6), and

$\begin{pmatrix}C_{p}\\\ C_{m}\end{pmatrix}=M(N_{3},\kappa_{2},3,k)\cdot
M(N_{2},3,\kappa_{2},k)\begin{pmatrix}1\\\ i\end{pmatrix},$ (2.29)

for the negative-friction “pulse” (upper curve in fig. 6). As we discussed in
the previous subsection, the fact that the integral of the function $f(N)-3$
over $N$ does not vanish for these cases means that the quantity
$\varphi_{,N}^{2}$ changes across the “pulse” by a factor equal to the
exponential of this integral. The two slow-roll regimes are quite distinct in
this case and the effect of the “pulse” is not clear. We display the spectra
because they provide intuition on the features appearing in the two-“pulse”
spectrum, for which the integral of $f(N)-3$ vanishes. Details for the latter
are presented in the next two plots of fig. 6, for two successive $k/H$ ranges
on a linear horizonal axis. Notice the huge difference in the scale of the
vertical axis in the two plots.

Figure 6: Middle curve: Spectrum resulting from a double “pulse” with
$N_{1}=0$, $N_{2}=0.2$, $N_{3}=2.77$, $\kappa_{1}=80$, $\kappa_{2}=-3$ (Ex.
1). Upper curve: Spectrum resulting from a negative-friction single “pulse”
with $\kappa_{2}=-3$ between $N_{2}=0.2$, $N_{3}=2.77$. Lower curve: Spectrum
resulting from a positive-friction single “pulse” with $\kappa_{1}=80$ between
$N_{1}=0$, $N_{2}=0.2$.

Figure 7:  Middle curve: Spectrum resulting from a double “pulse” with
$N_{1}=0$, $N_{2}=1$, $N_{3}=3$, $\kappa_{1}=15$, $\kappa_{2}=-3$ (Ex. 2).
Upper curve: Spectrum resulting from a negative-friction single “pulse” with
$\kappa_{2}=-3$ between $N_{2}=1$, $N_{3}=3$. Lower curve: Spectrum resulting
from a positive-friction single “pulse” with $\kappa_{1}=15$ between
$N_{1}=0$, $N_{2}=1$.

Figure 8:  Middle curve: Spectrum resulting from a double “pulse” with
$N_{1}=0$, $N_{2}=1.95$, $N_{3}=2.05$, $\kappa_{1}=-3$, $\kappa_{2}=120$ (Ex.
3). Upper curve: Spectrum resulting from a negative-friction single “pulse”
with $N_{1}=0$, $N_{2}=1.95$, $\kappa_{1}=-3$. Lower curve: Spectrum resulting
from a positive-friction single “pulse” with $\kappa_{2}=120$ between
$N_{2}=1.95$, $N_{3}=2.05$.

Several features of the spectra are apparent in these plots:

  1. 1.

The two-“pulse” spectrum has a first minimum at a value of $k/H$ well
approximated by the positive root of the polynomial of eq. (2.22).

  2. 2.

The subsequent strong increase of the spectrum results from the effect of the
negative-friction “pulse”. The spectrum reaches a maximal value comparable to
that of the negative-friction single-“pulse” spectrum. A rough estimate can be
obtained from the asymptotic value of the single-“pulse” spectrum, which is
$\exp((N_{3}-N_{2})(3-\kappa_{2}))={\cal O}(10^{7})$.

  3. 3.

The envelope of the positive-friction single-“pulse” spectrum (lower curve)
displays a sharp drop to almost zero at a characteristic value of $k/H$. As we
discussed earlier, we expect that the positive friction will affect most
strongly the high-$k$ modes. A more quantitative estimate can be made by
observing that the matrix $M$ of eqs. (LABEL:Mab) involves the Bessel
functions $J_{\pm\kappa_{1}/2}$ and $J_{\mp 1\pm\kappa_{1}/2}$. For large
$\kappa_{1}$ these functions have a zero at a value of their argument roughly
equal to $\kappa_{1}/2$. The relevant argument in our case is
$e^{-\bar{N}}k/H$, with $\bar{N}\simeq(N_{1}+N_{2})/2$. Thus, we expect the
spectrum to approach zero at $k/H\simeq e^{\bar{N}}\kappa_{1}/2\simeq 44$,
consistently with what is observed.

  4. 4.

For $k/H\to\infty$, all three spectra become asymptotically constant, with
values given by the exponential of the integral of $f(N)-3$ over $N$. For the
middle spectrum, we have fine-tuned this integral to zero, so that the
spectrum returns to the value 1 to which we have normalized the spectrum for
$k\to 0$.

  5. 5.

Apart from the main features that we described above, which are consistent
with the general expectations [111], the spectra display oscillations with
characteristic scales. As we discussed in the previous subsection, the
asymptotic expansions of eqs. (2.20) indicate that the spectrum should
oscillate with periods $\delta k/H\simeq e^{N_{1}}\pi=3.1$, $\delta k/H\simeq
e^{N_{2}}\pi=3.8$ and $\delta k/H\simeq e^{N_{3}}\pi=50$. These characteristic
modes, as well as interference patterns between them, are visible in the
bottom plots of fig. 6.

The most important conclusion that can be drawn from this example is that
strong features in the background evolution can induce a spectrum of
fluctuations which displays, apart from an enhancement by several orders of
magnitude, strong oscillatory patterns. This is clearly visible in the bottom
left plot of fig. 6.

