Lensing constraints on ultradense dark matter halos

Cosmological observations precisely measure primordial variations in the density of the Universe at megaparsec and larger scales, but much smaller scales remain poorly constrained. However, sufficiently large initial perturbations at small scales can lead to an abundance of ultradense dark matter minihalos that form during the radiation epoch and survive into the late-time Universe. Because of their early formation, these objects can be compact enough to produce detectable microlensing signatures. We investigate whether the EROS, OGLE, and HSC surveys can probe these halos by fully accounting for finite source size and extended lens effects. We find that current data may already constrain the amplitudes of primordial curvature perturbations in a new region of parameter space, but this conclusion is strongly sensitive to yet undetermined details about the internal structures of these ultradense halos. Under optimistic assumptions, current and future HSC data would constrain a power spectrum that features an enhancement at scales $k \sim 10^7/{\rm Mpc}$, and an amplitude as low as $\mathcal{P}_\zeta\simeq 10^{-4}$ may be accessible. This is a particularly interesting regime because it connects to primordial black hole formation in a portion of the LIGO/Virgo/Kagra mass range and the production of scalar-induced gravitational waves in the nanohertz frequency range reachable by pulsar timing arrays. These prospects motivate further study of the ultradense halo formation scenario to clarify their internal structures.


I. INTRODUCTION
The spectrum of primordial curvature perturbations on large scales has been precisely constrained by a variety of observations, including the cosmic microwave background (CMB) [1], the Lyman-alpha forest [2], the UV galaxy luminosity function [3], and strong gravitational lensing [4].These measurements bring information about the energy content of our Universe and allow us to constrain models of inflation (see for example Ref. [5]).While current data are only able to constrain scales above roughly a megaparsec, an increasing number of new probes have been proposed to constrain primordial perturbations at smaller scales.Among these probes, the formation of primordial black holes (PBHs) [6][7][8] and nonlinearly induced stochastic gravitational waves (GWs) [9] are of particular recent interest.
However, a model that features a sufficiently enhanced power spectrum at small scales to give rise to significant GWs and PBHs would also lead to the formation of an abundant population of highly dense dark matter minihalos [53][54][55][56][57][58][59][60], as long the dark matter is collisionless and capable of clustering on the relevant scales (e.g.[61]).While PBHs arise from O(1) initial density perturbations, perturbations as small as O(10 −2 ) would still be sufficient to form halos already before matter-radiation equality [62][63][64][65][66], which occurred at redshift z 3400 [1].The smaller perturbations from which these halos arise would be far more common than the extreme perturbations necessary to produce PBHs, as illustrated in Fig. 1.Since these halos form long before the redshift 30 to 50 at which halos would begin to form in a standard cold dark matter scenario (e.g.[67]), they would be characterized by extraordinarily high internal density.These ultradense halos would be extended compact objects that can survive to the present day, giving rise to new observational signatures.
Lensing observations have already been used extensively to search for evidence of PBHs by directly looking for the signatures of those compact objects on the magnification of observed stars (e.g.[68][69][70][71][72][73][74][75][76]).In this work, we address a different question: can we constrain scenarios with boosted small-scale power (whether they produce PBHs or not) by searching for the lensing signatures of the ultradense compact halos formed by the enhanced perturbations? 1 Borrowing the analytic description of ultradense halo formation developed in Ref. [66], we will show that large primordial curvature perturbations at O(0.1) pc scales, which correspond to the formation of solar mass PBHs and nanohertz stochastic GW backgrounds, can lead to observable lensing signatures.

II. ULTRADENSE DARK MATTER HALOS FROM AN ENHANCED CURVATURE SPECTRUM
In this section, we describe the abundance and properties of the ultradense dark matter halos formed in scenarios with an enhanced power spectrum at small scales.For concreteness, we consider the scenario described by model A of Ref. [30], which produces PBHs around 10 M comprising roughly ≈ 0.05% of the dark matter and maximize the current upper bound set by LVK observations.Again, the ultradense halos are not connected to PBHs directly; they only emerge from the same cosmological scenario (see Fig. 1).Ultradense halos can arise with or without PBHs, and we will discuss implications for alternative (and agnostic) scenarios in Sec.IV.
Figure 2 shows the primordial curvature power spectrum P ζ (k) (dashed curve) in this model, which grows as P ζ ∝ k 4 at scales smaller than those constrained by CMB and large-scale structure data until it reaches a peak around P ζ ∼ 10 −2 near the pc −1 scale.The PBHs form around that scale.We also show the matter power spectrum P(k, a) (solid curve) at a = 10 −5 , approximated as with I 1 6.4 and I 2 0.47 [81].Here a eq 3 × 10 −4 and k eq 0.01 Mpc −1 are the scale factor and horizon wave number at matter-radiation equality, respectively.We assume for simplicity that dark matter is infinitely cold.In practice, for many dark matter models, the matter power should be truncated at some small scale due to kinetic coupling with the radiation and the resulting thermal free streaming (e.g.[82]).Our approach is thus limited to dark matter models that are capable of clustering at the relevant scales of about a comoving parsec.FIG.1: Illustration of the ultradense halo scenario.We show a cartoon picture of the primordial density field.A region of extreme excess density is necessary to exceed the collapse threshold and produce a PBH (e.g.[83]), but more modest density excesses are much more common and form ultradense halos.Note that ultradense halos are not significantly correlated with PBHs spatially; they simply arise from the same cosmological scenario.Also, scenarios with too little primordial power to produce a significant number of PBHs can still yield abundant ultradense halos, as we will see in Sec.IV.

