Density profiles of ultracompact minihalos: Implications for constraining the primordial power spectrum

I Introduction

Ultracompact dark matter minihalos have emerged as a powerful probe of early-Universe physics. Overdense regions with $\delta\equiv\delta\rho/\rho\gtrsim 10^{-3}$ at horizon entry seed the formation of dark matter minihalos near the time of recombination ($z\simeq 1000$) Ricotti and Gould 2009, and such early formation yields highly compact structures potentially visible through dark matter annihilation Scott and Sivertsson 2009; Saito and Shirai 2011; Yang et al. 2011a; Berezinsky et al. 2013; Yang and Su; Nakama et al. 2018; Josan and Green 2010; Bringmann et al. 2012; Yang et al. 2013a; Yang and Qin 2017; Yang et al. 2011b; Zhang 2011; Yang 2016; Clark et al. 2017a; Yang et al. 2011c; Yang et al. 2013b; Beck and Colafrancesco or by their gravitational signatures Ricotti and Gould 2009; Zackrisson et al. 2013; Clark et al. 2016a; *clark2016erratumI; Li et al. 2012. The nondetection of these structures thus constrains the amplitude of primordial density fluctuations, making it a probe of the primordial power spectrum Josan and Green 2010; Bringmann et al. 2012; Li et al. 2012; Yang et al. 2013a; Yang et al. 2013c; Yang 2014; Clark et al. 2016b; *clark2017erratumII; Yang and Qin 2017; Nakama et al. 2018 and hence of inflationary models Aslanyan et al. 2016 and the thermal history of the Universe Choi and Takahashi 2017.

However, with one recent exception Nakama et al. 2018, all constraints derived from UCMHs have been calculated assuming they develop the $\rho\propto r^{-9/4}$ density profile, which is drawn from analytic radial infall theory Fillmore and Goldreich 1984; Bertschinger 1985 and taken to apply to halos forming at $z\gtrsim 1000$ due to the small velocity dispersion at those times Ricotti and Gould 2009. This profile has a much steeper inner form than is typically observed in simulations (e.g., Frenk and White 2012), a property that enhances the observational signatures of UCMHs. The applicability of this profile was first called into question in Ref. Gosenca et al. 2017, and in Ref. Delos et al. 2018, hereafter Paper I, we showed by means of N-body simulations that halos forming in a Gaussian field—even UCMHs forming as early as $z=1000$ from fluctuations as extreme as $6.8\sigma$—do not develop the $\rho\propto r^{-9/4}$ density profile. Instead, they develop shallower inner density profiles: $\rho\propto r^{-\gamma}$ with $1\leq\gamma\leq 3/2$. In this paper, we present our results in greater detail and discuss the implications of this discovery. In addition to the $\rho\propto r^{-9/4}$ assumption, previous UCMH-derived power spectrum constraints employed only the minihalos that form at $z\geq 1000$. Since we have shown that all minihalos develop shallower density profiles, there is no need to make this restriction. We show that the resulting new bounds on the power spectrum are stronger than the previous UCMH constraints.

The $\rho\propto r^{-9/4}$ density profile has been taken to be a consequence of nearly radial mass infall onto a halo that formed at $z\gtrsim 1000$ Ricotti and Gould 2009, so we replicate this scenario as closely as possible in our simulations by finding extremely rare $6.8\sigma$ density peaks that collapse near $z=1000$. The UCMH formation scenario is tested in two power spectra at the opposite extremes that are motivated by inflationary phenomenology. First, we use a spiked power spectrum with fluctuations enhanced over a narrow range of scales. Second, we use a stepped power spectrum with fluctuations enhanced over all scales accessible to the simulation. In the narrowly enhanced power spectrum, halos develop in relative isolation, a situation that might be expected to reproduce the radial infall solution. However, we find that all halos, even the UCMHs forming at $z\simeq 1000$, develop $\rho\propto r^{-3/2}$ inner density profiles. In fact, this profile also appears in another context: it is the density profile seen in the smallest halos forming above a free-streaming cutoff Ishiyama et al. 2010; Anderhalden and Diemand 2013a; *anderhalden2013erratum; Ishiyama 2014; Polisensky and Ricotti 2015; Angulo et al. 2017; Ogiya and Hahn 2018. Meanwhile, the broadly enhanced power spectrum builds halos hierarchically from smaller halos and yields density profiles of the Navarro-Frenk-White (NFW) form Navarro et al. 1995; Navarro et al. 1996; Navarro et al. 1997,

with a $\rho\propto r^{-1}$ inner profile. This is the same form that appears in simulations of galaxy-scale structure. Evidently, UCMHs, which we define as halos forming at $z\geq 1000$, develop the same density profiles as halos that form at much later times.

We also introduce a new model for predicting the density profiles of minihalos that form from spiked power spectra based on their formation times. Spectral spikes can arise from steps in the inflaton potential Salopek et al. 1989; Starobinskij 1992; Ivanov et al. 1994; Starobinsky 1998 or from particle production during inflation Chung et al. 2000; Barnaby and Huang 2009; Barnaby 2010. Near the free-streaming cutoff, the power spectrum imprinted by an early matter-dominated era is also similar to a spike Erickcek and Sigurdson 2011; Barenboim and Rasero 2014; Fan et al. 2014; Erickcek 2015. Moreover, spiked power spectra are less well constrained than flatter power spectra by CMB spectral distortions, which limit the power integrated over a broad range in $k$-space Chluba et al. 2012. In this paper, we begin an investigation of halos forming from spiked power spectra that we will expand upon in the next paper of this series Delos et al. tion, hereafter Paper III (in preparation).

Finally, we discuss the impact of our result on the capacity for minihalos to constrain the primordial power spectrum. We use our model to calculate an upper bound on the amplitude of spiked power spectra that incorporates the new shallower minihalo density profiles. This upper bound is based on limits from Fermi-LAT Atwood et al. 2009 on gamma rays from dark matter annihilation. Despite the reduced annihilation rate implied by the shallower profile, this constraint is stronger than an equivalent UCMH constraint derived using the $\rho\propto r^{-9/4}$ density profile. Our model provides a stronger constraint because it accounts for all halos, whereas the old UCMH model only counted halos forming at $z\gtrsim 1000$. Our calculation demonstrates the continued viability of minihalos as probes of the small-scale power spectrum, and we discuss future avenues for improvement.

This paper is organized as follows. In Sec. II, we select power spectra and detail the setup of our simulations. Section III presents the simulation results. The UCMH density profile is the main result, but we also make remarks on UCMH growth and the effects of mergers. In Sec. III.4, we sample later-forming minihalos to develop a general model for minihalo density profiles, and Sec. IV employs this model to calculate new constraints on the primordial power spectrum. In Sec. IV.4, we discuss how the new minihalo picture differs from the old UCMH picture. Section V concludes and outlines the ways in which our calculation can be improved in future work. Appendices A and B contain additional information about our simulations, including numerical convergence tests. Appendices C–E contain further details about our calculation of the power spectrum constraint.

