Cannibal Dark Matter and Large Scale Structure

Cannibals are dark matter particles with a scattering process that allows three particles to annihilate to two. This exothermic process keeps the gas of the remaining particles warm long after they become non-relativistic. A cannibalizing dark sector which is decoupled from the Standard Model naturally arises from a pure-glue confining hidden sector. It has an effective field theory description with a single massive interacting real scalar field, the lightest glueball. Since warm dark matter strongly suppresses growth of structure cannibals cannot be all of the dark matter. Thus we propose a scenario where most dark matter is non-interacting and cold but about 1 percent is cannibalistic. We review the cannibals' unusual scaling of the temperature and energy and number densities with redshift and generalize the equations for the growth of matter density perturbations to the case of cannibals. We solve the equations numerically to predict the scaling of the Hubble parameter and the characteristic shape of the linear matter power spectrum as a function of model parameters. Our results may have implications for the $\sigma_8$ and $H_0$ problems.


I. INTRODUCTION
Dark matter could be a single species of particles with only gravitational interactions as in the cosmological standard model, ΛCDM. Alternatively, it might have multiple components.
If there is a dominant non-interacting component then other components can have interesting non-gravitational interactions. Recent observations of the Cosmic Microwave Background (CMB) and matter power spectrum (MPS) are already sensitive to non-standard dark matter components which comprise only a few % of the total, stage 4 experiments will be able to push the sensitivity below the percent level. Interestingly, precision fits with current cosmological data show some tension with predictions of ΛCDM for the expansion rate of the universe H 0 [1,2] and the amplitude of fluctuations in the MPS on galaxy cluster scales, σ 8 [3][4][5][6][7][8] 1 .
Motivated by the significant projected improvement in measurements of the MPS we propose and explore the possibility that a small component of dark matter is "cannibalistic".
Cannibal dark matter consists of massive particles with an efficient number-changing selfinteraction [29]. The most important process that such interactions mediate is from three particles in the initial state to two particles in the final state. In such a 3 → 2 process mass is turned into kinetic energy of the outgoing particles which heats the gas of particles 2 .
If there are also rapid 2 → 2 interactions the cannibalizing particle gas remains in thermal and chemical equilibrium, and can be described by the Boltzmann distribution with a temperature T (a) and vanishing chemical potential. Because of the cannibalization process the temperature drops only logarithmically with the scale factor T /m ∼ 1/ log a. This is very different from the case of non-relativistic matter which cools very quickly, T /m ∼ 1/a 2 .
Cannibal matter also has an unusual scaling of its number and energy densities. The number density dilutes like n can ∼ 1/(a 3 log a) where the 1/a 3 is the usual volume dilution while the 1/ log a comes from the cannibalization. Ignoring kinetic energy, the energy density is then simply ρ can ≈ m n can . Thus the energy density of cannibals scales intermediate between 1 For recent work motivated by these discrepancies see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. 2 Cannibals cannot constitute the entirety of the dark matter in the Universe precisely because they are heated up by their self-interactions and that interferes with the formation of structure [30,31]. Proposed solutions to this problem are to let cannibalism end much before matter domination [32][33][34][35] or to cool it through couplings to the Standard Model, like in the ELDER [36] or SIMP [37] paradigms. The SIMP mechanism has been the object of intense study in recent years [38][39][40][41][42][43][44][45][46]. ordinary matter for which ρ m ∼ 1/a 3 and radiation where ρ r ∼ 1/a 4 . Note that for these scalings to hold it is necessary that the cannibal particles are isolated from all other sectors, i.e. no significant interactions, so that any heat produced from cannibalization does not dissipate to other sectors.
We now discuss the impact of the cannibal fluid on cosmology with particular attention to the MPS. First, note that since the cannibal temperature decays very slowly the cannibal fluid has significant pressure P/ρ ≈ T /m. This pressure prevents growth of density perturbations in the cannibal fluid, instead one obtains "cannibal acoustic oscillations".
Overdensities in the cannibal fluid remain small and make only negligible contributions to the gravitational potential. On the other hand, the cannibal fluid does contribute to the overall energy density of the universe which determines the Hubble expansion rate. Since the gravitational potential drives the growth of structure whereas the Hubble expansion acts to slow it ("Hubble friction") the net effect of the cannibal fluid is to suppress the MPS. This is the main result of our paper.
