###### Abstract

We revisit the question whether the running–mass inflation model allows the formation of Primordial Black Holes (PBHs) that are sufficiently long–lived to serve as candidates for Dark Matter. We incorporate recent cosmological data, including the WMAP 7–year results. Moreover, we include “the running of the running” of the spectral index of the power spectrum, as well as the renormalization group “running of the running” of the inflaton mass term. Our analysis indicates that formation of sufficiently heavy, and hence long–lived, PBHs still remains possible in this scenario. As a by–product, we show that the additional term in the inflaton potential still does not allow significant negative running of the spectral index.

Running-Mass Inflation Model and Primordial Black Holes

Manuel Drees and Encieh Erfani

Physikalisches Institut and Bethe Center for Theoretical Physics, Universität Bonn,

Nussallee 12, 53115 Bonn, Germany

and

School of Physics, KIAS, Seoul 130–722, Korea

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^{0}0

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## 1 Introduction

###
The Cosmic Microwave Background (CMB) is very smooth. Full–sky observations
allow to expand the measured CMB temperature in spherical harmonics . One can then determine the size of the anisotropies as a function of
, with larger corresponding to smaller angles, and hence smaller
length scales. Current CMB observation probe this power spectrum down to
(comoving) length scales of about one Mpc. These observations imply very small
primordial density perturbations at such large length scales, characterized by
the power .

However, it is possible that the primordial density perturbations become much
larger at smaller length scales, beyond the range probed by cosmological
observations. Indeed, it is conceivable that these perturbations are so large
that overdense regions collapse to form Primordial Black Holes (PBHs) just
after the end of inflation [1, 2, 3]. They are called
“primordial” since they do not originate from the gravitational collapse of
burnt–out stars; they could thus have any mass, including masses well below
or well above stellar masses. Here we assume that the fluctuations which
sourced the PBHs are also generated during inflation, specifically towards the
end of inflation, well after the length scales probed by conventional
cosmological observations exited the horizon.

There are various constraints on PBH formation. For example, the density of
roughly stellar mass black holes has to satisfy limits from searches for
microlensing. Very light black holes could have evaporated (via Hawking
radiation) in the epoch of Big Bang nucleosynthesis, altering the predicted
isotope abundances. These and other constraints have recently been compiled in
[4]. They can be translated into upper limits on the amplitude of the
power spectrum at the length scales relevant for PBH formation, typically
with some scale dependence.

Clearly the power spectrum has to change dramatically towards the end of
inflation for PBH formation to occur. In the framework of standard slow–roll
inflation, this implies that the slow–roll parameters, and hence the inflaton
potential, also should show large variations. One simple, and yet well
motivated, model that can feature large variations of the slow–roll
parameters is the running-mass model [5, 6], a type of
inflationary model which emerges naturally in the context of supersymmetric
extensions of the Standard Model. The model is of the single–field type, but
nevertheless it has relatively strong scale–dependence (running) of the
spectral index. Previous studies of this model [7] showed that it can
indeed accommodate sufficiently large positive running of the spectral index
to allow for PBH formation. However, the recent 7–year WMAP data
[8] prefer negative running of the spectral index.

In this paper we therefore revisit the running mass model. We expand on
refs.[7] by including the running of the running of the spectral
index, which is only weakly constrained by existing CMB data. Similarly, we
include the running of the running of the inflaton mass parameter in the
potential. We show that for some (small) range of parameters, this model can
still accommodate sufficiently large density perturbations near the end of
inflation to allow for the formation of PBHs with mass larger than
g, which are sufficiently long–lived to be candidates for Cold Dark Matter
(CDM). As a by–product we show that even with the additional term in the
potential, the model does not allow for significant negative running of
the spectral index.

The remainder of this paper is organized as follows: In section 2 we briefly
review the formalism of PBH formation. In section 3 we discuss the
running–mass inflation model, including the running of the running of the
mass. We compute the slow–roll parameters which allow us to determine the
spectral index, its running, and the running of the running. In section 4 we
perform a numerical analysis where we compute the spectral index at PBH scales
exactly, by numerically solving the equation of motion of the inflaton field.
Finally, we conclude in section 5.

The Cosmic Microwave Background (CMB) is very smooth. Full–sky observations allow to expand the measured CMB temperature in spherical harmonics . One can then determine the size of the anisotropies as a function of , with larger corresponding to smaller angles, and hence smaller length scales. Current CMB observation probe this power spectrum down to (comoving) length scales of about one Mpc. These observations imply very small primordial density perturbations at such large length scales, characterized by the power .

However, it is possible that the primordial density perturbations become much larger at smaller length scales, beyond the range probed by cosmological observations. Indeed, it is conceivable that these perturbations are so large that overdense regions collapse to form Primordial Black Holes (PBHs) just after the end of inflation [1, 2, 3]. They are called “primordial” since they do not originate from the gravitational collapse of burnt–out stars; they could thus have any mass, including masses well below or well above stellar masses. Here we assume that the fluctuations which sourced the PBHs are also generated during inflation, specifically towards the end of inflation, well after the length scales probed by conventional cosmological observations exited the horizon.

There are various constraints on PBH formation. For example, the density of roughly stellar mass black holes has to satisfy limits from searches for microlensing. Very light black holes could have evaporated (via Hawking radiation) in the epoch of Big Bang nucleosynthesis, altering the predicted isotope abundances. These and other constraints have recently been compiled in [4]. They can be translated into upper limits on the amplitude of the power spectrum at the length scales relevant for PBH formation, typically with some scale dependence.

Clearly the power spectrum has to change dramatically towards the end of inflation for PBH formation to occur. In the framework of standard slow–roll inflation, this implies that the slow–roll parameters, and hence the inflaton potential, also should show large variations. One simple, and yet well motivated, model that can feature large variations of the slow–roll parameters is the running-mass model [5, 6], a type of inflationary model which emerges naturally in the context of supersymmetric extensions of the Standard Model. The model is of the single–field type, but nevertheless it has relatively strong scale–dependence (running) of the spectral index. Previous studies of this model [7] showed that it can indeed accommodate sufficiently large positive running of the spectral index to allow for PBH formation. However, the recent 7–year WMAP data [8] prefer negative running of the spectral index.

In this paper we therefore revisit the running mass model. We expand on refs.[7] by including the running of the running of the spectral index, which is only weakly constrained by existing CMB data. Similarly, we include the running of the running of the inflaton mass parameter in the potential. We show that for some (small) range of parameters, this model can still accommodate sufficiently large density perturbations near the end of inflation to allow for the formation of PBHs with mass larger than g, which are sufficiently long–lived to be candidates for Cold Dark Matter (CDM). As a by–product we show that even with the additional term in the potential, the model does not allow for significant negative running of the spectral index.

