Dark Matter Microhalos in the Solar Neighborhood:Pulsar Timing Signatures of Early Matter Domination

M. Sten Delos1, ∗ and Tim Linden2, †

1Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany2Stockholm University and The Oskar Klein Centre for Cosmoparticle Physics, Alba Nova, 10691 Stockholm, Sweden

Pulsar timing provides a sensitive probe of small-scale structure. Gravitational perturbations arising from aninhomogeneous environment could manifest as detectable perturbations in the pulsation phase. Consequently,pulsar timing arrays have been proposed as a probe of dark matter substructure on mass scales as small as10−11 M�. Since the small-scale mass distribution is connected to early-Universe physics, pulsar timing cantherefore constrain the thermal history prior to Big Bang Nucleosynthesis (BBN), a period that remains largelyunprobed. We explore here the prospects for pulsar timing arrays to detect the dark substructure imprintedby a period of early matter domination (EMD) prior to BBN. EMD amplifies density variations, leading to apopulation of highly dense sub-Earth-mass dark matter microhalos. We use recently developed semianalyticmodels to characterize the distribution of EMD-induced microhalos, and we evaluate the extent to which thepulsar timing distortions caused by these microhalos can be detected. Broadly, we find that sub-0.1-µs timingnoise residuals are necessary to probe EMD. However, with 10-ns residuals, a pulsar timing array with just 70pulsars could detect the evidence of an EMD epoch with 20 years of observation time if the reheat temperature isof order 10 MeV. With 40 years of observation time, pulsar timing arrays could probe EMD reheat temperaturesas high as 150 MeV.

I. INTRODUCTION

Pulsar timing presents an exquisitely sensitive probe ofgravitational perturbations. Due to the highly regular period-icity of many pulsars’ emissions, tiny disturbances can man-ifest as detectable shifts in their pulsation phases. For thisreason, pulsar timing is indispensable in the search for low-frequency gravitational waves, for which pulsar timing arrays(PTAs)—which search for correlated phase shifts—representthe most sensitive detection methodology [1]. However, PTAscan also probe perturbations arising from an inhomogeneousmatter environment. In particular, pulsar timing holds the po-tential to detect primordial black holes [2–4] or dark mattersubhalos [5–12] as small as 10−11 M�, which lie far beyondthe reach of other astrophysical probes of dark matter.

The small-scale distribution of cold dark matter is closelylinked to the Universe’s early history. The cosmic microwavebackground (CMB) and light element abundances establishthat the Universe was dominated by the Standard Model ra-diation bath by the time it cooled to a temperature of about5 MeV [13]. However, there is little reason to assume that ra-diation domination extends to higher temperatures (i.e., earliertimes); see Ref. [14] for a review. In particular, a well moti-vated possibility is that the Universe was dominated at earliertimes by an unstable heavy field. Examples of species thatcould drive this period of early matter domination (EMD) in-clude hidden-sector particles [15–27], moduli fields in stringtheory [28–35], inflationary spectator fields [36–39], and theinflaton itself [40–49]. EMD leaves a marked imprint on thesmall-scale mass distribution because the gravitational clus-tering of the early matter particle boosts density variations onscales that were subhorizon (i.e., causal) during this time [50].

∗ sten@mpa-garching.mpg.de, ORCID: orcid.org/0000-0003-3808-5321† linden@fysik.su.se, ORCID: orcid.org/0000-0001-9888-0971

In this article, we explore the potential for pulsar timingto probe an early matter dominated epoch. EMD’s boost todensity variations causes a large fraction of the dark matterto become bound into highly dense sub-earth-mass microha-los [50]. If any of these microhalos pass near a pulsar (orthe Earth) they perturb the body’s motion and potentially leadto a detectable pulsar timing distortion. Sufficiently compactmicrohalos that cross the line of sight to a pulsar can also per-turb the incoming light directly. Recently, Ref. [11] includedan exploration of detection prospects for EMD-induced mi-crohalos; that study concluded that an EMD could be probedif it ends at a reheat temperature TRH below about 1 GeV. Ourresearch builds on Ref. [11]’s analysis. In particular, we em-ploy recently developed semianalytic descriptions of micro-halo formation and evolution to more precisely characterizethe microhalo distribution arising from an early matter dom-inated epoch. We also explore the parameter space for earlymatter dominated epochs more exhaustively.

To describe microhalo formation, we employ the approachdeveloped in Ref. [51]. This prescription maps peaks in theinitial density field onto collapsed halos at later times; theproperties of the peak predict the internal structure of the re-sulting halo. Since microhalos detectable by pulsar timing arenear the Galactic disk, it is also essential to accurately modeltheir response to tidal forces and high-speed encounters. Forthis purpose we employ the model presented in Ref. [52] todescribe subhalo tidal evolution and that presented in Ref. [53]to describe the impact of encounters. We previously employedthese models in Ref. [54] to characterize another potential sig-nature of EMD—boosted dark matter annihilation—and wealso employ here the model refinements developed for thatwork. PYTHON codes that implement the microhalo modelsemployed in this work are publicly available.1

1 https://github.com/delos/microhalo-models

arX

iv:2

109.

0324

0v1

[as

tro-

ph.C

O]

7 S

ep 2

021

2

In this paper, we parametrize the impact of EMD in termsof its reheat temperature TRH, which sets the scales at whichdensity variations are boosted, and the cutoff ratio kcut/kRH,which sets the amplitude of the boost. On the observationalside we consider the impact of the pulsar count NP and ob-servation duration tobs as well as the rms timing noise residualtres. We find that, roughly, timing residuals tres . 0.1 µs areneeded to detect EMD epochs. If tres = 10 ns, EMD-inducedmicrohalos could be detected with about NP = 70 pulsarsand tobs ' 20 years if TRH . 20 MeV and kcut/kRH & 30.Higher TRH and lower kcut/kRH can be reached with largertobs and NP , although microhalo detection prospects are rel-atively insensitive to NP . With hundreds to thousands ofpulsars, reheat temperatures as high as TRH ' 150 MeV orcutoff ratios as low as kcut/kRH ' 8 could be probed intobs = 40 years.

This article is organized as follows. Section II describes thematter power spectrum that arises from an EMD scenario. InSec. III, we characterize the microhalo distribution resultingfrom such a power spectrum. In Sec. IV, we treat the evolutionof microhalos due to tidal forces and encounters within theGalactic potential. Section V outlines our method for assess-ing the detection prospects of a given microhalo populationusing pulsar timing, which is largely the same as the methodpresented in Ref. [11]. These detection prospects, which con-stitute the main results of this article, are presented in Sec. VI.Section VII presents our conclusions. Finally, Appendix Afurther details Sec. IV’s computation of the microhalo-stellarencounter distribution, while Appendix B refines the model inRef. [53] to better describe a halo’s long-term response to ahigh-speed encounter.

II. DENSITY VARIATIONS ARISING FROM EMD

We first review the spectrum of density variations that re-sults from an early matter dominated phase. During EMD,the dominant species, which we call φ, gravitationally clustersand creates gravitational potential wells. If the dark matter isboth nonrelativistic and kinetically decoupled from any rela-tivistic species (such as the Standard Model) by a temperatureof about 2TRH [54], then it falls into these potential wells andinherits the density variations in φ. After reheating, these den-sity variations persist in the dark matter.

Reference [50] developed a prescription to quantify thepower spectrum P(k) of density fluctuations imprinted ontothe dark matter by EMD. The form of P(k) is influencedby two main parameters: the reheat temperature TRH associ-ated with the end of EMD; and the dark matter free-streamingscale, which sets a cutoff wavenumber kcut. Figure 1 showsthe dark matter power spectrum for several different EMDscenarios.2 These power spectra are computed in linear the-ory, i.e., assuming P(k)� 1. A key observation is that if the

2 We neglect the possibility of a radiation-dominated (or other) epoch priorto EMD; in particular we assume that kcut is smaller than the wavenumberthat enters the horizon at the beginning of EMD.