In Section III we derive this result quantitatively. The connection to the physical explanation in the previous paragraph will become clear after we derive the Mészáros equation for the growth of cold dark matter (CDM) perturbations δ cdm in the presence of the cannibal fluid: This equation is valid during matter domination and for perturbations which are deep inside the horizon. 3 Here the derivatives are with respect to the scale factor a, and ρ cdm and ρ can are the background (average) energy densities of the cold dark matter and the cannibals, respectively. For zero cannibal energy density this has the usual linear growth of the matter perturbations δ cdm ∼ a as a solution. Expanding for small energy density in cannibals ρ can ρ cdm one finds a suppressed rate of growth: δ cdm ∼ a 1−γ with γ = 3 5 ρ can /ρ cdm . Given that current data suggest a suppression of matter perturbations by ∼ 5% and that the universe expands by a factor of a today /a equality ∼ 10 3 during matter domination we see that the preferred parameter space should have on the order of 1% of matter in cannibals, i.e. a fraction ρ can /ρ cdm ∼ 1% which slowly changes in time due to the extra 1/ log a in ρ can .
The minimal field theoretic model which exhibits cannibalism has a real scalar field with the Lagrangian In this minimal cannibal (MC) model m is the mass of the particle, λ denotes the overall strength of φ-interactions and κ 3,4 are numbers which we will take to be of order 1. The interactions mediate φ-number preserving φφ → φφ processes as well as φ-number changing processes such as φφφ → φφ (with a rate proportional to λ 6 ). At temperatures above the φ mass the φ particles can be described by an interacting relativistic fluid in equilibrium. Once the φ-fluid cools below the mass of the particles the 3 → 2 cannibalism interaction starts processing mass into temperature. This slows the cooling of the fluid. In Fig. 1 we plot the temperature-to-mass ratio as a function of scale factor for an example point in parameter space of the minimal cannibal model. Note the transition from relativistic behavior to cannibalism at T /m ∼ 1/3 ↔ a can ∼ 10 −6 and the decoupling transition to non-relativistic matter at a nr ∼ 10 −1 . The ratio of scale factors between start and end of the cannibalistic phase a nr /a can ∼ 10 5 depends on the strength of the interaction λ. We will be interested in models where λ is strong (between 1 and 4π); then the duration of cannibalism a nr /a can is between 10 −4 and 10 −5 with only a mild dependence on other model parameters. From preceding discussions it is clear that we can choose parameters in the cannibal sector such that the cannibalistic phase overlaps with the matter-dominated era of the universe.
This choice of parameters is the most interesting because then the cannibals suppress the matter power spectrum. We dedicate most of this paper to its study. In Fig. 2  (2) one also obtains higher-dimensional couplings of the form λ n−2 φ n /m n−4 which contribute to scattering with the same parametrics as the renormalizable couplings.
The cannibalism phase is not sensitive to the precise form of the interactions: what matters is that the number-changing transitions are faster than the Hubble expansion. Then the cannibal fluid satisfies thermal and chemical equilibrium and its evolution becomes independent of the details of the spectrum of glueballs and interactions. Note also that this UV completion very naturally explains the absence of couplings between φ and the Standard Model. In the UV theory gauge invariance forbids any renormalizable coupling between the two sectors. We describe such UV completions and study the dependence of our results on the UV completion in Section IV.
Finally, we do not consider but cannot resit mentioning the possibility that the cold dark matter required in our model might be "Baryons" or "Mesons" made of heavy dark quarks charged under the dark gauge group [47] although important details of the confining phase transition and entry into the cannibal phase would change from what we study in this paper.
We study the MC model of Eq. (2) and its thermal history in Sec. II where we also estimate the boundaries of the preferred parameter space. Within these boundaries we compute the effects of cannibalism on the matter power spectrum in Sec. III. Sec. IV gives possible UV realizations of the MC model in terms of simple confining (pure-glue) non-Abelian gauge theories. We also study the depedence of our results on the UV completion of the MC model.
In the Conclusions (Sec. V) we discuss the shape of the predicted MPS as a function of model parameters. We review the derivation of the background and perturbation equations for the cannibal fluid starting from the Boltzmann equation in an Appendix; our results agree with those given in [48].

II. THE MINIMAL CANNIBAL: THERMAL HISTORY AND PARAMETERS
In this Section we study the thermal history of the MC model fluid, identify the most useful parameters to describe it, and explore their parameter space. In order to do this we need to consider what the properties of the cannibal fluid are.