II Simulation preparation

We carry out simulations of halos forming at ${z\simeq 1000}$ from extreme peaks in the density field. This picture is intended to match the UCMH formation scenario Ricotti and Gould 2009, and we aim to show conclusively that the ${\rho\propto r^{-9/4}}$ single-power-law density profile does not arise in halos forming due to an enhancement of the primordial power spectrum.

In order to perform numerical experiments on such early-forming minihalos, we must start with an enhanced power spectrum. Inflationary models supply a rich phenomenology in this respect. Steps, kinks, or second-derivative jumps in the inflaton potential would imprint spikes, steps, or bends, respectively, on the primordial power spectrum Salopek et al. 1989; Starobinskij 1992; Ivanov et al. 1994; Starobinsky 1998; Joy et al. 2008. Particle production during inflation can produce a spike in the power spectrum Chung et al. 2000; Barnaby and Huang 2009; Barnaby 2010, while multifield inflation can imprint steps Silk and Turner 1987; Polarski and Starobinsky 1992; Adams et al. 1997 or oscillations Achúcarro et al. 2011; Cespedes et al. 2012. Inflation aside, an early matter-dominated era enhances perturbations that enter the horizon prior to the onset of radiation domination Erickcek and Sigurdson 2011; Barenboim and Rasero 2014; Fan et al. 2014; Erickcek 2015, and an era dominated by a fast-rolling scalar field generates a similar enhancement Redmond et al. 2018.

For our simulations, we consider two examples from these possible power spectrum enhancements. First, we consider a narrow spike in the power spectrum. This shape has possible inflationary origins, as discussed above, and is also qualitatively similar to the enhancement generated by an early matter-dominated era at scales close to the free-streaming cutoff. Next, we consider a step in the power spectrum, intended to represent the opposite extreme where fluctuations are enhanced over a broad range of scales. The two power spectra are plotted in Fig. 1. We superpose these modifications on a conventional power spectrum with amplitude $\mathcal{A}_{s}=$2.142\text{\times}{10}^{-9}$$and spectral index$n_{s}=0.9667$Ade et al. 2016. The spike contains 90$k$, while the step amplifies fluctuations over the full range of scales accessible to the simulation. We will focus on the spiked power spectrum for most of Sec. III and return to the step in Sec. III.5.

Halos forming from more extreme density contrasts are both more spherically symmetric Bardeen et al. 1986 and less affected by nearby structure. To best simulate the UCMH formation scenario, we tune our power spectra so that halos forming by $z=1000$ are exceedingly rare. In particular, the spiked power spectrum is tuned so that a $6.8\sigma$ fluctuation is necessary to seed such early collapse, and we generate a large number of random fields in order to obtain a handful of boxes to use as initial conditions for our simulations. This procedure may be contrasted with that of Ref. Gosenca et al. 2017, who simulated a typical box whose most extreme peak was $4.3\sigma$. UCMHs forming from peaks as extreme as $6\sigma$ are employed to derive observational constraints Bringmann et al. 2012, so we wish to exceed this amplitude to conclusively rule out the $\rho\propto r^{-9/4}$ profile.

II.2 Simulation setup

A matter power spectrum is calculated at $z=1000$ from the primordial power spectrum using the Boltzmann code Camb Sources Challinor and Lewis 2011; Lewis and Challinor 2007. To match simulation behavior, this power spectrum is evolved back to an earlier time using the Mészáros equation Meszaros 1974

which provides a convenient prescription for calculating the evolution of a density contrast $\delta$ at linear order during mixed matter-radiation domination.

For the spiked power spectrum shown in Fig. 1, we generate 2.3 million random density fields. Nine of them meet the collapse criterion, so we use these as the initial density fields and simulate them to $z=50$. We also pick out one such density field, which we label the primary, to simulate at higher resolution and perform convergence tests; a slice of its initial density field is shown in Fig. 2. Notice how extreme the most overdense region is compared to its surroundings: this is indeed a rare event.

We also resimulate each of these density fields with increased simulation-particle density by resampling the initial field at higher resolution and including only a sphere of radius 0.93 kpc (with vacuum boundary conditions) around the most overdense point. This cut-out region is drawn in Fig. 2. This procedure allows us to probe smaller scales, and in Appendix B, we demonstrate that it does not change the density profile of the UCMH in the primary density field at $z=100$. This convergence does not hold for the UCMHs in all nine fields: some of them begin to be influenced by structure outside the sphere as early as $z\sim 200$. Consequently, we only carry out these cut-out simulations up to $z=400$ for the other eight density fields.

II.3 N-body code

We use the cosmological simulation code Gadget-2 Springel 2005; Springel et al. 2001 for our numerical experiments. Gadget-2 is a hybrid N-body code that computes short-range forces using a tree method and long-range forces using Fourier techniques on a mesh. A discussion on our choices of simulation parameters can be found in Appendix B, along with convergence studies. We also model all matter as collisionless dark matter with $\Omega_{m}=0.3089$ Ade et al. 2016: at the scales we study, dark matter halos cannot capture significant baryon content.

In order to accurately simulate a halo collapse at $z\simeq 1000$, our experiments must begin during radiation domination, so our N-body code must account for radiation. However, fluctuations in the radiation density field decay rapidly after horizon entry (see e.g. Dodelson 2003), so it is only necessary to model the effect of a smooth radiation component on the expansion rate. We modified the publicly available release of Gadget-2 to include such a radiation component. Tests of the accuracy of this code can be found in Appendix A.

III Simulation results

A visual inspection of the primary simulation box yields some key insights. First, we note that our criterion for early collapse, that the linear density contrast be $\delta>1.686$ by $z=1000$, has worked as expected. Figure 3 shows a slice of the density field evolving from $z=1255$ to $z=941$ at the location of the extreme density peak where we expect the UCMH to form, and we see that the density at the central point grows astronomically around $z=1000$, an indication of collapse. To emphasize the rarity of this event, we also show the density field at $z=715$: the UCMH is still the only halo to have collapsed by this redshift.

Next, we look at the density field at a much later redshift. Figure 4 shows the density field at $z=100$ projected along one axis. The imprint of the spike in the power spectrum is evident, for we see an almost uniform distribution of halos with no large-scale structure. This is quite unlike a hierarchical growth picture (cf. Fig. 10). There is also minimal small-scale structure: these halos appear generally isolated and are only linked by filaments. These points are emphasized in the enhanced pictures of the main halo, where we see more clearly the lack of small-scale structure. We also see the beginning of fragmentation of the filaments into halos, but this fragmentation is a numerical artifact; see Appendix B.