10−2 100 102 104 106 108 1010

k (Mpc−1)

10−6

10−4

10−2

100

102

104

P(k)

10M

eV100M

eV

1GeVTRH =

kcut/kRH = 10kcut/kRH = 20kcut/kRH = 40

10−1310−710−110510111017M (M⊙)

FIG. 1. EMD-induced (dimensionless) dark matter power spectra atz = 300, computed using linear theory. We show different values ofthe reheat temperature TRH and the dark matter free-streaming cut-off scale kcut, expressed as the ratio kcut/kRH. TRH sets the scalesat which EMD boosts density variations, while kcut/kRH sets themaximum amplitude of the boost. For comparison, the solid blackcurve indicates the cold dark matter power spectrum without earlymatter domination. We also show the mass scale M associated witheach wavenumber, which roughly corresponds to the masses of halosforming from density variations of that scale. For kcut/kRH & 20,density variations are already nonlinear (P(k) & 1), implying mi-crohalos are already forming by z = 300.

ratio kcut/kRH is larger than about 20 between kcut and thewavenumber, kRH, that enters the horizon at reheating, thendensity variations are already becoming nonlinear (P & 1) bythe redshift z = 300 at which the power spectra are plotted.

We focus in this study on the reheat-temperature range3 MeV < TRH < 180 MeV. According to Ref. [13], the ef-fective number of neutrino species reflected in the CMB con-strains TRH > 4.7 MeV, which motivates our lower bound.Meanwhile, high reheat temperatures TRH are associated withsmaller mass scales (see Fig. 1), which are more difficult todetect through pulsar timing, and we will see later that re-heat temperatures TRH & 180 MeV are broadly inaccessi-ble. With respect to the cutoff parameter kcut/kRH,3 we limitour consideration to 5 < kcut/kRH < 40. We will see thatmicrohalos arising in scenarios with kcut/kRH < 5 are notdense enough to be detected. On the other hand, larger values

3 Note that the two scales kRH and kcut are naturally connected despitebeing associated with entirely different physics (kRH with the φ decayrate and kcut with dark matter’s interaction properties). To achieve theobserved dark matter abundance, a thermal relic must thermally decouplefrom the radiation bath not too long before reheating; see Ref. [54]. Thefree-streaming scale kcut is set by the the time of kinetic decoupling, whichis in turn related to the time of thermal decoupling through the particulardark matter microphysics. Further details of the kcut–kRH connection arelaid out in Refs. [55, 56], the latter of which suggests that kcut/kRH ashigh as 200 can be plausible for supersymmetric dark matter. However,the presence of a radiation-dominated epoch prior to EMD can break thekcut–kRH connection and allow kcut � kRH (see, e.g., Ref. [57]).

3

kcut/kRH > 40 lead to the collapse of overdense regions (andlikely halo formation [57]) deep in the radiation-dominatedepoch, a scenario for which our modeling of halo populationsis not calibrated. Nevertheless, as we discuss in Section VI,our conclusions are likely applicable to many scenarios withkcut/kRH > 40.

III. THE INITIAL MICROHALO DISTRIBUTION

To understand the present-day imprint of EMD, the nextstep is to advance from the linear-theory power spectrum tothe population of (nonlinear) collapsed dark matter halos. Tomodel halo populations in this way, it is common to use Press-Schechter theory [58, 59]. In this model, a halo of mass M isassociated with a region of Lagrangian space (the space ofinitial comoving particle positions) of mass M inside whichthe average linear-theory density contrast δ(x) ≡ δρ/ρ ex-ceeds some critical value of order 1. Mathematically, thelinear-theory density field δ(x) is convolved with a filter thatsmooths it over the mass scale M .

However, the smallest dark matter microhalos form directlyfrom the collapse of peaks in the linear-theory density fieldδ(x). To study them, it is more natural to consider the un-filtered density field and simply map each peak therein to acollapsed halo. This procedure carries the added benefit thateach halo’s density profile can be predicted from the proper-ties of the associated peak [51], so we do not need to rely onempirical concentration-mass relations (e.g., Ref. [60]) thatcan be difficult to generalize to arbitrary power spectra.

To characterize the microhalo population, we exploit thestatistics of Gaussian fields to generate a random sample ofpeaks along with their density profiles δ(q), where q is the co-moving radius. This procedure is described in Ref. [51] andbased on the theory of Ref. [61]. With these peak density pro-files, we estimate the density profiles of the resulting collapsedhalos using a secondary infall model [62, 63]. Specifically,we use the simulation-tuned “turnaround” model of Ref. [51](refined as described in Appendix A of Ref. [54]) to predictthe radius rmax at which the circular velocity is maximizedand the corresponding enclosed mass Mmax. Some peaks donot have a predicted collapse time due to their high elliptic-ity, and we neglect these peaks.4 Additionally, some peaksproduce halos whose predicted rmax values are so large as tobe populated by material initially lying well beyond the scalesboosted by early matter domination. These predictions repre-sent conventional CDM halos unboosted by EMD, so they areirrelevant to pulsar timing observations (e.g., Ref. [11]), andwe neglect them for computational convenience.5

4 Collapse time is predicted using the approximation in Ref. [64], whichis not guaranteed to have a solution. Whether these peaks actually failto collapse is likely immaterial, as ellipticity is anticorrelated with peakheight, so the alternative is that these peaks collapse late and produce halosof low density.

5 The rmax and Mmax predictions involve the covariance matrix betweenthe field values at all different radii about the peak, so it is computationallybeneficial to limit the radial extent explored.

While microhalos form with density profiles that asymptoteto ρ ∝ r−3/2 at small radii [51, 65–73], successive mergersdrive their inner density cusps toward a shallower ρ ∝ r−1

scaling [51, 72–75]. Therefore, we assume microhalos de-velop the Navarro-Frenk-White (NFW) profile [76, 77],

ρ(r) =ρs

(r/rs)(1 + r/rs)2, (1)

where the scale parameters rs and ρs are set to reproduce thepredicted rmax and Mmax. No precise description of the im-pact of microhalo mergers has yet been presented; the aboveprocedure yields a conservative estimate, as mergers betweenmicrohalos are understood to raise rmax and Mmax [51].

The above prescription sets the distribution of microhalodensity profiles. We set their total number density in thefollowing way. The cosmological microhalo number den-sity n is simply the number density of initial density peaks(which can be computed from P(k) using, e.g., the methodsof Ref. [61]) scaled to account for the aforementioned removalof some peaks from consideration. The number density of mi-crohalos in the Sun’s vicinity is then n = (ρlocal/ρ)n, where(ρlocal/ρ) ' 3 × 105 is the factor by which the local darkmatter density exceeds the cosmological mean.

These procedures do not fully account for the impact ofmicrohalo-microhalo mergers, a topic that we leave for fu-ture study. These mergers tend to raise microhalo masses, asnoted above, while also reducing their number count. Whilethese two effects have opposite implications on observationalprospects, we argue that neglecting both of them represents aconservative choice. Mergers have two relevant effects:

(1) Mergers boost individual microhalo masses while pre-serving their overall mass content (i.e., the fraction oftotal dark matter mass residing in microhalos).

(2) Mergers tend to approximately preserve halos’ internaldensity values (i.e., concentrations) [51, 78].

Reference [11], exploring constraints from pulsar timing onmonochromatic halo mass functions, found that when condi-tion (2) holds—concentrations are held fixed—the parameterchange (1)—increasing the halo mass while preserving thefraction of total dark matter mass residing in microhalos—can only strengthen pulsar timing constraints. Consequently,mergers between microhalos can only boost their detectabilityin pulsar timing. We also note that the preservation of micro-halo internal densities implies that their susceptibility to dis-ruptive effects within the Galaxy, which we treat in the nextsection, is unaltered by mergers.

Figures 2 and 3 show (as dashed lines) the initial microhalodistributions, computed as described above, for two differentEMD scenarios. We show the distributions in NFW scale den-sity ρNFW

s and halo massM , where we takeM to be the virialmass at z = 2, roughly the time at which microhalos might beexpected to accrete onto the Galaxy.6 The bulk of the micro-

6 For the initial halo population, we set M = M200,m, the mass of theregion that has 200 times the background matter density. We compute Msolely for illustrative purposes; it has no impact on our later results.