During its relativistic and cannibalistic phases the φ-fluid is in both thermal and chemical equilibrium. This means that its phase space distribution function f (p, a) is entirely parameterized by the mass of the particles m and the temperature T of the fluid 4 : where E = m 2 + p 2 is the energy of the φ particles. Here we only consider the homogeneous and isotropic background of the cannibal fluid which means that f does not depend on position. We will study x-dependent perturbations about this background in the following Section. The time dependence of f , encoded in the scale factor a(t), arises solely from that of the temperature. All other background quantities that describe the φ-fluid (such as energy and number densities) are momentum integrals of f , and therefore they depend on the two parameters m and T (a).
Since the cannibal fluid has no interactions with other fluids its (comoving) entropy S can is conserved. This makes S can a useful parameter of the MC model. We now derive formulae for the temperature and energy density of the φ-fluid in terms of the model parameters m and S can . From the Second Law of Thermodynamics: In the relativistic limit, T m, the phase space distribution function, Eq. (3), is easily integrated to obtain expressions for the energy density ρ = π 2 30 T 4 and pressure P = ρ/3 so that: Solving for T we find: Note that T ∼ 1/a and ρ can ∼ 1/a 4 , as expected for radiation components. 5 Once T ∼ m the φ-fluid enters its cannibalistic phase. After the temperature drops sufficiently far below the mass an expansion in T /m becomes appropriate, and the dominant contribution to the energy density comes from the mass of the particles, ρ can ≈ mn can , where 4 Here T (a) denotes the temperature of the cannibal fluid which may be different from the temperature of the Standard Model (e.g. photons). 5 Eq. (5) contains a factor of g that accounts for the degrees of freedom of the dark sector. This factor is 1 in the φ cannibal model but will be different in UV completions. n can = ( mT 2π ) 3/2 e −m/T is the equilibrium number density of φ. The contribution of the pressure P can ≈ T n can to the entropy in Eq. (4) is smaller by T /m relative to ρ can so that: and solving for T and ρ can in a leading-log approximation we have: Note that T ∼ 1/ log a and ρ can ∼ 1/(a 3 log a) as stated in the previous Section.
Having written T and ρ can as functions of a and the parameters m and S can , we now study the parameter space. Our goal is to estimate the values of the parameters for which the cannibal sector suppresses the matter power spectrum by about the amount that is preferred by the σ 8 measurements. As shown in the Introduction this requires a fraction of dark matter energy density in the φ-fluid f can ≡ ρ can /ρ cdm ∼ O(1%). Of course, since , this fraction evolves as f can ∼ 1/ log a. But the change in f can during matter domination is small enough (of order of a few) that we ignore it for the purpose of estimating the rough region of m -S can parameter space where we can expect to find good fits. The good region of parameter space is the one in which the cannibalism phase overlaps with matter domination, which corresponds to conditions on a c and a nr , and in which f can ∼ O(1%). In the remainder of this Section we use these conditions to derive A reader who is not interested in the following somewhat tedious derivation of these boundaries of the relevant parameter space may skip ahead to Sec. III where we derive and solve the density perturbation equations.
We first derive the lower bound on m. Define the scale factor a = a can at which T (a can ) ≡ m/3, i.e. where the φ-fluid stops being relativistic and starts cannibalizing. From Eq. (8) we obtain a can ∼ 10 S 1/3 can /m. Since we want cannibalism to act during matter domination, we require the start of cannibalism to be before matter-radiation equality, i.e. a can < a eq .
Ignoring the log a dependence (for simplicity) and using a can ∼ 10 S We solve this for m, substitute ρ can = f can ρ cdm , evaluate it today (a = 1) and impose a can < a eq to obtain: which for a eq ≈ 3 × 10 −4 gives the lower bound: At the edge of the preferred parameter space, when m saturates the bound, the φ-fluid enters its cannibalistic phase right at matter-radiation equality. Then the UV completion of the φ model is needed to determine the cannibal sector energy density for a < a eq . Thus in this case the matter power spectrum is sensitive to details of the UV completion such as the glueball spectrum and the size of the UV gauge group. We will study this model dependence in Sec. IV. For masses much smaller than the bound the cannibal sector is still relativistic at a eq . In that case the cannibal fluid behaves like extra radiation (∆N eff ) at the time of the CMB. Imposing observational bounds on ∆N eff bounds the energy density in the cannibal fluid at a eq and by the time cannibalism turns on at a can > a eq the energy density in the cannibal fluid has already become negligible compared with that in ΛCDM (orange curve of Fig. 2). Thus this is not a region in parameter space that we are interested in.
