# (Non-)Thermal Production of WIMPs during Kination

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Boltzmann Equations for the Model

## 3. Production of WIMPs during Kination

**Thermal production with chemical equilibrium (“freeze-out”)**. We first focus on the case $b=0$, corresponding to the negligible decay with respect to the annihilation into SM particles and approximated by the red dot-dashed and blue solid lines in Figure 2. We assume that the temperature ${T}_{\mathrm{kin}}$ is lower than the freeze-out temperature ${T}_{\mathrm{f}.\mathrm{o}}$, which is defined as the temperature at which $n\langle \sigma v\rangle =H$, or$${n}_{\mathrm{EQ}}\left({T}_{\mathrm{f}.\mathrm{o}}\right)={\left(\frac{{T}_{\mathrm{f}.\mathrm{o}}}{{T}_{\mathrm{kin}}}\right)}^{4}\frac{H\left({T}_{\mathrm{kin}}\right)}{\langle \sigma v\rangle}.$$However, in modified cosmologies where the expansion rate is faster than during radiation (such as kination), WIMP annihilation persists even after the departure from chemical equilibrium (i.e., freeze-out) has occurred, actually ceasing when the Universe transitions to the standard radiation-dominated scenario [78]. For this reason, the WIMP number density in the kination cosmology is not fixed at ${T}_{\mathrm{f}.\mathrm{o}}$ and annihilation continues until the temperature drops to ${T}_{\mathrm{kin}}$. We assume that the freeze-out is reached at ${x}_{\mathrm{f}.\mathrm{o}.}$, when $X\left({x}_{\mathrm{f}.\mathrm{o}.}\right)={X}_{\mathrm{f}.\mathrm{o}.}$. At later times, using the approximation in Equation (18) and the non-relativistic regime for the WIMPs, Equation (13) reads$${X}^{\prime}=-\frac{{\rho}_{\mathrm{kin}}\langle \sigma v\rangle}{{\mathsf{\Gamma}}_{\varphi}{m}_{\chi}}\frac{{X}^{2}}{x\sqrt{{\mathsf{\Phi}}_{I}}},$$$$\begin{array}{ccc}\hfill {X}_{\mathrm{kin},\mathrm{Th}}& =& {\left[\frac{\langle \sigma v\rangle {T}_{\mathrm{kin}}^{2}}{\sqrt{{\mathsf{\Phi}}_{I}}}\frac{{M}_{\mathrm{Pl}}}{{m}_{\chi}}\left(ln\frac{{x}_{\mathrm{kin}}}{{x}_{\mathrm{fo}}}+1\right)+\frac{1}{{X}_{\mathrm{f}.\mathrm{o}.}}\right]}^{-1},\phantom{\rule{1.em}{0ex}}\mathrm{or},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {n}_{\mathrm{kin},\mathrm{Th}}& =& {\left[\frac{\langle \sigma v\rangle {T}_{\mathrm{kin}}^{2}\phantom{\rule{0.166667em}{0ex}}{M}_{\mathrm{Pl}}}{{\rho}_{\mathrm{kin}}}\left(ln\frac{{x}_{\mathrm{kin}}}{{x}_{\mathrm{fo}}}+1\right)+\frac{{x}_{\mathrm{kin}}^{3}}{{x}_{\mathrm{f}.\mathrm{o}.}^{3}}\frac{1}{{n}_{\chi ,\mathrm{f}.\mathrm{o}.}}\right]}^{-1}.\hfill \end{array}$$This expression has been obtained, for example, in Ref. [77], since during kination ${a}_{I}^{3}{H}_{I}={a}_{\mathrm{kin}}^{3}H\left({T}_{\mathrm{kin}}\right)$. Neglecting ${X}_{\mathrm{f}.\mathrm{o}.}$ and setting ${T}_{\mathrm{fix}}={T}_{\mathrm{kin}}$ in Equation (19), the present abundance from the freeze-out mechanism gives$${\mathsf{\Omega}}_{\chi ,\mathrm{Th}}=\frac{{g}_{S}\left({T}_{0}\right)}{{g}_{S}\left({T}_{\mathrm{kin}}\right)}\frac{{\rho}_{\mathrm{kin}}}{{\rho}_{c}}\frac{{m}_{\chi}}{\langle \sigma v\rangle}\frac{{T}_{0}^{3}}{{M}_{\mathrm{Pl}}{T}_{\mathrm{kin}}^{5}}{\left(ln\frac{{x}_{\mathrm{kin}}}{{x}_{\mathrm{fo}}}+1\right)}^{-1}\propto \frac{1}{{T}_{\mathrm{kin}}}.$$The solution describes, for example, the lines with negative slopes in Figure 2 for $b=0$ and for the cross sections $\langle \sigma v\rangle =2\times {10}^{-6}\phantom{\rule{0.166667em}{0ex}}$GeV${}^{-2}$, $\langle \sigma v\rangle =2\times {10}^{-9}\phantom{\rule{0.166667em}{0ex}}$GeV${}^{-2}$, and $\langle \sigma v\rangle =2\times {10}^{-12}\phantom{\rule{0.166667em}{0ex}}$GeV${}^{-2}$. In Figure 2, we have not approximated the numerical results by neglecting ${X}_{\mathrm{f}.\mathrm{o}.}$.**Thermal production without ever reaching chemical equilibrium (“freeze-in”).**If the cross section is sufficiently low [77], WIMPs never reach thermal equilibrium and their number density freezes in at a fixed quantity. Since the number density of particles is always smaller than their value at thermal equilibrium, we neglect $X\ll {X}_{\mathrm{EQ}}$ so Equation (13) with $b=0$ reads$${X}^{\prime}=\frac{{\rho}_{\mathrm{kin}}\langle \sigma v\rangle}{{\mathsf{\Gamma}}_{\varphi}{m}_{\chi}}\frac{\left({X}_{\mathrm{EQ}}^{2}\right)}{x\sqrt{{\mathsf{\Phi}}_{I}}}={d}_{1}{x}^{11/4}exp\left(-2{d}_{2}{x}^{3/4}\right),$$$${d}_{1}=\frac{{g}^{2}\langle \sigma v\rangle {m}_{\chi}^{4}{T}_{\mathrm{kin}}^{3}}{{\left(2\pi \right)}^{3}{\mathsf{\Gamma}}_{\varphi}{\rho}_{\mathrm{kin}}{\mathsf{\Phi}}_{I}^{1/8}},\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{d}_{2}=\frac{{m}_{\chi}}{{T}_{\mathrm{kin}}{\mathsf{\Phi}}_{I}^{1/8}}.$$The solution to Equation (25) reaches the asymptotic value of X at freeze-in$${X}_{\mathrm{kin},\mathrm{f}.\mathrm{i}.}=\frac{{d}_{1}}{{d}_{2}^{5}}=\frac{{g}^{2}\langle \sigma v\rangle {T}_{\mathrm{kin}}^{8}\sqrt{{\mathsf{\Phi}}_{I}}}{{\left(2\pi \right)}^{3}{\mathsf{\Gamma}}_{\varphi}{\rho}_{\mathrm{kin}}{m}_{\chi}},$$$${\mathsf{\Omega}}_{\chi ,\mathrm{f}.\mathrm{i}.}=\frac{{g}_{S}\left({T}_{0}\right)}{{g}_{S}\left({T}_{\mathrm{kin}}\right)}\frac{{g}^{2}\langle \sigma v\rangle {T}_{0}^{3}{T}_{\mathrm{kin}}^{5}}{{\left(2\pi \right)}^{3}{\mathsf{\Gamma}}_{\varphi}{\rho}_{c}{m}_{\chi}}\propto {T}_{\mathrm{kin}}^{3}.$$The solution describes, for example, the lines with positive slopes in Figure 2, for $b=0$ and the cross sections $\langle \sigma v\rangle =2\times {10}^{-12}\phantom{\rule{0.166667em}{0ex}}$GeV${}^{-2}$, $\langle \sigma v\rangle =2\times {10}^{-15}\phantom{\rule{0.166667em}{0ex}}$GeV${}^{-2}$, and $\langle \sigma v\rangle =2\times {10}^{-18}\phantom{\rule{0.166667em}{0ex}}$GeV${}^{-2}$.

