Dark Matter from Exponential Growth

We propose a novel mechanism for the production of dark matter (DM) from a thermal bath, based on the idea that DM particles $\chi$ can transform heat bath particles $\psi$: $\chi \psi \rightarrow \chi \chi$. For a small initial abundance of $\chi$ this leads to an exponential growth of the DM number density, in close analogy to other familiar exponential growth processes in nature. We demonstrate that this mechanism complements freeze-in and freeze-out production in a generic way, opening new parameter space to explain the observed DM abundance, and we discuss observational prospects for such scenarios.

We propose a novel mechanism for the production of dark matter (DM) from a thermal bath, based on the idea that DM particles χ can transform heat bath particles ψ: χψ → χχ. For a small initial abundance of χ this leads to an exponential growth of the DM number density, in close analogy to other familiar exponential growth processes in nature. We demonstrate that this mechanism complements freeze-in and freeze-out production in a generic way, opening new parameter space to explain the observed DM abundance, and we discuss observational prospects for such scenarios.
Introduction.-While the identity and underlying properties of the dark matter (DM) in our Universe remain mysterious, its energy density has been precisely inferred by a series of satellite missions studying the Cosmic Microwave Background (CMB). Any theoretical description of DM must therefore include a DM production mechanism which leads to the observed DM relic abundance of Ω DM h 2 0.12 [1].
A particularly appealing framework for the genesis of DM, minimizing the dependence on initial conditions, is its creation out of a thermal bath. The most commonly adopted paradigm falling into this category is thermal freeze-out from the primordial plasma of Standard Model (SM) particles in the early Universe [2]. However, given the increasingly strong constraints on this setup, a plethora of alternate production scenarios with DM initially in thermal equilibrium have recently been proposed, including hidden sector freeze-out [3][4][5][6][7][8][9], Forbidden DM [10,11], Cannibal DM [12,13], Coscattering DM [14][15][16], Zombie DM [17], Elder DM [18], Kinder DM [19], and SIMP DM [20][21][22]. Another possibility is that DM never entered thermal equilibrium at all, in which case it can be produced via a 'leakage' out of a thermal bath, often referred to as freeze-in [23,24]. While a large number of variants of the freeze-out paradigm have been suggested, less model building has been performed around the freeze-in idea (see however Refs. [25][26][27][28][29][30][31][32]).
In this letter we propose a novel and generic DM production scenario between these two polarities, based on the idea that a DM particle χ can 'transform' a heat bath particle ψ into another χ, cf. Fig. 1. For a small initial abundance n χ , as shown below, this results in an exponential growth of the DM abundance. To be in accord with observations the exponential growth is required to shut off before the DM particle χ is fully thermalized, so one can also think of this mechanism as a 'failed ther- The transformation process leading to exponential production of DM (χ) from the heat bath (ψ).
malization'. Interestingly, the exponential growth of n χ comes to an end naturally in our framework, so that the observed DM abundance is readily obtained.
A novel DM production mechanism.-Quantitatively, the evolution of the DM number density n χ is governed by the Boltzmann equatioṅ Here, n eq ψ is the number density of ψ in equilibrium, H is the Hubble rate, and σv tr ( σv an ) is the cross-section for the process χψ → χχ (χχ → χψ), averaged over the phase-space of the initial state. We assume that ψ is in equilibrium with the SM heat bath; this equilibrium can be maintained for example by rapid annihilations of ψψ to SM states. We note that the zombie collisions of Refs. [17,33] involve a similar process to Fig. 1, but the roles of the DM and bath particles are reversed.
As long as n eq χ n χ , we can neglect the second term on the r.h.s. of the above equation. Introducing x ψ ≡ m ψ /T and Y χ ≡ n χ /s, with s being the entropy density of the heat bath, the solution of the Boltzmann equation arXiv:2103.16572v2 [hep-ph] 12 Jan 2022 is given by where Here Y 0 χ denotes the DM abundance at some initial 'time' x 0 ψ , and we have definedH , where g s eff encodes the entropy degrees of freedom.
