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In this paper we present a new scenario where massive Primordial Black Holes (PBH) are produced from the collapse of large curvature perturbations generated during a mild waterfall phase of hybrid inflation. We determine the values of the inflaton potential parameters leading to a PBH mass spectrum peaking on planetary-like masses at matter-radiation equality and producing abundances comparable to those of Dark Matter today, while the matter power spectrum on scales probed by CMB anisotropies agrees with Planck data. These PBH could have acquired large stellar masses today, via merging, and the model passes both the constraints from CMB distortions and micro-lensing. This scenario is supported by Chandra observations of numerous BH candidates in the central region of Andromeda. Moreover, the tail of the PBH mass distribution could be responsible for the seeds of supermassive black holes at the center of galaxies, as well as for ultra-luminous X-rays sources. We find that our effective hybrid potential can originate e.g. from D-term inflation with a Fayet-Iliopoulos term of the order of the Planck scale but sub-planckian values of the inflaton field. Finally, we discuss the implications of quantum diffusion at the instability point of the potential, able to generate a swiss-cheese like structure of the Universe, eventually leading to apparent accelerated cosmic expansion.
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arXiv:1501.07565v1 [astro-ph.CO] 29 Jan 2015
Massive Primordial Black Holes from Hybrid Inflation
as Dark Matter and the seeds of Galaxies
ebastien Clesse1, and Juan Garc´ıa-Bellido2,
1Namur Center of Complex Systems (naXys), Department of Mathematics,
University of Namur, Rempart de la Vierge 8, 5000 Namur, Belgium
2Instituto de F´ısica Torica UAM-CSIC, Universidad Auton´oma de Madrid, Cantoblanco, 28049 Madrid, Spain
(Dated: January 30, 2015)
In this paper we present a new scenario where massive Primordial Black Holes (PBH) are produced
from the collapse of large curvature perturbations generated during a mild waterfall phase of hybrid
inflation. We determine the values of the inflaton potential parameters leading to a PBH mass
spectrum peaking on planetary-like masses at matter-radiation equality and producing abundances
comparable to those of Dark Matter today, while the matter power spectrum on scales probed by
CMB anisotropies agrees with Planck data. These PBH could have acquired large stellar masses
today, via merging, and the model passes both the constraints from CMB distortions and micro-
lensing. This scenario is supported by Chandra observations of numerous BH candidates in the
central region of Andromeda. Moreover, the tail of the PBH mass distribution could be responsible
for the seeds of supermassive black holes at the center of galaxies, as well as for ultra-luminous X-rays
sources. We find that our effective hybrid potential can originate e.g. from D-term inflation with a
Fayet-Iliopoulos term of the order of the Planck scale but sub-planckian values of the inflaton field.
Finally, we discuss the implications of quantum diffusion at the instability point of the potential, able
to generate a swiss-cheese like structure of the Universe, eventually leading to apparent accelerated
cosmic expansion.
PACS numbers: 98.80.Cq
I. INTRODUCTION
A major challenge of present-day cosmology is the un-
derstanding of the nature of dark matter, accounting for
about thirty percent of the total energy density of the
Universe. Among a large variety of models, it has been
proposed that dark matter is composed totally or par-
tially by Primordial Black Holes (PBH) [1–6]. These
are formed in the early Universe when sufficiently large
density fluctuations collapse gravitationally. A threshold
value of δρ/ρ ∼ O(1) is a typical requirement to ensure
that gravity overcomes the pressure forces [7–15].
Several mechanisms can lead to the formation of PBH,
e.g. sharp peaks in density contrast fluctuations gen-
erated during inflation [16], first-order phase transi-
tions [17], resonant reheating [18], tachyonic preheat-
ing [19] or some curvaton scenarios [20–22]. Large curva-
ture perturbations on smaller scales than the ones probed
by CMB anisotropy experiments can also be generated
during inflation [5, 6, 23–28], e.g. for hybrid models end-
ing with a fast (in terms of e-folds of expansion) wa-
terfall phase. In this case, exponentially growing modes
of a tachyonic auxiliary field induce order one curvature
perturbations [16, 29, 30] and PBH can be formed when
they re-enter inside the horizon during the radiation era.
However, in the standard picture of hybrid inflation, the
corresponding scales re-enter into the horizon shortly af-
ter the end of inflation, leading to the formation of PBH
sebastien.clesse@unamur.be
juan.garciabellido@uam.es
with relatively low masses, MPBH <
O(10) kg. These
PBH evaporate in a very short time, compared to the age
of the Universe, and cannot contribute to dark matter to-
day. This process can nevertheless eventually contribute
to the reheating of the Universe [16].
Tight constraints have been established on PBH mass
and abundance from various theoretical arguments and
observations, like the evaporation through Hawking radi-
ation, gamma-ray emission, abundance of neutron stars,
microlensing and CMB distortions. It results that PBH
cannot contribute for more than about 1% of dark mat-
ter, except in the range 1018 kg <
MPBH <
1023 kg,
as well as for masses larger than around a solar mass,
M>
M1030 kg, under the condition that they do
not generate too large CMB distortions. It is also unclear
whether some models predicting a broad mass spectrum
of PBH can be accommodated with current constraints,
while generating the right amount of dark matter when
integrated over all masses.
In this paper, we present a new scenario in which the
majority of dark matter consists of PBH with a relatively
broad mass spectrum covering a few order of magnitudes,
possibly up to O(100) solar masses. The large curvature
perturbations at the origin of their formation are gen-
erated in the context of hybrid inflation ending with a
mild waterfall phase. This is a regime where inflation
continues for several e-folds (up to 50 e-folds) of expan-
sion during the final waterfall phase. Compared to the
standard picture of fast waterfall, important curvature
perturbations are generated on larger scales, that reen-
ter into the horizon at later times and thus lead to the
formation of PBH with larger masses.
2
More precisely, we consider for inflation an effective
hybrid-like two-field potential, having a nearly flat valley
where field trajectories are slowly evolving when scales
relevant for CMB anisotropies exit the Hubble radius. In
order to avoid a blue spectrum of scalar perturbations,
which is a generic prediction of the original hybrid model
in the false vacuum regime [31, 32], the effective poten-
tial has a negative curvature close to the critical point
below which the tachyonic instability develops [33]. The
slope of the potential at this critical point is nevertheless
sufficiently small for the final waterfall phase to be mild
and to last for typically between 10 and 50 e-folds of ex-
pansion. In this scenario, the second slow-roll parameter
gives the dominant contribution to the scalar spectral in-
dex, which can accommodate the recent constraints from
Planck [34, 35]. This scenario is similar to the case of a
mild waterfall phase with more than 50 e-folds of expan-
sion during the waterfall [36–40], but it must be seen as a
transitory case because observable scales exit the horizon
prior to the waterfall. The present scenario has therefore
strong similarities with the ones studied in Refs [16, 41].
In the context of a long waterfall phase, during the
first stage of the waterfall when the field dynamics is still
governed by the slope of the potential in the direction
of the valley, entropy perturbations due to the presence
of the auxiliary field grow exponentially and source the
power spectrum of curvature perturbations whose ampli-
tude can grow up to values larger than unity [37]. Then in
the second phase of the waterfall, field trajectories reach
an attractor and are effectively single field, so that curva-
ture perturbations becoming super-Hubble at this time
fall back to much lower values. Depending on the model
parameters, one predicts a broad peak in the power spec-
trum of curvature perturbations, whose maximal ampli-
tude can exceed the threshold value for the formation of
PBH.
By numerically solving the multi-field homogeneous
and linear perturbation dynamics during the waterfall,
and by cross-checking our result using the δN formal-
ism (both analytically and numerically), we calculate
the power spectrum of curvature perturbations at the
end of inflation. Particular care is given to take into
account the effect of quantum stochastic fluctuations of
the auxiliary field at the instability point, which can sig-
nificantly change the classical evolution during the wa-
terfall. From the spectrum of curvature perturbations,
the formation process of PBH is studied and their mass
spectrum is evaluated. Finally, the contribution of PBH
to the density of the Universe at matter-radiation equal-
ity is calculated, and we determine the parameter values
of the inflationary potential leading to the right amount
of dark matter. As mentioned later, we find that those
parameters can fit simultaneously with the observational
constraints on the curvature power spectrum from CMB
experiments. We argue that micro-lensing and CMB dis-
tortion constraints can be naturally evaded if PBH are
initially sub-solar and grow by merging after recombina-
tion until they acquire a stellar mass today. As a result,
the mass distribution of PBH could explain the excess of
few solar mass black holes in Andromeda that has been
observed recently [42–46]. Finally we discuss whether
PBH in the tail of the mass spectrum can serve as the
seeds of the supermassive black holes observed in galaxies
and quasars at high redshifts [47–52].