4

0 100 200

dn

d log ρNFWs

(1012 kpc−3)

1010

1011

1012

1013

ρNFW

s(M

⊙/k

pc3)

initialonly tidal evolutiontidal evolution andstellar encounters

d2n

d log ρNFWs d logM

/(1012 kpc−3

)=

TRH = 32 MeV, kcut/kRH = 15

1296532168.142

10−11 10−10 10−9 10−8 10−7

M (M⊙)

0

50

100

150

200

250

300

dn

dlogM

(101

2kp

c−3)

initialonly tidal evolutiontidal evolution andstellar encounters

FIG. 2. The distribution of microhalos near the Sun resulting from the EMD scenario with TRH = 32 MeV and kcut/kRH = 15. The mainplot shows contours of the differential halo number density distributed in massM and NFW-equivalent scale radius ρNFW

s (see the text), whilethe left and bottom plots show the projected distributions in ρNFW

s and M separately. In all plots, the dashed curves show the initial microhalopopulation before disruptive effects are accounted for (equivalently, before accretion onto a larger halo). The bulk of these halos are tightlyclustered in ρNFW

s -M space because they form from the smallest-scale density variations above the free-streaming cutoff, but a tail of halosforming from larger-scale fluctuations is visible. Tidal forces from a host halo strip off the microhalos’ weakly bound outskirts, resulting in ahalo populations with similar ρNFW

s but significantly reducedM (dotted lines). Also including the impact of stellar encounters greatly spreadsout the distribution (solid curves), producing a long tail of “unlucky” low-mass halos. Note that stellar encounters raise ρNFW

s only in the sensethat stripping off a halo’s low-density outskirts raises its average density. Stellar encounters still reduce a halo’s density at any given radius.

halos form from density variations that are just above the free-streaming scale, and these halos comprise the main cluster inρ −M space. Beyond this, there is a tail of larger and less-dense halos that are associated with increasingly large-scaledensity variations.

IV. DISRUPTION OF MICROHALOS

In isolation, microhalos arising from EMD maintain staticinternal density profiles, only growing outward as they accretematerial. In that case, the above predictions would remain

accurate today. However, microhalos relevant to pulsar tim-ing are in the vicinity of the Galactic disk at the present time.These microhalos accreted onto the Galactic halo and weresubjected to evolution due to tidal forces and encounters withother objects.

A. Tidal evolution

A common treatment for tidal evolution (e.g., Ref. [11]) isto simply truncate a subhalo’s density profile at its “tidal ra-dius”, the radius beyond which tidal forces from the host are

5

0 500 1000 1500 2000

dn

d log ρNFWs

(1012 kpc−3)

1012

1013

1014

1015

ρNFW

s(M

⊙/k

pc3)

initialonly tidal evolutiontidal evolution andstellar encounters

d2n

d log ρNFWs d logM

/(1012 kpc−3

)=

TRH = 32 MeV, kcut/kRH = 30

1197598299150753719

10−12 10−11 10−10 10−9 10−8

M (M⊙)

0

500

1000

1500

2000

2500

3000

dn

dlogM

(1012

kpc−

3)

initialonly tidal evolutiontidal evolution andstellar encounters

FIG. 3. Same as Fig. 2 but assuming a smaller free-streaming scale. Here, kcut/kRH = 30. The initial halo population (dashed lines) exhibitscoherently lower halo masses and higher internal densities but is otherwise the same as Fig. 2. However, the higher internal halo densitiesmake this population less susceptible to disruptive tidal effects and stellar encounters (dotted and solid lines).

stronger than the subhalo’s self-gravity. While this proceduresupplies a simple approximation, in reality tidal stripping is acontinuous process as the subhalo revirializes in response tomass loss (e.g., Ref. [79]). Instead, we use the model fromAppendix E of Ref. [52], which predicts the time evolutionof rmax and Mmax for an NFW subhalo orbiting an NFWhost and is tuned to match idealizedN -body simulations. TheGalactic potential is not NFW, owing to the contribution ofbaryons, so as an approximation, we pick the NFW potentialthat best reproduces the observationally modeled Galactic ro-tation curves in Ref. [80] for Galactocentric radii smaller than25 kpc. This NFW potential has scale radius rs = 3.4 kpc andρs = 4.0× 108 M�/kpc3.7

7 We emphasize that we employ the NFW profile solely to describe theGalactic potential and not its stellar distribution. We treat the Galaxy’sstellar distribution more accurately later.

This choice of host potential represents an approxima-tion, and we note the following limitations. Galactic rota-tion curves are valid within the disk plane, but the dark mat-ter halo—and hence microhalo orbits—typically extend farabove and below this plane, where the potential is weaker.In this respect our choice is likely to overestimate the impactof tidal stripping. Meanwhile, abrupt features within the hostpotential heat subhalos, increasing the rate at which they losemass. While we account for encounters with individual diskstars below, the encounter with the Galactic disk’s bulk po-tential can itself be a significant factor in microhalo disrup-tion. This concern suggests we may risk underestimating theimpact of tidal stripping. However, Ref. [81] estimates thatthe impact of individual stellar encounters dominates over thatof coherent disk shocking for halos smaller than earth mass.As Fig. 1 indicates, EMD-induced microhalos are sub-Earth-mass if TRH & 10 MeV. In any event, since our tidal evolutionmodel is only calibrated for NFW host potentials, we leave

6

to future work more precise modeling of microhalo survivalwithin the Galactic potential.

We now sample the orbital parameters of microhalos at thesolar radius r0 = 8 kpc under the approximation that the mi-crohalo velocity distribution is isotropic. Specifically, we userejection methods to sample the orbital energy E from the un-normalized distribution

√E − Φ(r0)f(E), where f(E) is the

phase-space distribution function and Φ is the Galactic poten-tial. A fitting form for f(E) is drawn from Ref. [82]. Themicrohalo velocity’s magnitude |V | immediately follows. Todescribe the direction of the microhalo’s velocity V , we uni-formly sample µ ≡ cos θ, where θ is the angle between V andthe Galactic radius vector, and the angle φ between the micro-halo’s tangential velocity projection and the velocity VLSR ofthe local standard of rest. Since faster microhalos are encoun-tered more frequently, we additionally use rejection methodsto weight our orbit sample by Vrel ≡ |V − VLSR|.8

In this way, we associate a randomly sampled orbit witheach initial microhalo from Sec. III. We now apply the tidalevolution model in Ref. [52] assuming a duration of 10 Gyr,which corresponds to accretion onto the Galaxy at a redshiftof about z = 2. This approximation does not have a ma-jor impact; large changes in the microhalo accretion redshiftcorrespond to only modest changes in the tidal evolution du-ration.

B. Stellar encounters

The next step is to consider encounters with other objects,such as stars or other microhalos. Reference [54] found thatthe impact of encounters with other microhalos was subdomi-nant to tidal evolution and stellar encounters even for the darkmatter-dominated Draco dwarf spheroidal galaxy, so we canreasonably assume it is even more subdominant for microha-los in the solar neighborhood. However, stellar encounters arelikely to have a significant, and perhaps even dominant, im-pact on microhalo evolution.

To handle stellar encounters, a common approximation(e.g., Ref. [84]) is to compare the energy injected into a mi-crohalo by these encounters to the halo’s total binding energy.However, as noted by Ref. [79], this computation is not clearlyconnected to the question of halo survival because these en-ergy injections are inefficient: the most weakly bound parti-cles receive the most energy in an encounter. Instead, we usethe model in Ref. [53] to treat the impact of stellar encoun-ters; this model precisely predicts the evolution of a halo’sdensity profile due to impulsive point-particle encounters. To

8 In principle, we should weight the encounter frequency by the relative ve-locity with respect to each pulsar under consideration. However, the mi-crohalo velocity dispersion is much larger than the dispersion in pulsarvelocities, so pulsar motions are neglected in the analysis of timing dis-tortions in Sec. V and Ref. [11]. In particular, the rms microhalo velocitywith respect to the local standard of rest turns out to be about 330 km/s inour calculation, while the mean two-dimensional velocity of millisecondpulsars is measured to be about 87 km/s [83].

apply this model it is necessary that for each microhalo wesample a series of stellar encounters, and we do so as follows.Let n∗(R) be the number density of stars at the galactocentricposition R, and suppose a microhalo’s orbital position andvelocity are described by R(t) and V (t), respectively. Thedifferential number of stellar encounters, per impact parame-ter b and time t, is

d2N∗πbdbdt

= n∗[R(t)] Vrel[V (t),R(t)], (2)

where

Vrel(V ,R) ≡∫

d3V∗ |V∗ − V | f(R;V∗) (3)

and f(R;V∗) is the distribution of stellar velocities V∗ at po-sition R. Appendix A shows how Eq. (3) can be rapidly eval-uated.

We use the Galactic model in Ref. [85] to describe the dis-tributions, n∗(R) and f(R;V∗), of stellar positions and ve-locities, but for simplicity we cylindrically symmetrize theGalactic bar and assume that the stellar velocity dispersionis isotropic.9 With these distributions established, we use in-verse transforms to sample stellar encounter times t and im-pact parameters b from Eq. (2). We then sample the relativevelocities |V∗ −V | of these encounters using the microhalo’sorbital information and the stellar velocity dispersions. Fi-nally, we sample the masses of the stars that our microhalosencounter from a Kroupa initial mass function with minimummass 0.01 M� (we include brown dwarfs) and a high-massindex of 2.7 [86].

