Collapsed Dark Matter Structures

Matthew R. Buckley and Anthony DiFranzo Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA (Dated: February 14, 2018) The distributions of dark matter and baryons in the Universe are known to be very different: the dark matter resides in extended halos, while a significant fraction of the baryons have radiated away much of their initial energy and fallen deep into the potential wells. This difference in morphology leads to the widely held conclusion that dark matter cannot cool and collapse on any scale. We revisit this assumption, and show that a simple model where dark matter is charged under a “dark electromagnetism” can allow dark matter to form gravitationally collapsed objects with characteris- tic mass scales much smaller than that of a Milky Way-type galaxy. Though the majority of the dark matter in spiral galaxies would remain in the halo, such a model opens the possibility that galaxies and their associated dark matter play host to a significant number of collapsed substructures. The observational signatures of such structures are not well explored, but potentially interesting.

Though dark matter outmasses the baryons five-to-one Outside it, the characteristic cooling time is longer than [1], we typically assume the baryonic components are far the infall time. As a result, little of the kinetic and po- more complex than their dark matter equivalents. While tential energy in the halo is lost. Thus, it is completely some of this is certainly due to baryonic chauvinism, it possible for compact objects to form at the scale of, say 6 is clear from rotation curves and lensing measurements 10 M and below, while above this scale, no significant 9 that, at the mass scales of dwarf galaxies (∼10 M ) and deviation from CDM would be seen, despite 100% of the above, dark matter resides in approximately spherical dark matter having the same set of non-gravitational in- “halos” whose shapes are consistent with primordial over- teractions. densities evolving solely under gravity. The baryons on This proposed cooling mechanism is in exact analogy the other hand can form objects with a wide variety of to the excitational cooling of bound hydrogen by elec- shapes (including the disks in Milky Way (MW)-type spi- trons, which sets the characteristic mass of the largest ral galaxies). This non-trivial structure continues down collapsed baryonic structures in the Universe to be 12 to smaller scales (stars and planets). ∼10 M [15] (see also [16, 17]). The key observation −1/2 These collapsed baryonic structures are made possible is that this cooling rate goes as TV , while the en- through the existence of a cooling mechanism: the elec- ergy of the particles increases as the virial temperature trons (and the coupled protons) can bleed away the pro- TV . As halo mass (and thus TV ) increases, cooling be- tons’ initial kinetic and potential energy and sink deep comes proportionally less efficient, setting an upper limit into the gravitational well. For baryons, the primary on the mass of halos that could cool and collapse. This cooling mechanism (prior to the formation of the first is why collections of collapsed, fragmented baryons exist 12 stars) is collisional excitation of hydrogen by electrons on mass scales below 10 M , while above this scale the [2]. Collapsed dark matter structures have been hypoth- galaxy clusters are roughly spherical collections of viri- esized, but only if they are created in the early Universe alized baryonic gas in which the galaxies are embedded (e.g. primordial black holes [3–5] or axion stars [6]), or in – the baryons in the galaxy cluster have too much ki- some subdominant component of the dark matter [7–11]. netic energy and the cooling mechanism cannot remove Even models of self-interacting dark matter, which can an O(1) fraction within the characteristic free-fall time. alter the morphology of dark matter halos [12], do not al- Going to smaller halo masses, the ionization fraction de-

