Inflationary Primordial Black Holes as All Dark Matter

TL;DR

This paper revisits the hypothesis that primordial black holes could account for all dark matter, incorporating new microlensing constraints and proposing an inflation model that produces the required PBH mass distribution.

Contribution

It updates observational constraints on PBH mass functions and introduces an inflation model that generates PBHs matching these constraints as all dark matter.

Findings

Viable PBH mass peak at ~10^{20} g with narrow width

Updated constraints restrict PBH mass function

Proposed inflation model successfully produces PBHs as all dark matter

Abstract

Following a new microlensing constraint on primordial black holes (PBHs) with $\sim10^{20}$--$10^{28}\,\mathrm{g}$~[1], we revisit the idea of PBH as all Dark Matter (DM). We have shown that the updated observational constraints suggest the viable mass function for PBHs as all DM to have a peak at $\simeq 10^{20}\,\mathrm{g}$ with a small width $\sigma \lesssim 0.1$, by imposing observational constraints on an extended mass function in a proper way. We have also provided an inflation model that successfully generates PBHs as all DM fulfilling this requirement.

Equations

$$M(k)$$

$$\simeq 10^{20}\mathrm{g}\left(\frac{\gamma }{0.2}\right){\left(\frac{g_{\ast }}{106.75}\right)}^{-\frac{1}{6}}{\left(\frac{k}{7\times 10^{12}{\text{Mpc}}^{-1}}\right)}^{-2}.$$

$$\beta (M)={\int }_{{\delta }_c}\frac{\mathrm{d}\delta }{\sqrt{2\pi {\sigma }^2(M)}}{e}^{-\frac{{\delta }^2}{2{\sigma }^2(M)}}\simeq \frac{1}{\sqrt{2\pi }}\frac{1}{{\delta }_c/\sigma (M)}{e}^{-\frac{{\delta }_c^2}{2{\sigma }^2(M)}}.$$

$${\sigma }^2(M(k))=\int \mathrm{d}\ln q{W}^2(q{k}^{-1})\frac{16}{81}{\left(q{k}^{-1}\right)}^4{\mathcal{P}}_{\zeta }(q),$$

$$f_{\text{PBH}}(M)\equiv \frac{{\Omega }_{\text{PBH}}(M)}{{\Omega }_c}={\frac{{\rho }_{\text{PBH}}}{{\rho }_m}|}_{\text{eq}}\frac{{\Omega }_m{h}^2}{{\Omega }_c{h}^2}=\left(\frac{T_M}{T_{\text{eq}}}\frac{{\Omega }_m{h}^2}{{\Omega }_c{h}^2}\right)\gamma \beta (M)$$

$$\simeq \left(\frac{\beta (M)}{8.0\times 10^{-15}}\right)\left(\frac{0.12}{{\Omega }_c{h}^2}\right){\left(\frac{\gamma }{0.2}\right)}^{\frac{3}{2}}{\left(\frac{106.75}{g_{\ast }(T_M)}\right)}^{\frac{1}{4}}{\left(\frac{M}{10^{20}\mathrm{g}}\right)}^{-\frac{1}{2}}.$$

$${\Omega }_{\text{PBH,tot}}=\int \mathrm{d}\ln M{\Omega }_{\text{PBH}}(M).$$

$$V(\varphi ,\phi )=$$

$$V_{\text{new}}(\phi )=$$

$$V_{\text{stb}}(\varphi ,\phi )=$$

$${\mathcal{P}}_{\zeta }$$

$$\simeq {\left[\frac{v^2}{2\sqrt{3}\pi M_{\mathrm{Pl}}^2}\frac{1}{\epsilon +\kappa \frac{\phi }{M_{\mathrm{Pl}}}+2ng\frac{{\phi }^{n-1}}{v^2{M}_{\mathrm{Pl}}^{n-3}}}\right]}^2.$$