A. Ultradense halo formation
As Fig. 2 shows, the linear matter power spectrum P(k, a) ∼ 10 2 at k 10 6 Mpc −1 by a = 10 −5 , implying that matter perturbations are already deeply nonlinear.During the radiation epoch, initially overdense regions exert gravitational attraction as they enter the horizon, which subsequently ceases as the radiation becomes homogeneous.However, dark matter particles set in motion by the initial pull continue drifting toward the initially overdense region, boosting its density further.In this scenario, Ref. [66] showed that the linear density contrast threshold for collapse is where σ is the rms density contrast, if we approximate that the ellipticity e of the initial tidal field within each region is equal to its most probable value, (e.g.[84]).As derived in Ref. [66], the collapse threshold in Eq. ( 2) leads to the excursion set mass function (e.g.[85]) describing the differential dark matter mass fraction in collapsed regions of mass M , where ν ≡ 3/σ M .Here σ M is the rms density contrast in spheres of mass M , i.e.
with W (x) ≡ 3(sin x−x cos x)/x 3 and M = (4π/3)ρ m,0 r 3 , where ρ m,0 33 M kpc −3 is the comoving dark matter FIG.2: Dimensionless power spectrum of primordial curvature perturbations (dashed curve) and matter perturbations (solid curve) in model A of Ref. [30].The matter power spectrum is evaluated at a = 10 −5 , and the dark matter is assumed to be infinitely cold.Note that matter perturbations are already deeply nonlinear by this time.
density. 2nsofar as the collapse of matter density perturbations results in halo formation, df /d log M describes the ultradense halo mass distribution.We evaluate df /d log M for the power spectrum in Fig. 2 and plot it in the upper panel of Fig. 3.This distribution integrates to 28 percent of the dark matter at a = 10 −5 (black curve), a much greater contribution than the sub-10 −4 fraction in PBHs.The peak of the distribution is around ∼ 10 −6 M , a mass scale that differs from that of the O(10) M PBHs that emerge from the same scenario.When dark matter halos and PBHs arise from perturbations of the same scale, the PBH masses M PBH are larger than the halo masses by a factor of about ∼ (M eq /M PBH ) 1/2 [66], where M eq 3×10 17 M is the horizon mass at matter-radiation equality.This scaling arises because while halos form out of matter, PBHs form primarily out of radiation, the density of which far exceeds that of matter during the radiation epoch.This relation suggests that dark matter halos accompanying O(10) M PBHs should have masses around ∼ 10 −7 M .In practice, halo masses somewhat exceed the above expectation because the halo formation threshold is much lower than that for PBHs, which allows halos to form from larger-scale initial perturbations.
However, the mass distribution df /d log M derived above really describes the distribution of collapsed regions, and it cannot be taken for granted that these regions form halos.During the radiation epoch, bound, virialized halos can only form within regions that are locally matter dominated [65].Reference [66] showed that in the continued absence of gravitational forces, an initially overdense region achieves an overdensity ρ/ρ m , with ρ m being the average matter density, as high as where e is the ellipticity of the initial tidal field and with ξ(r) being the correlation function.The functional dependencies in Eq. ( 6) arise because a collapsing mass shell will simply overshoot and expand back outward if it does not produce a high enough matter overdensity to halt the expansion.The following trends can be highlighted.
(1) The tidal field ellipticity e enters because, without gravity, particle drifts must be highly focused to produce regions of significant overdensity.
(2) The correlation function ξ(r) enters because it describes the average radial profile of the density contrast about an arbitrary point.Ξ(r) is then the average profile of the mean density enclosed within r.If Ξ drops steeply (high |d log Ξ/d log r|), then the collapse of successively larger mass shells is gradual, and so the density contributed by later shells is not able to efficiently build on top of the density contributed by earlier shells before the early shells disperse away (after overshooting the collapse).If instead Ξ drops shallowly (low |d log Ξ/d log r|), then the collapse of successively larger infalling mass shells occurs rapidly, and the density contributed by these shells builds up efficiently.
Due to Eq. ( 6), a collapsed region of mass M becomes locally matter dominated, resulting in a halo formation, at roughly the scale factor where M = (4π/3)ρ m,0 r 3 again and W is the derivative of W .The fraction in Eq. ( 8) is just d log Ξ/d log r.Meanwhile, Eqs. ( 2) and (3) imply that the ellipticity of the initial tidal field for a region of mass M , at its collapse time, is typically We show a f (M ), evaluated using Eqs.( 8) and (9), in the second panel of Fig. 3.Note that, as formulated, a f depends on the scale factor a at which we evaluate the matter power spectrum P(k).This dependency is not physically meaningful, and in principle, a f should instead depend on the full history of P(k).However, we will adopt the values of a f and df /d log M at a = 10 −5 (black curves in Fig. 3) for simplicity.Both a f and the halo mass function df /d log M only vary significantly with a for masses near 10 −6 M , and since a f ∼ 10 −5 there, the halo distributions evaluated at a = 10 −5 are expected to be approximately correct.
We also remark that the scaling of a f with halo mass M is easy to understand.In regimes where halos are rare, any halo must have formed from an extreme outlier peak in the initial density field.But outlier peaks are more spherical (e.g.[87]), which implies lower ellipticity e and hence earlier halo formation according to Eq. (8).

