¹

1 Introduction↩︎

The Standard Model of particle physics (SM) and General Relativity (GR) have proven to be extremely successful at describing a huge range of terrestrial, astrophysical, and cosmological observations [1], [2]. However, their successes are limited by a number of shortcomings, potentially (especially in the case of SM) pointing towards the need for new physics which may better describe the matter and gravity sectors. On the more observational/phenomenological side, the SM lacks a candidate for the dark matter (DM) which accounts for \(\simeq 25\%\) of the energy budget of the Universe [3], [4]. On the more theoretical side, continuous gravitational collapse in GR leads to the pathological appearance of curvature singularities [5], [6]. The nature of DM and the singularity problem are arguably two among the most important open questions in theoretical physics.

The solution to the former problem could reside in the physics of some of the most peculiar objects in the Universe: black holes (BHs). It has long been realized that primordial BHs (PBHs), hypothetical relics from the primordial Universe formed from the collapse of large density perturbations upon horizon re-entry, are indeed excellent DM candidates [7]–[56] (see e.g.Refs. [57]–[63] for reviews): in fact, PBHs are the only viable DM candidate which does not invoke new particles surviving to the present day. Once believed to merely be objects of mathematical speculation, observational effects associated to BHs are now routinely detected [64], turning these objects into extraordinary probes of fundamental physics [65]–[114]. As a result, the possibility of PBHs accounting for the entire DM budget is severely constrained by a wide range of considerations and constraints. The only (not entirely undebated) remaining open window of parameter space where PBHs could make up all the DM is the so-called “asteroid mass window”, roughly for PBH masses \(10^{14}\,{\text{kg}} \lesssim M_{\text{pbh}} \lesssim 10^{20}\,{\text{kg}}\) [115]–[129]: lighter PBHs would have evaporated fast enough to either have disappeared by now or overproduced \(\gamma\)-rays in the \({\text{MeV}}\) range, whereas heavier PBHs would have been detected through the microlensing of background stars.

Almost all works on PBHs assume that these are Schwarzschild or Kerr BHs [57]–[63]. All constraints and considerations on DM potentially being in the form of PBHs are therefore subject to this underlying assumption. The existence of the asteroid mass window, and the extension thereof, is of course no exception. The assumption in question is not at all unreasonable at least from the phenomenological point of view, given that there are at present no signs of tension between astrophysical observations and the Kerr-Newman family of metrics, and more generally the no-hair theorem. Nevertheless, from the theoretical point of view such an assumption might stir some unease, given the appearance of singularities in the Schwarzschild and Kerr metrics. The above considerations naturally lead to the following question: “what if PBHs are non-singular”? It is our goal in the present work to systematically address this question, which naturally merges the DM and singularity problems.

We entertain the possibility that PBHs are “regular”, i.e.free of curvature singularities [130]–[134], and therefore that DM may be in the form of primordial regular BHs (PRBHs). For concreteness, we consider so-called tr (time-radius)-symmetric metrics, for which the product of the coefficients for the \(dt^2\) and \(dr^2\) components of the line element in four-dimensional Boyer–Lindquist coordinates is equal to \(-1\), and the function which multiplies the angular part of the line element is \(r^2\), i.e.\(r\) is the areal radius. More specifically, we focus on the following four regular, static spherically symmetric space-times, all of which are characterized by an additional regularizing parameter \(\ell\) and recover the Schwarzschild space-time in the \(\ell \to 0\) limit: Bardeen BHs [135], Hayward BHs [136], Ghosh-Culetu-Simpson-Visser BHs [137]–[139], and a Hayward-like BH which, to the best of our knowledge, is introduced for the first time in this work. These four space-times present a rich phenomenology, including both de Sitter and Minkowski cores, and temperatures which can both decrease and increase with increasing regularizing parameter \(\ell\). We focus our attention on observational constraints from PRBH evaporation, which set the lower limit of the asteroid mass window, discussing in detail how the evaporation process is modified with respect to that for Schwarzschild PBHs. We show that, as a result, the phenomenology of PRBHs can be quite different from that of Schwarzschild PBHs, with a larger range of masses where PRBHs could make up the entire DM component, opening up the asteroid mass window by up to an extra decade in mass. Keeping in mind that the metrics in question are phenomenological in nature, our results demonstrate that a common solution to the DM and singularity problems in the form of primordial regular BHs is one which is worth taking seriously, and warrants further investigation, and more generally the interface of these two problems provides a promising arena. We stress that our work should not be intended as a comprehensive analysis of primordial regular BHs, but rather as a pilot study, pointing towards a direction which has thus far received very little attention and indicating promising directions for further work.

The rest of this paper is then organized as follows. In Sec. 2 we briefly introduce the regular space-times studied in the rest of the work. Various aspects of our methodology are discussed in Sec. 3, with Sec. 3.1 devoted to the calculation of the so-called greybody factors, Sec. 3.2 to the computation of photon spectra resulting from Hawking evaporation, and Sec. 3.3 to the comparison against observations. The resulting limits on the fraction of DM which may be in the form of primordial regular BHs are then critically discussed in Sec. 4. Finally, in Sec. 5 we draw concluding remarks. A number of more technical aspects concerning the greybody factors computation are discussed in Appendix 6. Unless otherwise specified, we adopt units where \(G=c=1\). In closing, we note that a related study is being presented in a companion paper [140]: this focuses on non-tr-symmetric metrics, which also include loop quantum gravity-inspired metrics, but at the same time complicate the study of the evaporation process. We recommend that the interested reader go through the present work prior to consulting our companion paper [140].

2 Regular black holes↩︎

It is well known, thanks to the Penrose-Hawking singularity theorems, that continuous gravitational collapse in GR sourced by matter contents satisfying reasonable energy conditions leads to the appearance of singularities [5], [6]. These are regions of space-time where curvature invariants, i.e.sets of independent scalars constructed from the Riemann tensor and the metric, diverge (with the archetypal example being the central singularity in the Kerr-Newman family of metrics). These singularities are arguably unsatisfactory as they lead to a potential breakdown in predictivity. For this reason, they are oftentimes regarded as a manifestation of our lack of knowledge of (new) physics in the high-energy/high-curvature regime. A widespread belief is that quantum gravity effects on these scales would ultimately cure the singularity problem (and potentially lead to observable effects), although this is more of a hope supported only by a few first-principles studies [141]–[155].

Even in the absence of a widely agreed upon theory of quantum gravity, one can still hope to make progress in understanding and taming singularities, while also potentially gaining intuition about the possible features of such a theory, through a more phenomenological approach. Under the assumption that a metric description holds valid, one can introduce metrics which are free of singularities in the entire space-time, and describe so-called regular BHs (RBHs) [156]. It is often (albeit not necessarily always) the case that RBH metrics are controlled by an extra parameter, which in what follows we shall refer to as regularizing parameter (and denote by \(\ell\)), typically recovering the Schwarzschild metric (for non-rotating RBHs) in the limit \(\ell \to 0\). Several RBH metrics have been studied over the past decades, see e.g.Refs. [157]–[191] for an inevitably incomplete selection of examples, as well as Refs. [130], [132], [134] for recent reviews on the subject. ² While most of these metrics have been introduced on purely phenomenological grounds it is known that possible sources for several RBH metrics lie in theories of non-linear electrodynamics [195]–[199].

As alluded to earlier, our interest in this work is to explore the possibility that primordial RBHs may play the role of DM. As a proof-of-principle in this sense we will establish constraints on \(f_{\text{pbh}}\), the fraction of DM in the form of PRBHs, focusing on the asteroid mass window, whose extent we will show can be either opened or tightened depending on the features of the RBH metric. To the best of our knowledge, we are aware of only four works in this direction [200]–[203]. Ref. [200] studied the thermodynamics of primordial regular BHs, focusing however on the case where they do not evaporate, and therefore did not study constraints on \(f_{\text{pbh}}\). Ref. [201] studied the evaporation of a loop quantum gravity-inspired BH, and weakened constraints on \(f_{\text{pbh}}\) were reported in a later proceeding (which however does not appear to be widely known). Finally, Ref. [203] studied signatures of primordial BHs with magnetic charge, which could be (as is often but necessarily the case) regular. The aim of this pilot study and our companion paper [140] is instead to provide a more comprehensive investigation of primordial regular BHs, considering a more diverse set of metrics and investigating constraints on \(f_{\text{pbh}}\) in detail.

