This paper is available on arxiv under CC 4.0 license.

**Authors:**

(1) Nicolas Bernal, New York University Abu Dhabi;

(2) Partha Konar, Physical Research Laboratory;

(3) Sudipta Show, Physical Research Laboratory.

## Table of Links

- Abstract and Intro
- S-matrix: Unitarity and its Consequences
- Dark Matter Annihilation and Unitarity Bound
- Low-temperature Reheating
- Freeze-out with a Low-temperature Reheating
- Summary and Conclusion
- Acknowledgments and References

## Abstract

Model-independent theoretical upper bound on the thermal dark matter (DM) mass can be derived from the maximum inelastic DM cross-section featuring the whole observed DM abundance. We deploy partial-wave unitarity of the scattering matrix to derive the maximal thermally-averaged cross-section for general number-changing processes r → 2 (with r ≥ 2), which may involve standard model particles or occur solely within the dark sector. The usual upper limit on the DM mass for s-wave annihilation is around 130 TeV (1 GeV) for r = 2 (3), only applies in the case of a freeze out occurring in the standard cosmological scenario. We consider the effects of two nonstandard cosmological evolutions, characterized by low-temperature reheating: i) a kination-like scenario and ii) an early matter-dominated scenario. In the first case, the early freeze-out strengthens the unitarity bound to few TeVs for WIMPs; while in the second case, WIMP DM can be as heavy as ∼ 1010 GeV due to a large entropy dilution.

## 1. Introduction

Consideration of a specific DM production paradigm in the early stage of the universe may further constrain the mass range for a viable DM candidate. For instance, the number-changing pair annihilation of DM to SM particles determines its present mass density, where it maintains the chemical and kinetic equilibrium with the thermal soup in the early universe. Interestingly, the requirement of the unitarity of the S-matrix sets a model-independent upper bound on the DM mass for this scenario [12, 13]. The implication of unitarity offers the maximum inelastic cross-section, which fixes the minimum number density of the frozen-out DM. Using this number density, one can establish the maximum allowed DM mass by fulfilling the observed relic density of it. In the theories of DM with long-range forces, bound states of DM can form and therefore relax the unitarity bound by suppressing the inelastic annihilation rate [14–16]. In addition, dark sectors with particle-antiparticle asymmetry enforce the apprehend of the nonzero equilibrium chemical potential for DM, further constraining the unitarity limits due to the demand for an increased effective DM number density at the time of freeze-out [15, 17]. Furthermore, different indirect searches for DM may put a lower limit on DM mass for some specified scenarios. A strong model-independent lower bound for the thermal DM that is annihilating to visible states through an s-wave process is about 20 GeV [18]. In addition, a more restrictive lower limit has recently been found. It has been shown that the lower bound is 200 GeV, considering H.E.S.S. and other updated observational data [19].

In particular, all the DM scenarios mentioned so far pay attention to the 2 → 2 number-changing process where a DM pair annihilates into a pair of SM particles, that is, the Weakly Interacting Massive Particle (WIMP) paradigm [20–22].[1] Moreover, it is not necessary that the number-changing processes involve SM particles, so they may also occur within the dark sector. The minimalistic realization of this scenario is the 3 → 2 process, where this kind of number-changing reaction involves a single DM species. In general, such processes arise in DM theories with new sizable self-interactions, and in several contexts as self-interacting DM [29–31], the Strongly Interacting Massive Particle (SIMP) paradigm [32–49], or even the ELastically DEcoupling Relic (ELDER) scenario [50, 51].

It is essential to mention that the early history of the universe plays a crucial role in DM genesis, since the decoupling of thermal DM occurred at that time. Generally, the studies of DM consider the standard cosmological picture in which the radiation energy density is assumed to dominate the energy budget before the Big Bang Nucleosynthesis (BBN). However, there is no direct evidence for the energy content at very high temperatures. Therefore, it is vital to look at the effects of modified cosmology on the production of DM. In recent times, the evolution of the DM in the period of non-standard expansion usually triggered by the decay of a long-lived massive particle [48, 52–71] or by Hawking evaporation of primordial black holes [72–95] is receiving increasing attention.[2] All such studies point towards the fact that non-standard cosmology alters the value of the thermally averaged cross section needed to satisfy the observed relic of DM. Such a modification in the thermally averaged cross-section may also change the unitarity mass bound of DM. In a recent article, the authors studied the impact of early matter domination on unitarity limits [112].

This article is decorated as follows. In Sections 2 and 3, we present the detailed derivation of the maximum thermally-averaged cross-section allowed by the unitarity of the Smatrix. We discuss two different non-standard cosmological pictures: kination-like and late-time reheating in Section 4. Section 5 shows the analytical expressions for freeze-out and cross sections for the radiation-dominated universe and the mentioned modified cosmologies, and we also demonstrate our results. Finally, we summarize our findings in Section 6.

[1] Alternatively, one can also have in the final state a DM and a SM particle (semi-annihilations) [23–27], or in the initial state a DM and another particle of the dark sector (coannihilations) [28].

[2] For studies on baryogenesis with a low reheating temperature or during an early matter-dominated phase, see Refs. [52, 96–100] and [101–104], respectively. Furthermore, the production of primordial gravitational waves in scenarios with an early matter era has recently received particular attention [105–111].