Revealing Dark Matter Dress of Primordial Black Holes by Cosmological Lensing

Stellar-mass primordial black holes (PBHs) from the early Universe can directly contribute to the gravitational wave (GW) events observed by LIGO, but can only comprise a subdominant component of the dark matter (DM). The primary DM constituent will generically form massive halos around seeding stellar-mass PBHs. We demonstrate that gravitational lensing of sources at cosmological ($\gtrsim$Gpc) distances can directly explore DM halo dresses engulfing PBHs, challenging for lensing of local sources in the vicinity of Milky Way. Strong lensing analysis of fast radio bursts detected by CHIME survey already starts to probe parameter space of dressed stellar-mass PBHs, and upcoming searches can efficiently explore dressed PBHs over $\sim 10-10^5 M_{\odot}$ mass-range and provide a stringent test of the PBH scenario for the GW events. Our findings establish a general test for a broad class of DM models with stellar-mass PBHs, including those where QCD axions or WIMP-like particles comprise predominant DM. The results open a new route for exploring dressed PBHs with various types of lensing events at cosmological distances, such as supernovae and caustic crossings.


I. INTRODUCTION
The predominant form of matter in the Universe, the dark matter (DM), has been detected only through gravitational interactions and its nature remains mysterious (e.g.[1]).The existence of black holes and their principal role in astronomy have been definitively established, including observations of supermassive black holes residing within galactic centers (e.g.[2][3][4][5]).Primordial black holes (PBHs) formed the early Universe prior to galaxies and stars could constitute all or a fraction of the DM abundance (e.g. ).Depending on the formation, PBHs can span decades in orders of magnitude in the mass range.
Constituting overdensities in an expanding Universe, PBHs will seed the growth of DM halos [47,48] following the theory of spherical gravitational collapse [49].Hence, stellar-mass PBHs forming a subdominant DM component will be generically engulfed in massive halo dresses composed of the primary DM component.
A well motivated possibility is that DM is predominantly composed of axions, which can naturally arise from fundamental theory [50] and directly connected with the strong CP problem [51][52][53][54].While significant experimental efforts are underway (see [55] for review), axions remain elusive.In general, DM halos can be composed of a slew of motivated DM candidates (see [56] for review), including also weakly interactive massive particles (WIMP-like particles) with typical mass in the GeV to 100 TeV range that often arise in models addressing the hierarchy problem.With DM concentrations significantly exceeding that of ambient cold DM, dark halo dresses surrounding PBHs make for an excellent laboratory to explore DM beyond conventional techniques.
In this work we propose cosmological lensing of fast radio bursts (FRBs) as a novel direct probe for exploring dark halo dresses around PBHs and a way to distinguish them with bare PBHs, allowing for unique tests of a wide range of DM models where stellar-mass PBHs can contribute to the detected GW events.In contrast to some of the previously discussed signatures associated with specific DM models comprising dark halos around PBHs (e.g.[57,58]), our analysis establishes a general direct test of dressed PBHs in models where QCD ax-ions and WIMP-like particles could be the primary DM contributors, among others.

II. DARK MATTER HALO FORMATION
Stellar-mass PBHs will generically seed DM halos from accretion of surrounding smooth background of the primary DM component, until dressed PBHs reside within galactic halos.The halo surrounding a PBH will grow of order unity in mass during radiation-dominated era, and as ∝ (1 + z) −1 during matter-dominated era at redshift z.Thus, following numerical and analytic calculations the dark PBH halo grows with redshift as [47,48] The description of halo growth holds until z c ∼ 30, when PBHs start to interact with nonlinear cosmic structures.We note that the high density of DM in halos surrounding PBHs protects them against tidal stripping.These effects are not expected to be significant unless PBHs are orbiting very close to center of galaxy [58].
The resulting dark halo develops a density profile ρ ∝ r −9/4 [49,59], as confirmed by N -body simulations [40,60,61].From Eq. ( 1) and parametrization we find ρ 0 ≃ 8701(1000/(1 + z c )) −3 M ⊙ pc −3 .Here we consider the characteristic general case that DM is collisionless.Depending on assumptions and DM model, the dark halo profile could have behavior distinct from ∼ r −9/4 at inner radii (e.g.[61][62][63]).However, these effects appear at the innermost radii where the enclosed dark halo mass is comparable to that of seeding PBH [60].Hence, we do not expect that this will significantly affect our results.Our analysis can be readily extended to accommodate these effects and we leave their detailed evaluation in the context of specific scenarios for future work.Further, we do not focus here on WIMPs with significant annihilation channel or DM decaying on time-scales shorter than halo formation that due to increased density in DM halo have already been constrained by indirectdetection observations such as gamma-rays (e.g.[64]).

