On the slope of the curvature power spectrum
in non-attractor inflation
Ogan
¨
Ozsoy
♣♥
Gianmassimo Tasinato
♣
♣ Department of Physics, Swansea University, Swansea, SA2 8PP, United Kingdom
♥ Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5,
Warsaw, Poland
Abstract
The possibility that primordial black holes constitute a fraction of dark matter motivates a
detailed study of possible mechanisms for their production. Black holes can form by the collapse
of primordial curvature fluctuations, if the amplitude of their small scale spectrum gets amplified
by several orders of magnitude with respect to CMB scales. Such enhancement can for example
occur in single-field inflation that exhibit a transient non-attractor phase: in this work, we make
a detailed investigation of the shape of the curvature spectrum in this scenario. We make use of
an analytical approach based on a gradient expansion of curvature perturbations, which allows
us to follow the changes in slope of the spectrum during its path from large to small scales.
After encountering a dip in its amplitude, the spectrum can acquire steep slopes with a spectral
index up to n
s
− 1 = 8, to then relax to a more gentle growth with n
s
− 1 . 3 towards its
peak, in agreement with the results found in previous literature. For scales following the peak
associated with the presence of non-attractor phase, the spectrum amplitude then mildly decays,
during a transitional stage from non-attractor back to attractor evolution. Our analysis indicates
that this gradient approach offers a transparent understanding of the contributions controlling
the slope of the curvature spectrum. As an application of our findings, we characterise the
slope in frequency of a stochastic gravitational wave background generated at second order from
curvature fluctuations, using the more accurate information we gained on the shape of curvature
power spectrum.
arXiv:1912.01061v3 [astro-ph.CO] 31 Mar 2020
Contents
1 Introduction 2
2 Enhanced curvature perturbations in single field inflation:
the role of the decaying mode 3
2.1 Growing and decaying mode: general considerations 5
2.2 Solving the mode equations at super-horizon scales 7
2.3 The spectrum of curvature perturbations on super-horizon scales 10
2.4 General comments on the results so far 11
3 Enhancement of R
k
in Single Field Inflationary Scenarios 12
3.1 Model 1: an instant transition between a slow-roll and a non-attractor phase 14
3.2 Model 2: an intermediate phase between attractor and non-attractor 18
3.2.1 On the steepest slope of the power spectrum 19
3.2.2 The slope of the power spectrum towards the peak in realistic models 21
3.3 On the final transition between non-attractor and slow-roll phase 24
4 Implications for stochastic gravitational wave backgrounds 27
5 Summary 31
A The solution for R
k
with more general initial conditions 32
B The curvature perturbation R
k
and fractional velocity v
R
34
C Model 1: Calculation of D(τ
k
), F
k
(τ
k
) and G(τ
k
) 37
D Model 2: Calculation of D(τ
k
), F
k
(τ
k
) and G(τ
k
) 41
E Coefficients to determine the location of k
dip
52
F Remarks on the high slopes after k
dip
and c
k
= −kτ
k
53
References 54
1
1 Introduction
Primordial black holes might constitute a fraction of dark matter [1–3]. This possibility has
been reinvigorated by the works [4–6], after the LIGO-Virgo detection of gravitational waves
from black hole merging events. See e.g. [7] for a thoughtful discussion of arguments in favour
of this hypothesis. The production of primordial black holes (PBHs) is associated with the
collapse of curvature perturbations at small scales. Starting with [8, 9], many works over the
years investigated the possibility to produce PBHs during cosmological inflation. One needs
dedicated mechanisms to amplify the curvature power spectrum by several orders of magnitude
between the large CMB scales, and the smaller scales associated with PBH formation. See e.g.
[10, 11] for reviews. Given that CMB observations are consistent with single-field inflation, it
is important to clarify the conditions to obtain a growth of the curvature power spectrum in
single-field scenarios, see e.g. [12–16]. A possibility is that the scalar field driving inflation first
rolls through a slow-roll phase – while it produces curvature fluctuations at the scales probed by
the CMB observations – to then pass over a inflection (or near-inflection) point of its potential,
leading to a transient non-attractor inflationary phase associated with PBH production. The
inflationary dynamics during such phases are dubbed as ultra or constant-roll inflation and have
been explored for example in [17–25].