The above considerations determine the stellar encounterdistribution for each microhalo, and the next step is to applythe stellar encounter model of Ref. [53]. We additionally fol-low the suggestions of Ref. [54], which carried out combinedtidal evolution and stellar encounter simulations. To wit, wedo not model any relaxation of microhalos between encoun-ters, instead simply summing the (relative) energy injectionsfrom all encounters, and we apply the full sequence of stellarencounters after the tidal evolution. The tidal evolution modelpredicts for each halo its maximum circular velocity vmax andassociated radius rmax, and we relate these quantities back toa density profile by assuming that the profile after tidal evo-lution resembles the profile after stellar encounters, which, asdiscussed in Ref. [53], is (almost) universal. We then apply thestellar encounter model with this density profile as a startingpoint. The post-encounter density profile given in Ref. [53] isρ = ρs(rs/r) exp [(−1/α) (r/rs)

α] with α = 0.78, but as we

discuss in Appendix B, this density profile becomes unphysi-cally steep at large radii. We instead propose the form

ρ = ρsrsr

exp

[− 1

α

(r

rs

)α(1+q)1− 1

β2F1

(1, 1; 1+

1

β;−q)]

(4)

9 In principle, we should account for the time evolution of the Galaxy’s stel-lar distribution. For instance, the bar should rotate and the stellar massfunction should evolve. But for simplicity we approximate the Galaxy asstatic.

7

with q ≡ [(1/3)(r/rs)α]β , α = 0.78, and β = 5. Here,

2F1 is the hypergeometric function. This profile transitions toρ ∝ r−4 at large radii instead of becoming arbitrarily steep,and it agrees with the simulation we carry out in Appendix B.

Figures 2 and 3 show the impact of tidal effects (dot-ted lines) and stellar encounters (solid lines) on the micro-halo population for TRH = 32 MeV and two different free-streaming scales. The post-encounter density profile, Eq. 4,has finite total mass, so we do not assume any radial truncationin determining each halo’s mass. Additionally, to fairly com-pare characteristic density values between the disrupted andinitial halo populations, we convert the post-encounter densityprofile’s scale density ρs into the equivalent NFW scale den-sity ρNFW

s = ρs/1.17 (see Ref. [53]). Microhalos evidentlysuffer a large drop in mass, which can be attributed to theloss of the halo’s weakly bound outskirts. Much of this massloss actually reflects the shift in density profile from NFW toEq. (4), but highly dense microhalos might not fully transitiontheir density profiles, so we likely overestimate microhalo dis-ruption when kcut/kRH is large.10 Meanwhile, the character-istic density ρNFW

s of microhalos increases because when thelow-density outskirts are stripped, the average density of thehalo rises.11 This change is relatively modest and arises pre-dominantly due to stellar encounters and not tidal stripping.The stochastic nature of stellar encounters also spreads outthe microhalo distribution, producing a long tail of low-masshalos that suffered more disruption by chance.

V. PULSAR TIMING SIGNALS

With microhalo populations established, the remaining stepis to determine the extent to which they can be detected byPTAs. In particular, we consider here the timing distortionsthat arise from the Doppler effect when microhalos perturbpulsar motions. While there is also a Doppler effect associatedwith perturbations to the Earth’s motion, Ref. [11] found thatthis effect generally produces weaker constraints (although ithas the potential to probe smaller masses). Timing distor-tions could also arise from the Shapiro effect when microhaloscross the line of sight to a pulsar, but this effect is most sensi-tive to mass scales M & 105 M� [10], which are larger thanthe masses of EMD-induced microhalos (see Fig. 1).

10 In particular, in the kcut/kRH = 30 case the tidal-stripping model predictsessentially no evolution in the scale parameters of the microhalo densityprofiles, so almost all of the change between the dashed and dotted linesin Fig. 3 arises from the change in the density profile’s form. Thus, it islikely that for kcut/kRH & 30 the assumption that density profiles fullytransition into the new form represents an overestimate. However, it shouldbe noted that the total microhalo mass M does not enter into the pulsar-timing analysis in Sec. V except for halos that remain very distant fromthe pulsar. The pulsar-timing signal is dominated by the closest encounters[11], for which only the halo mass interior to the pulsar’s separation fromthe halo is relevant.

11 As can be seen in Ref. [53], the density at any given radius within a haloalways drops due to stellar encounters even as the halo’s characteristic den-sity rises.

We follow the procedure of Ref. [11] and use a modifiedversion of the associated Monte Carlo simulation code.12 Thiscode prepares a random array of pulsar positions, and for eachpulsar, it randomly samples encounters with microhalos. Pul-sar velocities (relative to the local standard of rest) are ne-glected because they are much smaller than microhalo veloc-ities (see footnote 8). We modified the code to sample mi-crohalos from the distribution computed in Secs. III and IV.The remainder of this section constitutes a review of the com-putational procedure described in Ref. [11] through which weevaluate the the pulsar timing signal that arises from such amicrohalo distribution.

A pulsar’s Doppler shift due to microhalo encounters man-ifests into an accumulated shift δφ in the pulse phase. For apulsar with frequency ν and earth-pulsar direction vector d,the phase shift due to an encounter with a point object (like aprimordial black hole) with mass M , velocity v, and impactparameter b, is

δφpt =GMν

v2cd ·(√

1 + x2b− sinh−1(x)v)

(5)

(where c is the speed of light). Here, x ≡ (b/v)(t− t), wheret is the time of closest approach when the vector from thepulsar to the object is b. The phase shift due to an extendedobject must in general be computed numerically, but to avoidthe computational expense, we follow Ref. [11] and conser-vatively approximate the phase shift due to a microhalo en-counter as

δφ ' δφptF(rmin). (6)

Here, rmin ≥ b is the microhalo’s closest approach to the pul-sar during the observing time, and the “form factor”

F(r) ≡M(r)/M ≤ 1, (7)

where M(r) is the enclosed mass profile, is the factor bywhich the microhalo’s finite extent scales its gravitational fieldat the distance r, relative to a point object.

We sum the microhalo-induced phase shift over all of therandomly sampled microhalos in the pulsar’s neighborhood.Next, we subtract a quadratic fit, φfit(t) = φ0 + φ1t + φ2t

2,reflecting that these low-order terms are degenerate with thepulsar’s intrinsic behavior.13 We plot in Fig. 4 an examplephase-shift signal from a single pulsar due to a microhalo dis-tribution arising from the EMD scenario with TRH = 32 MeVand kcut/kRH = 30. A range of observation durations tobs

are shown (for the same microhalo encounters); the signal is

12 The original code resides at the URL https://github.com/szehiml/dm-pta-mc, and our modified code is located at https://github.com/delos/dm-pta-mc.

13 In particular, we follow Refs. [9–11] in assuming that ν and higher deriva-tives of the pulsar frequency arise solely due to perturbations by substruc-ture. In practice contributions to ν can arise from the pulsar’s intrinsic spin-down, its unperturbed line-of-sight velocity, or a combination of the two;these contributions can be estimated and are of order ν/ν ∼ 10−31 s−2

[87].

8

0 10 20 30 40

time (years)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5(δϕ−

ϕfit)/ν

(ns)

tobs/yr =

71320273340

FIG. 4. An example pulsar phase-shift signal δφ − δφfit, as a func-tion of time, due to microhalos arising from an EMD scenario withTRH = 32 MeV and kcut/kRH = 30. A quadratic fit has beensubtracted as described in the text. Different curves show the sig-nals arising from the same set of microhalo encounters for a rangeof observation durations tobs. These curves do not overlap only be-cause the subtracted quadratic fit is different in each case. For the40-year curve (black) we also show, as a thin dotted line, the phaseshift computed exactly instead of using the approximation in Eq. (6).This curve is barely visible due to how little it differs from the ap-proximate result (solid line), which suggests that the approximationdoes not have a major impact.

different in each case due to the subtraction of the quadraticfit. While these phase shifts δφ are evaluated using the ap-proximate expression in Eq. (6), we also compare the ex-act phase shift computed using numerical integration (dottedblack curve, nearly overlapping the solid black curve). Thedifference is marginal, which suggests that the approximationdoes not have a large impact.

