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

## 3. Dark Matter Annihilation and Unitarity Bound

Now, the Boltzmann equation boils down to the following simplified form

Now, we can obtain the maximum thermally averaged rate for the r â†’ 2 process using Eq. (3.2), as

where g stands for the internal degrees of freedom of the DM. Therefore, the maximum value of the thermally averaged s-wave annihilation cross section for 3 â†’ 2 can be written as

Likewise, one can also express the maximum value of the thermally average s-wave annihilation cross section for 4 â†’ 2 as

[3] Although the expression of Eq. (3.5) is derived considering equal mass for all the particles involved in the interaction; it provides maximum thermally-averaged cross section for WIMPs in the case r = 2 where the mass of the final-state particles can be different from the same of the initial-state particles.