In our work, we shall consider four different non-rotating RBH metrics, as discussed in more detail in the following subsections. The static, spherically symmetric space-times we investigate are a subset of the Petrov type-D class of metrics. In four-dimensional Boyer-Lindquist coordinates, their line elements take the following general form: \[\begin{align} ds^2 = -f(r)dt^2 + g(r)^{-1}dr^2 + h(r)d\Omega^2\,, \label{eq:ds2} \end{align}\tag{1}\] where \(d\Omega^2=d\theta^2 +\sin^2(\theta) d\phi^2\) is the metric on the 2-sphere. We also require our space-times to be asymptotically flat, which amounts to the following requirements: \[\begin{align} f(r) \xrightarrow{r \to \infty} 1\,, \quad g(r) \xrightarrow{r \to \infty} 1\,, \quad h(r) \xrightarrow{r \to \infty} r^2\,. \label{eq:asflat} \end{align}\tag{2}\] In addition, as stated earlier, we require our space-times to be tr-symmetric (the non-tr-symmetric case is covered in a companion paper [140]), which imposes the following additional conditions: \[\begin{align} f(r) = g(r)\,, \quad h(r) = r^2\,, \label{eq:tr} \end{align}\tag{3}\] implying that the coordinate \(r\) is effectively the areal radius. With the conditions given by Eqs. (2 ,3 ) imposed upon Eq. (1 ), our most general line element therefore takes the following form: \[\begin{align} ds^2 = -f(r) dt^2 + f(r)^{-1}dr^2 + r^2 \left ( d\theta^2 +\sin^2(\theta) d\phi^2 \right ) \,. \label{eq:ds2trsymmetric} \end{align}\tag{4}\] In what follows, we refer to the function \(f(r)\) as being the “metric function”. The four different RBH solutions we consider, which we will discuss very shortly in Sections 2.1– 2.4, are characterized by different functional forms of \(f(r)\).

An important parameter characterizing the behaviour of BHs is their temperature \(T\). This is particularly crucial when evaluating evaporation constraints on PBHs, given that the temperature controls the strength of the emitted radiation, which in turn can be directly constrained by various observations. We treat the temperature of the RBHs as being the usual Gibbons-Hawking one, i.e.the one evaluated by Wick rotating the metric in the standard way and imposing regularity in the Euclidean period [204]. The cyclic imaginary time \(\to\) temperature identification is legitimate if one can formally identify the Euclidean action \(e^{-S}\) with the Boltzmann factor \(e^{-\beta H}\) in the partition function, as usually done in finite temperature quantum field theory: in turn, this can be done if one is assuming the standard Boltzmann-Gibbs distribution, but may not be the consistent if other entropies are assumed (see e.g.the recent discussion in Ref. [205]). Since, as we will reiterate later, the RBHs we study are introduced on phenomenological grounds and we remain agnostic as to their theoretical origin (which may in principle be rooted within alternative entropic frameworks), in what follows we assume the Boltzmann-Gibbs distribution, so that the temperature of the RBHs in question is the standard Gibbons-Hawking one, and is given by the following: \[\begin{align} T=\frac{\kappa}{2\pi}=\frac{f'(r)}{4\pi}\vert_{r_H}\,, \label{eq:temperature} \end{align}\tag{5}\] where the prime indicates a derivative with respect to \(r\), and \(\kappa\) is the BH surface gravity, given by the following: \[\begin{align} \kappa=\frac{f'(r)}{2}\vert_{r_H}\,. \label{eq:kappa} \end{align}\tag{6}\] In Eqs. (5 ,6 ), \(r_H\) is the horizon radius, which is the solution to the following equation: \[\begin{align} g^{-1}(r_H)=f(r_H)=0\,, \label{eq:rh} \end{align}\tag{7}\] with the first equality following from the choice of focusing on tr-symmetric space-times. In the case of Schwarzschild BHs, where the metric function is \(f(r)=1-2M/r\), one recovers the well-known expressions \(r_H=2M\) and \(T_{\text{Sch}}=1/8\pi M\). However, in more general space-times the horizon radius in Eq. (7 ) is not guaranteed to have a closed form expression, and the same therefore holds for the temperature in Eq. (5 ). In Fig. 1 we show the evolution of the temperatures (normalized to the temperature of Schwarzschild BHs, \(T_{\text{Sch}}=1/8\pi M\)) of the four regular space-times we will introduce shortly, as a function of the regularizing parameter \(\ell\) (normalized by the event horizon radius \(r_H\)). As the Figure clearly shows, for three of these space-times (Bardeen, Hayward, and Ghosh-Culetu-Simpson-Visser) the temperature is a monotonically decreasing function of \(\ell\), whereas for the fourth (Hayward-like), after a small initial dip, the temperature grows monotonically with growing \(\ell\). As we will see, these different behaviour turn out to have important phenomenological consequences.

A final caveat is in order before discussing the RBH metrics we consider. The latter are all regular in the sense of having finite curvature invariants \(R \equiv g^{\mu\nu}R_{\mu\nu}\), \(R_{\mu\nu}R^{\mu\nu}\), and \({\cal K} \equiv R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\). However, a more stringent criterion for regularity is that of geodesic completeness, which does not necessarily imply finiteness of curvature invariants and viceversa. A number of “popular” RBHs have indeed been shown to have finite curvature invariants but to be geodesically incomplete [206]. This includes the well-known Bardeen and Hayward RBHs, which are among the ones we shall consider here. However, given the significant interest in these metrics, the fact that they are widely taken as prototypes for RBHs, and our phenomenological goal of going beyond Schwarzschild PBHs, we will take these space-times into consideration, while cautioning the reader about the above issues, and therefore that these metrics should be considered nothing more than phenomenological toy models at this stage. Note, in addition, that the stability of RBHs featuring inner horizons is currently a matter of debate in the literature [207]–[211].

2.1 Bardeen black hole↩︎

The Bardeen BH is easily one of the best known RBHs, and one of the first ones to ever have been proposed [135]. It is characterized by the following metric function: ³ \[\begin{align} f_{\text{B}}(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}\,, \label{eq:frbardeen} \end{align}\tag{8}\] where, in terms of the BH mass \(M\), the regularizing parameter satisfies \(\ell \leq \sqrt{16/27} M \sim 0.77\,M\) in order for the metric to describe a BH and not a horizonless object. Note that the Schwarzschild metric function is recovered in the \(\ell \to 0\) limit. A perhaps physically more motivated choice is to express quantities in units of the horizon radius \(r_H\), defined as the largest root of the equation \(f(r_H)=0\), in which case the regularizing parameter is subject to the constraint \(\ell \lesssim 0.70\,r_H\). In order to obtain this limit we have computed the solution to \(f(r_H)=0\) fixing \(M=1\), in order to extrapolate \(r_H(\ell)\), before analyzing for which real values of the parameter \(n\) the equation \(\ell=nr_H(\ell)\) admits solutions.

It is worth noting that the Bardeen BH possesses a de Sitter (dS) core which replaces the central singularity of the Schwarzschild BH. This is evident by noting that, in the limit \(r \to 0\), the metric function goes as \(f_B(r) \propto r^2\), exactly as expected for an asymptotically dS space-time. Although originally introduced on phenomenological grounds, it is now known that the Bardeen RBH can emerge from a magnetic monopole source [212], potentially within the context of a specific non-linear electrodynamics theory [213]. Another possible origin for the Bardeen RBH are quantum corrections to the uncertainty principle [214]. Irrespective of its origin, and consistently with the approach pursued for the other space-times, we consider this solution as a model-agnostic phenomenological toy model.

