Compact Dark Matter Objects Via $ N $ Dark Sectors

Compact Dark Matter Objects via N Dark Sectors

Gia Dvali,1, 2 Emmanouil Koutsangelas,1, 2, ∗ and Florian Kühnel3 1 Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,Theresienstraße 37, 80333 München,Germany, 2 Max-Planck-Institut fürPhysik, FöhringerRing 6, 80805 München,Germany 3 The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova University Center, Roslagstullsbacken 21, SE-10691 Stockholm, Sweden (Dated: Monday 27th April, 2020, 12:34am) We propose a novel class of compact dark matter objects in theories where the dark matter consists of multiple sectors. We call these objects N-MACHOs. In such theories neither the existence of dark matter species nor their extremely weak coupling to the observable sector represent additional hypotheses but instead are imposed by the solution to the Hierarchy Problem and unitarity. The crucial point is that particles from the same sector have non-trivial interactions but interact only gravitationally otherwise. As a consequence, the pressure that counteracts the gravitational collapse is reduced while the gravitational force remains the same. This results in collapsed structures much lighter and smaller as compared to the ordinary single-sector case. We apply this phenomenon to a dark matter theory that consists of N dilute copies of the Standard Model. The solutions do not rely on an exotic stabilization mechanism, but rather use the same well-understood properties as known stellar structures. This framework also gives rise to new microscopic superheavy structures, for example with mass 108 g and size 10−13 cm. By confronting the resulting objects with observational constraints, we find that, due to a huge suppression factor entering the mass spectrum, these objects evade the strongest constrained region of the parameter space. Finally, we discuss possible formation scenarios of N-MACHOs. We argue that, due to the efficient dissipation of energy on small scales, high-density regions such as ultra-compact mini-halos could serve as formation sites of N-MACHOs.

I. INTRODUCTION In the present paper we shall focus on such framework which goes under the name of many-species solution to To the present day the fundamental nature of the dark the Hierarchy Problem [4]. The key point is that in any matter (DM) remains one of the major mysteries in cos- low-energy effective theory that includes Ntot particle mology and particle physics. At the same time, another species, the quantum-gravitational cutoff is given by, major unanswered question in particle physics is what stabilizes the mass term of the Higgs boson against quan- MP tum corrections at a value more than 32 orders of magni- M∗ ≡ √ , (1) Ntot tude below the Planck mass. This puzzle is known as the Hierarchy Problem. Not surprisingly, the above two mys- where MP is the Planck mass. This is a fully non- teries are the main motivation behind many attempts to perturbative bound that follows from black-hole physics extend the Standard Model (SM). Obviously, most inter- and its derivation can be found in Refs. [4–8]. Hence, if 32 esting are the theories that address both puzzles simul- the theory would contain Ntot ∼ 10 different species, taneously and establish an underlying connection among the fundamental gravity scale would be M∗ ∼ 1 TeV. them. Such scenarios are extremely rare. In most of the This fact nullifies the Hierarchy Problem, since the UV- models, new long-lived particles are introduced by hand sensitive radiative corrections to the Higgs mass arising with sole purpose to account for the dark matter with- from Standard Model loops must be cutoff at the scale out addressing any other problem. Even in low-energy M∗. supersymmetry, which solves the Hierarchy Problem, the stability of the dark matter particle is not automatic and Notice, new species must be extremely weakly inter- arXiv:1911.13281v2 [astro-ph.CO] 23 Apr 2020 requires the postulation of an additional symmetry such acting with the observable particles from the SM. It is as R-parity. Of course, we cannot exclude that Nature important that this is not an additional assumption but could provide a dark matter candidate without any ad- is a constraint coming from unitarity [5,6]. This fact ditional purpose but as a theoretical guideline we wish makes the hidden species into natural dark matter can- to give priority to models that address more than one didates. In a particularly minimalistic scenario the new problem.1 species represent the exact copies of the Standard Model [4,9–11].

∗Electronic address: [email protected] 1 An example of the theory that obeys such a standard is that of tion [3] to strong-CP problem and as a byproduct provides an the axion [1,2], which is motivated by the Peccei-Quinn solu- excellent dark matter candidate. 2