**Non-thermal production without chemical equilibrium.**We now discuss the non-thermal production of DM, in the case in which the particle has never reached the chemical equilibrium. For a sufficiently large branching ratio b and for ${T}_{\mathrm{kin}}\ll {m}_{\chi}$, the abundance of DM is set by the decay of the $\varphi $ field, with an energy density at ${T}_{\mathrm{kin}}$ given by [53,55,58,61,63,81]$${\rho}_{\chi}\left({T}_{\mathrm{kin}}\right)\approx b\phantom{\rule{0.166667em}{0ex}}{\rho}_{\varphi}\left({T}_{\mathrm{kin}}\right).$$Deriving the result from directly integrating Equation (13) with $\langle \sigma v\rangle =0$ and neglecting the contributions from R and X in the denominator gives an extra logarithmic dependence on ${x}_{\mathrm{kin}}$, as$${X}_{\mathrm{kin},\mathrm{decay}}=b\sqrt{{\mathsf{\Phi}}_{I}}ln\frac{{x}_{\mathrm{kin}}}{{x}_{I}}.$$The present WIMP abundance when the non-thermal production dominates is given by Equation (19) with ${T}_{\mathrm{fix}}={T}_{\mathrm{kin}}$, corresponding to the moment at which WIMPs are produced from the decay of the $\varphi $ field with the initial amount in Equation (29),$${\mathsf{\Omega}}_{\chi ,\mathrm{decay}}=\frac{b{\rho}_{\mathrm{kin}}}{{\rho}_{c}}\frac{{g}_{S}\left({T}_{0}\right)}{{g}_{S}\left({T}_{\mathrm{kin}}\right)}{\left(\frac{{T}_{0}}{{T}_{\mathrm{kin}}}\right)}^{3}ln\frac{{x}_{\mathrm{kin}}}{{x}_{I}}.$$This latter expression predicts the behavior ${\mathsf{\Omega}}_{\chi ,\mathrm{decay}}\propto {T}_{\mathrm{kin}}$ corresponding to $b=0.001$ (blue solid line) with $\langle \sigma v\rangle =2\times {10}^{-18}\phantom{\rule{0.166667em}{0ex}}$GeV${}^{-2}$ in Figure 2, for ${T}_{\mathrm{kin}}\lesssim 10\phantom{\rule{0.166667em}{0ex}}$GeV. Notice that, except for the logarithmic dependence which is present in the kination cosmology, the result in Equation (30) is independent of the cosmology used.

