Production of high stellar-mass primordial black holes in trapped inflation
Production of high stellar-mass primordial black holes
Shu-Lin Cheng 0 1 4
Wolung Lee 0 1 4
Kin-Wang Ng 0 1 2 3
Open Access 0 1
c The Authors. 0 1
0 Taipei 11529 , Taiwan
1 Taipei 11677 , Taiwan
2 Institute of Physics , Academia Sinica
3 Institute of Astronomy and Astrophysics , Academia Sinica
4 Department of Physics, National Taiwan Normal University
Trapped inflation has been proposed to provide a successful inflation with a steep potential. We discuss the formation of primordial black holes in the trapped inflationary scenario. We show that primordial black holes are naturally produced during inflation with a steep trapping potential. In particular, we have given a recipe for an inflaton potential with which particle production can induce large non-Gaussian curvature perturbation that leads to the formation of high stellar-mass primordial black holes. These primordial black holes could be dark matter observed by the LIGO detectors through a binary black-hole merger. At the end, we have given an attempt to realize the required inflaton potential in the axion monodromy inflation, and discussed the gravitational waves sourced by the particle production.
Black Holes; Cosmology of Theories beyond the SM
in trapped inflation
The LIGO detectors have firstly observed gravitational waves predicted in general
relativity . The discovery opens up a new era of gravity-wave astronomy and
The detected gravity-wave source GW150914 provides an evidence for the
existhat inspiral and merge within the age of the Universe, with an inferred merger rate of
confirmed, implying that gas accretion onto the BH binary is insignificant at the time of
In astrophysics, the observation of GW150914 has constrained the theory of
stellarmass BBH formation. The merger rate inferred from the observation of GW150914 has
eximply either BBH formation in a local dwarf galaxy followed by a prompt merger, or
formation at high redshift with a long delayed merger. There remain two prevailing formation
categories for BBH mergers: dynamical gravitational interactions between BHs and stars
in dense globular clusters (ref.  and references therein) and isolated stellar binaries in
galactic fields (refs. [5, 6] and references therein).
Recently, inspired by LIGO results, the authors in ref.  have considered a possibility
that the BH binary associated with GW150914 may be of primordial origin, different from
more traditional astrophysical sources - the two BHs are primordial black holes (PBHs)
formed in the early Universe. Interestingly, massive PBHs being non-relativistic and weakly
interacting gravitationally may constitute the cold dark matter, if they are formed prior
to the big bang nucleosynthesis. The estimated merger rate for these PBHs, if they are
clustered in compact sub-halos, can span a range that overlaps the BH merger rate inferred
from GW150914 [7–9]. As such, GW150914 can be considered as an indirect signal of dark
gravitational waves in a massive PHB coalescence. This may explain why GW150914 has
no optical counterpart, because sub-halo PBHs deplete gas and quench star formation.
Furthermore, PBH sub-halos should manifest as faint dwarf galaxies, possibly alleviating
the missing satellite and too-big-to-fail problems.
Although the PBH alternative for GW150914 seems plausible, many challenging
questions such as their formation and subsequent evolution into sub-halos, the mass fraction
in dark matter, and the distinction between PBHs and astrophysical BHs, remain to be
answered. In fact, there are astrophysical and cosmological constraints that have already
excluded PBHs as the main component of dark matter over a vast range of PBH masses.1,2
observations, are controversial, because they require modeling of several complex physical
1See for example .
processes inherent with significant uncertainties. The existence of high stellar-mass PBHs
not only solves the dark matter problem, but also provides an important test on the theory
of PBH formation in the early Universe.
There are various scenarios for forming PBHs over a wide range of masses in the early
Universe . Apart from those spontaneously formed in phase transitions, PBHs can
be arisen from the collapse of horizon-sized large matter inhomogeneities. This matter
density perturbation may originate in quantum fluctuations during inflation that re-enter
the horizon in the subsequent expanding Universe. If a seed to form a PBH is created
during inflation, the mass of the PBH can be estimated as follows. The energy contained
within the comoving seed volume that leaves the horizon N e-foldings before the end of
inflation is governed by
where H is the Hubble constant during inflation and Mp the reduced Planck mass. After
be the scale factor at the onset of inflation, which will eventually span N0 e-foldings over
radiation-dominated epoch. The comoving volume re-enters the horizon when its scale
k = HeN0−N satisfies the condition kη ∼ 1, i.e., when a = eN0+N or the temperature of
the thermal bath is red-shifted by a factor of eN . Therefore, the mass of the PBH
presumably formed at this time is
In single-field slow-roll inflation models, the matter density perturbation is generally
well below the threshold to form PBHs though they can be formed at rare density peaks.