According to the Rockstar halo finder Behroozi et al. 2013, there are 530 halos with masses above $1.5M_{\odot}$ at $z=100$, and these halos contain 24% of the total mass of the simulation box within their virial radii. Such an abundance of halos is clearly expected in any picture that can produce a halo that collapses by $z\simeq 1000$, but later halos have been neglected in prior UCMH treatments because they are expected to be less compact. We will explore in Sec. III.4 whether younger halos have the same structure as the oldest ones.

We now study the spherically averaged density profiles of the UCMHs. We simulated the UCMH in the primary simulation box at the highest particle density (see Appendix B for details), so we first focus our study on that halo. This halo has mass $M=31M_{\odot}$ at $z=100$, and Fig. 5 shows its density profile at $z=50$, $z=100$, $z=200$, and $z=400$ plotted in physical (not comoving) coordinates. We first note that this halo clearly does not follow a $\rho\propto r^{-9/4}$ or similar single-power-law form, contradicting the assumption made in prior UCMH treatments. We have conducted extensive convergence testing to confirm the validity of this result, as described in Appendix B. The actual density profile is shallower, which will substantially reduce the observational signals of these halos, as we discuss in Sec. IV. However, the inner profile is still steeper than the $\rho\propto r^{-1}$ behavior of the NFW profile given by Eq. (1). In fact, the inner density profile approaches $\rho\propto r^{-3/2}$, and the full density profile is fit well by the double-power-law form

which scales as $\rho\propto r^{-3/2}$ at small $r$ and $\rho\propto r^{-3}$ at large $r$. We will call Eq. (4) the Moore profile due to its similarity to the form in Ref. Moore et al. 1999.

Inner profiles $\rho\propto r^{-\gamma}$ with index $\gamma$ ranging from 1.3 to 1.5 have previously been observed in the smallest halos forming above a cutoff in the power spectrum Angulo et al. 2017; Anderhalden and Diemand 2013a; Ishiyama et al. 2010; Ishiyama 2014; Polisensky and Ricotti 2015, and Ref. Ogiya and Hahn 2018 found that the emergence of $\rho\propto r^{-3/2}$ is connected to the presence of a uniform-density core in the precursor density peak. In this light, it is not surprising that $\rho\propto r^{-3/2}$ arises in our spiked power spectrum, since like a cutoff power spectrum, it lacks power below the scale of the spike and produces cored peaks in the primordial density field. The physical origin of the $\rho\propto r^{-3/2}$ profile is not well understood, but it is known to be markedly less rotationally supported than the NFW profile Ogiya and Hahn 2018.

Finally, we remark on the fitting parameters $\rho_{s}$ and $r_{s}$ of the Moore profile [Eq. (4)] for the UCMH shown in Fig. 5. The scale radius $r_{s}$ that separates the $\rho\propto r^{-3/2}$ behavior from the $\rho\propto r^{-3}$ behavior appears to be set by the physical scale associated with the spike in the power spectrum at $z=1000$, obeying $r_{s}\simeq 0.7[(1+z)k_{s}]^{-1}$. Similarly, the scale density $\rho_{s}$ is close to the background physical density at $z=1000$, obeying $\rho_{s}\simeq 30(1+z)^{3}\bar{\rho}_{0}$, where $\bar{\rho}_{0}$ is the background matter density today. These correlations suggest that the $\rho\propto r^{-3/2}$ inner profile is set during the earliest stages of the halo’s growth while the $\rho\propto r^{-3}$ outer profile grows during late accretion. We will develop these ideas in more detail in Sec. III.4.

All of these results come from the UCMH in the primary simulation run. We also simulated eight other UCMHs, and we show the density profiles of all nine of them in Fig. 6. All of these halos collapsed near $z=1000$, and there is clearly little deviation in the structure of these halos. In particular, all of them exhibit the same $\rho\propto r^{-3/2}$ inner density profile, providing further evidence that the $\rho\propto r^{-9/4}$ pure power law density profile does not arise in a realistic formation scenario.

III.2 Mass accretion

We briefly remark on the mass accretion history of the UCMHs. UCMHs have been previously assumed to grow as $M\propto a$ Ricotti and Gould 2009, but this is a result from radial infall theory Bertschinger 1985. This theory describes an overdense region in an unperturbed background, which is very different from the Gaussian random field from which a realistic halo would form.

III.3 Mergers

Three of our nine UCMHs underwent mergers between $z=100$ and $z=50$, with another two impending. One such event occurring at $z\simeq 86$ is depicted in the upper panel of Fig. 8. The UCMH had concentration $c=12$ at this time. The lower panel shows the density profile of this halo at $z=100$ and $z=50$ before and after the merger takes place, and we see that this event has been energetic enough to disperse mass out of the center of the halo and make the inner profile shallower. Unfortunately, we do not have the resolution at these redshifts to determine the slope of the inner profile after the merger, but the fact that the density profile at $r<r_{s}$ is altered indicates that the stability we observed in Sec. III.1 does not hold after mergers.

III.4 Other minihalos

So far, we have studied only the exceptionally rare halos that form at $z\simeq 1000$. In this section, we explore a sample of other halos in the simulation box shown in Fig. 4. We pick 10 halos, including the UCMH, with masses evenly distributed between 3 $M_{\odot}$ and 32 $M_{\odot}$ at $z=100$. Figure 9 shows the density profiles of these halos. As we discussed in Sec. III.1, we expect that each halo will obey

($\rho_{s}r_{s}^{3/2}$ is the $r\ll r_{s}$ asymptote of $\rho r^{3/2}$ for a Moore profile). The two halos lying farthest below this line have formed only slightly before the time $z=100$ at which we are seeing them, so it is plausible that their inner profiles are still growing.

We do not attempt to study the scaling of $\rho_{s}$ and $r_{s}$ separately because this requires fitting functional forms to the density profiles, which is unreliable with the resolution to which we are limited here. However, $\rho_{s}r_{s}^{3/2}$ alone is a useful combination because it determines most of the annihilation signal of the halo (see Sec. IV). Our ultimate goal is to predict halo density profiles from the power spectrum in order to place constraints thereon, and we find the spread in $\rho_{s}r_{s}^{3/2}$ to be well within a factor of 2 of Eq. (7), which is promising. However, our halo sample is small and we are biased by resolution toward larger halos. We are also limited to a single power spectrum. We will carry out in Paper III a more systematic study of the density profiles of halos forming from spiked power spectra.

III.5 Power spectrum with step

We finally step away from the spiked power spectrum to verify that a picture with power evenly distributed across scales still produces NFW halos even in the UCMH scenario involving the early collapse of rare extreme overdensities. We used the step power spectrum shown in Fig. 1 and prepared a set of initial conditions in a (7.4 kpc)3 periodic box using the procedure described in Sec. II. Boxes were repeatedly generated until the $z=1000$ collapse criterion was met, which occurred after about $2300$ boxes. We began the simulation run at $z=8\times 10^{6}$ and ended it at $z=100$; the resulting UCMH at $z=100$ is shown in Fig. 10. It is evident from the density field that this is a very different picture from what we have seen with our spiked power spectrum. The large-scale power has caused much of the mass within the box to collapse into the UCMH, while at the same time, the small-scale power has given this halo abundant substructure.