$$\frac{\mathrm{d}\log {\mathcal{P}}_{\zeta }}{\mathrm{d}\log k}=n_{{}_{\mathrm{S}}}-1=-6{ϵ}_V+2{\eta }_V\simeq 2{\eta }_V\simeq 2\kappa ,$$

$${\mathcal{P}}_{\zeta }\sim \{\begin{array}{ll}{{\mathcal{P}}_{\zeta }|}_{\text{ch}}\sim 10^{-9}& \text{for}{k}_i\gtrsim k,\\ \frac{v^4}{12{\pi }^2{M}_{\mathrm{Pl}}^4{\epsilon }^2}{\left(\frac{k_i}{k}\right)}^{2\kappa }& \text{for}k\gtrsim k_i,\end{array}$$

$$\frac{\mathrm{d}}{\mathrm{d}M}\frac{{\Omega }_{\text{PBH}}(M)}{{\Omega }_c}=N\exp \left[\frac{-(\log M-\log M_{\ast }{)}^2}{2{\sigma }^2}\right],$$

$$n$$

$$g$$

$$\bar{N}>N(f)\to 1>\frac{1}{{\bar{f}}_{i:i+1}}{\int }_{M_i}^{M_{i+1}}\mathrm{d}\ln Mf(M).$$

$$1>\frac{1}{{\bar{f}}_{i-1:i}}{\int }_{M_{i-1}}^{M_i}\mathrm{d}\ln Mf(M)+\frac{1}{{\bar{f}}_{i:i+1}}{\int }_{M_i}^{M_{i+1}}\mathrm{d}\ln Mf(M).$$

$$1>\sum _{i=1}^{N-1}{\int }_{M_i}^{M_{i+1}}\mathrm{d}\ln M\frac{f(M)}{f_{i:i+1}^{(x)}}\to 1>{\int }_{M_1}^{M_N}\ln M\frac{f(M)}{f^{(x)}(M)}.$$

Full text

††preprint: IPMU17-0009

I Introduction

Dark matter (DM) is one of the outstanding problems, which motivates us to seek for new particle physics models beyond the Standard Model (SM). Its existence is well established by the astrophysical and cosmological observations. However, we still do not know most of its property. For instance, its possible mass scale ranges large order of magnitude from $10^{-31}$ GeV to $10^{50}$ GeV. Primordial black hole (PBH) Hawking (1971); Carr and Hawking (1974); Carr (1975) resides in the heavy end of various candidates of DM. It behaves as cold matter, is stable for sufficiently heavy ones, and thus is a perfect candidate of DM. In particular, we do not need new particles beyond the SM. Therefore, whether or not PBHs can be a dominant component of DM is an important issue for particle physics.

PBHs are formed if large density perturbations collapse overcoming the pressure forces. Cosmic inflation can be a source of such large density perturbations. For instance, if the potential has a plateau during the inflationary epoch, large superhorizon fluctuations are produced, and they eventually collapse to form PBHs at the horizon reentry. Although the amplitude of scalar perturbations is strictly constrained at the large scale, say $0.0002$–$1$ Mpc*-1* Ade et al. (2016), by the cosmic microwave background (CMB) observation, they can be significant at the smaller scale, which opens up a possibility to produce a sizable amount of PBHs comparable to the current DM density.

Recently, making use of the Subaru Hyper Suprime-Cam (HSC) dense cadence data, Niikura et al. Niikura et al. put the most stringent upper bounds on the PBH abundance in the mass range $10^{20}\text{--}10^{28}\,\mathrm{g}$.111

Throughout this paper, we assume a conservative value for the abundance of DM inside the globular clusters, and neglect the NS constraints Kusenko and Rosenberg (2013); Carr et al. (2016); Clesse and García-Bellido (2016).

In this paper, we show that when we combine it with other observational constraints, only the mass region $\simeq 10^{20}\,\mathrm{g}$ remains for PBHs as all DM, by imposing observational constraints on the extended mass function in a proper way. We also show that the PBHs, which have an inflationary origin, can still be a dominant component of DM of mass $\simeq 10^{20}\,\mathrm{g}$, taking the model Kawasaki et al. (1998); Kawasaki and Yanagida (1999); Frampton et al. (2010); Kawasaki et al. (2012, 2016a, 2016b); Inomata et al. (2016) as an example.