2.2 Hayward black hole↩︎

Another widely known RBH space-time is the Hayward RBH [136], characterized by the following metric function: \[\begin{align} f_{\text{H}}(r)=1-\frac{2Mr^2}{r^3+2M\ell^2}\,. \label{eq:frhayward} \end{align}\tag{9}\] If expressed in terms of BH mass \(M\), the regularizing parameter for the Hayward BH is subject to the same limit as that of the Bardeen BH, i.e.\(\ell \leq \sqrt{16/27}\,,M\). On the other hand, if expressed in terms of the more physically motivated horizon radius, the limit is instead \(\ell \lesssim 0.57\,r_H\). We note that the Schwarzschild metric is function is recovered in the \(\ell \to 0\) limit.

Just as the Bardeen RBH possesses a dS core, so does the Hayward RBH. Indeed, introducing a dS core characterized by a (positive) cosmological constant \(\Lambda= 3/\ell^2\) in order to prevent the central singularity was precisely the original justification for the Hayward BH which, just like its Bardeen counterpart, was introduced on purely phenomenological grounds. Nevertheless, potential theoretical origins for the Hayward BH have been investigated, and range from corrections to the equation of state of matter at high density [215], [216], finite density and finite curvature proposals [217]–[219], theories of non-linear electrodynamics [220], [221], and more generally as the result of corrections due to quantum gravity [222], [223]. Just as with the Bardeen RBH, we shall here treat the Hayward RBH as a model-agnostic phenomenological toy model for a singularity-free space-time.

2.3 Ghosh-Culetu-Simpson-Visser black hole↩︎

The regular space-times considered so far feature dS cores, which in itself is a very common feature of several RBH metrics. Nevertheless, another interesting phenomenological possibility consists in considering “hollow” RBHs in which the central singularity is replaced by an asymptotically Minkowski core, where the associated energy density and associated pressure asymptote to zero. This is quite unlike the case of the dS core where the energy density asymptotes to a finite value associated to a positive cosmological constant, and the pressure asymptotes to an equal but opposite value. Possible theoretical/mathematical motivations for considering RBHs with Minkowski cores include the fact that the vanishing energy density can significantly simplify the physics in the deep core, whereas the otherwise messy solutions to polynomical equations (which often cannot be written down in closed form) can be traded for arguably more elegant special functions, resulting in the space-time being more tractable. Our physical motivation in considering this class of BHs is instead to broaden the range of physical properties and phenomenological implications of PRBHs, going beyond the dS core RBHs studied thus far.

With this in mind, we consider a RBH featuring a Minkowski core, independently studied by Ghosh [137], Culetu [138], as well as Simpson and Visser [139]. Although such a RBH does not have any particular name associated to it in the literature, here we shall conform to the name introduced in Ref. [224], referring to it as GCSV BH (from the initials of the four authors above). This space-time is characterized by the following metric function: \[\begin{align} f_{\text{GCSV}}(r)= 1-\frac{2M}{r}\exp \left ( -\frac{\ell}{r} \right ) \,. \label{eq:frgcsv} \end{align}\tag{10}\] The horizon radius \(r_H\), for which a closed form expression is not available in the Bardeen and Hayward cases, is given by: \[\begin{align} r_H=-\frac{\ell}{W \left ( -\frac{\ell}{2 M} \right ) }\,, \label{eq:rhgcsv} \end{align}\tag{11}\] where \(W\) denotes the Lambert function. Considering the principal branch \(W_0\), a real and positive horizon radius is present for: \[\begin{align} W_0 \left ( -\frac{\ell}{2 M} \right ) \leq 0 \implies 0 \leq \ell < \frac{2M}{e}\,, \label{eq:lambhert} \end{align}\tag{12}\] or, alternatively, \(0 \leq \ell < r_H\). While the GCSV BH was original introduced purely on phenomenological/mathematical grounds, it was shown in Refs. [225], [226] that such a space-time can emerge within the context of GR coupled to a specific non-linear electrodynamics source. In this case, denoting by \(g\) the non-linear electrodynamics coupling constant/charge, the regularizing parameter \(\ell\) is given by \(\ell=g^2/2M\), with \(M\) the BH mass. Nevertheless, as with all the other RBHs considered, here we shall treat the GCSV RBH as a toy model for a regular space-time possessing a Minkowski core.

2.4 Hayward-like black hole↩︎

For all of the three RBHs discussed earlier, the temperature decreases with increasing regularizing parameter, while converging to the Schwarzschild value \(T_{\text{Sch}}=1/8\pi M\) as \(\ell \to 0\), see Fig. 1. It is therefore not unreasonable to expect that evaporation constraints on \(f_{\text{pbh}}\) may be weakened if the DM consists of these PRHBs (this will of course be checked explicitly later). We have verified that such a behaviour of temperature versus regularizing parameter holds for most PRBHs of phenomenological interest considered in the literature. ⁴ For phenomenological reasons, especially in light of our expectation (verified a posteriori) that such a behaviour would lead to weaker bounds on \(f_{\text{pbh}}\) and thereby the asteroid window further opening, it could therefore be desirable to also contemplate a case where the BH temperature actually increases with increasing regularizing parameter.

With the previous considerations in mind, here we introduce a new RBH solution metric which somewhat resembles the Hayward BH. This Hayward-like RBH is characterized by the following metric function: \[\begin{align} f_{\text{H-l}}(r)=1 - \frac{2 M r^2}{r^3+\ell \left ( 1-\ell r \right ) }\,. \label{eq:frhaywardlike} \end{align}\tag{13}\] To the best of our knowledge, this Hayward-like RBH has never been proposed before. Just as the Hayward BH, it is evident that this space-time possesses a dS core, with an effective cosmological constant given by \(\Lambda=6M/\ell\). We explicitly verify the finiteness of the curvature invariants, which take the following very simple forms: \[\begin{align} R&=&g^{\mu\nu}R_{\mu\nu}=\frac{24 M}{\ell}\,, \quad R_{\mu \nu}R^{\mu \nu}=\frac{144 M^2}{\ell^2}\,,\nonumber \\ {\cal K} &=& R_{\mu \nu \sigma \rho}R^{\mu \nu \sigma \rho}=\frac{96 M^2}{\ell^2} \label{eq:curvaturescalarshaywardlike} \end{align}\tag{14}\] We later explicitly verify that the intensity of Hawking radiation of the Hayward-like RBH increases with increasing regularizing parameter, therefore tightening constraints on \(f_{\text{pbh}}\), and further closing the asteroid mass window. This behavior is similar to that of Kerr BHs, which display an enhancement of the primary photon emission spectrum with increasing spin parameter [230]–[232]. A similar behaviour has also been observed in the context of the Kazakov-Solodukhin BH [233] in Ref. [234], consistently with the expectation laid out in footnote 3 (note, in fact, that the shadow of the Kazakov-Solodukhin BH increases with increasing regularizing parameter, see e.g.Fig. 1 of Ref. [235], Fig. 18 of Ref. [236], and Fig. 9 of Ref. [224]). Finally, we have checked that the allowed limit for the regularizing parameter in terms of the horizon radius is \(0\leq\ell\leq r_H\).

3 Methodology↩︎

3.1 Greybody factors↩︎

A set of parameters playing a key role in describing the Hawking radiation spectra emitted from evaporating BHs are the so-called greybody factors (GBFs). These are functions of energy/frequency and angular momentum which govern the deviation of the emitted spectrum from that of a blackbody [237]–[239]. Although the emitted Hawking radiation at the horizon takes the blackbody form, the potential barrier due to space-time geometry will attenuate the radiation, so that an observer at spatial infinity will measure a spectrum which differs from that of a blackbody by a frequency-dependent function \(\Gamma(\omega)\). GBFs can be characterized by setting up a classical scattering problem around the BH potential barrier, with boundary conditions allowing for incoming wave packets from infinity or equivalently, due to the symmetries of the scattering problem, originating from the horizon. The scattering problem is governed by the so-called Teukolsky equation, which is a partial differential equation describing the propagation of perturbations of given spin in the BH background [240].

