Constraining Nonthermal Dark Matter's Impact on the Matter Power

Home , Mixed dark matter, Warm dark matter

arXiv:1908.10369v2 [astro-ph.CO] 3 Mar 2020 ‡ † ∗ iru ihS atce n hti rz u during out froze it that equi- and thermal particles in SM was matter with dark librium that assumptions the indirect or [2–4] commonly grows. this direct scenario to considered alternatives any in fail- from interest while searches, signals [5–10] properties, receive matter to dark more ing place on continually elec- bounds we the as stringent However, of [1]. “miraculously” scale is troweak section required the cross The generates annihilation abundance. that matter rate allow dark annihilation observed assumptions currently the These calculate to domination. one pe- a radiation during of that occurred riod process is freeze-out assumption matter abundance common dark second matter this A dark the constant. an- and point became which ceased, at anni- also rate, its Hubble nihilations the until equaled dark decreasing rate dark The began hilation of thus continued. production abundance Universe. annihilations matter thermal while early cooled, ceased the plasma matter in SM with equilibrium the particles thermal As (SM) is in assumptions Model once common was Standard most matter the dark of that One cosmology. in lcrncades [email protected] address: Electronic erickcek@physics.unc.edu address: Electronic lcrncades [email protected] address: Electronic lentvst h omnseai fe challenge often scenario common the to Alternatives question pressing a remains matter dark of nature The osriigNnhra akMte’ mato h Matter the on Impact Matter’s Dark Nonthermal Constraining temn ftedr atrprilscnas rvn h f Lyman- the the in prevent observed also structures can the particles and Way perturbation matter Milky the dark the dur erases the produced streaming of particles free streaming matter their dark so of reheating, relati majority distributi vast velocity with the resulting particles that the veloci matter determining relativistic by dark with spectrum of bot produced ma production particles, dark be daughter nonthermal of would and source parent particles primary the p Model the of Model be masses also the Standard between could into process decayed do decay that was This Universe field, the scalar (EMDE), con oscillating era observable matter-dominated and early interesting this has domination radiation of o xml,frarha eprtr f1 e,dr atrm matter dark MeV, 10 d of the temperature of velocity reheat factor the a on for limit example, upper For an put observations these h nlso fapro f(ffcie atrdmnto fo domination matter (effective) of period a of inclusion The FU nttt o udmna hsc fteUies,vi Universe, the of Physics Fundamental for Institute IFPU, .INTRODUCTION I. eateto hsc n srnm,Uiest fNrhCa North of University Astronomy, and Physics of Department γ . 550. 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We further consider under what conditions the free Thus, an early matter-dominated era (EMDE) arises in streaming of relativistically produced dark matter could many theories of the early Universe. suppress the structures we observe. The Lyman-α for- The occurrence of an EMDE can profoundly affect est provides information on the matter power spectrum dark matter phenomenology, notably its resulting relic at the smallest observable scales, 0.5Mpc/h < λ < abundance [31–47]. The entropy generated by the decay 20Mpc/h [55–57]. The Milky Way’s (MW) satellite of the dominant matter component during the EMDE galaxies also constrain the small-scale power spectrum dilutes the relic abundance of existing particles; if dark [58]. Preventing the suppression of power at these scales matter thermally decoupled during the EMDE, a smaller provides us with constraints on the allowed dark matter annihilation cross section σv is required to compensate velocity at its production for a given reheat temperature. for this dilution and provideh i the observed dark matter This paper is organized as follows. We begin in Section abundance. Contrarily, if dark matter is a decay prod- II by introducing our model for reheating and nonther- uct of the dominating component, its abundance can be mal dark matter production and the resulting evolution significantly enhanced, requiring a larger σv to com- of the average dark matter velocity. In Section III we de- pensate for the excess, a scenario already underh i pressure rive a distribution function for the dark matter and use by γ-ray observations [48, 49]. The correct relic abun- it to examine the fraction of dark matter that is nonrel- dance can almost always be obtained with the appropri- ativistic at reheating and the fraction whose velocity is ate combinations of σv , dark matter branching ratio, sufficiently low to preserve the EMDE-enhanced struc- and temperature at reheatingh i [35, 36, 40, 45]. In many ture formation. In Section IV we examine conditions scenarios, the dominant production mechanism for dark under which the dark matter velocity is high enough to matter is by decay, rather than thermal production. run afoul of constraints from Lyman-α forest observa- Another interesting consequence of an EMDE is tions and observed MW satellites. We conclude in Sec- the growth of small-scale structure. Subhorizon den- tion V. Throughout this paper we will use natural units: ~ sity perturbations in dark matter grow linearly with c = = kB = 1. the scale factor during an EMDE, as opposed to the much slower logarithmic growth experienced during a radiation-dominated era [50–52]. This linear growth can II. NONTHERMAL PRODUCTION OF DARK provide an enhancement to dark matter structure on ex- MATTER tremely small scales (λ . 30 pc for temperature at reheating > 3 MeV), providing observable consequences to In the scenario we consider, the energy density of the this scenario if dark matter is a cold thermal relic [52–54]. Universe is dominated by an oscillating scalar field (or a However, if the dark matter is relativistic at reheating, massive particle species). As previously mentioned, for the perturbation modes that enter the horizon during sufficiently rapid oscillations within a quadratic poten- −3 the EMDE will be wiped out by the free streaming of tial, the field’s energy density scales as ρφ a , and it dark matter particles [50, 51]. For this reason, Ref. [50] exhibits the same dynamics and perturbation∝ evolution assumed that the dark matter particles were born from as a pressureless fluid [16, 59, 60]. The Universe experi- the decay process with nonrelativistic velocities or had a ences an early “matter”-dominated era until the expan- way of rapidly cooling in order for the enhancement to sion rate equals the decay rate of the field, H Γφ, at substructure to be preserved. Assuming a nonrelativistic which point the Universe transitions from scalar≃ to ra- initial velocity for the dark matter requires a small, finely diation domination. We use this transition to define the tuned mass splitting between the parent and daughter reheat temperature, TRH: particles, and it is more natural to assume any daughter particles are produced relativistically. 4π3G g T 4 =Γ , (1) Reference [51] claimed that the large free-streaming 45 ∗,RH RH φ length of dark matter produced relativistically from r scalar decay would washout any enhancement to struc- where G is the gravitational constant and g∗,RH is the ture growth. However, Ref. [51] reached this conclusion number of relativistic degrees of freedom at TRH. For a by assuming that all dark matter particles were created scalar field that decays into both dark matter and rela- at reheating, neglecting those particles created during tivistic particles, the equations for the evolution of the the EMDE. The momenta of particles born prior to re- energy densities of the scalar field ρφ, relativistic SM par- heating decreased throughout the EMDE. Consequently, ticles ρr, and dark matter ρχ are given by particles born earlier will be slower at reheating. We investigate the extent to which the redshifting of the par- ρ˙φ = 3Hρφ Γφρφ, (2) ticles’ momenta affects their velocity distribution at re- − − ρ˙ = 4Hρ + (1 f)Γ ρ , heating, focusing on the average particle velocity and the r − r − φ φ ρ˙ = 3H(1 + w )ρ + fΓ ρ . fraction of particles below a given velocity, to determine χ − χ χ φ φ 3

