Dark Matter from freeze-in and its inhomogeneities

We consider generic freeze-in processes for generation of Dark Matter, together with the consequent re-thermalization of the Standard Model fluid. We find that Dark Matter inherits the Standard Model adiabatic inhomogeneities on the cosmological scales probed by current observations, that were super-horizon during freeze-in. Thereby, freeze-in satisfies the bounds on iso-curvature perturbations.


Introduction
Freeze-in is a possible mechanism that could have generated the Dark Matter (DM) cosmological abundance [1].It assumes that the Standard Model (SM) cosmological thermal plasma was not initially accompanied by any DM abundance.Since all SM components self-interact thermalising to a common temperature, cosmological inhomogeneities were initially adiabatic.
Next, 'freeze-in' particle physics processes produce DM particles with mass M out of the SM plasma.For example, one can have decays SM → DM DM or scatterings SM SM → DM DM, dominated either at large temperatures T ≫ M ('UV-dominated freeze-in') or at low temperatures T ∼ M ('IR-dominated freeze-in).In order to match the observed cosmological DM density [2], the rate of freeze-in processes must be much smaller than the Hubble rate H. Freeze-in automatically generates DM inhomogeneities out of SM inhomogeneities.
Observations are consistent with dominant adiabatic inhomogeneities (namely, the SM/DM fluid is the same everywhere), while iso-curvature inhomogeneities (namely, DM inhomogeneities different from SM inhomogeneities) are constrained, on cosmological scales, to be below a few % level [2].
We consider if freeze-in leads to acceptable DM inhomogeneities.
Weinberg answered positively this issue for thermal freeze-out: since freeze-out dominantly happens in the non-relativistic regime, computing inhomogeneities in the DM number density was enough [3].On the other hand, freeze-in can be relativistic, and the iso-curvature issue started being considered recently: [4] claims that a specific freeze-in model is excluded because it generates too large scale-independent iso-curvature perturbations.The authors of [4] argue that all freeze-in models are similarly problematic.In the model considered in [4] DM has a small electric charge and is thereby produced by IR-dominated scatterings of two SM particles, such as e − e + → DM DM.This generates, at any given time, a contribution to the DM density ρ DM proportional to the square of the SM density ρ SM , and DM inhomogeneities might be not proportional to SM inhomogeneities.However, one must consider the cumulative cosmological process taking into account that all regions of the Universe undergo a similarly diluting ρ SM .As we will see, this leads to negligible iso-curvature effects.A simple argument is presented in section 2, and the general formalism is used in section 3. Section 4 presents our conclusions.