**Non-thermal production with chemical equilibrium.**If the branching ratio b is sufficiently high, the evolution of the WIMP number density attains a secular equilibrium in which the rate at which WIMPs are produced from the decay of the $\varphi $ field equates that from WIMP annihilation. In this regime, the quantity X is fixed to the value obtained by setting to zero the right-hand side of Equation (13),$${X}_{\mathrm{kin},\mathrm{sec}}=\sqrt{\frac{b{\mathsf{\Phi}}_{I}{\mathsf{\Gamma}}_{\varphi}{m}_{\chi}}{{\rho}_{\mathrm{kin}}\langle \sigma v\rangle}}.$$The result in Equation (32), confirmed numerically in Figure 3 below, can be alternatively derived by considering the balancing between the decay rate of the $\varphi $ field into WIMPs and the annihilation rate of WIMPs, valid at ${T}_{\mathrm{kin}}$ when ${\rho}_{\varphi}={\rho}_{\mathrm{kin}}$, as$$\left({m}_{\chi}\phantom{\rule{0.166667em}{0ex}}{n}_{\chi}\right)\left(\langle \sigma v\rangle {n}_{\chi}\right)=b{\mathsf{\Gamma}}_{\varphi}{\rho}_{\varphi}.$$The value of ${X}_{\mathrm{kin},\mathrm{sec}}$ remains constant until ${T}_{\mathrm{kin}}$, without experiencing the additional depletion obtained in the freeze-out regime with a faster-than-radiation expansion rate [77,78]. However, when the temperature of the plasma falls below ${T}_{\mathrm{kin}}$, the secular equilibrium is no longer maintained since the energy density in the $\varphi $ field drops to zero and WIMPs are no longer produced. In this new regime, radiation evolves as $R\left(x\right)={\mathsf{\Phi}}_{I}{\left({x}_{I}/{x}_{\mathrm{kin}}\right)}^{2}$ and Equation (13) reads$$\frac{dX}{d(x/{x}_{\mathrm{kin}})}=-\frac{{\rho}_{\mathrm{kin}}\langle \sigma v\rangle}{{\mathsf{\Gamma}}_{\varphi}{m}_{\chi}}\frac{{X}^{2}}{{\left(\frac{x}{{x}_{\mathrm{kin}}}\right)}^{2}\sqrt{{\mathsf{\Phi}}_{I}}},$$$${\mathsf{\Omega}}_{\chi ,\mathrm{nonTh}}\propto \frac{{m}_{\chi}{\mathsf{\Gamma}}_{\varphi}}{{\rho}_{c}\phantom{\rule{0.166667em}{0ex}}\langle \sigma v\rangle}\frac{{g}_{S}\left({T}_{0}\right)}{{g}_{S}\left({T}_{\mathrm{kin}}\right)}{\left(\frac{{T}_{0}}{{T}_{\mathrm{kin}}}\right)}^{3}\propto \frac{1}{{T}_{\mathrm{kin}}}.$$