Figure 10 also shows the radial density profile of this halo. It follows the NFW form well, and does not fit the Moore form at all. Moreover, we resolve an inner density profile that is at least as shallow as $\rho\propto r^{-1}$. Even halos that collapse near $z\simeq 1000$ still possess the shallow inner profiles characteristic of hierarchical clustering.

A natural question to ask is how the density profiles behave in the transition between a spiked and a scale-invariant power spectrum. A careful treatment is beyond the scope of this paper, but we will see in Paper III that the answer is ultimately related to mergers. As we discussed in Sec. III.3, mergers induce shallowing of the inner density profile toward $\rho\propto r^{-1}$. Meanwhile, mergers occur more frequently when the power spectrum spike is wider, culminating in the hierarchical clustering characteristic of conventional power spectra. These concepts explain, at least qualitatively, the shift from $\rho\propto r^{-3/2}$ to $\rho\propto r^{-1}$ inner profiles when the spike in the power spectrum is replaced by a step.

As a final remark, we have found between the spiked and step power spectra that UCMHs develop the same density profiles as halos that form at much later times. The spike produces UCMHs with similar density profiles to those of the smallest halos forming above a free-streaming cutoff, while the step produces UCMHs with density profiles resembling those of the galactic halos created by hierarchical clustering. In retrospect, this is not surprising. Ref. Ricotti and Gould 2009 conceived of UCMHs as the late stage of rare non-Gaussian density fluctuations, so they assumed a conventional (unenhanced) power spectrum when calculating the velocity dispersion at $z\simeq 1000$. The small velocity dispersion that resulted was the basis for the argument that radial infall theory would apply, but this velocity dispersion would be increased by any power spectrum enhancement. There is no difference, aside from the emerging dominance of a radiation or dark energy component, between halos forming from a boosted power spectrum at early times and halos forming from a conventional power spectrum at late times. However, the velocity dispersion is not the only obstacle to the $\rho\propto r^{-9/4}$ profile. As we noted in Paper I, this profile results specifically from the collapse of an overdense region in an unperturbed background, which is not an instance of a peak that forms in a Gaussian random field. We will revisit this topic in Paper III when we explore the relationship between a collapsed halo and its precursor density peak.

IV Constraining the power spectrum

UCMHs have been employed to constrain the primordial power spectrum through nonobservation of their predicted signals in a variety of contexts. For thermal-relic dark matter models, such as the weakly interacting massive particle (WIMP) model Jungman et al. 1996; Bergström 2000; Bertone et al. 2005, the dark matter annihilation rate is greatly increased by the compactness of the assumed $\rho\propto r^{-9/4}$ density profile. The strongest constraints therefore come from nonobservation of the strong gamma-ray Josan and Green 2010; Bringmann et al. 2012; Nakama et al. 2018 or neutrino Yang et al. 2013a; Yang and Qin 2017 signals that are expected from WIMP annihilation within such dense clumps. These annihilation signals would also lead to other observable effects, such as heating of the intergalactic medium Zhang 2011; Yang 2016 and galactic gas Clark et al. 2017a and interactions with the CMB or other background photons Yang et al. 2011c; Yang et al. 2013b; Beck and Colafrancesco. The primordial power spectrum can also be constrained by searching for UCMHs using astrometric microlensing Li et al. 2012 or macrolens distortions Zackrisson et al. 2013 or by constraining the UCMH abundance using pulsar timing arrays Clark et al. 2016a; *clark2016erratumI; Clark et al. 2016b; *clark2017erratumII.

However, with the exception of Ref. Nakama et al. 2018, all of these works used only minihalos that form at $z\gtrsim 1000$ and assumed these halos possessed the $\rho\propto r^{-9/4}$ density profile. We showed in Sec. III that minihalos forming in an enhanced power spectrum, even UCMHs forming at $z\gtrsim 1000$, develop significantly shallower profiles. We also found that younger minihalos possess the same density profiles as the oldest ones. In this section, we explore the impact of this discovery. The observational signatures of UCMHs forming at $z\gtrsim 1000$ are weakened by the shallower density profile, but our analysis is now able to include minihalos forming at $z<1000$. As we saw in Fig. 4, these younger minihalos are far more abundant than the rare UCMHs.

Broadening to the entire population of minihalos brings new challenges. Minihalo-minihalo mergers will reduce the minihalo count and alter their density profiles Ogiya et al. 2016, and tidal interactions within galactic structures will have more impact on the shallower density profiles Berezinsky et al. 2008. These considerations are beyond the scope of this paper, but to motivate further study, we calculate in this section how the new minihalo picture directly alters previous constraints on the power spectrum derived from UCMHs. To this end, we focus on the upper bound derived by Bringmann, Scott, and Akrami Bringmann et al. 2012 (hereafter BSA) based on the gamma-ray signal from WIMP annihilation within UCMHs. In our calculation, we adopt our new minihalo model from Sec. III.4 but otherwise replicate BSA’s calculation as closely as possible. In particular, we employ the same Fermi-LAT data and, like BSA, when deriving bounds from diffuse emission, we consider only a Galactic contribution and neglect the possibility of improving constraints by including extragalactic sources (e.g., Refs. Yang et al. 2011a; Nakama et al. 2018).

The derivation of a constraint on the power spectrum using WIMP annihilation in minihalos proceeds in three steps:

The annihilation signal of a minihalo is calculated.

A constraint on the number density of minihalos is calculated from the nonobservation of such an annihilation signal.

The number density constraint is converted into a constraint on the primordial power spectrum using the statistics of Gaussian random fields.

In past studies, such as BSA, UCMHs are assumed to collapse at $z\simeq 1000$, so the UCMH luminosity is solely a function of its size. We now have the machinery to study minihalos collapsing at any redshift, so we calculate the minihalo luminosity $L$ as a function of both its formation time and the scale of the density fluctuation that sourced it.

We assume that the density profile of a minihalo follows the Moore fitting form given by Eq. (4). In addition, we found in Sec. III that the Moore fitting parameters $r_{s}$ and $\rho_{s}$ can be predicted from the halo formation time as

The gamma-ray signal $L$ of a halo with density profile $\rho(r)$ may be calculated as

For now, we keep our calculations model-independent and return to Eq. (9). Equation (9) diverges for a $\rho\propto r^{-3/2}$ profile, but this implies that annihilations would have smoothed out the central cusp within some small radius. We use the standard estimate Berezinsky et al. 1992

IV.2 Halo abundance

We use observational detection limits to constrain the minihalo number density based on the luminosity we computed above. Following BSA, we employ two approaches that utilize different observations and yield different constraints. First, we treat the minihalos as point sources and use their nonobservation to constrain their number density. Next, we consider the diffuse background flux from minihalos within the Milky Way and use the observed background gamma-ray flux to constrain the minihalo number density.

which gives us the prescription for constraining the local number density of minihalos based on the nonobservation of point sources. Due to the dependence of a minihalo’s luminosity on its formation time, we constrain a formation time-weighted density instead of a total UCMH density.