Intuitive argument based on 'separate universes'
We start presenting an intuitive argument.Working in the Newtonian gauge, the primordial adiabatic perturbations δρ α (t, x) in the density ρ α (t) of a fluid α can be characterised in a simple geometric way as [3,[5][6][7]] working at first-order in the small δρ α ≪ ρ α .In eq. ( 2) δt(t, x) is some universal function common to all fluids that can be intuitively thought as a delay in the time evolution of the different regions.
Observation constrain iso-curvature perturbations only on scales comparable to the horizon today, while the freeze-in DM density was generated before matter/radiation equality (much before in most freeze-in models).This means that we only need to worry if freeze-in generated iso-curvature perturbations on scales much larger than the small horizon at freeze-in.
We can thus apply the 'separate universes' picture (see e.g.[5]): the very early Universe at freeze-in can be thought as many homogeneous regions without causal contact, given that inhomogeneities on different scale evolve independently in first-order approximation.Freeze-in dynamics produces DM with adiabatic perturbations because all regions undergo the same dynamics, up to the delay δt.So eq. ( 2) holds for the DM density, no matter how complicated the freeze-in dynamics is.Explicitly, the Boltzmann equation for the homogeneous small DM number density n DM is d(n DM /s)/d ln T ≃ γ/Hs, where s is the entropy density, H is the Hubble rate, and γ(T ) is the space-time density rate of freeze-in processes that produce one DM particle out of the SM plasma at temperature T .Integrating this equation leads to Interpreting eq. ( 3) in the 'separate universes' picture implies that, in regions where the SM plasma was denser, freeze-in initially produced more DM by some amount that depends on the freeze-in model, but in these region the DM average density changed more rapidly leading to adiabatic DM inhomogeneities.The above discussion explicitly verifies how, in the special freeze-in case, the 'separate universes' regions undergo the same evolution, up to the time delay.
The next section substantiates the above intuitive reasoning by explicit computations.
A general formalism to compute the cosmological evolution of inhomogeneities in interacting fluids was developed in [8,9].We adopt its presentation as summarized in [10], that makes more explicit the sources of iso-curvature inhomogeneities.Simple first-order evolution equations for the various densities are obtained by combining the Einstein gravity equations into the conservation of the energy-momentum tensor T µν = α T µν (α) .The energy-momentum tensor T µν (α) of fluid α only is not conserved because interactions transfer energy-momentum Q ν (α) to other fluids.So one has because of total energy conservation.In the homogeneous limit, this implies that the average densities evolve as ρα + 3H( [10] so that α δQ α = 0 by total energy conservation.The total density is ρ = α ρ α .
It is useful to write equations in terms of the curvature perturbation ζ = −H[Ψ/H + δρ/ ρ], which is the relative displacement between uniform-density and uniform-curvature surfaces.This curvature perturbation can be defined for each fluid and it evolves as [10] ζα = − H ρα δQ intr,α + 3 where δQ intr,α and δ℘ intr,α will be defined later.As usual, small perturbations are conveniently expanded in comoving Fourier modes k, and the 'separate universe' argument amounts to consider the limit k → 0 of the full equations.We focus on large superhorizon scales, thereby omitting the label k and neglecting Laplacians and other terms suppressed by k 2 /a 2 H 2 .Such terms are indeed negligible whenever freeze-in occurs way before matter/radiation equality, for relevant cosmological scales k.
The equations ( 6) can be written in a slightly more convenient form by avoiding using the total density ρ and defining instead the iso-curvature relative perturbations S αβ between two fluids α and β that evolve as We again ignore the terms suppressed by k 2 .We can also ignore the 'multiplicative' terms (namely, those proportional to combinations of S α ′ β ′ terms) [10] Ṡmul because we are only concerned in understanding if non-zero iso-curvature perturbations are generated by the 'source' terms explicitly shown in eq. ( 8).The formalism summarized in [10] makes clear that, in the long-wavelength limit k → 0, iso-curvature perturbations are only sourced by the non-adiabatic energy transfer δQ intr,α and by the non-adiabatic pressure δ℘ intr,α intrinsic in each fluid α.These terms will be now be defined and evaluated.

Intrinsic non-adiabatic energy transfer
One source of iso-curvature perturbations is the intrinsic non-adiabatic energy transfer, the part of energy transfer δQ α from fluid α 'biased' with respect to its energy density ρ α [10]: We next consider its value during freeze-in, where the relevant fluids are α = {SM, DM}.
The rate of freeze-in particle collisions can be computed, in any given particle-physics model, as a function of the local temperature of the SM fluid, that also controls its density.Thereby the energy transfer from the SM fluid only depends on its local density, Q SM (ρ SM ).Consequently δQ intr,SM = δQ SM − δρ SM dQ SM /dρ SM = 0 vanishes in a generic freeze-in model.
Next, energy conservation demands δQ SM + δQ DM = 0, so that the intrinsic nonadiabatic energy transfer to the DM fluid can be written as This potential 'source' terms thereby becomes a 'multiplicative' term, proportional to the relative entropy S SM,DM .Since this is assumed to be initially vanishing, δQ intr,DM generates no isocurvature perturbation.

