On the merger rate of primordial black holes: effects of nearest neighbours distribution and clustering

Abstract

One of the seemingly strongest constraints on the fraction of dark matter in the form of primordial black holes (PBH) of O(10) M relies on the merger rate inferred from the binary BH merger events detected by LIGO/Virgo. The robustness of these bounds depends however on the accuracy with which the formation of PBH binaries in the early Universe can be described. We revisit the standard estimate of the merger rate, focusing on a couple of key ingredients: the spatial distribution of nearest neighbours and the initial clustering of PBHs associated to a given primordial power spectrum. Overall, we confirm the robustness of the results presented in the literature in the case of a narrow mass function (which constrain the PBH fraction of dark matter to be f PBH 0.001 − 0.01). The initial clustering of PBHs might have an effect tightening the current constraint, but only for very broad mass functions, corresponding to wide bumps in the primordial power spectra extending at least over a couple of decades in k-space.The recent LIGO/Virgo discoveries of binary black hole (BH) and neutron star (NS) mergers have ushered us in the era of gravitational wave (GW) astronomy [1][2][3][4], leading in a short time to interesting implications on several branches of physics. In particular, even though only a single NS-NS merger has been detected so far [4], its multi-messenger observation [5] has permitted to draw a number of inferences, ranging from constraints on the nuclear equation of state [6,7] and some alternative models of gravity at large scales [8-11] (a possibility anticipated in [12]) to the identification of an engine of short gamma-ray bursts [13] and r-process (and thus the nucleosynthesis of heavy elements) [14].On the other hand, the origin of the progenitors of the few observed binary BH mergers has yet to be determined. The high value of some of their mean component masses (above 30 M ) are suggestive of progenitor formation in low-metallicity environments; see e.g. [15]. Besides, classical isolated binary evolution via mass transfer including a common envelope phase has been shown to be capable of reproducing the first detected events with a single channel [16]. Yet, many uncertainties remain in the assumptions and parameters of astrophysical models of BH (and binary BH) formation; for a compact review, see [17]. For instance, a dynamical origin of the binaries, notably in young massive and open stellar clusters, is a viable (albeit uncertain) possibility, see e.g. [18,19]. Furthermore, a multi-channel origin, in particular the idea that some or all of the mergers could be primordial -a proposal dating back to [20]; see [21] for a recent review-cannot be disregarded at the moment [17].Soon after the first merger was detected, some authors [22,23] went as far as conjecturing that the population of BHs responsible for these mergers is, in fact, not only primordial, but also accounts for the still unidentified dark matter (DM) which constitutes about one quarter of the energy density of the Universe [24]. The plausi...

Primordial black hole merger rates: distributions for multiple LIGO observables

“…This has been a topic of some debate, but it is now generally agreed that, for Gaussian initial conditions, the spatial distribution of PBHs is Poissonian. [35][36][37][38]. Primordial non-Gaussianity can strongly change the initial clustering of PBHs [39][40][41][42][43] and the subsequent merger rate [44] (see also [45]).…”

From primordial black holes abundance to primordial curvature power spectrum (and back)

“…For distances greater than the decoherence length, the density fluctuations become "uncorrelated". In this regime, an estimate of the two-point correlation function Ξ s (τ, r) = ξ s (τ, r > r dec ) can be obtained, for instance, by studying the effects of primordial clustering of PBHs [193,[196][197][198][199][200][201], however this goes beyond the scope of this article, hence we leave it for future work. Equivalently, one can also work with the Fourier transform of equation (5.14), i.e., the smoothed power spectrum, which reads as Here we can neglect the second integral in the second line of equation (5.15) because of the sin(x)/x suppression factor, as long as we consider modes k k dec = r −1 dec , i.e., modes that play a role in the gravitational collapse.…”

“…At cosmological scales (k 1 Mpc −1 ) the primordial curvature power spectrum is very well constrained to be almost scale invariant, namely P ζ (k) = A s (k/k pivot ) ns−1 , where A s is the scalar perturbations amplitude, n s is the scalar tilt and k pivot is the pivot scale 18 . At intermediate scales (1 Mpc −1 k k t ), the primordial curvature power spectrum determines the clustering properties of CPBHs [193,[196][197][198][199][200][201]. Since a full modelling of this goes beyond the scope of this paper, we phenomenologically parametrise the power spectrum in this range of scales using the formula P (m,n) ζ (k) = Bk m log n (k)+C, where B and C are fitting parameters.…”

“…8. While here we have concentrated on scales comparable to the typical peak size, in principle our method can be extended to constrain larger scales, in the intermediate regime between standard cosmological scales and typical peaks size, via primordial clustering of PBHs [193,[196][197][198][199][200][201]. We have illustrated this in figure 8 and will be presented elsewhere.…”

Enhanced cosmological perturbations and the merger rate of PBH binaries

“…With these assumptions, the observational upper bound on the merger rate limits the fraction of PBHs in DM to f 1%. Several refinements to this estimate have been considered, including initial spatial correlations of the PBHs [14][15][16][17], tidal forces from non-relativistic matter perturbations [10,11,18], as well as the effect of a dark matter dress around the PBHs [19], with similar results for the bound on the PBH abundances. In Ref.…”