Here dots represent diﬀerentiation with respect to proper 100 time, f is the fraction of the scalar’s energy that is trans- -5 ferred to the dark matter, and wχ is the dark matter 10 equation-of-state parameter. 10-10

10-15 crit,i

A. Dark Matter Abundance ρ -20 / 10 ρ -25 In the above system of equations, we do not allow for 10 scattering interactions between the dark matter and SM 10-30 particles. Also, we neglect both the thermal production -35 and self-annihilation of dark matter particles, effectively 10 assuming that the velocity-averaged annihilation cross 0 2 4 6 8 10 10 10 10 10 10 10 section is small enough that any amount of dark mat- a ter lost to annihilations is negligible and any produced thermally is negligible compared to that produced from FIG. 1: The energy densities of the scalar (black, solid), ra- scalar decay. However, if dark matter annihilations are diation (grey, solid), and dark matter for particles born rela- s-wave, neglecting annihilations does not change the re- tivistic (vD = 0.99; red, dotted) and nonrelativistic (vD = 0.1; sults of our conclusions because we are interested in the blue, dashed). During the EMDE, both the dark matter and average dark matter velocity and the fraction of dark radiation densities scale as a−3/2 while they are being sourced matter that has lost sufficient momentum to participate by the decaying scalar field. Here f = 10−7 and the scalar −10 in structure formation. These quantities are dependent decay rate is Γ˜φ = Γφ/Hi = 10 . Reheating is marked by ˜−2/3 on the velocity distribution of dark matter. For s-wave the thin vertical dashed line at aRH = Γφ . annihilations, the velocity-averaged cross section is inde- pendent of particle velocity, and the distribution of particle velocities would be unaffected by the inclusion of matter particle created at aD, and we integrate over aD annihilations. with a = 1 setting the onset of dark matter production. Without annihilations, constraining the reheat temper- When evaluating Eq. (3) we use the fact that the co- ature to be above 3 MeV, as required by BBN, leads to moving number density of the dark matter evolves ac- a direct constraint on the fraction of the scalar’s energy cording to imparted to the dark matter: f . 10−7 [50] for nonrela- dnˆ tivistic dark matter. This branching ratio is quite small χ = bΓ nˆ , (4) and it would be more natural to expect the energy im- dt φ φ parted in the decay of the scalar to be more evenly allo- wheren ˆ is the comoving number density of φ particles, cated to both the dark matter and the SM. The inclusion φ and b is the number of dark matter particles produced of annihilations, however, significantly reduces the ratio per scalar decay. We can then (following a procedure of dark matter to radiation. This can allow for a more similar to that in Ref. [61]) express the term dnˆ /da balanced transfer of energy, f 0.5, while still achiev- χ D as ing a sufficiently small dark matter∼ abundance through ρφ 3 annihilations [35, 51]. For relativistic dark matter, its bΓφ a dnˆχ dnˆχ dt bΓφnˆφ mφ D ρφ 2 abundance is also dependent on the velocity imparted to = = = aD. (5) daD dt daD a˙ D aDHD ∝ HD the particles at decay (vD). In our model, we consider the scenario in which the The constants b, Γφ, and mφ appear in both integrals in dark matter is produced via a two-body decay so that Eq. (3) and consequently do not affect E . We numer- all dark matter particles are born with the same velocity, ically evaluate Eq. (3) to obtain the averageh i energy as vD. The energy density and decay rate of the scalar then a function of the scale factor; this is made even simpler govern the evolution of the equation of state wχ for the by noting that the contribution of the dark matter to the dark matter particles during the EMDE. The rates at expansion rate at the time of decay, HD, is entirely neg- which new particles are produced and the momentum of ligible compared to both the scalar and radiation energy existing particles redshifts away determine the average densities. The calculation of the average energy then in- energy per particle of the dark matter: forms how the dark matter equation of state evolves:

a 2 2 dnˆχ m + (γmχv(a,aD)) daD 1 d E 1 χ daD w = h i. (6) E = , (3) χ −3H E dt h i R q a dnˆχ da h i 1 daD D The mass of the dark matter particle can be pulled from R wheren ˆχ is the comoving number density of the dark both the average energy and its derivative, and so wχ at matter particles, v(a,aD) is the velocity at a of a dark any given time depends only on the average velocity. 4