For the static, spherically and tr-symmetric metrics given by Eq. (4 ) which we consider, the Teukolsky equation in spherical coordinates is separable. A key role in computing the GBFs is played by the radial Teukolsky equation, which we now report in full generality for the class of metrics in question. Using the Newman-Penrose (NP) formalism [241], the Teukolsky equation governing the evolution of (massless) perturbations of different spin can be condensed into a single master equation [240]: \[\begin{align} & \left [ - \frac{r^2}{f} \partial_t^2 + s \left ( r^2 \frac{f'}{f} -2 r \right ) \partial_t \right ] \Upsilon_s \nonumber \\ &+ \left [ (s+1) (r^2 f' + 2 r f) \partial_r \right ] \Upsilon_s \nonumber \\ &+\left [ \frac{1}{\sin{\theta}} \partial_\theta (\sin{\theta}\partial_\theta) + \frac{2 i s \cot{\theta}}{\sin{\theta}} \partial_\phi \right. \nonumber \\ &\left. + \frac{1}{\sin^2{\theta}} \partial^2_\phi -s -s^2 \cot^2{\theta} \right ] \Upsilon_s \nonumber \\ &+\left [ s r^2 f'' + 4 s r f' + 2 s f \right ] \Upsilon_s=0\,. \label{eq:teukolsky} \end{align}\tag{15}\] Here, \(\Upsilon_s\) represents a general perturbation of spin \(s\), defined by the NP scalars relative to the respective perturbation. To not make the notation too heavy, we drop the \(l\) and \(m\) indices labelling the field mode, so \(\Upsilon_s\) is understood to really mean \(\Upsilon^{lm}_s\). We note that Eq. (15 ) is separable if one makes the following wave ansatz: \[\begin{align} \Upsilon_s= \sum_{l,m} e^{-i \omega t } e^{i m \phi} S^{l}_s(\theta) R_s(r)\,, \label{eq:upsilon} \end{align}\tag{16}\] where \(\omega\) is the perturbation frequency, \(l\) is the angular node number, and \(m\) is the azimuthal node number.

The functions \(S^l_s(\theta)\) contribute to defining the so-called spin-weighted spherical harmonics \(S^s_{l,m}(\theta, \phi)=\sum S^l_s(\theta) e^{im\phi}\), satisfying the following equation [242]–[245]: \[\begin{align} &\left ( \frac{1}{\sin\theta}\partial_\theta(\sin\theta\,\partial_\theta)+\csc^2\theta\,\partial_\phi^2 \right. \nonumber \\ &\left. + \frac{2is\cot\theta}{\sin\theta}\partial_\phi+s-s^2\cot^2\theta+\lambda_l^s \right ) S_{l,m}^s=0\,, \label{eq:slms} \end{align}\tag{17}\] where \(\lambda_l^s\equiv l(l+1)-s(s+1)\) is the separation constant. For the spin 0 case, these functions reduces to the usual spherical harmonics \(S_{l,m}^0=Y_{l,m}\).

Analogously to the Schwarzschild and Kerr BH cases [240], the decoupled radial Teukolsky equation takes the following form [246], [247]: \[\begin{align} &\frac{1}{\Delta^s}\big(\Delta^{s+1}R'_s\big)' \nonumber \\ &+\left(\frac{\omega^2r^2}{f}+2i\omega sr-\frac{is\omega r^2f'}{f}+s(\Delta''-2)-\lambda_l^s\right)R_s=0\,, \label{eq:radialteukolsky} \end{align}\tag{18}\] where \(\Delta(r)\equiv r^2f(r)\) and \(' \equiv \partial_r\). We set in purely ingoing boundary conditions, so the asymptotic solutions of Eq. (18 ) are given by: \[\begin{align} & R_s \sim R^{\text{in}}_s \frac{e^{-i\omega r^{\star}}}{r}+ R^{\text{out}}_s \frac{e^{i\omega r^{\star}}}{r^{2s+1}} \quad (r\rightarrow \infty) \nonumber \\ & R_s \sim R^{\text{hor}}_s \Delta^{-s} e^{-i \omega r^{\star}} \quad \quad \quad \quad \, \, (r \rightarrow r_H)\,, \label{eq:asymptotic} \end{align}\tag{19}\] where \(r^{\star}\) is the tortoise coordinate defined by: \[\begin{align} \frac{dr^{\star}}{dr}=\frac{1}{f(r)}\,. \label{eq:rstar} \end{align}\tag{20}\] We note that \(r^{\star} \to r\) for large values of \(r\), given that the metrics we consider are asymptotically flat.

In general, numerical integration methods are required to compute GBFs for general space-times, and this holds for our tr-symmetric RBHs as well. In our work, we make use of the so-called shooting method (see Appendix 6 for further details), which has already been successfully applied to these types of calculations in several earlier works [248]–[255].

To begin with, we rewrite Eq. (18 ) in terms of the rescaled coordinate \(x\), given by the following: \[\begin{align} x \equiv \frac{r-r_H}{r_H}\,, \label{eq:varx} \end{align}\tag{21}\] where \(r_H\) is the largest real root of \(f(r)=0\). With this substitution Eq. (18 ) is rewritten as follows: \[\begin{align} & x^2(x+1)^3 f \ddot{R_s} \\ &+ (s+1) x(x+1) \left ( 2(x+1)f+(x+1)^2 \dot{f} \right ) \dot{R_s} \\ &+ V(\omega,x)R_s=0\,, \end{align} \label{eq:teukolsky2}\tag{22}\] where \(\dot{ } \equiv \partial_x\), and \(V(\omega,x)\) is given by: \[\begin{align} &V(\omega,x) = \left ( \frac{\omega^2 r_H^2 (x+1)^2}{f} + 2 i s (x+1) \omega \right. \\ & - i s r_H (x+1)^2 \frac{\dot{f} }{f} \omega + s \left ( 2 f + 4 (x+1) \dot{f} \right.\\ &\left. + (x+1)^2 \ddot f -2 \right ) -l(l+1) + s(s+1) \biggr) x (x+1)\,. \end{align}\] In order to further simplify the problem, we work in units of horizon radius and therefore set \(r_H=1\), so that \(r=x+1\). In these units, the metric functions of the four RBHs under consideration are given by the following: \[\begin{align} & f_{\text{B}}(x)=1-\frac{(1+\ell^2)^{3/2}(x+1)^2}{ \left ( \ell^2+(x+1)^2 \right ) ^{\frac{3}{2}}}\,, \\ & f_{\text{H}}(x)=1-\frac{(x+1)^2}{(1-\ell^2) \left( (x+1)^3 - \frac{\ell^2}{\ell^2-1} \right ) } \,, \\ & f_{\text{GCSV}}(x)=1- \frac{e^{\ell - \frac{\ell}{x+1}}}{x+1}\,,\\ & f_{\text{H-l}}(x)=1-\frac{(1+\ell-\ell^2)(x+1)^2}{ \left ( (x+1)^3 + \ell - \ell^2 (x+1) \right ) }\,, \end{align}\] for the Bardeen, Hayward, GCSV, and Hayward-like space-times respectively.