III. PBH-HALO LENSING
We now estimate the impact of the surrounding dark halo on PBH lensing.A central quantity characterizing gravitational lensing is the surface mass density Σ, which is the density profile of the lens projected along the line-of-sight.The corresponding dimensionless quantity is called convergence κ = Σ/Σ cr , with critical surface density being (e.g.[65]) Σ cr = (1/4πG)(D os /(D ol D ls )) , where we take G to be the gravitational constant, D os is the angular diameter distance between the observer and the source, D ol between the observer and the lens, D ls between the lens and the source.In Fig. 1 we schematically illustrate gravitational lensing by a dressed PBH.Throughout, we assume natural units, c = 1.The critical surface density corresponds to the mean surface density enclosed within the Einstein radius r E related to Einstein angle as r E = θ E D ol (tangential critical radius, defining the size of the Einstein ring), with enclosed mass being M (< r E ) = πΣ cr r 2 E .For the dark halo around a PBH we find its surface density to be Here we have ignored the truncation of the density profile beyond R h , as well as the truncation at the inner boundary.While this leads to a slight overestimation of the effects of dark halo, we expect these effects to be insignificant as long as r E ≪ R h , which is indeed the case for our parameters of interest in this work (see Appendix).From Eq. ( 3), we obtain the average halo surface density as In contrast, the surface density of PBH with mass M PBH can be modelled simply by ΣPBH (< r) = M PBH /(πr 2 ) assuming that its density profile is described by the Dirac's delta function at the origin.
We can now directly estimate the average convergence κ(< r), which for the Einstein radius satisfies κ(< r E ) = 1 and delineates the regimes of strong and weak gravitational lensing (e.g.[65]).From Eq. ( 4) we obtain the average convergence for the halo κh as well as the average convergence for PBH κPBH to be Comparing the average convergence of halo within PBH Einstein radius, i.e. κh (< r E,PBH ) = 1, allows to estimate the effect of dark halo on PBH lensing.Adding the contribution of the dark halo to a PBH, the new Einstein radius r E,tot is found by computing We can now compare results of Eq. ( 6) to that of isolated PBH r E,PBH , which is typically employed for PBH microlensing searches (e.g.EROS [66], OGLE [67] and Subaru HSC [68]).From the Einstein radius for a PBH with the dark halo, we can define the "effective PBH mass" The Einstein radius of an isolated point mass lens with M eff PBH matches the Einstein radius of a PBH with mass M PBH with the dark halo.For a given PBH with mass M PBH , its the lensing cross section is on the order to π{r E,tot (M PBH )} 2 , and by using Eq. ( 7) we can rewrite it as π{r E,PBH (M eff PBH )} 2 = M eff PBH /Σ cr , suggesting that a halo of a PBH with mass M PBH enhances its lensing cross section by M eff PBH /M PBH .