We turn next to our second example (Ex. 2). The oscillatory features in the
spectrum are less pronounced for different forms of the “pulses”. Reducing the
height of the positive-friction “pulse” leads to a suppression of the spectrum
at smaller values of $k/H$. As a result the oscillatory patterns may be
confined within the high-$k$ part of the spectrum, which does not get
enhanced. This is visible in fig. 7, where we plot the spectrum in the
$k$-range, $k/H=10^{-2}-10^{3}$ or from $N\simeq-4.6$ up to $6.9$. The
spectrum results from a positive-friction “pulse” with $\kappa_{1}=15$ in the
interval between $N_{1}=0$ and $N_{2}=1$, followed by a negative-friction
“pulse” with $\kappa_{2}=-3$ in the interval between $N_{2}=1$ and $N_{3}=3$.
The drop of the spectrum arising from only the positive-friction “pulse” is
expected to appear at $k/H\simeq e^{\bar{N}}\kappa_{1}/2\simeq 12$, where
$\bar{N}\simeq(N_{1}+N_{2})/2$. Indeed, the small-$k$ region displays a large
enhancement, but the oscillations appear only at large values of $k/H$, at
which the spectrum is suppressed. The bottom left plot of fig. 7 shows that
the enhanced part of the spectrum is smooth in this case. The expected
oscillatory modes with periods $\delta k/H\simeq e^{N_{1}}\pi=3.1$, $\delta
k/H\simeq e^{N_{2}}\pi=8.5$ and $\delta k/H\simeq e^{N_{3}}\pi=63$, as well as
interference patterns between them, are visible in the bottom right plot of
fig. 7.

Our third example (Ex. 3) demonstrates that spectra with a different structure
can result from different forms of the function $f(N)$. More specifically, the
positive- and negative-friction “pulses” may occur in the reverse order
compared to the one we assumed up till now. This is possible if the inflaton
encounters a region of the potential with almost vanishing slope, as displayed
in the second line of plots in fig. 1. The reduction of the field “velocity”
results in a period of positive values for the parameter $\eta$. When the
inflaton moves beyond this region its “velocity” grows again, with $\eta$
taking negative values. In fig. 8 we plot the resulting spectra in the range
$k/H=10^{-2}-10^{3}$ ($N\simeq-4.6$ up to $6.9$) considering an effective
friction function $f(N)$ composed of a negative-friction “pulse” with
$\kappa_{1}=-3$ in the interval between $N_{1}=0$ and $N_{2}=1.95$, followed
by a strong positive-friction “pulse” with $\kappa_{2}=120$ in the interval
between $N_{2}=1.95$ and $N_{3}=2.05$. The reduction of the spectrum is
expected at a scale $k/H\simeq e^{\bar{N}}\kappa_{2}/2\simeq 440$, where
$\bar{N}\simeq(N_{2}+N_{3})/2$. The oscillatory patterns have characteristic
periods $\delta k/H\simeq e^{N_{1}}\pi=3.1$, $\delta k/H\simeq
e^{N_{2}}\pi=22.1$ and $\delta k/H\simeq e^{N_{3}}\pi=24.4$. All these
features, as well as strong interference patterns arising from the proximity
of two characteristic periods, are visible in fig. 8.

##  3 Primordial black holes and induced gravitational waves

###  3.1 Specific inflationary models

Based on the discussion of section 1.2 that motivated the use of the framework
of $\alpha$-attractors, we assume the following form for the function $F$:

$F(x)=F_{0}\left(x+\sum_{i=1}^{n}c_{i}\tanh\left(d(x-x_{i})\right)\right)\,.$
(3.1)

The corresponding inflationary potential for the field $\varphi$ in the
Einstein frame

$V(\varphi)=F^{2}\left(\tanh\frac{\varphi}{\sqrt{6}}\right)\ $ (3.2)

features $n$ step-like transitions. (All dimensionful quantities are given in
units of $M_{\rm Pl}$.) Such a potential can lead to an enhancement of the
power spectrum of scalar perturbations [79] at particular scales, which can be
sufficiently large to trigger PBH formation and induce detectable GWs. In
addition, the shape of the scalar power spectrum around its peak is
characterized by an oscillatory pattern that can be inherited by the tensor
power spectrum. We will discuss these notable phenomenological implications of
potentials with steps in the next subsections.

Figure 9:  Left panel: The inflationary potentials described by eq. (3.2),
arbitrary placed on the $\varphi$ axis in order to make the step-like
structure visible. Right panel: The function $f(N)$ of eq. (2.8) in terms of
the number of efoldings for inflationary potentials with one and four steps.

The enhancement induced by a step has an upper bound corresponding roughly to
a multiplicative factor $\exp({-\Delta N(\kappa-3)})$, see eq.(2.21). Here
$\Delta N$ is the interval during which the value $\kappa$ of the effective-
friction term (2.8) is smaller than the value $\kappa=3$ that results in a
scale-invariant spectrum. Negative values of $\kappa$ are realized when the
background inflaton “decelerates” on the lower plateau, after a sharp
transition through a step in the potential. During this stage, which lasts a
few efoldings, we have $\varphi_{,NN}\simeq-3\varphi_{,N}$ and
$\kappa\simeq-3$. As a result, a single step generally enhances the scalar
power spectrum by roughly two or three orders of magnitude. However, it is
possible that the potential includes several step-like features. In fig. 9 we
plot a set of specific examples of inflationary potentials with steps,
described by eq. (3.2). These potentials yield 50 to 60 efoldings after the
crossing of the CMB scale ($k=0.05\,\text{Mpc}^{-1}$) and a spectral index
value $n_{s}=0.969$, within the $68\%$ CL range of Planck [112]. The
parameters of these models are $c_{i}=7\times 10^{-3}$. We consider from one
($n=1$) up to five steps ($n=5$) in eq. (3.2), placed at $\varphi_{1}=5.7$,
$\varphi_{2}=5.55$, $\varphi_{3}=5.4$, $\varphi_{4}=5.25$, $\varphi_{5}=5.1$,
respectively. The value of $F_{0}$ is adjusted each time in order to be
consistent with the measured amplitude of the spectrum at the CMB scale. For
the initial value of the inflaton field we choose $\varphi_{\text{CMB}}=6.33$,
so as to obtain appropriate values for the spectral index $n_{s}$ and the
number of efoldings $N$. The choice of the value of the parameter $d$ is not
crucial, as long as it is taken sufficiently large for the transition through
the steps to occur quickly, but continuously. Typical values are of order
$10^{3}-10^{5}$.