II Formation of PBHs during Inflation

The property of PBH is characterized by its mass and abundance. In the following, we briefly summarize the formation of PBHs by large superhorizon fluctuations. We adopt the conventional analysis for the formation of PBHs Carr (1975); Green and Liddle (1997). See Niemeyer and Jedamzik (1999); Shibata and Sasaki (1999); Musco et al. (2005); Musco and Miller (2013); Harada et al. (2013); Nakama et al. (2014) for attempts to refine the simple analysis.

When an over-dense region above the threshold, $\delta\rho/\rho>\delta_{c}$, reenters the horizon, with the threshold being $\delta_{c}$, it may overcome the pressure and collapse to form the PBHs. There exist many attempts to pin down the threshold value, but here we adopt the conventional one $\delta_{c}=1/3$ as a reference. In the simple analysis, the mass of PBH is proportional to the horizon mass at that time. Thus, it is estimated by

[TABLE]

Here we used an approximation that the effective degrees of freedom for energy density $g_{*}$ is almost equal to that for entropy density $g_{*s}$. $\gamma$ represents the ratio between the PBH mass and the horizon mass, which is estimated as $\gamma\simeq 3^{-3/2}$ in the simple analytical result Carr (1975). $M(k)$ denotes the mass of PBH that is formed when the comoving momentum $k$ reenters the horizon. $k_{\text{eq}}$ is the comoving momentum at the matter-radiation equality, i.e., $k_{\text{eq}}=a_{\text{eq}}H_{\text{eq}}$. $M_{\text{eq}}$ is the horizon mass at the same time.

The formation rate of PBHs with mass $M$, $\beta(M)$, is given by the probability of exceeding the threshold $\delta_{c}$. We assume that the density perturbation is governed by the Gaussian statistics. Then, the formation rate is given by

[TABLE]

$\sigma^{2}(M)$ represents the standard deviation of the coarse-grained density contrast for the PBH mass of $M$ Young et al. (2014)

[TABLE]

where $W$ is the Fourier transform of the window function smearing over $k^{-1}$, and we adopt the Gaussian window $W(x)=e^{-x^{2}/2}$. At the horizon reentry, a fraction of the total energy of the Universe, $\gamma\beta(M(k))\rho|_{k=aH}$, turns into PBHs. After their formation, $\rho_{\text{PBH}}/\rho$ grows inversely proportional to the cosmic temperature until the matter-radiation equality, since PBHs behave as matter. Thus, the abundance of PBHs with mass $M$ over logarithmic mass interval $\mathrm{d}\ln M$ may be estimated as

[TABLE]

$T_{\text{eq}}$ indicates the temperature at the matter-radiation equality and $T_{M}$ is the temperature at the formation of a PBH with a mass $M$. $\Omega_{m}$ ($\Omega_{c}$) is the current density function of matter (DM) where we used the recent Planck’s result $\Omega_{c}h^{2}\simeq 0.12$ Ade et al. (2016). The total abundance of PBH is obtained from

[TABLE]

As one can see from obtained equations, a sizable amount of PBHs is produced if the scalar perturbations are significant $\mathcal{P}_{\zeta}\sim 10^{-2}$. Thus, the problem is how to generate such large scalar perturbations without conflicting the CMB observation.

Double Inflation.

To make our discussion concrete, we adopt the double inflation scenario proposed in Ref. Kawasaki et al. (1998) as an example, which involves new inflation as the second inflation. (See also Refs. Kawasaki and Yanagida (1999); Frampton et al. (2010); Kawasaki et al. (2012, 2016a, 2016b).) We assume chaotic inflation as the pre-inflation before new inflation. The pre-inflation dynamically determines the initial condition of the new inflation, which solves the crucial drawback of the new inflation, namely the initial condition problem Izawa et al. (1997). In addition, the chaotic inflation is responsible for the large-scale perturbations, $k\lesssim 1$ Mpc*-1*, observed by Planck, and hence the new inflation can be free from the COBE normalization, which allows much larger scalar perturbations at the smaller scale.

