# PBH Dark Matter in Supergravity Inflation Models

###### Abstract

We propose a novel scenario to produce abundant primordial black holes (PBHs) in new inflation which is a second phase of a double inflation in the supergravity frame work. In our model, some preinflation phase before the new inflation is assumed and it would be responsible for the primordial curvature perturbations on the cosmic microwave background scale, while the new inflation produces only the small scale perturbations. Our new inflation model has linear, quadratic, and cubic terms in its potential and PBH production corresponds with its flat inflection point. The linear term can be interpreted to come from a supersymmetry-breaking sector, and with this assumption, the vanishing cosmological constant condition after inflation and the flatness condition for the inflection point can be consistently satisfied.

^{†}

^{†}preprint: IPMU 16-0087

## I Introduction

The primordial black holes (PBHs), which might be formed in the early radiation dominated universe, have attracted scientists for more than 40 years and are recently refocused on more and more. Theoretically they can be formed by the gravitational collapse of the Hubble patch if the mean energy density in that patch is higher by than its surroundings Hawking:1971ei ; Carr:1974nx ; Carr:1975qj . One of the main motivations of PBHs is dark matter (DM). Since they behave as a matter component, they can be a main component of DM without introducing other elementary particles. However PBHs are still not detected by any observation, and their abundance is constrained Josan:2009qn ; Carr:2009jm ; Griest:2013esa ; Graham:2015apa ; Tisserand:2006zx ; Ricotti:2007au ; Barnacka:2012bm ; Carr:2016drx , except in a mass window of , where PBHs can still account for all dark matter. Some recent attempts to close this window relied on assumptions that are difficult to justify. For example, the existence of neutron stars in globular clusters could help constrain the remaining window, if the density of dark matter exceeded the average by more than two orders of magnitude Capela:2013yf . However, observations of globular clusters show no evidence of significant dark matter content in such systems Bradford:2011aq ; Ibata:2012eq . Furthermore, it was suggested that tidal deformation of a neutron star could lead to an efficient energy dissipation and capture of a black hole, leading to stronger constraints Pani:2014rca , but such energy losses are uncertain, and they are likely to be suppressed for realistic parameters and velocities in excess of the speed of sound Capela:2014qea ; Defillon:2014wla , so that no new constraints can be derived, and the window for PBH dark matter remains open.

As one of the series of the PBH production works, several authors have studied the PBH formation in the double inflation
in the supergravity frame work Kawasaki:1997ju ; Kawasaki:1998vx ; Kawasaki:2006zv ; Kawaguchi:2007fz ; Frampton:2010sw ; Kawasaki:2012kn ; Kawasaki:2016ijp .^{1}^{1}1For other PBH production models, see Refs. of recent review Carr:2016drx .
Following this stream, we propose a novel new inflation model as the second phase of the double inflation,
where PBHs can be produced on mass enough to constitute the bulk of DM in this paper.^{2}^{2}2
In the supersymmetric frame work, the lightest supersymmetric particle (LSP) can be a candidate of WIMP DM. Here we assume that
the parity is broken and LSP is not stable, and then we need PBH DM instead.
As the preinflation before the new inflation, which is responsible for the curvature perturbations on the CMB scale, any specific model
is not required to be supposed. The potential of our new inflation model consists of the linear, quadratic, and -th moment terms
and we found that PBHs can be produced on enough to be a main component of DM.
Our model is based on the discrete symmetry and
we find that only the case could produce the desired spectrum of PBHs.
It is remarkable that the linear term can be consistently interpreted to come from the supersymmetry (SUSY) breaking sector in the case of
under the flatness condition at the inflection point and the vanishing cosmological constant condition.
While we propose the abundant PBHs of as DM,
we can make another peak in the PBH mass spectrum on . Those PBHs would have contributed to
gravitational waves which recently detected by LIGO/Virgo collaboration Abbott:2016blz ; Abbott:2016nhf
as discussed in Bird:2016dcv ; Clesse:2016vqa ; Sasaki:2016jop ; Eroshenko:2016hmn .