Eq. (2) describes exponential growth of the DM abundance, with growth rate P , as long as P (x) > 0. For highly relativistic heat bath particles (with n eq ψ ∝ x −3 ψ ) this is automatically achieved for σv tr = (σv) 0 tr x k ψ with (σv) 0 tr const. and k > 1, i.e. infrared (IR) dominated transformation processes since H ∝ x −2 ψ . Later, once the heat bath particles become non-relativistic, exponential growth will inevitably come to an end for any value of k due to the Boltzmann suppression of n eq ψ , leading to an asymptotically flat Y χ (x ψ ). Parametrically, we thus find for the final DM abundance, where x ψ,NR ∼ 3 and λ * ∼ eff m ψ m Pl (σv) 0 tr , with g eff the energy degrees of freedom and m Pl the Planck mass.
We confirm this expectation in Fig. 2 where we show (with solid lines) the full solution of Eq. (1), adopting for illustration a constant amplitude |M tr | 2 = λ 2 tr , which is realized if χ and ψ are real scalars with interaction L ⊃ (λ tr /3!)ψχ 3 . We calculate, for simplicity, σv tr for a Maxwell-Boltzmann distribution [34], leaving a determination of its precise phase space distribution for future work; this leads to σv tr ∝ T −2 for T m χ , m ψ . Starting from an initial value of the DM abundance (indicated by the dashed lines), the onset of exponential growth is clearly visible as P becomes larger than ∼ 0.2 for x ψ 0.01 -until it stalls because P is heavily suppressed again for x ψ 5. The figure also illustrates an attractive feature of exponential growth from a phenomenological point of view: the coupling strength required to match the observed DM relic abundance is only logarithmically sensitive to the initial abundance. In the examples shown here, e.g., decreasing the initial abundance by four orders of magnitude (from the green to the red line) is compensated by a mere increase of about 22 % in λ tr .
Exponential growth in nature.-It is intriguing how closely the evolution of the DM abundance in Fig. 2 mimics other well-known examples of exponential growth in nature, like for example the progression of an illness after an initial outbreak. In fact, we can formalize this analogy by referring to the SIR epidemiological model [35], where with S the number of susceptible individuals and β and γ the infection and recovery rates, respectively. We recognize that this is simply Eq. (1), in the limit n χ n eq χ , after identifying: I ↔ n χ , S ↔ n eq ψ , β ↔ σv tr , and γ ↔ 3H. This mathematically exact correspondence motivates us to further introduce where the final equality follows from Eq. (3). R measures the number of transformation processes that each DM particle undergoes per Hubble time and is, through Eqs. (2) and (6), directly related to the final DM abundance.
Initial abundance.-In the above discussion we have deliberately remained agnostic about the origin of Y 0 χ , and simply treated this quantity as a free input parameter. We now outline various physical mechanisms that could generate such an initial DM abundance.
The first class of initial DM production mechanisms takes place much earlier than the typically rather short period where χψ → χχ processes dominate. This includes well-studied examples such as UV-dominated freeze-in [36,37] or direct production from the decay of the inflaton or other heavy particles [38] -but could also be related to more exotic examples like false vacua after a phase transition in the dark sector [39,40] or by the evaporation of black holes. Common to all these scenarios is that the final DM abundance is independent of how exactly the initial abundance is set: the only phenomenologically relevant input is the DM abundance at the onset of the era of exponential growth, thus providing a direct map to the generic situation depicted in Fig. 2.