Using Eq. (6), we numerically solve the set of equa- In the absence of annihilations, the dark matter density tions in Eq. (2) with the initial condition a a(t ) = 1, during the EMDE is also determined by f: if v 1 i ≡ i D ≪ and we assume there is no dark matter in existence prior then ρχ/ρr (5/3)f [50], whereas if vD 1 then to this time. Figure 1 shows the evolution of the scalar, ρ /ρ f. In≃ the latter case, ρ /ρ f continues≃ until χ r ≃ χ r ≃ radiation, and relativistic (vD = 0.99) and nonrelativis- the dark matter is no longer relativistic or changes in g∗ −4 tic (vD =0.1) dark matter energy densities in our model. disrupt the a scaling of ρr. Therefore, for γD 1, The energy densities in the figure are given as fractions obtaining the observed dark matter abundance requires≫ −7 of the initial critical energy density ρcrit,i. that ρχ/ρr 2.3 10 (3MeV/TRH)γD shortly after re- In Fig. 1, we have chosen to fix f = 10−7, which heating. This≃ requirement× applies regardless of whether directly sets the relative abundance of dark matter to or not annihilations alter the dark matter abundance dur- radiation during the EMDE to be 10−7, and we have ing the EMDE. After reheating, but while the dark mat- ∼ −10 chosen a scalar decay rate Γ˜φ Γφ/Hi = 10 , which ter is still relativistic, the relative dark matter abundance −2/3 6 ≡ sets aRH Γ˜ 5 10 . It is worth noting that, will evolve as by this definition,≡ a≃ ×= a . The numerical so- RH T =TRH ρ 3MeV lutions to Eq. (2) show6 that| by a scale factor of 3a χ 2.3 10−7 γ (8) RH ρ ≃ × T D enough of the scalar has decayed away that it is a neg- r T RH 4/3 ligible source of radiation. As a result, the radiation g∗S(T ) g∗(0.34TRH) energy density then evolves as in the usual radiation- , × g∗S(0.34TRH) g∗(T ) dominated era from a temperature T (3a ) 0.34T RH ≃ RH onward. This sets the relation between aRH and TRH in either case. −1/3 to be aRH/a0 =1.54(T0/TRH)g∗S (0.34TRH), where g∗S If dark matter is still relativistic at neutrino decou- is the number of relativistic degrees of freedom in the pling, it could affect the predictions of BBN. Dark mat- entropy density. ter produced relativistically at reheating would still be The effects of the dark matter particles’ velocities can relativistic (γ & 2) when T = 10 MeV if already been seen in Fig. 1. During the EMDE, the energy density of any species, relativistic or nonrelativis- 1/3 TRH γD & 2.4g∗S (0.34TRH) . (9) tic, sourced by scalar decay evolves as ρ a−3/2. At 10MeV ∝ the end of the EMDE, when the scalar field is no longer Relativistic dark matter behaves as an additional radi- sourcing new particles, the energy densities of the de- ation component, and can be characterized as a change cay products will begin to scale as ρ a−3(w+1). For a ∝ in the number of effective neutrinos, ∆Neff . The energy scenario in which dark matter is born relativistic, the av- density in relativistic particles can be written as erage value of wχ during reheating is close to 1/3, and the dark matter behaves more like radiation. In Fig. 1 we can 2 4 π 7 Tν 4 see that, following reheating, the energy density of rela- ρr + ρχ,rel = g∗ + 2 ∆Neff T , (10) 30 8 × × T tivistic dark matter (dotted) redshifts away faster than " # its nonrelativistic counterpart (dashed). Once the scalar π2 7 ρ = 2 ∆N T 4, field has decayed completely and there is no creation of χ,rel 30 8 × × eff ν new, hot particles, the existing particles’ momenta con- tinue to redshift until the average particle is no longer where Tν is the neutrino temperature, which we as- −1 relativistic, and after that, the dark matter density scales sume evolves as a after T = Tν = 10 MeV, when −3 as a . g∗ = g∗S = 10.75. Thus, the fractional component of Increasing the velocity imparted to the dark matter the energy density in dark matter at 10 MeV is related particles upon their creation increases the time it takes to ∆Neff by after reheating for the dark matter energy density to be- − ρ 7 gin scaling as a 3, thus increasing the duration of radia- f χ = ∆N g−1(10MeV) 10MeV ≡ ρ 4 eff ∗S tion domination for a given value of f. The temperature r 10MeV at matter-radiation equality is Teq = 0.796 0.005 eV 0.074∆Neff. (11) [62], and so a longer radiation-dominated era± implies a ≃ higher temperature at reheating. For a fixed value of f, Bounds from BBN constrain Neff = 2.88 0.54 at 95% ± the reheat temperature that matches the observed dark C.L. [63], yielding an upper bound f10MeV < 0.028. matter abundance in a scenario of relativistically pro- If the dark matter is still relativistic at BBN, then duced dark matter is a factor of γD greater than that of achieving the observed relic abundance requires −7 −1/3 nonrelativistic dark matter, where γD is the Lorentz fac- f10MeV 5 10 γD(3MeV/TRH)g∗S (0.34TRH), tor of the relativistic dark matter particle at production. as given≃ × by Eq. (8). Therefore, the upper For a given reheat temperature, the value of f required bound on f10MeV translates to a bound on to produce the observed dark matter abundance is γ : γ . 5.4 104(T /3MeV)g1/3(0.34T ). D D × RH ∗S RH This upper bound on γD implies an upper bound on f 2.3 10−7(3MeV/T )γ . (7) ≃ × RH D the value of f in Eq. (7) that can result in the observed 5

1 which can be evaluated in the same manner as Eq. (3). Figure 2 shows the evolution of the average dark mat- 0.8 ter particle velocity throughout the EMDE until just after reheating; the various curves in Fig. 2 represent dif- 0.6 ferent values of vD. We can see that the average ve-

1/2 locity is initially dominated by the few particles born 〉 2 v

〈 immediately with the imparted velocity. The velocity of 0.4 these particles begins to redshift away, pulling the average down, until a steady state is reached between the 0.2 redshifting of the velocity of existing particles and the creation of new, hot particles. 0 In the regime where the average velocity has reached 100 101 102 103 104 105 106 107 108 a constant value - deep into the EMDE and well before a reheating - we can simplify our calculation of the average velocity even further and analytically solve the integrals FIG. 2: The average velocity of the dark matter particles as of Eq. (13). During the EMDE, the energy density of a function of scale factor a throughout the EMDE for vD = 0.1, 0.5, 0.9, 0.99 (bottom to top). Reheating is marked by the Universe is dominated by the scalar ﬁeld, and our the thin vertical dashed line. expression in Eq. (5) becomes

relic abundance We will see in Section IV that restric- −3 ρφ 2 ρφ,iaD 2 ρφ,i tions from small-scale structure on the parameter space aD −3/2 aD = √aD. (14) HD ≃ Hi of γD and TRH provide much stronger bounds. As such, HiaD annihilations are still necessary to reduce the dark matter abundance to its required value following reheating without ﬁne-tuning f. Rewriting the expression for v2 to make its dependence on the integration variable, aD, more apparent, we have B. The Adiabatic Cooling of Dark Matter

2 Given that the momentum of a particle scales as 2 aD −1 v (a,aD)= , (15) p a , a particle born from a decay at a scale factor a 2 2 ∝ X + aD aD, with a physical velocity vD, has a velocity at some later time given by

where X γDvD. And so, deep in the EMDE, our v2 expression≡ for the average velocity, given by Eq. (13), v2(a,a )= D . (12) D 2 takes the form (1 v2 ) a + v2 − D aD D The average velocity over all the dark matter particles at a 5/2 a −1 2 aD any given time is then v (a) = daD √aDdaD . a 2 h i 1 2 1 "Z X + aD # Z (16) a dnˆ a dnˆ −1 v2(a) = v2(a,a ) χ da χ da , h i D da D da D Z1 D Z1 D (13) The solution to the integral in the numerator is given by

a 5/2 3/2 a daD 2 a 1+ √2X + X a √2aX + X D = (a3/2 1) + ln − (17) a 2 1 + a2 3 − 2X 1 √2X + X a + √2aX + X ! Z X D − a 3/2 2X 2X + 2 tan-1(1 √2X) tan-1(1 + √2X) tan-1 1 + tan-1 1+ . 2X " − − − − a ! a !# r r 6