B. Structures of ultradense halos
We now discuss how we model the internal structures of the ultradense halos.Halo formation during the radiation epoch has not been simulated in detail, so there is considerable uncertainty in this treatment.Nevertheless, it is a general consequence of mass accretion in a cosmological context that the characteristic density ρ h of a collisionless dark matter halo is closely linked to the mean density of the universe at its formation time a f (e.g.[88][89][90]).For halos that form during the radiation epoch, this motivates where ρ r,0 0.012 M kpc −3 is the radiation density today and α is a proportionality factor.Simulations during the matter epoch suggest that the density of material within a halo is about 10 3 times the density of the universe at the time that the material became part of the halo [90,91].This consideration suggests α 10 3 , but halo formation dynamics may be significantly different during the radiation epoch.In the following, we will bracket the uncertainty in Eq. (10), as well as that in Eqs. ( 6) and (8), by allowing α to vary.Generally, one expects that at least the matter density does not drop during the formation process, which suggests α ∼ 1 as a lower limit.We will explore the consequences of assuming α ∈ [1 ÷ 10 3 ].The third panel of Fig. 3 shows ρ h as a function of halo mass M for α = 10 3 and α = 1.We assume ultradense halos form with density profiles similar to the Navarro-Frenk-White (NFW) form [92,93] with scale radius ρ h and scale density r h .Since Fig. 3 shows halo mass functions evaluated close to their formation times, we set a halo's outer virial radius at this time to be R vir = 2r h , roughly the smallest value found in simulations of the smallest halos during the matter epoch [94].Integrating the NFW profile out to radius 2r h leads to Given M and ρ h , this expression determines r h .These choices are only intended to give an approximate representation of the inner structure of an ultradense halo, and their impact is secondary to the uncertainty in α in The typical scale factor a f at which a collapsed region of mass M becomes matter dominated, resulting in halo formation.That a f depends on a is an artifact of our simplified computation, but we note that a f and df /d log M are mostly independent of a.This independence breaks down when M 10 −6 M , but a f ∼ 10 −5 in that regime, so for our analysis, we simply adopt the values of a f and df /d log M at a = 10 −5 (black curves).Lower middle panel: Halo's characteristic density ρ h , derived from a f as described in the text, as a function of mass M .We show both the conservative (dashed) and optimistic (solid) estimates; see Eq. (10).Bottom panel: Likewise, the conservative (dashed) and optimistic (solid) estimates of a halo's scale radius r h .
Eq. (10).For example, the precise choice of R vir /r h has only a minor impact since the numerical coefficient in Eq. ( 11) grows only logarithmically with the virial radius.The bottom panel of Fig. 3 shows the halo scale radii r h evaluated through this approach.We also comment that M here and in Fig. 3 thus represents the halo mass near the formation time and not the mass today.Put in another way, M is the mass of the densest central part of the halo, which is the part that contributes the most to microlensing and is the least susceptible to destruction during later evolution.While we assume that the NFW profile is approximately accurate near the halo's center, we cannot expect it to remain valid at distances much larger than r h .The density profile in that regime is set long after halo formation by the details of the halo's accretion history [88,89,91,95].It has been noted that the long-term accretion rates of galaxy cluster-scale halos [96] are slow enough that ρ ∝ r −4 is predicted after sufficiently long times [95], as opposed to the NFW profile's ρ ∝ r −3 .For ultradense halos in our scenario, the linear power spectrum has a similar P ∝ k 4 scaling to the spectrum of density variations relevant at cluster scales, and we also anticipate that radiation domination during the early evolution of ultradense halos will significantly suppress their growth.Due to these considerations, we approximate the late-time density profile of an ultradense halo with the Hernquist form [97], where the numerical factor is tuned so that this profile closely matches the NFW profile up to a few times r h .At large radii, the profile in Eq. ( 12) scales as ρ ∝ r −4 in accordance with the accretion rate considerations above.For a typical ultradense halo with mass M = 1.9 × 10 −6 M and characteristic radius r h = 1100α −1/3 R , the upper panel of Fig. 4 shows the density profile in Eq. ( 12) as well as the corresponding NFW profile.
When integrated out to infinity, the mass associated with the density profile in Eq. ( 12) converges to a value of about 3.5M , where M is the halo mass at formation given by Eq. (11).We noted in Section II A that the ultradense halos contain about 28 percent of the dark matter at early times a ∼ 10 −5 .Thus, adoption of this density profile implies that ultradense halos eventually come to contain essentially all of the dark matter.This outcome is consistent with reasonable expectations.For example, Ref. [99] found that in the standard picture of cosmological structure formation, 90 to 95 percent of the dark matter is expected to lie in halos today.A similar conclusion should apply to ultradense halos, which can accrete material up until around redshift 30, when CMB-level primordial perturbations (corresponding to k 10 4 Mpc −1 in Fig. 2) finally become nonlinear and much larger cosmic structures begin to dominate.One limitation to our analysis, however, is that we assume every halo grows by the same factor of 3.5.A more detailed treatment would account for the spread in ultradense halo accretion histories.
The central structures of collisionless dark matter halos remain mostly unaltered by later evolution (e.g.[90]).This is why the characteristics ρ h and r h that we fix at the ultradense halos' formation epoch are expected to remain largely accurate today.There are several ways in which this concordance can fail, however.First, ultradense halos can merge.While the merger rate is expected to be low for a power spectrum that grows as steeply as The density profile that we adopt [Hernquist form, Eq. ( 12)], in blue, for the optimistic and conservative assumptions of α = 10 3 (solid curves) and α = 1 (dashed curves), respectively.As Eq. ( 10) shows, α is the factor by which the halo's characteristic density exceeds that of the Universe at the halo's formation time.We also show in orange the corresponding NFW density profiles, for which the density is excessively high at large radii.Middle panel: v circ /r as a function of radius r for the same density profiles, where v circ is the circular orbit velocity.We show in green the median and central 68% band for tidal shocking from stellar encounters for halos around 8 kpc from the Galactic Center, derived in Ref. [98].The interpretation is that where v circ /r drops below the tidal shocking parameter, which is roughly the velocity kick ∆v/r imparted onto halo particles, disruption by stellar encounters is likely to become significant.We also show in red where the Galactic tidal forces become important.Lower panel: The deflection angle for light passing the halo at closest distance r.
that of Fig. 2, this effect would reduce the number of ultradense halos moderately.We note however that the fraction of dark matter in these halos is not altered, and simulations suggest that the characteristic density of a merger remnant is typically not lower than that of the progenitors [89,100].Thus, mergers are only expected to shift the ultradense halo distribution to slightly higher masses.We will neglect this effect here, leaving a more careful analysis for future work.Ultradense halos also accrete onto much larger halos at later times, such as the halos that surround galaxies.Indeed, they are expected to contain a fraction of the dark matter inside galactic halos that is comparable to the fraction of dark matter that resides in ultradense halos initially.The extremely high density of these objects makes them essentially unaffected by tidal forces inside a much larger host.For example, the middle panel of Fig. 4 shows v circ /r = F/r as a function of radius r for a typical ultradense halo, where v circ is the circular orbit velocity and F is the halo's central force.We compare this quantity to dF MW /dR (red line), where dF MW /dR 300 (km/s) 2 /kpc 2 is the radial component of the Milky Way's tidal tensor at the Sun's galactocentric radius of 8 kpc, which we evaluate using the mass model in Ref. [101].The interpretation is that the impact of Galactic tidal forces only becomes important when v circ /r approaches dF MW /dR, which occurs over 10 6 R from the halo center.
However, encounters with individual stars represent the more serious concern for ultradense halos inside the Galaxy.These become important when the velocity ∆v that they inject into halo particles approaches v circ .In the middle panel of Fig. 4, we also show the distribution of tidal shocking parameters B ∼ ∆v/r, as defined and evaluated by Ref. [98], for halos orbiting the Galaxy at about 8 kpc.Shocks by stellar encounters are expected to significantly alter the structures of ultradense halos at radii beyond roughly 104 to 10 5 R .However, it is unclear what impact they have on ρ ∝ r −4 density profiles, because ρ ∝ r −4 is already the limiting density profile arising from such tidal shocks [102,103]. 3We will neglect stellar encounters in this work, with the justification that they are not expected to alter ultradense halo density profiles until radii significantly beyond r h are reached. 4 Finally, the lower panel of Fig. 4 shows how our typical ultradense halo deflects passing light.We plot the deflection angle (4G/c 2 )M 2D (r)/r as a function of the distance r of closest approach, where is the mass within an infinite cylinder of radius r.