Intrinsic non-adiabatic pressure
The second kind of source term, the non-adiabatic part of the pressure perturbation intrinsic of each fluid α, is given by [10] δ℘ intr,α = δ℘ α − c 2 α δρ α where c 2 α = ℘α / ρα (12) is its adiabatic speed of sound.This term vanishes when the pressure and energy inhomogeneities respect the equation of state of the fluid, ℘ α (ρ α ).
Freeze-in particle-physics processes contribute as δ℘ intr,SM = 0 because they convert SM particles into DM particles, thereby inducing an energy and momentum loss of the SM fluid, as dictated by the specific freeze-in interaction, that generically does not follow the equation of state of the SM fluid.
As a simple example of this unbalance, freeze-in via the decay into DM particles of some SM particle (or, in SM extensions, of some speculative new-physics particle tightly coupled to the SM) transfers more energy than pressure ( ρSM / ℘SM > ρ SM /℘ SM ) because the decaying particles must be massive and thereby they decay slower when they have higher relativistic energy.An unbalance also generically occurs in freeze-in scatterings, described by a cross-section σ(SM SM → DM DM) that only depends on the invariant energy √ s at leading order in the couplings (the motion with respect to the plasma enters at higher orders).The sign of δ℘ intr,SM is not fixed, as the energy dependence of σ can either result in a larger energy transfer when the colliding SM particles have higher energy E > ∼ T (this can happen in UV-dominated freeze-in, via non-renormalizable interactions, for example gravitational [11]) or when the colliding SM particles have lower energy E < ∼ T (this can happen in IR-dominated freeze-in, via renormalizable interactions).As a possibly relevant special case, δ℘ intr,SM is nearly-vanishing in freeze-in models that only lead to the disappearance of ultra-relativistic SM particles, as they (on angular average) satisfy the same equation of state ℘ = ρ/3 as the radiation-dominated SM fluid.
However, the fact that freeze-in processes (decays and scatterings) can contribute as δ℘ intr,SM = 0 is inconsequential, as we must also take into account the self-interactions of the SM fluid.A multitude of SM particle processes allow the SM fluid to locally rethermalize to its equation of state with rates Γ much faster than the Hubble rate and than the freeze-in rate.Typically Γ ∼ g 2 T where g ∼ 1 is a typical SM coupling, such as a gauge coupling.The re-thermalizion processes conserve the SM energy ρ SM : δQ SM remains given by freeze-in processes only, so that δQ intr,SM = 0 remains as in section 3.1.On the other hand, the SM pressure ℘ SM changes such that the combination of the two processes (freeze-in and re-thermalization) leads to δ℘ intr,SM = 0.This leaves δ℘ intr,DM as a possible source of iso-curvatures.A self-thermalization argument parallel to what just discussed for the SM plasma implies δ℘ intr,DM = 0 if DM has significant self-interactions just after being produced during freeze-in.This happens, for example, if DM is a multiplet under a dark gauge group [12] that confines at a scale Λ and if freeze-in happens at T ≫ Λ.If instead DM self-interactions are negligible, a formalism extended to higher moments may be needed, but the physics is simple: DM particles free stream on sub-horizon scales, but not on large scales k → 0. The non-thermal DM distribution f ( x, t, q) = f 0 (q) + δf ( x, t, q) produced by freeze-in redshifts with scale factor a as [13] ρ DM = 1 a 4 where q and E = √ q 2 + a 2 M 2 are the comoving momentum and energy of the DM particle with mass M. Two limits are of special interest.If freeze-in is IR-dominated, DM is only mildly relativistic, so that DM motion is soon red-shifted down to negligible pressure, ℘ DM ≪ ρ DM .UV-dominated freeze-in can produce ultra-relativistic DM with ℘ DM /ρ DM ≃ ℘DM / ρDM ≃ 1/3, that becomes non-relativistic only later when the SM cools down to temperatures comparable to the DM mass M, while the horizon reaches larger scales.

Conclusions
We considered generic models of freeze-in (from decays, from scatterings, IR-dominated, UV-dominated...) finding that the generated Dark Matter inherits the Standard Model adiabatic inhomogeneities on the cosmological scales probed by current observations, that were super-horizon during freeze-in.In section 2 we presented an intuitive argument based on the well-known 'separate universe' picture.This was substantiated in section 3 by checking the explicit sources of iso-curvature perturbations on super-horizon scales.Iso-curvature perturbations can only be generated on small scales that were subhorizon during freeze-in: this effect can perhaps be relevant in models where freeze-in happens at the lowest possible temperature T ∼ M ∼ keV, possibly in the presence of dark long-range forces.
In conclusion, freeze-in appears a viable mechanism for generation of the cosmological DM abundance.Similar arguments hold for other particle-physics mechanisms such as 'cannibalism' [14] or 'freeze-out and decay'.Furthermore, baryogenesis mechanisms that involve elements similar to freeze-in (such as leptogenesis from right-handed neutrinos with initially negligible abundance) are similarly compatible with iso-curvature bounds.