The rest of this paper is organized as follows. In Section II, we describe the constitution method of new inflation in supergravity and introduce our model. Then we concretely evaluate the current PBH abundance in Section III. We make our conclusions in Section IV. In Appendix A, we describe the treatments for the modes which exit the horizon at near the beginning of the second new inflation.

## Ii New inflation with inflection point in supergravity

Let us briefly review the composition of new inflation in supergravity at first. Here we adopt the model proposed in Ref. Kumekawa:1994gx ; Izawa:1997df . In this model, we assume a discrete symmetry which is broken down to a discrete during and after the inflation. The inflaton superfield has an charge 2. These assumptions lead the following effective superpotential at the leading order:

(1) |

Here and hereafter we use the Planck units where the reduced Planck mass is set to be unity. The -invariant effective Kähler potential can be written as,

(2) |

For these super and Kähler potentials, the inflaton potential in supergravity is given by,

(3) |

where . Defining the inflaton by the real part of , namely , it can be expanded as,

(4) |

and the slow-roll inflation can be driven either by the quadratic or -th moment term. Also it has a negative minimum at as,

(5) |

This negative energy would be canceled out after inflation with a positive contribution due to a SUSY-breaking effect. Therefore the gravitino mass can be related with the new inflation scale as,

(6) |

The inflaton mass around the potential minimum is given by,

(7) |

Therefore, if the inflaton decays into standard model particles simply by Planck suppressed operators, the reheating temperature can be evaluated by,

(8) |

In the next section, we will use this reheating temperature to calculate the PBH mass spectrum.^{3}^{3}3
For the parameters which we will use in the next section, the reheating temperature can be estimated as
with the above assumption. This value may be marginal to realize the thermal leptogenesis Fukugita:1986hr .
However we have checked that desired PBH mass spectra can be achieved even for a higher reheating temperature e.g.
.

Now let us consider the initial condition for this inflation.
Small field new inflation generally suffers from a severe initial condition problem.
That is, both the inflaton initial field value and its time derivative should be extremely small to
have a sufficiently long inflation, but originally there is no reason to stabilize the inflaton field to the potential origin
since the inflaton potential should be flat enough to satisfy the slow-roll conditions.
Moreover, even if one can introduce some stabilizing term in the potential, new inflation realizes eternal inflation
if the inflaton’s initial field value
is much smaller than the Hubble fluctuation and it should continue much longer than 60 e-folds (we want new inflation to
contribute only to small scale perturbations as we will mention).
As proposed in Izawa:1997df , these problems can be naturally solved in the supergravity frame work
by introducing a preinflation phase before the new inflation and adding a constant term to superpotential .^{4}^{4}4Note that,
in the original model Izawa:1997df ,
hybrid inflation is assumed as a preinflation, which gives a non-zero superpotential and leads the linear term in the potential of the new inflation.
However, since we have already introduced the constant term
in the superpotential, the preinflation does not need to give a non-zero superpotential in our model.
During the preinflation, the inflaton of the new inflation can have a Hubble induced mass term
of the potential.
Moreover the constant term in the superpotential leads the linear term in the potential, which shifts the potential minimum
from zero to .
The Hubble induced mass keeps stabilizing the inflaton even after the preinflation until the beginning of the second new inflation
, and therefore the initial field value of is given by, through the coefficient

(9) |

The new inflation can avoid eternal inflation as long as is sufficiently larger than the Hubble fluctuation at the beginning of the new inflation.