III. MICROLENSING CONSTRAINTS ON ULTRADENSE HALOS
In this section, we summarize the computation of the gravitational microlensing of light sources, applying the technique to constrain the ultradense minihalos described above.As dark matter halos are intrinsically extended objects, it is important to account for their size when deriving their potential lensing signatures.We follow the works of Refs.[109,110] (see also [111][112][113][114][115]) to derive the rate of lensed events at the EROS-2 [69], OGLE-IV [72] and Subaru-HSC [70] surveys while accounting for finite source sizes and extended lenses.

A. Detectability of a microlensing event
The light coming from a source is deflected by the gravitational field of an object (lens).For low-mass lenses, the deflection cannot be resolved, but only a modification of the flux F, defined as may be detected, where F 0 is the flux in the absence of lensing.It is convenient to define the observer-lens, lens-source, and observer-source distances as D L , D S , and D LS ≡ D S − D L , respectively.With respect to the axis passing through the lens center and the source, one can also define the angle β as the true source position angle and θ as the angle of the observed lensed image of the source.We depict this geometrical setup in Fig. 5.As we noted above, the lensing halos are assumed to have Hernquist density profiles defined in Eq. ( 12).These profiles have total mass M 0 = 3.5M , as we noted above, where M is the formation mass discussed in Sec.II.Analogously, we define the late-time mass fraction, after halos have gained mass, as f 0 = 3.5f .Also, we define R 90 = 32r h as the radius at which 90% of the total mass is contained.These definitions mirror the choices used by Refs.[109,110].
The lensing equation determines the path of light rays after a deflection and can be written as where M 0 (θ) = M 2D (D L θ) is the lens mass projected onto the lens plane, as defined in Eq. (13).It is convenient to introduce the Einstein angle [116] defined as the solution of the lensing equation when β(θ E ) = 0 and obtained as the value of θ for a pointlike lens M 0 (θ) → M 0 .We also introduced the adimensional ratio x ≡ D L /D S .Correspondingly, we denote the Einstein radius as the distance R E ≡ D L θ E on the lens plane.
< l a t e x i t s h a 1 _ b a s e 6 4 = " < l a t e x i t s h a 1 _ b a s e 6 4 = " x X G h e 7 3 j R K t s 5 3 g 8 X q 9 9 L R v 7 < l a t e x i t s h a 1 _ b a s e 6 4 = " G 8 m w W q B p T + P p b U l r 1 T q 5 U h N N 7 r Q y X t E 5 q 7 n n t 7 P 6 0 W r 8 q 4 i i j A 3 S I j p G L L l A d 3 a I G a i K K R u g J v a B X K 7 e e r T f r f d 5 a s o q Z f f Q L 1 s c 3 i q S T n Q = = < / l a t e x i t > D L < l a t e x i t s h a 1 _ b a s e 6 4 = " R M x W H F O K s c 3 R + X q + T S O P N p F e + g A O e g U V d E V q q E 6 o u g B P a N X 9 G Y 9 W S / W u / U x a c 1 Z 0 5 k d 9 A f W 5 w 8 z v p P 6 < / l a t e x i t >

D LS
< l a t e x i t s h a 1 _ b a s e 6 4 = " Q g

It takes values
In units of R E , the source radius in the lens plane is r S ≡ xR /R E .In units of θ E , the angular distance from the lens center to the source center is u = β/θ E and to a point on the edge of the source is ū(ϕ) = u 2 + r 2 S + 2ur S cos ϕ (18) (see Fig. 5).One can then rewrite the lensing Eq. ( 15) for each infinitesimal point on the edge of the source as to find the positions of images at t i (ū(ϕ)) ≡ θ i /θ E with i labeling the, in general, multiple solutions.For a spherically symmetric density profile ρ(r) we assume throughout this work, one can write We neglect limb darkening and model the source star as having a uniform intensity in the lens plane.It follows that the magnification produced by an image i is given by the ratio of the image area to the source area [73,114] where η = sgn(dt 2 i /dū 2 | ϕ=π ) while the angular measure is defined from the angle ψ as Finally, we can compute the overall magnification µ tot as the sum of the individual contributions In this treatment, following Refs.[109,117], we ignore the wave optics effects that are relevant when computing the magnification from lenses whose size is smaller than the wavelength of the detected light.For the masses considered in this work, the finite source size effect dominates the suppression of lensing signatures below M ≈ 10 −11 M [118].Therefore, wave effects can be neglected.
If one takes the limit of negligible source size (i.e.r S u) and pointlike lens (i.e.R 90 R E ), one can derive analytical solutions to the lens equation, and find In the opposite limit of a very large source r s u, one finds that the lensing solutions give a large suppression of µ.This is because the lens only affects a negligible fraction of light rays coming from the source.
Lensing surveys (such as EROS, OGLE, and HSC that we will consider later on) define as detectable a microlensing event whose temporary magnification of the source star exceeds the threshold value µ th = 1.34.Following this criterion, we will require µ tot > µ th .It is therefore convenient to generalize this criterion and define the threshold impact parameter u 1.34 as such that the magnification is above 1.34 for all smaller impact parameters.In the limit of a pointlike lens and negligible source size, one can directly derive from Eq. ( 24) the maximum impact parameter that satisfies this condition, which is u = 1.We show in Fig. 6 the threshold impact parameter u 1.34 as a function of both the source size r S projected on the lens plane and the lens size r 90 .