IV.2.2 Diffuse flux

where $s$ is the line-of-sight distance. Here we have inserted the factor $\rho(s)/\bar{\rho}_{0}$ to account for the Milky Way density field at distance $s$ from Earth. Following BSA, we are only interested in the Galactic contribution to the diffuse flux, so we truncate the density field beyond the Milky Way, eliminating the need to redshift distant sources. The minihalo abundance constraint from the diffuse flux at angle $\theta$ to the Galactic center now becomes

IV.3 The power spectrum

For our calculation, we employ a more direct approach. Bardeen, Bond, Kaiser, and Szalay Bardeen et al. 1986, hereafter BBKS, formulated a description of the statistics of peaks in a Gaussian random field. In this approach, each peak in the primordial density field is to be identified with a halo at late times. Spiked power spectra are very natural arenas for peak theory because they possess finite integrated power and generate peaks around a particular scale, so it is not necessary to use any smoothing filter. To represent a spike centered at wave number $k_{s}$, we consider a delta-function matter power spectrum of the form ${\mathcal{P}(k)\propto D(a)^{2}k_{s}\delta(k-k_{s})}$, where $D(a)$ is the linear growth function. We will see in Sec. IV.4 that the minihalos contributing to our power spectrum constraint form in matter domination, so $D(a)=a$, and we may write

where $\mathcal{A}$ parametrizes the integrated area of the spike. We use the BBKS formalism to calculate the number density of peaks with $\delta>\delta_{c}$, where ${\delta_{c}=1.686}$ is the linear collapse threshold. The identification of these peaks with halos leads to a number density $n$ that increases in time solely due to halo formation, implying we can differentiate it with respect to scale factor $a$ to obtain the distribution of halos by formation time. A disadvantage to this procedure is that minihalo-minihalo mergers are not automatically accounted for and must be handled separately, a task that is beyond the scope of this paper However, as we will see in Paper III, mergers become rare as the power spectrum spike is narrowed. With a delta-function spike in the primordial power spectrum, we suspect that they are negligible..

with amplitude $\mathcal{A}_{0}$. The transfer function given by Eq. (40) converts the bound on $\mathcal{A}$ into a bound on $\mathcal{A}_{0}$.

Figure 12 shows the resulting upper bound on the integrated area $\mathcal{A}_{0}$ of a spike in the primordial curvature power spectrum if the spike is located at wave number $k_{s}$. We show the constraints from point sources and diffuse flux separately, and the shaded regions are forbidden. We wish to compare this constraint to the upper bound derived in BSA under the UCMH picture, but BSA assumed a locally scale-invariant power spectrum for their analysis. Therefore, we employ the UCMH abundance constraints in BSA to derive a constraint on the spiked power spectrum of Eq. (22). This calculation is detailed in Appendix E, and the results are plotted on Fig. 12 as dashed lines. Evidently, new minihalo constraints can be stronger than old UCMH constraints despite employing shallower density profiles. For comparison, we also show as dotted lines the upper bounds that employ the shallower density profiles while restricting to UCMHs forming by $z\geq 1000$. These bounds are calculated by altering the upper limit of the integrals in Eqs. (17) and (20). We noted in Paper I that the shallower density profiles reduce the signal from each halo by a factor of 200, and we see now that this reduction weakens the upper bound on the power spectrum by roughly a factor of two In the next section, we discuss why power spectrum constraints derived from UCMHs are so insensitive to reductions in the UCMH signal. This feature is a consequence of the restriction to halos forming at $z\geq 1000$ and is no longer applicable once all minihalos are included.. The inclusion of all minihalos, instead of only the rare UCMHs that form by $z=1000$, more than compensates for this loss.

IV.4 Discussion

for point sources in a uniform field with $\mu$ times the background density and

for diffuse sources. Here, $I_{3/2}=0.228$, $J_{3/2}=0.370$, $I_{1}=0.0477$, and $J_{1}=0.0478$ are different moments of the peak height distribution $h(\nu)$.

We first note that if we neglect logarithms Using Fig. 13, $\beta\sim 10^{5}\gg\mathcal{A}^{1/2}$, so the logarithmic dependence of Eqs. (IV.4) and (26) on $\mathcal{A}$ is weak., the constraint on $\mathcal{A}$ is proportional to $B^{-2/3}$ and hence to the $-2/3$ power of the WIMP annihilation rate within minihalos [see Eq. (13)]. This relationship implies that the upper bound on $\mathcal{A}$ is highly sensitive to the WIMP model. For example, if the annihilation cross section $\langle\sigma v\rangle$ were increased by a factor of 8, the upper bound on $\mathcal{A}$ would be reduced by a factor of 4. This behavior is a stark contrast to that of constraints in the old UCMH picture, which exhibit a very weak dependence on WIMP model (see BSA Fig. 5).

The influence of the peak population on the power spectrum constraint is encoded in the moments $I_{3/2}$, $J_{3/2}$, $I_{1}$, and $J_{1}$ of the peak distribution. These are integrals over peak height $\nu=\delta/\sigma$, and their integrands exhibit most of their support between $\nu=2$ and $\nu=4$ (see Appendix D). Consequently, the integrals in Eqs. (17) and (20) that determine the upper bounds on the power spectrum are dominated by peaks with amplitudes between $2\sigma$ and $4\sigma$, which confirms the difference in statistics from the old UCMH picture. We can also use this information to find the formation times of the corresponding halos. The upper bound on $\mathcal{A}$, which parametrizes the matter power spectrum, is shown in Fig. 13 and lies between $3\text{\times}{10}^{2}$ and $6\text{\times}{10}^{4}$. The root-mean-squared density variance of the spiked power spectrum is $a\mathcal{A}^{1/2}$ at scale factor $a$, implying that the collapse time $a_{c}$ of a peak with amplitude $\nu$ is $a_{c}=\delta_{c}/(\nu\mathcal{A}^{1/2})$. It follows that peaks contributing significantly to the power spectrum constraint would have formed between $z=20$ and $z=600$, confirming that matter domination was a valid approximation.

Finally, we remark on a similar constraint that was recently published by Nakama, Suyama, Kohri, and Hiroshima in Ref. Nakama et al. 2018 (hereafter NSKH). Unlike previous UCMH works, this work did not employ the $\rho\propto r^{-9/4}$ density profile. Instead, NSKH assumed that minihalos developed NFW density profiles, and like us, they constrained a delta-spiked power spectrum. Thus, a comparison is warranted: despite assuming shallower density profiles, NSKH were able to derive comparable or stronger constraints on the integrated area $\mathcal{A}_{0}$ ($\mathcal{A}^{2}$ in their paper) of the power spectrum spike.