The remainder of this paper is organized as follows: In section 2 we briefly review the formalism of PBH formation. In section 3 we discuss the running–mass inflation model, including the running of the running of the mass. We compute the slow–roll parameters which allow us to determine the spectral index, its running, and the running of the running. In section 4 we perform a numerical analysis where we compute the spectral index at PBH scales exactly, by numerically solving the equation of motion of the inflaton field. Finally, we conclude in section 5.

## 2 Formation of Primordial Black Holes

###
PBHs may have formed during the very early universe, and if so can have
observational implications at the present epoch, e.g. from effects of their
Hawking evaporation [1] for masses g, or by
contributing to the present “cold” dark matter density if they are more
massive than g [3]. (These PBHs would certainly be massive
enough to be dynamically ‘‘cold’’^{1}^{1}1It has been argued
[9] that BH evaporation might leave a stable remnant with mass of
order of the Planck mass, which could form CDM; we will not pursue this
possibility here.).

The traditional treatment of PBH formation is based on the Press-Schechter
formalism [10] used widely in large–scale structure
studies. Here the density field is smoothed on a scale . In the case at
hand, is given by the mass enclosed inside radius when crossed
the horizon. The probability of PBH formation is then estimated by simply
integrating the probability distribution over the range of
perturbations which allow PBH formation: , where the upper limit arises since very large
perturbations would correspond to separate closed ’baby’ universes
[3, 11]. We will show that in practice is such a
rapidly decreasing function of above that the upper
cutoff is not important. The threshold density is taken as , where is the equation of state parameter describing the epoch
during which PBH formation is supposed to have occurred [3]. Here we
take , characteristic for the radiation dominated epoch which should
have started soon after the end of inflaton. However the correct value of the
threshold is quite uncertain. Niemeyer and Jedamzik
[12] carried out numerical simulations of the collapse of the
isolated regions and found the threshold for PBH formation to be . We
will show that PBHs abundance is sensitive to the value of .

The fraction of the energy density of the Universe in which the density
fluctuation exceeds the threshold for PBH formation when smoothed on scale
, , which will hence end up in PBHs with
mass ,^{2}^{2}2Throughout we assume for simplicity that the PBH mass is a fixed fraction of the horizon mass corresponding to the smoothing scale. This is not strictly true. In general the mass of PBHs is expected to depend on the amplitude, size and shape of the perturbations [12, 13]. is given as in Press--Schechter theory by^{3}^{3}3We follow ref.[14] in including a factor of two on the right–hand side; this factor is not very important for delineating the inflationary model parameters allowing significant PBH formation. Moreover, we set to infinity.

(2.1)

Here is the probability distribution function (PDF) of the
linear density field smoothed on a scale , and is the
fraction of the total energy within a sphere of radius that ends up inside
the PBH.

For Gaussian fluctuations, the probability distribution of the smoothed
density field is given by^{4}^{4}4This PDF is often written as
. However, we think it is more transparent to consider to
be the PDF of , which is just an integration variable in
eq.(2.1). Eq.(2.2) shows that the functional form of
depends on the parameter , which in turn depends on the
horizon mass .

(2.2)

This PDF is thus uniquely determined by the variance of
, which is given by

(2.3)

In order to compute the variance, we therefore have to know the power spectrum
of , , as well as the volume–normalized Fourier
transform of the window function used to smooth , .

It is not obvious what the correct smoothing function is; a top–hat
function has often been used in the past, but we prefer to use a Gaussian
window function^{5}^{5}5Bringmann et al. [15] argued
that a top–hat window function predicts a larger PBH abundance.,

(2.4)

Finally, on comoving hypersurfaces there is a simple relation between the
density perturbation and the curvature perturbation
[16]:

(2.5)

The density and curvature perturbation power spectra are therefore related by

(2.6)

The mass fraction of the Universe that will collapse into PBHs can now be
computed by inserting eqs.(2.6) and (2.4) into eq.(2.3) to
determine the variance as function of . This has to be used in
eq.(2.2), which finally has to be inserted into eq.(2.1). Since
we assume a Gaussian in eq.(2.2), the integral in eq.(2.1) simply gives an error function.

In order to complete this calculation one needs to relate the mass to the
comoving smoothing scale . The number density of PBHs formed during the
reheating phase just after the end of inflation will be greatly diluted by the
reheating itself. We therefore only consider PBHs which form during the
radiation dominated era after reheating is (more or less) complete. The
initial PBHs mass is related to the particle horizon mass
by^{6}^{6}6Throughout the paper we put

(2.7)

when the scale enters the horizon, . Here the coefficient
, which already appeared in eq.(2.1), depends on the details of
gravitational collapse. A simple analytical calculation suggest that during the radiation era [3]. During
radiation domination , and expansion at constant entropy
gives [17] (where
is the number of relativistic degrees of freedom, and we have approximated the
temperature and entropy degrees of freedom as equal). This implies that

(2.8)

where the subscript “eq” refers to quantities evaluated at matter–radiation
equality. In the early Universe, the effective relativistic degree of freedom
is expected to be of order , while
and [8]). The horizon mass at matter-radiation equality
is given by

(2.9)

where and (assuming three species
of massless neutrinos) . Then
it is straightforward to show that

(2.10)

Note that , not as one might naively
have expected. Recall that is related to the horizon mass at the
time when the comoving scale again crossed into the horizon. Larger scales
re–enter later, when the energy density was lower; this weakens the
dependence of on . Moreover, the lightest black holes to form
are those corresponding to a comoving scale that re--enters the horizon
immediately after inflation.^{7}^{7}7In fact, PBH formation might also occur
on scales that never leave the horizon [18]. We do not consider
this contribution here.

The Gaussian window function in eq.(2.3) strongly suppresses
contributions with . At the same time, the factor in
eq.(2.6) suppresses contributions to the integral in eq.(2.3)
from small . As a result, this integral is dominated by a limited range of
values near . Over this limited range one can to good approximation
assume a power–law primordial power spectrum with fixed power
^{8}^{8}8“Fixed” here means that does not depend on ; however,
does depend on , since a large range of values of has to be
considered for PBH formation of different masses.
, with . With this ansatz, the variance of
the primordial density field at horizon crossing is given by

(2.11)

for .

The power is known accurately at CMB scales; for
example,
at the “COBE scale” Mpc [8]. In order to
relate this to the scales relevant for PBH formation, we parameterize the
power spectrum as

(2.12)

It is important to distinguish between and at this
point. describes the slope of the power spectrum at scales , whereas fixes the normalization of the spectrum
at . The two powers are identical if the spectral index is
strictly constant, i.e. if neither nor depend on . However, in this
case CMB data imply [8] that is close to
unity. Eqs.(2.11) and (2.12) then give a very small variance,
leading to essentially no PBH formation.