## 4. Discussion and Summary

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DM | Dark Matter |

BBN | Big Bang Nucleosynthesis |

WIMP | Weakly Interacting Massive Particle |

LRTS | Low Reheat Temperature Scenario |

KS | Kination Scenario |

## References

- Jungman, G.; Kamionkowski, M.; Griest, K. Supersymmetric dark matter. Phys. Rep.
**1996**, 267, 195–373. [Google Scholar] [CrossRef] [Green Version] - Bertone, G.; Hooper, D.; Silk, J. Particle dark matter: Evidence, candidates and constraints. Phys. Rep.
**2005**, 405, 279–390. [Google Scholar] [CrossRef] - Ade, P.A.R.; Aghanim, N.; Ahmed, Z.; Aikin, R.W.; Alexander, K.D.; Arnaud, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barkats, D.; et al. Joint Analysis of BICEP2/Keck Array and Planck Data. Phys. Rev. Lett.
**2015**, 114, 101301. [Google Scholar] [CrossRef] [PubMed] - Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; Bartlett, J.G.; et al. Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys.
**2016**, 594, A13. [Google Scholar] [CrossRef] - Vysotsky, M.I.; Dolgov, A.D.; Zeldovich, Y.B. Cosmological Restriction on Neutral Lepton Masses. JETP Lett.
**1977**, 26, 188–190. [Google Scholar] - Hut, P. Limits on Masses and Number of Neutral Weakly Interacting Particles. Phys. Lett. B
**1977**, 69, 85–88. [Google Scholar] [CrossRef] - Sato, K.; Kobayashi, M. Cosmological Constraints on the Mass and the Number of Heavy Lepton Neutrinos. Prog. Theor. Phys.
**1977**, 58, 1775–1789. [Google Scholar] [CrossRef] [Green Version] - Lee, B.W.; Weinberg, S. Cosmological Lower Bound on Heavy Neutrino Masses. Phys. Rev. Lett.
**1977**, 39, 165–168. [Google Scholar] [CrossRef] - Dicus, D.A.; Kolb, E.W.; Teplitz, V.L. Cosmological Implications of Massive, Unstable Neutrinos: New and Improved. Astrophys. J.
**1978**, 221, 327–341. [Google Scholar] [CrossRef] - Steigman, G. Cosmology Confronts Particle Physics. Ann. Rev. Nucl. Part. Sci.
**1979**, 29, 313–338. [Google Scholar] [CrossRef] - Bernstein, J.; Brown, L.S.; Feinberg, G. The Cosmological Heavy Neutrino Problem Revisited. Phys. Rev. D
**1985**, 32, 3261. [Google Scholar] [CrossRef] - Kolb, E.W.; Olive, K.A. The Lee-Weinberg Bound Revisited. Phys. Rev. D
**1986**, 33, 1202. [Google Scholar] [CrossRef] - Profumo, S.; Sigurdson, K.; Kamionkowski, M. What mass are the smallest protohalos? Phys. Rev. Lett.
**2006**, 97, 031301. [Google Scholar] [CrossRef] [PubMed] - Griest, K.; Seckel, D. Three exceptions in the calculation of relic abundances. Phys. Rev. D
**1991**, 43, 3191–3203. [Google Scholar] [CrossRef] - Edsjo, J.; Gondolo, P. Neutralino relic density including coannihilations. Phys. Rev. D
**1997**, 56, 1879–1894. [Google Scholar] [CrossRef] [Green Version] - D’Agnolo, R.T.; Ruderman, J.T. Light Dark Matter from Forbidden Channels. Phys. Rev. Lett.
**2015**, 115, 061301. [Google Scholar] [CrossRef] [PubMed] - Cline, J.; Liu, H.; Slatyer, T.; Xue, W. Enabling Forbidden Dark Matter. Phys. Rev. D
**2017**, 96, 083521. [Google Scholar] [CrossRef] - Chang, S.; Pierce, A.; Weiner, N. Momentum Dependent Dark Matter Scattering. J. Cosmol. Astropart. Phys.
**2010**, 1001, 006. [Google Scholar] [CrossRef] - Fan, J.; Reece, M.; Wang, L.T. Non-relativistic effective theory of dark matter direct detection. J. Cosmol. Astropart. Phys.
**2010**, 1011, 042. [Google Scholar] [CrossRef] - Fitzpatrick, A.L.; Haxton, W.; Katz, E.; Lubbers, N.; Xu, Y. The Effective Field Theory of Dark Matter Direct Detection. J. Cosmol. Astropart. Phys.