Hereafter, we phenomenologically take the following potential:222 In general, one also expects the Planck-suppressed operators for kinetic terms. Here we suppressed them for simplicity. See Ref. Inomata et al. (2016) for their possible effects on the formation of PBHs.

[TABLE]

$\phi$ and $\varphi$ are the inflatons responsible for chaotic inflation and the new inflation respectively. $v$ is the scale of the new inflation. $c_{\text{pot}}$, $g$, $\kappa$, and $\varepsilon$ are dimensionless parameters. $V_{\text{ch}}(\phi)$ and $V_{\text{new}}(\varphi)$ are the potentials for chaotic inflation and the new inflation respectively. $V_{\text{stb}}(\phi,\varphi)$ stabilizes $\varphi$ during chaotic inflation for $c_{\text{pot}}\gtrsim\mathcal{O}(1)$. For simplicity, we assume $V_{\text{ch}}(\phi)\simeq m_{\phi}^{2}\phi^{2}/2$. A slight modification of the large field value regime to accommodate the CMB observation is straightforward.

Let us briefly sketch the dynamics of this model. First, chaotic inflation takes place. The field value of $\varphi$ is determined by the balance between the linear term $\varepsilon v^{4}\varphi/M_{\rm Pl}$ and $V_{\text{stb}}$ as $\varphi\sim\varepsilon v^{4}M_{\rm Pl}/V_{\text{ch}}$. After the pre-inflation, $\phi$ starts to oscillate and keeps stabilizing $\varphi$ until the energy of chaotic inflaton becomes small enough as $V_{\text{ch}}\sim v^{4}$ and the second new inflation starts. Thus, the initial field value of the new inflation is roughly $\varphi_{i}\sim\varepsilon M_{\rm Pl}$. The new inflation continues until the slow roll condition is violated at $\varphi_{e}\sim(v^{2}M_{\rm Pl}^{n-4}/(2n(n-1)g))^{1/(n-2)}$. Typical sizes of parameters to achieve successful inflation are $|\kappa|\lesssim\mathcal{O}(0.1)$ and $\varepsilon\ll(v^{2}/(2n(n-1)gM_{\rm Pl}^{2}))^{1/(n-2)}$. After the end of the new inflation, $\varphi$ oscillates around its potential minimum with a mass scale of $m_{\varphi}\sim n(v^{2}/M_{\rm Pl})(v^{2}/gM_{\rm Pl}^{2})^{-1/n}$. Assuming that the inflaton decays via a dimension-five Planck-suppressed operator, one can estimate the reheating temperature as $T_{\text{R}}\sim(90/\pi^{2}g_{\ast})^{1/4}\sqrt{m_{\varphi}^{3}/M_{\rm Pl}}$. We adopt this value of the reheating temperature in the following analysis.

Finally, let us briefly describe the power spectrum of scalar perturbations during the new inflation. The curvature perturbation during the new inflation may be evaluated by

[TABLE]

The curvature perturbation becomes large at the beginning of the new inflation. For $\varphi\sim\varphi_{i}$, the value can be estimated as $\mathcal{P}_{\zeta}\sim v^{4}/(12\pi^{2}M_{\rm Pl}^{4}\varepsilon^{2})$. One can see that the scalar perturbations can be sizable for $\varepsilon=\alpha v^{2}/M_{\rm Pl}^{2}$ with $\alpha\sim\mathcal{O}(1)$. In the following discussion, we consider the case with $\kappa>0$.333 For $\kappa<0$, the curvature perturbations can also be large at $\varphi\sim\varphi_{\ast}$ if there exists the flat inflection point $V^{\prime}(\varphi_{\ast})\simeq V^{\prime\prime}(\varphi_{\ast})\simeq 0$. In this paper, we do not consider this case. See Ref. Kawasaki et al. (2016b).