Significant PBH formation can therefore only occur in scenarios with running
spectral index. We parameterize the scale dependence of as [19]:

(2.13)

recall that we are interested in , i.e. .
The parameters and denote the running of the effective
spectral index and the running of the running, respectively:

(2.14)

Eq.(2.13) illustrates the difference between and . The
latter has an expansion similar to eq.(2.13), but with the usual
Taylor–expansion coefficients, in front of and in front
of . One therefore has

(2.15)

Setting for simplicity, eq.(2.15) implies for , and for . We
will compute the variance , and hence the PBH fraction , for
these two relations; they represent extreme cases if neither nor
is negative.

The result of this calculation is shown in figure 1. Here we have
fixed , and show results for two choices of the threshold
and three choices of . We see that scenarios where
(or smaller) are safe in the SM, because there is no
model–independent limit on for g [4]. As
noted earlier, PBHs contributing to Dark Matter today must have g; at this mass, they saturate the DM relic density if .^{9}^{9}9Note that describes the fraction of the
energy density in PBHs at the time of their formation. Since they behave
like matter at all times, their fractional contribution to the energy
density increases during the radiation–dominated epoch, and stays
essentially constant during the subsequent matter–dominated epoch.
Figure 1 shows that this requires for
. The dependence on is much milder.

Figure 1 also illustrates a serious problem that all scenarios that
aim to explain the required CDM density in terms of post–inflationary PBH
formation face. We just saw that this can happen only if the spectral index
increases significantly between the scales probed by the CMB and other
cosmological observations and the scale pc relevant for the
formation of g PBHs. However, must then decrease rapidly
when going to slightly smaller length scales, since otherwise one would overproduce lighter PBHs. For example, successful Big Bang Nucleosynthesis
requires [4] , about 12
orders of magnitude below that predicted by keeping fixed at the value
required for having g PBHs as CDM candidates. We will come back to
this point at the end of our paper.

Another problem is that the expansion of in eq.(2.13) will
generally only be accurate if is not too large. This is the
case for cosmological observations, which probe scales Mpc. The
expansion becomes questionable for the scales probed by PBH formation. For
example, fixing Mpc, eq.(2.10) gives for g. Fortunately within the framework of a
given inflationary scenario this second problem can be solved by computing
and exactly, rather than using the expansion (2.13).

Observational bounds on and can e.g. be found in
ref.[8]. By requiring that these parameters remain within the ranges for all scales between Mpc and
Mpc, we find . Here we used the error bars derived
from the analysis of WMAP7 data plus data on baryonic acoustic oscillations
(BAO), ignoring possible tensor modes (as appropriate for small–field
models), and including independent measurements of the Hubble constant as a
prior (i.e. we used the “WMAP7+BAO+” data set). Here we use the pivot
scale Mpc where and are
uncorrelated. On the other hand, eq.(2.13) shows that for
we only need in order to generate sufficiently
large density perturbations to allow formation of g PBHs. Even if we
set equal to its central value, , we
only need . Including the running of the running
of the spectral index thus easily allows to accommodate PBH formation in
scenarios that reproduce all current cosmological observations at large
scales.

Of course, this kind of model–independent analysis does not show whether
simple, reasonably well–motivated inflationary models exist that can generate
a sufficiently large . One scenario that quite naturally
accommodates strong scale dependence of the spectral index is the
running–mass inflation model [5, 6]. We now apply our
results to this model.

PBHs may have formed during the very early universe, and if so can have
observational implications at the present epoch, e.g. from effects of their
Hawking evaporation [1] for masses g, or by
contributing to the present “cold” dark matter density if they are more
massive than g [3]. (These PBHs would certainly be massive
enough to be dynamically ‘‘cold’’^{1}^{1}1It has been argued
[9] that BH evaporation might leave a stable remnant with mass of
order of the Planck mass, which could form CDM; we will not pursue this
possibility here.).

The traditional treatment of PBH formation is based on the Press-Schechter formalism [10] used widely in large–scale structure studies. Here the density field is smoothed on a scale . In the case at hand, is given by the mass enclosed inside radius when crossed the horizon. The probability of PBH formation is then estimated by simply integrating the probability distribution over the range of perturbations which allow PBH formation: , where the upper limit arises since very large perturbations would correspond to separate closed ’baby’ universes [3, 11]. We will show that in practice is such a rapidly decreasing function of above that the upper cutoff is not important. The threshold density is taken as , where is the equation of state parameter describing the epoch during which PBH formation is supposed to have occurred [3]. Here we take , characteristic for the radiation dominated epoch which should have started soon after the end of inflaton. However the correct value of the threshold is quite uncertain. Niemeyer and Jedamzik [12] carried out numerical simulations of the collapse of the isolated regions and found the threshold for PBH formation to be . We will show that PBHs abundance is sensitive to the value of .

The fraction of the energy density of the Universe in which the density
fluctuation exceeds the threshold for PBH formation when smoothed on scale
, , which will hence end up in PBHs with
mass ,^{2}^{2}2Throughout we assume for simplicity that the PBH mass is a fixed fraction of the horizon mass corresponding to the smoothing scale. This is not strictly true. In general the mass of PBHs is expected to depend on the amplitude, size and shape of the perturbations [12, 13]. is given as in Press--Schechter theory by^{3}^{3}3We follow ref.[14] in including a factor of two on the right–hand side; this factor is not very important for delineating the inflationary model parameters allowing significant PBH formation. Moreover, we set to infinity.

(2.1) |

Here is the probability distribution function (PDF) of the linear density field smoothed on a scale , and is the fraction of the total energy within a sphere of radius that ends up inside the PBH.

For Gaussian fluctuations, the probability distribution of the smoothed
density field is given by^{4}^{4}4This PDF is often written as
. However, we think it is more transparent to consider to
be the PDF of , which is just an integration variable in
eq.(2.1). Eq.(2.2) shows that the functional form of
depends on the parameter , which in turn depends on the
horizon mass .

(2.2) |

This PDF is thus uniquely determined by the variance of , which is given by

(2.3) |

In order to compute the variance, we therefore have to know the power spectrum of , , as well as the volume–normalized Fourier transform of the window function used to smooth , .

It is not obvious what the correct smoothing function is; a top–hat
function has often been used in the past, but we prefer to use a Gaussian
window function^{5}^{5}5Bringmann et al. [15] argued
that a top–hat window function predicts a larger PBH abundance.,

(2.4) |

Finally, on comoving hypersurfaces there is a simple relation between the density perturbation and the curvature perturbation [16]:

(2.5) |

The density and curvature perturbation power spectra are therefore related by

(2.6) |

The mass fraction of the Universe that will collapse into PBHs can now be computed by inserting eqs.(2.6) and (2.4) into eq.(2.3) to determine the variance as function of . This has to be used in eq.(2.2), which finally has to be inserted into eq.(2.1). Since we assume a Gaussian in eq.(2.2), the integral in eq.(2.1) simply gives an error function.