**2013**, 1302, 004. [Google Scholar] [CrossRef] - Hisano, J.; Matsumoto, S.; Nojiri, M.M.; Saito, O. Non-perturbative effect on dark matter annihilation and gamma ray signature from galactic center. Phys. Rev. D
**2005**, 71, 063528. [Google Scholar] [CrossRef] - Lattanzi, M.; Silk, J.I. Can the WIMP annihilation boost factor be boosted by the Sommerfeld enhancement? Phys. Rev. D
**2009**, 79, 083523. [Google Scholar] [CrossRef] - Cirelli, M.; Kadastik, M.; Raidal, M.; Strumia, A. Model-independent implications of the e+-, anti-proton cosmic ray spectra on properties of Dark Matter. Nucl. Phys. B
**2009**, 813, 1–21. [Google Scholar] [CrossRef] - Arkani-Hamed, N.; Finkbeiner, D.P.; Slatyer, T.R.; Weiner, N. A Theory of Dark Matter. Phys. Rev. D
**2009**, 79, 015014. [Google Scholar] [CrossRef] - Pospelov, M.; Ritz, A. Astrophysical Signatures of Secluded Dark Matter. Phys. Lett. B
**2009**, 671, 391–397. [Google Scholar] [CrossRef] - Fox, P.J.; Poppitz, E. Leptophilic Dark Matter. Phys. Rev. D
**2009**, 79, 083528. [Google Scholar] [CrossRef] - Iengo, R. Sommerfeld enhancement: General results from field theory diagrams. J. High Energy Phys.
**2009**, 05, 024. [Google Scholar] [CrossRef] - Hall, L.J.; Jedamzik, K.; March-Russell, J.; West, S.M. Freeze-In Production of FIMP Dark Matter. J. High Energy Phys.
**2010**, 03, 080. [Google Scholar] [CrossRef] - Co, R.T.; D’Eramo, F.; Hall, L.J.; Pappadopulo, D. Freeze-In Dark Matter with Displaced Signatures at Colliders. J. Cosmol. Astropart. Phys.
**2015**, 1512, 024. [Google Scholar] [CrossRef] - Bernal, N.; Heikinheimo, M.; Tenkanen, T.; Tuominen, K.; Vaskonen, V. The Dawn of FIMP Dark Matter: A Review of Models and Constraints. Int. J. Mod. Phys.
**2017**, A32, 1730023. [Google Scholar] [CrossRef] - Kamionkowski, M.; Turner, M.S. Thermal relics: Do we know their abundances? Phys. Rev. D
**1990**, 42, 3310–3320. [Google Scholar] [CrossRef] [Green Version] - Kofman, L.; Linde, A.D.; Starobinsky, A.A. Reheating after inflation. Phys. Rev. Lett.
**1994**, 73, 3195–3198. [Google Scholar] [CrossRef] [PubMed] - Kofman, L.; Linde, A.D.; Starobinsky, A.A. Towards the theory of reheating after inflation. Phys. Rev. D
**1997**, 56, 3258–3295. [Google Scholar] [CrossRef] [Green Version] - Kawasaki, M.; Kohri, K.; Sugiyama, N. Cosmological constraints on late time entropy production. Phys. Rev. Lett.
**1999**, 82, 4168. [Google Scholar] [CrossRef] [Green Version] - Kawasaki, M.; Kohri, K.; Sugiyama, N. MeV scale reheating temperature and thermalization of neutrino background. Phys. Rev. D
**2000**, 62, 023506. [Google Scholar] [CrossRef] - Hannestad, S. What is the lowest possible reheating temperature? Phys. Rev. D
**2004**, 70, 043506. [Google Scholar] [CrossRef] - Ichikawa, K.; Kawasaki, M.; Takahashi, F. The Oscillation effects on thermalization of the neutrinos in the Universe with low reheating temperature. Phys. Rev. D
**2005**, 72, 043522. [Google Scholar] [CrossRef] - De Bernardis, F.; Pagano, L.; Melchiorri, A. New constraints on the reheating temperature of the universe after WMAP-5. Astropart. Phys.
**2008**, 30, 192–195. [Google Scholar] [CrossRef] - Gelmini, G.B.; Gondolo, P. Ultra-cold WIMPs: Relics of non-standard pre-BBN cosmologies. J. Cosmol. Astropart. Phys.