The paper is organized as follows: In Sec. II we re-
view the principal constrains on PBH abundances. The
effective model of hybrid inflation with a mild waterfall
is introduced in Sec. III. The field dynamics is described
and the curvature power spectrum is calculated in this
section. The formation of PBH from large curvature
perturbations produced during the waterfall is studied
in Sec. IV, where we also derive their mass spectrum
and their contribution to the density of the Universe
at matter-radiation equality. In Sec. V, we identify the
model parameter ranges leading to the right amount of
dark matter, with mass spectra evading the observational
constraints. In Sec. VI, we discuss how to embed our
effective inflationary potential in a realistic high-energy
framework. The level of CMB distortions induced by an
excess of power on small scales is evaluated in Sec. VIII
and compared to the limits reachable by future exper-
iments like PIXIE and PRISM. In Sec. VII we discuss
how the most massive PBH can be identified to the seeds
of the supermassive black holes in quasars at high red-
shifts, and in the center of galaxies today. We conclude
and present some interesting perspectives in Sec. IX.
II. PRIMORDIAL BLACK HOLES AND
OBSERVATIONAL CONSTRAINTS
Because primordial black holes are non-relativistic and
effectively nearly collisionless, they are good candidates
for dark matter. The mass spectrum of primordial black
holes is nevertheless severely constrained by several types
of observations, which are listed and briefly explained
below (for a review, see Ref. [53]).
1. Lifetime of primordial black holes: Due to the
Hawking radiation, PBH evaporate on a time scale
of the order of tev (M)G2M3/¯hc4, so that PBH
with a mass MPBH 5×1011 kg evaporate in
a time much shorter than the age of the Uni-
verse [2, 53] and therefore cannot contribute sig-
nificantly to dark matter today.
2. Light element abundances: PBH evaporation may
also have an effect on Big Bang Nucleosynthesis
(BBN). Only PBH of masses MPBH <
107kg evap-
orate before the BBN. More massive ones can af-
fect light element abundances through the emis-
sion of mesons and anti-nucleons; and through
the hadro-dissociation and photo-dissociation pro-
cesses. Strong bounds have been established on
PBH abundance from BBN [53], but those only
concern PBH of masses MPBH <
1011 kg which are
3
not good candidates for dark matter due to early
evaporation.
3. Extragalactic photon background: Evaporating
PBH at the present epoch can emit observable ex-
tragalactic gamma-ray radiation. The photon in-
tensity spectrum can be calculated. For instance,
the Hawking radiation produced by PBH with mass
MPBH 1013 kg is responsible for the emission
of 100 MeV radiation, which should have been
observed with the EGRET and Fermi Large Area
telescopes if ΩPBH >0.01 [53]. PBH cannot ac-
count for the totality of dark matter if MPBH <
7×1012 kg. Note however that this limit as-
sumes a uniform distribution of PBH throughout
the Universe, which is not realistic since dark mat-
ter clusters like galaxies, galaxy clusters and super-
clusters.
4. Galactic background radiation: PBH are expected
to have clustered with galactic halos, and thus there
should be also a galactic background of gamma ra-
diation, which should be anisotropic and in princi-
ple separable from the extra-galactic emission. The
constraints from galactic background radiation are
close but less competitive than BBN and extra-
galactic ones. A distinctive signature of PBH could
also be seen in the ratio of antiprotons to protons,
in the energy range between 100 MeV and 10 GeV,
in the galactic cosmic ray spectrum [53]. This gives
typically similar constraints on the abundance of
PBH.
5. Femtolensing of gamma-ray bursts: Compact ob-
jects can induce gravitational femto-lensing of
gamma-ray bursts. The lack of femto-lensing de-
tection in the Fermi Gamma-Ray Burst Monitor
experiment has provided evidence that in the mass
range 5 ×1014 1017 kg, PBH cannot account for
a large fraction of dark matter [54].
6. Star Formation: If star formation occurs in an en-
vironment dominated by dark matter, constituted
partially or totally of PBH, these can be captured
by stars, they sink to the center, and at the end of
the star evolution they destroy in a very short time
by accretion the compact remnant (a white dwarf
or a neutron star). The constraints resulting from
the observation of neutron stars and white dwarves
in globular clusters does not allow PBH to con-
stitute totally the dark matter in the mass range
1013 kg <
MPBH <
1019 kg [55].
7. Capture of PBH by neutron stars: In a similar
way, PBH can be captured by neutron stars which
are then accreted onto the PBH and destroyed in
a short time. Assuming large dark matter densi-
ties and low velocities, conditions that can be ful-
filled in the cores of the globular clusters, PBH can-
not account entirely for dark matter in the range
1015 kg < MPBH <1021 kg [56]. However, the
dark matter density inside globular clusters is not
known, and those constraints (as well as constraints
from star formation) are evaded if the dark matter
density is ρGlob.Cl.
DM <
102GeVcm3.
8. Microlensing surveys: If the dark matter galactic
halo is mostly composed of PBH, one expects grav-
itational micro-lensing events of stars in the Magel-
lanic clouds. The EROS micro-lensing survey and
the MACHO collaboration did not observe such
events and have put a limit on PBH abundance in
the range 1023 kg < MPBH <1031 kg [57, 58]. The
Kepler dada have permitted to extend this range
down to 4 ×1021 kg <
MPBH <
2×1023 kg [59].
9. CMB spectral distortions: X-rays emitted by gas
accretion near PBH modify the recombination his-
tory, which generates CMB spectral distortions and
CMB temperature anisotropies [60]. Distortions
are strongly constrained by the COBE/FIRAS ex-
periment. It results that PBH of MPBH >10M
cannot contribute to more than a few percent of
the dark matter; whereas solar mass PBH cannot
contribute to more than about 10% of dark matter.
However these bounds assume that PBH masses do
not change with time. But merging and accretion
since recombination can in principle lead to PBH
with masses significantly larger than 10Mtoday
while being sub-solar before recombination, thus
evading both micro-lensing and CMB constraints.
Another source of distortions comes from super ra-
diant instabilities of non-evaporating PBH, due to
the release with the expansion of the energy as-
sociated to exponentially growing photon density
around PBH. The non-detection of distortions puts
limits in the range 1022 kg <
MPBH <
1029 kg
for dark matter consisted of maximally-spinning
PBH [61].
The principal constraints on the fraction of dark mat-
ter due to massive PBH are summarized on Fig. 1. The
bounds from star formation and capture by neutron stars
are displayed as brown dotted lines, since they are easily
evaded if the dark matter density inside globular clus-
ters is sufficiently low. In this case, there exists a gap
between 1017 kg <
MPBH <
1023 kg where there is no
constraint, and thus where PBH can be identified to the
dark matter component. Finally, note that most lim-
its have been obtained under the implicit assumption of
a mono-chromatic distribution of PBH. In the scenario
proposed in this paper, the PBH mass spectrum is very
broad, covering typically five orders of magnitudes, and
it is still rather unclear how the current constraints must
be adapted for such broad mass spectra.
In the future, those constraints should be improved
by other experiments and observations: PBH passing
through stars can lead to detectable seismic signatures
(in the form of solar oscillations), induced by the squeeze
4
10-18
10-13
10-8
0.001
100
107
10-5
10-4
0.001
0.1
1
MPBHM
WPBHWDM
femto-lensing
EGB
NS capture
MACHO
EROS
FIRAS
WMAP3
PBH
FIG. 1. Limits on the abundance of PBH today, from ex-
tragalactic photon background (orange), femto-lensing (red),
micro-lensing by MACHO (green) and EROS (blue) and
CMB distortions by FIRAS (cyan) and WMAP3 (purple).
The constraints from star formation and capture by neu-
tron stars in globular clusters are displayed for ρGlob.Cl.
DM =
2×103GeVcm3(brown). The black dashed line corresponds
to a particular realization of our scenario of PBH formation.
Figure adapted from [56].
of the star in presence of the PBH gravitational field.
PBH of masses larger than 1018 kg are potentially ob-
servable [62]. Even if highly unlikely (1 event in 107
years for ρPBH =ρDM with MPBH 1012 kg), the
transit of PBH of masses MPBH >
1012 kg through or
nearby the Earth could be detected because of the seis-
mic waves they induce [63]. X-rays photons emitted by
non-evaporating PBH should ionize and heat the nearby
intergalactic medium at high redshifts. This produces
specific signatures in the 21cm angular power spectrum
from reionization, which could be detected with the SKA
radio-telescope [64]. For PBH of masses from 102M
to 108M, densities down to ΩPBH >
109could be
seen. A PBH transiting nearby a pulsar gives an impulse
acceleration which results in residuals on normally or-
derly pulsar timing data [65, 66]. Those timing residuals
could be detected with future giant radio-telescope like
the SKA. The signal induced by PBH in the mass range
1019 kg <
MPBH <
1025 kg and contributing to more
than one percent to dark matter should be detected [66].