B. Number of detectable microlensing events
The number of detectable lensing events can be computed by integrating the rate of overthreshold signals.For a single source star and unit exposure time, the differential event rate with respect to the halo mass distribution, FIG.6: Threshold impact parameter for detection as a function of both the source size r S projected on the lens plane and the lens size r 90 , both normalized with respect to the Einstein radius.In the limit of negligible source and lens sizes, the threshold impact parameter converges toward the pointlike limit where u 1.34 → 1.
x = D L /D S , and event timescale t E (i.e. the time the magnification remains larger than the threshold), can be written as ) where v 0 is the circular velocity in the galaxy.The differential density distribution of lenses ρ lenses (x) can be derived by multiplying the galactic overdensity by the ultradense halo mass fraction, which means We also introduced Q(x) ≡ 4 (u 1.34 (x)R E (x)/t E ) 2 , and ε(t E ) is the efficiency of telescopic detection.The total number of detectable events N events is where N is the number of observed source stars in the survey, T obs is the total observation time, and dn/dR is the distribution of source star radii.As we will see, the finite source size is only relevant for the HSC survey of M31.When considering the other surveys, we will simply marginalize over the stellar radius distribution, as it does not affect the lensing signatures.In Appendix A, we summarise the setup of the three surveys we consider in this study, i.e.EROS-2 [69], OGLE-IV [72] and Subaru-HSC [70].
To gain intuition on what is the reach of current constraints, in Fig. 7 we show the upper bound on the fraction of dark matter in the form of ultradense halos by taking a simplified monochromatic halo mass distribution.We show the constraint by assuming different values of the average density ρ ≡ 3M 0 /(4πR 3 90 ) = ρ h /7300 of the halos.At the lowest masses, the dominant constraint comes from HSC data, peaking around M 0 10 −9 M .OGLE dominates the bounds at intermediate masses around M 0 5 × 10 −5 M , while the EROS survey constrains the heavier portion of the plot.As one can see, for dense enough halos, the constraint converges to the one for pointlike lenses shown in Ref. [109].On the other hand, assuming less dense halos and correspondingly larger lens sizes relaxes the constraint, and the M 0 < 10 −2 M regime becomes entirely unconstrained for ρ 10 −15 M /R 3 .This shows how crucial the halo density (or, equivalently, its physical extension) is for setting constraints via microlensing.
In Fig. 7 we superimpose the final halo mass distribution df 0 /d log M 0 obtained in Sec.II considering model A from Ref. [30].We see that this distribution crosses the constraint coming from HSC and OGLE lensing surveys, but only if one assumes the average density to be at least as high as ρ = 10 −12 M /R 3 .However, translating the formation properties of the halos from Fig. 3 into the late-time Universe mass M 0 and size R 90 , one discovers that halos in this scenario are too large to be constrained by the current experiment, having an average density of ρ ≈ 3 × 10 −17 M /R 3 at M 0 = 10 −5 M and ρ ≈ 7 × 10 −15 M /R 3 at M 0 = 10 −8 M , assuming α = 10 3 .Even smaller average densities, and correspondingly larger lens sizes, are attained by assuming smaller values of α.We conclude, therefore, that current lensing surveys are not able to constrain enhanced power spectra peaked around k ≈ 10 6 /Mpc.In particular, we verified by considering the full halo distributions from Fig. 3 (as opposed to the monochromatic cases studied in Fig. 7) that the specific PBH formation scenario in the LVK detection window from Ref. [30] (model A) is not currently constrained by lensing surveys.Also, due to the strong relationship between a halo's density and its mass in Fig. 3, the strongest prospects for the detection of ultradense halos in such a PBH scenario involve using HSC to constrain the low-mass tail of the halo distribution.Unfortunately, since this tail is a highly model-dependent feature (related to how shallowly the primordial power spectrum in Fig. 2 decays at large k), this approach is unlikely to yield generally applicable constraints on the PBH scenario.In the next section, we will explore the constraints that can be set by current and future HSC observations on a narrow enhancement to the power spectrum, for which this low-mass halo tail does not contribute.FIG.7: Upper bound on the fraction of dark matter in ultradense halos [with the density profile of Eq. ( 12)] for the simplified cases where all halos have the same mass and characteristic density.The colored curves show the maximum mass fraction today, f 0 = 3.5f , assuming a monochromatic mass distribution at mass M 0 = 3.5M and different values of the average halo density ρ = ρ h /7300 (f , M , and ρ h are the formation-time halo parameters considered in Sec.II).For each mass and density, the corresponding radius R 90 can be computed as R 90 = (3M 0 /4π ρ) 1/3 .The upper side of the frame indicates the R 90 assuming ρ = 10 −6 M /R 3 .In this case, the halo sizes are always much smaller than the Einstein radius in Eq. ( 17), and the constraints coincide with those derived in the point-mass limit (see Ref. [109]).We also show, for visual comparison, the halo mass distribution df 0 /d log M 0 obtained in Sec.II considering model A from Ref. [30] as well as a few realizations of the narrowly peaked power spectrum in Eq. ( 29).Note that when deriving constraints on these models, we will account for the full halo distributions arising therefrom rather than the monochromatic distributions probed here.