Taking above things, we consider the PBH formation in the following model:

(10) |

Here note that, if , the linear, quadratic, and -th moment terms can have an inflection point for . Since is locally maximized at that inflection point, if the maximum of is still negative but quite close to zero, the slow-roll inflation are not spoiled and moreover very large curvature perturbations can be produced. The inflection point can be obtained as,

(11) |

Then we require the flat inflection condition as,

(12) |

neglecting and the higher order term. With this condition, the curvature perturbations generated around would be large enough to produce abundant PBHs, and the concrete successful parameters will be shown in the next section. Here note that the inflection point can be written as . That is because the initial field value is determined by the balance between the linear term and the preinflaiton-induced mass term , while the flat inflection is the point where the linear term is comparable with the self-induced mass term which is smaller than the preinflation-induced mass term by a factor . under this condition and therefore it is automatically set to be slightly larger than the initial field value for

If the new inflation is assumed to realize both small perturbations like as those on the CMB scale and large perturbations which would cause the formation of PBHs, it generally takes too many e-folds in the transition from small to large perturbations since the time derivative of the slow-roll parameter itself is suppressed by the slow-roll parameters, where denotes the e-folding number and and represent the slow-roll parameters respectively. However, since we have already introduced double inflation, the new inflation can be free from the COBE normalization by simply assuming that the preinflation is responsible for the CMB scale perturbations, and then the new inflation can end in sufficiently short time in that case. Indeed we will show in the next section that there are parameter regions where PBHs can be produced enough to be a main component of DM and the constraints for large scale perturbations like the CMB spectral distortion can be avoided. Here if one assume that the curvature perturbations generated in the new inflation are large even apart from the inflection point, the linear term itself is required to be small enough. Letting denote the amplitude of the power spectrum of the curvature perturbations during the linear term in the potential dominantly contributes to the perturbations, the constant superpotential is determined through the following relation: and

(13) |

Therefore, for , is required to be as small as .

The above two conditions (II) and (13) are required for PBH formation. Now combining them clarifies the -dependence of as .
Therefore, in the case of , does not depend on the new inflation scale and could be smaller than unity.
On the other hand, for , becomes much larger than unity for and such a large would spoil unitarity of the theory
for example Harigaya:2013pla .
Moreover for large the duration of the new inflation after the inflection point will be short due to its steep potential
and it tends to make the PBH mass small. Indeed we have checked that the PBH mass spectrum are tilted to
the lighter mass and conflicts with the constraints on the PBH abundance for .
For those reasons, we will concentrate on the case of hereafter.^{5}^{5}5In addition,
is uniquely favored by the anomaly free conditions for supersymmetric standard gauge groups
with the discrete symmetry Evans:2011mf .

As an interesting fact, the small constant term in the superpotential can be interpreted to come from the SUSY-breaking sector
in the case of Takahashi:2013cxa .^{6}^{6}6Note that the case of is considered in Ref. Takahashi:2013cxa
since their motivation is not to produce PBHs but to modify the spectral index in new inflation and therefore the required condition is different.
The SUSY-breaking F-term order naively arise from the term like,

(14) |

If obtains a vev , this term can lead the constant superpotential . Indeed it can be realized in the dynamical SUSY breaking models proposed in Ref. Izawa:1996pk ; Intriligator:1996pu if the origin of is destabilized due to a large Yukawa coupling (), and the estimated constant term is given by where the dynamical scale is related with by . On the other hand, under the flat inflection condition (II) and the large curvature perturbation condition (13), the vanishing cosmological constant condition (6) gives the parameter dependence of the SUSY-breaking scale as is consistent with the above assumption if and only if as, , neglecting numerical factors. Therefore the scale dependence of the constant term

(15) |

It can be checked that this consistency is retained for the concrete parameter values which we will show in the next section even if numerical factors are included.

## Iii Formation of primordial black holes

In this section, we will calculate the PBH abundance and exemplify parameter sets where PBHs constitute the main component of DM. At first, given the inflation scale and the reheating temperature , the perturbation scale can be related with the backward e-folds by Lyth:2009zz ,

(16) |

We will use Eq. (8) as the reheating temperature hereafter. Then, with the potential (4) and the Hubble induced mass , we can calculate the power spectrum of the curvature perturbations in the standard linear theory and the result is shown in Fig. 1 as the black thick line for the following parameter values:

(17) |

In this calculation, we have simply assumed that the preinflation potential behaves as a matter component after the preinflation. For these parameters, the vanishing cosmological constant (6) and the SUSY-breaking assumption (15) consistently predict which suggests the pure gravity mediation Ibe:2011aa . In this figure, we also show the constraints from the CMB spectral -distortion as the red region. The CMB -distortion from the Silk damping of a single -mode can be approximated by Kohri:2014lza ,

(18) |

where represent the wavenumber in . We have used the current 2 upper limit by the COBE/FIRAS experiment Fixsen:1996nj . Finally, the modes actually reenter the horizon between the preinflation and the second new inflation. We will describe the treatments of these modes in Appendix A.

With use of this power spectrum, the PBH abundance can be calculated as follows. At first, the mass of PBH is almost given by the horizon mass when the overdensity reenters the horizon, and let denote the ratio between them here. That is, the PBH mass corresponding with the scale is given by,

(19) |

where denotes the horizon mass at the matter-radiation equality calculated as,

(20) |

Also we have used an approximation that the effective d.o.f. for energy density is almost equal to that for entropy density . Using , , , and , we can finally obtain Josan:2009qn ,

(21) |

where is the solar mass. In the simple analytic calculation Carr:1975qj , is evaluated as and we will use this value hereafter.

The formation rate of PBHs is given by the probability of excess over the threshold.
That is, under the assumption that the density perturbations follow the Gaussian distribution,^{7}^{7}7
Note that the non-Gaussianity (NG) is expected to be small since the second new inflation is driven almost only by a single scalar field
around the inflection point. Indeed we have briefly checked that the local non-linearity parameter is as small as
and for such a small NG it is known that the predicted PBH abundance is hardly affected Saito:2008em ; Byrnes:2012yx ; Young:2013oia .
However the second peak on which we will show later might be modified by NG effects since it corresponds with
the phase of the beginning of the new inflation. We leave this problem for future works.

(22) |

Here represents the threshold density perturbation and we will adopt the simple analytic estimation Carr:1975qj . is the variance of the comoving density perturbations coarse-grained on , which is given by Young:2014ana ,

(23) |

represents the Fourier transformed window function and we will adopt the Gaussian window in the following calculations. Note that the PBH mass and coarse-graining scale are related by Eq. (21). The formation rate directly gives the ratio of the PBH energy density to the total energy density at the horizon reentering, . Therefore the current PBH fraction to DM for a single mass mode can be derived as,

(24) |

where represents the current DM abundance Ade:2015xua . The resultant PBH fraction is plotted in Fig. 2 as the black thick line. We also show several observational constraints for PBH abundance as red regions Carr:2009jm ; Griest:2013esa ; Graham:2015apa ; Tisserand:2006zx ; Ricotti:2007au ; Barnacka:2012bm ; Carr:2016drx . Here we have not used the constraints from existence of neutron stars Capela:2013yf ; Pani:2014rca and white dwarfs Capela:2012jz in globular clusters since they require the high DM-density assumption whose validity seems questionable as mentioned in introduction. Although the shown PBH fraction does not reach unity in each logarithmic mass bin, the total PBH fraction:

(25) |

reaches unity with the current parameters. Therefore our model can describe the formation of sufficient PBHs to be a main component of DM. One might think that the PBH mass spectrum can be shifted to the more massive direction and even the neutron stars constraints could be avoided for some parameters. However we have checked that the -distortion constraints shown in Fig. 1 as the red line cannot be satisfied in that case.