Setting purely ingoing boundary in proximity of the horizon, the solutions to Eq. (22 ) can be expressed in the form of a Taylor expansion as follows [248], [249], [256]–[261]: \[\begin{align} R_s(x)= x^{-s- \frac{i \omega}{\tau}} \sum_{n=0}^\infty a_n x^n\,, \label{eq:near} \end{align}\tag{23}\] where \(i \omega / \tau\) is a function of the field spin and the regularizing parameter, and also depends on the space-time in question. We refer the reader to Appendix 6 for further details. The \(a_n\) coefficients can be determined by substituting Eq. (23 ) in Eq. (22 ) and iteratively solving the resulting algebraic equations. The near-horizon solution is then used to set the boundary conditions and numerically integrate the radial equation up to large distances, where the general form of the solution is the following: \[\begin{align} R(x) \xrightarrow{r \to \infty} R^{\text{in}}_s \frac{e^{-i \omega x}}{x}+R^{\text{out}}_s \frac{e^{i \omega x}}{x^{2s+1}}\,. \label{eq:far} \end{align}\tag{24}\] The GBFs can then be computed from the \(_s R^{l m}_{\text{in}}(\omega)\) coefficient. More specifically, the normalization of the scattering problem is set by requiring \(a_0=1\). With this normalization, the GBFs then read: \[\begin{align} \Gamma^s_{l m}(\omega)=\delta_s \vert _s R^{l m}_{\text{in}}(\omega)\vert^{-2}\,, \label{eq:gamma} \end{align}\tag{25}\] where the coefficient \(\delta_s\) is given by: \[\begin{align} \delta_s = \tau \frac{ i e^{i \pi s} (2 \omega)^{2s-1} \Gamma \left ( 1-s- \frac{2 i \omega}{\tau} \right ) }{\Gamma \left ( s-\frac{2 i \omega}{\tau} \right ) } \label{eq:delta} \end{align}\tag{26}\] Using the method discussed above, we compute the GBFs for perturbations of different spin on the backgrounds of the four RBH space-times discussed earlier, for different values of the regularizing parameter \(\ell\). In the specific case \(s=1\), we have checked that calculating the GBFs up to \(l=4\) is sufficient for our purposes. The GBFs we calculate are then used to characterize the Hawking evaporation spectra, as we will discuss shortly.

3.2 Evaporation spectra↩︎

We now discuss our computation of the photon spectra resulting from Hawking evaporation of the regular BHs discussed previously. In what follows, we only account for the primary photon spectrum. Nevertheless, we have checked that in the mass region of interest the impact of the secondary component of the spectrum, i.e.that resulting from the decay into photons of other unstable particles which are also produced during the evaporation process, is negligible.

The Hawking radiation rate (number of particles emitted per unit time per unit energy) of a given particle species \(i\) with spin \(s\), as a result of Hawking evaporation, is given by the following [262]–[266]: ⁵ \[\begin{align} \tag{27} \frac{d^2N_i}{dtdE_i}=\frac{1}{2\pi}\sum_{l,m}\frac{n_i\Gamma^s_{l,m}(\omega)}{ e^{\omega/T}\pm 1}\,, \tag{28} \end{align}\] where \(n_i\) is the number of degrees of freedom of the particle in question, \(\omega=E_i\) is the mode frequency (in natural units), \(\Gamma^s_{l,m}\) are the GBFs discussed previously, and we have implicitly set \(k_B=1\). Note that the plus (minus) sign in the denominator is associated to fermions (bosons). Following the methodology discussed in Sec. 3.1, we calculate the GBFs within all the BH space-times in question for photons (\(s=1\)), up to \(l=4\) (note that the angular node number \(l\) should not be confused with the regularizing parameter \(\ell\)). We have checked that adding higher \(l\) modes does not appreciably improve the resulting spectra.

We show examples of the resulting evaporation spectra in Figs. 2, 3, 4, and 5. The spectra obviously depend on the mass of the evaporating PBH, which we have set to \(M_{\text{pbh}}=10^{13}\,{\text{kg}}\), as it sits roughly in the middle of the mass range of interest. Nevertheless, we stress that the features we discuss below do not depend on the chosen mass. The resulting spectra all peak approximately between \(5\,{\text{MeV}}\) and \(10\,{\text{MeV}}\). For the Bardeen, Hayward, and GCSV RBHs we observe that an increase in the regularizing parameter \(\ell\) leads to a decrease in the intensity of the spectra at all energies. These behaviours are consistent with the temperature evolution shown in Fig. 1, although we stress that studying the temperature alone is not sufficient to draw these conclusions as the GBFs also play a key role in determining the shape and intensity of the resulting spectra, as Eq. (28 ) makes very clear. For the Bardeen and Hayward RBHs the position of the peak in the spectrum is only mildly affected by the regularizing parameter, an increase in which pushes the peak towards slightly lower energies. On the other hand, for the GCSV BH an increase in the regularizing parameter pushes the peak towards higher energies. We do not exclude that this different behaviour may be related to the type of core being considered, and we defer a more detailed investigation of this point to future work. At any rate, we expect that the behaviour of the spectra discussed above should lead to constraints on \(f_{\text{pbh}}\) which are loosened for these classes of primordial RBHs.

Contrarily to what we observe for these three RBHs, we see that the intensity of the spectra for Hayward-like RBHs substantially increase with increasing \(\ell\) for sufficiently large energies \(E \gtrsim {\cal O}({\text{MeV}})\), whereas for lower energies they very slightly decrease. This behaviour agrees qualitatively with the expectations which motivated us to construct the Hayward-like RBH in first place, and with the temperature evolution shown in Fig. 1. In addition, we observe that an increase in the regularizing parameter leads to a much richer structure in the spectra: the observed multi-peak structure is the result of the contribution of different \(l\) modes in the emission being disentangled more clearly. A similar behaviour is actually observed in the Kerr BHs evaporating spectra which, by virtue of their rotation, lead to a more complex spectrum where the contribution of high-\(l\) modes, while still subdominant with respect to the \(l=1\) one, emerges more clearly (although we have explicitly checked that adding \(l>4\) modes has virtually no effect on our results). In the case of Hayward-like RBHs, we remain agnostic as to the physical reason why such a structure is observed, and leave a more detailed study to follow-up work.

3.3 Evaporation constraints↩︎

The spectra calculated in Sec. 3.2 are then used to set evaporation constraints on \(f_{\text{pbh}}(M) \equiv \Omega_{\text{pbh}}/\Omega_{\text{dm}}\), the fraction of DM in the form of PBHs, where \(\Omega_{\text{pbh}}\) and \(\Omega_{\text{dm}}\) are the PBH and DM density parameters respectively. Specifically, the computed spectra are used to obtain predictions for the flux of photons resulting from Hawking evaporation, which are then directly compared against measurements of the extragalactic photon background across a wide range of energies (see e.g.Ref. [267] for a recent review). Evaporation constraints are the dominant ones in the \(10^{10}\,{\text{kg}} \lesssim M_{\text{pbh}} \lesssim 10^{15}\,{\text{kg}}\) mass range: the lower limit of the range is set by the requirement that PBHs have not yet evaporated at the time of recombination, whereas the upper limit is defined by measurements of the diffuse extragalactic \(\gamma\)-ray background (EGRB) in the energy range \(100\,{\text{keV}} \lesssim E_{\gamma} \lesssim 5\,{\text{GeV}}\), given that the intensity of the Hawking radiation flux is inversely proportional to the mass of the evaporating BH. In what follows, we will direct our attention exclusively to PBHs for which \(M_{\text{pbh}} \gtrsim 10^{12}\,{\text{kg}}\): these have yet to fully evaporate today and, having formed deep during the radiation domination era, are therefore excellent non-baryonic DM candidates.

We work under the commonly adopted assumption that PBHs are isotropically distributed on sufficiently large scales. Therefore, the flux resulting from their evaporation and reaching us today is given by the redshifted sum of the contributions from all evaporating PBHs in our Universe, and can be used to constrain the average extragalactic distribution of DM in the form of PBHs. We also work within the (also commonly adopted) approximation of monochromatic mass distributions (which can be expected if the formation mechanism arises from an amplification of the power spectrum at a very specific scale), although the effect of extended mass distributions is the subject of active research [268]–[282]. Finally, as discussed earlier, we only consider the primary photon contribution, as the secondary component resulting from the decay into photons of other unstable particles is verified a posteriori to be negligible given the mass range of interest. While all these are clearly approximations, albeit widely adopted ones, we are confident that they are appropriate given the aim of our work. Our main goal is to examine how the limits on \(f_{\text{pbh}}\) change when moving from the Schwarzschild PBH framework to that of the regular metrics presented in Sec. 2, potentially opening or closing the asteroid mass window. It is more than reasonable to expect that the shift in constraints relative to the Schwarzschild case \(\delta f_{\text{pbh}}\) is only weakly affected by the above approximations. In other words we expect these approximations to have similar impacts on the contraints on \(f_{\text{pbh}}\) relative to the Schwarzschild BHs and the RBHs discussed in Sec. 2, therefore leading to negligible effects on the shift \(\delta f_{\text{pbh}}\), in which we are ultimately interested. In any case, such approximations also allow for a more direct comparison to several previous works and therefore we consider them appropriate for our pilot study, but their impact should definitely be explored in follow-up works. Finally, note that we are tacitly assuming that PBHs cluster in the galactic halo in the same way as other forms of DM (unless they are extremely large, which is not the case for the mass range of interest).