In order to complete this calculation one needs to relate the mass to the
comoving smoothing scale . The number density of PBHs formed during the
reheating phase just after the end of inflation will be greatly diluted by the
reheating itself. We therefore only consider PBHs which form during the
radiation dominated era after reheating is (more or less) complete. The
initial PBHs mass is related to the particle horizon mass
by^{6}^{6}6Throughout the paper we put

(2.7) |

when the scale enters the horizon, . Here the coefficient , which already appeared in eq.(2.1), depends on the details of gravitational collapse. A simple analytical calculation suggest that during the radiation era [3]. During radiation domination , and expansion at constant entropy gives [17] (where is the number of relativistic degrees of freedom, and we have approximated the temperature and entropy degrees of freedom as equal). This implies that

(2.8) |

where the subscript “eq” refers to quantities evaluated at matter–radiation equality. In the early Universe, the effective relativistic degree of freedom is expected to be of order , while and [8]). The horizon mass at matter-radiation equality is given by

(2.9) |

where and (assuming three species of massless neutrinos) . Then it is straightforward to show that

(2.10) |

Note that , not as one might naively
have expected. Recall that is related to the horizon mass at the
time when the comoving scale again crossed into the horizon. Larger scales
re–enter later, when the energy density was lower; this weakens the
dependence of on . Moreover, the lightest black holes to form
are those corresponding to a comoving scale that re--enters the horizon
immediately after inflation.^{7}^{7}7In fact, PBH formation might also occur
on scales that never leave the horizon [18]. We do not consider
this contribution here.

The Gaussian window function in eq.(2.3) strongly suppresses
contributions with . At the same time, the factor in
eq.(2.6) suppresses contributions to the integral in eq.(2.3)
from small . As a result, this integral is dominated by a limited range of
values near . Over this limited range one can to good approximation
assume a power–law primordial power spectrum with fixed power
^{8}^{8}8“Fixed” here means that does not depend on ; however,
does depend on , since a large range of values of has to be
considered for PBH formation of different masses.
, with . With this ansatz, the variance of
the primordial density field at horizon crossing is given by

(2.11) |

for .

The power is known accurately at CMB scales; for example, at the “COBE scale” Mpc [8]. In order to relate this to the scales relevant for PBH formation, we parameterize the power spectrum as

(2.12) |

It is important to distinguish between and at this point. describes the slope of the power spectrum at scales , whereas fixes the normalization of the spectrum at . The two powers are identical if the spectral index is strictly constant, i.e. if neither nor depend on . However, in this case CMB data imply [8] that is close to unity. Eqs.(2.11) and (2.12) then give a very small variance, leading to essentially no PBH formation.

Significant PBH formation can therefore only occur in scenarios with running spectral index. We parameterize the scale dependence of as [19]:

(2.13) |

recall that we are interested in , i.e. . The parameters and denote the running of the effective spectral index and the running of the running, respectively:

(2.14) |

Eq.(2.13) illustrates the difference between and . The latter has an expansion similar to eq.(2.13), but with the usual Taylor–expansion coefficients, in front of and in front of . One therefore has

(2.15) |

Setting for simplicity, eq.(2.15) implies for , and for . We will compute the variance , and hence the PBH fraction , for these two relations; they represent extreme cases if neither nor is negative.

The result of this calculation is shown in figure 1. Here we have
fixed , and show results for two choices of the threshold
and three choices of . We see that scenarios where
(or smaller) are safe in the SM, because there is no
model–independent limit on for g [4]. As
noted earlier, PBHs contributing to Dark Matter today must have g; at this mass, they saturate the DM relic density if .^{9}^{9}9Note that describes the fraction of the
energy density in PBHs at the time of their formation. Since they behave
like matter at all times, their fractional contribution to the energy
density increases during the radiation–dominated epoch, and stays
essentially constant during the subsequent matter–dominated epoch.
Figure 1 shows that this requires for
. The dependence on is much milder.

Figure 1 also illustrates a serious problem that all scenarios that aim to explain the required CDM density in terms of post–inflationary PBH formation face. We just saw that this can happen only if the spectral index increases significantly between the scales probed by the CMB and other cosmological observations and the scale pc relevant for the formation of g PBHs. However, must then decrease rapidly when going to slightly smaller length scales, since otherwise one would overproduce lighter PBHs. For example, successful Big Bang Nucleosynthesis requires [4] , about 12 orders of magnitude below that predicted by keeping fixed at the value required for having g PBHs as CDM candidates. We will come back to this point at the end of our paper.

Another problem is that the expansion of in eq.(2.13) will generally only be accurate if is not too large. This is the case for cosmological observations, which probe scales Mpc. The expansion becomes questionable for the scales probed by PBH formation. For example, fixing Mpc, eq.(2.10) gives for g. Fortunately within the framework of a given inflationary scenario this second problem can be solved by computing and exactly, rather than using the expansion (2.13).

Observational bounds on and can e.g. be found in ref.[8]. By requiring that these parameters remain within the ranges for all scales between Mpc and Mpc, we find . Here we used the error bars derived from the analysis of WMAP7 data plus data on baryonic acoustic oscillations (BAO), ignoring possible tensor modes (as appropriate for small–field models), and including independent measurements of the Hubble constant as a prior (i.e. we used the “WMAP7+BAO+” data set). Here we use the pivot scale Mpc where and are uncorrelated. On the other hand, eq.(2.13) shows that for we only need in order to generate sufficiently large density perturbations to allow formation of g PBHs. Even if we set equal to its central value, , we only need . Including the running of the running of the spectral index thus easily allows to accommodate PBH formation in scenarios that reproduce all current cosmological observations at large scales.

Of course, this kind of model–independent analysis does not show whether simple, reasonably well–motivated inflationary models exist that can generate a sufficiently large . One scenario that quite naturally accommodates strong scale dependence of the spectral index is the running–mass inflation model [5, 6]. We now apply our results to this model.

## 3 Running-Mass Inflation Model

###
This model, proposed by Stewart [5, 6], exploits the
observation that in field theory the parameters of the Lagrangian are scale
dependent. This is true in particular for the mass of the scalar inflaton,
which can thus be considered to be a ‘‘running’’ parameter.^{10}^{10}10Note that the physical, or pole, mass of the inflaton is not “running”; however, at the quantum level the physical mass differs from the parameter appearing in eq.(3.1) even if . The running of the mass
parameter can be exploited to solve the “problem” of inflation in
supergravity [20]. This problem arises because the vacuum energy driving
inflation also breaks supersymmetry. In “generic” supergravity models the
vacuum energy therefore gives a large (gravity–mediated) contribution to the
inflaton mass, yielding . However, this argument applies to the
scale where SUSY breaking is felt, which should be close to the (reduced)
Planck scale GeV. In the running mass model,
renormalization group (RG) running of the inflaton mass reduces the inflaton
mass, and hence , at scales where inflation actually happens. There
are four types of model, depending on the sign of the squared inflaton mass at
the Planck scale, and on whether or not that sign change between
and the scales characteristic for inflation [21].