Binaries of PBH forming a fraction of DM should emit
gravitational waves; this results in a background of grav-
itational waves that could be observed by LIGO, DE-
CIGO and LISA [67, 68].
Finally, the recent discovery by CHANDRA of tens of
black hole candidates in the central region of the An-
dromeda (M31) galaxy [42–46] provides a hint in favor
of models of PBH with stellar masses. As detailed later
in the paper, such massive PBH can be produced in our
model. The CMB distortions and micro-lensing limits
could be evaded if PBH were less massive at the epoch of
recombination and then have grown mostly by merging
to form black holes with stellar masses today.
III. HYBRID-WATERFALL INFLATION
It has been shown recently that the original non-
supersymmetric hybrid model [31, 32] and its most well-
known supersymmetric realizations, the F-term and D-
term models [69, 70], own a regime of mild waterfall [36–
40]. Initially the field tra jectories are slowly rolling along
a nearly flat valley of the multi-field potential. When tra-
jectories cross a critical field value, denoted φc, the po-
tential becomes tachyonic in the orthogonal direction to
the valley. In the mild-waterfall case, inflation continues
for more than 50 e-folds of expansion after crossing the
critical instability point and before tachyonic preheat-
ing [33] is triggered. This scenario has the advantage
that topological defects formed at the instability point
are stretched outside our observable patch of the Uni-
verse by the subsequent inflation.
According to Refs. [37–39], the mild waterfall can be
decomposed in two phases (called phase-1 and phase-2).
During the first one, inflation is driven only by the infla-
ton, whereas the terms involving the auxiliary field can be
neglected. At some point, these terms become dominant
and trajectories enter in a second phase. When the wa-
terfall lasts for much more than 50 e-folds, the observable
scales exit the Hubble radius in the second phase, when
trajectories are effectively single field and curvature per-
turbations are generated by adiabatic modes only. For
the three hybrid models mentioned above (original, F-
term and D-term), the observable predictions are conse-
quently modified and a red scalar spectral index is pre-
dicted (instead of a blue one for the original model fol-
lowed by a nearly instantaneous waterfall). If one denotes
by Nthe number of e-folds between horizon exit of the
pivot scale k= 0.05Mpc1and the end of inflation, the
scalar spectral index is given by ns= 1 4/N, too low
for being within the 95% C.L. limits of Planck. Only
a low, non-detectable, level of local non-gaussianitiy is
produced, characterized by fNL ≃ −1/N[37].
When inflation continues during the waterfall for a
number of e-folds close but larger than 50 e-folds, the
pivot scale becomes super-Hubble during the phase-1.
Trajectories are not effectively single-field, and entropic
perturbations source the curvature perturbations [37].
This leads to a strong enhancement in the scalar power
spectrum amplitude, whose thus cannot be in agreement
with observations.
In this paper, we focus on an intermediate case, be-
tween fast and mild waterfall. We consider the regime
where inflation continues for a number of e-folds be-
tween about 20 and 40 after crossing the instability point.
There is a major difference with the previous case: ob-
servable scales leave the Hubble radius when field tra-
jectories are still evolving along the valley, when the
usual single-field slow-roll formalism can be used to de-
5
rive accurately the observable predictions. Nevertheless,
in order to study the waterfall phase, we have also inte-
grated numerically the full multi-field dynamics, both a
the background and linear perturbation level.
In the following, we introduce the field potential and
derive the scalar power spectrum amplitude and spectral
index on scales that are relevant for CMB anisotropies.
Then we study the waterfall phase and calculate the
power spectrum of curvature perturbations on small
scales, and we show that for some parameters, the en-
hancement of power is so important that it leads later to
the formation of massive PBH.
A. Along the valley
Since the original hybrid potential with a constant plus
a quadratic term in φpredicts a blue-tilted scalar spec-
trum, we have considered a more general form for the
effective potential close to the critical point of instabil-
ity. Contrary to the original hybrid model, it exhibits a
negative curvature in order to generate a red spectrum,
plus a linear term in φto control the duration of the
waterfall phase. The embedding of this model in some
high-energy frameworks will be discussed in Sec. VI. The
considered two-field potential reads
V(φ, ψ) = Λ "1ψ2
M22
+(φφc)
µ1
(φφc)2
µ2
2
+2φ2ψ2
M2φ2
c.(1)
Initially, inflation takes place along the valley ψ= 0. As
shown in [71–73], there is no fine-tuning of initial fields
hidden behind this assumption. Below the critical value
φc, this potential develops a tachyonic instability, forcing
the field trajectories to reach one of the global minima,
located at φ= 0, ψ=±M. Apart from the negative
curvature and the additional linear term, the potential is
identical to the one of the original hybrid model.
The slope and the curvature of the potential at the
critical point are thus controlled respectively by the mass
parameters µ1and µ2. We assume that µ1is sufficiently
large compared to µ2for the slope along the valley to
be constant over the range of scales going from scales
relevant to CMB anisotropies down do scales that exit
the Hubble radius at the critical instability point. At
φ=φc, the slow-roll approximation is valid and the first
and second Hubble-flow slow-roll parameters are given
by
ǫ1φc=M2
P
2V
V2
=M2
P
2µ2
1
,(2)
ǫ2φc= 2M2
P"V
V2
V
V#= 2M2
P1
µ2
1
+2
µ2
2.(3)
where MPis the reduced Planck mass and a prime de-
notes the derivative with respect to the field φ. In the
regime of interest, µ1µ2and the scalar spectral index,
given by
ns= 1 2ǫ1ǫ2,(4)
is dominated by the contribution of the second slow-
roll parameter. The star index denotes quantities eval-
uated at the time twhen k=a(t)H(t) with k=
0.05Mpc1being the pivot scale used by Planck. One
thus gets
ns14M2
P
µ2
2
.(5)
If the scalar spectral index is given by the best fit value
from Planck [34], ns0.9603, one obtains
µ2=2MP
1ns10MP.(6)
The scalar power spectrum amplitude is also measured
by Planck and is given at the pivot scale by
Pζ(k) = H2
8π2M2
Pǫ1
(7)
Λµ2
1
12π2M6
Pk
kφcns1
(8)
= 2.21 ×109.(9)
The second equality is derived by using the Friedmann
equation in slow-roll H2V /3M2
P. This leads to a rela-
tion between the Λ and µ1parameters,
Λ = 2.21 ×109×12π2M6
P
µ2
1kφc
kns1
.(10)
Practically, since the duration of inflation depends on the
waterfall dynamics, k/kφccannot be known before a first
integration of the background dynamics. One needs also
to assume a reheating history. We consider for simplicity
the case of instantaneous reheating [74]. For the numeri-
cal implementation of Λ we use the following procedure:
(i) we first guess its value assuming k=kφc, (ii) we solve
numerically the 2-field dynamics and find the correspond-
ing k/kφc, (iii) Eq. (10) is used to guess a better value
of Λ. We proceed iteratively until the scalar power spec-
trum amplitude corresponds to the Planck measurement.
Usually three iterations give a good agreement, because
the waterfall dynamics in influenced only slightly by Λ,
as discussed thereafter.
B. Waterfall phase
1. Quantum diffusion at the critical point
Once the critical instability point has been crossed, the
waterfall phase takes place. We have assumed that the
6
classical 2-field dynamics is valid from the critical point,
given a tiny initial displacement from ψ= 0. In a realistic
scenario the quantum diffusion close to φcplays the role
of displacing the auxiliary field. This process can be more
or less efficient in different patches of the Universe, hav-
ing an effect on the subsequent classical dynamics during
the waterfall and on the resulting scalar power spectrum.
It is therefore important to take properly into account the
quantum diffusion of ψ.
The auxiliary field distribution over a spatial region is
a Gaussian whose width at the critical point of instability
can be calculated by integrating the quantum stochastic
dynamics of ψ[16, 36, 37]. It reads
ψ0phψ2i=Λ2φcµ1M
96π3/21/2
,(11)
where the brackets denote averaging in real space. It
must be also noticed that quantum diffusion only plays
a role very close to the critical instability point and that
the classical dynamics is quickly recovered. Quantum
effects taking place after crossing φcactually influence
only marginally the waterfall dynamics, and for simplic-
ity these have been neglected. But we have taken into ac-
count that different classical evolutions can emerge from
various values of ψc.
To do so, for each parameter set we consider, the clas-
sical two-field dynamics is integrated numerically over
200 realizations of ψc, distributed according to a Gaus-
sian of width ψ0. Then the mean scalar power spectrum
is obtained by averaging over all realizations. Since each
realization can be more or less efficient in producing PBH
(which is a non-linear process), the same averaging pro-
cedure is applied for the calculation of the fraction of the
Universe that collapses into PBH. In order to illustrate
the importance of this effect, the scalar power spectrum
for each of these realizations has been plotted on Fig. 3in
addition to the averaged spectrum and the one obtained
assuming ψc=ψ0.