We can also derive an upper bound on m. To do so we first solve for the scale factor a nr when the 3 → 2 interactions decouple and the φ-fluid transitions from cannibal behavior to standard non-relativistic behavior. Dimensional analysis allows us to estimate the nonrelativistic 2 → 2 and 3 → 2 scattering cross sections in the φ theory from Eq. (2): where α ∼ λ 2 /(4π), Γ ij are the i → j interaction rates, and we have been cavalier with factors of order 1 and π. Keeping in mind a strongly coupled UV completion of the cannibal sector we expect α somewhere between 1 and 4π.
Eventually Γ 32 cannot keep up with the rate of expansion of the Universe H and the 3 → 2 interactions decouple and cannibalism stops at a nr . Setting Γ 32 = H and using Eqs. (14) and (10) we can solve for the duration of the cannibalistic phase Note the small exponent of 1/6. This shows that the duration of the cannibalism phase is only weakly dependent on the model parameters m and S c . In particular, the duration of the cannibalism phase is rather insensitive to when the decoupling occurs. For example, if cannibalism ends at matter-radiation equality (a nr = a eq ) then (H(a eq )/eV) 1/6 ∼ 10 −5 ; whereas if it ends today (a nr = 1), then (H 0 /eV) 1/6 ∼ 10 −6 ; a change of only one order of magnitude. The duration of the cannibalistic phase is therefore between 4 and 5 decades in the scale factor: We will use the approximation a can ∼ 10 −5 a nr . Substituting this in Eq. (10) yields: In order to find an upper bound on the interesting range of m we impose a condition on a nr , the scale factor when cannibalism stops. Demanding that cannibalism lasts throughout matter domination and does not end before today so as to maximize the suppression of the MPS is a possibility. But this is really too aggressive because even when cannibalism stops midway through matter domination the MPS is suppressed relative to ΛCDM. We impose -admittedly somewhat arbitrarily -that a nr > ∼ 10 −2 . This together with Eq. (17) implies For masses much larger than this bound the end of cannibalism occurs too close to (or before) matter-radiation equality, so that the φ-fluid clusters like cold dark matter during matter domination as discussed in the previous section (blue curve in Fig. 2). Comparing Eq. (18) and Eq. (12) we see the range of masses, eV < m < keV, which satisfies both constraints.
Having restricted the mass of the φ particles to a range for which cannibalization has an interesting effect on the MPS we now focus our attention on the other parameter of the MC model, the entropy. Starting again from the relationship between the energy density and the entropy in Eq. (8), approximating log a −1 can ∼ 8, demanding that the energy density in cannibals be a small fraction f of that in the the ΛCDM sector, and evaluating energy densities today we obtain where clearly bigger than H 0 ∼ 10 −33 eV. This is not surprising because 2 → 2 interactions are much more rapid than 3 → 2 interactions which are suppressed by an additional power of the particle number density.
In summary, in order for the cannibalistic phase to overlap with matter domination and suppress the matter perturbations at galaxy cluster scales by about 5% we need f can, 0 ∼ 0.01 and a can < ∼ a eq and a nr > ∼ 10 −2 . This corresponds to the parameter range in Eq. (9).

III. DENSITY PERTURBATIONS IN THE CANNIBAL MODEL
With the thermal history and parameter space of the MC model determined we now study the effects of the cannibal fluid on density perturbations. In particular, we derive the suppression of the matter power spectrum (MPS) and solve for the region in parameter space with the correct amount of suppression to address the large-scale structure (LSS) discrepancy on σ 8 . We start from the equations governing the evolution of the cosmological perturbations in the energy density and velocity of the different components of the Universe, focusing on the dark matter and cannibal fluids. In this Section we simply state the equations and study their solutions, first numerically and then analytically using simplifying approximations. We review the derivation of the perturbation equations in Appendix A.
The equations for the cannibal and CDM perturbations in Fourier space are [48]: where the dots represent derivatives with respect to conformal time η; k is the Fourier momentum mode, H ≡ aH =ȧ/a, δ ≡ δρ/ρ and θ are the density contrast and the velocity divergence perturbations, while ϕ and ψ are the scalar perturbations of the metric. 6 Finally, w can ≡ P can /ρ can is the equation of state of the φ-sector, while c 2 s ≡Ṗ can /ρ can = w can −ẇ can 3H(1+wcan) is the speed of sound of the φ-fluid. Recall that during the cannibalistic phase ρ can ≈ mn can and P can ≈ T n can and therefore w can ≈ T /m ∼ 1/ log a.