The solution to the integral in the denominator is simply a 0.6 2 3/2 √aDdaD = (a 1). (18) 3 − 0.5 Z1

Long after the decays have started (a 1), both inte- ) 0.4 grals scale as a3/2 and v2(a) is constant≫ until just prior RH /a to reheating, at whichh pointi our approximation in Eq. D 0.3 (14) is no longer valid. f(a The steady state between the cooling of old particles 0.2 and the creation of new, hot ones is maintained until 0.1 just before reheating, and the average dark matter velocity at reheating is not reduced signiﬁcantly from the 0 velocity imparted at the scalar’s decay. Relativistic-born 0 1 2 3 4 5 a / a dark matter, vD = 0.99, is still considerably relativis- RH tic at reheating, v2 0.93. At reheating, the av- h i ≃ erage dark matter particle is nonrelativistic only if vD p 2 FIG. 3: The birth time distribution function of dark mat- is already largely nonrelativistic: v . 0.01 requires ter for our model. The peak production of dark matter oc- v < 0.017. h i D p curs just prior to reheating and f(aD/aRH) is maximized at If the comoving size of the horizon at reheating, aD ≃ 0.68aRH. −1 −1 kRH = (aRHHRH) , is smaller than the dark matter −1 free-streaming horizon, kfs , then the random drift of dark matter particles will erase the growth of density The fraction, ε, of dark matter particles born within a perturbations that occurred during the EMDE. Refer- particular interval of scale factor, aD,1 > aD > aD,2, is ence [50] found that kRH/kfs < 1 required the dark mat- given by ter velocity at reheating to be vRH . 0.06. Preserving aD,2 enhanced structure growth requires an even smaller av- εa = f(aD)daD. (20) erage velocity. Achieving such a small average velocity D,12 ZaD,1 at reheating would require the dark matter particles to be born with a similarly small velocity. This fraction can also be directly computed by

aD,2 dnˆ aD,2 dnˆχ da aD,1 χ aD,1 daD D εa = = . (21) III. DARK MATTER DISTRIBUTION D,12 ∞ ∞ dnˆχ dnˆχ da FUNCTION R 1 R 1 daD D R Equating the two expressions forRε and considering Although the average dark matter particle may have aD,12 small intervals in the scale factor, we can derive an ex- too large a velocity to participate in enhanced structure pression for the distribution function: formation, we would like to investigate what fraction of the dark matter population has a sufficiently low velocity aD,2 dnˆ aD,2 aD,1 χ to do so. Instead of considering the average particle ve- ∞ = f(aD)daD (22) dnˆ locity at reheating, we will consider what fraction of the R 1 χ ZaD,1 dark matter has a velocity at reheating that is less than f(a )∆a , R ≃ D D a percent of the speed of light. To this end, we begin aD,2 dnˆχ by deriving the dark matter distribution function. At a aD,1 f(aD) ∞ . given time, a, the fraction of dark matter with velocities ≃ ∆RaD 1 dnˆχ below a particular threshold equals the fraction of dark matter born before the correspondingly required “birth Numerically evaluating Eq. (22),R we obtain the distribu- time,” aD. tion function of dark matter birth times seen plotted in From the fact that v/√1 v2 a−1, we know that Fig. 3 as f(a /a )= a f(a ). From Fig. 3, one can − ∝ D RH RH D in order for a particle born with velocity vD to have a see that approximately half of the dark matter is born velocity less than vRH at reheating, that particle must after reheating, a/aRH > 1. have been born from decay at a scale factor, From Eq. (19) one can find that, even for dark matter particles imparted with a velocity only half of the speed 2 vRH 1 vD of light, only those born before aD . 0.017aRH will have aD < − aRH. (19) 2 a velocity at reheating vRH < 0.01. Integrating our dis- vD s1 vRH − tribution function over this interval in aD, we find the Obtaining the distribution function in birth times of the fraction of dark matter born before this time to be ap- dark matter particles, f(aD), will then allow us to com- proximately 0.15%. Figure 4 shows the fraction of dark pute the fraction of dark matter born before this time. matter that has a velocity below v =0.01 at reheating as 7

100 100

10-1 10-1 /10) ) -2 -2 10 RH 10 (a < 0.01) hor λ RH 10-3 10-3 < (v ε fs λ ( 10-4 ε 10-4