Before ending this section, let us concentrate on the sharp peaks of the power spectrum around shown in Fig. 1. This peak comes from the fact that at the beginning of the new inflation the Hubble induced mass cannot be neglected yet and the inflaton potential is being slightly flattened due to its stabilizing effect at first. For a slightly smaller (namely a slightly flatter linear potential), this peak can be enlarged and another peak of the fraction for more massive PBHs can be shown. For the following parameters:

(26) |

the resultant power spectrum and PBH fraction are plotted in Fig. 1 and 2 as the black dashed lines.
Again the total PBH fraction is about unity () for them,
but they show another peak at . Recently LIGO/Virgo collaboration succeeded in the first direct detection of gravitational waves GW150914,
which came from an inspiral and merger of a black hole binary Abbott:2016blz . The masses of these black holes are estimated as and .
Taking this, the binary formation rate of PBHs whose masses are around has been evaluated by several
authors Bird:2016dcv ; Clesse:2016vqa ; Sasaki:2016jop ; Eroshenko:2016hmn ,
and the authors of Sasaki:2016jop claimed that the binary formation rate of PBHs
satisfying current constraints () would be high enough to be consistent with
the black hole merger rate inferred from LIGO observations Abbott:2016nhf
(however it was claimed in Eroshenko:2016hmn that the binary formation rate might be smaller than that evaluated in Sasaki:2016jop ).
Therefore our model might explain both of DM whose main component is as PBHs and
GW150914 from the merger of the PBH binary.^{8}^{8}8Recently, in Ref. Cheng:2016qzb , it has also been proposed that
-PBH could be produced by the gauge field production with the Chern-Simons coupling .

However, the existence of the peak around in our model should be examined more carefully since the corresponding fluctuation modes reenter the horizon between the preinflation and the new inflation and in such a case effects of the metric perturbations that are not included in the present analysis might be important . So we need to solve full evolutions of fluctuations of scalar fields and metric perturbations, which will be investigated in future work.

## Iv Conclusions

In this paper, we proposed a new inflation model as a second phase of a double inflation consistently constituted in supergravity frame work, where sufficient primordial black holes (PBHs) can be produced to be a main component of dark matter (DM). Any specific inflationary model is not required for the preinflation which is responsible for the large scale curvature perturbations, as long as it is consistent with the observations of, e.g., CMB. The potential of the new inflation in our model consists of the linear, quadratic, and cubic terms and has a flat inflection point where PBHs can be produced. The specific power spectra of the curvature perturbations and the resultant PBH mass spectra are shown in Fig. 1 and 2 for two parameter sets (17) and (26). The inflection point corresponds with PBHs and they constitute the bulk of DM in our model. In addition, we can make another peak for the PBH mass spectrum on as indicated by the black dashed line in Fig. 2. Such a peak corresponds with the beginning of the new inflation and those PBHs might cause the gravitational waves detected by LIGO/Virgo collaboration.

###### Acknowledgements.

We thank Keisuke Inomata for useful comments and discussion. This work is supported by MEXT KAKENHI Grant Number 15H05889 (M.K.), 16H02176 (T.T.Y.), JSPS KAKENHI Grant Number 25400248 (M.K.), 26104009 and 26287039 (T.T.Y.), and also by the World Premier International Research Center Initiative (WPI), MEXT, Japan. A.K. is supported by the U. S. Department of Energy Grant DE-SC0009937. Y.T. is supported by JSPS Research Fellowship for Young Scientists.## Appendix A Multiple horizon crossing modes

In this appendix, we will describe the treatments for the modes which exit the horizon at the end of the preinflation once, enter the horizon between two inflations, and then reexit the horizon at the beginning of the second new inflation. They correspond with the modes for in our parameters. Their dynamics are non-trivial due to their multiple horizon crossing and the amplitudes of them at the second horizon exit which determine the curvature perturbations in the new inflation have to be evaluated carefully. To do so, one must solve EoM for perturbations continuously over the two inflations and connection phase, including the effects of the metric perturbations. However here let us roughly estimate them in the super- and subhorizon limit without metric perturbations.