IV. CONSTRAINTS ON THE POWER SPECTRUM AT SMALL SCALES
We now test the sensitivity of microlensing to the ultradense halos arising from a broader family of primordial curvature power spectra.We want to consider realistic narrow spectra as a benchmark.Therefore, we consider which is parametrized by the peak amplitude A 0 and wave number k 0 such that the maximum is achieved at This spectrum grows as P ζ (k) ∝ k 4 for k < k 0 , the characteristic growing slope that can arise in simple ultraslow-roll inflation models (see e.g.[119][120][121]), while it is Gaussian suppressed for k > k 0 .The precise form of the small-scale (large k) suppression turns out to be unimportant for our constraint.We explicitly check this by also considering the functional form This spectrum similarly peaks at P PL+PL ζ (k 0 ) = A 0 and grows as k 4 for k < k 0 , but at larger k it decays as k −4 .
We repeat the procedure in Sec.II to generate the ultradense halo distribution for these scenarios.We make one change, however.Press-Schechter mass functions like Eq. ( 4), when evaluated using the real-space tophat window function, are not well behaved when the power spectrum decays rapidly at small scales.They predict a halo count that diverges at small mass scales, even when there is no power on such scales [86].This difficulty arises from the assumption of uncorrelated steps in the excursion set formulation of Press-Schechter theory [85], which corresponds to the use of a sharp k-space window function, W (x) = θ H (c−x), instead of the top-hat window.Here c is a constant, which fixes the connection between the wavenumber and the mass scale, and θ H is the Heaviside step function.Therefore, we adopt the sharp k-space window when evaluating σ M in Eq. ( 5).Following Refs.[86,122], we set c = 2.5.
Figure 8 shows the resulting halo distributions.One noteworthy feature is that, even though the overall mass fraction f decreases with smaller A 0 , the corresponding halo density ρ h increases.This is a consequence of the important role of ellipticity in the collapse; see Sec.II for more details.As the amplitudes of primordial perturbations decrease, the overdensities that can collapse to form ultradense halos are rarer and hence increasingly spherical (e.g.[87]).This allows the halos to form at earlier epochs and therefore possess larger internal density.
We derive current constraints and forecast what is accessible with future HSC observations (see for example forecasts in Ref. [123]) by following the same steps as in the previous section.In particular, we assume T obs = 7 h to derive the constraint using available HSC observations [70] and T obs = 70 h to forecast future reach (extrapolating the same detection efficiency and assuming the same number of stars in the survey).
We first test the impact of different parametrizations of the spectral shape, Eqs. ( 29) and (30).Figure 9 shows the current HSC constraint for both cases under the optimistic assumption that α = 10 3 [see Eq. (10)].The close match between the two outcomes confirms that the particular form of the low-mass tail of the power spectrum is not important.
Figure 10 shows the HSC constraints for the P PL+Exp ζ (k) power spectrum parametrization [Eq.(29)].Under the optimistic assumption that α = 10 3 , current measurements force the spectral amplitude to be A 0 2 × 10 −3 around k 0 = 10 7 (solid blue curve).Increasing the observation time T obs by a factor of 10 would give rise to slightly more stringent constraints (solid red curve) in a wider range of k 0 .We also note that for larger spectral amplitudes A 0 , the constraint degrades.This is a consequence of the more diffuse halos that arise if the amplitudes of initial density perturbations are too large, as discussed above.
As we discussed in Sec.II, α parametrizes theoretical uncertainty about the internal structures of ultradense halos.If we adopt a more moderate assumption, α = 30, then the current constraint disappears and only future observations can constrain a smaller portion of parameter space (red dashed curve).With the most conservative assumption of α = 1 instead, both current and future constraints disappear, as the lenses are then too diffuse to generate observable signatures within the HSC survey.This outcome motivates further study of the ultradense halo formation scenario, likely with numerical simulations, to understand their internal structures and hence determine whether they can produce solid constraints through microlensing.(31) between the cosmological horizon mass M H , which relates to the PBH mass m PBH by an order unity factor dictated by the critical collapse parameters [124], and the comoving wave number k.Here, g * is the effective number of degrees of freedom of relativistic particles and κ ≡ kr m relates k to the characteristic perturbation size at horizon crossing r m .For example, if we consider a spectrum centered around k 0 = 6 × 10 6 Mpc −1 one finds M H ≈ 2.5 M , corresponding to the formation of PBHs of around a solar mass.
We compute the PBH abundance f PBH as [6] where M H is the horizon mass at the time of horizon reentry, M eq 3 × 10 17 M the horizon mass at matterradiation equality, and Ω CDM is the dark matter density  29) and ( 30) that peak at amplitude A 0 at the scale wave number k 0 .We shade the region excluded by current HSC observations with T obs = 7 h under the optimistic assumption α = 10 3 about ultradense halos' internal structures.Evidently, the two parametrizations yield comparable results.today (in units of the critical density).Adopting threshold statistics, we compute the mass fraction assuming Gaussian primordial curvature perturbations5 and accounting for the nonlinear relationship between curvature and density perturbations [127][128][129].One obtains where δ l is the linear (i.e.Gaussian) component of the density contrast and the integration boundaries are dictated by having overthreshold perturbations and type-I PBH collapse (see e.g.[130]).We indicate with σ(r m ) the variance of the linear density field computed at horizon crossing time and smoothed on a scale r m (see e.g.Ref. [30] for more details), while δ c is the threshold for collapse.We also introduced the parameters K and γ to include the effect of critical collapse, while Φ controls the relationship between the density contrast and the curvature perturbations.We adopt the technique of Ref. [83] to compute the threshold δ c for PBH formation.The PL+Exp spectrum [Eq.(29)] gives rise to collapsing peaks for which the characteristic comoving size is κ ≡ r m k = 2.51, the shape parameter is α c = 4.14 (see Ref. [130] for more details) and the threshold for collapse is δ c = 0.572 in the limit of perfect radiation domination.When considering the PL+PL spectrum [Eq.( 30)], we find instead that κ = 2.36, α c = 3.25, and δ c = 0.558.
We include the effect of the softening of the equation of state of the Standard Model plasma due to the QCD transition as done in Ref. [30] and based on the numerical simulations of Ref. [131] (see also [31,132,133]).This generates the slight dip in the black lines of Fig. 10 around M H ≈ M .
The gray shaded region at the top of Fig. 10 is immediately ruled out because PBHs would be produced with an abundance larger than all of the dark matter in our Universe (f PBH ≥ 1).The two lines below indicate PBH mass fractions f PBH = 10 −3 and f PBH = 10 −5 respectively, following the logarithmic scaling with A 0 .We conclude from the plot that, provided that dark matter can cluster on the relevant scales and that halos are dense enough with α ≈ 10 3 , current HSC data exclude the possibility that a narrow population of PBHs with stellar mass comprise a non-negligible fraction of the dark matter.

Induced stochastic GW background
We also compute the GW signal sourced by scalar perturbations at second order to show how constraints on ultradense dark matter halos may have interesting consequences for induced GWs within the nanohertz frequency range [134][135][136][137][138]. The frequency f GW of the stochastic GW background (SGWB) is related to the comoving wave number k = 2πf GW by the relation k 6.47 × 10 14 f GW Hz Mpc −1 .( For example, a spectrum centered around k 0 = 6 × 10 6 Mpc −1 corresponds to f GW ≈ 9 × 10 −9 Hz.This falls within the range of frequencies corresponding to the putative signal recently reported by the NANOGrav Collaboration [44] (and also independently supported by other pulsar timing array data [45][46][47]).The current energy density of GWs is given by [38][39][40][41][42][43] Ω GW,0 = 0.39 Ω r,0 g * (T H ) 106.75 g * ,s (T H ) 106.75 as function of their frequency f GW , with (see e.g.Ref. [9] for a recent review).Here, b ≡ (1 − 3w)/(1 + 3w) with w being the Universe's equation of state at the emission time, Ω r,0 is the density fraction of radiation, g * (T ) and g * ,s (T ) are the temperature-dependent effective number of degrees of freedom for energy density and entropy density, respectively, and T (u, v) is the transfer function [139,140].We denote with the subscript "H"  29) that peaks at amplitude A 0 at the scale wave number k 0 .The optimistic assumption α = 10 3 about the internal density of ultradense halos results in the blue region being ruled out currently and the larger red region in the forecast.For the more moderate assumption α = 30, current data do not set constraints, but the 70-h observations are forecasted to rule out the region enclosed by the dashed line.If we assume α = 1, then neither current nor the forecasted observations can set any constraint.At the top, we indicate the horizon mass M H corresponding to k 0 [assuming g * = 25 and κ = 2.5; see Eq. ( 31)], which is close to the mass scale of the PBHs that would result from primordial power at that scale.We also indicate the SGWB peak frequency f GW corresponding to power at that scale.We shade in gray the region already ruled out by requiring no overproduction of PBHs, while the solid black horizontal lines correspond to the PBH mass fractions f PBH = [1, 10 −3 , 10 the time when induced GWs of the given wave number k fall sufficiently within the Hubble horizon to behave as a radiation fluid in an expanding universe.
In Fig. 10, we show the GW abundance produced at second order by the curvature power spectrum in Eq. ( 29).Current and future constraints would be able to rule out the scalar-induced interpretation of the SGWB potentially hinted by the PTA data, again provided that the internal density of ultradense halos is sufficiently high (α ≈ 10 3 ) and that dark matter can cluster on the relevant ∼ 10 7 Mpc −1 scales.