For the rest of this Section we make the following simplifications: i. ignore the baryons, adding their energy density to that of CDM, ii. ignore the anisotropic stress of the neutrinos, taking ϕ = ψ, and iii. add the neutrino energy density to that of the photons. Since we are only interested in the effects of cannibals on the MPS, we will compare the MPS in the theory with cannibals to the MPS in ΛCDM, evaluated today, and denote the ratio by R(k): where the index c denotes the value in the theory with cannibals, while Λ means ΛCDM.
With the assumptions mentioned above, we solved Eqs. (21)-(24) numerically and calculated R(k). We now describe the solutions for δ can and δ cdm , and the resulting R(k).  matter. This can be seen in the large-a behavior of the green curve in Fig. 3.
The cannibal fluid affects the perturbation equations for the CDM in two ways: through its contributions to the gravitational potential term k 2 ψ in Eq. (24) and through its contribution to the energy densities in the Hubble friction term −Hθ cdm in Eq. (24); theφ term in Eq. (23) is negligible for the modes of interest. Since δ can oscillates and does not grow during the cannibalistic phase its contributions to the gravitational potential ψ remain negligible and do not enhance the growth of CDM perturbations. On the other hand, the contribution of ρ can to the Hubble expansion rate during matter domination and therefore to the Hubble friction term is significant. The net effect, no enhancement of the potential but more friction, is to slow the growth of CDM perturbations relative to ΛCDM. Thus the MPS is suppressed in theories with cannibals. This is the main result of our paper.  of the preferred value of σ 8 according to [4], about a 10% suppression (R(0.2h Mpc −1 ) = 0.9).
We see that this roughly corresponds to f can, 0 ∼ 1%. The orange region corresponds to the lower bound on m we estimated in Sec. II, made up of those parameter values for which a can > a eq . Deep inside this region the φ-fluid behaves just like radiation. The blue region corresponds to the upper bound also estimated in Sec. II, for which a nr < 10 −2 . Deep inside this region the φ-fluid behaves like ordinary CDM. Finally, the red band corresponds to a region in parameter space in which the φ-fluid would contribute too much radiation (∆N eff > 0.66) to the energy density of the Universe at the time of Big Bang Nucleosynthesis [49]. However, as we will show in Sec. IV this constraint is relaxed in UV completions of the MC model because the energy density in radiation in the UV is reduced in such models. The green band is an estimate for the suppression that gives a σ 8 within 1σ of the value quoted in [4]. The orange region corresponds to MC models that enter the cannibalistic phase after matterradiation equality, while the blue one corresponds to those for which cannibalism ends before a = 10 −2 . In red are those models whose ρ can contributes to ∆N eff | BBN > 0.66 [49] when they are in their radiation phase. The red, green, and blue points correspond to the three choices of m and S can in Fig. 4.
Having shown that the presence of cannibals suppress the MPS by numerically solving the equations for the perturbations, we devote the rest of this Section to understanding this result from Eqs. (21)- (24). We will only be interested in k modes which are well inside the horizon during matter domination, i.e. modes for which k 1/η eq ∼ 0.01Mpc −1 .
Let us start with the cannibal perturbations. For modes deep inside the horizon the gravitational potential is approximately constant so that we can ignore derivatives of ψ. In addition, we can use w can 1, c 2 s 1 to drop all subleading terms in Eqs. (21) and (22).
Then taking the second derivative of δ can and substituting Eq. (22) into Eq. (21) yields: where the term on the right-hand-side is the solution of the Poisson equation Anticipating that the CDM contribution dominates the sum during matter domination We now turn our attention to the CDM perturbations. Following the same procedure as before, combining Eqs. (23) and (24) gives: where ψ is given by Eq. (27) but only keeping the CDM contribution ρ cdm δ cdm in the sum.
Using this, rewriting the Hubble parameter in terms of the energy density during matter domination ρ tot ρ cdm + ρ can , and changing variables from η to a we can write: Were it not for the cannibals, this would be the Mészáros equation during matter domination, whose growing solution is δ cdm ∼ a. Eq. (29) shows that cannibal dark matter increases the Hubble friction (δ cdm term) felt by the CDM perturbations but does not contribute to the gravitational pull from the Poisson term. This explains the smaller rate of growth of δ cdm we discovered in our numerical solutions.