We repeat the above calculation for all NP pulsars underobservation. Additionally, to account for stochasticity, werepeat this process for 1000 “universes” each with indepen-dently sampled microhalo distributions. From each pulsar’sphase-shift signal, we compute the signal-to-noise ratio (SNR)as described in Ref. [11] assuming optimal filtering, and theSNR associated with each “universe” is taken to be the largestSNR associated with any pulsar therein. The overall SNR isthen the 10th percentile of the SNRs of the 1000 “universes”.Finally, we evaluate the significance level σ associated withthat SNR. That is, σ is the significance level with which wecan exclude the possibility that the given SNR arose fromnoise alone. These choices are identical to those made inRef. [11].

VI. RESULTS AND DISCUSSION

We initially fix the observational cadence to 1 week and therms timing residual to 10 ns, and we assume that the timingnoise is white. These are the choices made in Ref. [11], butwe discuss their impact and validity later. However, we varythe observing time tobs and the number of pulsars NP underobservation, and we also vary the cosmology parameters TRH

and kcut/kRH. We thus have a four-dimensional parameterspace, and each point in this space is associated with a partic-ular pulsar timing SNR and hence statistical significance, asdescribed in Sec. V. From a more practical point of view, forany given pulsar count NP and observation duration tobs, wemay target a minimum significance level for a pulsar timingsignal. There is a two-dimensional subspace of cosmologicalparameters TRH and kcut/kRH that satisfy this constraint andhence can be probed by the given observational parameters.

A. Impact of pulsar count and observation time

We first explore the impact of the observation durationtobs and pulsar count NP on the range of EMD scenariosthat can be probed. We begin by varying NP with fixedtobs = 20 years. The upper panel of Fig. 5 shows, as col-ors, the number of pulsars NP needed to detect microhalosarising from the given TRH and kcut/kRH at 2σ significance.With this tobs, the detection of microhalos arising from theEMD scenario with TRH = 3 MeV and kcut/kRH = 40 re-quires at least NP ' 70 pulsars, a number comparable to thecount in existing PTAs (e.g., Ref. [88]). Meanwhile, as NP israised, higher reheat temperatures TRH and lower cutoff ratioskcut/kRH can be probed. This trend is easy to understand:higher TRH leads to microhalos of lower mass, while lowerkcut/kRH leads to microhalos of lower internal density; bothof these trends make the microhalos more difficult to detect.Scenarios with TRH & 70 MeV or kcut/kRH . 13 requiremore than 2000 pulsars if tobs = 20 years.

We next explore what observation duration tobs is neces-sary if we are able to observe NP = 200 pulsars. The lowerpanel of Fig. 5 shows, again as colors, the tobs needed to de-tect microhalos arising from the EMD scenario with the givenTRH and kcut/kRH at 2σ significance. With this pulsar count,at least tobs ' 15 years (black) are required to detect micro-halos arising from the EMD scenario with TRH = 3 MeVand kcut/kRH = 40. Similarly to NP , raising tobs allowshigher TRH and lower kcut/kRH to be probed. EMD scenar-ios with kcut/kRH . 10 or TRH & 100 MeV require morethan 40 years of observation time if NP = 200.

Naıvely one might expect NP and tobs to have a similarquantitative impact, because the effective spatial volume thatpulsar timing probes is proportional to both. That is, the num-ber of microhalos passing near any given pulsar is propor-tional to tobs, so the total number of microhalos passing nearpulsars is proportional NP tobs. However, by comparing thetwo panels of Fig. 5 we notice that raising the observation du-ration tobs has a much greater impact than raising the pulsarcount NP on the range of EMD scenarios that can be probed.For instance, raising tobs from 20 years to 30 years has aboutas much impact as boosting NP by a factor of 10 from 200 to2000.

The reason for observation time’s power is that beyondboosting the number of microhalo encounters, tobs has an-other advantage that is illustrated in Fig. 4. This figure sug-gests that raising tobs increases not only the duration of thesignal but also its amplitude. The mathematical reason for

9

10

20

30

40kcut/k

RH

tobs = 20 yr

200

500

101 102

TRH (MeV)

10

20

30

40

kcut/kRH

NP = 200

20

30

102

103

NP

15

20

25

30

35

40

t obs

(yr)

FIG. 5. Top: The pulsar count NP required to detect microhalosarising from EMD scenarios with each TRH and kcut/kRH at 2σsignificance, if the observation duration is fixed at tobs = 20 years.In this case at least 70 pulsars are needed to probe any EMD scenariowith TRH > 3 MeV and kcut/kRH < 40. Bottom: Similarly, theobservation duration tobs required to probe each EMD scenario if weinstead fix NP = 200. In this case at least 15 years of observationare required. In both cases, more pulsars or observation time allowhigher TRH and smaller kcut/kRH to be probed. The hatched regionrequires NP > 2000 (top) or tobs > 40 years (bottom).

this effect is the subtraction of the quadratic fit. Physically,more observation time allows us to better distinguish the im-pact of microhalos—which induce perturbations to the phaseshift δφ at higher-than-quadratic order—from the pulsar’s in-trinsic quadratic-order phase shift.14 Put another way, the timescale associated with microhalo encounters is of order years(see Fig. 4), so tens of years of observation time are needed tofully resolve them.

B. Detection prospects for EMD reheat temperatures andcutoff ratios

Figure 5 indicates that when kcut/kRH & 30, the detec-tion prospects of an EMD scenario are nearly independentof kcut/kRH, and when when TRH . 20 MeV, the detec-tion prospects are nearly independent of TRH. Both of these

14 In practice, long observation durations can also increase the impact of tim-ing noise if the timing residuals are correlated over long time periods. Forinstance, the contribution from any intrinsic frequency second-derivative,ν, could lead to such correlations. The analysis here does not reflect thispossibility because we follow Ref. [11] in assuming, for simplicity, thatpulsars’ intrinsic timing noise is white.

trends can be understood in light of Fig. 2 of Ref. [11]. LowTRH corresponds to large microhalo mass scales, and this fig-ure indicates that when the mass scale of microhalos exceedsroughly 10−7 M�, their detection prospects become mass-independent. Meanwhile, high kcut/kRH leads to microha-los of high internal density, which means that these halos arehighly centrally concentrated. In this case we approach thepoint-mass limit, in which microhalo detection prospects be-come independent of their level of central concentration.

Thus, in these regimes we can express EMD detectionprospects as a function of TRH alone and kcut/kRH alone, re-spectively. Accordingly, the upper panel of Fig. 6 shows thenumber NP of pulsars and observation duration tobs neededto detect microhalos (at 2σ significance) arising from an EMDscenario with a given TRH in the case where kcut/kRH &30.15 Evidently, microhalos arising from reheat temperaturesas high as TRH ' 170 MeV can be detected if NP = 2000and tobs = 40 years. We also remark that to the extent that adirect comparison can be made, our results are similar to theEMD detection prospects presented in Ref. [11].16

Similarly, the lower panel of Fig. 6 shows the NP and tobs

required to detect microhalos (at 2σ significance) arising froman EMD scenario with a given kcut/kRH if TRH . 20 MeV.In this case, microhalos associated with cutoff ratios as lowas kcut/kRH ' 8 can be detected if NP = 2000 and tobs =40 years.

C. Impact of timing noise and cadence

Up to this point, we have fixed the observational ca-dence at tcad = 1 week and the rms timing noise residual attres = 10 ns. This is an optimistic timing-noise assumption;timing noise residuals in present PTAs are on the order of µsor greater (e.g., Ref. [88]), although precision modeling canmitigate this noise (e.g., Ref. [90]). In Fig. 7 we test howEMD detection prospects change if tres is increased. This fig-ure shows, for tres ranging from 10 ns to 50 ns, the maximumTRH and minimum kcut/kRH detectable for given tobs andNP . Evidently, the rms timing residual tres is a highly impor-tant variable, and tres . 0.1 µs is necessary to detect microha-los arising from EMD using pulsar timing if tobs . 40 yearsand NP . 800.

Under the assumptions made in Section V (and Ref. [11]),decreasing the observational cadence tcad can counteract theimpact of timing noise because the signal-to-noise ratio scales

15 For kcut/kRH & 40, microhalos can form significantly before the onsetof the last matter-dominated epoch. Halos can form during radiation dom-ination and would be even denser than those that form later [57], so ourconclusions still apply in this case. However, if kcut/kRH is sufficientlylarge that halos form during the EMD epoch, their evaporation at reheatingcan suppress small-scale structure [57, 89]. Further analysis is needed todetermine microhalo detection prospects in these cosmologies.