The simplest running–mass model is based on the inflationary potential

(3.1)

where is a real scalar; in supersymmetry it could be the real or
imaginary part of the scalar component of a chiral superfield. The natural
size of in supergravity is of order . Even
for this large value of , which gives , the
potential will be dominated by the constant term for ;
as mentioned above, running is supposed to reduce even more at
lower .

The potential (3.1) would lead to eternal inflation. One possibility to
end inflation is to implement the idea of hybrid inflation [22]. To that end, one introduces a real scalar “waterfall field”
, and adds to the potential the terms^{11}^{11}11The parameters of this
potential will in general also be scale dependent; however, this is
immaterial for our argument.

(3.2)

Here and are real couplings, and the coefficient of the
term has been chosen such that has a
minimum with if . As long as
, remains frozen
at the origin. Once falls below this critical value, quickly
approaches its final vacuum expectation value, given by , while quickly goes to
zero, thereby “shutting off” inflation. However, in this paper we focus on
the inflationary period itself; the evolution of perturbations at length
scales that left the horizon a few e–folds before the end of inflation should
not be affected by the details of how inflation is brought to an
end.^{12}^{12}12It has recently been pointed out that the waterfall phase might
contribute to PBH formation [23].

During inflation, the potential is thus simply given by eq.(3.1).
Here is obtained by integrating an RG equation of the form

(3.3)

where is the function of the inflaton mass parameter. If
is a pure SUSY–breaking term, to one loop can be
schematically written as [5, 24]

(3.4)

where the first term arises from the gauge interaction with coupling
and the second term from the Yukawa interaction . and
are positive numbers of order one, which depend on the representations of the
fields coupling to , is a gaugino mass parameter, while
is the scalar SUSY breaking mass–squared of the scalar particles
interacting with the inflaton via Yukawa interaction .

For successful inflation the running of must be sufficiently strong
to generate a local extremum of the potential for some nonvanishing
field value, which we call . The inflaton potential will obviously be
flat near , so that inflation usually occurs at field values not very
far from . We therefore expand around . The potential we work with thus reads:

(3.5)

Here function of eq.(3.4), and 3.4). In
contrast to earlier analyses of this model [21, 24, 25],
we include the term in the potential. This is a two--loop
correction, but it can be computed by ‘‘iterating’’ the one--loop
correction.^{13}^{13}13There are also “genuine” two–loop corrections, which
can not be obtained from a one–loop calculation, but they only affect the
term linear in . They are thus formally included in our
coefficient . Since the coefficient of this term is of fourth order
in couplings, one will naturally expect . However, this need not
be true if “happens” to be suppressed by a cancellation in
eq.(3.4). Including the second correction to seems
natural given that we also expanded the running of the spectral index to
second (quadratic) order. is given
by the scale dependence of the parameters appearing in eq.( is
given by the

Recall that we had defined to be a local extremum of ,
i.e. . This implies [21] ; this relation is not affected by the
two–loop correction .

In order to calculate the spectral parameters and
defined in eqs.(2), we need the first four slow–roll parameters,
defined as [16]:^{14}^{14}14The powers on and are
purely by convention; in particular, could be negative.

(3.6)

where primes denote derivatives with respect to . All these parameters
are in general scale–dependent, i.e. they have to be evaluated at the value
of that the inflaton field had when the scale crossed out of the
horizon. The spectral parameters are related to these slow–roll parameters by
[16]:

(3.7)

Slow–roll inflation requires . Combining
eqs.(3.5) and (3) we see that we need , where we have introduced the short–hand notation

(3.8)

In other words, the inflaton potential has to be dominated by the constant
term, as noted earlier. Eqs.(3) then imply two strong
inequalities between (combinations of) slow–roll parameters:

(3.9)

The first relation means that is essentially determined by
. Similarly, both relations together imply that is basically
fixed by , while only the last two terms in the expression for
are relevant; these two terms are generically of similar order of
magnitude. Finally, the factors of appearing in the denominators of
eqs.(3) can be replaced by . The spectral parameters are
thus given by:

(3.10)

where has been defined in eq.(3.8). Clearly the spectral index is not
scale–invariant unless and are very close to zero. Note that
appears in eqs.(3) only in the dimensionless combination , while only appears via , i.e. only the ratio
appears in these equations.

In contrast, the absolute normalization of the power spectrum is given by (in
slow–roll approximation)

(3.11)

This normalization is usually quoted at the “COBE scale”
Mpc, where
[8]. Applying our potential (3.5) to eq.(3.11),
replacing by in the numerator, we see that
not only depends on and , but also on the ratio
. We can thus always find parameters that give
the correct normalization of the power spectrum, for all possible combinations
of the spectral parameters.

We want to find out whether the potential (3.5) can accommodate
sufficient running of to allow PBH formation. There are strong
observational constraints on and . It is therefore preferable
to use these physical quantities directly as inputs, rather than the model
parameters and . To this end we rewrite the
first eq.(3) as:

(3.12)

Inserting this into the second eq.(3) gives:

(3.13)

We thus see that the running of the spectral index is “generically” of order
; similarly, the running of the running can easily be
seen to be . This is true in nearly all
inflationary scenarios that have a scale–dependent spectral index.

Eq.(3.13) can be solved for . Bringing the denominator to the
left–hand side leads to a quadratic equation, which has two solutions. They
can be written as:

(3.14)

where we have introduced the quantity

(3.15)

Since and are real quantities, eq.(3.14) only makes sense if the
argument of the square root is non–negative. Note that the coefficients
multiplying and inside the square root are both
non–negative. This means that the model can in principle accommodate any
non–negative value of . However, small negative values of cannot
be realized. It is easy to see that the constraint on is weakest for
. The argument of the square root is then positive if , which implies either or . Recalling the definition
(3.15) we are thus led to the conclusion

(3.16)

this bound should hold on all scales, as long as the potential is described by
eq.(3.5). Note that it is identical to the bound found in
ref.[25], i.e. it is not affected by adding the term
to the inflaton potential. This is somewhat disappointing, since recent data
indicate that is negative at CMB scales. Even the generalized
version of the running mass model therefore cannot reproduce the current range of .