We would like to note that, as explained in [9], the lat- Universe both with ordinary SM particles as well as with ter scenario provides an additional bonus by addressing the hidden sector species. the strong-CP problem without need of an axion. The Of course, other sectors could posses their own infla- idea is that the sum of CP-violating order parameters ton candidates. In particular, in the scenario with exact 32 over all 10 copies of QCD is bounded from above by copies of the SM, this is imposed by the permutation the cutoff scale. This sets the value of the QCD θ-angle symmetry. In this case, a model builder has to decide in the observable sector in an experimentally interesting whether the inflaton field is a singlet under the permu- range below the current phenomenological bound. tation group or transforms under it non-trivially. Before proceeding, we cite some other implications of The former possibility would imply that the inflaton this framework studied in Refs. [12–21]. We also note couples to all the copies with the universal strength that a possible solution with less number of copies was which by unitarity must be suppressed by 1/Ntot. Conse- discussed in Ref. [22]. Below, we shall try to keep our quently, in such a scenario after reheating all the sectors discussion maximally general and to distill the model- would end up being equally densely populated, which is independent predictions. We shall assume a framework excluded observationally. The alternative possibility is in which there exists a large number of hidden sectors that inflaton transforms under the permutation symme- that interact with the Standard Model as well as among try and thus shares the label j with the SM copies. That each other extremely weakly. At the same time, the inter- is, each SM copy possesses its own inflaton candidate. actions within a given sector are assumed to be relatively However, the permutation symmetry is spontaneously stronger than among different sectors. broken by the cosmological evolution and the sector that The solution of the Hierarchy Problem demands that drives the latest stage of inflation is the one that mat- 32 the total number of species is Ntot ∼ 10 [4]. How- ters for the reheating. By default, the sector that this ever, what matters for their astrophysical consequences particular inflaton populates most efficiently is the one is the number of species that become significantly pop- that we today call “our” SM copy. Of course, equally ulated during the cosmological evolution. We shall refer legitimate are the scenarios in which the hidden sectors to these as actualized species and denote their number are not related by any permutation symmetry with our by N. Obviously, theoretically N 6 Ntot. However, in sector and are very different. All the above-mentioned practice the number of actualized species could be much unitarity constraints, however, hold even in such cases smaller. and the reheating process must obey them. The reason is the following. As was shown in Ref. [5] The question now is: how densely are the other sectors and in more details in Ref. [6], unitarity imposes severe populated? The relevant decay and scattering processes constraints on the strengths of the couplings among dif- are: ferent species. The point is that for any particle the maximal possible strength of interaction with the major- • Inflaton → Us, ity of species is suppressed by M 2/M 2 = 1/N . That ∗ P tot • Inflaton → Them, is, the particles from a given sector, e.g. the SM, are allowed to couple relatively strongly only with the few • Us → Them, “privileged neighbors”, while the coupling to the rest is extremely weak (i.e., 1/Ntot-suppressed). In the opposite where obviously “us” and “them” denote the SM and hid- case, unitarity would be violated by loop expansions and den sectors, respectively. Notice, the process “them → scattering amplitudes. us” can be dismissed due to very strong dilution of the In other words, if we think of the species label j = individual hidden sectors. Thus, there exist two mechanisms for populating the dark sectors: 1,...,Ntot as a discrete coordinate in an imaginary space of species, the unitarity imposes a certain well-defined sense of locality and distance in this space [6]. 1. Direct: Through the decay of the inflaton into the dark species (Inflaton → Them). The above knowledge has important implications once the many-species framework is embedded in the cosmo- 2. Indirect: Through re-scattering of particles of the logical context of the inflationary scenario [11] (see also visible sector into the dark ones Refs. [18–22] for some related analysis and [23] for grav- (Inflaton → Us → Them). itational inflaton decay). As it is well-known, in the latter scenario, the key Notice, although the latter mechanism of populating player is a slowly-rolling inflaton field that serves as a dark sector can also be operative in standard single-sector clock controlling the end of inflation. Through its oscil- dark matter scenarios, it is the many-species framework lations and subsequent decay, this field also ensures the that makes it especially efficient due to enormous 1/Ntot- conversion of the vacuum energy density into radiation. suppression of the inverse process. Since, as the fact of Nature, our sector is most densely The direct population process is determined by the de- PNtot populated, the inflaton field must couple strongest to our cay rate Γinfl → them = j=1 Γinfl → j, where Γinfl → j rep- sector. After inflation, this field decays and populates the resents the decay rate of the inflaton into the j-th hidden 3 species. Likewise, the indirect process “us → them” is Intuitively, it is clear that the mass spectrum of N- controlled by the total re-scattering of SM quanta into MACHOs must be different from that of a single-sector the hidden sectors that is given by summing over the object. Putting the particles from the N sectors into a PNtot certain volume, gravity will try to collapse the system individual scattering rates Γus → them = j=1 Γus → j. The unitarity puts certain generic constraints such as while the pressure from each sector will try to counteract Γinfl ≡ Γinfl → them + Γinfl → us . minfl, where minfl is the the collapse. Eventually, the total pressure can balance mass of the inflaton. This constraint means that during gravity, leading to an equilibrium configuration. The fact the reheating process the inflaton represents a weakly- that in this case each particle can only interact with a interacting degree of freedom and can be consistently small fraction of the present particles, leads to a strong treated within effective field theory without the risk of reduction of the pressure as compared to the single sector violating perturbative unitarity. case. At the same time, gravitational forces remain un- affected. In other words, each particle feels the pressure The second constraint is Γus → them . TSM, where TSM is the temperature in the SM sector at which the re- only from the species of its own sector, while it expe- scattering into hidden species takes place. Furthermore, riences a universal gravitational attraction from all the in order not to destroy the success of standard big bang other sectors. Consequently, an equilibrium configura- nucleosynthesis, we assume that the reheating tempera- tion can only be achieved for much smaller masses and thus much smaller radii than in ordinary single-sector ture TR is higher than the big bang nucleosynthesis temperature, ∼ MeV. Then, at the moment of reheating, models. the rates must satisfy: Γinfl → them Γinfl → us. This is Compact DM structure not only represent a power- because we know that during nucleosynthesis Universe’s ful tool to discriminate between classes of DM models, energy balance was dominated by the energy density of but could also provide new information on the spectrum radiation of the SM species. Correspondingly, the total of inflation. For instance, this is the case for primordial energy density deposited by the inflaton into the hidden black holes (PBHs) (see Ref. [25] for a review). The stan- sector must be negligible as compared to the energy den- dard scale-invariant inflationary power spectrum results sity deposited into the SM particles. Of course, later, in essentially zero PBHs in the present day [26]. There- after the Universe cools down and crosses into matter fore, the discovery of such an object would require an domination, the integrated energy density of matter from additional feature in the standard power spectrum. N- the hidden sector must become a few times larger than MACHOs behave similar but are easier to produce than the density in the observable sector, in order to account PBHs since they do not require that high primordial over- for current dark matter density. densities. Subject to the above constraints we can consider differ- Moreover, in light of recent discoveries by the Laser ent cosmological scenarios. One regime is achieved when Interferometer Gravitational Wave Observatory (LIGO) [27, 28], a promising signature resides in the form of grav- only a relatively small number of sectors N Ntot is actualized while the rest remains empty. Another pos- itational waves. If future generations of gravitational- sibility is to assume that no privileged neighbours exist wave experiments would observe a signal from a com- and that all the dark sectors are equally populated. In pact object merger with neutron-star-like compactness the latter case, each dark sector is ∼ 10−32 times more but different mass, then a new kind of compact object dilute than our sector. Of course, more involved scenarios must exist. This object would most likely consist of non- in which the partial density is some non-trivial function baryonic DM, due to the lack of alternatives in the SM. of species index j are also possible. Additionally, if the number of violent astrophysical processes happening today in the dark sector is of the same In the present paper, we shall try to keep our discussion order as those happening in ours, then the total num- maximally open, while keeping in mind the above general ber of gravitational-wave sources increases by the ratio constraints. For a more detailed discussion of concrete of DM to ordinary matter [24]. cosmological realizations of many-species dark matter, we refer the reader to Ref. [11]. This paper is structured as follows: In Sec.II the Ein- stein equations are solved, resulting in the solution we The crucial difference between these theories and tra- call N-MACHOs. The calculations are first performed ditional DM models is that DM is able to dissipate energy for equal particle content and equal particle density in due to the intrinsic interactions in each sector, possibly each sector. Later they are generalized to non-equal par- enabling the formation of non-baryonic compact objects. ticle content and particle density. SectionIII discusses Analogous possibilities were already realized in earlier some novel stable bound-states in the particle spectrum papers [10, 24] without any detailed analysis. that emerge in the many species scenario. In Sec.IV In the current paper we shall investigate how various we elaborate on the possibility of radiation into dark dark sectors can collectively form compact objects. We photons during merger processes involving N-MACHOs. shall call these objects N-MACHOs. Furthermore, we In Sec.V the constraints on N-MACHOs are presented, will remain agnostic about the origin of the different dark while Sec.VI discusses potential formation scenarios. Fi- sectors. We simply consider N gravitationally-coupled nally, in Sec.VII we summarize our results and give an dark sectors with intrinsic SM-strength interactions. outlook. 4

II. N-MACHOS supplied by the three brackets. In order to solve it, an equation of state (EOS), i.e., the pressure Ptot(Stot, ρtot) A. Hydrostatic Equilibrium as a function of the entropy Stot and the density ρtot, needs to be speciﬁed.

In this Subsection, we discuss the equations of hydro- In this article, we will only consider polytropes in the static equilibrium in General Relativity in the presence Newtonian limit – so called Newtonian polytropes – since of N matter sectors. these provide a simplified model for the stellar interior. First, it should be noted that the Einstein field equa- They are characterized by a polytropic EOS, i.e., tions only depend on the total energy-momentum ten- P = K ργ , (6) sor. By assumption, the different sectors interact only tot tot tot via gravity, which at the energies of interest is extremely 2 where γ and Ktot are constants. Physically speaking, a weak . Therefore, with a good approximation, the total polytropic EOS describes an object with uniform entropy, energy-momentum tensor can be decomposed as, which, for example, is the case if the latter has effectively N zero temperature or is in convective equilibrium. These X properties may sound highly idealistic, but they often (Ttot) = T , (2) µν µν,j give a right order-of-magnitude estimate. For example, j=1 the former condition is given in white dwarfs or neutron where the index j labels the individual sectors (or stars, while the latter is realized in supermassive stars. species). Following the standard approach, we will as- Furthermore, we assume that the chemical composition is sume that each sector behaves as a perfect fluid, which uniform, so that one polytropic EOS describes the whole in the rest frame takes the following form, object.