**2008**, 0810, 002. [Google Scholar] [CrossRef] - Visinelli, L.; Gondolo, P. Kinetic decoupling of WIMPs: Analytic expressions. Phys. Rev. D
**2015**, 91, 083526. [Google Scholar] [CrossRef] - Waldstein, I.R.; Erickcek, A.L.; Ilie, C. Quasidecoupled state for dark matter in nonstandard thermal histories. Phys. Rev. D
**2017**, 95, 123531. [Google Scholar] [CrossRef] [Green Version] - Waldstein, I.R.; Erickcek, A.L. Comment on “Kinetic decoupling of WIMPs: Analytic expressions”. Phys. Rev. D
**2017**, 95, 088301. [Google Scholar] [CrossRef] [Green Version] - Dine, M.; Fischler, W. The Not So Harmless Axion. Phys. Lett. B
**1983**, 120, 137–141. [Google Scholar] [CrossRef] - Steinhardt, P.J.; Turner, M.S. Saving the Invisible Axion. Phys. Lett. B
**1983**, 129, 51–56. [Google Scholar] [CrossRef] - Turner, M.S. Coherent Scalar Field Oscillations in an Expanding Universe. Phys. Rev. D
**1983**, 28, 1243. [Google Scholar] [CrossRef] - Scherrer, R.J.; Turner, M.S. Decaying Particles Do Not Heat Up the Universe. Phys. Rev. D
**1985**, 31, 681. [Google Scholar] [CrossRef] - Lyth, D.H.; Stewart, E.D. Thermal inflation and the moduli problem. Phys. Rev. D
**1996**, 53, 1784–1798. [Google Scholar] [CrossRef] [Green Version] - Chung, D.J.H.; Kolb, E.W.; Riotto, A. Production of massive particles during reheating. Phys. Rev. D
**1999**, 60, 063504. [Google Scholar] [CrossRef] [Green Version] - Giudice, G.F.; Kolb, E.W.; Riotto, A. Largest temperature of the radiation era and its cosmological implications. Phys. Rev. D
**2001**, 64, 023508. [Google Scholar] [CrossRef] - Moroi, T.; Randall, L. Wino cold dark matter from anomaly mediated SUSY breaking. Nucl. Phys. B
**2000**, 570, 455–472. [Google Scholar] [CrossRef] [Green Version] - Fujii, M.; Hamaguchi, K. Nonthermal dark matter via Affleck-Dine baryogenesis and its detection possibility. Phys. Rev. D
**2002**, 66, 083501. [Google Scholar] [CrossRef] [Green Version] - Fujii, M.; Ibe, M.; Yanagida, T. Thermal leptogenesis and gauge mediation. Phys. Rev. D
**2004**, 69, 015006. [Google Scholar] [CrossRef] [Green Version] - Gelmini, G.B.; Gondolo, P. Neutralino with the right cold dark matter abundance in (almost) any supersymmetric model. Phys. Rev. D
**2006**, 74, 023510. [Google Scholar] [CrossRef] - Gelmini, G.; Gondolo, P.; Soldatenko, A.; Yaguna, C.E. The Effect of a late decaying scalar on the neutralino relic density. Phys. Rev. D
**2006**, 74, 083514. [Google Scholar] [CrossRef] - Acharya, B.S.; Kane, G.; Watson, S.; Kumar, P. A Non-thermal WIMP Miracle. Phys. Rev. D
**2009**, 80, 083529. [Google Scholar] [CrossRef] - Grin, D.; Smith, T.; Kamionkowski, M. Thermal axion constraints in non-standard thermal histories. AIP Conf. Proc.
**2010**, 1274, 78–84. [Google Scholar] - Harigaya, K.; Kawasaki, M.; Mukaida, K.; Yamada, M. Dark Matter Production in Late Time Reheating. Phys. Rev. D
**2014**, 89, 083532. [Google Scholar] [CrossRef] - Baer, H.; Choi, K.Y.; Kim, J.E.; Roszkowski, L. Dark matter production in the early Universe: Beyond the thermal WIMP paradigm. Phys. Rep.
**2015**, 555, 1–60. [Google Scholar] [CrossRef] [Green Version] - Monteux, A.; Shin, C.S. Thermal Goldstino Production with Low Reheating Temperatures. Phys. Rev. D
**2015**, 92, 035002. [Google Scholar] [CrossRef] - Reece, M.; Roxlo, T. Nonthermal production of dark radiation and dark matter. J. High Energy Phys.