However, at the level significantly positive values
are still allowed. Let us therefore continue with our analysis, and search for
combinations of parameters within the current range that might
lead to significant PBH formation. Using eqs.(3.12) and (3.14) we
can use and
as input parameters in the last eq.(3) to evaluate
. This can then be inserted into eq.(2.13) to see how
large the density perturbations at potential PBH scales are.

This numerical analysis is most easily performed at the “pivot scale”, where
the errors on and are essentially uncorrelated; at values
above (below) this scale, and are correlated (anticorrelated)
[26]. The pivot scale for the “WMAP7+BAO+” data set we
are using is Mpc
[8].^{15}^{15}15This is the smaller of two values given
in ref.[8]. The difference between these two values is not
important for our numerical analysis. At this scale, observations give
[8]:

(3.17)

We saw above that requiring the correct normalization of the power spectrum at
CMB scales does not impose any constraint on the spectral parameters. However,
the model also has to satisfy several consistency conditions. To begin with,
it should provide a sufficient amount of inflation. In the slow–roll
approximation, the number of e–folds of inflation following from the
potential (3.5) is given by:

(3.18)

where we have introduced the dimensionless quantity

(3.19)

recall that it can be traded for using eq.(3.12). Moreover,
can be related to the scale through

(3.20)

This can be inverted to give

(3.21)

where we have introduced

(3.22)

The problem is that the denominator in eq.(3.21) vanishes for some
finite value of . This defines an extremal value of :

(3.23)

A negative value of is generally not problematic, since
only a few e–folds of inflation have to have occurred before our pivot scale
crossed out of the horizon. However, a small positive value of would imply insufficient amount of inflation after the scale
crossed the horizon. In our numerical work we therefore exclude scenarios with
, i.e. we (rather conservatively) demand that at
least 50 e–folds of inflation can occur after crossed out of the
horizon.

A second consistency condition we impose is that should not become too
large. Specifically, we require for all scales between and
the PBH scale. For (much) larger values of our potential (3.5)
may no longer be appropriate, i.e. higher powers of may need to be
included.

Note that eq.(3.21) allows to compute the effective spectral index
exactly:

(3.24)

This in turn allows an exact (numerical) calculation of the spectral
index :

(3.25)

In our numerical scans of parameter space we noticed that frequently the exact
value for at PBH scales differs significantly from the values predicted
by the expansion of eq.(2.13); similar statements apply to
. This is not very surprising, given that for
our value of the pivot scale and g. In fact, we noticed
that even if and are both positive,
may not grow monotonically with increasing . In some cases
computed according to eq.(3.24) even becomes quite large at values of
some 5 or 10 e–folds below the PBH scale. This is problematic, since our
calculation is based on the slow–roll approximation, which no longer works if
becomes too large. We therefore demanded for all scales
up to the PBH scale; the first eq.(3) shows that this corresponds to
. This last requirement turns out to be the most constraining one
when looking for combinations of parameters that give large .

This model, proposed by Stewart [5, 6], exploits the
observation that in field theory the parameters of the Lagrangian are scale
dependent. This is true in particular for the mass of the scalar inflaton,
which can thus be considered to be a ‘‘running’’ parameter.^{10}^{10}10Note that the physical, or pole, mass of the inflaton is not “running”; however, at the quantum level the physical mass differs from the parameter appearing in eq.(3.1) even if . The running of the mass
parameter can be exploited to solve the “problem” of inflation in
supergravity [20]. This problem arises because the vacuum energy driving
inflation also breaks supersymmetry. In “generic” supergravity models the
vacuum energy therefore gives a large (gravity–mediated) contribution to the
inflaton mass, yielding . However, this argument applies to the
scale where SUSY breaking is felt, which should be close to the (reduced)
Planck scale GeV. In the running mass model,
renormalization group (RG) running of the inflaton mass reduces the inflaton
mass, and hence , at scales where inflation actually happens. There
are four types of model, depending on the sign of the squared inflaton mass at
the Planck scale, and on whether or not that sign change between
and the scales characteristic for inflation [21].

The simplest running–mass model is based on the inflationary potential

(3.1) |

where is a real scalar; in supersymmetry it could be the real or imaginary part of the scalar component of a chiral superfield. The natural size of in supergravity is of order . Even for this large value of , which gives , the potential will be dominated by the constant term for ; as mentioned above, running is supposed to reduce even more at lower .

The potential (3.1) would lead to eternal inflation. One possibility to
end inflation is to implement the idea of hybrid inflation [22]. To that end, one introduces a real scalar “waterfall field”
, and adds to the potential the terms^{11}^{11}11The parameters of this
potential will in general also be scale dependent; however, this is
immaterial for our argument.

(3.2) |

Here and are real couplings, and the coefficient of the
term has been chosen such that has a
minimum with if . As long as
, remains frozen
at the origin. Once falls below this critical value, quickly
approaches its final vacuum expectation value, given by , while quickly goes to
zero, thereby “shutting off” inflation. However, in this paper we focus on
the inflationary period itself; the evolution of perturbations at length
scales that left the horizon a few e–folds before the end of inflation should
not be affected by the details of how inflation is brought to an
end.^{12}^{12}12It has recently been pointed out that the waterfall phase might
contribute to PBH formation [23].

During inflation, the potential is thus simply given by eq.(3.1). Here is obtained by integrating an RG equation of the form

(3.3) |

where is the function of the inflaton mass parameter. If is a pure SUSY–breaking term, to one loop can be schematically written as [5, 24]

(3.4) |

where the first term arises from the gauge interaction with coupling and the second term from the Yukawa interaction . and are positive numbers of order one, which depend on the representations of the fields coupling to , is a gaugino mass parameter, while is the scalar SUSY breaking mass–squared of the scalar particles interacting with the inflaton via Yukawa interaction .

For successful inflation the running of must be sufficiently strong to generate a local extremum of the potential for some nonvanishing field value, which we call . The inflaton potential will obviously be flat near , so that inflation usually occurs at field values not very far from . We therefore expand around . The potential we work with thus reads:

(3.5) |

Here function of eq.(3.4), and 3.4). In
contrast to earlier analyses of this model [21, 24, 25],
we include the term in the potential. This is a two--loop
correction, but it can be computed by ‘‘iterating’’ the one--loop
correction.^{13}^{13}13There are also “genuine” two–loop corrections, which
can not be obtained from a one–loop calculation, but they only affect the
term linear in . They are thus formally included in our
coefficient . Since the coefficient of this term is of fourth order
in couplings, one will naturally expect . However, this need not
be true if “happens” to be suppressed by a cancellation in
eq.(3.4). Including the second correction to seems
natural given that we also expanded the running of the spectral index to
second (quadratic) order. is given
by the scale dependence of the parameters appearing in eq.( is
given by the

Recall that we had defined to be a local extremum of , i.e. . This implies [21] ; this relation is not affected by the two–loop correction .