(Ttot)µν = diag(−ρtot,Ptot,Ptot,Ptot) All properties of the N-MACHOs are then determined by the central density ρ(0) and the only free parameter in N X (3) our theory, N. The non-relativistic TOV equation in the = diag(−ρj,Pj,Pj,Pj) , presence of N sectors has exactly the same overall form j=1 as in case of a single sector. Therefore, the N-sector solution will be of the same form as the single-sector one. where ρ and P are the total energy density and the tot tot In other words, the N-sector solution is also a Newtonian total pressure, respectively. polytrope. Since these are well-studied astrophysical ob- Solving Einstein’s equations with the standard spher- jects, we summarize respective details in App.A (see for ical ansatz (i.e., neglecting rotation) yields the Tolman- instance Ref. [30] for a pedagogical treatment). Oppenheimer-Volkoﬀ (TOV) equation dPtot(r) M(r) Ptot(r) = − 2 2 ρtot 1 + dr MP r ρtot(r) 3 −1 4π r Ptot(r) 2M(r) B. Solution for Equal Sectors × 1 + 1 − 2 , M(r) MP r (4) As stated previously, a polytropic EOS can be thought where of as describing an isentropic stellar object. The key Z r point now is, that if an object composed of N exclusively 0 02 0 M(r) ≡ dr 4π r ρtot(r ) . (5) gravitationally-interacting components is isentropic, then 0 each component has to be isentropic as well. Conse- The TOV equation is the fundamental equation of New- quently, each sector and the full system possess a poly- tonian astrophysics with General Relativity corrections tropic EOS with the same exponent γ. Let us, for now, assume an equal particle content in each sector, or equivalently, the universal polytropic constant K. That is,

γ Pj = Kρj . (7) 2 This is true even though the gravitational self coupling and/or coupling among few nearest neighbour species become strong In the presence of N purely gravitationally-interacting around energies M∗ ∼ TeV. The reason is the same as the one explained in [29] in the context of large extra dimensions: Strength- sectors, the total pressure satisfies, −1 −17 ening gravity at distance of M∗ ∼ 10 cm has a negligible effect on the gravitational dynamics of a macroscopic object of N N −1 X X γ γ size R M∗ . For such objects, the treatment of gravity within Ptot = Pj = K ρj ≡ Ktot ρtot . (8) standard Newton-Einstein theory is a justified approximation. j=1 j=1 5

From this we see that the inﬂuence of the N sectors is where ρcr is the critical density at which pF = mn, and encoded in the quantity the EOS is given by

PN γ 2 5/3 K j=1 ρj 5/3 1 3π Ktot = γ . (9) PF = KF ρ ,KF ≡ 2 . (13) ρtot 15π mn mn Assuming that the sectors have equal particle densities, This is the EOS of a Newtonian polytrope with poly- i.e., tropic constant KF and polytropic index γ = 5/3. How- ever, since the quantities appearing in the non-relativistic N X TOV equation are the total pressure and the total energy ρtot = ρj ≡ Nρ , (10) density, the total EOS of the system is needed. Summing j=1 over all the copies gives we have 5/3 1 5/3 Ptot = NPF = NKF ρ = KF ρtot , (14) K N 2/3 Ktot = . (11) N γ−1 which, as expected, again yields a Newtonian polytrope. Inserting this in the expressions for the radius and For the latter (using again App.A) we obtain the mass of a generic Newtonian polytropic object (see ρ(0)−1/6 1 Eqs. (A3) and (A4), respectively in App.A) we ﬁnally R ∼ 106 cm √ , (15a) arrive at ρcr N 1 and R ∼ R0 √ , (12a) N ρ(0)1/2 1 M ∼ M √ . (15b) and ρcr N

1 With this, the total density is given by M ∼ M0 √ , (12b) N M g ρ(0) ρ = ∼ 1015 N, (16) tot 4π 3 3 R cm ρcr where R0 and M0 depend on the particle content and the 3 nature of pressure in each sector. Thus, increasing the while the density per sector is number of sectors√ reduces the mass of a stellar object by a factor 1/ N . What is remarkable is that this factor ρ g ρ(0) is independent of γ and thus is universal for all possi- ρ = tot = 1015 . (17) N cm3 ρ ble stellar dark matter structures. Given the available cr range for N motivated by the Hierarchy Problem [11], The central density of each sector should be around nu- the potential impact is dramatic. 14 3 clear density, ρnucl ≈ 10 g/ cm , in order for the degen- Let us now apply the above results to the case of N eracy pressure to be dominant. This roughly sets exact SM copies. In this theory the interactions among the particles of a given copy are identical to those of the ρ(0) −1 . 10 . (18) SM. At the same time, the particles from different copies ρcr talk to each other only through gravity. As in the single-sector case, which would reproduce an In this framework the simplest example of N-MACHO ordinary neutron star, the aforementioned total density is provided by the polytropic model of an “N-neutron is close to that of a black hole. For N = 1, the usual star”. The pressure is dominated by the Fermi pressure neutron stars are retrieved, while for N → ∞ the mass P of N identical sets of non-relativistic copies of neu- F and radius vanish, showing that these objects would trons. The masses of hidden neutrons are of course equal not form. For “N-neutron stars”, the mass spectrum to the mass m of the ordinary SM neutron. Thus, each n depends only on the parameter N. For example, for neutron is kept in the bound-state due to the gravita- the cosmological scenario proposed in Ref. [11] modi- tional attraction from the rest. However, it is balanced fied with non-annihilating electrons, the window for N against the collapse due to the Fermi pressure exclusively is 1016 < N < 1032. Correspondingly, the ranges for the from the fellow neutrons belonging to the same SM copy. mass and the size are respectively given by At the same time, due to the effectively vanishing temperature of the degenerate neutron gas, the EOS is isen- 1017 g < M < 1025 g , (19a) tropic and therefore PF is only a function of ρ. In the non-relativistic limit the Fermi momentum pF = 10−10 cm < R < 10−2 cm . (19b) 2 1/3 (3π ρ/mn) satisfies pF mn or equivalently ρ ρcr, 6

We would like to stress that other cosmological scenarios sector dominates within a given stellar object, the situ- would lead to different windows. ation effectively reduces to Neff ' N ' 1 case. This is trivially exhibited by the above calculation. Thus, for instance, the physics of ordinary stellar objects that are dominated by SM baryons is unchanged. C. N-MACHO Solution for Unequal Sectors