We also examine the inflationary dynamics of models that feature both a step
and a near-inflection point. The production of PBHs and induced GWs due to the
presence of a near-inflection point in the framework of $\alpha$-attractors
has been studied in [34, 37, 93]. Such models result in a significant
enhancement of the scalar power spectrum, while the presence of a step-like
feature adds a prominent oscillatory pattern around the peak value. In fig. 10
we plot an example of such a potential, within the $\alpha$-attractor
framework, with parameters $c_{1}=8.70\times 10^{-2}$, $c_{2}=-2.77\times
10^{-4}$. The step is placed at $\varphi_{1}=5.4$ and a shallow nearly-
inflection point exists at $\varphi_{2}=4.8$. The spectral index value for
this model is $n_{s}=0.968$, within the $68\%$ CL region of Planck [112]. The
number of efoldings after the crossing of the CMB scale is $N=51$ for an
initial field value $\varphi_{\text{CMB}}=6.17$. In figs. 9 and 10 we also
plot the function $f(N)$ that determines key characteristics of the scalar
power spectrum, such as the amplitude and the oscillatory pattern, as
discussed in section 2.

In the following subsections we examine the cosmological implications for PBH
formation and GW production arising from the amplification of the scalar
spectrum by the step-like features in the potential (3.2). Remarkably, models
of this type yield striking predictions for the induced GWs that render them
testable by the forthcoming GW detection experiments.

Figure 10:  Left panel: The inflationary potential with one step and an
inflection point as a function of $\varphi$ in Planck units. The model gives
$n_{s}=0.968$ for an initial field value
$\phi_{\text{CMB}}=6.17\,M_{\text{Pl}}$. The inflection point at
$\phi_{\text{inflec}}=4.8\,M_{\text{Pl}}$ is clearly visible through the
magnification of the potential in the box. Right panel: The effective-friction
function $f(N)$ during the part of the evolution in which it deviates from the
standard value $f(N)\simeq 3$.

###  3.2 Primordial black holes

Inflationary potentials with steps enhance the amplitude of the primordial
density perturbations at particular scales and might lead to gravitational
collapse and PBH production. We review briefly observational bounds on the PBH
abundance, relevant for our analysis.

In the largest part of the mass spectrum there are stringent upper bounds on
$\Omega_{\text{PBH}}/\Omega_{\text{DM}}$ arising from observational
constraints, see fig. 11 for monochromatic PBH spectra. Light PBHs are
constrained by the extra-galactic gamma ray background (EGB); black holes of
mass above $10^{17}$g are subject to constraints from gravitational lensing of
stars by Subaru (HSC), Ogle (O), EROS (E) and MACHO (M), microlensing of
supernova (SN) and other experiments. The CMB anisotropies measured by Planck
(PA) constrain the PBHs with masses above $10^{33}$g. In the large-mass region
there are also constraints from accretion limits in X-ray and radio
observations and X-ray binaries (XB), and dynamical limits from disruption of
wide binaries (WB) and survival of star clusters in Eridanus II (Er). Advanced
LIGO/Virgo searches for compact binary systems with component masses in the
range $0.2-1M_{\odot}$ find no GW events. For a detailed discussion and
references on the PBH constraints we refer the reader to [126].

The maximal value of the PBH abundance can be achieved in the mass range
$M_{\text{PBH}}\sim 10^{-15}-10^{-10}M_{\odot}$. In this work we focus on this
mass window that can be tested by near-future GW experiments, such as LISA.
Nonetheless, the parameters of the same inflationary model with step-like
features can be adjusted in order to generate PBHs in other mass windows, such
as the ${\cal O}(10)$ solar mass window that is relevant for the LIGO/Virgo
observed events.

The theoretical framework for the PBH formation that we shall follow next is
based on the traditional Press-Schechter formalism [113]. Large density
perturbations can create overdense regions that may collapse to form black
holes after the horizon reentry. We examine separately the two most
interesting cosmological scenarios for the very early universe: the radiation
(RD) and matter domination (MD) scenarios.

####  3.2.1 Radiation-dominated era

For a Gaussian distribution function of the primordial density perturbations
and for spherically symmetric regions, the mass fraction of PBHs at formation
is

$\beta_{\text{}}(M)=\int_{\delta_{c}}d\delta\frac{1}{\sqrt{2\pi\sigma^{2}(k)}}e^{-\frac{\delta^{2}}{2\sigma^{2}(k)}}\,\simeq\,\frac{1}{2}\text{erfc}\left(\frac{\delta_{c}}{\sqrt{2}\sigma(k)}\right)\,\simeq\,\frac{1}{\sqrt{2\pi}}\frac{\sigma(k)}{\delta_{c}}e^{-\frac{\delta^{2}_{c}}{2\sigma^{2}(k)}}\,.$
(3.3)

The parameter $\delta_{c}$ is the threshold density perturbation and erfc$(x)$
is the complementary error function. For $\delta>\delta_{c}$ density
perturbations overcome internal pressure and collapse. The $\beta$ parameter
can be regarded as the probability that the density contrast is larger than
$\delta_{c}$. The PBH abundance is exponentially sensitive to the threshold
$\delta_{c}$. Different values for $\delta_{c}$ are quoted in the literature,
see e.g. [4, 114, 115, 116, 117, 118, 119], so that its precise value seems to
be rather uncertain. In the comoving gauge, ref. [118] finds $\delta_{c}=0.41$
for $w=1/3$. Numerical simulations demonstrate that there is no unique value
for the threshold, because it depends on the density profile.