10-5 10-5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

vD vD

FIG. 4: Fraction of dark matter whose velocity at reheating FIG. 5: Fraction of dark matter whose free-streaming length is vRH < 0.01 as a function of the assumed velocity imparted is smaller than the scale of the perturbation mode that ex- to the dark matter at its production. periences a factor of 10 in growth during the EMDE as a function of the assumed dark matter velocity at production. The different lines represent values of f = 10−7 (solid), 10−5 −3 ≃ a function of the given value of v . Even for dark matter (dashed), and 10 (dotted), or equivalently, TRH 3γD D ≃ ≃ born at only one tenth of the speed of light, only 2% MeV, TRH 0.03γD MeV, and TRH 0.0003γD MeV, re- of the dark matter has the required v < 0.01. ∼ spectively, in order to produce the observed relic abundance RH of dark matter without annihilations. One of the intriguing consequences of an EMDE is that density perturbations in matter grow linearly with scale factor during an EMDE, which is faster than the logarithmic growth expected during the typically assumed radiation-dominated epoch. We now examine what fraction of the dark matter is able to retain an appreciable perturbation enhancement from this linear growth. Since energy). The contribution coming from the radiation- density perturbations grow linearly during the EMDE, a dominated era is dependent on the duration of the era, mode that enters the horizon at a scale factor of 0.1aRH which, in our formalism, is set by the relative abundance will grow by a factor of 10 during the EMDE, which we of dark matter and radiation following reheating, and this will consider “appreciable”.∼ The comoving wavelength of is directly related to the reheat temperature. The de- such a mode is given by the horizon size at this time: pendence on TRH, however, is only logarithmic and large variations in TRH do not result in significant changes in 1 λ = λ = . (23) the resulting fraction. Due to the interdependency dis- hor aRH/10 aRH aRH | 10 H 10 cussed in Section II between the parameters f, TRH, and vD required to obtain the appropriate dark matter abun- Any modes that enter the horizon prior to 0.1aRH will ex- dance, we plot the fraction of dark matter able to pre- perience even more growth. The comoving free-streaming serve enhanced structure growth both as a function of length of a dark matter particle born at aD is given by vD for various f in Fig. 5 and as a function of TRH for various vD in Fig. 6. Figure 6 shows how insensitive the a0 da fractional component of dark matter that experiences en- λfs = v(a) , (24) a2H(a) hanced structure growth is to the reheat temperature. ZaD where a0 is the value of the scale factor today. Similar to our approach in the previous evaluation, there is a value Unfortunately, for dark matter particles born relativis- of aD for which the dark matter free-streaming length is tically throughout an EMDE, the redshifting of their mo- less than the horizon size at 0.1aRH (λfs < λhor aRH/10). mentum is not enough to allow an appreciable fraction of The resulting fraction of dark matter that is| born be- the particles to participate in enhanced structure forma- fore this time, and thus that preserves a factor of 10 or tion. Studies of mixed dark matter, in which there are more growth in perturbation amplitude, is shown in Fig. both cold and warm dark matter components, show that 5. The integral in Eq. (24) can be broken into three the small-scale matter power spectrum is suppressed by separate contributing integrals, representing the scalar- 99% even when up to half of the dark matter is cold [64]. , radiation-, and matter-dominated eras (because the Therefore, the fraction of dark matter that is cold enough dark matter free-streaming length does not change signif- to benefit from the growth of perturbations during the icantly after matter-radiation equality, we neglect dark EMDE is far too small for these structures to form. 8

100 dance of small structures, and the known abundance of substructures in the vicinity of the MW provides a bound 10-1 on the allowed suppression [58].

-2 /10) ) 10 RH (a 10-3 A. Free-Streaming Length hor λ

< -4 fs 10 λ We begin by calculating the physical streaming length ( ε today of a particle born at reheating: 10-5

a0 -6 phys da 10 λ = a0 v(a) . (25) 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 fs,0 a2H(a) ZaRH TRH (GeV) The choice of aD = aRH is beneficial in that the fraction of dark matter born before reheating, found by our previ- FIG. 6: Fraction of dark matter whose free-streaming length ous analysis of the distribution of birth times, is 0.51, and is smaller than the scale of the perturbation mode that experiences a factor of 10 in growth during the EMDE as a function we can say that approximately half of the dark matter of the reheat temperature. The different lines represent, from will have a free-streaming length above or below our cal- top to bottom, values of vD = 0.1, 0.5, and 0.99. culated value. Another benefit of this choice is that our calculations of the free-streaming length are made simpler by neglecting the contribution to the free-streaming IV. LYMAN-ALPHA AND MW SATELLITE length coming from the EMDE. CONSTRAINTS Calculating the free-streaming length using Eq. (25) shows that > 75% of this distance is covered after We have shown that the redshifting of the momen- the dark matter particle has become nonrelativistic, tum of dark matter particles prior to reheating does not (γ 1.01). For the highly relativistic initial velocities we ≤ cool the dark matter enough to preserve the enhanced would like to consider, the dark matter particles remain structure growth on scales that enter the horizon dur- relativistic well after reheating, and are still relativistic ing the EMDE (λ . 30 pc for TRH > 3 MeV). In this after changes in the number of relativistic degrees of free- section we consider if the dark matter is too hot, i.e. if dom have ceased. Beginning the integral in Eq. (25) at its free-streaming length large enough to prevent the for- a∗, the value of the scale factor after which g∗ remains mation of the smallest observed structures. Analysis of constant, captures most (& 90%) of the free-streaming Lyman-α data can be used to probe the matter power length and illuminates the important features of this sce- spectrum on small scales, 0.5Mpc/h < λ < 20Mpc/h nario by allowing us to assume that H a−2 during ∝ [55–57], or 12.6h/Mpc >k> 0.06h/Mpc, and we com- radiation domination. pare the degree of gravitational clustering at these scales We begin by breaking the integral into two separate in our model to that of the traditional model of cold dark contributing integrals, representing the radiation- and matter. The existence of MW satellite galaxies provides matter-dominated eras (again we neglect dark energy) 2 another probe of small-scale structure formation. Sup- and introducing the variable Y (γDvDaD) . The free- pression of the power spectrum leads to an underabun- streaming length is then ≡

a0 Y da 1 aeq Y 1 a0 Y da λ = da + fs Y + a2 a2H(a) ≃ H a2 Y + a2 3/2 Y + a2 √a Za∗ r ∗ ∗ Za∗ r Heqaeq Zaeq r a0 2 √Y aeq + Y + aeq 2 i√Y i√Y = ln + F i sinh−1 , 1 , (26) H a2  q 2  3/2  s a −  ∗ ∗ a∗ + Y + a∗ Hpeqaeq aeq     p where F (φ, m) is the elliptic integral of the first kind. By choosing aD = aRH (contained in the variable Y ), the first term in the above expression can be simplified under the assumption that a a , which is reasonable eq ≫ RH considering that matter-radiation equality occurs at a temperature of Teq 0.8 eV and we require TRH > 3 MeV. The expression for the contribution to the physical free-streaming length today≃ coming from the radiation-dominated 9 era then simplifies to

RD γDvDaRHa0 aeq 2 λfs,0 2 ln   ; ≃ H∗a∗ a∗ 2 1+ γ v aRH +1  D D a∗     r  a T a 2 T 2 = 4.66 1011pc γ v RH ln 2 ∗ ln 1+ γ v RH ∗ +1 , (27) × D D a  T −  D D a T  0 eq s 0 0    −5 where we have used the fact that g∗ remains constant after T∗ = 2 10 GeV to set a∗T∗ = aeqTeq = a0T0. An important feature of this calculation is that the parameters of our model,× the dark matter velocity at its production and the reheat temperature, only enter into this expression through the combination

1/3 γDvDaRH T0g∗S,0 µ = γDvD , (28) ≡ a0 1/3 3 Tg∗S T =0.34TRH h i where g∗S is the number of relativistic degrees of freedom in the entropy density and again we assume entropy is conserved after a = 3aRH. Expressed in terms of the variable µ, the physical free-streaming length calculated from the contribution from both the radiation- and matter-dominated eras is