To get a rough idea of what this change in the growth rate is let us further simplify
Eq. (29) by taking ρ can /ρ cdm 1 and dividing by ρ cdm + ρ can to arrive at Eq. (1). This is easily integrated in an approximation where we neglect the slow log a dependence of ρ can . In fact, this equation for the growth of perturbations without the log a dependence applies to a model with CDM and a subdominant component of dark plasma [11,19]. The solution for the growing mode is the power law δ cdm ∼ a 1− 3 5 ρcan/ρ cdm [11,19,50], a growth rate smaller than the linear one from the usual Mészáros equation. For the decaying mode, one finds a − 3 2 + 3 5 ρcan/ρ cdm . In the cannibal case the exponent is a slowly varying integral function of f can that depends on a (because of the slow logarithmic decay of f can ), which explains the change in the slope of the suppression we saw in Fig. 4.

IV. NATURAL UV COMPLETIONS FROM SECLUDED GAUGE SECTORS
In this Section we discuss our favorite UV completion of the MC model, a simple non-Abelian "pure-glue" gauge sector which confines at low energies and produces canniballistic glueballs.
Consider an SU (N ) gauge theory with no light matter fields. Such a theory has two marginal operators, the gauge kinetic term (30) and the CP-violating θFF term. We set θ = 0 mostly because it makes no qualitative difference but also because it is zero if the dark sector preserves CP. All other operators as well as couplings to the SM are irrelevant (in the sense of their scaling with energy) and therefore do not impact the confining dynamics and cannibalism. The dark sector could be coupled to the SM in the UV by heavy matter fields which are charged under both the SM and dark SU (N ) gauge group. Then it would be natural for the two sectors to have a common temperature in the UV. However if inflation and reheating occur at temperatures below the coupling of the two sectors or if there is a phase transition or there are heavy particles with associated entropy production then the two sectors may end up with very different temperatures. We take the temperature of the cannibal sector to be a free parameter T .
Assuming that the SU (N ) gauge coupling in the UV is not too small the coupling runs strong in the IR and the theory confines at temperatures below some scale Λ c . The confining gauge theory has a spectrum of stable glueball states with varying spin and parity quantum numbers [51][52][53][54]. The most important of these glueballs for cosmology is the lightest glueball φ with mass m ∼ Λ c which is a parity even scalar and carries no conserved quantum number.
It has number-changing interactions and its low energy effective description is the Lagrangian Note that the number density of heavier glueballs φ H is exponentially suppressed relative to φ at low temperatures even if they are stable because they can efficiently annihilate Since the φ particles have no conserved quantum number they are unstable to decay.
φ has no other particles to decay to in the dark sector but it can decay to gravitions or SM particles through higher dimensional operators. For example, the width to decay into gravitons is roughly m 5 /M 4 P lanck ∼ 10 −108 eV[m/eV] 5 . This is much smaller than the Hubble constant today for the masses we consider. In fact, even decays mediated by a dimension 6 operator suppressed by a scale of 1 GeV are too slow to be cosmologically relevant for m ∼ 1 eV. This justifies treating the φ particles as stable.
This completes our description of the UV completion of the MC model. In most of the interesting parameter space, Fig. 5, the UV completion is not needed for the computation of the MPS. This is because either i. the confining transition happens well before matterradiation equality and the energy density in the cannibal sector is negligible during and before the transition or ii. because the confining transition happens well after matterradiation equality. In the latter case the cannibal Sector is "gluon" radiation well into matter domination and its energy density redshifts to being negligible before cannibalism even starts. Thus only in models where the confinement transition happens close to matter-radiation equality (red dot in Fig. 5) is the UV completion needed for the computation of the MPS. We study this special case in the remainder of this Section.
Computing the cosmological evolution of the cannibal fluid through the confining phase transition exactly is very difficult as one would have to solve for the dynamics of a strongly coupled thermal gauge theory [55]. We take a simplified approach and match the UV theory with N 2 − 1 weakly interacting gluons onto the confined theory of the lightest glueball φ.