16 For instance, in Fig. 8 of Ref. [11] the observation duration is fixed attobs = 30 years, and when NP = 1000, 2σ significance is achievedwhen TRH is slightly over 100 MeV, a result that is also evident in ourFig. 6.

10

102

103

NP

kcut/kRH = 30

2550

75

100

125

10 15 20 25 30 35 40

tobs (year)

102

103

NP

TRH = 10 MeV

10

15

202530

35

0

25

50

75

100

125

150

175

TRH

(MeV

)5

10

15

20

25

30

35

40

kcut/kRH

FIG. 6. The maximum reheat temperature TRH (top) or the min-imum cutoff ratio kcut/kRH (bottom) for which the resulting mi-crohalo population is detectable at 2σ significance with the givenpulsar count NP and observation duration tobs. In the upper panelwe fix kcut/kRH = 30, but the result applies broadly to the casekcut/kRH & 30 since microhalo detection prospects are fairly insen-sitive to kcut/kRH in this regime. In the lower panel we fix TRH =10 MeV, but the result applies generally to the case TRH . 20 MeVsince detection prospects are insensitive to TRH in this regime.

as SNR ∝ t−1rest−1/2cad (see Eq. 14 of Ref. [11]). That is, since

the noise is assumed to be white (uncorrelated), its impact canbe reduced to an arbitrary extent by simply taking more mea-surements. In practice, however, a significant contribution tothe timing noise is red noise associated with the pulsar’s spin(e.g., Refs [91, 92]), which exhibits temporal correlations. Fora red noise spectrum there is less advantage to raising the ob-servational cadence.

We also remark that the timescale associated with a typicalmicrohalo encounter is about a year if TRH ∼ 200 MeV and islonger for lower TRH. Any cadence tcad much shorter than ayear is sufficient to temporally resolve these encounters. Con-sequently, the cadence plays no other role in our calculationthan to control the noise level per the above discussion.

VII. CONCLUSION

In this article we explored the prospects for pulsar timingarrays to probe early matter dominated eras. We found thatin order to detect EMD-induced dark matter microhalos, pul-sar timing noise residuals must be brought below roughly the0.1 µs level. Additionally, due to the long timescale associ-

0

50

100

150

TRH

(MeV

)

tres= 10

ns

tres= 20 ns

tres =50 ns

kcut/kRH = 30

NP = 800

NP = 200

10 15 20 25 30 35 40

tobs (year)

0

10

20

30

40

kcut/kRH

tres = 10 nstres = 20 nstres = 50 ns

TRH = 10 MeV

NP = 200

NP = 800

FIG. 7. Impact of the rms timing residual tres on the maximum re-heat temperature TRH (top) or the minimum cutoff ratio kcut/kRH

(bottom) that can be probed for the given NP and tobs. Specifically,these curves are the slices of Fig. 6 at fixed NP = 200 (solid curves)and NP = 800 (dashed curves), but we consider several differentvalues of tres: 10 ns (blue), 20 ns (orange), and 50 ns (green).

ated with microhalo-pulsar encounters (of order years), tensof years of observing time are necessary. However, with 10 nstiming residuals, EMD scenarios can begin to be probed withabout 20 years of observation time and as few as 70 pul-sars. This is roughly the pulsar count in existing arrays [88].With 40 years of observing time and hundreds to thousands ofpulsars, EMD reheat temperatures TRH up to approximately150 MeV can be reached. Note that detection prospects areonly weakly sensitive to the pulsar count.

Figure 6 shows our main results. In addition to the re-heat temperature TRH, which sets the mass scale for EMD-induced microhalos, we also explore the detection prospectsfor kcut/kRH, which sets the internal density scale (seeFigs. 1–3). For kcut/kRH & 30 microhalos are close to thepoint-mass limit, for the purpose of pulsar timing distortionscaused by the Doppler effect. In this case the detectable rangeof TRH is insensitive to kcut/kRH. For smaller kcut/kRH mi-crohalos’ spatial extent matters, and kcut/kRH as small asabout 8 can be probed given again hundreds to thousands ofpulsars and 40 years of observing time (see Fig. 6).

This work refines the EMD detection prospects presentedin Ref. [11]. The major new feature of our analysis is theuse of the recently developed semianalytic microhalo modelsput forth in Refs. [51–53]. These models describe the forma-tion and evolution of the first and smallest dark matter halos,and they were specifically built for the purpose of understand-ing the microhalo populations that arise from EMD scenariosand other cosmologies that feature boosted small-scale inho-mogeneity. The application of these models is discussed in

11

Secs. III and IV, and PYTHON codes that implement them arepublicly available.17 Compared to Ref. [11], we also exploremore extensively the space of cosmological and observationalparameters. Consequently, we are able to clarify EMD detec-tion prospects more precisely and to do so in terms of bothTRH and kcut/kRH.

We end with a general remark about the capacity for PTAsto probe the Universe’s early history. The results of a simpli-fied analysis in Ref. [11] suggest that pulsar timing is sensitiveto halo masses above approximately 2× 10−8 M� (see Fig. 2of that article). Through the size of the cosmological horizon,this mass scale corresponds to a time when the temperatureof the Universe was roughly 150 MeV. Consequently, one canmake a general statement that PTAs are able to probe the earlyUniverse up to a temperature of about 150 MeV. EMD is notthe only early-Universe scenario that results in boosted small-scale power. For instance, pulsar timing could also probe thepossibility of domination by a cannibal species with number-changing interactions, which leaves a different spectral im-print [93, 94], as long as it occurred below 150 MeV. If thedark matter is an axion and its Peccei-Quinn phase transitionoccurred below 150 MeV, then the resulting axion miniclus-ters [95–102] could also be detected using pulsar timing.

ACKNOWLEDGEMENTS

We thank Simon White for useful discussions. TL is par-tially supported by the Swedish Research Council under con-tract 2019-05135, the Swedish National Space Agency un-der contract 117/19 and the European Research Council undergrant 742104.

Appendix A: Integrating the stellar encounter velocitydistribution

In this appendix we evaluate the integral in Eq. (3) that ex-presses the average relative velocity Vrel(V ,R) between amicrohalo at position R and velocity V and the stars it en-counters. Since we assume a spherically symmetric potential,a microhalo’s orbit is planar with a static angle φ to the Galac-tic disk plane. The microhalo’s position and velocity withinthis plane are described by the radius R, angle θ, and asso-ciated velocity components Vr and Vθ, all of which are func-tions of time. We can decompose the halo’s velocity insteadinto components

V‖ =cosφ

cos θ

(1 + tan2 θ cos2 φ

)−1/2Vθ (A1)

parallel to the mean stellar motion, which is assumed to becircular within the Galactic plane, and

V⊥ =

(V 2r +

sin2 φ

1 + tan2 θ cos2 φV 2θ

)1/2

(A2)

17 https://github.com/delos/microhalo-models.

10−1 100 101

r/rNFWs

10−3

10−2

10−1

100

ρr/(ρ

NFW

srN

FW

s)

t = 0t = 57tdyn

FIG. 8. The density profile of a simulated microhalo (initially NFWwith parameters ρNFW

s and rNFWs ) as it responds to an impulsive

stellar encounter at time t = 0. Colored curves show the profile intime intervals of 3tdyn, with tdyn defined as in Ref. [53]. The solidblack line shows Eq. (B1), the expression proposed in Ref. [53] asa universal fit to the post-encounter density profile. This expressionis fitted (with ρs and rs allowed to vary) to the late-time simulateddensity profile up to r = 4rNFW

s . The dashed line shows the mod-ified profile, Eq. (B2), with the same ρs and rs (not an independentfit); it is identical to Eq. (B1) at small radii but approaches ρ ∝ r−4

at large radii, as it should [103]. Moreover, tuned such that β = 5,it also accurately matches the temporally converged part of the sim-ulated density profile at all radii.

perpendicular thereto. If the stars have mean velocity V∗ andisotropic velocity dispersion σ (both of which can be functionsof position), then Eq. (3) can be written

Vrel(V ,R) = σ F

(σ−1

√(V‖ − V∗

)2+ V 2⊥

). (A3)

Here,

F (a) ≡∫

d3x

(2π)3/2

√(x− a)2 + y2 + z2 e−|x|

2/2 (A4)

with x ≡ (x, y, z), which can be rapidly evaluated using aninterpolation table. We also note that F (a) ' a + 1/a towithin one part in 104 when a > 3.

