Thermalization in the presence of a time-dependent dissipation and its impact on dark matter production

In standard cosmological scenarios, a heavy meta-stable field dominates the energy density of the universe after inflation. The dissipation of this field continuously sources high-energy particles. In general, the dissipation rate of this meta-stable field can have a non-trivial time dependence. We study the impact of this time-dependent dissipation rate on the thermalization of the high-energy decay products of the meta-stable field. These energetic particles can contribute substantially to dark matter production in addition to the usual production from the thermal bath particles during reheating. We investigate the impact of this generalized dissipation on dark matter production in a model-independent way. We illustrate the parameter space that explains the observed dark matter relic abundance in various cosmological scenarios. We observed that dark matter having a mass larger than the maximum temperature attained by the thermal bath can be produced from the collision of the high-energy particles which are not yet thermalized.


Introduction
In standard cosmological theories existence of a meta-stable field dominating the energy density of the universe at the end of inflation is postulated.This meta-stable field can be e.g., an inflaton field [1,2], a curvaton [3], a dilaton [4], or string theory inspired modulus field [5][6][7][8][9] etc.The meta-stable field dissipates its energy and produces a thermal bath of relativistic particles or radiation.The universe reheats into a temperature T Reh via this dissipation and enters into a radiation dominated era at the end of reheating.
In recent years, there have been works [1,[10][11][12][13][14][15][16][17][18][19][20][21] which envisages the possibility of a stable relic production during this matter dominated era after the end of inflation.Ref. [11] has shown that during reheating the thermal bath can attain a significantly higher temperature than T Reh , the temperature at the onset of radiation dominated universe.It has also shown that after reaching the maximum temperature T max , the temperature of the thermal bath falls off differently (T ∝ a −3/8 ) in comparison with the usual radiation dominated epoch (T ∝ a −1 ).Ref. [11] has also shown that how the relic abundance of dark matter depends on T max and there can be situations in which significant amount of dark matter can be produced at T ∼ T max [15,[22][23][24][25].
These recent works assume one crucial assumption that the radiation created from the decay of the meta-stable scalar field instantaneously thermalizes among themselves and creates a dominating thermal bath of temperature T .Recently the validity of this assumption has been questioned.There have been a few works [26][27][28][29][30][31] which relaxes this assumption and tries to address the finiteness of the timescale for thermalization.Taking this into account ref. [26][27][28][29] have shown that UV-sensitivity of the scattering cross-section significantly enhances dark matter production both before and after thermalization.The timescale for thermalization is vital for determining the abundance of dark matter produced during reheating.
In the very early stages of reheating, these decay products are highly energetic and less populated compared to their thermal counterpart.These high-energy particles loose their energy and form a thermal distribution of particles having energies much smaller than their progenitors.The thermalization of these high-energy particles, produced via dissipation of the meta-stable field, proceeds via 2 → N scattering with N ≥ 2 [12,27,28,30,32,33]. 2 → 2 scattering hardly plays any role in attaining a chemical equilibrium, but only redistribute the energy among these particles.It has been shown in ref. [12,27,28,32] that 2 → 3 scattering process is more efficient for the thermalization process in comparison with 2 → 2 scattering.This 2 → 3 process can be realized as a 2 → 2 process followed by a 1 → 2 process, in which any one of the final-state particle splits nearly-collinear in momentum into two low-energy daughter particles.Due to this collinearity, these daughter particles experience multiple scattering before their formation which suppresses the splitting rate [34,35].This destructive interference effect is known as Landau-Pomeranchuk-Migdal(LPM) effect.This suppressed splitting rate delays the thermalization, but happens at a time-scale smaller than the time-scale associated with a 2 → 2 scattering process.