In what follows, we therefore assume that PBHs all have the same initial mass \(M_{\text{pbh}}\). Following Ref. [283] we approximate the number of emitted photons in the logarithmic energy bin \(\Delta E_{\gamma} \simeq E_{\gamma}\) as being given by \(\dot{N}_{\gamma}(E_{\gamma}) \simeq E_{\gamma}(d\dot{N}_{\gamma}/dE_{\gamma})\). The emission rate of photons from Hawking evaporation per volume at a cosmological time \(t\) is then given by [283]:⁶ \[\begin{align} \frac{dn_{\gamma}}{dt}(E_{\gamma},t) \simeq n_{\text{pbh}}(t)E_{\gamma}\frac{d^2{N}_{\gamma}}{dt dE_{\gamma}}(M_{\text{pbh}},E_{\gamma})\,, \label{eq:dngammadt} \end{align}\tag{29}\] where \(n_{\text{pbh}}(t)\) is the PBH number density at time \(t\). By integrating and taking into account the redshift scaling of the photon energy and density we end up with: \[\begin{align} \begin{aligned} &n_{\gamma 0}(E_{\gamma 0})\\ &= n_{\text{pbh}}(t_0) E_{\gamma 0}\int^{t_0}_{t_{\star}} dt(1 + z) \frac{d^2{N}_{\gamma}}{dt dE_{\gamma}}(M_{\text{pbh}},(1+z)E_{\gamma 0})\\ & = n_{\text{pbh}}(t_0) E_{\gamma 0}\int^{z_{\star}}_{0} \frac{dz}{H(z)}\frac{d^2{N}_{\gamma}}{dt dE_{\gamma}}(M_{\text{pbh}},(1+z)E_{\gamma 0}) \label{eq:npbh1} \end{aligned} \end{align}\tag{30}\] where \(t_0\) denotes the present time, \(t_{\star}\) and \(z_{\star}\) are respectively the cosmic time and redshift at recombination, and \(H(z)\) is the expansion rate. Finally, \(n_{\gamma 0}(E_{\gamma 0})\) is the present number density of photons with energy \(E_{\gamma 0}\). The resulting photon flux (more properly, the rate of photons per unit time per unit area per unit solid angle) is then given by: \[\begin{align} I(E_{\gamma 0}) \equiv \frac{c}{4\pi}n_{\gamma 0}(E_{\gamma 0})\,. \label{eq:flux} \end{align}\tag{31}\] It is this quantity which can then be directly compared against observations.

We assume a spatially flat \(\Lambda\)CDM cosmological model in specifying the expansion rate entering into Eq. (30 ), with the same cosmological parameters as in Ref. [283]. This allows us to robustly cross-check our Schwarzschild constraints on \(f_{\text{pbh}}\) against those reported in the seminal Ref. [283], although we stress that our constraints are very stable against reasonable changes in the values of the cosmological parameters. Once the cosmological model is fixed, all the relevant quantities in Eq. (30 ) are known except for the present-day PBH number density, \(n_{\text{pbh}}(t_0)\), which can be constrained from EGRB observations and is ultimately related to \(f_{\text{pbh}}\). More specifically, for any given value of the PBH mass \(M_{\text{pbh}}\), through Eqs. (28 ,30 ,31 ) we can compute the unnormalized photon flux \(I(E_{\gamma 0})/n_{\text{pbh}}(E_{\gamma 0})\), and adjust the normalization \(n_{\text{pbh}}(E_{\gamma 0})\) by comparing against EGRB observations (as we will explain shortly). This procedures gives us an upper limit on \(n_{\text{pbh}}(E_{\gamma 0})\), which can be translated into an upper limit on \(f_{\text{pbh}}\) as follows: \[\begin{align} f_{\text{pbh}}(M_{\text{pbh}}) \equiv \frac{\Omega_{\text{pbh}}}{\Omega_{\text{dm}}} = \frac{n_{\text{pbh}}(t_0)M_{\text{pbh}}}{\rho_{\text{crit},0}\Omega_{\text{dm}}}\,, \label{eq:npbhtofpbh} \end{align}\tag{32}\] where \(\rho_{\text{crit},0}=3H_0^2/8\pi G\) is the present-day critical density, with \(H_0\) the Hubble constant, and we recall that this procedure is done for various values of \(M_{\text{pbh}}\).

We compare our theoretical predictions against various measurements of the EGRB. Specifically, we use observations of the EGRB from the HEAO-1 X-ray telescope in the \(3\)-\(500\,{\text{keV}}\) range [284], the COMPTEL imaging Compton \(\gamma\)-ray telescope in the \(0.8\)-\(30\,{\text{MeV}}\) range [285], and the EGRET \(\gamma\)-ray telescope [286]. A few comments are in order concerning the adopted datasets. While these are by now a couple of decades old, they basically still represent the state-of-the-art in the energy range of interest. One could entertain other observations, including local galactic measurements of the galactic \(\gamma\)-ray background [287], positron flux [288], \(0.511\,{\text{MeV}}\) annihilation radiation [289], [290], and various other sources. While these galactic observations could give potentially stronger limits, they depend strongly on the form of the PBH mass function (assumed to be monochromatic in our study), as well as the clustering properties of these PBHs. On the other hand, our limits on \(f_{\text{pbh}}\) are effectively testing the average extragalactic distribution of DM. Finally, other measurements of the EGRB are available, e.g.from Fermi-LAT [291], but these are mostly important for energy ranges larger than the ones of interest, and therefore for PBHs lighter than the ones we are considering. Therefore, we believe the choice of datasets (which is the one adopted in several works estimating evaporation limits on PBHs) is appropriate given the objective of our study.

To set upper limits on \(n_{\text{pbh}}(t_0)\) – and therefore \(f_{\text{pbh}}\) through Eq. (32 ) – we adopt the simple method first explained in the seminal Ref. [283], and later adopted in most of the works examining constraints on PBHs from the EGRB. Specifically, for each value of \(M_{\text{pbh}}\), and for given values of the regularizing parameter \(\ell\), the maximum allowed value of \(f_{\text{pbh}}\) is determined by requiring that the predicted photon flux does not overshoot any of the ERGB datapoints by more than \(1\sigma\). An example is shown in Fig. 6 for a Bardeen PRBH with regularizing parameter \(\ell=0.3\,r_H\): for each of the mass values \(M_{\text{pbh}}\) represented, the upper limit on \(f_{\text{pbh}}\) is set as soon as the first datapoint is overshot. As is clear from the Figure, different PBH masses result in different datapoints being overshot. For each of the PRBHs considered, we use this procedure to determine upper limits on \(f_{\text{pbh}}\) for fixed, representative values of \(\ell\) (\(\ell/r_H=0.15\), \(0.3\), and \(0.45\) for all BHs, as well as \(0.6\) and \(0.75\) only for the Hayward-like BH), comparing the results to the Schwarzschild case which is recovered when \(\ell=0\). ⁷ We note that the exact origin of the EGRB is currently a matter of debate [292]: although it is believed that distant astrophysical sources such as blazars give a major contribution to the EGRB, there is no complete consensus on the level of this contribution. In this light, our approach of simply requiring that the PRBH evaporation contribution to the EGRB does not exceed any observed datapoint is rather conservative (given that there could in principle be a PBH contribution to the EGRB, should it be conclusively determined that known astrophysical sources cannot fully account for the latter).

