Primordial Black Holes Mass Function Research Articles

Merger history of primordial black-hole binaries

As a candidate of dark matter, primordial black holes (PBHs) have attracted more and more attentions as they could be possible progenitors of the heavy binary black holes (BBHs) observed by LIGO/Virgo. Accurately estimating the merger rate of PBH binaries will be crucial to reconstruct the mass distribution of PBHs. It was pointed out the merger history of PBHs may shift the merger rate distribution depending on the mass function of PBHs. In this paper, we use 10 BBH events from LIGO/Virgo O1 and O2 observing runs to constrain the merger rate distribution of PBHs by accounting the effect of merger history. It is found that the second merger process makes subdominant contribution to the total merger rate, and hence the merger history effect can be safely neglected.

Initial clustering and the primordial black hole merger rate

If the primordial curvature perturbation followed a Gaussian distribution, primordial black holes (PBHs) will be Poisson distributed with no additional clustering. We consider local non-Gaussianity and its impact on the initial PBH clustering and mass function due to mode coupling between long and short wavelength modes. We show that even a small amount of non-Gaussianity results in a significant enhancement on the PBH initial clustering and subsequent merger rate and that the PBH mass function shifts to higher mass PBHs. However, as the clustering becomes strong, the local number density of PBHs becomes large, leading to a large theoretical uncertainty in the merger rate.

A novel formulation of the primordial black hole mass function

Abstract Computations of the primordial black hole (PBH) mass function discussed in the literature have conceptual issues. They stem from the fact that the mass function is a differential quantity and the standard criterion of the PBH formation from the seed primordial fluctuations cannot be directly applied to the computation of the differential quantities. We propose a new criterion of the PBH formation, which is the addition of one extra condition to the existing one. By doing this, we derive a formal expression of the PBH mass function without introducing any ambiguous interpretations that exist in the previous studies. Once the underlying primordial fluctuations are specified, the PBH mass function can be in principle determined by the new formula. As a demonstration of our formulation, we compute the PBH mass function analytically for the case where the perturbations are Gaussian and the space is 1 dimension.

Primordial black holes from cosmic domain walls

We investigate the formation of primordial black holes (PBHs) from the collapse of spherically symmetric domain wall bubbles, which spontaneously nucleate via quantum tunneling during inflation. Since the tension of domain walls changes with time and so domain walls nucleate in a short time interval, the mass function of PBHs in general has a spike-like structure. In contrast to models in which PBHs produced from overdense regions, our model avoids the uncertainties of PBHs production mechanism. PBHs from domain walls with mass around $10^{20}\mathrm{g}$ may constitute all dark matter, those with mass around $10^{34}\mathrm{g}$ can explain the merger events of binary black holes detected by LIGO.

Primordial Black Holes as a Dark Matter Candidate Are Severely Constrained by the Galactic Center 511 keVγ-Ray Line

We derive the strongest constraint on the fraction of dark matter that can be composed of low mass primordial black holes by using the observation of the Galactic Center 511keV γ-ray line. Primordial black holes of masses ≲10^{15} kg will evaporate to produce e^{±} pairs. The positrons will lose energy in the Galactic Center, become nonrelativistic, then annihilate with the ambient electrons. We derive robust and conservative bounds by assuming that the rate of positron injection via primordial black hole evaporation is less than what is required to explain the SPI/INTEGRAL observation of the Galactic Center 511keV γ-ray line. Depending on the primordial black hole mass function and other astrophysical uncertainties, these constraints are the most stringent in the literature and show that primordial black holes contribute to less than 1% of the dark matter density. Our technique also probes part of the mass range which was completely unconstrained by previous studies.

Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation

The formation and abundance of primordial black holes (PBHs) arising from the curvature perturbation ζ is studied. The non-linear relation between ζ and the density contrast δ means that, even when ζ has an exactly Gaussian distribution, significant non-Gaussianities affecting PBH formation must be considered. Numerical simulations are used to investigate the critical value and the mass of PBHs which form, and peaks theory is used to calculate the mass fraction of the universe collapsing to form PBHs at the time of formation. A formalism to calculate the total present day PBH abundance and mass function is also derived. It is found that the abundance of PBHs is very sensitive to the non-linear effects, and that the power spectrum \U0001d4abζ must be a factor of \U0001d4aa (2) larger to produce the same number of PBHs as if using the linear relation between ζ and δ (where the exact value depends on the critical value for a region to collapse and form a PBH). This also means that the derived constraints on the small-scale power spectrum from constraints on the abundance of PBHs are weaker by the same factor.