Given the importance of this subject for connecting the physics of the early universe with the
dark matter problem, it is important to develop a reliable formalism, which allows one to acquire
an analytical understanding of the features of the growing curvature power spectrum during non-
attractor inflation. Ideally, such formalism should be physically transparent, and flexible enough
to be applied also to cases where the inflationary potential is non-monotonic, being characterised
by local maxima and minima that can lead to a rich inflationary dynamics. The appearance of
these kind of features in the scalar potential are motivated and explored in explicit string theory
constructions [26–28]. A possible framework to study these possibilities in string theory is axion
monodromy [29, 30], which realise natural inflation potentials [31] (see also [32–38]). In these sce-
narios, the inflationary evolution can pass through non-attractor phases, during which one of the
slow-roll conditions are not satisfied [16], and the amplitude of curvature perturbation becomes
several orders of magnitude larger than its amplitude at CMB scales. A consistent approach for
analysing the behaviour of curvature fluctuations during transitions between attractor and non-
attractor phases has been implemented in the interesting work [39], making use of Israel junction
conditions [40, 41] to match the mode functions of curvature perturbation at different epochs.
The study performed in [39] provides an analytical understanding for the presence of a spiky
dip, that usually precedes a rapid growth in the power spectrum of curvature perturbation in
inflationary scenarios where the transition to a transient non-attractor phase is realised through
monotonic potentials [39] then finds that the maximal slope of the curvature power spectrum
growth, well far from the dip, is characterised by a spectral index n
s
− 1 = 4, before reaching
a peak associated with the non-attractor era. The more recent work [42] found that a slightly
steeper growth is possible if a prolonged intermediate stage of non-slow-roll expansion occurs
between the standard slow-roll attractor, and the non-attractor epochs.
In this work we propose to study the problem adopting a method based on a gradient expansion
for solving the mode equation of curvature perturbation. This method, first introduced in [43] and
2
extended here, is especially appropriate to accurately investigate the behaviour of the curvature
spectrum in models where the inflationary dynamics is characterised by non-attractor phases of
evolution. In particular:
• In Section 2 we present and develop our formalism, and we make use of it for describing the
behaviour of the would-be decaying mode of curvature fluctuations. We show that in models
characterised by non-attractor epochs during inflation the would-be decaying mode actually
grows, and influences the spectrum of curvature fluctuations at super-horizon scales. The
spectrum of fluctuations acquires interesting features as a spiky dip followed by a rapid
growth in the amplitude as a function of the scale, which can be qualitatively understood
in terms of the formulas we provide.
• In Section 3 we apply our general formalism to various concrete scenarios. We show that
our method allows us to accurately describe the behaviour of the curvature spectrum. For
a range scales following the dip in the spectrum – whose extension usually depends on the
model parameters – growth of the spectrum is characterized by a large spectral index (up
to n
s
− 1 = 8), that then reduces to smaller values, (i.e. n
s
− 1 . 3) towards the peak. A
good analytic control on these regimes can be important to fully characterise the growth
rate of the curvature spectrum in generic scenarios of inflation with non-attractor epochs.
Furthermore, the method on gradient expansion we adopt here allows us to re-derive the
results of [39, 42] on the asymptotic slope of the spectrum well after the dip occurs, in
terms of easy-to-handle analytic formulas.
• Our findings can find several applications. For definiteness, in Section 4 we investigate how
the new features that further characterise the slope of the curvature spectrum (the steeper
growth right after the dip, the gentle decrease after the peak in the spectrum) affect the
spectrum of gravitational waves generated at second order by the strong amplification of
curvature fluctuations [44–46]. Making use of duality arguments developed in [43, 47–51],
we also get analytic control, in representative scenarios, on the behavior of the scalar power
spectrum after the peak in the power occurs: i.e. for modes associated with the transition
from non-attractor to final attractor phase that we dub as graceful exit epoch.
We conclude in Section 5. Four technical Appendixes contain details of our calculations.