After the beginning of the new inflation, the scale dependence of the power spectrum is given by, in the slow-roll limit,

[TABLE]

where $\epsilon_{V}=(M_{\rm Pl}^{2}/2)\left(V^{\prime}/V\right)^{2}$ and $\eta_{V}=M_{\rm Pl}^{2}V^{\prime\prime}/V$ are the slow-roll parameters. Therefore the power spectrum can be approximated by

[TABLE]

where $k_{i}$ is a comoving momentum which exits the horizon at the beginning of the new inflation. One can see that a sizable $\kappa\sim\mathcal{O}(0.1)$ yields a sharp spectrum, while the slow roll condition enforces $\kappa\ll 1$.444 Planck-suppressed corrections on kinetic terms allow a much steeper spectrum. See Ref. Inomata et al. (2016).

Here we estimate the scalar power spectrum in the analytic slow-roll approximation, but we show the full numerical result in the linear order in Fig. 1.

III Observational Constraints

In this section, we briefly summarize observational constraints imposed on PBHs. See Refs. Carr et al. (2010, 2016) for a review. We may roughly split them into two classes: (a) constraints related to the current abundance of PBHs, and (b) constraints involving physical processes which might accompany the PBH formation or its time-evolution. See also Fig. 1 as an illustration, but be careful that constraints shown in Fig. 1 assume a monochromatic mass function. Thus, we have to extend the treatment so as to put constraints on an extended mass function as concretely discussed in Appendix. See also discussion in the next section and Fig. 2.

Let us start with the constraints of the class (a) from the lighter ones.

Extra-galactic gamma-ray background (green).

First of all, PBHs lighter than $6\times 10^{14}\,\mathrm{g}$ evaporate within the current age of the Universe, which is out of our interest. Though PBHs with $6\times 10^{14}\lesssim M_{\text{PBH}}\lesssim 10^{17}\,\mathrm{g}$ remain by now, they emit a sizable amount of photons that contribute to the extragalactic photon background. EGRET and Fermi LAT put constraints on this mass range Carr et al. (2010). This constraint is shown by the green line with shade in Fig. 1.

Gravitational lensing (blue).

Above $10^{17}\,\mathrm{g}$, gravitational lensing constraints come into play, which is caused when a compact object passes through our line of sight towards known sources. PBHs with $10^{17}\,\mathrm{g}\lesssim M_{\text{PBH}}\lesssim 10^{19}\,\mathrm{g}$ are constrained, since we do not see any femtolensing events from gamma-ray bursts of known redshift observed by the Fermi Gamma-ray Burst Monitor Barnacka et al. (2012). The other three constraints from $10^{20}\,\mathrm{g}$ to $10^{35}\,\mathrm{g}$ basically stem from null observation of micro/millilensing events. The difference comes from the sources they used and the sensitive time scale of lensing events, which results in different mass ranges of PBHs. The constraint from Kepler satellite utilizes near stars $\sim 1\,\textrm{kpc}$ slightly out of our galaxy plane Griest et al. (2013). The MACHO/EROS collaboration observed stars in Large and Small Magellanic Clouds, $\sim 50$ and $60$ kpc Tisserand et al. (2007). Recently, a very severe constraint ranging from $10^{20}$ to $10^{28}$ g has come out by using the Subaru HSC data Niikura et al. . Source stars are in Messier 31 (M31), $\sim 770$ kpc. The large distance of M31 and its short cadence data enable us to probe PBHs of smaller masses. These constraints are shown by blue lines with shades in Fig. 1.

Dynamical constraints (orange).