Modifications of the inflation potential to achieve blue-tilted matter power spectra or
running spectral indices may lead to large density perturbation at the end of inflation.3 When
inflaton is coupled to quantum fields, copious quanta production can induce large
perturbation at small scales too [14–18]. However, the resulting PBH masses in most of these
cal and even cosmological mass scales, several scenarios involving multi-field inflation have
been proposed, such as the hybrid inflation,4 the double inflation [21–24], and the
curvaton models [25, 26], in which small-scale density perturbation can be inflated to a scale
ranging from the size of a stellar-mass PBH to a supermassive PBH. The astrophysical and
cosmological bounds on PBHs have been used to exclude or constrain all these models.
In this paper, we propose a mechanism for the production of high stellar-mass PBHs in a
single-field trapped inflation [27–32].
Trapped inflation has been proposed to provide a successful inflation with a steep
potential. Due to a coupling of the inflaton with other massless particles, the inflaton in
3See for example .
4See for example [19, 20].
the trapped inflation rolls slowly down a steep potential by dumping its kinetic energy
into particle production [27–29]. As shown in ref. , a viable trapped inflation model
can be achieved even on a potential which is too steep for slow-roll inflation; in addition,
the inflaton fluctuations induced by the backreaction of the particle production dominate
over the vacuum inflaton fluctuations. In ref. , the authors considered an axion-like
inflaton coupled to massless gauge fields through a Chern-Simons term. They found that
the inflation with a steep potential is effectively a slow-roll inflation but unfortunately the
induced inflaton fluctuations become too large to satisfy CMB constraints. Here we adopt
the main idea of trapped inflation but we modify it in the following manner. In order
to explain the CMB anisotropy data, we still use a flat potential for driving a standard
slow-roll inflation for the first 15 e-foldings after the beginning of inflation. After that, the
inflaton slides into a steeper potential well, experiencing a trapped type of inflation. Large
inflaton fluctuations will be produced by profuse particle production in this trapped period
until the inflaton exits the potential well and rolls towards the end of inflation. These large
inflaton fluctuations during the trapped inflation are the seed for the formation of PBHs
when the fluctuations exit and re-enter the horizon.
To achieve our goal, we consider a modified version of the trapped inflation driven by
1 F μν Fμν − 4f
an energy scale, Fμν = ∂μAν −∂ν Aμ is the field strength tensor, and F˜μν = 21 ǫμναβFαβ/
is its dual. The potential V (ϕ) is a standard flat potential superimposed with a shallow
To calculate the production of gauge quanta, we separate the inflaton into a mean field
and its fluctuations:
1 hE~ 2 + B~ 2i =
hE · Bi = −
dη2 + k ∓ 2aHkξ A±(η, k) = 0, ξ ≡ 2f H dt
This equation implies that either one of the two modes, satisfying the condition k/(aH) <
term of the produced gauge quanta are given by the vacuum expectation values of the
electric and magnetic fields, respectively,
On the other hand, the production of gauge quanta gives rise to a backreaction on the
background. The background evolution is therefore governed by
Then, the generation of inflaton fluctuations during inflation can be calculated by
∂t − a
where the dissipation effect is depicted as [33, 34]
E~ · B~ − hE~ · B~ i ,
It was shown that the solution to this equation can be well approximated by [15, 34]
which leads to a contribution to the power spectrum of the curvature perturbation given by
Following the numerical scheme in our previous paper , we compute the power
spectrum (13) by solving numerically the coupled differential equations of motion for
inflaton and photon mode functions in eqs. (5), (8), and (9) from the onset of inflation to the
end of inflation, included self-consistently with the backreaction due to photon production.
We employ a modified quadratic potential,
V (ϕ) =
7.39 × 10−6.
We set α = 17, f = 1, and φ0 = −13.5 and (dφ/dt)0 = 6.02 × 10−6
respectively for the initial position and speed of the inflaton. The number of e-foldings
after the beginning of inflation is defined by R0 H(t′)dt′. Note that a0 = 1 and we find
functions of the e-foldings N before the end of inflation are shown in figure 2. Figure 4
depicts the total power spectrum of the curvature perturbation, which is the sum of the
the inflaton rolls slowly in the quadratic part of the potential in a manner as that in the
dynamical variables in this figure and in the following figures are rescaled by the reduced Planck
mass, Mp = 2.435 × 1018 GeV.
end of inflation. Inflation starts at N0 ≃ 61.
induced by photon production is denoted by the dotted line and the vacuum contribution by the
dashed line. The e-folding N denotes the time when the k-mode leaves the horizon. The primordial
black hole bound is the short-dashed line.