Primordial black holes with multimodal mass spectra

A mechanism for generating primordial black-hole mass spectra with many spikes is proposed and investigated. This mechanism relies on the choice of non-Bunch-Davies vacua, these leading to oscillatory features in the inflationary power spectrum. This in turn generates oscillations in the primordial black-hole mass function with exponentially enhanced spikes. This ``multimodal'' effect is demonstrated for most of the well-studied models of primordial black-hole formation.

Primordial black hole formation during slow reheating after inflation

We study the formation of primordial black holes (PBHs) in the early Universe during a period of slow reheating after inflation. We demonstrate how the PBH formation mechanism may change even before the end of the matter-dominated phase and calculate the expected PBH mass function. We find that there is a threshold for the variance of the density contrast, $\sigma_c \simeq 0.05$, below which the transition occurs even before reheating, with this having important consequences for the PBH mass function. We also show that there is a maximum cut-off for the PBH mass at around $100\,M_{\odot}$, below which the subdominant radiation bath affects PBH production, making the scenario particularly interesting for the recent LIGO observations of black hole mergers.

The maximal-density mass function for primordial black hole dark matter

The advent of gravitational wave astronomy has rekindled interest in primordial black holes (PBH) as a dark matter candidate. As there are many different observational probes of the PBH density across different masses, constraints on PBH models are dependent on the functional form of the PBH mass function. This complicates general statements about the mass functions allowed by current data, and, in particular, about the maximum total density of PBH. Numerical studies suggest that some forms of extended mass functions face tighter constraints than monochromatic mass functions, but they do not preclude the existence of a functional form for which constraints are relaxed. We use analytical arguments to show that the mass function which maximizes the fraction of the matter density in PBH subject to all constraints is a finite linear combination of monochromatic mass functions. We explicitly compute the maximum fraction of dark matter in PBH for different combinations of current constraints, allowing for total freedom of the mass function. Our framework elucidates the dependence of the maximum PBH density on the form of observational constraints, and we discuss the implications of current and future constraints for the viability of the PBH dark matter paradigm.

Hidden Universality in the Merger Rate Distribution in the Primordial Black Hole Scenario

Abstract It has been proposed that primordial black holes (PBHs) form binaries in the radiation dominated era. Once formed, some fraction of them may merge within the age of the universe by gravitational radiation reaction. We investigate the merger rate of the PBH binaries when the PBHs have a distribution of masses around , which is a generalization of the previous studies where the PBHs are assumed to have the same mass. After deriving a formula for the merger time probability distribution in the PBH mass plane, we evaluate it under two different approximations. We identify a quantity constructed from the mass distribution of the merger rate density per unit cosmic time and comoving volume , , which universally satisfies 0.97 ≲ α ≲ 1.05 for all binary masses independently of the PBH mass function. This result suggests that the measurement of this quantity is useful for testing the PBH scenario.

Single field double inflation and primordial black holes

Within the framework of scalar-tensor theories, we study the conditions that allow single field inflation dynamics on small cosmological scales to significantly differ from that of the large scales probed by the observations of cosmic microwave background. The resulting single field double inflation scenario is characterised by two consequent inflation eras, usually separated by a period where the slow-roll approximation fails. At large field values the dynamics of the inflaton is dominated by the interplay between its non-minimal coupling to gravity and the radiative corrections to the inflaton self-coupling. For small field values the potential is, instead, dominated by a polynomial that results in a hilltop inflation. Without relying on the slow-roll approximation, which is invalidated by the appearance of the intermediate stage, we propose a concrete model that matches the current measurements of inflationary observables and employs the freedom granted by the framework on small cosmological scales to give rise to a sizeable population of primordial black holes generated by large curvature fluctuations. We find that these features generally require a potential with a local minimum. We show that the associated primordial black hole mass function is only approximately lognormal.

Gravitational waves from primordial black hole mergers

We study the production of primordial black hole (PBH) binaries and the resulting merger rate, accounting for an extended PBH mass function and the possibility of a clustered spatial distribution. Under the hypothesis that the gravitational wave events observed by LIGO were caused by PBH mergers, we show that it is possible to satisfy all present constraints on the PBH abundance, and find the viable parameter range for the lognormal PBH mass function. The non-observation of a gravitational wave background allows us to derive constraints on the fraction of dark matter in PBHs, which are stronger than any other current constraint in the PBH mass range 0.5−30M⊙. We show that the predicted gravitational wave background can be observed by the coming runs of LIGO, and its non-observation would indicate that the observed events are not of primordial origin. As the PBH mergers convert matter into radiation, they may have interesting cosmological implications, for example in the context of relieving the tension between high and low redshift measurements of the Hubble constant. However, we find that these effects are suppressed as, after recombination, no more that 1% of dark matter can be converted into gravitational waves.