It was noted in [36,37] that the dissipation rate of a meta-stable field oscillating in a thermal background depends on the field and its mass or the background temperature.If the background temperature is smaller than the mass of the field, then the dissipation rate only depends on the field and its mass.For a quadratic potential, the amplitude of the field redshifts as a −3/2 .Thus, the evolution of the field induces a time dependence in the dissipation rate of the field [37].Motivated by these scenarios, in this work, we focus on the effect of having a generalized dissipation rate [38,39] on the thermalization of high-energy particles.Presence of such generalized dissipation rate alters the temperature evolution of the universe during reheating as compared to the scenario of a constant dissipation rate.Thus, we study the impact of such generalized dissipation rate on dark matter production via the freeze-in mechanism [22,23,[40][41][42][43][44][45][46][47][48][49][50][51][52][53], while considering non-instantaneous thermalization, in a model independent way.We delineate the parameter space satisfying observed dark matter relic for a few specific models.We find that dark matter can still be produced from the scattering of non-thermal high-energy particles in the region where production from the thermalized bath particles is kinematically forbidden.the abundance of dark matter produced during reheating in the presence of a generalized dissipation rate.In Sec. 4, we calculate the total dark matter relic density before and after the onset of radiation dominated era considering non-instantaneous thermalization.We study the viable parameter space for specific models in Sec. 5, and compare various contributions that contributes to the dark matter relic in Sec. 6.Finally, we summarize our results in Sec. 7.
2 Reheating via a time-dependent dissipation rate of a meta-stable field During inflation, the universe goes through a period of accelerated exponential expansion.After the end of inflation, a heavy meta-stable field ϕ coherently oscillates around the minimum of its potential and then decays into relativistic particles.The meta-stable field dissipates its energy density into its decay products.The dynamics of this energy transfer is described by the following set of Boltzmann equations [25,54,55] ) where ρ ϕ is the average energy density of the meta-stable field, ρ R is the energy density of its decay products, Γ ϕ denotes the dissipation rate of the meta-stable field, ω is the equation-of-state of the meta-stable field, H(≡ ȧ/a) is the Hubble expansion rate of the universe, where a indicates the scale factor of the Friedmann-Robertson-Walker (FRW) Universe.Using the Friedmann equation, one can express H in terms of ρ ϕ and ρ R and is given by where GeV is the reduced Planck mass.The conventional assumption is that after inflation, a heavy meta-stable field decays via a time-independent (constant) dissipation rate [26,[56][57][58][59][60] and reheats the universe.In principle, the dissipation rate of the meta-stable field Γ ϕ can depend on the scale factor a(t) of evolution.The dissipation rate of the meta-stable field can evolve with time through its dependence on energy density and field-dependent mass [25,37,55,56,[61][62][63]. Depending upon the ratio of Γ ϕ /H, there can be situations in which reheating is incomplete (see ref. [25,37,55]).It was shown in ref. [36,37] that the dissipation rate can be expressed where m ϕ is the mass of the decaying field, φ is its amplitude and M is a UV mass-scale associated with the theory.This dissipation rate results from an effective operator obtained through integrating out a heavy particle.The effective operator is valid only when the mass of the integrated particle is larger than the mass of ϕ, known as the large field limit [36].The large field limit can be realized in various cosmological scenarios e.g., in Affleck-Dine baryogenesis [64,65], modulated reheating in the presence of a spectator field [66][67][68], and axionic dark matter production within the paradigm of kinetic misalignment [69,70].Therefore, it is plausible that the universe reheats within the region of validity of the above-mentioned dissipation rate.Following this, we parametrize the dissipation rate of the meta-stable field as where Γ 0 is the time-independent part of the dissipation rate of the inflaton, and a end is the scale factor at the end of inflation.In the very early stage of reheating, we assume that Γ ϕ (a)/H(a) ≪ 1, and the energy density of the meta-stable field dominates the energy density of the universe.The solution of the Eq.(2.1) and Eq.(2.2) with the initial conditions where, z is a dimensionless variable (z ≡ a/a end ) and H end is the Hubble parameter at the end of inflation.The particle produced from the decay of the meta-stable field has energy m ϕ /2 with m ϕ being the mass of the meta-stable field.These high-energy decay products do not thermalize instantaneously, rather take a finite amount of time to form a thermal bath.For complete understanding of the kinematics of the thermalization process, one needs the evolution of the phase-space distribution of these high-energy decay products.The Boltzmann equation for the process ϕ → γ + γ to compute the phase space distribution of these decay products is given by [26,71,72] (2.7) The general solution to the Eq.(2.7) is given by (for a detailed derivation see appendix A) where we have used n R = 6(1+ω) 2k 1 +3(1+ω) n ϕ (t)Γ ϕ (t)t in the regime Γ ϕ t ≪ 1.For ω = 0, the above expression reduces to with the condition a(t) a( t) = , where t is the cosmic time when p is equal to m ϕ /2.n ϕ (t) is the number density of inflaton that has not decayed upto that instant and n R (t) is the number density of the decay products.The expression in Eq. (2.9) matches with the phase-space distribution of ref. [26] in the k 1 → 0 limit.In the next section we will discuss the thermalization of the decay products of the meta-stable field.