IV. COSMOLOGICAL LENSING WITH FAST RADIO BURSTS
In order to efficiently distinguish between dressed and bare PBHs, for a given PBH mass the ratio of mass in Eq. ( 7) should satisfy M eff PBH /M PBH ≫ 1.As we explicitly demonstrate in Appendix, gravitational lensing of sources at cosmological distances with redshifts z ∼ O(1) can efficiently reveal dark dresses of PBHs.On the other hand, as we verify, it is challenging to distinguish dressed and bare PBHs through gravitational lensing of local sources located at z ≪ 1.
Particularly promising sources for cosmological lensing are FRBs (see [69] for review).These are luminous transient radio signals of millisecond time-scales at cosmological distances, able to provide favorable sources for lensing by bare stellar-mass PBHs [70][71][72].Strong lensing of FRBs by stellar-mass PBHs will generate two distinct images of the bursts whose time delays are sufficiently large to resolve them in the time domain [70].More than several hundred FRBs, including some repeating, have been identified by the Canadian Hydrogen Intensity Mapping Experiments (CHIME) experiment [73].Analysis of lensing of 536 FRBs [74] detected by CHIME, as well as 593 FRBs with cross-correlation [75], constrained stellar-mass PBH DM abundance at the level of f PBH ∼ O(10 −2 ).Additional details for the framework of FRB strong lensing we employ can be found in Appendix.We note that the CHIME collaboration has already carried out an extremely detailed related lensing search and constraints involving interferometric lensing, which yields time resolution of up to ∼ 100 nanoseconds [76,77].
With an extended halo, for dressed PBHs we can compute the lensing optical depth as where χ(z) is the comoving distance at redshift z, n PBH = f PBH Ω cdm ρ crit /M PBH is the comoving number density of PBHs, with f PBH being the fraction PBH DM abundance, ρ crit is the critical density, Ω cdm = 0.24 is the Universe's cold DM density and σ is the lensing crosssection characterizing the lens and dependence on the impact parameters (see Appendix).The lensing optical depth τ w/o h denote the traditional PBH lensing cross section without the halo contribution as described in Appendix.Optical depth for point source lensing by bare PBHs can be recovered from Eq. ( 8) by substituting M PBH for M eff PBH .Prefactors in the last expression in Eq. ( 8) originate from the fact that for bare PBH lensing, n PBH remains the same while the lensing cross-section is modified.While here we derive the lensing cross-section assuming the point mass lens with mass M eff PBH , the computation involving lensing cross section could be further refined by employing the point mass added to the model of dark halo mass distribution is possible, analysis of which we leave for future work [78].
For a given redshift-distribution function of N (z) FRB sources, the integrated optical depth is determined by For the optically thin regime τ ≪ 1, relevant here, the lensing probability is P l = 1 − e −τ ≃ τ .Thus, for N o observed FRB events, the number of lensed events follows N o τ .For N (z), we adopt the constant comovingdensity model with cutoff z cut = 0.5 as in Ref. [70] (see Appendix).
Given that τ ∝ f PBH , non-detection of lensed FRBs by dressed PBHs from N o observations translates into 2σ (95% confidence level) Poisson statistics of .
(10) The constraint on dressed PBHs of Eq. ( 10) translates into constraint on bare PBHs when M eff PBH = M PBH .In Fig. 2 we depict our computed projections for nondetection of bare and dressed PBHs for upcoming detection of 5×10 4 FRB events.Unlike the case of bare PBHs, we stress that sensitivity to dressed PBHs significantly increases with mass due to growth of dark halo dress around PBHs.We adopt the threshold of the flux ratio for the lensing pair search to 5, as in analysis of Ref. [70].We note that as the PBH mass increases, eventually the sensitivity becomes limited to due to maximum time delay in CHIME data (taken to be 1 minute) [73].At lower mass range, below ∼ 10M ⊙ , sensitivity decreases due to requirement that time delay is larger than critical value ∆ t = 1 ms.We have verified that assuming ∼ 550 FRBs our estimated limits are in general agreement with dedicated detailed analyses of bare PBHs using CHIME cat-alogue of Ref. [74,75].Since above f PBH ∼ O(10 −2 ) dressed PBHs might not fully develop a DM halo, we depict these limits with dashed line and leave a more detailed study of this complication for future work [78].As we demonstrate, with upcoming observations of ∼ 10 4 FRB events, CHIME or other FRB surveys will be able to probe dressed PBHs in the region relevant for observed GWs and have the capability to definitely distinguish with bare PBHs.Put another way, if the PBH scenario for the GW events is correct, we should be able to detect FRB lensing events from observations of ≳ 10 4 FRBs, indicating that we may confirm or exclude the PBH scenario of GW events by the near-future FRB observations.Dressed PBHs are especially relevant in this region of interest, while the region expanding to the right in PBH mass-range is already under significant pressure from multiple constraints (e.g.[39][40][41][42][43]79]).
Our findings establish novel opportunities for exploring directly models of dressed PBHs with dark halos through cosmological lensing, and can be applied to a variety of PBH lensing events at cosmological distances, including caustic crossings [80] and Type Ia supernovae [81].The analysis of lensing of these sources involves additional complexity and a detailed study is a topic of separate work [78].

V. CONCLUSIONS
Stellar-mass PBHs contributing to the DM abundance have been intimately linked with the detected GW events by LIGO.However, such PBHs can comprise only a subdominant fraction of the DM and are expected to be dressed in DM halos composed of a distinct primary DM component, such as QCD axion or WIMP-like particles.We advance cosmological lensing as a novel general method for directly identifying dressed PBHs and distinguishing them with bare PBHs, challenging for lensing of local sources.As we have demonstrated, strong lensing of observed FRBs from CHIME already begin testing dressed PBHs when stellar-mass PBHs constitute at the sub-percent level to the DM abundance.Upcoming observations by CHIME, or other FRB surveys, can explore the parameter space associated with detected GW events.Our findings open a new route for studying dressed PBHs with other cosmological lensing sources, such as caustic crossings and supernovae.To obtain a favorable configuration for lensing where dressed PBHs can be efficiently detected and distinguished from bare PBHs, we analyze key lensing quantities for local as well cosmological sources.For local sources, we assume D ol = 100 kpc and D os = 770 kpc for microlensing of Andromeda M31 galaxy, as employed for analysis of PBHs by Subaru HSC [68].For cosmological sources, we consider location at z S = 1 as characteristic of FRBs and PBHs located at z L = 0.5.
In Fig. 3, we display the average convergence κ(< 1) for bare PBHs as well as their surrounding dark halo computed following Eq.( 5) of the main text.With κ = 1 corresponding to the Einstein radius and delineating the boundary between weak and strong lensing, we find that for M PBH ≳ O(1)M ⊙ strong lensing by cosmological sources can efficiently distinguish bare PBHs with halodressed PBHs.On the other hand, the contribution of the dark halo to κ(< r) is subdominant for local sources even at around the Einstein radius.
We also display in Fig. 4 computed Einstein radius with and without the dark halo to explicitly verify the effect of the dark halo.We observe that, for cosmological lensing, dressed PBHs will have a significantly distinct Einstein radius compared to bare PBHs for M PBH ≳ 10 −2 M ⊙ .The difference of the Einstein radii are larger for higher PBH mass, indicating that PBH lensing with higher PBH masses can probe DM dress more efficiently.The figure also confirms that the halo size R h is typically much larger than the Einstein radius.
Using Eq. ( 7), in Fig. 5 we compute effective PBH mass M eff PBH that reproduces the Einstein radius of dressed PBHs with mass M PBH , for lensing by local (M31) and cosmological sources.Again, we observe that, for lensing of cosmological sources, dressed PBHs can be readily distinguished from bare PBHs with M eff PBH /M PBH ≫ 1 when M PBH ≳ 10 −2 M ⊙ .On other other hand, we confirm that probing DM dress is difficult for PBH lensing of local sources, as the difference between M eff PBH and M PBH is small even for PBHs with high M PBH .