In this Section we relax the assumption of universality III. N-PROTONIUM of parameters for different species. Therefore, we assign to a sector j an energy density ρj and a polytropic constant Kj. In order to find a generalization of Eq. (11) for Not surprisingly, having a large number of hidden sec- such a case, we need to assume a particular j-dependence tors gives rise to a plethora of new exotic stable bound- of these parameters. That is, we need to assume a par- states in the particle spectrum. This is because vari- ticular profile in species space. ous stable particles from different sectors can become gravitationally bound to each other. These structures Consider first the case with K ≡ K, but ρ 6= ρ. Then, j j can be viewed as the extreme microscopic versions of it is useful to introduce the following parameterization, N-MACHOs. As an example, let us consider a bound- 32 ρ = ρ f , (20) state formed out of N ∼ 10 species of stable baryons, j max j one from each sector. Namely, in this bound-state each Standard-Model-like sector is represented either by a pro- where ρmax is a maximal density and fj is some positive ton or an anti-proton. Each baryon “sees” the rest of discrete function of j with fj=jmax = 1 and fj6=jmax ≤ 1. Next, introducing the following notation, particles exclusively through gravity and is confined by their collective gravitational potential. N X Such a state, which can be called N-protonium, has ρtot = ρmax fj ≡ ρmax Neff , (21) the following characteristics. Its mass is given by M ∼ j=1 32 32 8 10 mp ∼ 10 GeV ∼ 10 g, whereas its radius is given by the radius of a proton, R ∼ 10−13 cm. Although mi- we can rewrite the r.h.s of Eq. (9) as croscopic, this length exceeds the corresponding gravitational Schwarzschild radius R ∼ 10−20 cm by seven K PN ργ PN f γ PN f G j=1 j γ j=1 j γ j=1 j orders of magnitude. Correspondingly, the N-protonium γ = K ρmax γ ≤ K ρmax γ ρtot ρtot ρtot is stable with respect to collapse into a black hole. The γ−1 gravitational binding energy experienced by each proton ρmax K 2 = K = . (22) is approximately V ∼ mp RG/R ∼ 10 eV. Notice, this is ρ γ−1 tot Neff close to an attractive gravitational potential experienced by a proton near the surface of the earth and is much Since Neff ≤ N, the total pressure can be larger for non- less than the nuclear binding energy. Of course, such universal sectors. Consequently, by replacing N with Neff binding energy is not sufficient for stabilizing a neutron in Eqs. (15a,b), it becomes clear that the N-MACHOs against the beta-decay. Because of this reason, the stable are heavier and larger in this scenario. configuration involves a charged baryon (a proton or an Consider now the case with Kj 6= K and ρj 6= ρ. This antiproton) from each sector, rather than a neutron. corresponds to the case in which particles of different Notice that because of the microscopic size of the ob- sectors have different masses. The total pressure now ject, the gravitation binding force near the surface of N- becomes protonium is 24 orders of magnitude stronger than the PN γ analogous gravitational force at the surface of the earth. j=1 Kj ρj γ During a passage through an ordinary macroscopic stellar Ptot = γ ρtot . (23) ρtot object, the N-baryonium will remain essentially undis- rupted. This is because the tidal forces that could poten- Repeating the previous approach, we find tially destroy the bound-state are much weaker that the

N γ attractive force that keeps the baryon species together. P K ρ K j=1 j j ≤ max , (24) For example, the tidal acceleration caused by solar grav- ργ γ−1 ity near the surface of the sun is 47 orders of magnitude tot Neﬀ weaker than the binding acceleration, whereas the tidal where Kmax is deﬁned by Kmax ≥ Kj for all j. This case acceleration caused by a neutron star is 32 orders of mag- also results in heavier and larger objects. nitude weaker. Finally, we must note that even in non-universal case, Therefore, the only likely loss that N-baryonium can the most interesting regimes are the ones in which the experience during the pass of a stellar object is that distribution in j is more or less uniform over the actual- it will be stripped of “our” ordinary proton due to a ized N species. In the opposite case, i.e., when a single non-gravitational interaction with the stellar medium. 7

The probability of loss of hidden baryons, due to non- they source different photon species – instead of averag- gravitational encounter with their own species, is negli- ing out to zero, add up in the emission rate and sharply gible because of extreme dilution of the hidden sectors. enhance it. Now, taking N-baryonium as a starting object, we can In order to perform a simple qualitative estimate, con- scan the landscape of states that can be obtained by some sider a merging binary system with the participation of simple variations of it. For instance, places of protons an N-MACHO. We shall be interested in the radiation from some of the species may be “vacant”, which would emitted during the latest stages of merger. Let the char- result in a lighter bound-state with less number of con- acteristic mass of the system be M and its size be R. stituents, i.e., N < 1032. Correspondingly, the binding We shall assume that dark matter is predominantly rep- potential for each constituent proton, V ∼ N 10−30 eV, resented by dark protons (anti-protons). While electrons becomes more shallow for smaller N. Also, some of the (positrons) do exist in equal numbers, the encounter is baryons may be substituted by the electrons or positrons unlikely due to an extreme dilution. We also assume from the same sectors. Obviously, it is possible to triv- that the net charge in each sector (i.e., the number dif- ially obtain much more complex structures by putting ference between the protons and anti-protons) is signifi- together more composites from different sectors, such as, cantly larger than one and that the separation amongst e.g., the neutral atoms or the entire chemical elements. the charges exceeds the size of the proton. We shall not discuss them further. Since each proton is attracted by the total mass of the In summary, the theory with a large number of dark system, the maximal net charge in each copy is given by sectors opens a possibility for the existence of new types −1 of simple and stable composite structures that are mi- Q ∼ α RG mp , (25) croscopic in size but macroscopic in mass. In contrast, −2 in theories with a single non-interacting dark matter where α ∼ 10 is the electromagnetic fine-structure con- species, such composite structures are hard to stabilize stant of the respective SM copy and is universal for all the against a decay or a collapse. sectors. The maximal power of electromagnetic radiation per copy is then given by