In the second class of relevant scenarios, the initial and exponential phases of DM production are intertwined. This is particularly relevant for IR dominated freeze-in rates [23] that are too small to explain the observed DM abundance without a subsequent phase of exponential growth. A nice feature of the mechanism proposed here is in fact that a non-vanishing freeze-in contribution due to the transformation coupling λ tr is automatically built-in, as discussed below. In general, the Boltzmann equation including 2 → 2 freeze-in processes becomeṡ n χ + 3Hn χ σv tr n χ n eq ψ + σv fi (n eq ψ ) 2 , where σv fi is the total cross-section for ψψ → χχ and ψψ → χψ. Since n eq ψ n χ , a necessary condition for transformation processes to be non-negligible compared to traditional freeze-in is thus σv fi σv tr . Once the two terms on the r.h.s. of the above equation are of a similar size, on the other hand, χψ → χχ will very quickly take over due to the exponential growth of n χ . Gray regions would require a new heat bath particle ψ too light to be compatible with constraints from BBN [41]. In the light blue region an additional 2 → 4 freeze-in contribution is expected.
We show the evolution of the DM abundance for this scenario in Fig. 3, assuming for simplicity that all amplitudes are constant. We also choose the same masses and transformation couplings as in Fig. 2, to facilitate comparison. Instead of fixing the initial abundance, however, we now fix the freeze-in coupling to result in the correct relic abundance (thus taking a vanishing DM abundance as the initial condition). The above discussed three phases -freeze-in, transformation, and the final flattening of the abundance evolution curve -are clearly visible in the figure. We stress that this brings a new perspective to the widely studied freeze-in mechanism, which can trigger a subsequent phase of exponential growth in a rather natural way. It therefore becomes possible to satisfy the relic density constraint with significantly smaller couplings λ fi than generally assumed, opening up new parameter space where freeze-in is relevant for setting the DM energy density.
To further illustrate the last point we show in Fig. 4 a full 'phase diagram' of the combination of couplings λ tr and λ fi that allow the production of DM from the heat bath, for a fixed mass ratio of m ψ /m χ = 1.2. At each point in this plane, we thus adjust the mass m χ such that Ω DM h 2 = 0.12 (with dashed lines indicating isocontours of m χ ). Depending on the couplings, the relic abundance can be set via different mechanisms. In the green (red) region, the relic is mainly set via freeze-in (freeze-out) of the process ψψ ↔ χχ. In the blue (yellow) region, the relic density is instead mainly set via freeze-in (freeze-out) of the process ψχ ↔ χχ. In regions where two colors overlap, more than one production mechanism can lead to the correct relic abundance, albeit for different masses. In the gray regions, either of the production mechanisms would require fully thermalized scalars with m ψ < 0.5 MeV -for the mass ratio m ψ /m χ = 1.2 adopted for the purpose of this figurewhich is in conflict with constraints from big bang nucleosynthesis (BBN) [41]. The blue region is bounded towards large values of λ tr , since such couplings would -for exponential production -require masses above the unitarity limit of a thermal particle, m ψ > 140 TeV [42].
There is an irreducible 2 → 4 freeze-in contribution ψψ → 4χ, with cross-section scaling as λ 4 tr , which we estimate to dominate over 2 → 2 freeze-in within the light blue region. We neglect 2 → 4 processes in Fig. 4, which only have a logarithmic effect on the value of λ tr that results in the observed relic density. Finally, we note that λ tr generates λ fi radiatively and, in the absence of fine-tuning, we expect λ fi λ 2 tr /(4π) 2 . This bound is satisfied except in the light blue region where 2 → 4 processes are relevant.
Discussion.-We stress that exponential growth due to processes as depicted in Fig. 1 is by no means restricted to specific model realizations, but is a general mechanism of DM production that essentially interpolates between the traditionally considered freeze-in and freeze-out regimes. At first glance it may seem worrisome that the dark matter density depends exponentially on the transformation cross-section, implying that the crosssection must be carefully chosen to match observation. But this can be turned around: in fact it implies that this mechanism is highly predictive, as manifested by the logarithmic sensitivity of the necessary cross-section to the initial conditions. We also note that exponential sensitivity is quite common in nature and for example can result from renormalization group flows, where IR parameters can be exponentially sensitive to UV parameters: the proton mass, e.g., depends exponentially on the size of the strong coupling at high energies [43].