Their assumption of a faster concentration growth rate likely explains why the constraints in NSKH are so strong. The annihilation rates within such concentrated minihalos would be greatly enhanced. However, NSKH also employed the diffuse gamma-ray flux from extragalactic sources, whereas we, for parity with BSA, assumed only Galactic sources. This could contribute to the strength of their constraints: as we discuss above, the upper bound on the power spectrum is now highly sensitive to observational limits on the gamma-ray flux.

V Conclusion

Expanding on the results of Paper I, we have shown that the minihalos that form due to a power spectrum enhancement do not develop the single-power-law ${\rho\propto r^{-9/4}}$ density profile even when they form by $z=1000$ from extremely rare ($6.8\sigma$) peaks. Instead, they develop density profiles with inner power-law indices between $-3/2$ and $-1$, depending on the range of scales that are enhanced. This finding contradicts the assumption made by previous UCMH work Josan and Green 2010; Bringmann et al. 2012; Li et al. 2012; Yang et al. 2013a; Yang et al. 2013c; Yang 2014; Clark et al. 2016b; *clark2017erratumII; Yang and Qin 2017; Aslanyan et al. 2016; Choi and Takahashi 2017, throwing into question power spectrum constraints that have been derived from this theory. However, we have also offered hope. We constructed a new model based on our simulation results for minihalos that form from a spiked power spectrum, and we calculated a new power spectrum constraint in this model using Fermi-LAT constraints on gamma rays from WIMP annihilation. The resulting upper bound on the primordial power spectrum is stronger than an equivalent constraint derived in the old UCMH picture. It turns out that the drop in signal from each early-forming halo is more than compensated by the vast increase in the number of halos that contribute to the expected gamma-ray signal.

Our constraint is specialized to a power spectrum enhanced over a narrow range of scales. Such spiked power spectra have motivations in inflationary phenomenology Salopek et al. 1989; Starobinskij 1992; Ivanov et al. 1994; Starobinsky 1998; Chung et al. 2000; Barnaby and Huang 2009; Barnaby 2010 and in nonstandard thermal histories of the Universe Erickcek and Sigurdson 2011; Barenboim and Rasero 2014; Fan et al. 2014; Erickcek 2015, but halos forming from these spectra have not been numerically studied prior to this work and Refs. Gosenca et al. 2017; Delos et al. 2018. Our model for halos forming from spiked power spectra predicts halo density profiles based on their formation time: the characteristic density is set by the background density at formation, while the characteristic scale is set by the spike scale at formation. However, we developed this model based a single power spectrum. We also neglected any scatter in the density profiles of halos forming at the same time. In Paper III, we will extend this model by quantifying its scatter and its applicability to different power spectra.

Inflationary phenomenology also includes less scale-localized power spectrum boosts such as steps or bends Salopek et al. 1989; Starobinskij 1992; Ivanov et al. 1994; Starobinsky 1998; Joy et al. 2008; Silk and Turner 1987; Polarski and Starobinsky 1992; Adams et al. 1997. We have found that the halos forming from a stepped power spectrum develop the same density profiles as later-forming galaxy-scale halos. Consequently, there is already a vast body of literature on modeling the density profiles of these halos (e.g. Navarro et al. 1996; Navarro et al. 1997; Jing 2000; Eke et al. 2001; Bullock et al. 2001; Zhao et al. 2003; Avila-Reese et al. 2005; Neto et al. 2007; Gao et al. 2008; Duffy et al. 2008; *duffy2011erratum; Macciò et al. 2008; Zhao et al. 2009; Muñoz-Cuartas et al. 2011; Klypin et al. 2011; Giocoli et al. 2012; Prada et al. 2012; Ludlow et al. 2012; Ludlow et al. 2013; Bhattacharya et al. 2013; Meneghetti and Rasia; Dutton and Macciò 2014; Ludlow et al. 2014; Diemer and Kravtsov 2015; Correa et al. 2015; Okoli and Afshordi 2016; Klypin et al. 2016; Angel et al. 2016), and we expect that these results may be adapted toward constraining steps or bends in the power spectrum.

Our constraint also employed only gamma rays from WIMP annihilation in Galactic or near-Galactic sources. We made this restriction in order to facilitate a direct comparison between an upper bound on the power spectrum derived in the shallower minihalo picture and an equivalent bound derived using the results of Ref. Bringmann et al. 2012, which made the same restriction, used UCMHs forming at $z\geq 1000$, and assumed the $\rho\propto r^{-9/4}$ profile. As a result, we have left open the possibility of immediately improving the power spectrum constraints by considering the diffuse annihilation signal from extragalactic minihalos as Refs. Yang et al. 2011a; Nakama et al. 2018 do. We have also not explored the impact of the shallower density profiles on gravitational probes such as astrometric microlensing Li et al. 2012 and pulsar timing arrays Clark et al. 2016a; *clark2016erratumI; Clark et al. 2016b; *clark2017erratumII.

Most pressingly, we neglected the influence of disruptive events on the minihalo abundance and their density profiles. Minihalo-minihalo mergers are one such disruptive event. They can be counted by means of Press-Schechter theory Lacey and Cole 1993 However, the self-consistency of Press-Schechter merger rates is questioned in Ref. Benson et al. 2005; see also Ref. Neistein and Dekel 2008 for a counterpoint. with a sharp $k$-space filter Barkana et al. 2001; Benson et al. 2013; Schneider et al. 2013, but their physical impact, especially on minihalos with $\rho\propto r^{-3/2}$ inner profiles, is not yet well understood. Reference Ogiya et al. 2016 simulated controlled halo mergers and observed that successive mergers cause the inner density profiles of these halos to relax toward shallower forms, an effect that we confirmed. However, they also found that the merger product can have a higher central density than its progenitor halos. Moreover, for highly unequal-mass mergers, a remnant of the smaller halo is expected to survive within the larger one Berezinsky et al. 2008.

Figure 14 illustrates the possible impact of mergers on the minihalo-derived constraints on the primordial power spectrum. If we naively assume that minihalos develop NFW profiles with the same scale parameters $r_{s}$ and $\rho_{s}$, then the shallower inner profiles weaken the power spectrum bound by roughly a factor of 10. If mergers additionally halve the minihalo count, the constraint is weakened by another factor of $1.6$. We suspect that this latter constraint, depicted as the red curve in Fig. 14, represents a pessimistic estimate of how mergers may weaken the upper bound on the power spectrum. First, we neglected the increased central densities that can result from mergers. Second, we assumed that all halo profiles fully relax to NFW form, while Ref. Ogiya et al. 2016 showed that such relaxation is a gradual process occurring over multiple mergers. Finally, mergers are relatively rare in spiked power spectra, and even if more than half of the minihalo population is ultimately destroyed by mergers (not even becoming subhalos), smaller halos, which contribute less to observational signals, are preferentially destroyed. Hence, we expect that a careful accounting of mergers will produce a result between the black and red curves of Fig. 14.