PBHs may collide with astrophysical objects and could leave observational signatures. Here we introduce three constraints, which are relevant in the following discussion. The others that we will not mention here are basically less stringent than different constraints. In Refs. Capela et al. (2013a, b), it is claimed that neutron stars (NSs) in the globular clusters may capture PBHs which destroys NSs immediately. The existence of NSs in the globular clusters puts constraints on PBHs with $10^{16}$ g to $10^{25}$ g. However, it is argued for instance in Refs. Kusenko and Rosenberg (2013); Carr et al. (2016); Clesse and García-Bellido (2016) that the amount of DM inside globular clusters can be much smaller than their assumption, and the constraints are evaded for dark matter densities below $\sim 10^{2}\,\textrm{GeV\,cm}^{-3}$. Thus, we will not show the constraint from the NS-capture, for it can be avoided for conservative values of DM density. Another constraints come from a dynamical heating of stars in the cluster or ultra-faint dwarf galaxies Brandt (2016). It is claimed that a conservative limit comes from the entire sample of compact ultra-faint dwarf galaxies, which may exclude PBHs with $10^{34}$–$10^{39}$ g. Also, the dynamical heating around the trajectory of PBHs may lead to the explosion of white dwarfs as supernovae Graham et al. (2015). It is argued that the shape of the observed distribution of white dwarfs rules out PBHs with $10^{19}$–$10^{20}\,\textrm{g}$. Ref. Gaggero et al. (2016) claims that PBHs cannot constitute all the DM for $M\sim\mathcal{O}(10)M_{\odot}$ by using a new accretion constraint on PBHs at the galactic center via the radio and X-ray. These constraints are shown by orange lines with shades in Fig. 1.

Then we discuss observational constraints (b).

Cosmic Microwave Background (red).

Massive PBHs may accrete the gas during the recombination epoch. The emission of radiation associated with the accretion could provide observable signatures in CMB spectrum and anisotropies. By using FIRAS/WMAP data, Ricotti et al. (2008) puts severe constraints on PBHs above $10^{33}$ g.555 See also a recent update discussed in Ref. Chen et al. (2016).

However, it is claimed in Refs. Bird et al. (2016); Clesse and García-Bellido (2016); Carr et al. (2016) that the constraints require modeling of complicated physical processes and thus the associated uncertainties are difficult to estimate. Moreover, recent conservative analyses Ali-Haïmoud and Kamionkowski (2016); Blum et al. (2016); Horowitz (2016) show that the bound can be much weaker, allowing PBHs with masses of $M\lesssim\mathcal{O}(10)M_{\odot}$ – $\mathcal{O}(10^{2})M_{\odot}$. This constraint is shown by a red line in Fig. 1.

Gravitational Waves from second order effects.

In order to generate a sizable amount of PBHs, large scalar perturbations during the inflation are required, $\mathcal{P}_{\zeta}\sim 10^{-2}$, as discussed in the previous section. While such large scalar perturbations may lead to the PBH-formation when they reenter the horizon, they can simultaneously generate a substantial amount of GWs as second order effects Saito and Yokoyama (2009, 2010); Bugaev and Klimai (2010, 2011). This is because the energy-momentum tensor of scalar perturbations acts as the source term in the equation of motion for GWs. Since GWs are produced when the scalar perturbations reenter the horizon, the momentum scale of GWs is necessarily related to the PBH mass, Eq. (2). Also, the amount of GWs is roughly proportional to the square of the scalar perturbation, which may be conveniently estimated as $\Omega_{\text{GW}}\sim 10^{-9}({\mathcal{P}}_{\zeta}/0.01)^{2}$. The current pulsar timing array experiments Arzoumanian et al. (2016); Lentati et al. (2015); Shannon et al. (2015) put severe constraints on $k\sim 10^{6}$ Mpc*-1* corresponding to $M\sim 0.75\gamma M_{\odot}$–$75\gamma M_{\odot}$. If one would like to interpret the LIGO events as PBH-mergers, these constraints play important roles Inomata et al. (2016); Nakama et al. (2016); Orlofsky et al. (2016).