arXiv:1707.03829v2 [hep-ph] 12 Feb 2018 low for the loss of energy, but only the transfer of energy creases with the virial temperature, as does the scattering between dark matter particles. In general, we assume rate of free electrons off of bound hydrogen. This sets a all of the dark matter cannot possess the necessary in- lower limit on the mass of collapsed objects.1 teractions to dissipate energy, because such interactions The range of masses of baryon halos which can collapse would seem to violate the known shape of dark matter is set by the masses and couplings of protons and elec- halos at the scale of dwarf galaxies and larger, which are trons; we can hypothesize that the dark sector could have largely consistent with the predictions of non-interacting, a similar mechanism with different parameters, leading non-cooling, cold dark matter (CDM) [13, 14]. to a different mass range that does not extend up to the However, this is not the case: cooling mechanisms generically operate efficiently only within a relatively nar- row range of halo masses set by the parameters of the 1 This assumes no additional source of reionization is present. For dark sector. Inside this mass range, primordial dark baryons, stars will eventually provide ionizing photons, allowing matter halos can cool and collapse into compact objects, the baryons in small halos to collapse, but we will not appeal to fragmenting down to smaller mass objects as they do. a similar mechanism in the dark sector. 2 scale of spiral galaxies. As a proof of principle, we intro- whereρ ¯(z) is the average density of dark matter at red- 9 duce a simple model, where dark matter is composed of shift z. For the mass range of interest (.10 M ), we equal numbers of a heavy particle H and a light particle expect the dark matter halos to be virialized by z ∼ 10 L, with opposite charges under an unbroken U(1)X with [36, 37], which we adopt as our benchmark. The lighter L fine-structure constant αD (a “dark electromagnetism” particles are kept in equilibrium with the H via Coulomb [18]). Such dark matter is asymmetric (see [19] for a scattering [9]; the timescale of which is much shorter than review), and so must have additional non-trivial interac- other processes, so we treat both particles as having a tions in the early Universe to annihilate away the ther- common virial temperature TV . mal component [20], which we will take as given. Models In the Standard Model, there are four primary mech- of dark matter charged under a hidden U(1) have been anisms for the baryons to lose energy: inverse Compton considered in the literature, e.g. [21–23]. In this Let- scattering of Cosmic Microwave Background (CMB) pho- ter, our goal is to show that this simple model contains tons, bremsstrahlung, free-bound scattering where the the necessary components to allow the formation of col- electrons bind to a proton, and collisional excitation of lapsed structures in the dark sector and evades other con- hydrogen by a free electron. As the temperature of the straints. We note that dissipative cooling of dark matter “dark CMB” must be lower than our own to evade con- has been considered previously [24–27], and for a sub- straints on the number of light degrees of freedom in the dominant component of dark matter in [7–9, 28, 29], but early Universe [38], we treat “dark inverse Compton” as we believe this is the first work that emphasizes that subdominant for the purposes of our proof of principle. the cooling mechanism selects a critical mass scale above Of the remaining cooling mechanisms, the most impor- which cooling is inefficient. tant for us (and for the earliest baryonic halos) is col- Our model has three free parameters: the coupling αD lisional excitation from the 1s → 2p state, which has a and two masses, mH and mL. We are most interested in volumetric cooling rate of [27]: the parameter space mH  mp, in order to avoid con- 16 r straints from elastic scattering. Assuming the visible and 2 2π 2 Π = 9 αDnL(z)nB(z) (3) dark sectors were once in thermal equilibrium, the photon 3 mLTV 2 −6 bath and dark sector temperatures today differ accord- Z ∞ ue−u Z x+ dx  4x2  × du 1 + , ing to the ratio ξ = TD/Tγ set by the effective number √ 7y2 3y/2 1 + x x 9 of relativistic degrees of freedom today as compared to 4u2 − when the dark and visible sectors decoupled from each   p 2 u p 2 2 other [9, 18]: where y = αDmL/2TV , x± = y 1 ± 1 − 3y /4u , and nL(z)(nB(z)) is the number density of free L (bound 1/3  gvis(T ) gDM(T ) states) at redshift z. Clearly, for this cooling to be effec- ξ = ∗s ∗s dec (1) vis DM tive, there must be a non-negligible number density of g∗s (Tdec) g∗s (T ) both bound and free states, which occurs over a rela- If both H and L are relativistic when the sectors decou- tively narrow temperature range. The ionization frac- pled and there are no additional degrees of freedom in tion is set by the relative rates of collisional ionization the visible sector, then ξ ∼ 0.5, dropping to ∼0.4 if H and recombination: nL/nB = hσcivi/hσrecvi [39]. Using is already non-relativistic at decoupling. This continues the binary-encounter Bethe model [27, 40] and setting to decrease if more particles are added to the visible sec- the bound-free Gaunt factor [41] to one: tor. A more complicated thermal history could further s 2 reduce ξ, for example, if the reheating mechanism favors 27π Z ∞ ue−u hσcivi = du × (4) one sector over the other [30–33]. As it depends on un- 3 2y2 mLTV y 1 + specified high energy physics which are beyond the scope u2 " y2 # of this Letter, we treat ξ as a free parameter, noting that 2  4  2 y2 log 1 − y − 1 1 − y log y + u2 , ξ ∼ 0.2 − 0.5 are typical values for minimal dark matter u2 2 u4 u2 u2+y2 models with a standard thermal history. s ∞ 2 In the early Universe, nearly scale-free primordial den- 211π Z ∞ X ue−u hσ vi = α5 du . (5) sity fluctuations are seeded by inflation [34]. These over- rec D 33m T 3 u2n3 + y2n L V 0 n=1 densities will increase in size due to gravitational col- lapse, departing from linear growth when the fractional The kinetic energy density of the dark matter is 3 overdensity is δ ∼ 1.7, and virializing at δV ∼ 178 [35]. 