**2016**, 09, 096. [Google Scholar] [CrossRef] - Kane, G.L.; Kumar, P.; Nelson, B.D.; Zheng, B. Dark matter production mechanisms with a nonthermal cosmological history: A classification. Phys. Rev. D
**2016**, 93, 063527. [Google Scholar] [CrossRef] [Green Version] - Erickcek, A.L. The Dark Matter Annihilation Boost from Low-Temperature Reheating. Phys. Rev. D
**2015**, 92, 103505. [Google Scholar] [CrossRef] - Kim, H.; Hong, J.P.; Shin, C.S. A map of the non-thermal WIMP. Phys. Lett. B
**2017**, 768, 292–298. [Google Scholar] [CrossRef] - Barrow, J.D. Massive Particles as a Probe of the Early Universe. Nucl. Phys. B
**1982**, 208, 501–508. [Google Scholar] [CrossRef] - Ford, L.H. Gravitational Particle Creation and Inflation. Phys. Rev. D
**1987**, 35, 2955. [Google Scholar] [CrossRef] - Spokoiny, B. Deflationary universe scenario. Phys. Lett. B
**1993**, 315, 40–45. [Google Scholar] [CrossRef] - Joyce, M. Electroweak Baryogenesis and the Expansion Rate of the Universe. Phys. Rev. D
**1997**, 55, 1875–1878. [Google Scholar] [CrossRef] [Green Version] - Salati, P. Quintessence and the relic density of neutralinos. Phys. Lett. B
**2003**, 571, 121–131. [Google Scholar] [CrossRef] - Profumo, S.; Ullio, P. SUSY dark matter and quintessence. J. Cosmol. Astropart. Phys.
**2003**, 0311, 006. [Google Scholar] [CrossRef] - Feng, J.L.; Rajaraman, A.; Takayama, F. SuperWIMP dark matter signals from the early universe. Phys. Rev. D
**2003**, 68, 063504. [Google Scholar] [CrossRef] - Feng, J.L.; Su, S.f.; Takayama, F. SuperWIMP gravitino dark matter from slepton and sneutrino decays. Phys. Rev. D
**2004**, 70, 063514. [Google Scholar] [CrossRef] - Pallis, C. Quintessential kination and cold dark matter abundance. J. Cosmol. Astropart. Phys.
**2005**, 0510, 015. [Google Scholar] [CrossRef] - Gomez, M.E.; Lola, S.; Pallis, C.; Rodriguez-Quintero, J. Quintessential Kination and Thermal Production of SUSY e-WIMPs. AIP Conf. Proc.
**2009**, 1115, 157–162. [Google Scholar] - Lola, S.; Pallis, C.; Tzelati, E. Tracking Quintessence and Cold Dark Matter Candidates. J. Cosmol. Astropart. Phys.
**2009**, 0911, 017. [Google Scholar] [CrossRef] - Lewicki, M.; Rindler-Daller, T.; Wells, J.D. Enabling Electroweak Baryogenesis through Dark Matter. J. High Energy Phys.
**2016**, 06, 055. [Google Scholar] [CrossRef] - Artymowski, M.; Lewicki, M.; Wells, J.D. Gravitational wave and collider implications of electroweak baryogenesis aided by non-standard cosmology. J. High Energy Phys.
**2017**, 03, 066. [Google Scholar] [CrossRef] - Redmond, K.; Erickcek, A.L. New Constraints on Dark Matter Production during Kination. Phys. Rev. D
**2017**, 96, 043511. [Google Scholar] [CrossRef] - D’Eramo, F.; Fernandez, N.; Profumo, S. When the Universe Expands Too Fast: Relentless Dark Matter. J. Cosmol. Astropart. Phys.
**2017**. [Google Scholar] [CrossRef] - Pallis, C. Kination-dominated reheating and cold dark matter abundance. Nucl. Phys. B
**2006**, 751, 129–159. [Google Scholar] [CrossRef] [Green Version] - Allahverdi, R.; Drees, M. Production of Massive Stable Particles in Inflaton Decay. Phys. Rev. Lett.
**2002**, 89, 091302. [Google Scholar] [CrossRef] [PubMed] - Choi, K.Y.; Kim, J.E.; Lee, H.M.; Seto, O. Neutralino dark matter from heavy axino decay. Phys. Rev. D
**2008**, 77, 123501. [Google Scholar] [CrossRef] - Ackermann, M. Updated search for spectral lines from Galactic dark matter interactions with pass 8 data from the Fermi Large Area Telescope. Phys. Rev. D
**2015**, 91, 122002. [Google Scholar] [CrossRef] [Green Version] - Gondolo, P.; Gelmini, G. Cosmic abundances of stable particles: Improved analysis. Nucl. Phys. B
**1991**, 360, 145–179. [Google Scholar] [CrossRef] - Kim, Y.G.; Lee, K.Y. The Minimal model of fermionic dark matter. Phys. Rev. D
**2007**, 75, 115012. [Google Scholar] [CrossRef]