The linear EoM for perturbations which have a Hubble induced mass is given by,

(27) |

In the second line, we have used the super- and subhorizon limit. The subhorizon EoM can be always rewritten, with use of the conformal time and in the subhorizon limit, as,

(28) |

and therefore it only has oscillating solutions whose amplitudes decrease as . On the other hand, assuming that the background EoS is given by (), the superhorizon EoM reads,

(29) |

It can be easily solved by assuming the power-law solution and the real part of the power is given by . That is, the amplitude of the solutions damps as . Also, in the exact de Sitter background, the Hubble parameter is constant and the two solutions are soon found as

Now let us evaluate the concrete amplitude of the multiple horizon crossing modes at the horizon exit during the new inflation. Letting and denote the horizon scale at the end of the preinflation and the beginning of the new inflation, such modes correspond with . We illustrate an schematic image about the relation between the wavelength and the horizon scale in Fig. 3 as the solid and dot-dashed lines respectively. At the first horizon exit during the preinflation, the perturbation amplitude is given by the standard one . After that, the amplitude decreases until the second horizon exit during the new inflation as,

(30) |

following the scale factor dependences which we showed previously. Here the subscript 1, 2, and 3 represent each horizon crossing time during the preinflation, between two inflations, and during the new inflation respectively. Also we have referred to the EoS between two inflations as . Noting that the horizon scale is proportional to , it can be rewritten as,

(31) |

Finally, using , we can obtain,

(32) |

That is, the resultant amplitude is eventually identical with the standard one even though they experienced a complicated process.

The results obtained in this appendix would be changed if the effects of, e.g., the metric perturbations or the resonance. For example, if one assumes the mass term chaotic inflation as the preinflation, the Hubble induced mass represents the inflaton of the preinflation, leads the parametric resonance Kofman:1994rk . However this problem would be solved by introducing a Kähler coupling where is the preinflaton superfield, since this Kähler potential brings coupling for example and it cancels and reduces the oscillation of the Hubble induced mass. Also the metric perturbations would also grow density perturbations on subhorizon scales Jedamzik:2010dq ; Easther:2010mr . However the peak of the PBH mass spectrum on is mainly caused by the largest scale mode which is hardly affected by the subhorizon effect. Moreover, even if the amplitude of the perturbations would be slightly modified, it could be absorbed into the parameter tuning. Anyway more strict analysis for those modes is postponed to future works. where