T T 2 λphys = 4.66 105Mpc µ ln 2 ∗ ln 1+ µ2 ∗ +1 fs,0 ×  T −  T  eq s 0    T T + 4.66 105Mpc 0 2iµ F i sinh−1 iµ F i sinh−1 iµ eq . (29) T T × s eq " − r 0 !# p p

The above equation gives the scale at which the power Acquiring the power spectrum for our scenario re- spectrum of our model begins to differ from that of the quires us to determine the momentum distribution func- standard ΛCDM power spectrum. Figure 7 shows the tion of our dark matter model. Shortly after reheating free-streaming length calculated by Eq. (29) as a func- (a 3a ), the scalar field has decayed almost entirely, ∼ RH tion of the Lorentz factor at decay, γD, for different val- and essentially no new dark matter particles are being phys −1 ues of the reheat temperature. We define kfs = (λfs,0 ) , produced. After this point, the distribution of the co- and above the horizontal dashed line, the free-streaming moving momenta of the dark matter particles does not length of the dark matter reaches scales probable by the change. We scale the comoving momenta of the dark phys matter particles by the comoving momentum of a par- Lyman-α forest: k . 12.6 h/Mpc, λfs,0 & 0.08 Mpc/h. Since the free-streaming lengths of our model enter the ticle born at the scale factor that maximizes f(aD), observable regime, we consider a more precise determi- amax = 0.68aRH, and express our distribution function nation of the effects of the dark matter free-streaming in terms of length in the next section. ap a q = D , (31) ≡ amaxpD amax

B. Transfer Function where pD is the physical momentum of a particle with velocity vD. Since we assume that all dark matter particles We use the Cosmic Linear Anisotropy Solving System are produced with the same velocity, the distribution in (CLASS) [65] to obtain the dark matter transfer function, momentum for particles in our scenario can be entirely determined from the distribution in the particles’ scale factor at production, which we have already determined. P (k) T 2(k) nCDM , (30) The two distribution functions can be related through ≡ PCDM(k) 2 daD aD which describes suppression of structure due to non-cold 4πq f(q)= f(aD) =0.68 f . (32) dq aRH dark matter (nCDM) compared to that of the standard CDM scenario; PnCDM(k) and PCDM(k) are the matter This distribution function is shown in Fig. 8, and we also power spectra in each respective case. show for comparison the Fermi-Dirac distribution that is 10

2 10 100 101 100 10-1 10-2 (Mpc) -1 T(k) fs 10 λ 10-3 10-4 10-5 10-6 100 101 102 103 104 105 106 107 108 109 10-2 10-1 100 101 102 103 γD k (h/Mpc)

FIG. 7: The free-streaming length of the dark matter as calcu- FIG. 9: The transfer function for several values of the dark lated by Eq. (29) as a function of the Lorentz factor at decay, matter velocity at production, vD. Right to left (cool to warm γD, for (left to right) TRH = (3, 30, 300, 3000) MeV. The hori- colors), the solid lines represent dark matter produced with phys zontal dashed line marks λfs,0 & 0.08 Mpc/h, approximately increasing Lorentz factor γD = 100, 300, 900, 2700, 8100, re- the lower limit to scales probed by Lyman-α observations. spectively. In all cases TRH = 3 MeV. Vertical dashed lines represent the scale of the free-streaming horizon calculated using Eq. (29). The red dotted line represents the typical 0.6 transfer function for dark matter with a thermal distribution (βWDM = 2.24 and γWDM = −4.46) with a cutoﬀ parameter 0.5 αWDM ≃ 0.16 Mpc/h in order to match the same half-mode scale khm as our far left curve with α = 0.31 Mpc/h. 0.4

f(q) 0.3 2 greater velocities result in the suppression of larger scales q (smaller k). The vertical dashed lines in Fig. 9 mark the 0.2 phys −1 free-streaming horizon kfs = (λfs,0 ) given by Eq. (29) for each of the different velocities at production, confirm- 0.1 ing that it is the scale at which our model begins to show deviation, T (k) 0.95, from the CDM scenario. 0 ≃ 0 0.5 1 1.5 2 2.5 3 3.5 4 Transfer functions in nCDM models, such as this, can q be well described by a fitting formula [66]:

FIG. 8: The distribution function of dark matter for our T (k) = [1 + (αk)β]γ . (34) model (solid) and a Fermi-Dirac distribution (dashed) for comparison. Using Lyman-α data, the fitting parameters α, β, and γ can be constrained [66, 67], and the parameters of our maximized at q = 1. We can see that, compared to the model, vD and TRH, can be constrained as well. The thermal case, we have a broader distribution function. typical scale of the suppression is set by α, whereas the With our distribution function f(q), we are able to general shape is determined by β and γ. When fitting our use CLASS to obtain transfer functions for any combi- transfer function at values T (k) > 0.01, the overall shape nation of the velocity imparted to the dark matter and of our transfer function varies little across wide ranges of the reheat temperature by also providing the present-day our parameter space, and β and γ can be expressed as physical momentum of a dark matter particle with q = 1: functions of α, as seen in Fig. 11. The cutoff parameter a p a γ v α is then our only free parameter, and it can be robustly p = max D RH D D . (33) constrained using Lyman-α data. 0 a ∝ a 0 0 We find typical values of β and γ for our model to be Again we find that, just as in our calculations of the free- approximately 2.4 and 1.1 respectively, for α near the streaming length, the relevant combination of parameters constrainable regime. These− values are noticeably dif- is µ = γDvDaRH/a0. In Figs. 9 and 10 we show the ferent from those that describe the thermal warm dark transfer functions for dark matter produced at different matter (WDM) transfer function, βWDM = 2.24 and velocities, but in scenarios with the same reheat temper- γWDM = 4.46. If we compare our transfer functions ature, 3 MeV. As expected, dark matter particles born at to those of− WDM with the same value of the half-mode 11

the transfer function fit between the two models, match- 1 ing their half-mode scales requires the cutoff parameter in the WDM transfer function to be roughly a factor 0.8 of 2 smaller than that in the corresponding nonthermal transfer function, α 2α . The cutoff in the transfer ≃ WDM 0.6 function of our model is not as sharp as that of WDM, but they only begin to differ significantly at scales at T(k) which the power in the nCDM model is already greatly 0.4 suppressed, T (k) . 0.1. Reference [67] provided marginalized bounds on all 0.2 three fitting parameters in Eq. (34). The predictable shape of our transfer function determines the values of β 0 and γ as a function of α, as shown in Fig. 11, and allows -1 0 1 2 3 10 10 10 10 10 us to obtain a bound on the remaining free parameter, k (h/Mpc) the scale of the cutoff: α< 0.011Mpc/h (68% C.L.) and α < 0.026Mpc/h (95% C.L.). Following Ref. [67], these FIG. 10: The solid lines show the same transfer functions limits have been obtained by performing a comprehensive shown in Fig. 9. Dashed lines show the thermal WDM trans- Markov hain Monte Carlo (MCMC) analysis of the full fer functions with matched half-mode scales. Due to the dif- parameter space affecting the one-dimensional flux power ference in the shape parameters β and γ in the fitting form of Eq. (34) between the two models, matching the half-mode spectrum, which is the Lyman-α forest physical observ- scales requires the cutoff parameters, α, to be related by ap- able, with a data set consisting of the high-resolution and proximately α ≃ 2αWDM high-redshift (4.2