4 Results↩︎

For each of the PRBHs discussed in Sec. 2, we now proceed to derive upper limits on \(f_{\text{pbh}}\) as a function of the PRBH mass \(M_{\text{pbh}}\), for different values of \(\ell\), using the methodology presented in Sec. 3.3. The results are shown in Figs. 7,  8,  9, and 10 for the Bardeen, Hayward, GCSV, and Hayward-like BHs respectively. For each case, we also plot the constraints on \(f_{\text{pbh}}\) for the \(\ell=0\) case (blue solid curve in all the Figures), which correspond to the standard Schwarzschild PBH scenario widely studied in the literature. As a sanity check, we have verified that our \(\ell=0\) constraints exactly recover those of the seminal Ref. [283]. It is worth noting that, for any given value of \(\ell\), the value of \(M_{\text{pbh}}\) corresponding to the upper right edge of the \(f_{\text{pbh}}\) constraints (i.e.the value of \(M_{\text{pbh}}\) for which the limit reads \(f_{\text{pbh}}<1\)) marks the lower edge of the (modified – either enlarged or contracted) asteroid mass window.

We begin by discussing the Bardeen, Hayward, and GCSV PRBHs, for which we saw earlier that the temperature and photon spectra decrease in intensity with increasing regularizing parameter \(\ell\) (see the discussion in Sec. 3.2, and Figs. 2–5). As we could have expected, this behaviour leads to overall looser constraints on \(f_{\text{pbh}}\) (for any given \(M_{\text{pbh}}\)) relative to the standard limits reported for Schwarzschild PBHs in the literature. In the case of near-extremal Hayward PRBHs (\(\ell=0.45r_H\)) this behaviour is somewhat enhanced compared to the near-extremal Bardeen and GCSV PRBHs, with the upper limits on \(f_{\text{pbh}}\) approximately three orders of magnitude looser than the corresponding Schwarzschild ones: again, this is somewhat unsurprising when comparing Fig. 3 to Figs. 2 and 4. This could also have been expected from Fig. 1, noting that the temperature of Hayward BHs decreases more rapidly with increasing \(\ell/r_H\) relative the Bardeen and especially GCSV ones. Although the temperature is not the only factor at play in determining the resulting evaporation spectrum, given that the GBFs also play a key role as per Eq. (28 ), it is reassuring that the temperature behaviour observed in Fig. 1 is qualitatively reflected in the limits on \(f_{\text{pbh}}\) we derive.

For the Hayward-like BH, we observe exactly the opposite trend, with overall tighter constraints on \(f_{\text{pbh}}\) relative to the Schwarzschild ones. Again, such a behaviour is in line with expectations given the behaviour of the temperature observed in Fig. 1, and of the spectra as shown in Fig. 5. The relative shift in \(f_{\text{pbh}}\) constraints for a given value of \(M_{\text{pbh}}\) and \(\ell/r_H\) is somewhat less dramatic than what we observed for the Bardeen, Hayward, and GCSV BHs, with shifts of less than two orders of magnitude.

As a result of the shifts discussed above, the lower edge of the asteroid mass window where PBHs could make up the entire DM component is modified for all four metrics considered. We recall that in the Schwarzschild case, the lower edge of the window lies at \(M_{\text{pbh}} \simeq 10^{14}\,{\text{kg}}\). For the Bardeen, Hayward, and GCSV PRBHs, the looser constraints on \(f_{\text{pbh}}\) result in the asteroid mass window further opening up by approximately half a decade in mass or more. The maximum extension of the window is reached for the Hayward PRBH closer to extremality, in which case the lower edge decreases by about an order of magnitude to \(M_{\text{pbh}} \simeq 10^{13}\,{\text{kg}}\). The opposite behaviour is of course observed for Hayward-like PRBHs, in which case the asteroid mass window further closes down, although by a lesser extent with respect to the previous cases. For instance, for \(\ell/r_H=0.75\), the lower edge increases to \(M_{\text{pbh}} \simeq 2\times 10^{14}\,{\text{kg}}\). Overall, we therefore observe that considering primordial regular BHs in place of the standard Schwarzschild ones can move the resulting constraints on \(f_{\text{pbh}}\) in either direction, further opening up or closing down the asteroid mass window, with the allowed region for the window lower edge spanning over a decade in mass, at least for the PRBHs considered.

Three comments are in order before concluding. Firstly we note that, for a given PRBH space-time, the curves describing the \(f_{\text{pbh}}(M)\) limits are approximately, but not exactly parallel to the Schwarzschild ones (blue solid curves in Figs. 7–10). The reason is simply that, as \(\ell\) is increased, the datapoint shown in Fig. 6 which is first being overshot and therefore responsible for determining the \(f_{\text{pbh}}\) limit can potentially change (in part due to the spectrum slightly changing shape, especially in the Hayward-like case where we saw that an increase in \(\ell\) lead to a richer peak structure due to the effect of different \(l\) modes).

Next, the constraints we have determined on \(f_{\text{pbh}}\) at a fixed value of \(\ell/r_H\) implicitly assume that all PRBHs in the Universe carry the same value of “hair” parameter \(\ell\). However, particularly given our agnostic stance with regards to the origin of these space-times, in principle the value of \(\ell/r_H\) can vary from PRBH to PRBH. To make an analogy, let us assume for a moment that Reissner-Nordström BHs are astrophysically relevant. Then, since the electric charge \(Q\) is not tied to a universal parameter of the underlying Einstein-Maxwell Lagrangian, there is no reason to expect it to carry the same value across all BHs. In the language of Ref. [224], the regularizing parameter for all four RBHs considered is a “specific hair” rather than an “universal hair” (see Ref. [224] for various examples of BH solutions carrying universal hair), unless one were able to tie \(\ell\) to some fundamental parameter of the underlying theory, which however is not the case in the phenomenological approach we are following. In principle one should therefore account for the (non-monochromatic) \(\ell\) distribution for PRBHs across the Universe to determine constraints on \(f_{\text{pbh}}\). We see no obvious way of doing this, while noting that such a procedure would most likely result in upper limits on \(f_{\text{pbh}}\) lying between the Schwarzschild and extremal cases: this observation suffices for our pilot study, and we defer a more complete investigation to future work.

Our final comment concerns the fact that evaporation limits on the PBH abundance are not the only ones at play. Indeed, as recently summarized in Ref. [60], there are essentially four classes of limits, each of which is relevant in a different mass range: evaporation, lensing, dynamical, and accretion constraints. Constraints from the accretion of background gas at early times are relevant in a completely different mass range (\(10^{30} \lesssim M_{\text{pbh}}/{\text{kg}} \lesssim 10^{37}\) – see Fig. 7 of Ref. [60] and Fig. 10 of Ref. [293]). Although these have been derived assuming Schwarzschild PBHs, moving to the PRBH picture we have considered will not shift the relevant mass range by the \(\gtrsim 18\) orders of magnitude required for these constraints to compete with the evaporation ones, unless the physics of gas accreting around RBHs changes drastically with respect to the standard picture, which appears very unlikely. Dynamical constraints, most of which are associated to the destruction of different astronomical objects by the passage of nearby PBHs, are also relevant in a completely different mass range (\(10^{31} \lesssim M_{\text{pbh}}/{\text{kg}} \lesssim 10^{52}\) – see Fig. 7 of Ref. [60] and Fig. 10 of Ref. [293]), and considerations completely analogous to those we made for accretion constraints hold. ⁸

Of potentially more relevance to the present work are lensing constraints, which constrain the abundance of PBHs (and more generally massive compact halo objects) with masses \(M_{\text{pbh}} \gtrsim 10^{12}\,{\text{kg}}\). Indeed, it is lensing constraints which locate the upper edge of the asteroid mass window where PBHs can make up all the DM. Nevertheless, we expect that these constraints should not change when moving from the Schwarzschild PBH framework to the PRBHs considered in this work. Indeed, with all other quantities being fixed (mass of source, relative distances, and so on), lensing constraints only depend on the lens mass \(M\), and are unaffected by the metric structure of the lens. Therefore, at fixed mass \(M\), we can assume that the lensing limits on Schwarzschild PBHs hold for our PRBHs as well. Note that, as already pointed out in footnote 2, the parameter \(M\) appearing in the RBH metrics can be unambiguously identified with the RBH mass, just as with the parameter \(M\) in the Schwarzschild metric. We can therefore conclude that for the PRBHs we are considering it is only the lower edge of the asteroid mass window which is altered with respect to the Schwarzschild case, but not the upper edge. In other words, space-times for which the lower edge moves towards lower masses (as in the Bardeen, Hayward, and GCSV PRBH cases) genuinely corresponds to an enlarged asteroid mass window, and conversely for the Hayward-like case where the lower edge moves towards higher masses. Therefore, the window where Bardeen, Hayward, and GCSV PRBHs could account for all the DM is larger compared to that of Schwarzschild PBHs.