In order to calculate the spectral parameters and
defined in eqs.(2), we need the first four slow–roll parameters,
defined as [16]:^{14}^{14}14The powers on and are
purely by convention; in particular, could be negative.

(3.6) |

where primes denote derivatives with respect to . All these parameters are in general scale–dependent, i.e. they have to be evaluated at the value of that the inflaton field had when the scale crossed out of the horizon. The spectral parameters are related to these slow–roll parameters by [16]:

(3.7) |

Slow–roll inflation requires . Combining eqs.(3.5) and (3) we see that we need , where we have introduced the short–hand notation

(3.8) |

In other words, the inflaton potential has to be dominated by the constant term, as noted earlier. Eqs.(3) then imply two strong inequalities between (combinations of) slow–roll parameters:

(3.9) |

The first relation means that is essentially determined by . Similarly, both relations together imply that is basically fixed by , while only the last two terms in the expression for are relevant; these two terms are generically of similar order of magnitude. Finally, the factors of appearing in the denominators of eqs.(3) can be replaced by . The spectral parameters are thus given by:

(3.10) | |||||

where has been defined in eq.(3.8). Clearly the spectral index is not scale–invariant unless and are very close to zero. Note that appears in eqs.(3) only in the dimensionless combination , while only appears via , i.e. only the ratio appears in these equations.

In contrast, the absolute normalization of the power spectrum is given by (in slow–roll approximation)

(3.11) |

This normalization is usually quoted at the “COBE scale” Mpc, where [8]. Applying our potential (3.5) to eq.(3.11), replacing by in the numerator, we see that not only depends on and , but also on the ratio . We can thus always find parameters that give the correct normalization of the power spectrum, for all possible combinations of the spectral parameters.

We want to find out whether the potential (3.5) can accommodate sufficient running of to allow PBH formation. There are strong observational constraints on and . It is therefore preferable to use these physical quantities directly as inputs, rather than the model parameters and . To this end we rewrite the first eq.(3) as:

(3.12) |

Inserting this into the second eq.(3) gives:

(3.13) |

We thus see that the running of the spectral index is “generically” of order ; similarly, the running of the running can easily be seen to be . This is true in nearly all inflationary scenarios that have a scale–dependent spectral index.

Eq.(3.13) can be solved for . Bringing the denominator to the left–hand side leads to a quadratic equation, which has two solutions. They can be written as:

(3.14) |

where we have introduced the quantity

(3.15) |

Since and are real quantities, eq.(3.14) only makes sense if the argument of the square root is non–negative. Note that the coefficients multiplying and inside the square root are both non–negative. This means that the model can in principle accommodate any non–negative value of . However, small negative values of cannot be realized. It is easy to see that the constraint on is weakest for . The argument of the square root is then positive if , which implies either or . Recalling the definition (3.15) we are thus led to the conclusion

(3.16) |

this bound should hold on all scales, as long as the potential is described by eq.(3.5). Note that it is identical to the bound found in ref.[25], i.e. it is not affected by adding the term to the inflaton potential. This is somewhat disappointing, since recent data indicate that is negative at CMB scales. Even the generalized version of the running mass model therefore cannot reproduce the current range of .

However, at the level significantly positive values are still allowed. Let us therefore continue with our analysis, and search for combinations of parameters within the current range that might lead to significant PBH formation. Using eqs.(3.12) and (3.14) we can use and as input parameters in the last eq.(3) to evaluate . This can then be inserted into eq.(2.13) to see how large the density perturbations at potential PBH scales are.

This numerical analysis is most easily performed at the “pivot scale”, where
the errors on and are essentially uncorrelated; at values
above (below) this scale, and are correlated (anticorrelated)
[26]. The pivot scale for the “WMAP7+BAO+” data set we
are using is Mpc
[8].^{15}^{15}15This is the smaller of two values given
in ref.[8]. The difference between these two values is not
important for our numerical analysis. At this scale, observations give
[8]:

(3.17) |

We saw above that requiring the correct normalization of the power spectrum at CMB scales does not impose any constraint on the spectral parameters. However, the model also has to satisfy several consistency conditions. To begin with, it should provide a sufficient amount of inflation. In the slow–roll approximation, the number of e–folds of inflation following from the potential (3.5) is given by:

(3.18) | |||||

where we have introduced the dimensionless quantity

(3.19) |

recall that it can be traded for using eq.(3.12). Moreover, can be related to the scale through

(3.20) |

This can be inverted to give

(3.21) |

where we have introduced

(3.22) |

The problem is that the denominator in eq.(3.21) vanishes for some finite value of . This defines an extremal value of :

(3.23) |

A negative value of is generally not problematic, since only a few e–folds of inflation have to have occurred before our pivot scale crossed out of the horizon. However, a small positive value of would imply insufficient amount of inflation after the scale crossed the horizon. In our numerical work we therefore exclude scenarios with , i.e. we (rather conservatively) demand that at least 50 e–folds of inflation can occur after crossed out of the horizon.

A second consistency condition we impose is that should not become too large. Specifically, we require for all scales between and the PBH scale. For (much) larger values of our potential (3.5) may no longer be appropriate, i.e. higher powers of may need to be included.

Note that eq.(3.21) allows to compute the effective spectral index exactly:

(3.24) |

This in turn allows an exact (numerical) calculation of the spectral index :

(3.25) |

In our numerical scans of parameter space we noticed that frequently the exact value for at PBH scales differs significantly from the values predicted by the expansion of eq.(2.13); similar statements apply to . This is not very surprising, given that for our value of the pivot scale and g. In fact, we noticed that even if and are both positive, may not grow monotonically with increasing . In some cases computed according to eq.(3.24) even becomes quite large at values of some 5 or 10 e–folds below the PBH scale. This is problematic, since our calculation is based on the slow–roll approximation, which no longer works if becomes too large. We therefore demanded for all scales up to the PBH scale; the first eq.(3) shows that this corresponds to . This last requirement turns out to be the most constraining one when looking for combinations of parameters that give large .

## 4 Numerical Results

We are now ready to present some numerical results. We begin in
figure 2, which shows a scatter plot of the spectral parameters
and , which has been obtained by randomly
choosing model parameters [defined in eq.(3.19)], and
in the ranges^{16}^{16}16We actually only find acceptable solutions for
. . We require that and lie within
their ranges, and impose the consistency conditions discussed
above. The plot shows a very strong correlation between and
: if the latter is negative or small, the former is also small in
magnitude. Moreover, there are few points at large , and even there
most allowed combinations of parameters lead to very small . The
accumulation of points at small can be understood from our earlier
result (3.13), which showed that is naturally of order
within . Moreover, is naturally of
order . On the other hand, for values close to the
upper end of the current range, we do find some scenarios where
is sufficiently large to allow the formation of g PBHs.