Appendix B: Cosmological Lensing of FRBs by PBHs
Below we outline the details of cosmological lensing of FRBs by PBHs that we consider, following Ref.[70].
Strong lensing of an FRB by a PBH produces two images.For a PBH lens of mass M PBH located at redshift z L , the lensed images are located at distinct positions r ± = (l ± l 2 + 4r 2 E )/2, where l is the angular impact parameter and r E is the angular Einstein radius as be-fore.The images are separated in time with the delay ∆t = 4GM PBH (1+z L ) y 2 y 2 + 4+log y 2 + 4 + y y 2 + 4 − y , (B1) where y = l/r E is the normalized impact parameter.The corresponding lensing cross-section for a PBH point lens is given by where y min and y max set the range of the considered impact parameters.The flux ratio denoting absolute value of the ratio of the magnifications µ ± of the two lensed images can be defined as Requiring that both images are distinctly observed and none too dim by constraining R f to be smaller than a critical value Rf , the maximum impact parameter is constrained to be As in Ref. [70], we consider Rf = 5 throughout our analysis.
To distinguish lensed images as double peaks, separated in time, imposes that ∆t is larger than the typical millisecond temporal pulse width of observed FRB signal [73].Throughout the paper, we require that time delay is larger than critical value ∆ t = 1 ms.This restriction results in a value of y min , which we find by numerically solving Eq. (B1).
Provided lensing cross-section of Eq. (B2), for a given FRB at distance z S , the lensing optical depth for a (bare) PBH is where χ(z) is the comoving distance at redshift z, n PBH = f PBH Ω cdm ρ crit /M PBH is the comoving number density of PBHs, with f PBH being the fraction PBH DM abundance, Ω cdm = 0.24 is the Universe's cold DM density.
In order to calculate integrated lensing probability with optical depth, we account for FRB population distribution across redshifts.Following Ref. [70], we consider FRBs distributed with a constant comoving number density as where d L is luminosity distance, H(z) is the Hubble parameter at redshift z, N is normalization factor for in-tegration of N (z) to unity.We take the redshift model Gaussian cut-off z cut = 0.5, which represents potential instrumental signal-to-noise threshold.

FIG. 1 .
FIG. 1. Schematic depicting cosmological lensing of a source by a PBH dressed in a DM halo.

FIG. 2 .
FIG. 2. PBH DM parameter space as a function of bare PBH mass, depicting dressed and bare PBHs.Projections at 95% confidence level for dressed ("w/ halo") as well as bare ("w/o halo") PBHs considering 5 × 10 4 upcoming FRB observations are shown.Regions of "approx.CHIME data" denote estimated limits from 550 FRBs, an approximate amount already collected by CHIME.Dashed lines depict regions above fPBH = 10 −2 where large halo dress accumulation around PBHs could be affected.
Appendix A: Lensing by Dressed PBHs for Cosmological and Local Sources

FIG. 3 .FIG. 4 .
FIG. 3. Left: Comparison of the average convergence for a PBH and its surrounding halo for three different PBH masses, in case of lensing of local sources at M31 Andromeda galaxy.The average convergence profiles of halos are truncated at R h to indicate the halo size.The horizontal dotted line indicates κ(< r) = 1, corresponding to the Einstein radius.Right: Similar to the left panel, but for cosmological sources assuming zS = 1 and zL = 0.5.