This matching depends on the size of the gauge group, N , the details of the phase transition (it can be 1st or 2nd order), the glueball spectrum, and couplings in the strongly coupled regime. It is believed that for N = 2 the phase transition is 2nd order so that entropy is conserved in the phase transition, for N = 3 it is probably weakly 1st order and for higher N strongly 1st order [56,57]. To study the sensitivity of the MPS predictions to this model-dependence we look at two simplified cases: a smooth 2nd order phase transition with conserved entropy and a simplified glueball spectrum and a very strongly 1st order phase transition with a jump in entropy and temperature (see for example [58]).
To model the 2nd order phase transition we assume that the full theory is described by g * = 2(N 2 − 1) bosonic degrees of freedom. The lightest, φ, has mass m, and the others have a common mass M which we vary from 1.25m to 3m. We also assume entropy conservation and that the theory remains in chemical and thermal equilibrium throughout the phase transition. Then all distributions are simply given by Boltzmann distribution functions for the 2(N 2 − 1) degrees of freedom. In the UV, when all masses can be ignored, this reproduces the physics of free SU (N ) gluons. In the transition region where T ∼ m the "heavy glueballs" of mass M pair annihilate into the lightest glueballs φ. And in the IR, when the temperature drops below m, only the cannibals remain. For the very strongly 1st order phase transition we match a UV theory of N 2 − 1 massless gluons onto the IR theory with a jump in entropy at a scale factor a can . We choose the matching scale factor such that the temperature evaluated in the IR theory (the theory of the cannibal φ) equals m/3 at the matching scale. There we match onto the UV theory with g * = 2(N 2 − 1) massless bosonic degrees of freedom and a jump in entropy (increasing from the UV to the IR) by a multiplicative factor which we vary from 1 to 2. The discontinuity in degrees of freedom and entropy at the matching point also implies a discontinuity in other background quantities.

V. CONCLUSIONS
We have studied the possibility that a subdominant component of the dark matter might posses a cannibalistic phase. If this phase overlaps with matter domination then the most other two parameters characterizing the cannibal fluid are its entropy S can and the mass m of the cannibal particle. We conclude this Section with two plots which show the impact of these two parameters on the predicted matter power spectrum shape.  Note that this second dependence is similar to that of the MPS on neutrino mass [59].
However the smallest k affected by non-zero neutrino masses is constrained to within a factor of a few of k N R ∼ 0.01 Mpc −1 whereas for cannibals the onset of the suppression in the MPS can lie anywhere within k ∼ 0.001 − 0.1 Mpc −1 (see Fig. 8).
Finally, we wish to mention the other "anomaly" in cosmological precision fits: the dis- In this Appendix we derive the equations that describe the φ-fluid from the statistical description of its particles, and show that the self-interactions of the fluid do not appear. In particular we obtain Eqs. (21) and (22).

The Boltzmann equation
The fluid description for a fluid φ can be derived from its one-particle distribution function f (p, x, η) for a φ particle of mass m, momentum p, energy E = p 2 + m 2 , and position x, at a conformal time η. To zeroth order the Universe is isotropic and homogeneous, and thus it is described by the Friedmann-Lemaître-Robertson-Walker metric. Therefore, its distribution function is similarly homogeneous and isotropic to zeroth order. Expanding around this background to first order in small inhomogeneities: where Ψ is the small perturbation in the distribution function that codifies the inhomogeneities and anisotropies of the φ-fluid. Since we are ultimately interested in the matter power spectrum (MPS), we Fourier transform f and Ψ with regards to x to arrive at equivalent expressions for the Fourier transforms with x → k.
From the distribution function one can calculate the quantities that describe the φ-fluid ( [48]): with Dp ≡ g d 3 p (2π) 3 being the phase space element, assuming each φ particle has g degrees of freedom. From now on we take g = 1, as in our MC model. For brevity we also drop the overline denoting a background (average) quantity.
The Boltzmann equation describes the evolution of the distribution function: where ψ, ϕ are the Newtonian gauge scalar perturbations of the metric, a is the scale factor, and the dots are derivatives with respect to conformal time. The left hand side describes the free evolution of the distribution function, whereas the right hand side encodes the change in f due to collisions, and it is appropriately called the collision operator C[f ].