Finally, note that the inflaton field φremains classical
during the waterfall phase and drives the Universe ex-
pansion in the regime where the stochastic dynamics of
ψcan be important. For a more precise investigation of
the stochastic dynamics in hybrid inflation, the interested
reader can refer to Ref. [75].
2. Background classical dynamics
In order to get the classical field trajectories and the
expansion during the waterfall, one needs to solve the
two-field dynamics governed by the Friedmann-Lemaˆıtre
equation
H2=1
3M2
P"˙
φ2
2+˙
ψ2
2+V(φ, ψ)#,(12)
and the Klein-Gordon equations
¨
φ+ 3H˙
φ+∂V
∂φ = 0,(13)
¨
ψ+ 3H˙
ψ+∂V
∂ψ = 0.(14)
These exact equations have been implemented numeri-
cally. But it is also possible to derive accurate analytical
solutions, by considering the usual slow-roll approxima-
tion where the kinetic terms and the second field deriva-
tives can be neglected.
As already mentioned, the waterfall can be decom-
posed in two phases, called phase-1 and phase-2. The
slow roll dynamics of both fields can be integrated in
each of them. Let us first introduce the notation
φφceξ, ψ ψ0eχ.(15)
During the waterfall, as long as the slow-roll approxima-
tion is valid, |ξ| ≪ 1. One therefore has φφc(1 + ξ)
and the Klein Gordon equations for the scalar fields in
the slow-roll approximation reduce to
3H˙
ξ≃ −
µ2
11 + 2µ2
1ψ2
M2φ2
c,(16)
3H˙χ≃ −
M22ξ+ψ2
M2,(17)
whereas the Friedmann-Lemaˆıtre equation is given by
H2=Λ
3M2
P
.(18)
During phase-1, the second term of Eqs (16) and (17) are
by definition negligible. At some time, the second term in
the r.h.s. of Eq. (16) becomes larger than unity and the
dynamics enters into phase-2. In the following, we repro-
duce in a straightforward way the results of Refs. [36–38]
on the waterfall dynamics and refer the reader to them
for the detailed calculation.
In phase-1, integrating the slow-roll equations gives the
field trajectories
ξ2=M2
4φcµ1
χ. (19)
If we define the e-fold time Nby imposing that N= 0
at the critical point φc, one gets also
N=ξµ1φc
M2
P
,(20)
so that the total number of e-folds in phase-1 is given by
N1=χ2φ1/2
cµ1/2
1M
2M2
P
,(21)
7
where
χ2ln φ1/2
cM
2µ1/2
1ψ0!(22)
is the value of χat the transition point between phase-
1 and phase-2. We focus only on the regime where the
temporal minimum of the potential, defined by the ellipse
of local minima in the ψdirection, ξ=ψ2/M2, is not
reached by field trajectories before the end of inflation.
By integrating the slow-roll equations in phase-2, one
gets that trajectories satisfy
ξ2=ξ2
2+M2
8φcµ1he2(χχ2)1i(23)
where
ξ2≡ − Mχ2
2µ1φc
(24)
is the value of ξat the transition between the two phases.
Inflation ends when the slow-roll approximation breaks
for the field ψ, at
ξend =M2
8M2
P
.(25)
Assuming that |ξend| ≫ |ξ2|(which is valid in the mild
waterfall case with N50) the total number of e-folds
in phase-2 is well approximated by
N2Mµ1/2
1φ1/2
4M2
Pχ1/2
2
.(26)
3. Power Spectrum of Curvature Perturbations
The power spectrum of curvature perturbations has
been calculated numerically by integrating both the clas-
sical exact homogeneous dynamics and the linear per-
turbations. For this purpose, we have used the method
of [76], similarly to what was done in [37] in the case of a
long waterfall phase. As a cross check, we have used the
δN formalism [77–79] that allows to derive good analyt-
ical approximations for the power spectrum of curvature
perturbations.
In the δN formalism, the scalar power spectrum at a
wavelength mode kis given by
Pζ(k) = H2
k
4π2(N2
+N2
),(27)
where Nand Nare defined as
N∂N f
i
∂φi
k
, N∂N f
i
∂ψi
k
,(28)
and Nf
icorresponds to the number of e-folds realized be-
tween an initially flat hypersurface at the time tkdefined
such that k/a(tk) = H(tk), and a final surface of uniform
energy density, which we take to be the hypersurface of
ξ=ξend1.
One therefore needs to calculate how the number of
e-folds varies in order to reach ξ=ξend when the initial
field values at the time tkare slightly shifted. Actually,
one can show that the phase-2 is effectively single field
so that one can focus only on modes leaving the Hubble
radius during phase-1. It is useful to decompose N=
N1
+N2
and N=N1
+N2
, where the first and
second terms correspond to the variation of the number of
e-folds realized in phase-1 and phase-2 respectively. The
number of e-folds in phase-1 from arbitrary field values
(ξi, χi) is given by
N1=µ1φc(ξ2i ξi)
M2
P
(29)
where ξ2i ≡ −pξ2
i+M2(χ2χi)/(4µ1φc) is the value
of ξat the transition between phase-1 and phase-2. One
recovers ξ2i =ξ2when setting ξ2
i=M2χi/(4φcµ1). It
has been shown [37] that the number e-folds in phase-2
from field values (ξ2i, χ2) gives a subdominant contri-
bution to Nand N . The importance of the errors
induced by neglecting N2
and N2
will be nevertheless
discussed later. Under those conditions, one therefore
gets
Nµ1
M2
P
, NM2
8M2
Pξ2ψk
,(30)
with ψk=ψ0exp χk,χk= 4φcµ1ξ2
k/M2and ξk=
M2
P(N1+N2Nk)/(µ1φc). For 10 <
Nk<
50, it
follows that NN. The power spectrum amplitude
then can be approximated by
Pζ(k)ΛM2µ1φc
192π2M6
Pχ2ψ2
k
.(31)
For the mode exiting the Hubble radius exactly at the
transition between phase-2 and phase-1, one can obtain
Pζ(k, tkt1,2)Λµ2
1
48π2p2M6
Pχ2
,(32)
whereas for modes exiting the horizon deeper in phase-1,
one obtains an exponential increase of the power spec-
trum amplitude
Pζ(k) = Pζ(k, tkt1,2)
×exp "2χ2 1(Nend Nk)2
N2
1!#.(33)
1This choice does not correspond exactly to a hypersurface of
constant density ρend, but as explained in [37] the difference in
e-folds between the two hypersurfaces is marginal and can be
safely neglected.
8
It is maximal at the critical point of instability. The mild-
waterfall therefore induces a broad peak in the scalar
power spectrum for modes leaving the horizon in phase-1
and just before the critical point. The maximal ampli-
tude for the scalar power spectrum is given by
Pζ(kφc)ΛM2µ1φc
192π2M6
Pχ2ψ2
0
.(34)
Depending of the model parameters, the curvature per-
turbations can exceed the threshold value for leading to
the formation of PBH.
This calculation was performed assuming that ψc=
ψ0. It is important to remark that for values of ψ0
and Λ given by Eqs. (11) and (10), one gets that N1,
N2and the amplitude of the scalar power spectrum
depend on a concrete combination of the parameters,
ΠM(φcµ1)1/2/M2
P, plus some dependence in χ2. But
χ2itself depends only logarithmically on Π. As a result,
χ2varies by no more than 10% for relevant values of Π2.
The parameters φc,µ1and Mappear to be degenerate
and all the model predictions only depend on the value
given to Π. Nevertheless, Eq. (34) implicitly assumes
that field values are strongly sub-Planckian. In the op-
posite case, when φcMMP, we find important
deviations and the numerical results indicate that the wa-
terfall is longer by about two e-folds and that the power
spectrum is enhanced by typically one order of magni-
tude, compared to what is expected for sub-Planckian
fields and for identical values of Π2.
As a comparison between numerical and analytical
methods, we have plotted in Fig. 2the power spectrum of
curvature perturbations for Π2= 300 and sub-Planckian
fields, by using the analytical approximation given by
Eq. (31), by using the δN formalism including all terms
(i.e. the contributions from phase-1 and phase-2) in N
and N, and by integrating numerically the exact dy-
namics of multi-field perturbations. As expected we find
a good qualitative agreement between the different meth-
ods. Nevertheless, one can observe about 20% differences
when using the analytical approximation, which actually
is mostly due to the fact that N2
has been neglected. In
the rest of the paper, we use the numerical results for a
better accuracy.