γ -1.2 ing the astrophysical parameters, we model the redshift -1.25 evolution of the temperature of the intergalactic medium -1.3 as a power law, imposing flat priors on both its ampli- -1.35 γ = -(0.0055 ln(α)2 + 0.083 ln(α) + 1.35) tude and tilt (once again, see Ref. [67] for further de- -1.4 tails). Finally, we adopt conservative Gaussian priors on -5 -4 -3 -2 -1 0 the mean Lyman-α forest fluxes F (z) , with standard ln(α) h i deviation σ =0.04 [57], and a flat prior on fUV, which is an effective parameter accounting for spatial ultraviolet FIG. 11: The fitted values for the parameters β and γ for fluctuations in the ionizing background. transfer functions whose cutoff parameters, α, span from 0.005 to 1.5 Mpc/h. In the fitting functions for β and γ, Last but not least, we adopt a flat prior on α in the α has units of Mpc/h. interval [0, 0.1] Mpc/h, while the parameters β and γ are derived analytically, per each MCMC step, according to the expressions reported in Fig. 11. For further details on the data set, simulations, and methods that we have

1 used, we address the reader to any of the aforementioned We follow the convention of Ref. [68] and define the half-mode references [67, 70, 71]. scale via T (khm) = 0.5, noting that this convention is different k from the half-mode scale, 1/2, defined in Refs. [66] and [67], for For comparison, just as matching the half-mode scale T 2 k . which ( 1/2) = 0 5. of the thermal WDM transfer function with that of our 12

107 0.8 -1 f = 10 0.7 106 0.6

105 0.5

D -3 γ f = 10 0.4 4 (Mpc) 10 fs λ 0.3

103 0.2

f = 10-5 0.1 102 101 102 103 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 TRH (MeV) α (Mpc/h)