IV PBH as All DM

As one can infer from Fig. 1, there are very limited ranges of the PBH mass in which PBHs can be a dominant component of DM. The first viable region may lie between the white dwarf and HSC constraints around $\sim 10^{20}$ g.@footnotemark The next possibility would be between the MACHO/EROS and the dynamical heating constraints around $10^{34-35}$ g Carr et al. (2016), since the CMB constraints can be much weaker as claimed recently Ali-Haïmoud and Kamionkowski (2016); Blum et al. (2016); Horowitz (2016). This region is recently revisited because there is a possibility to explain the LIGO gravitational events simultaneously Bird et al. (2016); Clesse and García-Bellido (2016); Sasaki et al. (2016). However, in Ref. Green (2016), it is argued that PBHs as all DM in this region is disfavored if one uses the constraint from the dynamical heating of ultra-faint dwarf galaxies.666 Ref. Carr et al. (2016) varies the parameters of constraints from the dynamical heating of Eridanus II.

Ref. Gaggero et al. (2016) also claims that PBHs cannot constitute all the DM for $M\sim\mathcal{O}(10)M_{\odot}$ by using a new accretion constraint on PBHs at the galactic center via the radio and X-ray.777 Note that this constraint depends on the profile of PBH DM. For the Burkert profile, we can evade it as discussed in Ref. Gaggero et al. (2016).

In addition, for PBHs generated via superhorizon fluctuations, the pulsar timing array experiments Arzoumanian et al. (2016); Lentati et al. (2015); Shannon et al. (2015) set severe constraints on gravitational waves via the second order effects Saito and Yokoyama (2009, 2010); Bugaev and Klimai (2010, 2011) for $M\sim 0.75\gamma M_{\odot}$–$75\gamma M_{\odot}$ as mentioned previously. If the formation of PBHs is well approximated by the Gaussian statistics, the power spectrum of curvature perturbation should be sharp enough to avoid the constraints at $\mathcal{O}(10)M_{\odot}$ Inomata et al. (2016); Orlofsky et al. (2016). Inflation models with enhanced non-Gaussianity at small scales may evade this constraint since the same amount of PBHs can be produced by a smaller amplitude of the curvature perturbation than the Gaussian one Nakama et al. (2016). We will return to these issues elsewhere Inomata et al. .

Fig. 2 shows observational constraints shown in Fig. 1 on parameters of the following form of the extended mass function adopted in Ref. Green (2016):

[TABLE]

where $N$ is determined so that the integration of Eq. (13) becomes $f_{\text{PBH}}$. This results indicate that, if we employ constraints shown in Fig. 1, the viable region is $M\sim 10^{20}$ g. Moreover, one can see that the PBH mass spectrum should be sharp enough, $\sigma\lesssim 0.1$, to avoid the new constraint. A closer view of this region is displayed in Fig. 3 for $f_{\text{PBH}}=1$. Such a sharp peak in the curvature perturbation can be obtained for a sizable $\kappa$. See also discussion around Eq. (12). We explicitly provide parameters of our model:

[TABLE]

which satisfies $\Omega_{\text{PBH,tot}}=\Omega_{c}$. We have checked that the predicted extended mass function passes all the observational constraints. See also Fig. 1. One can see that our inflation model can produce PBHs with $\sim 10^{20}$ g responsible for all DM, if one takes the conservative bound given in Ref. Niikura et al. .

V Conclusions

In this paper, we have revisited the idea of PBH as all DM, confronting the new severe microlensing constraint in Ref. Niikura et al. . The new constraint puts the severest upper bounds on the abundance of PBHs from $10^{20}$ g to $10^{28}$ g. We have shown that the remaining mass window for PBH as all DM is around $\simeq 10^{20}$ g and also that the width of the mass function should be sharp enough $\sigma\lesssim 0.1$, by treating constraints on the extended mass function in a proper way. We have constructed an inflationary model which yields a sharp PBH mass spectrum at $10^{20}$ g and avoids the observational constraints. Although it is quite marginal, it has been shown that the PBH can still be responsible for all DM.