In Figure 3the power spectrum of curvature perturba-
tions has been plotted for different values of the param-
eters. This shows the strong enhancement of power not
only for the modes exiting the Hubble radius in phase-1,
but also for modes becoming super-horizon before field
trajectories have crossed the critical point. One can ob-
serve that if the waterfall lasts for about 35 e-folds then
the modes corresponding to 35 <
Nk<
50 are also af-
fected. As expected one can see also that the combi-
nation of parameters Π drives the modifications of the
power spectrum. We find that it is hard to modify inde-
pendently the width, the height and the position of the
peak in the scalar power spectrum.
Finally, for comparison, the power spectra assuming
ψc=ψ0and averaging over a distribution of ψcvalues
-70
-60
-50
-40
-30
-20
-10
0
10-10
10-8
10-6
10-4
0.01
1
Nk
PΖHkL
FIG. 2. Power spectrum of curvature perturbations for pa-
rameters M=φc= 0.1MP,µ1= 3 ×105MP. The solid curve
is obtained by integrating numerically the exact multi-field
background and linear perturbation dynamics. The dashed
blue line is obtained by using the δN formalism. The dotted
blue line uses the δN formalism with the approximation of
Eq. (31).
are displayed. They nearly coincide for Π2<
300 but we
find significant deviations for larger values.
FIG. 3. Power spectrum of curvature perturbations for pa-
rameters values M= 0.1MP,µ1= 3 ×105MPand φc=
0.125MP(red), φc= 0.1MP(blue) and φc= 0.075MP(green),
φc= 0.1MP(blue) and φc= 0.05MP(cyan). Those pa-
rameters correspond respectively to Π2= 375/300/225/150.
The power spectrum is degenerate for lower values of M, φ
and larger values of µ1, keeping the combination Π2con-
stant. For larger values of M, φcthe degeneracy is broken:
power spectra in orange and brown are obtained respectively
for M=φc=MPand µ1= 300MP/225MP. Dashed lines
assume ψc=ψ0whereas solid lines are obtained after av-
eraging over 200 power spectra obtained from initial con-
ditions on ψcdistributed according to a Gaussian of width
ψ0. The power spectra corresponding to these realizations are
plotted in dashed light gray for illustration. The Λ parame-
ter has been fixed so that the spectrum amplitude on CMB
anisotropy scales is in agreement with Planck data. The pa-
rameter µ2= 10MPso that the scalar spectral index on those
scales is given by ns= 0.96.
9
IV. FORMATION OF PRIMORDIAL BLACK
HOLES
In this section, we calculate the mass spectrum of PBH
that are formed when O(1) curvature perturbations orig-
inating from a mild-waterfall phase re-enter inside the
horizon and collapse. Furthermore we show that the
abundance of those PBH can coincide with dark matter
and can evade the observational constraints mentioned
in Sec. II.
The mass of a PBH whose formation is associated to a
density fluctuation with wavenumber k, exiting the Hub-
ble radius |Nk|e-folds before the end of inflation, is given
by [16]
Mk=M2
P
Hk
e2Nk,(35)
where Hkis the Hubble rate during inflation at time tk.
In our model, HkpΛ/3M2
P.
Assuming that the probability distribution of density
perturbations are Gaussian, one can evaluate the fraction
βof the Universe collapsing into primordial black holes
of mass Mat the time of formation tMas [7]
βform(M)ρPBH (M)
ρtot t=tM
(36)
= 2 Z
ζc
1
2πσ eζ2
2σ2dζ(37)
= erfc ζc
2σ.(38)
In the limit where σζc, one gets
βform(M) = 2σ
πζc
eζ2
c
2σ2.(39)
The variance σof curvature perturbations is related to
the power spectrum through hζ2i=σ2=Pζ(kM), where
kMis the wavelength mode re-entering inside the Hubble
radius at time tM.
For the study of PBH formation in our model, the
range of masses has been discretized and the value of
βform has been calculated for mass bins ∆M. This corre-
sponds to PBH formed by the density perturbations reen-
tering inside the horizon between tMand tM+∆M. We
have considered mass bins whose width corresponds to
one e-fold of expansion between these times, i.e. Nk=
∆ ln k= (∆ ln M)/2 = 1. This is sufficiently small for the
power spectrum to be considered roughly as constant on
each bin. Thus one gets σ(Mk) = pPζ(k)∆ ln k.
In the following, we use Eq. (38) instead of Eq. (39), so
that the results are accurate when σ>
ζc. Calculating
ζchas been the sub ject of intensive studies [7–15], using
both analytical and numerical methods. An analytical
estimate has been determined recently using a three-zone
model to describe the PBH formation process [7]. For
a given equation of state parameter wat the time of
formation, it is given by
ζc=1
3ln 3(χasin χacos χa)
2 sin3χa
,(40)
with χa=πw/(1 + 3w). In the present scenario, PBH
are formed in the radiation dominated era and one can
set w= 1/3, leading to ζc0.086. Different values have
been obtained with the use of numerical methods, but
it seems a general agreement that ζclies in the range
0.01 <
ζc<
1 (see Fig. 3 of [7] for a comparison between
different methods). For the sake of generality, ζcwill be
kept as a free parameter.
10-20
10-15
10-10
10-5
1
105
10-16
10-14
10-12
10-10
10-8
10-6
MPBHM
Βform
10-20
10-15
10-10
10-5
1
105
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
MPBHM
Βeq
FIG. 4. PBH abundances at the time of formation βform (M)
(top panel) and at matter-radiation equality β(M, Neq ) (bot-
tom panel). The color scheme corresponds to the parame-
ters given in Fig. 3. The blue dashed curve is obtained for
Π2= 300 but with M=φc= 0.01MPand µ1= 3 ×108MP,
illustrating that PBH masses can be made arbitrarily large
for a given value of Π2. The critical curvature ζchas been
set so that the right amount of dark matter has been pro-
duced at matter-radiation equality. Values of ζcare reported
in Table IV
In our scenario of mild waterfall, the peak in the power
spectrum of curvature perturbations is broad and cov-
ers several order of magnitudes in wavenumbers. There-
10
fore, instead of a distribution of black holes that would
be close to monochromatic, which is easy to evolve in
the radiation era, one expects that PBH have a broad
mass spectrum and form at different times in the radi-
ation era. Since the energy density associated to PBH
of mass Mdecreases like a3due to expansion, the
contribution of PBH to the total energy density in the
radiation era grows like a. As a result, at the end of
the radiation era, PBH with low masses, forming earlier,
contribute more importantly to the total energy density
than more massive ones, forming later, given identical
values of βform.
In addition, one must consider that a fraction the PBH
is absorbed by the formation of more massive ones at
later times. This merging process is subdominant as long
at β(M)1, but has nevertheless been considered in the
calculation of the ratio ΩPBH (zeq )ρPBH(zeq )tot (zeq )
at matter-radiation equality.
Taking account those considerations, during the
radiation-dominated era, the fraction of the Universe
that has collapsed into primordial black-holes of mass
Mkevolves as
dβ(Mk, N (t))
dN=β(Mk, N (t)) 1βform(Mt).(41)
The first term is due to cosmic expansion and the second
term represents the fraction of PBH of mass Mkthat are
absorbed by the formation of PBH with larger mass Mt
at the time t. Since we adopt a discretization in ln M
(and equivalently in ln kand Nk), it is convenient to use
the e-fold time N(t) instead of the cosmic time. Note
that we neglect evaporation through Hawking radiation
since it is relevant only for PBH with very low masses
that are formed immediately after inflation. These are
very subdominant in our model due to the duration of
the waterfall. In order to get βeq β(Mk, N (teq)), this
equation must be integrated over cosmic history, from
the time of PBH formation until matter-radiation equal-
ity. For all the considered curvature power spectra, the
formation of PBH stops before Neq (corresponding to
ln(aeq/a0)≃ −8), since the variance of curvature pertur-
bations can be close or overpass the threshold value only
in the range 40 <
Nk<
10.
The total density of PBH at radiation-matter equality
is obtained by integrating βeq over masses:
PBH(zeq ) = ZMteq
0
β(M, Neq )d ln M. (42)
Eqs. (41) and (42) have been solved numerically using
bins ∆N= 1, corresponding to ∆ ln M= 2. At matter-
radiation equality one has Ωmatter(teq ) = 0.5 and PBH
constitute the totality of the dark matter if ΩPBH(teq)
0.42, the rest coming from baryons. For simplicity we
have neglected the matter contribution to the Universe
expansion in the radiation era. This effect is only impor-
tant close to matter-radiation equality, when all PBH are
formed, and it is expected to be compensated by a small
variation of ζc.