Other potentially relevant constraints come from \(\mu\)-distortions in the Cosmic Microwave Background, and gravitational waves (either a stochastic background due to a population of coalescing PBHs or produced via second-order tensor perturbations generated by the scalar perturbations producing the PBHs, or associated to resolved events). The latter are expected to be relevant in a much higher mass range (again, see Fig. 7 of Ref. [60] and Fig. 10 of Ref. [293]), whereas the former are somewhat dependent on the PBH formation scenario from high-\(\sigma\) tails of density fluctuations, and in particular on the shape of the tail. At any rate, while the focus in the present pilot study has been solely on evaporation constraints from the ERGB, revisiting all these other important sources of constraints (including the ones discussed earlier) is a worthwhile endeavour which we plan to explore in upcoming works.

5 Conclusions↩︎

Over the past decade, primordial black holes have regained tremendous interest as viable dark matter candidates, with the so-called “asteroid mass window” (\(10^{14}\,{\text{kg}} \lesssim M_{\text{pbh}} \lesssim 10^{20}\,{\text{kg}}\)) where PBHs could potentially account for the entire DM currently still open. Nearly all works on PBHs assume that these are Schwarzschild or Kerr BHs. However, while phenomenologically perfectly valid, such an assumption may stir some unease on the theoretical side, due to the appearance of singularities in these metrics. In our work, we have conducted a pilot study aimed at addressing a question which naturally merges the DM and singularity problems, arguably two among the most important open problems in theoretical physics: “What if PBHs are non-singular”? Our study of primordial regular BHs (PRBHs) has focused on four so-called tr-symmetric metrics (including the well-known Bardeen and Hayward space-times), whereas our companion paper [140] considers non-tr-symmetric metrics, including various metrics inspired from loop quantum gravity.

We show that evaporation constraints on \(f_{\text{pbh}}\), the fraction of DM in the form of PRBHs, can be substantially altered in either direction when moving away from the Schwarzschild picture, leading to the asteroid mass window further opening up or closing down depending on the direction of the shifts in \(f_{\text{pbh}}\) limits. For three of the PRBHs we considered (the Bardeen, Hayward, and GCSV space-times) the lower edge of the asteroid mass window is shifted by nearly a decade in mass, leading to a larger region of parameter space where PRBHs could account for the entire DM component, which should be the target of the same probes proposed to test the standard window [118], [120], [122]–[124], [126], [127]. The opposite trend takes place with the Hayward-like BH we constructed, and we have argued that part of this different behaviour can be traced back to the evolution (increase or decrease) of the PRBH temperature as the regularizing parameter is increased. On the other hand, the nature of the regular BH core (de Sitter or Minkowski) does not appear to play a significant role in this sense. Overall, we have shown that the phenomenology of primordial regular BHs can be particularly rich, making the associated simultaneous solution to the DM and singularity problems one worthy of further study.

We remark that the present work (alongside our companion paper [140]) should be intended as a pilot study, and there are a huge number of interesting follow-up directions. One interesting avenue for further work involves systematically revisiting other sources of constraints which have been studied in the Schwarzschild PBH case, including but not limited to lensing, accretion, and dynamical constraints: while we have argued that these should not alter our considerations on the asteroid mass window, a detailed study which would allow us to extend our constraints over a much larger region of \(M_{\text{pbh}}\)-\(f_{\text{pbh}}\) plane is nevertheless in order. In addition, the metrics we have considered are inherently phenomenological in nature, and it would therefore be worth extending our study to non-singular metrics which enjoy a strong theoretical motivation (our companion paper [140] goes partially in this direction), including potentially metrics which are coupled to the cosmological expansion. Last but definitely not least, if PBHs are truly regular, one would hope to ascertain this via signatures complementary to those we have studied (for instance through gravitational wave signatures, VLBI imaging, particle motion, or energy extraction). We plan to address these and other related points in follow-up work.

6 Details on the computation of GBFs↩︎

Here we provide a few more details on the computation of GBFs. We recall that we expressed the solutions to the radial Teukolsky equation, Eq. (22 ), in the form of a Taylor expansion as given by Eq. (23 ). This is also known as a Frobenius series, being a by-product of a method for solving second-order differential equations named after Frobenius. The method applies to equations which take the following form \[\begin{align} u'' +p(x) u' + q(x) u=0\,, \label{eq:frobenius} \end{align}\tag{33}\] in proximity of its singular points, namely those where \(p(x)\) and \(q(x)\) diverge. One can notice that Eq. (22 ) can be rewritten in the form of Eq. (33 ), with one of its singular point being at \(x=0\), i.e.at the event horizon.

To solve the radial Teukolsky equation we therefore proceed as follows:

• We work in units of the event horizon and rewrite Eq. (22 ) in order to remove the denominators \[\begin{align} & A(x)R''_s + B(x)R'_s + C(x) R_s=0, \end{align} \label{Teuk3}\tag{34}\] where the functions \(A(x)\), \(B(x)\), and \(C(x)\) are given by the following: \[\label{Teuk4} \begin{align} & A(x)=f^2 (x+1)^2 \;\;,\\ & B(x)= (s+1)f^2(x) (2(x+2)+(x+1)^2 f'/f) \;\;,\\ & C(x)=+(x+1)^2 \omega ^2 +2 i s (x+1) \omega f -i s (x+1)^2 \omega f' \\ & \;\;\;\;\;\;\;\;\;\;\;\, +s f \left( (x+1)^2 f''+4 (x+1) f'+2 f-2 \right)\\ & \;\;\;\;\;\;\;\;\;\;\;\, -l (l+1) f +s (s+1) f \;\;,\nonumber \end{align}\tag{35}\]

• The lowest power term around \(x=0\) of each coefficient can be written in the following form: \[\begin{align} & A(x)\sim x^2 \tau^2 \;\;,\\ & B(x)\sim x (s+1) \tau^2 \;\;,\\ & C(x) \sim \omega^2 - i \omega s \tau \;\;,\nonumber \end{align}\] where \(\tau=\tau(\ell)\) depends on the choice of RBH.

• We then build the following characteristic equation: \[\begin{align} m(m-1) \tau^2 + m (s+1) \tau^2 + \omega (\omega - i s \tau)=0\,, \label{Eu-Cau} \end{align}\tag{36}\] whose solutions are the following: \[\begin{align} m_1=-s - \frac{i \omega}{\tau} \;\; ,\;\; m_2=\frac{i \omega}{\tau} \end{align}\]

• It is then possible to conclude that Eq. (22 ) admits solutions near the singular point \(x=0\) of the form given by Eq. (23 ).

Explicitly, for the four RBHs in question, \(\tau\) is given by the following: \[\begin{align} & \tau_{\text{B}}=\frac{1-2 \ell^2}{\ell^2+1} \;\;,\\ & \tau_{\text{H}}= (1-3 \ell^2)\;\;,\\ & \tau_{\text{GCSV}}=1-\ell\;\;,\\ & \tau_{\text{H-l}}= \frac{(\ell-1)^2}{1-\ell(\ell-1)}\;\;.\nonumber \end{align}\] We notice that in the Schwarzschild limit \(\ell \to 0\), all of the above reduce to \(\tau=1\) as one could expect.

References↩︎