## References

- (1) S. Hawking, Mon. Not. Roy. Astron. Soc. 152, 75 (1971).
- (2) B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron. Soc. 168, 399 (1974).
- (3) B. J. Carr, Astrophys. J. 201, 1 (1975). doi:10.1086/153853
- (4) A. S. Josan, A. M. Green and K. A. Malik, Phys. Rev. D 79, 103520 (2009) doi:10.1103/PhysRevD.79.103520 [arXiv:0903.3184 [astro-ph.CO]].
- (5) B. J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, Phys. Rev. D 81, 104019 (2010) doi:10.1103/PhysRevD.81.104019 [arXiv:0912.5297 [astro-ph.CO]].
- (6) P. Tisserand et al. [EROS-2 Collaboration], Astron. Astrophys. 469, 387 (2007) doi:10.1051/0004-6361:20066017 [astro-ph/0607207].
- (7) M. Ricotti, J. P. Ostriker and K. J. Mack, Astrophys. J. 680, 829 (2008) doi:10.1086/587831 [arXiv:0709.0524 [astro-ph]].
- (8) A. Barnacka, J. F. Glicenstein and R. Moderski, Phys. Rev. D 86, 043001 (2012) doi:10.1103/PhysRevD.86.043001 [arXiv:1204.2056 [astro-ph.CO]].
- (9) K. Griest, A. M. Cieplak and M. J. Lehner, Phys. Rev. Lett. 111, no. 18, 181302 (2013). doi:10.1103/PhysRevLett.111.181302
- (10) P. W. Graham, S. Rajendran and J. Varela, Phys. Rev. D 92, no. 6, 063007 (2015) doi:10.1103/PhysRevD.92.063007 [arXiv:1505.04444 [hep-ph]].
- (11) B. Carr, F. Kuhnel and M. Sandstad, arXiv:1607.06077 [astro-ph.CO].
- (12) F. Capela, M. Pshirkov and P. Tinyakov, Phys. Rev. D 87, no. 12, 123524 (2013) doi:10.1103/PhysRevD.87.123524 [arXiv:1301.4984 [astro-ph.CO]].
- (13) J. D. Bradford et al., Astrophys. J. 743, 167 (2011) Erratum: [Astrophys. J. 778, 85 (2013)] doi:10.1088/0004-637X/743/2/167, 10.1088/0004-637X/778/1/85 [arXiv:1110.0484 [astro-ph.CO]].
- (14) R. Ibata, C. Nipoti, A. Sollima, M. Bellazzini, S. Chapman and E. Dalessandro, Mon. Not. Roy. Astron. Soc. 428, 3648 (2013) doi:10.1093/mnras/sts302 [arXiv:1210.7787 [astro-ph.CO]].
- (15) P. Pani and A. Loeb, JCAP 1406, 026 (2014) doi:10.1088/1475-7516/2014/06/026 [arXiv:1401.3025 [astro-ph.CO]].
- (16) F. Capela, M. Pshirkov and P. Tinyakov, arXiv:1402.4671 [astro-ph.CO].
- (17) G. Defillon, E. Granet, P. Tinyakov and M. H. G. Tytgat, Phys. Rev. D 90, no. 10, 103522 (2014) doi:10.1103/PhysRevD.90.103522 [arXiv:1409.0469 [gr-qc]].
- (18) M. Kawasaki, N. Sugiyama and T. Yanagida, Phys. Rev. D 57, 6050 (1998) doi:10.1103/PhysRevD.57.6050 [hep-ph/9710259].
- (19) M. Kawasaki and T. Yanagida, Phys. Rev. D 59, 043512 (1999) doi:10.1103/PhysRevD.59.043512 [hep-ph/9807544].
- (20) M. Kawasaki, T. Takayama, M. Yamaguchi and J. Yokoyama, Phys. Rev. D 74, 043525 (2006) doi:10.1103/PhysRevD.74.043525 [hep-ph/0605271].
- (21) T. Kawaguchi, M. Kawasaki, T. Takayama, M. Yamaguchi and J. Yokoyama, Mon. Not. Roy. Astron. Soc. 388, 1426 (2008) doi:10.1111/j.1365-2966.2008.13523.x [arXiv:0711.3886 [astro-ph]].
- (22) P. H. Frampton, M. Kawasaki, F. Takahashi and T. T. Yanagida, JCAP 1004, 023 (2010) doi:10.1088/1475-7516/2010/04/023 [arXiv:1001.2308 [hep-ph]].
- (23) M. Kawasaki, A. Kusenko and T. T. Yanagida, Phys. Lett. B 711, 1 (2012) doi:10.1016/j.physletb.2012.03.056 [arXiv:1202.3848 [astro-ph.CO]].
- (24) M. Kawasaki, K. Mukaida and T. T. Yanagida, arXiv:1605.04974 [hep-ph].
- (25) B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett. 116, no. 6, 061102 (2016) doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]].