Gravitational wave signatures of inflationary models from Primordial Black Hole dark matter

Primordial Black Holes (PBH) could be the cold dark matter of the universe. They could have arisen from large (order one) curvature fluctuations produced during inflation that reentered the horizon in the radiation era. At reentry, these fluctuations source gravitational waves (GW) via second order anisotropic stresses. These GW, together with those (possibly) sourced during inflation by the same mechanism responsible for the large curvature fluctuations, constitute a primordial stochastic GW background (SGWB) that unavoidably accompanies the PBH formation. We study how the amplitude and the range of frequencies of this signal depend on the statistics (Gaussian versus χ2) of the primordial curvature fluctuations, and on the evolution of the PBH mass function due to accretion and merging. We then compare this signal with the sensitivity of present and future detectors, at PTA and LISA scales. We find that this SGWB will help to probe, or strongly constrain, the early universe mechanism of PBH production. The comparison between the peak mass of the PBH distribution and the peak frequency of this SGWB will provide important information on the merging and accretion evolution of the PBH mass distribution from their formation to the present era. Different assumptions on the statistics and on the PBH evolution also result in different amounts of CMB μ-distortions. Therefore the above results can be complemented by the detection (or the absence) of μ-distortions with an experiment such as PIXIE.

Primordial black holes as dark matter

The possibility that the dark matter comprises primordial black holes (PBHs) is considered, with particular emphasis on the currently allowed mass windows at $10^{16}$ - $10^{17}\,$g, $10^{20}$ - $10^{24}\,$g and $1$ - $10^{3}\,M_{\odot}$. The Planck mass relics of smaller evaporating PBHs are also considered. All relevant constraints (lensing, dynamical, large-scale structure and accretion) are reviewed and various effects necessary for a precise calculation of the PBH abundance (non-Gaussianity, non-sphericity, critical collapse and merging) are accounted for. It is difficult to put all the dark matter in PBHs if their mass function is monochromatic but this is still possible if the mass function is extended, as expected in many scenarios. A novel procedure for confronting observational constraints with an extended PBH mass spectrum is therefore introduced. This applies for arbitrary constraints and a wide range of PBH formation models, and allows us to identify which model-independent conclusions can be drawn from constraints over all mass ranges. We focus particularly on PBHs generated by inflation, pointing out which effects in the formation process influence the mapping from the inflationary power spectrum to the PBH mass function. We then apply our scheme to two specific inflationary models in which PBHs provide the dark matter. The possibility that the dark matter is in intermediate-mass PBHs of $1$ - $10^{3}\,M_{\odot}$ is of special interest in view of the recent detection of black-hole mergers by LIGO. The possibility of Planck relics is also intriguing but virtually untestable.

Microlensing and dynamical constraints on primordial black hole dark matter with an extended mass function

The recent discovery of gravitational waves from mergers of $\ensuremath{\sim}10\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ black hole binaries has stimulated interested in primordial black hole (PBH) dark matter in this mass range. Microlensing and dynamical constraints exclude all of the dark matter in compact objects with a delta function mass function in the range ${10}^{\ensuremath{-}7}\ensuremath{\lesssim}M/{M}_{\ensuremath{\bigodot}}\ensuremath{\lesssim}1{0}^{5}$. However it has been argued that all of the dark matter could be composed of compact objects in this range with an extended mass function. We explicitly recalculate the microlensing and dynamical constraints for compact objects with an extended mass function which replicates the PBH mass function produced by inflation models. We find that the microlensing and dynamical constraints place conflicting constraints on the width of the mass function, and do not find a mass function which satisfies both constraints.

Constraints on primordial black holes from the Galactic gamma-ray background

The fraction of the Universe going into primordial black holes (PBHs) with initial mass M_* \approx 5 \times 10^{14} g, such that they are evaporating at the present epoch, is strongly constrained by observations of both the extragalactic and Galactic gamma-ray backgrounds. However, while the dominant contribution to the extragalactic background comes from the time-integrated emission of PBHs with initial mass M_*, the Galactic background is dominated by the instantaneous emission of those with initial mass slightly larger than M_* and current mass below M_*. Also, the instantaneous emission of PBHs smaller than 0.4 M_* mostly comprises secondary particles produced by the decay of directly emitted quark and gluon jets. These points were missed in the earlier analysis by Lehoucq et al. using EGRET data. For a monochromatic PBH mass function, with initial mass (1+\mu) M_* and \mu << 1, the current mass is (3\mu)^{1/3} M_* and the Galactic background constrains the fraction of the Universe going into PBHs as a function of \mu. However, the initial mass function cannot be precisely monochromatic and even a tiny spread of mass around M_* would generate a current low-mass tail of PBHs below M_*. This tail would be the main contributor to the Galactic background, so we consider its form and the associated constraints for a variety of scenarios with both extended and nearly-monochromatic initial mass functions. In particular, we consider a scenario in which the PBHs form from critical collapse and have a mass function which peaks well above M_*. In this case, the largest PBHs could provide the dark matter without the M_* ones exceeding the gamma-ray background limits.