Appendix B: The impulsively stripped density profile

Based on simulations of microhalos undergoing impulsivestellar encounters, Ref. [53] proposed

ρ = ρsrsr

exp

[− 1

α

(r

rs

)α], (B1)

with α = 0.78, as the universal density profile of a halo ini-tially possessing an NFW profile after arbitrarily many en-counters. However, this profile becomes arbitrarily steep atlarge radii, whereas an analytic argument by Ref. [103] sug-gests that the post-encounter profile does not steepen beyondρ ∝ r−4. The simulations in Ref. [53] each covered a time

12

duration equal to only about 10 times the dynamical timescalewithin the halo’s scale radius, and within that duration thelarge-radius density profile did not stabilize sufficiently to testthis argument. Consequently, in this appendix we carry outa new impulsive encounter simulation using the methodol-ogy of Ref. [53]. The simulated microhalo initially has anNFW density profile with scale parameters rs,0 and ρs,0, andit undergoes an encounter with a star of mass M∗ at impactparameter b and relative velocity V such that M∗/(V b2) =

0.52√ρs,0/G and b = 16rs,0. This description completely

characterizes an encounter in the limit that the encounter isimpulsive. We follow the halo’s response for almost 60 dy-namical time intervals.

Figure 8 shows how the microhalo’s density profile evolvesin this simulation. To fit the time-stabilized part of this density

profile at late times, we propose the form

ρ = ρsrsr

exp

[− 1

α

(r

rs

)α(1+q)1− 1

β2F1

(1, 1; 1+

1

β;−q)]

(B2)with q ≡ [(1/3)(r/rs)

α]β , α = 0.78, and β = 5.18 Here,

2F1 is the hypergeometric function. This profile is identicalto Eq. (B1) at small radii but transitions to ρ ∝ r−4 at largeradii. The suddenness of the transition is controlled by theparameter β, and we fix it through comparison with our sim-ulation. As shown in Fig. 8, Eq. (B1) (solid black curve) failsto match the post-encounter density profile at large radii be-cause the latter does not steepen beyond ρ ∝ r−4 at largeradii. Instead, Eq. (B2) (dashed curve) accurately describesthe post-encounter density profile.

[1] J. P. W. Verbiest, S. Oslowski, and S. Burke-Spolaor,arXiv:2101.10081.

[2] N. Seto and A. Cooray, Astrophys. J. Lett. 659, L33 (2007),astro-ph/0702586.

[3] K. Kashiyama and N. Seto, MNRAS 426, 1369 (2012),1208.4101.

[4] K. Schutz and A. Liu, Phys. Rev. D 95, 023002 (2017),1610.04234.

[5] E. R. Siegel, M. P. Hertzberg, and J. N. Fry, Mon. Not. R.Astron. Soc. 382, 879 (2007), astro-ph/0702546.

[6] S. Baghram, N. Afshordi, and K. M. Zurek, Phys. Rev. D 84,043511 (2011), 1101.5487.

[7] H. A. Clark, G. F. Lewis, and P. Scott, Mon. Not. R. Astron.Soc. 456, 1394 (2016), 1509.02938; 464, 2468 (2017).

[8] K. Kashiyama and M. Oguri, arXiv:1801.07847.[9] J. A. Dror, H. Ramani, T. Trickle, and K. M. Zurek, Physical

Review D 100, 023003 (2019), 1901.04490.[10] H. Ramani, T. Trickle, and K. M. Zurek, Journal of Cos-

mology and Astroparticle Physics 2020, 033–033 (2020),2005.03030.

[11] V. S. Lee, A. Mitridate, T. Trickle, and K. M. Zurek, J. HighEnergy Phys. 2021, 1 (2021), 2012.09857.

[12] V. S. H. Lee, S. R. Taylor, T. Trickle, and K. M. Zurek,arXiv:2104.05717.

[13] P. F. de Salas, M. Lattanzi, G. Mangano, G. Miele, S. Pastor,and O. Pisanti, Phys. Rev. D 92, 123534 (2015), 1511.00672.

[14] R. Allahverdi, M. A. Amin, A. Berlin, N. Bernal, C. T. Byrnes,M. S. Delos, A. L. Erickcek, M. Escudero, D. G. Figueroa,K. Freese, et al., Open J. Astrophys. 4, 1 (2021), 2006.16182.

[15] M. Pospelov, A. Ritz, and M. Voloshin, Phys. Lett. B 662, 53(2008), 0711.4866.

[16] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, andN. Weiner, Phys. Rev. D 79, 015014 (2009), 0810.0713.

[17] D. Hooper, N. Weiner, and W. Xue, Phys. Rev. D 86, 056009(2012), 1206.2929.

[18] M. Abdullah, A. DiFranzo, A. Rajaraman, T. M. P. Tait,

18 Equation (B2) is obtained by integrating the expression d ln ρ/d ln r =

−1 −[(r/rs)−αβ + 3−β

]−1/β , which interpolates smoothly fromEq. (B1) at small radii to ρ ∝ r−4 at large radii.

P. Tanedo, and A. M. Wijangco, Phys. Rev. D 90, 035004(2014), 1404.6528.

[19] A. Berlin, P. Gratia, D. Hooper, and S. D. McDermott, Phys.Rev. D 90, 015032 (2014), 1405.5204.

[20] A. Martin, J. Shelton, and J. Unwin, Phys. Rev. D 90, 103513(2014), 1405.0272.

[21] Y. Zhang, J. Cosmol. Astropart. Phys. 05, 008 (2015),1502.06983.

[22] A. Berlin, D. Hooper, and G. Krnjaic, Phys. Lett. B 760, 106(2016), 1602.08490.

[23] A. Berlin, D. Hooper, and G. Krnjaic, Phys. Rev. D 94,095019 (2016), 1609.02555.

[24] J. A. Dror, E. Kuflik, and W. H. Ng, Phys. Rev. Lett. 117,211801 (2016), 1607.03110.

[25] T. Tenkanen and V. Vaskonen, Phys. Rev. D 94, 083516(2016), 1606.00192.

[26] J. A. Dror, E. Kuflik, B. Melcher, and S. Watson, Phys. Rev.D 97, 063524 (2018), 1711.04773.

[27] T. Tenkanen, Phys. Rev. D 100, 083515 (2019), 1905.11737.[28] G. Coughlan, W. Fischler, E. W. Kolb, S. Raby, and G. G.

Ross, Phys. Lett. B 131, 59 (1983).[29] B. De Carlos, J. Casas, F. Quevedo, and E. Roulet, Phys. Lett.

B 318, 447 (1993), hep-ph/9308325.[30] T. Banks, D. B. Kaplan, and A. E. Nelson, Phys. Rev. D 49,

779 (1994), hep-ph/9308292.[31] T. Banks, M. Berkooz, and P. J. Steinhardt, Phys. Rev. D 52,

705 (1995), hep-th/9501053.[32] T. Banks, M. Berkooz, S. H. Shenker, G. Moore, and P. J.

Steinhardt, Phys. Rev. D 52, 3548 (1995), hep-th/9503114.[33] B. S. Acharya, G. Kane, and E. Kuflik, Int. J. Mod. Phys. A

29, 1450073 (2014), 1006.3272.[34] G. Kane, K. Sinha, and S. Watson, Int. J. Mod. Phys. D 24,

1530022 (2015), 1502.07746.[35] J. T. Giblin, G. Kane, E. Nesbit, S. Watson, and Y. Zhao, Phys.

Rev. D 96, 043525 (2017), 1706.08536.[36] S. Mollerach, Phys. Rev. D 42, 313 (1990).[37] A. Linde and V. Mukhanov, Phys. Rev. D 56, R535(R) (1997),

astro-ph/9610219.[38] D. H. Lyth and D. Wands, Phys. Lett. B 524, 5 (2002), hep-

ph/0110002.[39] T. Moroi and T. Takahashi, Phys. Lett. B 522, 215 (2001), hep-

ph/0110096; 539, 303 (2002).[40] A. Albrecht, P. J. Steinhardt, M. S. Turner, and F. Wilczek,

13

Phys. Rev. Lett. 48, 1437 (1982).[41] M. S. Turner, Phys. Rev. D 28, 1243 (1983).[42] J. H. Traschen and R. H. Brandenberger, Phys. Rev. D 42,

2491 (1990).[43] L. Kofman, A. Linde, and A. A. Starobinsky, Phys. Rev. Lett.