V. CONCLUSIONS AND OUTLOOK
Constraining the primordial Universe is one of the fundamental endeavors of modern cosmology.While CMB and large-scale structure observations allow us to measure the amplitude of perturbations at megaparsec scales and higher, primordial fluctuations on smaller scales evade our observational capabilities.Developments in the theory of PBH formation and GW emission allow us to set conservative upper limits on the amplitudes of initial perturbations, while current and future GW data may allow for stronger constraints or, potentially, discoveries.
PBHs and the stochastic GW background probe scenarios where the amplitudes of perturbations are enhanced at small scales.In this paper we have discussed a different probe of these scenarios: the potential lensing signatures left by the formation of ultradense dark matter halos in the early Universe, as modeled in Ref. [90].Using data from the HSC, OGLE, and EROS surveys, we showed that it is possible to constrain the amplitude of perturbations at small scales.In particular, the Subaru-HSC observations may allow us to constrain primordial perturbations in a narrow range of scales around k ∼ 10 7 Mpc −1 .Such scales correspond to the formation of stellar mass PBHs, which are of interest for LIGO/Virgo/Kagra and next-generation gravitational-wave detectors.Perturbations at these scales could also source a stochastic GW background in the nanohertz range, which is of potential interest to PTA experiments.Microlensing of ultradense halos could probe primordial spectral amplitudes as low as P ζ 10 −4 .
The principal limitation to this approach is that it requires that the dark matter be capable of clustering, and remaining clustered, at the relevant k ∼ 10 7 Mpc −1 scales.If the dark matter is too warm [141,142] or too light [143,144], then its thermal motion or quantum pressure, respectively, could preclude the formation of the sub-Earth-mass halos that arise from perturbations on such scales.However, particle dark matter heavier than about 10 −7 eV [144] (including the QCD axion [145]) can carry density perturbations at the relevant O(0.1) pc scales as long as it was never in kinetic equilibrium with the Standard Model plasma.So can particles that were in thermal contact with the plasma as long as they are heavier than O(100) GeV (with the precise threshold being sensitive to when kinetic decoupling occurred [146]).Meanwhile, if the dark matter has significant nongravitational self-interactions, then collisional relaxation effects would likely suppress the density inside ultradense halos to a significant extent over cosmic time (e.g.[147]).A similar effect would arise from gravitational collisions if the dark matter is predominantly PBHs at mass scales not sufficiently smaller than the ultradense halo mass scale (e.g.[148]).However, collisionless particle dark matter would preserve the internal density of ultradense halos.PBH dark matter toward the lower end of the asteroidmass window (e.g.[18]) is likely also light enough to maintain the density within the relevant 10 −8 to 10 −7 M halos.
Finally, we emphasize that our current and prospective constraints are subject to significant theoretical uncertainty regarding the properties of halos that form during the radiation epoch, particularly their internal density values.While optimistic assumptions produce very interesting constraints on primordial power that are relevant to the interpretations of ongoing GW and PTA experiments, conservative assumptions yield no constraint from microlensing.This outcome motivates more detailed explorations, likely with numerical simulations of cosmological volumes, of halo formation during the radiation epoch.
The resulting constraint for a monochromatic halo mass distribution is shown in Fig. 7.The OGLE survey is dominant in the range of masses M ∈ [10 −6 ÷ 10 −3 ]M .Due to the lighter lenses considered here, and the consequent smaller Einstein radius, the constraint begins to degrade when ρ 10 −9 M R −3 .

HSC
The Subaru-HSC survey [70] observed stars of the M31 galaxy, whose distances from us are approximately D S 770 kpc.The lensing signatures may arise from the presence of compact structures in both the MW and M31.The circular speeds are taken to be approximately v 0 = 220 km/s for the MW [150] and v 0 = 250 km/s for M31 [154].The differential event rate is therefore the sum of two pieces dΓ = dΓ MW + dΓ M31 .We find that the contribution from lenses within the MW dominates the number of events.Following Ref [70], the spatial dark matter distribution of the MW is assumed to be given by an NFW profile with scale density 0.184 GeV/cm 3 , scale radius 21.5 kpc, and r determined from Earth's location as in Eq. (A2) with ( , b) = (121.2• , −21.6 • ) [155].The num-ber of stars in the Subaru-HSC survey is N = 8.7 × 10 7 , while the observation time is T obs = 7 h.Finally, the detection efficiency is given by Fig. 19 in Ref. [70].Following Ref. [109], we approximate the detection efficiency as = 0.5 in the regime with 2 min ≤ t E ≤ 7 h.
The constrained lens masses in the Subaru-HSC survey are much lighter than in the case of EROS and OGLE surveys.Therefore, they correspond to much smaller Einstein radii for which both extended lens size and the finite source size corrections are important.In particular, the latter effect was neglected in the first version of the analysis of Ref. [70] that lead to an overestimation of the number of detectable events below around 10 −11 M , which is instead drastically suppressed.Following Ref. [117], when integrating over the source star radii of M31 R in Eq. ( 28), we adopt the distribution derived using the Panchromatic Hubble Andromeda Treasury star catalog [156,157] and the MESA Isochrones and Stellar Tracks stellar evolution package [158,159] (see Fig. 4 of Ref. [117] and related discussion for more details).Also in this case, we consider the HSC single microlensing event candidate as foreground and require N exp 3.9, which corresponds to the 90% confidence level assuming Poisson statistics.

3 FIG. 3 :
FIG.3: Ultradense halos arising in a scenario characterized by the primordial power spectrum P ζ in Fig.2, at three different times during radiation domination.Top panel: The differential dark matter mass fraction in collapsed regions of mass M .These regions become halos once they are locally matter dominated.Upper middle panel: The typical scale factor a f at which a collapsed region of mass M becomes matter dominated, resulting in halo formation.That a f depends on a is an artifact of our simplified computation, but we note that a f and df /d log M are mostly independent of a.This independence breaks down when M 10 −6 M , but a f ∼ 10 −5 in that regime, so for our analysis, we simply adopt the values of a f and df /d log M at a = 10 −5 (black curves).Lower middle panel: Halo's characteristic density ρ h , derived from a f as described in the text, as a function of mass M .We show both the conservative (dashed) and optimistic (solid) estimates; see Eq.(10).Bottom panel: Likewise, the conservative (dashed) and optimistic (solid) estimates of a halo's scale radius r h .