2 nDMTV (where nDM is the total dark matter number For a halo of total mass M, the virial temperature of the density). The characteristic time for the dark matter to H particles will be: radiate away O(1) of its kinetic and potential energy is 3 therefore tc = nDMTV /Π. This must be compared to  1/3 2 4π 2/3 the free-fall time for the halo, which for an isothermal TV ∼ δV ρ¯(z) GN mH M , (2) p 3 sphere is tf = 3π/32GδV ρ¯(z). Thus our collapse con- 3 dition is: would be: 4π p 3 3 3π/32Gδ ρ¯(z) > n T /Π, (6) Mmin = ρDMrDAO, (9) V 2 DM V 3 −1 −1/2 1 Z (1+zdec) da  3aΩ  r = √ 1 + DM . which depends on the halo mass M via TV . Note that DAO 2 4 3 a H(a) 4ξ Ωγ our assumptions are conservative, as we assume the halo 0 density is constant and ignore the feedback effect of in- The ratio in brackets is the sound speed in the dark fluid. creasing densities as cooling begins. We therefore ex- Consistency requires that Mmin is smaller than the heav- pect a larger region of parameter space would allow for iest dark matter halos which satisfy Eq. (6), otherwise cooling, but definitive answers to these questions likely no surviving halos can cool. From Eq. (8), we see Mmin require detailed simulation. will also strongly depend on the temperature ratio ξ. In Figure 1, we show the regions of dark matter halo Given free rein to chose αD, mH , and mL, the range of collapsing halo masses could be dialed anywhere from mass M which can cool and collapse as a function of mL, galaxy-cluster scale down to sub-Earth masses. However, for two values of mH , 1.2 TeV and 250 GeV, and two val- −1 −2 these parameters are constrained by other measures of ues of αD, 10 and 10 . All cooling mechanisms save dark matter phenomenology. Compton scattering are included. Collisional excitation Charged dark matter is subject to self-scattering con- dominates for larger mL, whereas bremsstrahlung be- straints. For charged dark matter composed of only comes important for lower mL, resulting in a kink in the one species, the tightest constraint from self-scattering upper range of collapsing halo mass. For mH = 250 GeV, −1 is [18, 42]: the αD = 10 parameter space is excluded by the scat- tering limit, but all other parameter points are allowed. 2 As can be seen, increasing mH decreases the critical αD −11 −3 3 . 10 GeV . (7) mass, as does decreasing αD. For our mH = 1.2 TeV, mX −1 αD = 10 working point, ξ = 0.5 allows for minimum As the L particles are heated to the temperature of the halo masses at the collapse scale, even with our conser- H, hard scatterings must transfer the equivalent of the vative use of the DAO scale for Mmin. Such a value H’s kinetic energy in order to significantly alter the halo. for ξ is easily accommodated with the Standard Model degrees of freedom. Smaller m or α require signif- Thus, we can apply the scattering limit with mX = mH . H D In addition, we must consider the self-consistency of icantly lower ξ. Barring other mechanisms of entropy the model. By adding a massless dark photon, we not injection into the visible sector, ξ = 0.1(0.02) requires vis 4 6 only introduce new relativistic degrees of freedom in the g∗s (Tdec) ∼ 10 (10 ). early Universe which are constrained by CMB measure- Even with our conservative assumptions on Mmin, we ments, we also couple dark matter to a relativistic fluid see that the halo of dark matter that encompasses a in the early Universe. This will introduce “dark acoustic galaxy such as the MW could contain collapsed structures oscillations” (DAO) [23], in analogy to baryon acoustic with masses ranging from the equivalent of supermassive 2−3 oscillations. Primordial perturbations, which cross inside stars (10 M ) up to that of dwarf galaxies. the sound horizon before dark photon decoupling, will be In the Standard Model, a cloud of gas with the mass suppressed by energy transfer mediated by the dark pho- of a galaxy will cool and fragment into smaller masses. tons and can erase overdensities which would otherwise The exact process is greatly complicated by the existence later collapse. This “DAO scale” serves a conservative of the rich structure of baryonic bound states, but is ul- lower limit, as the true cut-off is the Silk damping scale timately stopped by nuclear fusion. Without it, the in- [43] which is lower than the DAO scale [44]. The dark falling gas is expected to form one or more black holes radiation will decouple from the dark matter at the fol- [45]. If the dark sector physics allows the existence of lowing redshift [9, 28]: something akin to fusion, one could speculate about the existence of pressure-supported dark matter conglomer-  2   ations, with mass scales set by additional physics which 8.5 × 108 αD
mL
0.5 , β ≥ 1  0.1 GeV ξ was not necessary to specify in our simple model. A zdec ≈    2 (8) 3.0 × 104 0.1 mL
p mH 0.5 , β < 1 new source of energy injection into the dark sector could αD GeV GeV ξ modify the density profile of small halos while leaving