Disruption of minihalos can also occur by the tidal influence of larger galactic potentials or by high-speed encounters with objects, such as other substructure or stars, within these galactic potentials. This topic has been studied in a number of previous works, such as Refs. Diemand et al. 2005; Angus and Zhao 2007; Berezinsky et al. 2006; Green and Goodwin 2007; Goerdt et al. 2007; Zhao et al. 2007; Schneider et al. 2010, and a recent overview of such disruptive processes can be found in Ref. van den Bosch et al. 2017. It is also possible to bypass the issue of galactic disruption by only considering minihalos that have not accreted onto galactic halos, as Ref. Nakama et al. 2018 does.

Our goal in this paper was to show that despite them not possessing the $\rho\propto r^{-9/4}$ density profiles that were previously assumed, minihalos are still able to yield competitive constraints on the primordial power spectrum. The plethora of minihalos that now contribute to observational signals counteracts the loss of signal from the rarest of these halos. This finding motivates their further study, and we have discussed avenues for future work. With a better understanding of disruptive processes, minihalos can become strong and robust cosmological probes.

Appendix A Simulations prior to the matter-dominated era

For our numerical experiments, we employ a modified version of Gadget-2 that includes a smooth radiation component, and we begin the simulations long before matter-radiation equality. The simulation starting redshift of $z=8\times 10^{6}$ is necessary so that our enhanced fluctuations are still in the linear regime with amplitude $\delta\lesssim 0.1$. However, the assignment of initial particle velocities is more complicated in this picture than during matter domination, and we use the Zel’dovich approximation to compute them in the following way.

We begin with the density contrast field $\delta(\mathbf{q})$ as a function of the comoving grid coordinate $\mathbf{q}$. We wish to convert this description into a comoving position field $\mathbf{x}(\mathbf{q})$ and velocity field $\dot{\mathbf{x}}(\mathbf{q})$ treating $\mathbf{q}$ as a Lagrangian coordinate assigned to each particle. The position calculation proceeds by writing

where the displacement vector $\mathbf{s}$ is related to $\delta$ at linear order by $\nabla\cdot\mathbf{s}=-\delta$. If we assume $\mathbf{s}$ is irrotational, then $\mathbf{s}\propto\nabla\delta$, and the Fourier-transformed quantities are related by

Equations (27) and (28) determine the initial positions and are valid regardless of the composition of the universe.

We next turn to the velocity field $\dot{\mathbf{x}}(\mathbf{q})$ or its Fourier transform $\dot{\mathbf{x}}(\mathbf{k})=\dot{\mathbf{s}}(\mathbf{k})$. Let $t_{0}$ be the time at which we are generating initial conditions, and write $\mathbf{s}$ as a function of time, using a new function $D(\mathbf{k},t)$ to encode its time dependence:

We define $\mathbf{s}(\mathbf{k})\equiv\mathbf{s}(\mathbf{k},t_{0})$ so that $D(\mathbf{k},t_{0})\equiv 1$. During matter domination, $D(\mathbf{k},t)=a(t)/a(t_{0})$ independent of $\mathbf{k}$, but radiation complicates the picture. However, it is evident from Eq. (28) that $\mathbf{s}$ and $\delta$ evolve identically in time, implying $D(\mathbf{k},t)=\delta(\mathbf{k},t)/\delta(\mathbf{k},t_{0})$. The initial velocity becomes

To test the modified simulation code and initial conditions, we compare simulation results to linear theory. We produce a matter power spectrum at $z=8\times 10^{6}$ using the procedure described in Sec. II.2, but we leave it unenhanced so that density contrasts near matter-radiation equality are well in the linear regime. We draw initial conditions from this power spectrum in a (7.4 kpc)3 periodic box and then evolve this box to $z=996$ using our modified version of Gadget-2 The fluctuations drawn from an unenhanced power spectrum have amplitude $\delta\sim 10^{-3}$ at $z=8\times 10^{6}$, which results in extremely small particle accelerations. To evade errors resulting from floating-point precision, we also set Gadget-2 to use double-precision arithmetic. All simulations in this paper employ this setting.. All simulation parameters are the same as those of the reference simulation described in Appendix B. Figure 15 shows the growth of the power spectrum during this simulation. It matches the linear-theory prediction of Eq. (II.2), including the scale-dependent growth. Note that without the radiation component, the power spectrum would have instead grown by a factor of about $6\times 10^{7}$.

As another demonstration, we also evolve the initial density field to $z=996$ using linear theory by applying the evolution specified by Eq. (II.2) to the Fourier-transformed density field; we may then compare the resulting density field to the one evolved using Gadget-2. In Fig. 16, we plot a slice of the density field at $z=996$ evolved using both methods. Our modified version of Gadget-2 with initial conditions described above successfully reproduces the results of linear theory.

Appendix B Convergence testing and simulation parameters

We carry out a reference simulation run and five convergence-testing runs: one with improved force accuracy, one with improved integration accuracy, two with respectively increased and reduced softening scales, and one with higher particle count. Our parameter choices for these runs are summarized in Table 1 and described below. We refer the reader to Ref. Springel 2005 for further detail on these parameters.

Gadget-2 computes the long-range force using a particle mesh and the short-range force using an octree. The parameter $r_{s}$ determines the splitting scale in units of mesh cells. In our reference run, we set $r_{s}=1.25$ mesh cells. In the force-accuracy run, we increase this to $r_{s}=2.5$ mesh cells, which effectively leaves the splitting scale unchanged since we also double the mesh frequency.

Gadget-2’s short-range (octree) force calculation only opens a tree node if the estimated force error from truncating it is less than $\alpha$ times the estimated total force. In our reference run, we set $\alpha=0.005$. In the force-accuracy run, we set $\alpha=0.002$ to increase the accuracy of the short-range force.

Gadget-2 uses individual adaptive time steps with an accuracy parameter $\eta$. Roughly, the time step is set so that the maximum displacement due to a particle’s acceleration over one time step is smaller than $\eta$ times the force-softening scale. In the reference run, we set $\eta=0.025$, while in the integration-accuracy run, we set $\eta=0.01$ to reduce the particle time steps.

B.2 Procedure, results, and discussion

We conduct convergence testing on the primary simulation run of Sec. III. The density field is generated with $1024^{3}$ cells but reduced to $512^{3}$ cells for all but the high-particle-count run by averaging 8 neighboring cells. Each convergence run is carried out as in Sec. III with only the simulation accuracy parameters changed. We study here the spherically averaged density profile $\rho(r)$ of the UCMH at $z=100$.

There is also an obvious lower limit to the range of radii over which we expect our results to be representative, and that is where the radius is equal to $2.8\epsilon$, where $\epsilon$ is the gravitational softening length. Below this radius, all forces are non-Newtonian, and the density profile will unphysically flatten out.