2 Enhanced curvature perturbations in single field inflation:
the role of the decaying mode
On a background described by the FRW line element, ds
2
= a
2
(τ)
−dτ
2
+ d~x
2
, the comoving
curvature perturbation obeys the following equation in Fourier space [52]:
1
z
2
(τ)
z
2
(τ)R
0
k
(τ)
0
= −k
2
R
k
(τ) , (2.1)
3
where for a scalar field minimally coupled to gravity, the ‘pump field’
1
is defined as z ≡ a
˙
φ/H.
It is a well known fact that in a standard slow-roll background, the growing mode solution of
eq. (2.1) is conserved on super horizon scales. This can be readily seen from the formal integral
solution of (2.1), which can be written up to order O(k
2
) for small but finite wave-numbers as
R
k
(τ) ' R
(0)
"
1 + C
2
Z
τ
dτ
0
z
2
(τ
0
)
− k
2
Z
τ
dτ
0
z
2
(τ
0
)
Z
τ
0
dτ
00
z
2
(τ
00
)
#
, (2.2)
where we obtained the last term by solving iteratively the inhomogeneous part of eq. (2.1) using
the leading growing mode which we identify as R
(0)
. The constant behavior of R
k
shortly after
its scale crosses the horizon can be seen from the solution (2.2), by realizing that – in a slow-roll
background where z ∝ (−τ )
−1
– the second and the third term in (2.2) decay respectively as (−τ)
3
and (−τ)
2
in the late time limit −τ → 0. Therefore, in a slow-roll background we can immediately
identify the second and third term in (2.2) as the decaying modes
2
. However, the form of the
“decaying” solutions in (2.2) can already guide us when the mode evolution does not follow
the slow-roll trajectory we described above. For example, if a mode experiences a background
evolution in which the pump field z(τ) quickly decays (i.e. non-attractor backgrounds) after the
horizon exit, we can no longer assume a constant R
k
on super-horizon scales as the would be the
“decaying” mode can contaminate the constant growing solution substantially. In this case, R
k
will not become constant until the background switches back to the slow-roll attractor regime
(where z(τ) grows) such that the decaying solutions in (2.2) die out. The important point in
this example is that the second and third term in (2.2) can only be identified as “decaying”
asymptotically in the far future but they do not have to monotonically decay right after horizon
exit.
In what follows, we take eq. (2.1) as our starting point to analytically investigate the spectral
behavior of comoving curvature perturbation in models that include phases of non-attractor in-
flation, where the would-be decaying mode starts to grow instead of rapidly decay after horizon
exit. This phenomenon can lead to a steep growth of the curvature spectrum which can reach
a sufficiently large amplitude for the production of primordial black holes (see e.g. [11] for a
review). On the other hand, accurate characterization the behavior of curvature perturbation in
these scenarios requires solving (2.1) in backgrounds where slow-roll conditions are strongly vio-
lated [16] and therefore is less amenable to analytic
3
descriptions. In this Section, we therefore
wish to develop a formalism to analytically capture the behaviour of the curvature spectrum in
scenarios that include a phase of non-attractor evolution. For this purpose, we structured the
following sub-sections as follows:
1
In more general single-field scenarios of inflation including a scalar field with non-canonical kinetic terms or
non-minimal couplings between φ and the metric, the mode equation for R
k
is identical to the eq. (2.1), but with
a more general definition for the pump field: see e.g. [53]. As we will comment further in Section 4, our findings
in this paper can be directly applied to such generalized models as well.
2
In fact, the standard decaying mode is given by the last term as it decays slowly, i.e. ∝ (−kτ )
2
, compared to
the second.
3
If certain conditions are satisfied, the statistics of curvature fluctuations both during the attractor and non-
attractor eras of inflation can be analytically related in terms of Wands’ duality [47].
4
![Figure 9. Scalar power spectrum in the bumpy axion model of [27] towards the peak. Large spectral slopes after the spiky dip, as well as smaller than k3 slopes obtained towards the peak is shown by dashed reference lines (brown,blue,black).](/figures/figure-9-scalar-power-spectrum-in-the-bumpy-axion-model-of-38mdbtug.webp)