Since we are interested in identifying our φ-sector with our cannibal fluid, we will assume that the particles have 2 → 2 and 3 → 2 interactions, and therefore write The change in f arising from collisions must be proportional to the distributions of the particles involved, as well as to the amplitude squared of the interactions. We can then write the collision term for f 1 ≡ f (p 1 , k, η) as: where dΠ i ≡ Dp i 2E i is the Lorentz-invariant momentum space element, while, for example, |M 32 (p 1 , p 2 , p 3 ; p 4 , p 5 )| 2 is the 3 → 2 scattering amplitude squared, and we have used that the matrix element squared is invariant under permutation of the identical initial particles, final particles, and time reversal (swap of initial and final states). Note that for brevity we have not included the Bose enhancement factors (1 + f i ), but this does not affect any of our arguments. When the φ particles are non-relativistic the Bose enhancement reduces to 1.
For the 2 → 2 scattering factor in Eq. (A9) we have labeled the momenta as p 1 + p 2 ↔ p 3 + p 4 . A plus sign denotes a "gain" in the distribution function of f 1 because of the appearance of a particle with momentum p 1 , while a minus sign denotes a "loss" of the same, due to the inverse process. The symmetry factors correspond to the different permutations of the initial and final state particles, once p 1 has been selected.
For the 3 → 2 scattering in Eq. (A10) there are two places for p 1 : it can either be part of the set of three particles, or of the set of two. For this reason we have two different amplitudes and energy-momentum conserving Dirac deltas: one corresponding to the scattering p 1 + p 2 + p 3 ↔ p 4 + p 5 , and another to p 4 + p 2 + p 3 ↔ p 1 + p 5 . Once again, each of these two options has also its reversed version, which translates into a "gain" or a "loss" of a particle of momentum p 1 . Finally, note that the symmetry factors are different depending on whether the momentum p 1 is part of the set of three or the set of two.

Continuity and Euler equations
We now have the necessary ingredients to obtain the equations governing the evolution of the φ-fluid: according to Eqs. (A2)-(A5) we can multiply Eq. (A6) by the appropriate functions of p 1 and integrate to obtain the equations for ρ, δ, θ, and so forth. As we will show in this Appendix, for ρ, δ, and θ the collision terms vanish, since their corresponding weights inside the integrals in Eqs. (A3) and (A5) are the energy E and momentum p, which are conserved by the interactions.
From Eq. (A6) we can obtain the evolution of ρ, δ, and θ using Eqs. (A3) and (A5), to zeroth or first order in Ψ. The left hand side gives the well known results ( [48]): Let us now focus on the right hand side of Eq. (A6), the collision operators. Starting with the 2 → 2 term we obtain: where W 1 ≡ W (p 1 ) is some weight function of the momentum p 1 , according to Eqs. (A2)-(A5): W 1 = 1 if we want the equations for n, W 1 = E 1 for ρ and δ, W 1 = ik · p 1 for θ, and so forth.
Since we are integrating over all the momenta we are free to relabel them at will. Changing 12 ↔ 34 in the second term of Eq. (A9) and making use of |M 22 (p 3 , p 4 ; p 1 , p 2 )| = |M 22 (p 1 , p 2 ; p 3 , p 4 )| takes Eq. (A14) to: Similarly the integrals remain the same if we exchange 1 ↔ 2, and 3 ↔ 4. Doing this and taking the half sum of these exchanges gives ( [60]): It is clear that if the weight W is a quantity conserved by the 2 → 2 interactions (such as energy, momentum, or particle number) this collision term vanishes.
Changing the labels 1 ↔ 4 in the second term: The right hand side is difficult to compute for general f (0) . Nevertheless, we can make some simplifying assumptions. In particular, if we assume very fast 2 → 2 self-interactions then the φ-fluid is in thermal equilibrium, i.e. there is a sensible temperature T that can be associated with it. Therefore, under this assumption we can write: for our bosonic cannibal dark matter particles, with µ their chemical potential.
We can further simplify the right-hand-side of the number density equation by concerning ourselves only with non-relativistic φ particles, since we know that when they are relativistic they are in chemical equilibrium. Indeed, in the relativistic regime dimensional analysis implies that the 3 → 2 interactions rate is Γ 32 ∼ α 3 T ∝ a −1 , which remains bigger than the Hubble expansion rate of the Universe H during both radiation and matter domination.
Therefore for the relativistic case Γ 32 H and the fluid is in chemical equilibrium where the number and energy densities can be determined from entropy conservation.
For non-relativistic particles E ∼ m T and therefore we can write: ch , with f With this in mind, we can finally write the background number density equation: n + 3Hn = −a σ 32 v 2 n 2 (n − n ch ) , where we have used f