It is worth pointing out that the mechanism proposed here works for a large range of different particle masses, not just the specific choice displayed in the examples above. Limits from BBN or CMB on new light degrees of freedom [41,44], as encountered in Fig. 4, could be significantly lowered by considering the possibility of ψ being a SM particle. Larger values of m ψ would not affect the DM velocity dispersion at T ∼ m χ (but may sharpen model-dependent constraints from the decay of ψ, see below). Even a reverted mass hierarchy is possible, m χ > m ψ , in which case it would be m χ rather than m ψ that determines when the evolution of n χ /s starts to flatten. For m χ > m ψ , DM in general becomes unstable -but not necessarily on cosmological time-scales. We fix λ hψ = 3.9 × 10 −4 , giving 20% of DM in ψ, and choose λ fi for each mχ and λtr such that χ provides 80% of the observed DM abundance. Green and blue regions indicate which production mechanism dominates the χ-abundance. Red lines (and hatched area) show where CMB signatures can naturally be expected. Within the light blue region, freeze-in via SM SM → ψχχχ dominates over χχ → ψψ in setting the initial χ abundance, thus generically leading to overproduction of DM.
Such highly suppressed decays may potentially be visible in late-time observables related to cosmological structure formation or cosmic rays, making corresponding scenarios even more attractive. Let us finish by considering the phenomenological consequences for a simple, concrete model realization, where ψ is a real scalar and couples to the SM via a quartic Higgs portal coupling λ hψ |H| 2 ψ 2 /2. The relic density of ψ is then determined by standard freeze-out, exactly as for scalar singlet DM [45] where the resonant region m ψ ∼ m h /2 features the best likelihood in globals fits [46]. For concreteness we use m ψ = m h /2 and λ hψ = 3.9 × 10 −4 , resulting in Ω ψ = 0.2 Ω DM when including the effect of early kinetic decoupling as implemented in DarkSUSY [47]. In Fig. 5, we show the plane m χ −λ tr after fixing λ fi such that the dominant DM component satisfies Ω χ = 0.8 Ω DM and is mostly produced by transformation (blue) or ψψ → χχ freeze-in (green) processes; in the white area Ω χ = 0.8 Ω DM may only be possible via freeze-out or semi-annihilations. Using FeynRules [48] and MadGraph5 [49], furthermore, we find that for λ fi λ hψ λ tr /[3(4π) 2 ] freeze-in production through SM SM → ψχχχ would dominate over that from ψψ → χχ (indicated by the light blue region). Finally, the decay of χ to SM particles can impact the observed CMB power spectrum; for an 80% DM subcomponent, this leads to a constraint of τ χ 0.8 × 10 24 s [50] (indicated by the hatched area in Fig. 5), projected to tighten by a factor of ∼3 with CMB-S4 [51] (red lines). Here we estimated the decay width as where v ≈ 246 GeV is the vacuum expectation value of the Higgs, and Γ h (q) is its off-shell decay width as evaluated with HDECAY [52]. For m χ < m ψ +m h = (3/2)m h , the Higgs can only be produced off-shell, resulting in a significant suppression of Γ χ and hence the absence of CMB limits in this parameter range.
Conclusions.-We have introduced a novel type of DM production mechanism, where an initially tiny DM abundance is enhanced due to a process where DM particles convert bath particles into more DM particles. The DM abundance grows exponentially with time, in stark contrast to models of freeze-in where the abundance grows only as a power law. Our mechanism complements both freeze-in and freeze-out thermal production scenarios in a generic way. Concrete model realizations lead, already in their simplest forms, to interesting phenomenological consequences. Further, and detailed, exploration of this new way of producing DM from the thermal bath thus appears highly warranted.
Note added. After our work appeared on the arXiv, the mechanism introduced here was independently suggested in Ref. [53]. In that reference, the authors study in detail the phenomenological consequences of another concrete model realization of Pandemic DM.