1 FIG. 4 :
FIG.4: Properties of an ultradense halo with formation mass M = 1.9 × 10 −6 M and characteristic radius r h = 1100α −1/3 R .Upper panel: The density profile that we adopt [Hernquist form, Eq. (12)], in blue, for the optimistic and conservative assumptions of α = 10 3 (solid curves) and α = 1 (dashed curves), respectively.As Eq. (10) shows, α is the factor by which the halo's characteristic density exceeds that of the Universe at the halo's formation time.We also show in orange the corresponding NFW density profiles, for which the density is excessively high at large radii.Middle panel: v circ /r as a function of radius r for the same density profiles, where v circ is the circular orbit velocity.We show in green the median and central 68% band for tidal shocking from stellar encounters for halos around 8 kpc from the Galactic Center, derived in Ref.[98].The interpretation is that where v circ /r drops below the tidal shocking parameter, which is roughly the velocity kick ∆v/r imparted onto halo particles, disruption by stellar encounters is likely to become significant.We also show in red where the Galactic tidal forces become important.Lower panel: The deflection angle for light passing the halo at closest distance r.
X e P O W 9 e O / e x 7 y 1 4 O U z h / A H 3 u c P M 3 C O 5 g = = < / l a t e x i t > ū < l a t e x i t s h a 1 _ b a s e 6 4 = " d S 9 x 8 q s s j j w c w T G c g g d X U I E 7 q E I d G C A 8 w y u 8 O Y / O i / P u f M x b c 0 4 2 c w h / 4 H z + A L F l j O E = < / l a t e x i t > S < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 u L A 9 0 o D 5 X 7 l z 1 t N d K b y t 3 2 W 1 w w = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y K r 2 P Q i z c T M A 9 I l j A 7 6 S R j Z m e X m V k h L P k C L x 4 U 8 e o n e f N v n C R 7 0 M S C h q K q m + 6 u I B Z c G 9 f 9 d n I r q 2 v r G / n N w t b 2 z u 5 e c f + g o a N E M a y z S E S q F V C N g k u s G 2 4 E t m K F N A w E N o P R 7 d R v P q H S P J I P Z h y j H 9 K B 5 H 3 O q L F S 7 b 5 b L L l l d w a y T L y

4 c 6 b w 4 7 8 7
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FIG. 5 :
FIG.5: Upper panel: Geometrical setup under consideration.The light source (S), the lens (L), and the observer (O) are separated by their respective distances D i .Lower panel: The lensing plane reporting the source and lens finite sizes, rescaled compared to the Einstein radius, as well as the integration angles.

FIG. 8 : 75 −
FIG.8: Ultradense halos arising from the PL+Exp power spectrum [Eq.(29), solid curves] and the PL+PL power spectrum [Eq.(30), dashed curves].We fix k 0 = 6 × 10 6 Mpc −1 but vary the amplitude A 0 from 10 −4 to 10 −1 (different colors).Upper panel: The differential dark matter mass fraction in halos as a function of formation mass M .Middle panel: Halo's characteristic density ρ h , as a function of formation mass M , for α ranging from 1 to 10 3 [shaded bands; see Eq. (10)].The lines correspond to α = 30; for the PL+PL power spectrum (dashed) we only show this line.The right-hand axis also denotes the halo formation time (the bands are not relevant here).Lower panel: Likewise, a halo's scale radius r h as a function of formation mass M .

1 FIG. 9 :
FIG.9: Comparison between HSC constraints on the PL+Exp or PL+PL power spectra in Eqs.(29) and (30) that peak at amplitude A 0 at the scale wave number k 0 .We shade the region excluded by current HSC observations with T obs = 7 h under the optimistic assumption α = 10 3 about ultradense halos' internal structures.Evidently, the two parametrizations yield comparable results.

2 FIG. 10 :
FIG.10: Parameter space excluded by current (7 h) and future (70 h) HSC observations, assuming a PL+Exp power spectrum of the form in Eq. (29) that peaks at amplitude A 0 at the scale wave number k 0 .The optimistic assumption α = 10 3 about the internal density of ultradense halos results in the blue region being ruled out currently and the larger red region in the forecast.For the more moderate assumption α = 30, current data do not set constraints, but the 70-h observations are forecasted to rule out the region enclosed by the dashed line.If we assume α = 1, then neither current nor the forecasted observations can set any constraint.At the top, we indicate the horizon mass M H corresponding to k 0 [assuming g * = 25 and κ = 2.5; see Eq. (31)], which is close to the mass scale of the PBHs that would result from primordial power at that scale.We also indicate the SGWB peak frequency f GW corresponding to power at that scale.We shade in gray the region already ruled out by requiring no overproduction of PBHs, while the solid black horizontal lines correspond to the PBH mass fractions f PBH = [1, 10 −3 , 10 −5 ] from top to bottom.The horizontal green lines indicate the energy density of the SGWB, Ω SGWB = [10 −10 , 10 −11 , 10 −12 , 10 −13 ].
FIG.10: Parameter space excluded by current (7 h) and future (70 h) HSC observations, assuming a PL+Exp power spectrum of the form in Eq. (29) that peaks at amplitude A 0 at the scale wave number k 0 .The optimistic assumption α = 10 3 about the internal density of ultradense halos results in the blue region being ruled out currently and the larger red region in the forecast.For the more moderate assumption α = 30, current data do not set constraints, but the 70-h observations are forecasted to rule out the region enclosed by the dashed line.If we assume α = 1, then neither current nor the forecasted observations can set any constraint.At the top, we indicate the horizon mass M H corresponding to k 0 [assuming g * = 25 and κ = 2.5; see Eq. (31)], which is close to the mass scale of the PBHs that would result from primordial power at that scale.We also indicate the SGWB peak frequency f GW corresponding to power at that scale.We shade in gray the region already ruled out by requiring no overproduction of PBHs, while the solid black horizontal lines correspond to the PBH mass fractions f PBH = [1, 10 −3 , 10 −5 ] from top to bottom.The horizontal green lines indicate the energy density of the SGWB, Ω SGWB = [10 −10 , 10 −11 , 10 −12 , 10 −13 ].