For the parameter sets considered in Fig. 3, we have
found the value of ζcthat give rise to the right amount of
dark matter. They are reported in Table IV. This must
not be seen as an accurate result, because the matter
contribution to the Universe’s expansion is not accounted
for in Eq. (41) whereas it is not negligible in the last few
e-folds before reaching matter-radiation equality. This
effect reduces the value of βeq, which must be compen-
sated by a lower value of ζcto get the right amount of
dark matter (thus values ζccfid of a few tens can still
be seen as realistic). The corresponding βform and βeq
functions are represented in Fig. 4. As expected, βform
is much smaller than βeq (remember that βa). It
takes larger values for massive black holes that are pro-
duced after a long waterfall phase, since they are formed
later and thus if we identify them to dark matter, βmust
grow up to values of O(0.1) in a fewer number of e-folds.
The amplitude of the peak in βform is controlled by Π2,
but also by the value of µ1. Indeed, to larger values of µ1
correspond lower energy scales for inflation, and it results
that PBH are formed at higher redshifs.
Π2(µ1, M, φcin MP)ζccfid µdist ydist
375 (3 ×105,0.1,0.125) 102.4 2.7×1031.3×105
300 (3 ×105,0.1,0.1) 22.05 1.6×1075.2×109
300 (3 ×108,0.01,0.01) 20.2 1.6×1075.2×109
300 (300,1,1) 49.60 2.6×1065.6×109
225 (3 ×105,0.1,0.075) 2.337 4.9×1095.4×109
225(225,1,1) 6.060 4.0×1094.6×109
150 (3 ×105,0.1,0.05) 0.0567 4.0×1094.6×109
TABLE I. Critical value ζcof curvature fluctuation (2nd
column) leading to PBH formation with ΩPBH (zeq) = 0.42
at matter radiation equality, for several sets of the model
parameters (1st column). The fiducial value is ζcfid = 0.086,
according to the three-zone model of [7]. In 3rd and 4th
columns are reported the corresponding distortion µand y
parameters (see Sec. VIII).
The mass range for PBH is very broad, 1020M<
MPBH <
105M. But given one set of parameter, the
mass spectrum typically covers 3-5 order of magnitudes
at matter-radiation equality. Given Π2, we find that
PBH can be made arbitrarily massive by increasing µ1
and reducing Mand φc. This lowers the energy scale of
inflation and thus increases PBH masses, but this does
not affect importantly the shape of the mass spectrum.
Therefore it is easy to find parameters for which the mass
spectrum peaks in the range where there is no solid ob-
servational constraints. It is also possible that the peak
in the mass spectrum is located on planet-like masses at
recombination (so that CMB distortion constraints are
satisfied), but evade micro-lensing limits of PBH abun-
dances if merging induces their growth by more than two
or three orders of magnitudes during cosmic history.
Finally, the width of the peak in βeq is reduced for
lower values of Π2, as expected given that it is related to
11
the broadness of the peak in the scalar power spectrum.
It is therefore possible, in principle, to control this width,
but note that the range where Π2can vary is rather lim-
ited by the value of ζc, which needs to be realistic.
V. CONSTRAINTS ON THE INFLATIONARY
POTENTIAL
Given the strong modifications of the power spectrum
of curvature perturbations when varying the potential
parameters, modifications that are amplified when con-
sidering the mass spectrum of PBH, the scenario we pro-
pose predicts that the inflation potential parameters obey
to very specific combinations. As already mentioned, µ1,
φcand M2are degenerate and we thus can consider their
combination within a unique parameter Π. Nevertheless,
this is only valid for sub-Planckian fields, and therefore
we have also calculated scalar power spectra and BPH
mass spectra for the specific values M=MP, φc= 0.1MP
and φ=M=MP.
First, we find that values
Π2M2φcµ1
M4
P
>
400 (43)
are not allowed observationnally, because the waterfall
phase is too long and the increase of power arises already
on CMB anisotropy scales. On the other hand, for values
Π2<
200 (44)
the curvature perturbations are not sufficiently enhanced
during the waterfall and the model cannot produce the
right abundances of PBH, except if ζctakes unrealisti-
cally low values. Then, for Π2>
350 we find that ζc
must be much larger than unity, which is also unreal-
istic. It results that the model works for very specific
values of Π2, within the range 200 <
Π2<
350.
Since Λ M6
P2
1, this bound on Π2gives a maximal
energy scale for inflation if the field values are restricted
to be sub-planckian, or close to the Planck mass (which
is expected in SUSY hybrid models to avoid important
corrections of supergravity),
Λ1/4<
8×103MP2×1016 GeV.(45)
This scale is close to the GUT energy scale. From the
scalar power spectrum only, there is no lower bound on
the energy scale of inflation (other than the usual BBN
constraints). It can be arbitrarily low if the parameters
Mand φcare much lower than the planck mass. The
maximal value for the tensor to scalar ratio (requiring
Planck-like values of M) is given by
r= 16ǫ18M2
P
µ2
1
<
0.08,(46)
it is somewhat below the present limits, but in the range
of detectability of future CMB polarization experiments
like COrE+[80].
A similar bound arises from the observational con-
straints on PBH abundances. Values of Π2>
400
combined with sub-Planckian fields will generate PBH
with masses larger than Mat matter-radiation equal-
ity, which is ruled out by the limits on CMB distortions.
Furthermore, as shown in Sec. VIII, additional distor-
tions are produced in this case because of higher power on
Silk-damped scales and they should have been detected
by FIRAS.
VI. EMBEDDING IN MORE REALISTIC
INFLATION MODELS
The effective potential given in Eq. (1) has a tiny slope
and a negative curvature in the valley direction, close to
the critical point of instability. This is opposite to the
original hybrid model where the curvature is positive.
This potential can nevertheless arise in the framework of
hybrid models having a flat direction lifted up by loga-
rithmic radiative corrections. The most famous examples
are the supersymmetric F-term [69] and D-term mod-
els [70, 81]. After a Taylor expansion around φc, the
Coleman-Weinberg logarithmic corrections indeed lead
to a positive linear term and a negative quadratic term
in the effective potential. In this section, we examine
whether our model can be embedded in realistic high-
energy framework, considering the parameter space pro-
ducing a mild waterfall phase lasting a few tens of e-folds.
A. F-term supersymmetric model
The F-term hybrid model is based on the superpoten-
tial
WFterm =κS ¯
ΦΦ M2,(47)
where Sis a singlet chiral superfield and ¯
Φ,Φ is a pair
of chiral superfields charged under U(1)BL. Along the
D-flat direction, the SUSY vacuum is reached at hSi= 0
and |hΦi| =|h¯
Φi| =M. One can introduce the real
canonically normalized inflaton field φ2|S|. The
effective potential along the flat direction ¯
Φ = Φ = 0 is
then given by
VFterm(φ) = κ2M41 + κ2¯
N
32π22 ln κ2M2x
Q2
+ (x+ 1)2ln 1 + 1
x+ (x1)2ln 11
x
aSφ
2κM2.(48)
The second term represents the radiative corrections to
the tree-level potential, xφ2/2M2,Qis a renormaliza-
tion scale and ¯
Nis the dimensionality of the representa-
tion to which ¯
Φ,Φ belong. The third term is a possible
soft SUSY-breaking term, which we first consider to be
12
negligible. We have neglected supergravity corrections to
the potential. They are negligible as long as the fields are
much lower than the reduced Planck mass but could lift
up the potential for Planck-like values. The instability
point is located at φc=2M.
After expanding in Taylor series the F-term potential
close to φc, one recovers the potential of Eq. (1) and one
can identify (in the limit of vanishing soft-term)
Λ = κ2M4,(49)
1
µ1
=κ2¯
Nln 2
8π2,(50)
1
µ2
2
=κ2¯
N
32π23
2ln 2.(51)
Two of the three F-term parameters can then be deter-
mined by requiring that the scalar power spectrum am-
plitude and spectral index are in agreement with CMB
observations. The third parameter can be determined
with the requirement of a mild waterfall, which is trans-
lated in constraints on Π2. However, we find that in order
to have Π 200 as well as M<
MPto avoid supergravity
corrections, then one must satisfy ¯
N>
O(105) which is
very unrealistic since this parameter denotes the dimen-
sionality of the representation of the superfields ¯
Φ,Φ.
Therefore it is not possible to embed our effective po-
tential in the standard version of the F-term scenario.
Nevertheless, if the potential is protected by a gauge
symmetry, one can allow the fields to take values of the
order of the reduced Planck mass. Then one can satisfy
Π2200 and ¯
N∼ O(1) simultaneously.
Another possibility is that the soft SUSY-breaking
term plays the role of reducing the slope of the poten-
tial close to the instability point, while the curvature is
unchanged. This extends the duration of the waterfall
phase, and one can identify
1
µ1
=κ2¯
Nln 2
8π2aS
2κM2(52)
However, if one imposes that Mto be lower than the
Planck mass and ¯
N∼ O(1), then the soft term in µ1
needs to be fine tuned to the term arising from radiative
corrections, at least at the percent level.