dE R2 ∼ α Q2 G . (26) IV. RADIATION INTO DARK PHOTONS dt R4 Inserting Eq. (25) and multiplying by N, we obtain the The large number of dilute dark sectors opens up the total power of dark electromagnetic radiation in all the possibility of enhanced radiation into hidden massless copies: species in various structure-formation processes with the participation of dark matter. An example of such a pro- m2 R4 P ∼ N p G . (27) cess would be provided by a merger of binaries with the D α R4 participation of N-MACHOs. The latter objects can pair-up and form binaries either with ordinary stellar ob- Now, taking into account that the power of ordinary grav- jects, such as neutron stars or black holes, or with other itational radiation is given by N-MACHOs. As a result of such merger, a certain frac- dE R4 tion of radiated energy will appear unaccounted by ordi- P ≡ ∼ M G , (28) gr 5 nary electromagnetic radiation and gravitational waves. dt gravity R For concreteness, we shall illustrate this phenomenon for we arrive to the following fraction the scenario with N exact Standard Model copies. 2 Because of the extreme dilution of particles in these PD mp R D ≡ ∼ N. (29) copies, the N-MACHOs can easily have an excess of Pgr α M charge with respect to some or all dark electromagnetic gauge symmetries. Existence of charges with respect to Defining the gravitational analog of the fine-structure 2 2 the hidden photons, opens up invisible channels of ra- constant for the proton, αgr ≡ mp/MPl, we can rewrite diation into the corresponding dark sectors. Because of the above expression in a much more transparent and this, the overall rate of energy loss into the dark matter universal form, is enhanced by N. αgr R This situation is very different from the ordinary stel- D ∼ N. (30) α RG lar structures in two respects. First, in such objects the large fluctuations of the ordinary electric charge are sup- This expression applies for charge carrying dark matter pressed due to its swift neutralization by readily-available species of arbitrary mass m, provided αgr is understood freely-floating charges. Secondly, there is no enhance- as the gravitational coupling. If the net charge is due −37 ment by N. In the multiple-copy scenario, on the con- to dark protons, mp ∼ GeV, we have αgr ∼ 10 and trary, the fluctuations of the individual charges – since correspondingly the maximal fraction of dark electromag- 8 netic radiation is given by, or an assumed modifications of Hawking radiation [39] the stability window of black holes can be expanded, the −37 R N-MACHOs are special in the sense that their stabil- D ∼ 10 N. (31) RG ity does not require going beyond standard semi-classical gravity. 32 For an N-MACHO with R close to RG and N ∼ 10 −4 In the following we will assume the previously men- this gives D ∼ 10 . For less compact objects this would of course be higher. Notice that the estimate (25) does tioned N-neutron star to be the main representative of not apply to objects in which the charge is stabilized N-MACHOs. Furthermore we will again use the window by other forces. For example, the size of N-protonium for N according to the proposed cosmological scenario is set by the size of the proton, R ∼ 10−13 cm, which in Ref. [11] modified by non-annihilating electrons, i.e., 1016 ≤ N ≤ 1032, so that the corresponding masses lie is determined by the strong interaction scale. Such N- −16 −8 protonium gravitationally merging with a neutral object between 10 M and 10 M . In this interval, there of approximately the same mass and size would radiate are lensing observations searching for compact objects: energy 12 orders of magnitude larger into dark photons than in gravitons. (i) femtolensing of gamma-ray bursts with the FERMI telescope [40], posing constraints in the range As an alternative example, consider the case when dark −17 −14 10 M and 10 M , and matter is represented by N ∼ 1032 species of m ∼ 1 TeV mass, each carrying a unit charge under the respective (ii) microlensing of M31 stars based on HSC/Subaru U(1) gauge symmetry, with gauge coupling αD. Now, data [41], which limits the abundance of compact −14 −5 consider an N-protonium-type object but instead com- DM objects from ∼ 10 M to ∼ 10 M . posed out of these TeV-mass species, one from each 1032 sectors placed on top of each other within the size of the Additionally, there are constraints coming from white- Compton wavelength R ∼ m−1. Such object will have a dwarf explosions. These are triggered by transit of com- mass M ∼ 1032 TeV ∼ 1011 g and unit charges (of either pact bodies (such as PBHs) through the wide dwarfs, sign) with respect to dark U(1)s. Notice that since m is which then initiate thermonuclear fusion [42]. This has very close to cutoff, the gravitational radius of such an been argued to rule out PBHs with masses 10−15 – −1 −17 −11 object is close to its radius, R ∼ RG ∼ m ∼ 10 cm. 10 M . Undoubtably, this raises the intriguing pos- During the merger or collision of two such objects we get sibility that a class of supernovae may be triggered in D ∼ αD. this way rather than via the conventional mechanism. Of course, one can play around with many analogous However, it has recently been argued that this argument structures but the general point we would like to make is inapplicable [43]. Hence, we show these constraints is that the multiplicity of charged dark matter species in Fig.1 with a broken line, reflecting the uncertainty allows to increase the intensity of dark radiation at the on those limits. Similarly, the mentioned femtolensiong expense of opening up new channels that are not in con- constraints recently got revised, where it has been argued flict with gravitational binding force. that most gamma-ray bursts are inappropriate for femtolensing searches due to their large sizes [44]. Therefore, we again display the corresponding limit with dashed line type. V. CONSTRAINTS & SIGNATURES In regard of the constraints coming from microlensing searches of M31 stars, a remark is in order: because of The question of how to pose constraints, or detection the black hole’s significantly higher compactness, the re- prospects, on N-MACHOs overlaps with that for general spective Schwarzschild radius is (much) smaller than the compact objects at similar masses, such as primordial wavelength of the lensed light used by the instruments. black holes (PBHs) [31, 32], ultra-compact mini-halos Therefore, Ref. [41] actually poses weaker constraints on [33], and other macroscopic objects, such as those of nu- PBHs, so that there are essentially no constraints below clear density [34, 35]. −12 masses of ∼ 10 M . Despite this fact, even for N- However, an important difference is that because N- MACHOs being (much) larger than black holes, there is MACHOs do not experience Hawking radiation [36] they an unconstrained mass gap in which these objects could, can be stable for arbitrarily small masses. This goes in in principle, constitute all the DM. The allowed fraction −18 contrast with PBHs with mass below 10 M whose of N-MACHOs is shown in Fig.1 for the mass range −16 −7 abundance has been very strongly constrained due to 10 M < M < 10 M . An interesting advantage Hawking evaporation. Thus, unlike microscopic black of DM in this scenario is that it does not need to be holes, light N-MACHOs can be stable in a much larger entirely in the form of N-MACHOs, since they are com- range of masses in which they could in principle consti- posed of elementary dark particles. tute the entirety of the dark matter. As we stressed previously, the mentioned interval of We must stress that although it is conceivable that ei- allowed N-MACHO masses is√ derived under the premise ther due to quantum-gravitational back reaction [37, 38] that these are given by M / N . However, even if this 9

As mentioned earlier, besides the lensing signatures discussed in this Section, N-MACHOs will lead to ��-�� -�� -�� -� ��-� ��-� gravitational-wave emission, allowing to constrain their � abundance. Similarly to PBHs, these objects would also �� - emit gravitational waves below the mass window of astro- �� physical black holes which are constraint to O(1) M . �� & The gravitational waves from N-MACHO mergers should �� [��/��] �� -�� be different in two ways. Firstly, they suffer from a lack �� of energy due to radiation into particles from other sectors (see the discussion in Sec.IV). Secondly, due to the �� N-MACHOs’ vastly different mass-to-extend ratio, their �� waveforms and merger time will be significantly different, allowing to distinguish these objects from PBHs. �� Also, similarly to PBHs, N-MACHOs will be subject to accretion around them. This concerns (a) SM particles from our sector, yielding the same accretion constraints from CMB distortions as PBHs (see Refs. [47, FIG. 1: Constraints on the allowed N-MACHO DM fraction 48]); (b) in case of extensions to the SM, particles above as a function of N. In the allowed range 1016 < N < 1032, the SM scale, implying enhanced annihilation signatures there are bounds from femtolensing of gamma-ray bursts in the accreted dark matter spikes (cf. Refs. [49, 50]); (c) with the FERMI telescope [40], from White-dwarf explosions particles from other sectors, which would lead to effec- [42] and from the microlensing search of M31 stars based tively more extended, puffier objects with, for instance, on HSC/Subaru data [41] (red contours; 95% C.L.).√ It is different lensing characteristics. Of course, the above assumed that the N-MACHO mass is given by M / N [see mentioned possibilities could occur in combination. Eq. (15b)].