We also explored the correlation between and (not shown). Here the only notable feature is the lower bound (3.16) on ; values of up to (and well beyond) its observational upper bound can be realized in this model for any value of within the presently allowed range. Similarly, we do not find any correlation between and . This lack of correlation can be explained through the denominator in eq.(3.13), which also appears (to the third power) in the expression for once eq.(3.12) has been used to trade for : this denominator can be made small through a cancellation, allowing sizable even if is very close to 1. Since the same denominator appears (albeit with different power) in the expressions for and , it does not destroy the correlation between these two quantities discussed in the previous paragraph.

Figure 2 indicates that large values of at the PBH scale can be achieved only if at the CMB scale is positive and not too small. In figure 3 we therefore explore the dependence of the potential parameters, and of the spectral parameters at the PBH scale on , for . Note that varying also changes the parameters and (or ), see eqs.(3.12) and (3.14). The latter in general has two solutions; however, for most values of , only one of them leads to sufficient inflation while keeping ; if both solutions are allowed, we take the one giving a larger spectral index at the PBH scale, taken to be Mpc corresponding to g. Note that and at the PBH scale are calculated exactly, using eqs.(3) and (3.25). We find that the expansion (2.13) is frequently very unreliable, e.g. giving the wrong sign for at the PBH scale for .

Figure 3 shows that is usually well below 1, as expected from the fact that , see the first eq.(3). Moreover, in most of the parameter space eq.(3.14) implies ; recall that this is also expected, since is a two–loop term. We find only if is small. In particular, the poles in shown in figure 3 occur only where vanishes; note that the spectral parameters remain smooth across these “poles”.

There are a couple of real discontinuities in figure 3, where the curves switch between the two solutions of eq.(3.14). The first occurs at . For smaller values of , the solution giving the smaller violates our slow–roll condition at scales close to the PBH scale. For larger this condition is satisfied. Just above the discontinuity, where is close to 2 at the PBH scale, we find the largest spectral index at the PBH scale, which is close to 0.47. Recall from figure 1 that this will generate sufficiently large density perturbations to allow the formation of PBHs with g. However, the formation of PBHs with this mass is possible only for a narrow range of , roughly .

At , goes through zero, giving a pole in as discussed above. Then, at , the second discontinuity occurs. Here the curves switch between the two solutions of eq.(3.14) simply because the second solution gives a larger spectral index at the PBH scale. Right at the discontinuity both solutions give the same spectral index, i.e. the curve depicting remains continuous; however, jumps from about to . The effective spectral index at the PBH scale also shows a small discontinuity. Recall from our discussion of eq.(2.15) that will generally be larger than at the PBH scale, but the difference between the two depends on the model parameters.

For very small values of , becomes very large; this region of parameter space is therefore somewhat pathological. For sizably positive , at the PBH scale increases slowly with increasing , while and both decrease. However, the spectral index at the PBH scale remains below the critical value for the formation of long–lived PBHs.

Note that always maintains its sign during inflation, since corresponds to a stationary point of the potential, which the (classical) inflaton trajectory cannot cross. For most of the parameter space shown in figure 3, decreases during inflation. If decreasing corresponds to , i.e. the inflaton rolls towards a minimum of the potential at . For we instead have , i.e. the inflaton rolls away from a maximum of the potential.

In fact, this latter situation also describes the branch of figure 3 giving the largest spectral index at the PBH scale; since here , increases during inflation on this branch. This is illustrated in figures 4, which show the (rescaled) inflaton potential as well as the effective spectral index as function of either the inflaton field (left frame) or of the scale (right frame). Note that all quantities shown here are dimensionless, and are determined uniquely by the dimensionless parameters defined in eq.(3.19), and . This leaves two dimensionful quantities undetermined, e.g. and ; one combination of these quantities can be fixed via the normalization of the CMB power spectrum, leaving one parameter undetermined (and irrelevant for our discussion).

The left frame shows that the (inverse) scale first increases quickly as rolls down from its initial value . This means that initially moves rather slowly, as can also be seen in the right frame. Since , the effective spectral index increases with increasing . The right frame shows that this evolution is quite nonlinear, although for , is to good approximation a parabolic function of . However, for even smaller scales, i.e. larger , the rate of growth of decreases again, such that reaches a value very close to at the scale relevant for the formation of PBHs with g. Recall that we only allow solutions where for the entire range of considered; figures 4 therefore illustrate our earlier statement that this constraint limits the size of the spectral index at PBH scales.

The left frame of figure 4 shows that the inflaton potential as written becomes unbounded from below for . This can be cured by introducing a quartic (or higher) term in the inflaton potential; the coefficient of this term should be chosen sufficiently small not to affect the discussion at the values of of interest to us. Note also that this pathology of our inflaton potential is not visible in the right frame, since assuming at , the inflaton field can never have been larger than : as noted above, it cannot have moved across the maximum of the potential.

Figure 2 indicated a strong dependence of the maximal spectral index at PBH scales on . This is confirmed by figure 5, which shows the maximal possible consistent with our constraints as function of , as well as the corresponding values of the parameters and . We saw in the discussion of eq.(3.14) that is only allowed for a narrow range of . In this very constrained corner of parameter space, remains less than , although the effective spectral index can exceed for ; recall that for the given choice , solutions only exist if , see eq.(3.16).

For the optimal set of parameters lies well inside the region of parameter space delineated by our constraints. therefore grows very fast with increasing . For these very small values of , the largest spectral index at PBH scales is always found for , which implies that the second derivative of the inflaton potential also vanishes at , i.e. corresponds to a saddle point, rather than an extremum, of the potential.

For slightly larger values of the choice allows less than 50 e–folds of inflation after exited the horizon. Requiring at least 50 e–folds of inflation therefore leads to a kink in the curve for . The optimal allowed parameter set now has considerable smaller , but larger .

The curve for the maximal shows a second kink at . To the right of this point the most important constraint is our requirement that at all scales up to , as discussed in connection with figures 3 and 4. Note that for , the optimal parameter choice leads to reaching its maximum at some intermediate close to, but smaller than, . This leads to a further flattening of the increase of .

We nevertheless see that for values of close to the upper end of the range specified in (3) the spectral index at the scale relevant for the formation of g PBHs can be well above the minimum for PBH formation found in section 2. Figure 1 then implies that the formation of considerably heavier PBHs might be possible in running mass inflation. However, larger PBH masses correspond to smaller , see eq.(2.10). This in turn allows for less running of the spectral index. In order to check whether even heavier PBHs might be formed during the slow–roll phase of running mass inflation, one therefore has to re–optimize the parameters for different choices of .