16   larger ones unaffected. This might have interesting con- where β ≡ 10 α6 ξ2 GeV . In the first case above, 3 D mH sequences for the structure of dwarf galaxies [46–55]. the dark radiation decouples due to H–L recombination. Taking the benchmark point, with mH = 1200 GeV, −3 8 In the latter case, dark radiation decouples due to Hub- αD = 0.1, mL = 10 GeV, and M = 10 M , we find ble expansion before H and L can recombine. Assum- tc ≈ 0.01 Gyr, tf ≈ 0.09 Gyr, and Hubble time tH ≈ 0.7 ing zdec is much earlier than matter-radiation equality Gyr at z = 10. Thus, without a significant source of en- (zeq ∼ 3500), the smallest dark matter halo mass today ergy, we should expect these objects to collapse in isola- 4

m = 1200 GeV m = 250 GeV 1013 H 1013 H 1 1 1012 αD = 10− 1012 αD = 10− 2 2 11 ξ αD = 10− 11 αD = 10− 10 = 10 0 1010 .5 1010

109 109 ) ) ξ

¯ 8 ¯ 8 = 10 10 0.1 M ξ M ( 7 = ( 7 10 0 10 ξ .1 = M M 0 106 106 .5

105 105 ξ = ξ = 4 0 4 10 . 10 0. 1 ξ = 1 3 ξ 3 0 10 = 0 10 .02 .02 102 102 10-5 10-4 10-3 10-2 10-1 10-5 10-4 10-3 10-2 10-1 mL (GeV) mL (GeV)

FIG. 1: Range of dark matter halo masses that cool as a function of mL for mH = 1.2 TeV (left) and 250 GeV −1 −2 (right). Colored regions result in halos cooling for αD = 10 (red) and αD = 10 (blue). Lower limits on the halo mass as a function of ξ are shown as solid, dashed, and dotted lines for ξ = 0.5, 0.1, and 0.02, respectively (red for −1 −2 αD = 10 , blue for αD = 10 ). For mH = 250 GeV, the αD = 0.1 region violates the self-scattering bound Eq. (7), but is included to demonstrate the cooling dependence on mH and αD. tion sufficiently quickly and to become sufficiently com- one of the massive compact halo object (MACHO) pact to avoid significant tidal stripping as they are ac- paradigm, and several of the same dynamical limits ap- creted onto larger, uncollapsed dark matter halos. They ply. In particular, tidal disruption of star clusters [59], may still be susceptible to stripping if they collapse into dynamical friction in the halo [60], and millilensing of individual disks, however dark-magnetic fields are likely quasars [61] can exclude the possibility of 10% of the dark to be generated in this model, providing an avenue for matter being in primordial black holes with a monochro- 3−9 angular momentum loss [18]. matic mass spectrum in the range of 10 M . However, Due to tidal stripping, most subhalos falling into a the limits on collapsed objects with a broad spectrum of MW-sized galaxy do not survive to the present day. N- masses, as would be expected in our model, are gener- body simulations of MW-mass galaxies finds 10 − 100 ally weaker but can vary in strength depending on the 5 8 4 halo masses [62–64]. Furthermore, it is unclear whether (10 ) substructures with masses around 10 (10 ) M present in the halo at z = 0. In all, less than 2% of the these objects will collapse down to a single object, or con- dark matter in the MW would be in substructure [56]. tinue to fragment down, with smaller fragments possibly However, only ∼5% of the MW was smoothly accreted stripped from the originating subhalo (perhaps forming from dark matter not contained in a halo [57]. Instead, a “dark disk” [7, 8]). much of the dark matter in the MW was originally ac- In addition to constraints from disruption of galac- creted as part of a smaller virialized halo; if those halos tic systems, searches for substructure using gravitational can cool, then we should expect those objects to have 8 lensing will eventually probe the mass range of 10 M collapsed and retained most if not all of their dark mat- [65], while Lyman-α constraints set the best current ter. For example, if dark matter halos in the mass range 7−8 bound at a similar mass [13, 14] (though care must be 10 M can cool and collapse (as realized in the param- taken when extrapolating these limits to models far re- eter point mH = 1.2 TeV, mL = 1 MeV, and αD = 0.1), moved from collisionless CDM). Being more compact then fully 10% of the MW’s dark matter would be in than the originating halo, a collapsed dark matter struc- ∼2000 objects in this mass range [58]. Though simula- ture moving through a stellar stream in the Galaxy would tions are resolution-limited for smaller dark matter ha- presumably have a very different signature than expected los, each decade of mass for smaller subhalos might be from CDM [66–69]. The presence of additional col- expected to contribute a similar fraction of the MW’s lapsed structure might also affect the evolution of bary- mass. 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