VI. FORMATION ��-� ��-� ��-� ��-� ��-� � �� A. Present-Day Formation �� In general it is assumed that DM cannot possess a mechanism to dissipate energy, because this would vi- �� /�� olate the known shape of DM halos at scales of dwarf ��+ �� galaxies. Therefore, it may seem surprising that in the �� many-species model the entirety of DM could dissipate. �� % �� However, given a large enough number of populated sec- �� tors, each sector becomes so dilute that the probabil- �� ity for the encounter of particles from the same species becomes extremely small. For example, in the extreme case of equally-populated N = 1032 dark SM copies, today’s mean partial density in any other sector would be 0 −32 0 −38 −3 FIG. 2: The blue-shading represents the 95% C.L. allowed nj ∼ 10 nb ∼ 10 cm , where nb denotes the bary- region of N-MACHO abundance using combined data from onic density. This number is so dilute that the particles OGLE [45] and HSC/Subaru [41]. (Figure adapted from from any given dark SM copy have practically no chance Ref. [46]) of meeting each other. Also, notice that the energy loss by hidden baryons due to emission of dark photons at the galactic scale is negligible. Indeed, the energy loss due was the case, there would still remain a possibility that to a dark electromagnetic radiation by a hidden proton not all sectors contribute (equally) to the N-MACHOs freely falling in a gravitational potential of the galaxy is, or that the sectors have diﬀerent abundances, leading to heavier objects. This option is particularly appealing 2 dE R −54 eV due to recent results using OGLE microlensing data [46]. ∼ α G ∼ 10 , (32) dt R4 sec Therein it has been found that PBHs actually provide the best ﬁt (!) to the data explaining a population of six where α is electromagnetic coupling constant, 17 ultrashort-timescale events. Here, we would like to point RG ∼ 10 cm is the gravitational radius of our galaxy out that these events could, of course, equally well be and R ∼ 50 kpc is its actual radius. Obviously, in such a explained by N-MACHOs. We depict this in Fig.2. case, the system would imitate collisionless dark matter. 10

Thus, the interesting regimes for N-MACHO forma- be primordial. This possibility shares similarities with tion are for larger over-densities or lower numbers of N the formation of so-called ultra-compact mini-halos [33]. with higher population. These are compact structures, whose origin are primor- In the latter scenario, since the other sectors are com- dial inflationary overdensities reentering the horizon dur- posed of SM copies, one possibility is that the formation ing radiation domination. The overdensities need to be of N-MACHOs happens similar to usual star formation relatively large but below the threshold for primordial – but time delayed. This was first pointed out in the con- black-hole formation [31, 32], meaning that the necessary −4 text of folded braneworlds [24] and later in the context overdensities δ ≡ δρ/ρ lie in the range δmin ∼ 10 < −1 of many SM copies [10]. In order to understand this, we δ < δmax ∼ 10 . Here, the lower threshold depends on the scale and on the value of the critical redshift z un- consider the time tcool it takes a plasma to lose an O(1) cr fraction of its kinetic energy. For a single sector this cool- til these structures are essentially unaltered and undergo −1 gravitational collapse. Very roughly, this holds true until ing time scales as tcool ∼ n . Since galaxy formation in our sector starts at a redshift z ∼ 4, the galaxy formation a non-linear structure formation starts. The conservative in the dark sector would start right now if scenarios assume zcr = 1000, but it may as well be an order of magnitude lower. In such a case, their fraction nj 1 can be many orders of magnitude higher at the present . . (33) nb 4 day. It can be shown [33] that during radiation domination those overdensities grow logarithmically, which later This scenario would require few dark sectors to be ex- during the matter domination changes to a linear growth tremely more populated than the others in order to keep till the collapse starts. ρ at the observed value. Then, according to App.IIC, DM While in the standard DM scenario the ultra-compact ρ ρ ρ , (34) mini-halos form virialized structures, the situation j max . tot changes fundamentally in the large N species framework. Due to an efficient dissipation of energy on those small leading to an Neff of O(1). The resulting DM objects scales, the ultra-compact mini-halos could cool and serve would hence have masses around M . This possibility, in particular, is very interesting, since it would lead to as formation sites of N-MACHOs. Their (primordial) an increase in the number of gravitational-wave sources fraction can be estimated assuming a Gaussian distribu- as a result of the violent collapse characteristic of star tion via formation. However, it does not enable the formation of 1 Z δmax δ2 low mass N-MACHOs. β(`) = √ dδ exp − 2 , (35) 2π σH(`) δmin 2 σH(`) which is the probability that a region of co-moving size B. Primordial Formation ` (at the time when 1/` = aH, i.e., when this scale reen- ters the Hubble horizon) will later undergo a collapse to 2 As discussed above, in the present-day scenario, the an N-MACHO. Above, σH is the DM variance at the N-MACHOs cool because the DM simply takes longer to time of horizon entry. The quantity β is exponentially dissipate enough energy. It is important to realise that sensitive to the value of the lower threshold δmin. Un- the cooling mechanisms only operate efficiently within less this becomes of the order 10−6 – which it does for a certain window of masses set by the parameters of zcr ∼ O(1) – the standard almost scale-invariant expres- the theory. This is why in our sector baryons col- sion for the primordial power spectrum with moderate lapse and fragment below the characteristic mass scale tilt, yields only a negligible fraction of primordial-origin 12 Mcr ∼ 10 M , while galaxy clusters are roughly spher- N-MACHOs. ical collections of virialized baryonic gas. A cooling mech- In order to make this fraction non-negligible, a mechanism is efficient if it is able to remove an O(1) fraction anism for increasing the primordial power spectrum has of the kinetic energy within the characteristic free-fall to be employed. Notice, this is the same condition that is time. Therefore, the baryonic cooling mechanism is not required for PBH formation (see Ref. [25] for a review). efficient on scales of clusters. In this sense, formation conditions for N-MACHOs are The same reasoning also applies for dissipative DM, not any less natural than the ones demanded by PBH. possibly resulting in a much smaller value of Mcr [51]. We want to stress, however, that the power spectrum re- In our scenario the only parameter that controls the effi- quired for obtaining a substantial fraction of N-MACHOs ciency of the cooling mechanism is the tiny partial den- is much easier to achieve than in the PBH case, due to a sity of the dark sectors, which in turn depends on the much lower threshold. number of populated copies N. Thus, for regions where An interesting alternative for increasing the N- a high concentration of individual sectors is achieved, the MACHO fraction is given by extending the matter- cooling can become efficient. dominated era due to reheating between inflation and With that in mind we propose that the necessary seed radiation domination. In this scenario as much as 50% overdensity for the formation of N-MACHOs could also of all DM could be in compact DM structures at z = 100 11

[52]. Coincidentally, since the reheating temperature is inflationary spectrum or a long matter-dominated phase proportional to N −1/2 according to Ref. [11], the values between inflation and radiation domination. The latter at the larger end of the range of N automatically lead to possibility is of particular interest, since such a phase such a phase. automatically appears for large values of N in the under- Here, a cautionary note is necessary. In the cosmolog- lying cosmological scenario [11]. ical scenario in Ref. [11], the dark particles never reach The case with N identical copies of the SM is par- thermal equilibrium due to their tiny densities. The ad- ticularly appealing due to the fact that structures com- vantage of this is that the predictions in the dark sectors posed of SM particles are relatively well understood. It are independent of the concrete mechanism of baryon would be interesting, however, to embed N-MACHOs in asymmetry generation in our sector. The other copies, other scenarios. Since the only real requirement for N- however, are composed of effectively an equal number MACHOs is a polytropic equation of state, a bosonic re- of baryons and anti-baryons. Within that scenario, the alization could be constructed, where the particles would right DM abundance is produced provided that all an- form a weakly interacting Bose gas. Moreover, for parti- nihilation channels are always frozen out. Regions of cles of higher mass, N-MACHOs could have masses be- −16 higher density could not only spoil this prediction, but low 10 M . Since they are not subject to Hawking also strongly suppress the fraction of N-MACHOs as a evaporation, a totally unconstrained region could hence consequence of favoring an annihilation over a dissipa- be entered, where such objects could make up all of the tion. dark matter.