chaotic inflation . As in standard slow-roll inflation, the power spectrum of the
curvature perturbation is dominated by vacuum fluctuations, being consistent with the amplitude
ζ ≃ 2 × 10−9 at large scales measured by Planck mission . The slow-roll parameters,
e-foldings, as shown in figure 3. The predicted spectral index of the curvature perturbation
This results in a huge production of photons and a largely induced inflaton fluctuations at
prolongs the duration of inflation. Near the end of inflation, the inflaton speeds up again
and hence produces a large amount of photons and inflaton fluctuations. Finally, inflation
ends up with a rather complicated dissipation-fluctuation process. However, because of
the strong backreaction to the inflaton motion, the inflaton gradually reduces to zero field
value and inflation ends gracefully without undergoing preheating or typical perturbative
reheating. During the full course of a hybrid type of slow-roll and trapped inflation, the
vacuum energy is being drained by the production of photons and converted into radiation.
nearly steady thermal bath during inflation and exhaust the vacuum energy to end the
inflation gracefully .
on the power spectrum coming from the non-detection of PBHs . Note that the bound
MBH < 108g are strongly model dependent. The power spectrum induced by the photon
NANOGrav, and PPTA, as well as the projected SKA sensitivity.
Near the end of inflation the power spectrum shows high spikes where PBHs of much smaller
masses are likely to be copiously formed as well. We warn that in the present consideration
the power spectrum is no longer valid when its value becomes near unity. In this situation,
one must take into account the effects of gravitation, which could presumably dampen the
spikes to a consistent level .
We have given a novel model associated with trapped inflation in which high
stellarmass PBHs can be formed. Although we have designed a modified quadratic potential to
achieve this, we anticipate that many similar potentials can fulfill the requirements. To
realize our model, we have studied a case in the context of axion monodromy inflation.5 The
potential in single-field axion monodromy inflation has a monomial form with superimposed
modulations whose size is model-dependent, given by 
axion decay constant f , which are in general functions of ϕ. We have tuned the model
parameters to obtain the potential,
V (ϕ) =
where a = 2.24 and b = 0.712. With α = 18.4, φ0 = −10.5, and (dφ/dt)0 = 6.02 × 10−6,
well as the power spectrum of the curvature perturbation as shown in figure 2 and figure 4,
respectively. It would be interesting to perform a systematic search for the parameter space
in the modulations that allow for the formation of high stellar-mass PBHs. Also, we can
use existing astrophysical and cosmological constraints on a wide range of PBH masses to
assess axion couplings and modulations in axion monodromy inflation.
The produced photons also source tensor metric perturbation or gravitational waves,
2 da ∂
~ 2 hij =
Mp2 (−EiEj − BiBj )T T ,
where T T denotes the transverse and traceless projection of the spatial components of the
energy-momentum tensor of the gauge field. This equation can be simultaneously solved
with the numerical solution for the photon mode function in eq. (5). Instead of pursuing
this computationally heavy task, we estimate the tensor perturbation by using the results
based on analytic photon mode functions found in refs. [40–42], where the present energy
density per logarithmic k interval of the gravitational waves relative to the critical density
0.002 Mpc−1. In figure 5, we plot ΩGW h2 against the frequency f = 3 × 10−18eN0−N Hz
figure also shows the current upper limits on gravitational wave background made by the
pulsar timing array experiments EPTA , NANOGrav , and PPTA , and the
projected sensitivity of SKA radio telescope . The gravitational waves associated with
timing array sensitivity. Slightly lighter PBHs may source gravitational waves whose peak
shifts to higher frequencies within the reach of the future SKA observation. Then, a full
numerical calculation over a wide range of PBH masses should be carried out to assess the
detectability or to constrain the formation of PBHs.
In a trapped inflation, the backreaction due to particle production damps the motion
of the inflaton so that a successful inflation can be realized in a steep potential. This may
solve the long standing problem in the slow-roll inflation that a flat potential is required.
Moreover, the curvature perturbation induced by the particle production has distinctive
non-Gaussianities [40, 41, 47]. In the case of a quadratic inflation, the measurements of
due to the photon production at the end of inflation has been discussed [14–18]. Here we
leading to the production of high stellar-mass PBHs. These PBHs could be the cold dark
matter observed by the LIGO detectors through a binary BH merger. We thus emphasize
that PBHs are naturally produced in inflation with a trapping potential, although the
PBH mass range depends on model parameters. Future observations of non-Gaussianities
in CMB and large scale structures as well as direct measurements of gravitational wave
background will be very important for testing the trapped inflationary scenario.
This work was supported in part by the Ministry of Science and Technology, Taiwan, ROC
under Grants No. MOST104-2112-M-001-039-MY3 (K.W.N.) and No.
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