**Figure 1.**The quantities $\mathsf{\Phi}$ and R, defined in Equation (8) and related to ${\rho}_{\varphi}$ and ${\rho}_{R}$ respectively, in units of the initial value ${\mathsf{\Phi}}_{I}$. The vertical dashed line marks the moment at which the transition to the standard scenario occurs, according to Equation (18). The green dot-dashed line shows the temperature $T\left(x\right)$, in units of its initial value $T\left({x}_{I}\right)$.

**Figure 2.**The present WIMP relic abundance for different values of $\langle \sigma v\rangle $ (see figure labels), and for different values of the branching ratio: $b=0$ (red dot-dashed line), $b=0.001$ (blue solid line), and $b=1$ (green dashed line), as a function of ${T}_{\mathrm{kin}}$. The horizontal dashed line shows the measured dark matter abundance ${\mathsf{\Omega}}_{\mathrm{DM}}{h}^{2}\sim 0.12$. The vertical yellow band defines the region excluded by BBN considerations, ${T}_{\mathrm{kin}}\ge 5\phantom{\rule{0.166667em}{0ex}}$MeV. Freeze-out occurs in the kination scenario in the region to the left of the black dot-dashed curve.

**Figure 3.**The solution to the Boltzmann Equation (13) for different values of the annihilation cross section. We have set ${T}_{\mathrm{kin}}=100\phantom{\rule{0.166667em}{0ex}}$MeV and, when non-thermal production is considered, $b=2\times {10}^{-4}$. See text for further details.

**Figure 4.**Top panel: the present relic density ${\mathsf{\Omega}}_{\chi}{h}^{2}$, in units of the critical density of the Universe, as a function of the quantity $\langle \sigma v\rangle $ for ${T}_{\mathrm{kin}}=1\phantom{\rule{0.166667em}{0ex}}$GeV, $b=0$, and for the value of the WIMP mass ${m}_{\chi}=100\phantom{\rule{0.166667em}{0ex}}$GeV (red line), ${m}_{\chi}=200\phantom{\rule{0.166667em}{0ex}}$GeV (orange line), ${m}_{\chi}=500\phantom{\rule{0.166667em}{0ex}}$GeV (green line), and ${m}_{\chi}=1000\phantom{\rule{0.166667em}{0ex}}$GeV (blue line). Bottom panel: same as the top panel for ${T}_{\mathrm{kin}}=1\phantom{\rule{0.166667em}{0ex}}$GeV, ${m}_{\chi}=100\phantom{\rule{0.166667em}{0ex}}$GeV and for the branching ratio $b={10}^{-11}$ (red line), $b={10}^{-8}$ (orange line), $b={10}^{-5}$ (green line), and $b={10}^{-2}$ (blue line).

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Visinelli, L.
(Non-)Thermal Production of WIMPs during Kination. *Symmetry* **2018**, *10*, 546.
https://doi.org/10.3390/sym10110546

**AMA Style**

Visinelli L.
(Non-)Thermal Production of WIMPs during Kination. *Symmetry*. 2018; 10(11):546.
https://doi.org/10.3390/sym10110546

**Chicago/Turabian Style**

Visinelli, Luca.
2018. "(Non-)Thermal Production of WIMPs during Kination" *Symmetry* 10, no. 11: 546.
https://doi.org/10.3390/sym10110546