In the comoving gauge, assuming a nearly scale-invariant curvature power
spectrum for a few e-folds around horizon crossing, the curvature perturbation
${R}$ can be related to the density perturbation $\delta$ as
$\delta(k,t)=2(1+w)/(5+3w)\left({k}/{aH}\right)^{2}{R}(k,t)$. The variance of
the density perturbations $\sigma(k)$, smoothed on a scale $k$ in the
radiation-dominated era, is given by [121]

$\sigma^{2}(k)=\left(\frac{4}{9}\right)^{2}\int\frac{dq}{q}W^{2}(qk^{-1})(qk^{-1})^{4}{\Delta^{2}_{R}}(q)\,,$
(3.4)

where ${\Delta^{2}_{R}}(q)$ is the power spectrum of the curvature
perturbations, usually calculated numerically. Here $W(z)$ represents the
Fourier transform of the Gaussian window function. In order to estimate the
mass spectrum of the PBHs, the horizon scale at the time of reentry of the
perturbation mode $k$ has to be related to the mass of formed PBHs. During the
radiation era, the wavenumber scales as $k_{\text{}}\propto
g^{1/2}_{*}g^{-2/3}_{s}S^{2/3}a^{-1}$ and the Hubble horizon as $H\propto
g^{1/2}_{*}g^{-2/3}_{s}S^{2/3}a^{-2}$, where $S$ denotes the entropy, and
$g_{*}$, $g_{s}$ count the total number of the effectively massless degrees of
freedom for the energy and entropy densities respectively. Assuming
conservation of the entropy between the reentry moment and the epoch of
radiation-matter equality, the relation between the PBH mass $M$ and the
comoving wavenumber $k$ is given by

$\displaystyle M(k)=\gamma\rho\frac{4\pi H(k)^{-3}}{3}\Big{|}_{k=aH}\simeq
2.4\times
10^{-16}M_{\odot}\Big{(}\frac{\gamma}{0.2}\Big{)}\left(\frac{g_{*}(T)}{106.75}\right)^{-\frac{1}{6}}\left(\frac{k}{10^{14}\,\text{Mpc}^{-1}}\right)^{-2},$
(3.5)

where we took the effective degrees of freedom $g_{*}$ and $g_{s}$
approximately equal. The factor $\gamma$ gives the fraction of the horizon
mass $M_{H}$ that collapses to form PBHs. Its value depends on the details of
the gravitational collapse and an analytical estimation [4] gives
$\gamma=0.2$. The present ratio of the abundance of PBHs with mass $M$ over
the total dark matter (DM) abundance,
$f_{\text{PBH}}(M)\equiv{\Omega_{\text{PBH}}(M)}/{\Omega_{\text{DM}}}$, can be
expressed as

$f_{\text{PBH}}(M)\equiv\frac{\Omega_{\text{PBH}}}{\Omega_{\text{DM}}}\,=\,\left(\frac{\beta_{\text{}}(M)}{3.3\times
10^{-14}}\right)\,\left(\frac{\Omega_{\text{DM}}h^{2}}{0.12}\right)^{-1}\Big{(}\frac{\gamma_{\text{}}}{0.2}\Big{)}^{\frac{3}{2}}\left(\frac{g_{*}}{106.75}\right)^{-\frac{1}{4}}\left(\frac{M}{10^{-12}M_{\odot}}\right)^{-1/2}\,.$
(3.6)

The abundance of PBHs produced during RD can be significant if the scalar
spectrum is amplified by roughly 7 orders of magnitude. In our single field
models, described by the the $\alpha$-attractors potential (3.2), such an
enhancement is achieved if the potential involves several steps or a step and
an inflection point. In fig. 11 we plot the PBH fractional abundance for a
potential with a step and and inflection point, for the parameter values
listed in section 3.1. The scalar power spectrum of this model is depicted in
fig. 13. For the estimation of the PBH abundance we assumed a threshold value
$\delta_{c}=0.45$. We see that, although the scalar power spectrum is
characterized by an oscillatory pattern around the peak of the PBH abundance,
it is predominantly monochromatic. However, the induced GW spectrum is much
more informative, as we will discuss in the following.

####  3.2.2 Matter-dominated era

PBHs might also form in the matter-dominated era (MD). In the absence of
pressure, even minute perturbations will evolve and deviations from spherical
configurations play an essential role. Refs. [122, 123, 124] examined the PBH
production in a matter-dominated universe and considered the non-spherical
effects during gravitational collapse. The PBH production rate $\beta$ tends
to be proportional to the fifth power of the variance $\sigma$ [124]:

$\beta(\sigma)\,=\,0.056\,\sigma^{5}\,.$ (3.7)

This expression has been derived with semi-analytical calculations and applies
to $0.005\lesssim\sigma\lesssim 0.2$, whereas for $\sigma\lesssim 0.005$ the
PBH production rate is modified if there is significant angular momentum in
the collapsing region [125]. The PBH fractional abundance is

$\displaystyle f_{\text{PBH}}\simeq\,1.3\times
10^{9}\,\gamma_{\text{}}\,\beta\,\frac{T_{\text{rh}}}{\text{GeV}}\,,$ (3.8)

with $T_{\text{rh}}$ is the reheating temperature.