- (26) B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], arXiv:1602.03842 [astro-ph.HE].
- (27) S. Bird, I. Cholis, J. B. Munoz, Y. Ali-Haimoud, M. Kamionkowski, E. D. Kovetz, A. Raccanelli and A. G. Riess, Phys. Rev. Lett. 116, no. 20, 201301 (2016) doi:10.1103/PhysRevLett.116.201301 [arXiv:1603.00464 [astro-ph.CO]].
- (28) S. Clesse and J. García-Bellido, arXiv:1603.05234 [astro-ph.CO].
- (29) M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama, arXiv:1603.08338 [astro-ph.CO].
- (30) Y. N. Eroshenko, arXiv:1604.04932 [astro-ph.CO].
- (31) K. Kumekawa, T. Moroi and T. Yanagida, Prog. Theor. Phys. 92, 437 (1994) doi:10.1143/PTP.92.437 [hep-ph/9405337].
- (32) K. I. Izawa, M. Kawasaki and T. Yanagida, Phys. Lett. B 411, 249 (1997) doi:10.1016/S0370-2693(97)01040-X [hep-ph/9707201].
- (33) M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986). doi:10.1016/0370-2693(86)91126-3
- (34) K. Harigaya, M. Ibe and T. T. Yanagida, Phys. Rev. D 89, no. 5, 055014 (2014) doi:10.1103/PhysRevD.89.055014 [arXiv:1311.1898 [hep-ph]].
- (35) J. L. Evans, M. Ibe, J. Kehayias and T. T. Yanagida, Phys. Rev. Lett. 109, 181801 (2012) doi:10.1103/PhysRevLett.109.181801 [arXiv:1111.2481 [hep-ph]].
- (36) F. Takahashi, Phys. Lett. B 727, 21 (2013) doi:10.1016/j.physletb.2013.10.026 [arXiv:1308.4212 [hep-ph]].
- (37) K. I. Izawa and T. Yanagida, Prog. Theor. Phys. 95, 829 (1996) doi:10.1143/PTP.95.829 [hep-th/9602180].
- (38) K. A. Intriligator and S. D. Thomas, Nucl. Phys. B 473, 121 (1996) doi:10.1016/0550-3213(96)00261-1 [hep-th/9603158].
- (39) D. H. Lyth and A. R. Liddle, Cambridge, UK: Cambridge Univ. Pr. (2009) 497 p
- (40) M. Ibe and T. T. Yanagida, Phys. Lett. B 709, 374 (2012) doi:10.1016/j.physletb.2012.02.034 [arXiv:1112.2462 [hep-ph]].
- (41) K. Kohri, T. Nakama and T. Suyama, Phys. Rev. D 90, no. 8, 083514 (2014) doi:10.1103/PhysRevD.90.083514 [arXiv:1405.5999 [astro-ph.CO]].
- (42) D. J. Fixsen, E. S. Cheng, J. M. Gales, J. C. Mather, R. A. Shafer and E. L. Wright, Astrophys. J. 473, 576 (1996) doi:10.1086/178173 [astro-ph/9605054].
- (43) R. Saito, J. Yokoyama and R. Nagata, JCAP 0806, 024 (2008) doi:10.1088/1475-7516/2008/06/024 [arXiv:0804.3470 [astro-ph]].
- (44) C. T. Byrnes, E. J. Copeland and A. M. Green, Phys. Rev. D 86, 043512 (2012) doi:10.1103/PhysRevD.86.043512 [arXiv:1206.4188 [astro-ph.CO]].
- (45) S. Young and C. T. Byrnes, JCAP 1308, 052 (2013) doi:10.1088/1475-7516/2013/08/052 [arXiv:1307.4995 [astro-ph.CO]].
- (46) S. Young, C. T. Byrnes and M. Sasaki, JCAP 1407, 045 (2014) doi:10.1088/1475-7516/2014/07/045 [arXiv:1405.7023 [gr-qc]].
- (47) P. A. R. Ade et al. [Planck Collaboration], arXiv:1502.01589 [astro-ph.CO].
- (48) F. Capela, M. Pshirkov and P. Tinyakov, Phys. Rev. D 87, no. 2, 023507 (2013) doi:10.1103/PhysRevD.87.023507 [arXiv:1209.6021 [astro-ph.CO]].
- (49) S. L. Cheng, W. Lee and K. W. Ng, arXiv:1606.00206 [astro-ph.CO].
- (50) L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. Lett. 73, 3195 (1994) doi:10.1103/PhysRevLett.73.3195 [hep-th/9405187].
- (51) K. Jedamzik, M. Lemoine and J. Martin, JCAP 1009, 034 (2010) doi:10.1088/1475-7516/2010/09/034 [arXiv:1002.3039 [astro-ph.CO]].
- (52) R. Easther, R. Flauger and J. B. Gilmore, JCAP 1104, 027 (2011) doi:10.1088/1475-7516/2011/04/027 [arXiv:1003.3011 [astro-ph.CO]].