Finally, let us mention the possibility of a next-to-
minimal form for the K¨ahler potential
K=|S|2+kS|S|4
4m2
P
,(53)
where the parameter kScan be either positive or neg-
ative. In the latter case, this induces a negative mass
term 3kSH2|S|2in the effective potential [82], and one
could identify µ2
2=M2
P/kS, the other contributions to
µ2being subdominant. This new parameter controls the
value of the spectral index. As a result, larger values
of µ1and sub-Planckian field values are allowed for the
F-term model. For Π2200, this gives
κ5×1017, M 1.6×1011 MP,(54)
but then the energy scale of inflation is very low, at the
GeV scale, and the PBH are much too massive, MPBH
1022M.
B. Loop/D-term inflation
Radiative corrections of a flat direction can give rise to
a potential of the form
Floop = Λ 1 + λ1ln λ2φp
Mp
P.(55)
D-term inflation, for instance, belongs to this class. After
expanding around φc, one can identify
1
µ1
=1MP
φc
,1
µ2
2
=1M2
P
2φ2
c
(56)
that does not depend on λ2. The effect of pcan also be
incorporated by a redefinition of λ11. Then the
relevant value of λ1,φcand Λ can be derived from the
measurements of the scalar spectrum amplitude and tilt
and by imposing Π2200 . In particular, one finds that
Π2=2M2
(ns1) (57)
and therefore if ns= 0.96 one gets Π2= 200 for
M2MP. We therefore conclude that, as for the F-
term model, one requires field values close to the Planck
scale. However, note that only the value of Mneeds
to be at the Planck scale, the critical value φccan take
much lower values, compensated by larger values of µ1.
Supergravity corrections along the valley can therefore
be subdominant. In the case of the D-term model, the
superpotential reads
WDterm =κS ¯
ΦΦ (58)
and M2is identified to the Fayet-Iliopoulos term ξF I in
the D-term
D=g
2Φ2¯
Φ2+ξF I .(59)
One can also identify λ1=g2/(16π2), φc=ξF I g/(2κ)
and Λ = g2ξF I /2. For the scalar power spectrum ampli-
tude to agree with CMB data, in addition to a Planck-
like Fayet-Iliopoulos term, one therefore needs a large
coupling κ105contrary to the usual regime where
κ102. The coupling gmust be sufficiently small to
keep φcsub-Planckian.
13
C. Some alternatives
One can mention some alternatives for the embedding
of the effective potential in a realistic scenario: dissipa-
tive effects can be invoked to reduce the scalar spectral
index, as proposed in [83], and to allow larger values
of µ2and µ1in loop inflation. Inflexion point models
where inflation terminates with a waterfall phase could
be good candidates if the position of the critical point is
tuned to be very close to a flat inflexion point. Another
interesting possibility is the natural inflation potential
V(φ) = Λ[1 + acos(φ/f )], coupled to an auxiliary sec-
tor that triggers a waterfall phase close to the maximum
of the potential, where the potential is flat with a neg-
ative curvature. Such a model has been studied in [84]
assuming instantaneous waterfall.
VII. THE SEED OF SUPERMASSIVE BLACK
HOLES
The center of galaxies is believed to host supermas-
sive black holes (SMBH) of mass going from 106Mto
109M. These are thought to descent from less massive
BH seeds in quasars at high redshifts. But the existence
of SMBH at redshift up to z>
8 [47–52] remains a mys-
tery. Their existence as fully formed galaxies before 500
Myr is a challenge for standard ΛCDM model. It is ex-
tremely difficult to form such a massive BH so quickly
from stellar evolution, and several proposals suggest that
they are built up from smaller BH that act as seeds of
the galactic SMBH [85–90].
Assuming uninterrupted accretion at the Eddington
limit, BH seeds of at least 103Mare needed at z15.
Our scenario provides a mechanism for the formation of
those seeds, in the tail of the BH mass distribution. As
already mentioned, abundance of stellar-mass PBH prior
recombination is severely constrained. However, sub-
stellar PBH can grow by merging and it is possible that
a significant fraction of the smaller mass BH grow to be-
come intermediate mass black holes (IMBH) at redshift
z>
15. We expect the mechanisms of accretion for an
initial broad spectrum of PBH masses to be very com-
plex, so here we have adopted for simplicity the naive
prescription that PBH grow by a factor fmerg between
matter-radiation equality and the late Universe, indepen-
dently of their mass. Typically we find that fmerg >
103
for the model to pass both CMB distortions and micro-
lensing constraints. If the PBH spectrum peaks on stel-
lar masses at late time, as for the scenario displayed on
Fig. 1, we find that β(104M)105. These rare seeds
then can merge and accrete matter to form SMBH. It is
also possible to form SMBH at high redshifts with even
more massive PBH seeds in the tail of the spectrum. So
in this scenario, it is easy to get a number of SMBH
roughly 1 for 1012 stellar-mass BH in galaxies, which is
expected for a PBH dark matter component. Our model
therefore predicts that i) supermassive black holes should
be observed at the center of galaxies at very early times,
and ii) their mass distribution should follow a gaussian
decrease.
Moreover, within a generic broad mass distribution
of PBH, as produced in our scenario, it is natural that
PBH formed in the early universe and cluster during the
radiation era [91, 92]. Furthermore it is possible that
a significant fraction of the smaller mass BH grow to
become intermediate mass black holes (IMBH), which
could be responsible for the observed ultra-luminous X-
ray sources [93–96].
VIII. CMB DISTORTIONS
By increasing the amplitude of the scalar power spec-
trum on scales in the range 8Mpc1<
k<
104Mpc1,
our model induce potentially observable spectral distor-
tions of the CMB black-body spectrum. This range of
scales corresponds to 55 <
Nk<
48 (assuming
N50). For some of the considered parameters, the
curvature perturbations are enhanced within that range
(see Fig. 3).
CMB distortions are produced when the thermal equi-
librium is broken due to some energy injection before the
recombination, even if electrons and ions can remain in
thermal equilibrium due to Coulomb collisions. The en-
ergy injection can be due to several processes, such as
the decay or the annihilation of relic particles [97]. The
Silk damping leads also to some energy injection from
the dissipating acoustic waves [98–103], and the mag-
nitude of this effects is related to the amplitude of the
density perturbations. The increase of power on small
scales induced by a mild waterfall can therefore result
in higher energy injection to the CMB monopole, result-
ing is enhanced spectral distortions. Distortion spectra
for a scenario of mild waterfall were calculated in [104]
and could be seen by the Primordial Inflation Explorer
(PIXIE) [105] and the Polarized Radiation Imaging and
Spectroscopy Mission (PRISM) [106, 107]. Those exper-
iments are expected to improve by several order of mag-
nitudes the present limits on the signal intensity in each
frequency bin they probe, with
δIPIXIE
ν= 5 ×1026 Wm2Sr1Hz1,(60)
and
δIPRISM
ν= 6.5×1027 Wm2Sr1Hz1.(61)
In this section, we have reproduced the results of [104]
for our effective potential and for relevant parameters
in the context of PBH production. For this purpose,
we have used a modified version of the idi stort tem-
plate [103], which solves the Kompannets equations and
calculates the spectrum of CMB-distortions. The code
has been modified to allow any shape of the primordial
scalar power spectrum.
Distortions can be of µ-type, y-type and intermediate
i-type depending on when during the cosmic history the
14
PIXIE
PRISM
1
10
100
1000
10-28
10-26
10-24
10-22
10-20
Ν@GHzD
ÈDIÈ @Wm-2Sr-1Hz-1D
FIG. 5. Total spectrum of CMB distortions for same param-
eters and colors as in Figs. 3 and 4 (brown and green curves
are superimposed, undistinguishable from standard inflation
with ns= 0.961 and no running). The 1σlimits for PIXIE
and PRISM, see Eqs. (60) and (61), are also represented.
thermal equilibrium is broken. The importance of dif-
ferent types is usually encoded in the so-called µand y
parameters. The present limits from COBE-FIRAS are
µ < 9×105and y < 1.5×105and the objective of
PIXIE/PRISM is to improve this limit by about three
orders of magnitudes.
In addition to distortion spectra, µand yvalues have
been calculated for the parameters in Table IV. The cor-
responding spectra are displayed in Fig. 5. We find that
the distortion signal can be enhanced by several order
of magnitudes compared to the standard case where the
nearly scale-invariant scalar power spectrum can be ex-
tended down to small scales. As expected the effect is
maximal for Π 300 whereas the spectrum cannot be
distinguished from the standard case when Π <
200.
Nevertheless, for Π2220 the enhancement is about
10%. We therefore conclude that if PBH are identified
to dark matter and if ζctakes reasonable values, corre-
sponding to Π2200, the induced spectral distortions
pass the present constraints but are sufficiently impor-
tant to be detected by a PRISM-like experiment. Our
model therefore has a very specific prediction and could
be tested with future observations.