VII. CONCLUSION & OUTLOOK Appendix A: Newtonian Polytropes The purpose of the present paper was to point out some astrophysical consequences of scenarios in which DM originates from species of a large number N of hid- For a polytrope, the TOV equation can be brought into den sectors. At the fundamental level, such theories are a dimensionless form by introducing new variables θ and motivated by the solution of the Hierarchy Problem [4,5] ξ, defined by but as a bonus they offer potentially interesting DM sce- 2 1/2 narios [10, 11]. In this picture, the DM is composed out 2MP Ktot γ (γ−2)/2 r = ρtot(0) ξ , (A1a) of particles belonging to many dark sectors. Each sec- γ − 1 tor is so dilute that particles belonging to it rarely meet 1/(γ−1) each other. This creates an effect of slow dissipation, ρtot(r) = ρtot(0) θ (r) , (A1b) despite the fact that at the fundamental level the DM p (r) = K ρ (0)γ θγ/(γ−1)(r) , (A1c) species can posses interactions that are as strong as the tot tot tot ones experiences by the Standard Model particles. This yielding interplay between low partial densities and unsuppressed forces results into unusual composite structures that can 1 d dθ ξ2 + θ1/(γ−1) = 0 , θ(0) = θ0(0) = 0 . be of both fundamental as well as astrophysical signifi- ξ2 dξ dξ cance. (A2) Using the Tolman-Oppenheimer-Volkoff equation, we The function θ(ξ) defined by this equation is called the showed that particles from different dark sectors can col- Lane-Emden-function of index (γ−1)−1. It can be shown lectively form stable compact objects, which we called N- [30], that for γ > 6/5, θ(ξ) vanishes at some finite ξ0. MACHOs. As a direct consequence of the segmentation This fact and Eq. (A1a) can be used to define the radius into N mutually non-interacting sectors, these objects of the object by have very low mass and a tiny radius as compared to the single-sector case. This can be intuitively understood 2 M 2 K γ 1/2 R ≡ P tot ρ (0)(γ−2)/2 ξ . (A3) by the reduction of the thermal pressure counteracting γ − 1 tot 0 the gravitational collapse, in contrast to the unchanged gravitational force. We showed under general assump- Using this radius, Eqs. (A1a,b), the mass of the polytrope tions that this holds for different abundances and masses can be calculated to be in the other sectors. Z R 2 For the scenario where the dark matter is represented M ≡ dr 4π r ρtot(r) by N hidden copies of the SM, these solutions do not 0 rely on a new mechanism, but rather use the same well- 2 3/2 (3γ−4)/2 2MP Ktot γ 2 0 understood properties of known stellar structure. There, = 4π ρtot(0) ξ0 θ (ξ0) . we argue that N-MACHOs could form in high-density γ − 1 regions as a consequence of an additional feature in the (A4) 12

2 0 The numerical values for ξ0 and − ξ0 θ (ξ0) for the case by the Deutsche Forschungsgemeinschaft (DFG, Ger- 2 0 of γ = 5/3 are given by ξ0 = 3.65375 and − ξ0 θ (ξ0) = man Research Foundation) under Germany’s Excellence 2.71406 (see for instance Ref. [30] for further details and Strategy - EXC-2111 - 390814868, and Germany’s Excel- more numerical values). lence Strategy under Excellence Cluster Origins. F.K. ac- knowledges support from the Swedish Research Council (Vetenskapsr˚adet)through contract No. 638-2013-8993 Acknowledgments and the Oskar Klein Centre for Cosmoparticle Physics. He also thanks the Arnold Sommerfeld Center at Ludwig- Maximilians-Universit¨atfor hospitality received during This work was supported in part by the Hum- part of this work. boldt Foundation under Humboldt Professorship Award,

[1] F. Wilczek, “Problem of Strong P and T Invariance in [15] S. Kovalenko, H. Pas, and I. Schmidt, “Lepton number, the Presence of Instantons,” Phys. Rev. Lett. 40 (1978) black hole entropy and 10 to the 32 copies of the 279–282. Standard Model,” J. Phys. Conf. Ser. 259 (2010) 012070, arXiv:1011.0410 [hep-ph]. [2] S. Weinberg, “A New Light Boson?,” Phys. Rev. Lett. 40 (1978) 223–226. [16] S. Kovalenko, H. Pas, and I. Schmidt, “Lepton number violation in theories with a large number of Standard [3] R. D. Peccei and H. R. Quinn, “Constraints Imposed by Model copies,” Phys. Rev. D83 (2011) 051901, CP Conservation in the Presence of Instantons,” Phys. arXiv:1011.4915 [hep-ph]. Rev. D 16 (1977) 1791–1797. [17] A. Addazi, “Limiting First Order Phase Transitions in [4] G. Dvali, “Black Holes and Large N Species Solution to Dark Gauge Sectors from Gravitational Waves the Hierarchy Problem,” Fortsch. Phys. 58 (2010) experiments,” Mod. Phys. Lett. A32 no. 08, (2017) 528–536, arXiv:0706.2050 [hep-th]. 1750049, arXiv:1607.08057 [hep-ph]. [5] G. Dvali and M. Redi, “Black Hole Bound on the [18] T. Cohen, R. T. D’Agnolo, and M. Low, “Freezing in Number of Species and Quantum Gravity at LHC,” the hierarchy problem,” Phys. Rev. D99 no. 3, (2019) Phys. Rev. D77 (2008) 045027, arXiv:0710.4344 031702, arXiv:1808.02031 [hep-ph]. [hep-th]. [19] A. del Rio, R. Durrer, and S. P. Patil, “Tensor Bounds [6] G. Dvali, “Nature of Microscopic Black Holes and on the Hidden Universe,” JHEP 12 (2018) 094, Gravity in Theories with Particle Species,” Int. J. Mod. arXiv:1808.09282 [gr-qc]. Phys. A25 (2010) 602–615, arXiv:0806.3801 [hep-th]. [20] Z.-L. Han, B. Zhu, L. Bian, and R. Ding, “Consistent origin of neutrino mass and freeze-in dark matter in [7] G. Dvali and C. Gomez, “Quantum Information and large N theories,” arXiv:1812.00637 [hep-ph]. Gravity Cutoff in Theories with Species,” Phys. Lett. B674 (2009) 303–307, arXiv:0812.1940 [hep-th]. [21] P. Archer-Smith, D. Linthorne, and D. Stolarski, “Gravitational Wave Signals from Multiple Hidden [8] G. Dvali and C. Gomez, “Species and Strings,” Sectors,” arXiv:1910.02083 [hep-ph]. arXiv:1004.3744 [hep-th]. [22] N. Arkani-Hamed, T. Cohen, R. T. D’Agnolo, A. Hook, [9] G. Dvali and G. R. Farrar, “Strong CP Problem with H. D. Kim, and D. Pinner, “Solving the Hierarchy 10**32 Standard Model Copies,” Phys. Rev. Lett. 101 Problem at Reheating with a Large Number of Degrees (2008) 011801, arXiv:0712.3170 [hep-th]. of Freedom,” Phys. Rev. Lett. 117 no. 25, (2016) [10] G. Dvali and M. Redi, “Phenomenology of 1032 Dark 251801, arXiv:1607.06821 [hep-ph]. Sectors,” Phys. Rev. D80 (2009) 055001, [23] Y. Watanabe and E. Komatsu, “Gravitational inflaton arXiv:0905.1709 [hep-ph]. decay and the hierarchy problem,” Phys. Rev. D77 [11] G. Dvali, I. Sawicki, and A. Vikman, “Dark Matter via (2008) 043514, arXiv:0711.3442 [hep-th]. Many Copies of the Standard Model,” JCAP 0908 [24] N. Arkani-Hamed, S. Dimopoulos, G. R. Dvali, and (2009) 009, arXiv:0903.0660 [hep-th]. N. Kaloper, “Many fold universe,” JHEP 12 (2000) 010, [12] G. Dvali and S. N. Solodukhin, “Black Hole Entropy arXiv:hep-ph/9911386 [hep-ph]. and Gravity Cutoff,” arXiv:0806.3976 [hep-th]. [25] B. Carr, F. Kuhnel, and M. Sandstad, “Primordial [13] G. Dvali and O. Pujolas, “Micro Black Holes and the Black Holes as Dark Matter,” Phys. Rev. D94 no. 8, Democratic Transition,” Phys. Rev. D79 (2009) 064032, (2016) 083504, arXiv:1607.06077 [astro-ph.CO]. arXiv:0812.3442 [hep-th]. [26] P. Ivanov, P. Naselsky, and I. Novikov, “Inflation and [14] R. Rahman, “Consequences of Gravity-Induced primordial black holes as dark matter,” Phys. Rev. D50 Couplings in Theories with Many Particle Species,” (1994) 7173–7178. Phys. Lett. B688 (2010) 71–74, arXiv:0901.2111 [27] LIGO Scientific, Virgo Collaboration, B. P. Abbott [hep-ph]. 13