Interestingly, a parameter set with $v\gtrsim 10^{15}$ GeV is allowed, which is compatible with a scenario in Ref. Kawasaki et al. (2016a). There, three of the authors proposed a mechanism, where the Higgs metastability problem during inflation Espinosa et al. (2008); Fairbairn and Hogan (2014); Hook et al. (2015); Herranen et al. (2014); Kamada (2015); Kearney et al. (2015); Espinosa et al. (2015); East et al. (2016) and preheating Ema et al. (2016); Kohri and Matsui (2016); Enqvist et al. (2016) can be avoided and PBHs are simultaneously generated, while the SM sector is kept intact. Therefore, a minimal scenario, involving the metastable electroweak vacuum, chaotic inflation, and PBHs as all DM, is still viable.

Since the new constraint only utilizes one-night data observed by Subaru HSC Niikura et al. , further observations might completely close the remaining window at $10^{20}$ g. As claimed in Ref. Graham et al. (2015), once populations of relatively heavy white dwarfs at the galactic center are confirmed, constraints from the existence of white dwarfs may also close the remaining window. Also, as revisited in Ref. Inomata et al. (2016) in the context of explaining LIGO events by PBH-mergers, the induced GW with $f\sim\mathcal{O}(10^{-2})$ Hz could be an interesting signature for future space-based detectors Seoane et al. (2013); Harry et al. (2006); Seto et al. (2001). PBH is one of leading non-particle candidates of DM. Thus, for particle physics, it is of quite importance to probe/exclude the remaining window at $\simeq 10^{20}$ g via several observations.

Acknowledgements

This work is supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No. 15H05889 (M.K.), No. 25400248 (M.K.), No. 26104009 (T.T.Y.), No. 26287039 (T.T.Y.) and No. 16H02176 (T.T.Y.), World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan (K.I., M.K., K.M., Y.T., and T.T.Y.), JSPS Research Fellowships for Young Scientists (Y.T., and K.M.), and Advanced Leading Graduate Course for Photon Science (K.I.).

Appendix A Constraints on extended mass functions

As emphasized in the main text, though a physical mass function of PBHs is smooth and continuous, all the constraints summarized in Fig. 1 assume that it takes a delta function. And thus, we need to be careful in applying these constraints on the extended mass function. In this section, we revisit this issue for the sake of completeness.

Let us start with the following simplest example. Suppose that we have a constant upper bound on the PBH fraction $\bar{f}_{i:i+1}$ between $M_{i}$ and $M_{i+1}$ assuming a monochromatic mass function. The question is how to apply it to the extended mass function $f(M)$. In this case, the constraint is translated into the following upper bound:

[TABLE]

Here $\bar{N}$ is an observational upper bound on the event number. We assume that the expected event number $N(f)$ is expressed as an integration of some function that is linear in $f$. The next simplest example may be the following. We have not only $\bar{f}_{i:i+1}$ in $[M_{i},M_{i+1}]$ but $\bar{f}_{i-1:i}$ in $[M_{i-1},M_{i}]$ which come from the same observation. Then, the upper bound on the extended mass function would be

[TABLE]

The above consideration suggests the following procedure for a more generic upper bound expressed as a continuous function. First, pick up one observational constraint $x$, take small enough mass bins, and split the upper bound $\bar{f}^{(x)}(M)$ into a set of constant upper bounds in each mass bin: $\{\bar{f}_{i:i+1}^{(x)}~{}\text{for}~{}[M_{i},M_{i+1}];i=1,\dots,N-1\}$. Then, the constraints of $x$ on the extended mass function $f(M)$ can be expressed as

[TABLE]

In the second equation, we assume small enough mass bins and approximate it by a continuous integration from $M_{1}$ to $M_{N}$, which gives a proper way to constrain PBHs with an extended mass function from a given continuous upper bound of an observation $x$, $\bar{f}^{(x)}(M)$, In the main text, we have used this second equation as a convenient way to constrain the extended mass function. Finally, by imposing this bound for all the constraints (all $x$), we can discuss the allowed region of the extended mass function parametrized in Eq. (13).