There are two very interesting implications of PBH production during the MD
era. Firstly, the PBH abundance is found to be larger compared to RD for a
given amplitude of the curvature power spectrum. Inflationary potentials with
steps, which enhance the curvature power spectrum by 4 or 5 orders of
magnitude, can have an observational effect by generating a significant
cosmological PBH abundance. Secondly, the PBH production during the MD era
yields a PBH mass spectrum that is not predominantly monochromatic. It has a
distribution over a few orders of the PBH mass values, which might reveal a
non-trivial shape for the underlying power spectrum of the primordial density
perturbations. Although the specific inflationary models that we examine here
do not have a very strong effect on the PBH mass spectrum, inflationary models
with steps can in principle produce mild modulations in the distribution of
PBHs. The blue curve in fig. 11 depicts the PBH abundance produced by an
inflationary potential given by eq. (3.2) with three steps, for amplitude
$\Delta^{2}_{R}\sim 10^{-4}$ and $T_{\text{rh}}\sim 10^{3}$ GeV. In the same
figure, the dashed curve depicts the PBH abundance produced by the
double-“pulse” model of fig. 8 for $\Delta^{2}_{R}\sim 10^{-3}$ and
$T_{\text{rh}}\sim 1$ GeV. The spectrum is sufficiently wide and oscillating
in order to have an observational impact on the PBH mass distribution.

Figure 11:  The red curve depicts the abundance of PBHs produced by an
inflationary model with a step and an inflection point during the RD era. The
blue curve depicts the abundance of PBHs produced by an inflationary model
with three steps during the MD era. The dashed curve is the PBH abundance
produced during the MD era for the double-“pulse” model (termed Ex. 3) of fig.
8, in which a negative-friction “pulse” is followed by a positive-friction
“pulse”.

###  3.3 Induced gravitational waves

Primordial density perturbations that seed PBHs also produce stochastic GWs
through the mode-mode coupling of the density perturbations beyond the linear
order in the perturbative expansion. The GW production takes place mainly at
the time when the perturbations reenter the Hubble horizon. If density
perturbations enter during the RD era, the stochastic spectrum of second order
GWs can be computed following cosmological perturbation theory [9, 10, 11, 12,
13, 14, 15, 16, 17, 18]. The same density perturbations will also produce PBHs
with abundance proportional to $\beta$ given by eq. (3.3). On the other hand,
if perturbations enter deep in an early MD era, a different analysis has to be
followed in order to find the GW energy density spectrum [128, 129]. The
corresponding PBH abundance will now be proportional to $\beta$ given by eq.
(3.7). In the following we will consider GW production only during the RD era,
leaving the study of the early MD era scenario for future work. We will also
assume that curvature perturbations are described by Gaussian
statistics111Non-Gaussian statistics may also generate modulations in the GW
energy density spectrum [130]..

The spectrum of the induced GWs is sourced and shaped by the curvature
perturbations. In section 2 we found that inflationary potentials with steps
generate a distinct oscillatory profile for the curvature power spectrum. We
expect that this profile is transmitted to GWs. In the following subsections
we will further elaborate on the modulations of the amplitude in the GW energy
density spectrum, which will be found to display a multiple peak structure. We
will show in particular that the amplitude and the frequency of the peaks in
the GW spectrum are determined by the position and the number of the steps in
the inflaton potential. The GW spectrum inherits the pattern characteristics
of the curvature power spectrum and, hence, serves as a portal to the
inflationary dynamics.

Different GW experiments are sensitive to different frequency bands. Curvature
power spectra with a prominent peak at the horizon mass range
$10^{-15}-10^{-10}M_{\odot}$ generate induced GWs at the frequency band
$1-10^{-4}$ Hz, and can be tested by space-based interferometers like LISA
[96], scheduled to operate in the following decade.

####  3.3.1 The formalism of induced GWs

GWs are described by the tensor perturbation $h_{ij}$ in the FRW spacetime

$\displaystyle
ds^{2}=a^{2}(\tau)\left[-(1+2\phi)d\tau^{2}+\left((1-2\psi)\delta_{ij}+\frac{1}{2}h_{ij}\right)dx^{i}dx^{j}\right],$
(3.9)

where $\phi$ and $\psi$ are scalar perturbations and vector perturbations are
neglected. In the absence of anisotropic stress, which is a good approximation
for our purposes, we have $\phi=\psi$. The Fourier components of the tensor
modes are

$h_{ij}(\tau,\textbf{x})\,=\,\sum_{\lambda}\int\frac{d^{3}k}{(2\pi)^{3/2}}h_{\lambda}(\tau,\textbf{k})e^{(\lambda)}_{ij}(\textbf{k})e^{i\textbf{k}\textbf{x}}$
(3.10)

where $e^{(\lambda)}_{ij}$, with $\lambda=+,\times$, are polarization tensors.
Through the definition of the dimensionless power spectrum

$\left\langle
h_{\lambda}(\tau,\textbf{k})h_{\lambda}(\tau,\textbf{k}^{\prime})\right\rangle=\delta_{\lambda\lambda^{\prime}}\delta^{3}(\textbf{k}+\textbf{k}^{\prime})\frac{2\pi^{2}}{k^{3}}{\cal
P}_{h}(\tau,k)$ (3.11)

we have

$\rho_{\text{GW}}(\tau,k)=\frac{M^{2}_{\text{Pl}}}{8}\,\frac{k^{2}}{a^{2}}\,\overline{{\cal
P}_{h}(\tau,k)}\,.$ (3.12)

The evolution of $h_{ij}$ is obtained by expanding the Einstein equations. At
second order in scalar perturbations, the equation of motion for the Fourier
components of the tensor perturbations is

$h^{\prime\prime}_{\lambda}+2{\cal
H}h^{\prime}_{\lambda}+k^{2}h_{\lambda}\,=\,4S_{\lambda}(\tau,\textbf{k})\,,$
(3.13)

where $S_{\lambda}(\tau,\textbf{k})$ is a source that consists of products of
scalar perturbations:

$S_{\lambda}(\tau,\textbf{k})=\int\frac{d^{3}k}{(2\pi)^{3/2}}e^{ij}(\textbf{k})q_{i}\,q_{j}\left(2\phi_{\textbf{q}}\phi_{\textbf{k}-\textbf{q}}+\frac{4}{3(1+w)}({\cal
H}^{-1}\phi_{\textbf{q}}+\phi_{\textbf{q}})({\cal
H}^{-1}\phi_{\textbf{k}-\textbf{q}}+\phi_{\textbf{k}-\textbf{q}})\right)\,.$
(3.14)

The evolution of $\phi_{\textbf{k}}$ is given in terms of the scalar transfer
function. For radiation domination, we have $\phi_{\textbf{k}}(\tau)=\phi(x)\
\Phi_{\textbf{k}}$, with

$\phi(x)=\frac{9}{x^{2}}\left(\frac{\sin(x/\sqrt{3})}{x/\sqrt{3}}-\cos(x/\sqrt{3})\right),$
(3.15)

where $x=k\tau$, and $\Phi_{\textbf{k}}$ is the primordial value, related to
the curvature perturbation as

$\left\langle\Phi_{\textbf{k}}\Phi_{\textbf{k}^{\prime}}\right\rangle=\delta^{3}(\textbf{k}+{\textbf{k}^{\prime}})\frac{2\pi^{2}}{k^{3}}\left(\frac{3+3w}{5+3w}\right)^{2}\Delta_{R}^{2}(\tau,k).$
(3.16)

The solution of eq. (3.13) reads

$h_{\lambda}(\tau,\textbf{k})=\frac{1}{a(\tau)}\int_{0}^{\tau}G_{k}(\tau,\bar{\tau})a(\bar{\tau})S_{\lambda}(\bar{\tau},\textbf{k})d\bar{\tau}.$
(3.17)

where $G_{k}(\tau,\bar{\tau})$ is the the Green function for eq. (3.13). The
power spectrum of induced GWs is expressed in a compact form as a double
integral involving the power spectrum of the curvature perturbations [127]

$\overline{\mathcal{P}_{h}(\tau,k)}=\int_{0}^{\infty}dt\int_{-1}^{1}ds\
\mathcal{T}(s,t,\tau,k)\ \Delta_{R}^{2}\left(\frac{t+s+1}{2}k\right)\
\Delta_{R}^{2}\left(\frac{t-s+1}{2}k\right)\,.$ (3.18)

The overline denotes the oscillation average. The $t$ and $s$ variables are
defined as $t=u+v-1$, $s=u-v$, where $v={q}/{k}$,
$u=|\textbf{k}-\textbf{q}|/{k}$. The integral kernel ${\cal T}$ is given by
the expression

$\displaystyle\lim_{x\to\infty}x^{2}\ \mathcal{T}(s,t,x)$ $\displaystyle=\
2\left(\frac{t(2+t)(s^{2}-1)}{(1-s+t)(1+s+t)}\right)^{2}\frac{288(-5+s^{2}+t(2+t))^{2}}{(1-s+t)^{6}(1+s+t)^{6}}$
(3.19)
$\displaystyle\times\left\\{\frac{\pi^{2}}{4}\left(-5+s^{2}+t(2+t)\right)^{2}\Theta\left(t-(\sqrt{3}-1\right)+\right.$
$\displaystyle\left.\left(-(t-s+1)(t+s+1)+\frac{1}{2}(-5+s^{2}+t(2+t))\log\left|\frac{-2+t(2+t)}{3-s^{2}}\right|\right)^{2}\right\\}\,,$

where $\Theta$ is the Heaviside step function. The fraction of the GW energy
density per logarithmic wavenumber interval is

$\displaystyle\Omega_{\text{GW}}(\tau,k)=\frac{1}{\rho_{\text{tot}}(\tau)}\frac{d\rho_{\text{GW}}(\tau,k)}{d\ln
k}=\frac{1}{24}\left(\frac{k}{a(\tau)H(\tau)}\right)^{2}\overline{{\cal
P}_{h}(\tau,k)}\,.$ (3.20)

At a certain time $t_{\text{c}}$ the production of induced GWs ceases, while
their propagation becomes free. In a RD background the energy density
parameter $\Omega_{\rm GW}$ remains constant. Its value at the current time
$t_{0}$ is given by eq. (3.20) times the current radiation density parameter,
$\Omega_{\gamma,0}h^{2}=4.2\times 10^{-5}$, modulo changes in the number of
the relativistic degrees of freedom $g_{*}$ in the radiation fluid:

$\displaystyle\Omega_{\textrm{GW}}(t_{0},f)h^{2}=0.39\times\left({g_{\ast}\over
106.75}\right)^{-{1/3}}\
\Omega_{\gamma,0}h^{2}\times\Omega_{\textrm{GW}}(t_{\text{c}},f).$ (3.21)

The total energy density parameter of induced GWs is obtained by integrating
the GW energy density spectrum over the entire frequency interval.

Figure 12:  Left: The scalar power spectrum produced by a double square
“pulse”, normalized to the CMB measured amplitude. The dashed line is a
lognormal power spectrum with width $\sigma_{\text{N}}$. Right: The induced GW
spectrum, along with a fitting harmonic function. Each row, from top to
bottom, corresponds to the “pulse” producing the spectrum of each of figs. 6,
7, 8, respectively.

Figure 13:  Left panels: The curvature power spectra produced by inflationary
models with potential given by eq. (3.2) and depicted in fig. 9, for parameter
values given in the text. Right panels: The generated GW density parameter
produced by each inflationary model. A zoom-in plot of the peak region has
been included in each panel. Note that the last row of panels corresponds to
an inflationary model that features both a step and an inflection point,
depicted in fig. 10.