et al., “Observation of Gravitational Waves from a primordial black holes with Subaru/HSC Andromeda Binary Black Hole Merger,” Phys. Rev. Lett. 116 no. 6, observations,” Nat. Astron. 3 no. 6, (2019) 524–534, (2016) 061102, arXiv:1602.03837 [gr-qc]. arXiv:1701.02151 [astro-ph.CO]. [28] LIGO Scientific, Virgo Collaboration, B. P. Abbott [42] P. W. Graham, S. Rajendran, and J. Varela, “Dark et al., “GW151226: Observation of Gravitational Waves matter triggers of supernovae,” Phys. Rev. D 92 no. 6, from a 22-Solar-Mass Binary Black Hole Coalescence,” (Sept., 2015) 063007, arXiv:1505.04444 [hep-ph]. Phys. Rev. Lett. 116 no. 24, (2016) 241103, [43] P. Montero-Camacho, X. Fang, G. Vasquez, M. Silva, arXiv:1606.04855 [gr-qc]. and C. M. Hirata, “Revisiting constraints on [29] N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, asteroid-mass primordial black holes as dark matter “The Hierarchy problem and new dimensions at a candidates,” JCAP 1908 (2019) 031, millimeter,” Phys. Lett. B429 (1998) 263–272, arXiv:1906.05950 [astro-ph.CO]. arXiv:hep-ph/9803315 [hep-ph]. [44] A. Katz, J. Kopp, S. Sibiryakov, and W. Xue, [30] S. Weinberg, Gravitation and Cosmology: Principles “Femtolensing by Dark Matter Revisited,” JCAP 1812 and Applications of the General Theory of Relativity. (2018) 005, arXiv:1807.11495 [astro-ph.CO]. New York: Wiley, 1972. [45] A. Udalski, M. K. Szymański,and G. Szymański, [31] Y. B. Zel’dovich and I. D. Novikov, “The Hypothesis of “OGLE-IV: Fourth Phase of the Optical Gravitational Cores Retarded during Expansion and the Hot Lensing Experiment,” Act. Astr. 65 no. 1, (2015) 1–38, Cosmological Model,” Sov. Astron. 10 (1967) 602–603. arXiv:1504.05966 [astro-ph.SR]. [32] B. J. Carr and S. W. Hawking, “Black holes in the early [46] H. Niikura, M. Takada, S. Yokoyama, T. Sumi, and Universe,” Mon. Not. R. Astron. Soc. 168 (1974) S. Masaki, “Constraints on Earth-mass primordial black 399–415. holes from OGLE 5-year microlensing events,” Phys. Rev. D99 no. 8, (2019) 083503, arXiv:1901.07120 [33] M. Ricotti and A. Gould, “A New Probe of Dark [astro-ph.CO]. Matter and High-Energy Universe Using Microlensing,” Astrophys. J. 707 (2009) 979–987, arXiv:0908.0735 [47] Y. Ali-Ha¨ımoudand M. Kamionkowski, “Cosmic [astro-ph.CO]. microwave background limits on accreting primordial black holes,” Phys. Rev. D95 no. 4, (2017) 043534, [34] E. Witten, “Cosmic Separation of Phases,” Phys. Rev. arXiv:1612.05644 [astro-ph.CO]. D 30 (1984) 272–285. [48] T. Nakama, B. Carr, and J. Silk, “Limits on primordial [35] B. W. Lynn, A. E. Nelson, and N. Tetradis, “Strange black holes from µ distortions in cosmic microwave Baryon Matter,” Nucl. Phys. B345 (1990) 186–209. background,” Phys. Rev. D97 no. 4, (2018) 043525, [36] S. Hawking, “Gravitationally collapsed objects of very arXiv:1710.06945 [astro-ph.CO]. low mass,” Mon. Not. Roy. Astron. Soc. 152 (1971) 75. [49] Yu. N. Eroshenko, “Dark matter density spikes around [37] G. Dvali and C. Gomez, “Black Hole’s Quantum primordial black holes,” Astron. Lett. 42 no. 6, (2016) N-Portrait,” Fortsch. Phys. 61 (2013) 742–767, 347–356, arXiv:1607.00612 [astro-ph.HE]. [Pisma arXiv:1112.3359 [hep-th]. Astron. Zh.42,no.6,359(2016)]. [38] G. Dvali and A. Gußmann, “Skyrmion Black Hole Hair: [50] S. M. Boucenna, F. Kuhnel, T. Ohlsson, and Conservation of Baryon Number by Black Holes and L. Visinelli, “Novel Constraints on Mixed Dark-Matter Observable Manifestations,” Nucl. Phys. B913 (2016) Scenarios of Primordial Black Holes and WIMPs,” 1001–1036, arXiv:1605.00543 [hep-th]. JCAP 1807 (2018) 003, arXiv:1712.06383 [hep-ph]. [39] M. Raidal, S. Solodukhin, V. Vaskonen, and [51] M. R. Buckley and A. DiFranzo, “Collapsed Dark H. Veermäe,“Light Primordial Exotic Compact Objects Matter Structures,” Phys. Rev. Lett. 120 no. 5, (2018) as All Dark Matter,” Phys. Rev. D97 no. 12, (2018) 051102, arXiv:1707.03829 [hep-ph]. 123520, arXiv:1802.07728 [astro-ph.CO]. [52] A. L. Erickcek and K. Sigurdson, “Reheating Effects in [40] A. Barnacka, J. F. Glicenstein, and R. Moderski, “New the Matter Power Spectrum and Implications for constraints on primordial black holes abundance from Substructure,” Phys. Rev. D84 (2011) 083503, femtolensing of gamma-ray bursts,” Phys. Rev.D 86 arXiv:1106.0536 [astro-ph.CO]. (2012) 043001, arXiv:1204.2056 [astro-ph.CO]. [41] H. Niikura et al., “Microlensing constraints on