73, 3195 (1994), hep-th/9405187.[44] L. Kofman, A. Linde, and A. A. Starobinsky, Phys. Rev. D

56, 3258 (1997), hep-ph/9704452.[45] J. F. Dufaux, G. N. Felder, L. Kofman, M. Peloso, and

D. Podolsky, J. Cosmol. Astropart. Phys. 07, 006 (2006), hep-ph/0602144.

[46] R. Allahverdi, R. Brandenberger, F.-Y. Cyr-Racine, andA. Mazumdar, Ann. Rev. Nucl. Part. Sci. 60, 27 (2010),1001.2600.

[47] K. Jedamzik, M. Lemoine, and J. Martin, J. Cosmol. As-tropart. Phys. 09, 034 (2010), 1002.3039.

[48] R. Easther, R. Flauger, and J. B. Gilmore, J. Cosmol. As-tropart. Phys. 04, 027 (2011), 1003.3011.

[49] N. Musoke, S. Hotchkiss, and R. Easther, Phys. Rev. Lett.124, 061301 (2020), 1909.11678.

[50] A. L. Erickcek and K. Sigurdson, Phys. Rev. D 84, 083503(2011), 1106.0536.

[51] M. S. Delos, M. Bruff, and A. L. Erickcek, Phys. Rev. D 100,023523 (2019), 1905.05766.

[52] M. S. Delos, Phys. Rev. D 100, 063505 (2019), 1906.10690.[53] M. S. Delos, Phys. Rev. D 100, 083529 (2019), 1907.13133.[54] M. S. Delos, T. Linden, and A. L. Erickcek, Phys. Rev. D 100,

123546 (2019), 1910.08553.[55] A. L. Erickcek, Phys. Rev. D 92, 103505 (2015), 1504.03335.[56] A. L. Erickcek, K. Sinha, and S. Watson, Phys. Rev. D 94,

063502 (2016), 1510.04291.[57] C. Blanco, M. S. Delos, A. L. Erickcek, and D. Hooper, Phys.

Rev. D 100, 103010 (2019), 1906.00010.[58] W. H. Press and P. Schechter, Astrophys. J. 187, 425 (1974).[59] J. Bond, S. Cole, G. Efstathiou, and N. Kaiser, Astrophys. J.

379, 440 (1991).[60] A. D. Ludlow, S. Bose, R. E. Angulo, L. Wang, W. A. Hell-

wing, J. F. Navarro, S. Cole, and C. S. Frenk, Mon. Not. R.Astron. Soc. 460, 1214 (2016), 1601.02624.

[61] J. M. Bardeen, J. R. Bond, N. Kaiser, and A. S. Szalay, Astro-phys. J. 304, 15 (1986).

[62] J. E. Gunn and J. R. Gott III, Astrophys. J. 176, 1 (1972).[63] J. R. Gott, Astrophys. J. 201, 296 (1975).[64] R. K. Sheth, H. Mo, and G. Tormen, Mon. Not. R. Astron.

Soc. 323, 1 (2001), astro-ph/9907024.[65] T. Ishiyama, J. Makino, and T. Ebisuzaki, Astrophys. J. Lett.

723, L195 (2010), 1006.3392.[66] D. Anderhalden and J. Diemand, J. Cosmol. Astropart. Phys.

04, 009 (2013), 1302.0003; 08, E02 (2013).[67] T. Ishiyama, Astrophys. J. 788, 27 (2014), 1404.1650.[68] E. Polisensky and M. Ricotti, Mon. Not. R. Astron. Soc. 450,

2172 (2015), 1504.02126.[69] G. Ogiya and O. Hahn, Mon. Not. R. Astron. Soc. 473, 4339

(2018), 1707.07693.[70] M. S. Delos, A. L. Erickcek, A. P. Bailey, and M. A. Alvarez,

Phys. Rev. D 97, 041303(R) (2018), 1712.05421.[71] M. S. Delos, A. L. Erickcek, A. P. Bailey, and M. A. Alvarez,

Phys. Rev. D 98, 063527 (2018), 1806.07389.[72] R. E. Angulo, O. Hahn, A. D. Ludlow, and S. Bonoli, Mon.

Not. R. Astron. Soc. 471, 4687 (2017), 1604.03131.[73] T. Ishiyama and S. Ando, Mon. Not. R. Astron. Soc. 492, 3662

(2020), 1907.03642.[74] G. Ogiya, D. Nagai, and T. Ishiyama, Mon. Not. R. Astron.

Soc. 461, 3385 (2016), 1604.02866.

[75] M. Gosenca, J. Adamek, C. T. Byrnes, and S. Hotchkiss, Phys.Rev. D 96, 123519 (2017), 1710.02055.

[76] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J.462, 563 (1996), astro-ph/9508025.

[77] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J.490, 493 (1997), astro-ph/9611107.

[78] N. E. Drakos, J. E. Taylor, A. Berrouet, A. S. Robotham,and C. Power, Mon. Not. R. Astron. Soc. 487, 1008 (2019),1811.12844.

[79] F. C. van den Bosch, G. Ogiya, O. Hahn, and A. Burkert,Mon. Not. R. Astron. Soc. 474, 3043 (2017), 1711.05276.

[80] J. Bland-Hawthorn and O. Gerhard, Annual Review of Astron-omy and Astrophysics 54, 529–596 (2016), 1602.07702.

[81] V. S. Berezinsky, V. I. Dokuchaev, and Y. N. Eroshenko,Physics-Uspekhi 57, 1–36 (2014), 1405.2204.

[82] L. M. Widrow, Astrophys. J. Suppl. Ser. 131, 39 (2000).[83] G. Hobbs, D. R. Lorimer, A. G. Lyne, and M. Kramer, Mon.

Not. R. Astron. Soc. 360, 974 (2005), astro-ph/0504584.[84] A. Schneider, L. Krauss, and B. Moore, Phys. Rev. D 82,

063525 (2010), 1004.5432.[85] A. C. Robin, C. Reyle, S. Derriere, and S. Picaud, Astronomy

& Astrophysics 409, 523 (2003), astro-ph/0401052; 416, 157(2004).

[86] P. Kroupa, Science 295, 82 (2002), astro-ph/0201098.[87] X. J. Liu, C. G. Bassa, and B. W. Stappers, Mon. Not. R.

Astron. Soc. 478, 2359 (2018), 1805.02892.[88] B. B. P. Perera, M. E. DeCesar, P. B. Demorest, M. Kerr,

L. Lentati, D. J. Nice, S. Osłowski, S. M. Ransom, M. J. Keith,Z. Arzoumanian, et al., Mon. Not. R. Astron. Soc. 490, 4666(2019), 1909.04534.

[89] G. Barenboim, N. Blinov, and A. Stebbins,arXiv:2107.10293.

[90] B. Goncharov, D. J. Reardon, R. M. Shannon, X.-J. Zhu,E. Thrane, M. Bailes, N. D. R. Bhat, S. Dai, G. Hobbs,M. Kerr, et al., Mon. Not. R. Astron. Soc. 502, 478 (2021),2010.06109.

[91] R. M. Shannon and J. M. Cordes, Astrophys. J 725, 1607(2010), 1010.4794.

[92] Y. Wang, in Journal of Physics Conference Series, Journalof Physics Conference Series, Vol. 610 (2015) p. 012019,1505.00402.

[93] A. L. Erickcek, P. Ralegankar, and J. Shelton, Phys. Rev. D103, 103508 (2021), 2008.04311.

[94] A. L. Erickcek, P. Ralegankar, and J. Shelton,arXiv:2106.09041.

[95] C. J. Hogan and M. J. Rees, Phys. Lett. B 205, 228 (1988).[96] E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994),

astro-ph/9403011.[97] E. W. Kolb and I. I. Tkachev, Astrophys. J. Lett. 460, L25

(1996), astro-ph/9510043.[98] M. Fairbairn, D. J. E. Marsh, and J. Quevillon, Phys. Rev.

Lett. 119, 021101 (2017), 1701.04787.[99] M. Fairbairn, D. J. E. Marsh, J. Quevillon, and S. Rozier,

Phys. Rev. D 97, 083502 (2018), 1707.03310.[100] L. Dai and J. Miralda-Escude, Astron. J. 159, 49 (2020),

1908.01773.[101] B. J. Kavanagh, T. D. P. Edwards, L. Visinelli, and

C. Weniger, arXiv:2011.05377.[102] H. Xiao, I. Williams, and M. McQuinn, Phys. Rev. D 104,

023515 (2021), 2101.04177.[103] W. Jaffe, in Structure and Dynamics of Elliptical Galaxies

(Springer, 1987) pp. 511–512.