FIG. 12: Limits on the Lorentz factor γD and the reheat FIG. 13: A plot of the relationship between the fitting pa- temperature. The shaded regions correspond to the 1σ and 2σ rameter α and the free-streaming length as calculated by Eq. bounds, α = 0.011Mpc/h and 0.026Mpc/h, respectively. The (29). Black dots represent points for which we have used −1 thick dashed line represents kfs = (λfs) = 12.6h/Mpc as our model parameters to calculate the free-streaming length calculated by Eq. (29). The discontinuity at TRH ≃ 170MeV and obtain the transfer function using CLASS. The grey line occurs due to the sudden change in g∗ during the QCD phase shows the fit to the data given by Eq. (35). transition. Thin solid lines show the contours of f required to obtain the observed dark matter abundance in the absence of annihilations. The thick solid line shows the bound on γD as a function of TRH imposed by BBN, derived in Section II A. tion depend on the same combination of our parameters, γDvDaRH/a0, there is a simple relationship between the scale of suppression, α, and the free-streaming length cal- nonthermal model requires α 2α , the α value culated by Eq. (29): ≃ WDM WDM of a 3 keV WDM particle, αWDM 0.015 Mpc/h [67], is approximately a factor of 2 smaller≃ than that of our phys 0.908 λfs,0 Mpc 95% C.L. bound on α. Our limits on γD and TRH cor- α 0.177 (35) responding to our 68% and 95% C.L. bounds on α are ≃ Mpc ! h shown in Fig. 12. Scenarios in which the dark matter is born at too high of a velocity (large γD) or in which We show this relationship in Fig. 13. In our model, the the radiation-dominated era is too short (low TRH) are bounds on α can be easily used to limit the free-streaming part of the excluded parameter space for our relativistic length, and thereby the parameters γD and TRH on which nonthermal dark matter model. The thin grey lines rep- it depends. resent contours of constant f, the fraction of the scalar’s energy imparted to the dark matter particle that is required to obtain the correct relic abundance for a given reheat temperature without dark matter annihilations. C. Milky-Way Satellites We can see that, in the absence of annihilations, the allowed values of γD are not large enough for the required value of f to be of order unity. In addition to structures inferred by Lyman-α data, we We also show the outline (dashed) of the parameter can also constrain our model using observed structures in space in which the free-streaming length, calculated by the Milky Way. Simulations of thermal warm dark mat- Eq. (29), is naively probable by Lyman-α data, i.e. ter provide an indication of how the suppression expected kfs < 12.6h/Mpc. As can be seen in Fig. 12, limiting in the matter power spectrum decreases the abundance the free-streaming length provides a bound that is com- of collapsed objects. The subhalo mass function in sim- parable to those obtained from the full consideration of ulations with WDM characterizes this underabundance effects to the matter power spectrum; the free-streaming [72], scales of a particle on the boundary of our 68% and 95% C.L. regions are kfs = 11.4h/Mpc and 28h/Mpc, respec- dN dN M −ε tively. While examining effects on the matter power spec- = 1+ δ hm , (36) dM dM M trum leads to more robust bounds within our parame- WDM CDM ter space, calculations of the free-streaming length are more readily performed. Fortunately, as both the free- where M is the subhalo mass, δ = 2.7 and ε = 0.99, streaming length and the dark matter distribution func- and Mhm is the mass scale associated with the half-mode 13 scale2: large free-streaming length will wipe out this enhancement to structure formation. 3 4π π By investigating the velocity evolution and distribution Mhm = ΩDM ρcrit,0 , (37) 3 khm of dark matter produced nonthermally from the decay of a massive scalar field, we have confirmed that retaining where ΩDM is the fraction of the critical density in dark the linear enhancement to structure growth requires the matter. dark matter to be produced largely nonrelativistic. De- Knowing the abundance of satellites of our own spite the early creation of many particles, and their loss galaxy, constraints can be placed on the amount of al- of momentum due to adiabatic cooling, the continuous lowed suppression in the subhalo mass function. Us- creation of new, hot particles prevents the average dark ing a probabilistic analysis of the MW satellite popula- matter velocity from decreasing appreciably during the tion and marginalizing over astrophysical uncertainties, EMDE. The average particle at reheating is nearly as Ref. [58] found an upper limit on Mhm in Eq. (36), relativistic as those newly produced from decay. And be- 8 M < 3.1 10 M⊙ (95% C.L.), which implies that the hm × cause a majority of the dark matter is created around half-mode scale must satisfy khm > 36h/Mpc. This reheating, essentially negligible fractions of dark matter bound on khm can be used to constrain any dark matter particles have velocities low enough to preserve enhanced model that has a transfer function comparable to WDM, structure formation. as we have shown ours to be in Fig. 10. Though the We next investigated the upper limit on the dark mat- transfer function of our model does differ slightly from ter velocity required to preserve the structures we ob- that of WDM, the differences occur only when the nCDM serve. Dark matter particles born with relativistic veloc- power spectrum is already greatly suppressed compared ities have free-streaming lengths that may also washout to that of CDM, T (k) . 0.1. The fitting form to our observed small-scale structures. Lyman-α forest data transfer function, Eq. (34), implies that the half-mode provides the best-known probes of inhomogeneity at scale is given by: small scales, and ensuring that structure formation at 1/β these scales is not observably suppressed constrains the 1 1 1/γ parameter space of nonthermal dark matter. khm = 1 , (38) α " 2 − # Using the software CLASS, we obtained the matter power spectrum resulting from our model of nonthermal and the constraint khm > 36 h/Mpc directly translates dark matter. A transfer function was used to compare to α< 0.026 Mpc/h, matching our 95% C.L. bound from our spectrum to that of the standard CDM scenario and Lyman-α constraints. showed a cutoff in the power at small scales in our nonthermal scenario similar to that due to WDM. We fit the form of our transfer functions using three free param- V. CONCLUSION eters, one of which, α describes the scale of the cutoff in the transfer function and the other two describe its The inclusion of a period of effective matter domina- overall shape. The shape of our transfer function varies tion between inflation and BBN is an amply motivated slightly with the cutoff scale and the two parameters de- alternative to the standard thermal history of the Uni- scribing its shape are well determined by analytic func- verse. If dark matter is produced nonthermally during tions of α. this era, the viable parameter space for the dark matter We obtained limits on the allowed scale of the cut- annihilation cross section widens greatly, as large ranges off in the transfer function by performing a compre- of production and annihilation efficiencies can combine hensive MCMC analysis using Lyman-α observations: to result in the correct relic abundance. α< 0.011Mpc/h (68% C.L.) and α< 0.026Mpc/h (95% Nonstandard thermal histories could potentially have C.L.). From this constraint, we were able to place limits observable consequences. Unlike the typically assumed on the allowed velocity imparted at scalar decay for a period of radiation domination following inflation, in given temperature at reheating, summarized in Fig. 12. which subhorizon density perturbations grow logarith- We also found a simple relation between α and the dark mically, EMDEs provide an era of linear growth. Linear matter free-streaming length that allows one to use the growth would enhance structure formation on scales that limits on α to limit the free-streaming length which can enter the horizon during this era, possibly leading to ob- be calculated analytically from γD and TRH. servable effects. However, in the absence of fine-tuning, it Observations of the abundance of MW satellite galax- is likely that dark matter produced nonthermally will be ies provide another probe of the small-scale power spec- imparted with relativistic velocities, and its subsequently trum. Using the halo-mass function obtained from WDM simulations, limits on the cutoff scale can also be placed on the WDM transfer function by requiring consistency between the decreased abundance of collapsed objects ex- 2 We have verified with the authors of Ref. [72] that they used the pected in WDM scenarios, compared to CDM, and the same definition of khm that we have presented here. abundance of satellites observed orbiting the MW. These 14 constraints are applicable to any model of dark matter model. We have assumed here that all dark matter par- with a transfer function comparable to that of WDM. ticles are born from the decay process with the same Comparison of the parameter values that fit the transfer velocity, though this need not necessarily be the case. function of our model to those that fit WDM naively im- Including a range of possible velocities could tighten or ply marked differences between the two models; however, relax our bounds, depending on the exact distribution of matching the transfer functions at the same half-mode the imparted velocity. We have also assumed that any scale shows the two models to be remarkably similar, dif- annihilations take place via s-wave processes. If annihi- fering significantly only on scales at which the power was lations occur preferentially for faster particles, this could already greatly suppressed. In our model, MW satellite shift the peak of our velocity distribution to lower veloci- considerations provide a practically identical bound on α ties. Finally, we have only considered the cooling of dark to those of Lyman-α data. matter due to the redshifting of its momentum. If dark Using the impact on the matter power spectrum ex- matter is allowed to exchange momentum with Standard pected in the model of nonthermal dark matter we have Model particles, this could provide an additional mecha- presented here, we have constrained the physical param- nism to reduce its momentum and lower the peak velocity eters of our model: the velocity imparted at the dark of the velocity distribution, perhaps allowing for the for- matter production, characterized by the Lorentz factor mation of microhalos from perturbations that grow lin- γD, and the temperature at reheating, TRH. Constraints early during the EMDE. We leave these investigations for in this parameter space also inform the allowed value of future work. f, the fraction of the decaying component’s energy allo- cated to the dark matter, that is required to obtain the correct relic abundance in the absence of dark matter annihilations. While naturalness would suggest a value Acknowledgments of f 0.5, our constraints show that f must be less than ∼ 10−4, implying that annihilations must be considered∼ to avoid finely tuning f. Our limits within the The authors would like to thank Kimberly Boddy, Ju- parameter space of TRH and γD can also be equivalently lian Muñoz, Jessie Shelton, and Matteo Viel for use- viewed as limits on the scalar decay rate Γφ [see Eq. ful discussions. This work made use of the Ulysses (1)] and the mass hierarchy between the scalar parent SISSA/ICTP supercomputer. C.M. and A.L.E. were and daughter dark matter particles for a two-body decay partially supported by NSF Grant No. PHY-1752752. (mφ =2γDmχ). R.M. is partially supported by the INFN-INDARK PD51 There are many opportunities for extensions to our grant.

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