###  3.4 Oscillations in the power spectrum of the induced GWs from potentials
with steps

We start the discussion of the pattern of induced GWs produced in inflationary
models with sharp features by looking at the spectrum characteristics of
analytically calculable models, such as those depicted in figs. 6-8. In
section 2 we performed a semi-analytic calculation of the curvature power
spectrum by modeling the function $f(N)$ of eq. (2.8), which captures the
dynamics of the inflaton field beyond the slow-roll regime, through a sequence
of square “pulses”. The amplitude of the produced curvature power spectrum is
enhanced by the factor $|C_{m}|^{2}$ of eq. (2.21), while it also displays
oscillatory patterns with characteristic periods $\delta k/H\sim e^{N}\pi$ in
$k$-space, where $N$ is the number of efoldings at which the function $f(N)$
varies strongly. (See the discussion below eq. (2.21).) Roughly the same
characteristic frequency is observed in the GW spectra. In fig. 12 the GW
spectra for the three examples studied in section 2.5 are plotted. We also
plot a harmonic function
$\Omega_{\text{GWmax}}(2+\sin(\alpha_{\text{GW}}\,k-\theta))/3$ that
highlights the periodic change of the GW amplitude around the peak through a
fit of the $k$-space period $\delta
k_{\text{GW}}\sim{2\pi}/{\alpha_{\text{GW}}}$. We find
$\alpha_{\text{GW}}={\cal O}(1)/H$, corresponding to the smallest period of
the oscillations in the curvature spectrum $\delta k/H\sim{\cal O}(1)\pi$. It
must be noted that larger periods of size $\delta k/H={\cal O}(10)\,\pi$
appearing in $\Delta^{2}_{R}$ can also be discerned in the modulations of the
amplitude of the GW spectrum at the corresponding scales. However, they are
less visible as they extend to regions in $k$-space far from the peak.

The short-period modulations of order $\pi$ imply that the peaks in the
curvature spectrum are narrow. In fig. 12 a lognormal distribution with a
certain width $\sigma_{\text{N}}<1$ is plotted together with $\Delta^{2}_{R}$.
It elucidates the prominent two-peak structure induced in the GW spectrum [17,
92, 91], which appears when the main peak of the curvature spectrum is
sufficiently narrow. In our first example, for a square pulse starting at
$N=0$, corresponding to a smallest period of oscillations $\delta
k/H\simeq\pi$, the peak of $\Delta^{2}_{R}$ is found at a wavenumber
$k_{\text{p}}/H\simeq 4.5$, comparable to $\pi$. The $k$-range of the fitting
lognormal distribution is determined by requiring that $\delta k\sim
k_{\text{p}}(e^{\sigma_{\text{N}}}-1)$, which implies that
$\sigma_{\text{N}}<1$. As a result, and in agreement with the analysis of ref.
[91], the GW spectrum is found to feature a major, relatively sharp peak at
$(2/\sqrt{3})k_{\text{p}}/H$. Additionally, in the low-$k$ side there is a
relatively flat local maximum, at a wavenumber near $k_{\text{p}}/e$. This
characteristic two-peak structure is evident in all three examples we studied.
In the second example in particular, in which $\Delta^{2}_{R}$ is dominated by
a single peak because the smallness of the positive “pulse” confines the
oscillatory patterns within the high-$k$ part of the spectrum, the two-peak
structure is practically the only observable feature.

Let us now turn to the inflationary models with step-like features described
by eq. (3.2). In these models, the effective-friction function $f(N)$ of eq.
(2.8), depicted in fig. 9, has a $2n$-“pulse” structure for $n$ steps, with
each positive “pulse” followed by a negative one. In the semi-analytically
calculable models that we studied before, the power spectrum was normalized
such that the step features started at $N=0$. We observed an enhancement by a
factor $|C_{m}|^{2}$, together with oscillations of period $\delta k/H\sim
e^{N}\pi$. In the inflationary models of eq. (3.2) we find numerically similar
patterns. The curvature spectrum $\Delta^{2}_{R}$ is to a good approximation
enhanced by $|C_{m}|^{2}$, with a main peak at a wavenumber $k_{\text{p}}$
characteristic of the step position in terms of the number of efoldings $N$,
which are now counted from the exit of the CMB scale. Oscillations are also
produced with an approximate period $\delta k\sim k_{p}$.

In fig. 13 we plot four curvature power spectra together with the GW density
spectra that they induce. For the three-step model, the curvature spectrum
displays strong modulations and one can read off an oscillatory pattern with
period $\delta k\simeq 2.5\times 10^{11}\,\text{Mpc}^{-1}$. The spectrum
$\Delta^{2}_{R}$ has three prominent peaks at comparable wavenumbers
$k_{\text{p}1}$, $k_{\text{p}2}$, $k_{\text{p}3}$. Each peak is well described
by a narrow lognormal distribution in the $k$-range with width
$\sigma_{\text{N}}<1$. The combination of these three peaks induces a
characteristic five-peak structure in the GW spectrum [131], along with the
rather flat local maximum at lower $k$, as can be seen in the first row of
panels in fig. 13. The first sharp peak in the GW spectrum is located at the
value $k_{\text{GW,1}}=2k_{\text{p}1}/\sqrt{3}$, the second at
$k_{\text{GW,2}}=(k_{\text{p}1}+k_{\text{p}2})/\sqrt{3}$, the third at
$k_{\text{GW,3}}=2k_{\text{p}2}/\sqrt{3}$, the fourth at
$k_{\text{GW,4}}=(k_{\text{p}2}+k_{\text{p}3})/\sqrt{3}$ and the fifth at
$k_{\text{GW,5}}=2k_{\text{p}3}/\sqrt{3}$. The rather flat local maximum at
lower $k$ is located near $k_{\text{p}2}/e$, where $k_{\text{p}2}$ is the
wavenumber of the highest peak. These values can be seen in fig. 13 in the
frequency spectrum and in Hz units through the conversion $f=k/(2\pi)$, where
$\text{Mpc}^{-1}\simeq 0.97\times 10^{-14}$ Hz. Similar conclusions can be
drawn for the next two inflationary models that feature four and five steps,
respectively.