IX. CONCLUSION
We have proposed a model where dark matter is com-
posed of massive primordial black holes formed in the
early Universe due to the collapse of large curvature fluc-
tuations generated during a mild-waterfall phase of hy-
brid inflation. This regime is transitory between the
usual fast-waterfall assumption and the mild-waterfall
case with more than 50 e-folds of expansion realized af-
ter the crossing of the critical instability point of the
potential. In our scenario, the waterfall lasts between
20 and 40 e-folds. The classical field trajectories and
the power spectrum of curvature perturbations have been
calculated both by using analytical approximations and
by solving numerically the exact background and linear
perturbation dynamics. The quantum diffusion close to
the instability point has been accounted for by consider-
ing and averaging over many possible realizations of the
auxiliary field at the instability point, distributed accord-
ingly to the quantum stochastic treatment of this field,
whereas the inflaton itself remains classical.
Once the potential parameters are chosen to fit with
CMB anisotropy observations, we have shown that a
quantity combining the position of the critical instability
point, the position of the global minima of the potential
and the slope of the potential at the critical point, con-
trols the duration of the waterfall, the peak amplitude
and its position in the power spectrum of curvature per-
turbations. This parameter therefore controls also the
shape of the PBH mass spectrum. An additional pa-
rameter comes from the threshold curvature fluctuation
from which gravitational collapse leads to PBH formation
when it reenters inside the Hubble radius during the ra-
diation phase. For realistic values, we have identified the
potential parameter ranges leading to the right amount of
PBH dark matter at matter-radiation equality. If PBH
masses then grow by merging or accretion, by at least
a factor 103, we find that the model can be in agree-
ment with the current constraints on PBH abundances.
In particular, we have identified a scenario where the
PBH spectrum peak is centered on sub-solar masses, thus
evading CMB distortion constraints, and is then shifted
up to stellar-like masses today, thus evading constraints
from micro-lensing observations. This scenario explains
the excess of BH candidates in the central region of the
Andromeda galaxy.
Our effective hybrid potential can be embedded in a
hybrid model where the slope of the potential in the val-
ley direction is due to logarithmic radiative corrections.
In particular, it was found that the above scenario works
well for D-term inflation with Planck-like values of Fayet-
Iliopoulos term.
Finally we discussed whether PBH in the tail of the
distribution can serve as the seeds of the supermassive
black holes observed at the center of galaxies and in high-
redshift quasars. Seeds having a mass larger than 104M
at redshift z15 are produced and can then accrete mat-
ter and merge until they form supermassive black holes
(SMBH). This does not require any specific additional
tuning of parameters and is obtained for free from our
model, whereas the formation of SMBH at high redshifts
is challenging is standard ΛCDM cosmology. PBH with
intermediate masses are also produced and could explain
ultra-luminous X-ray sources.
It is worth mentioning that our scenario leads to spe-
cific predictions that could help to distinguish it from
other dark matter scenarios in the near future. First, sig-
nificant CMB distortions are expected due to the increase
of power in the Silk-damped tail of the scalar power spec-
15
trum. We found that they could be detected by PIXIE
or PRISM. Then, a large number of stellar-mass BH can-
didates in nearby galaxies should be detected by X-rays
observations. Moreover, PBH binaries should emit grav-
itational waves that could be detected by future gravi-
tational waves experiments such as LIGO, DECIGO and
LISA [67, 68]. Finally, X-rays photons that are emitted
by PBH at high redshifts should affect the reionization
of the nearby intergalactic medium and leave imprints on
the 21cm signal from the reionization and the late dark
ages [64]. In the next decade, 21cm observations by the
Square Kilometre Array will constrain the abundance of
PBH in the range from 102Mto 108M, down to a level
β109, which is enough to rule out our model.
Further work is certainly needed to understand the role
and the importance of mergers between the time of for-
mation and the late Universe. The fact that the PBH
distribution is broad makes this process even more com-
plex than in simpler scenarios. The clustering of PBH
holes, and especially its relation with the quantum diffu-
sion during the inflationary phase, is an open question.
Qualitatively, the scalar power spectrum at the end of in-
flation varies in different patches of the Universe, having
emerged from different realizations of the quantum diffu-
sion of the auxiliary field at the instability point. In those
patches, the PBH production rate will be different, and
therefore will lead to different dark matter abundances.
One thus expects the Universe to be formed of a mixture
of large voids and high dark matter density regions, on
scales going from a few Mpc up to thousands of Mpc,
depending on the inflation potential parameters. This
picture is in some way similar to the Swiss-cheese model,
where inhomogeneities can lead to an apparent cosmic
acceleration mimicking dark energy. Holes larger than
35 Mpc are expected to leave distinguishable signatures
on the CMB, but smaller sizes are still viable [108]. It will
be very interesting but challenging to investigate whether
apparent local cosmic acceleration can be obtained from
the inhomogeneous structures in our model.
X. ACKNOWLEDGMENTS
We warmly thank A. Linde, J. Silk and F. Capela for
useful comments and discussion. The work of S.C. is sup-
ported by the Return Grant program of the Belgian Sci-
ence Policy (BELSPO). JGB acknowledges financial sup-
port from the Spanish MINECO under grant FPA2012-
39684-C03-02 and Consolider-Ingenio “Physics of the Ac-
celerating Universe (PAU)” (CSD2007-00060). We also
acknowledge the support from the Spanish MINECO’s
“Centro de Excelencia Severo Ochoa” Programme under
Grant No. SEV-2012-0249.
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... This form was first suggested by Dolgov & Silk [40] (see also References [41,42]) and has been demonstrated both numerically [43] and analytically [44] for the case in which the slowroll approximation holds. It is therefore representative of a large class of inflationary scenarios, including the axion-curvaton and running-mass inflation models considered by Kühnel et al. [45]. ...
... With the current Planck exclusion limits [103], this implies that the non-Gaussianity parameters f NL and g NL for a PBH-producing theory are both less than O(10 −3 ). For the curvaton and hybrid inflation models [42,104], this leads to the immediate exclusion of PBH dark matter. Non-sphericity has not yet been subject to extensive numerical studies but non-zero ellipticity may lead to large effects on the PBH mass spectra, as shown in Reference [30]. ...
... Some of them produce an extended plateau or dome-like feature in the power spectrum. For example, this applies for two-field models like hybrid inflation [42] and even some single-field models like Higgs inflation [190,191], although not for the minimal Higgs model [192]. Instead of focussing on any specific scenario, Reference where the spectral index n s and amplitude A are treated as free phenomenological parameters. ...
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We review the formation and evaporation of primordial black holes (PBHs) and their possible contribution to dark matter. Various constraints suggest they could only provide most of it in the mass windows 10^{17}\; 10 17 – 10^{23}\; 10 23 g or 10 10 – 10^2\;M_{\odot} 10 2 M ⊙ , with the last possibility perhaps being suggested by the LIGO/Virgo observations. However, PBHs could have important consequences even if they have a low cosmological density. Sufficiently large ones might generate cosmic structures and provide seeds for the supermassive black holes in galactic nuclei. Planck-mass relics of PBH evaporations or stupendously large black holes bigger than 10^{12}\;M_{\odot} 10 12 M ⊙ could also be an interesting dark component.
... Compact lowluminous massive objects are expected to lurk the CGM of galaxies. From lone low-mass stars (e.g., Helmi 2020), to BHs arising from stellar evolution (e.g., Fender et al. 2013), including remnants of Pop iii stars (e.g., Madau & Rees 2001;, and primordial BHs (e.g., Carr & Hawking 1974;Clesse & García-Bellido 2015). The number density of these objects is assumed to be high; for example, there are claims that primordial BHs account for all the dark matter in the Universe (Clesse & García-Bellido 2015). ...
... From lone low-mass stars (e.g., Helmi 2020), to BHs arising from stellar evolution (e.g., Fender et al. 2013), including remnants of Pop iii stars (e.g., Madau & Rees 2001;, and primordial BHs (e.g., Carr & Hawking 1974;Clesse & García-Bellido 2015). The number density of these objects is assumed to be high; for example, there are claims that primordial BHs account for all the dark matter in the Universe (Clesse & García-Bellido 2015). Even if this is not the case (e.g., Carr & Kühnel 2020), the expectations clearly overwhelm the density required to account for all the emission signals that we observe, of the order of one clump per central galaxy (Sect. ...
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... The next step is to extend PyFPT to solve multi-dimensional first-passage time problems. This would allow investigations beyond the single-field slow-roll approximation to be done, such as phases of ultra-slow-roll [54,55,104] or multiple-field setups [85,86,94,97,98,[105][106][107][108]. In principle, the noise amplitude is determined by the field mode equation when integrated along a given realisation, and this leads to non-Markovian effects [89,109] that we could also incorporate, together with other recently-proposed refinements of the stochastic-inflation equations [90,110]. ...
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