Our main results are shown in Fig. 17. The dotted lines indicate the density profile of the reference simulation run, with the gray shading representing the root-mean-squared variance in snapshots between ${z=100}$ and ${z=99}$. The solid lines with colored shading depict alternate runs.

B.2.2 Integration accuracy

B.2.3 Force softening

B.2.4 Particle count

These fluctuations may be related to the artificial fragmentation of filaments that occurs in simulations with a small-scale cutoff in the power spectrum Angulo et al. 2013; Lovell et al. 2014. We observe artificial fragmentation in our simulations using the spiked power spectrum, as evidenced in Fig. 20, which shows one of the filaments connected to the UCMH for different simulation parameters. Like the density fluctuations, the frequency and size of these fragments is correlated with the simulation particle resolution, while their positions vary with force-accuracy parameters. The fluctuations in the density profile could be caused by the accretion of artificial fragments.

B.3 The smallest resolved radius

We can double the resolution by employing the high-particle-count simulation run, but we would like to go still deeper into the halo. While it is computationally challenging to simulate the full box with more than $1024^{3}$ particles, it is also unnecessary. A common practice in N-body simulations is to resample the halo progenitor at higher particle resolution and embed this high-resolution region into the same periodic box. Because the halos we consider are much more isolated than halos in a hierarchical growth picture, we need not even go this far: we can simply isolate a sphere around the halo progenitor and use vacuum boundary conditions. Figure 21 shows the comparison between the periodic box with $1024^{3}$ particles and an otherwise identical simulation of a vacuum-bounded sphere of radius 0.92 kpc around the UCMH. The spherical region is depicted in Fig. 2. The sphere requires only $1/122$ as many particles for an identical result. We therefore exploit this method to simulate the UCMH in the primary simulation at $64\times$ particle density and the other UCMHs at $8\times$ particle density relative to the reference simulation with $512^{3}$ particles.

B.4 Summary of simulation choices

We now summarize the simulation parameters we use to study the UCMHs in Sec. III. Aside from the particle count $N$, all simulation runs use the reference parameters of Table 1. We employ two classes of simulation region: a comoving cube with periodic boundary conditions or an isolated comoving sphere with vacuum boundary conditions. Table 2 shows the particle counts and simulation region sizes for all simulation runs. The mass $m$ of the simulation particle is also shown for clarity.

All density profiles are averaged over 16 snapshots within $1/100$ of a Hubble time to suppress fluctuations. Density profiles are binned logarithmically in intervals separated by a factor of 1.1 (corresponding to an interval of 0.041 in $\log_{10}r$), but we have checked that the results depend negligibly on this choice.

Appendix C Constraining point-source abundance in the Milky Way

In Sec. IV, we defined $\mu(d)=3M(d)/(4\pi d^{3}\bar{\rho}_{0})$, where $M(d)$ is the dark matter mass contained within distance $d$ of Earth. If we are at distance $r_{0}$ from the Milky Way center, then we may write

Appendix D Constraining the power spectrum using the statistics of peaks

where the function $f(\nu)$ is defined by BBKS Eq. (A15).

For the general case of point sources with nonconstant $\mu(d)$, we have little choice but to numerically invert the integral in Eq. (17) to obtain an upper bound on $\mathcal{A}$ as a function of $k_{s}$. However, in a limiting case where $\mu$ is constant, or to derive a bound from the diffuse flux, we can fully extract the $\mathcal{A}$-dependence from the integral. In these cases, we have an integral of the form

with $p=3/2$ or $p=1$ for point sources or the diffuse flux respectively, and where $I_{p}$ and $J_{p}$ are defined as

In the first line, we use Eq. (35) with $\nu=\delta_{c}/(\mathcal{A}^{1/2}a)$. We saw in Sec. IV.4 that the integral in Eq. (D) is dominated by minihalos forming at $z\gtrsim 20$, so we exploit the negligible contribution to this integral of minihalos forming at $a>1$ to extend the lower limit of the integral on the right-hand side to $\nu=0$. In the second line, we specialize to the Moore density profile using Eq. (13). In the last line, we take advantage of the limited support of $h(\nu)$ to claim that $\ln\nu\ll\ln\left(B\delta_{c}/\mathcal{A}^{1/2}\right)$ so that we can use a binomial expansion (and this is exact for $p=1$). Note that the integrands in Eq. (38) tell us the range of peaks that are relevant to constraints: as we see in Fig. 22, most of their support lies in peaks between roughly $2\sigma$ and $4\sigma$.

These expressions follow from Eqs. (17), (20), and (D). The numbers $I_{3/2}$, $J_{3/2}$, $I_{1}$, and $J_{1}$ have approximate values

Equations (IV.4) and (26) are now algebraic equations for the upper bounds on $\mathcal{A}$.

The constraint on $\mathcal{A}$ is not our final goal, but it is close. $\mathcal{A}a^{2}$ is the power associated with the spike in the (linear) matter power spectrum during matter domination. We seek instead the power $\mathcal{A}_{0}$ associated with the spike in the primordial curvature power spectrum. These quantities are related by a transfer function such as Weinberg 2008

which relates the matter density contrast $\delta$ during matter domination to the primordial curvature fluctuation $\zeta$. Here

yields the desired constraint on the primordial power spectrum.

Appendix E The UCMH constraint on a spiked power spectrum

Bringmann, Scott, and Akrami Bringmann et al. 2012 (BSA) calculated an upper bound on the number density of UCMHs (as a function of scale wave number $k$) using the $\rho\propto r^{-9/4}$ density profile from Ref. Ricotti and Gould 2009. They then converted this constraint into an upper bound on the primordial power spectrum under the assumption of local scale invariance. We followed the calculation in BSA as closely as possible in Sec. IV so as to facilitate a direct comparison in constraining strength between our new minihalo model and the old UCMH model. However, because a spiked power spectrum does not exhibit local scale invariance, we must return to the UCMH abundance constraint in BSA and convert it into a constraint on the delta-spiked power spectrum given by Eq. (22). We show that calculation here.

Here we have employed $R=1/k$, where $R$ is the comoving radius of the precursor overdense region; this is the same relation BSA used. The UCMHs are taken to follow the dark matter distribution, so $n$ and $\rho_{m}$ are the comoving background UCMH number density and matter density respectively.

This procedure has given us a constraint $n$ on the comoving number density of halos of scale wavenumber $k$ forming at $z\gtrsim 1000$. In the delta-spiked power spectrum given by Eq. (22), all halos form from fluctuations with wavenumber $k_{s}$, the wavenumber of the spike, so we will take $k=k_{s}$ in Eq. (43). We can then use Eq. (34) with $a=10^{-3}$ to convert this upper bound on the abundance of halos that form at $z\geq 1000$ into a constraint on the primordial power spectrum, which is shown in Fig. 12.

References