For the inflationary model that features both an inflection point and a step,
the resulting spectra are quite different compared to the previously discussed
models that involve only steps. The inflection point is responsible for the
strong enhancement of the curvature power spectrum and its relatively wide
peak. Indeed, the envelope function that outlines the peak can be fitted by a
lognormal distribution with width $\sigma_{\text{N}}\simeq 0.4$. Hence the
characteristic two-peak structure in the GW spectrum [91] is not very
prominent here. The step-like feature has a minor contribution to the
enhancement of the curvature spectrum, but it is the source of the oscillatory
pattern around the peak with the characteristic period $\delta k\simeq 7\times
10^{-11}\text{Mpc}^{-1}$. These oscillations are also transferred to the GW
spectrum. As before, the GW oscillatory pattern is well described by a
harmonic function and reflects the pattern in the curvature spectrum.

It is important to emphasize that, even though it is not clearly visible in
the log-plot, the oscillations near the peak are substantial: The
$\Omega_{\text{GW}}$ spectrum displays variations in its amplitude that are
$25\%$ of its maximal value or larger. Such modulations in the amplitude are
likely to be detectable by the near-future space interferometers.

##  4 Conclusions

In this paper we studied single-field inflationary models with sharp, step-
like features in the potential. The evolution of the inflaton through such
features leads to the violation of the slow-roll conditions, and in some cases
even to the temporary interruption of inflation. The striking consequences of
the transition through a generic step-like feature are the enhancement of the
power spectrum of the curvature perturbations at certain scales by several
orders of magnitude and the production of distinctive oscillatory patterns. We
studied analytically and numerically the inflationary dynamics and we derived
the expressions that describe quantitatively the size of the enhancement, as
well as the profile of the oscillations. It is interesting that these features
impact the power spectrum in a distinctive and predictive way that reflects
their properties: the amplification and the oscillatory pattern are shaped by
the position, number and steepness of the features.

Our analysis has revealed the origin of the oscillations. They are generated
through the detuning of the phase difference between the real and imaginary
parts of the curvature perturbation, which evolves according to the Mukahnov-
Sasaki equation (2.2). When the inflaton moves through a step in the
potential, the background evolution deviates strongly from the standard slow-
roll for a small number of efoldings, in a way that the real and imaginary
parts of the solution are detuned. The detuning results in time-dependent
oscillations of the amplitude. When the perturbations asymptotically freeze at
superhorizon scales, an oscillatory pattern is induced on the wavenumber
dependence of the power spectrum. A detailed discussion of this point can be
found in ref. [79] and subsection 2.2. It must be emphasized that this not a
consequence of all features in the potential that violates slow roll. In
contrast to a step, an inflection point in general induces an enhanced, but
smooth, power spectrum.

From the model-building perspective, a steep step can appear if the
inflationary potential includes plateaus with different energy densities. A
nearly constant potential energy density can be associated with underlying
symmetries that are preserved in the plateau [106]. A deformation of the
symmetry results in energy splitting, so that a transition between different
energy levels can be induced. Such a behavior can be captured by the framework
of the inflationary models characterized as $\alpha$-attractors [104, 105],
which our investigation has utilized.

A strong motivation for analysing such models is that the induced tensor power
spectrum inherits the oscillating profile of the primordial curvature
spectrum. The combined pattern of an enhanced spectrum together with strong
oscillations is potentially detectable by near future space interferometers.
Through the detection of the GW spectrum, one can aim at inferring at least
some basic feature of the inflationary potential, such as whether step-like
transitions are present. Motivated by this possibility, we examined in detail,
numerically and analytically, the scalar and the induced tensor spectra and we
identified correlations between them. The main characteristic property of both
spectra, related to the transition through a step, is a series of peaks.
Through a more refined analysis of the spectrum, one can look for more
detailed information, such as the number of the steps, their position and
exact shape, and whether there is, in addition to a step, an inflection point.
We explored this possibility by studying several analytical examples, as well
as inflationary models in the $\alpha$-attractor framework, always imposing
consistency with the contraints for the spectral index $n_{s}$ and the
amplitude of the scalar spectrum arising from the CMB measurements.

The detection of GWs from inflationary models with sharp features may be
accompanied by the presence of PBHs as a significant fraction of dark matter.
The enhancement of the power specrtum due to the presence of step-like
features, though considerable, may be inefficient to trigger the production of
a sizeable number of PBHs, if radiation dominates the energy density of the
early universe. However, it can be sufficient to induce gravitational collapse
processes and PBH production if the universe energy density is dominated by
non-relativistic matter. We examined the profile of the PBH mass spectrum
produced either in a radiation or an early matter-dominated universe, looking
for deviations from the common monochromatic profile. For the latter scenario
we found that this is possible because of the multiple-peak structure of the
curvature power spectrum.

It is important to note that the oscillations near the peak of the GW spectrum
have a scale comparable to that of its maximal value. Such modulations in the
amplitude are likely to be detectable by the near future space
interferometers, such as LISA. This demonstrates that induced GWs can be used
as a powerful tool for probing the inflationary potential. The detection of
oscillatory patterns in the amplitude of the GW spectrum will be a strong
indication for sharp features in the potential of single-field inflation.

## Acknowledgments

We would like to thank V. Spanos for useful discussions. The work of I.
Dalianis, G. Kodaxis, I. Stamou and N. Tetradis was supported by the Hellenic
Foundation for Research and Innovation (H.F.R.I.) under the “First Call for
H.F.R.I. Research Projects to support Faculty members and Researchers and the
procurement of high-cost research equipment grant” (Project Number: 824).

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