Constraints on diffuse neutrino background from primordial black holes

We calculate the energy spectra and fluxes of electron neutrinos emitted by the evaporation of primordial black holes (PBHs) formed by a blue power-law spectrum of primordial density fluctuations in the early universe. In the calculations of the neutrino spectra, the spectral index of density fluctuations and the reheating temperature are used as free parameters and the absorption of neutrinos during propagation in space is taken into account. We obtain the bounds on the spectral index assuming the validity of the standard picture of gravitational collapse and using the available data of several atmospheric and solar neutrino experiments. The comparison of our results with previous constraints (which had been obtained using the diffuse photon background data) shows that such bounds are quite sensitive to the assumed form of the initial PBH mass function.

Viability of primordial black holes as short period gamma-ray bursts

It has been proposed that the short period gamma-ray bursts, which occur at a rate of $\sim 10 {\rm yr^{-1}}$, may be evaporating primordial black holes (PBHs). Calculations of the present PBH evaporation rate have traditionally assumed that the PBH mass function varies as $M_{{\rm BH}}^{-5/2}$. This mass function only arises if the density perturbations from which the PBHs form have a scale invariant power spectrum. It is now known that for a scale invariant power spectrum, normalised to COBE on large scales, the PBH density is completely negligible, so that this mass function is cosmologically irrelevant. For non-scale-invariant power spectra, if all PBHs which form at given epoch have a fixed mass then the PBH mass function is sharply peaked around that mass, whilst if the PBH mass depends on the size of the density perturbation from which it forms, as is expected when critical phenomena are taken into account, then the PBH mass function will be far broader than $ M_{{\rm BH}}^{-5/2}$. In this paper we calculate the present day PBH evaporation rate, using constraints from the diffuse gamma-ray background, for both of these mass functions. If the PBH mass function has significant finite width, as recent numerical simulations suggest, then it is not possible to produce a present day PBH evaporation rate comparable with the observed short period gamma-ray burst rate. This could also have implications for other attempts to detect evaporating PBHs.

Critical collapse and the primordial black hole initial mass function

It has normally been assumed that primordial black holes (PBHs) always form with mass approximately equal to the mass contained within the horizon at that time. Recent work studying the application of critical phenomena in gravitational collapse to PBH formation has shown that in fact, at a fixed time, PBHs with a range of masses are formed. When calculating the PBH initial mass function it is usually assumed that all PBHs form at the same horizon mass. It is not clear, however, that it is consistent to consider the spread in the mass of PBHs formed at a single horizon mass, whilst neglecting the range of horizon masses at which PBHs can form. We use the excursion set formalism to compute the PBH initial mass function, allowing for PBH formation at a range of horizon masses, for two forms of the density perturbation spectrum. First we examine power-law spectra with $n>1$, where PBHs form on small scales. We find that, in the limit where the number of PBHs formed is small enough to satisfy the observational constraints on their initial abundance, the mass function approaches that found by Niemeyer and Jedamzik under the assumption that all PBHs form at a single horizon mass. Second, we consider a flat perturbation spectrum with a spike at a scale corresponding to horizon mass $\sim 0.5 M_{\odot}$, and compare the resulting PBH mass function with that of the MACHOs (MAssive Compact Halo Objects) detected by microlensing observations. The predicted mass spectrum appears significantly wider than the steeply-falling spectrum found observationally.

Dynamics of primordial black hole formation

We present a numerical investigation of the gravitational collapse of horizon-size density fluctuations to primordial black holes (PBHs) during the radiation-dominated phase of the early Universe. The collapse dynamics of three different families of initial perturbation shapes, imposed at the time of horizon crossing, is computed. The perturbation threshold for black hole formation, needed for estimations of the cosmological PBH mass function, is found to be ${\ensuremath{\delta}}_{\mathrm{c}}\ensuremath{\approx}0.7$ rather than the generally employed ${\ensuremath{\delta}}_{\mathrm{c}}\ensuremath{\approx}1/3$ if $\ensuremath{\delta}$ is defined as $\ensuremath{\Delta}{M/M}_{\mathrm{h}},$ the relative excess mass within the initial horizon volume. In order to study the accretion onto the newly formed black holes, we use a numerical scheme that allows us to follow the evolution for long times after formation of the event horizon. In general, small black holes (compared to the horizon mass at the onset of the collapse) give rise to a fluid bounce that effectively shuts off accretion onto the black hole, while large ones do not. In both cases, the growth of the black hole mass owing to accretion is insignificant. Furthermore, the scaling of black hole mass with the distance from the formation threshold, known to occur in near-critical gravitational collapse, is demonstrated to apply to primordial black hole formation.