Thermalization of the decay products
We briefly discuss the thermalization of the highly energetic decay products of the metastable field in this section.After the end of inflation, the high-energy particles from the decay of the meta-stable field lose their energy either via large-angle scattering or through soft splittings.Ref. [35] has shown that large-angle scattering (scatterings with large momentum exchange) alone is not sufficient to thermalize the decay products of the meta-stable field.Following ref. [34], we will consider the thermalization of these high-energy decay products via splitting into low-energy particles mediated by a gauge interaction of coupling strength α.During thermalization, the high-energy particles with momenta p ∼ m ϕ dominate the number and energy densities of the decay products.Since, the mass of the meta-stable field m ϕ (z) = m ϕ (a end )z −3ω redshifts faster for nonzero ω, the decays products thermalize faster compared to ω = 0. So, from this section onwards, we only consider the case with ω = 0. We consider the case having a mono-energetic distribution function of the decay products where n R is the number density of the high-energy particles with momenta p ∼ m ϕ .Following ref. [34], we express the distribution function in terms of the gauge coupling strength From Eq. (2.10) -(2.11) we can write where c is an arbitrary constant.For the same energy density, systems with c > 0 is overoccupied and c < 0 is under-occupied compared to thermal occupancy.The number density of these high-energy particles can be written as (2.14) These high-energy decay products dissipate their energy continuously into low-energy particles via multiple collinear splittings.For a collinear splitting (small angle splitting) when the formation of the wave-packet takes a longer period compared to the time-scale associated with elastic scattering (t el ∼ Γ −1 el ), then there will be interference.This out-of-phase interference suppresses the splitting rate between the daughter and the mother particle, which is known as Landau-Pomeranchuk-Migdal(LPM) effect [73][74][75][76].The LPM suppressed splitting rate reads [34] as where t form is the formation time of a collinear splitting process given by where qelastic is mean-squared momentum transfer per unit time.These low-energy particles produce a thermal bath having an effective temperature T s .The temperature of the softthermal bath alone is sufficient to describe its dynamics.Momentum transfer in this process can be written in terms of T s as At this moment, the low-energy particles dominate the number density, whereas the highenergy particles dominate the energy density.We assume daughter particles with momentum k split can transfer their energy entirely to the soft-thermal bath.One can get an estimate about the scale k split by demanding that in time t, each high-energy particle produces O(1) such daughter particles i.e., Using Eq. (2.15) -(2.17) we can express k split as Particles with momentum k split can split into daughter particles having smaller momentum through a very high splitting rate.These particles with momentum k split transfer all their energy into the soft-thermal bath through multiple splittings in a period much shorter than the formation time of the initial daughter particle.The energy density of the soft-thermal bath is Solving Eq. (2.18) we get the expression for the temperature of the soft-thermal bath and These high-energy particles lose their energy completely when the splitting momentum becomes comparable to their maximum momentum, i.e., k split | z=z th ∼ m ϕ .At the time t th , the thermalization of the decay products concluded.The corresponding z is given by . (2.23) Table 2.1: Typical time-scale for thermalization in various scenarios In the above table, we compare the typical time-scale for the thermalization of high-energy decay products.We have set α = 0.03, m ϕ = 3 × 10 13 GeV, H end = 10 12 GeV to obtain z th for different k 1 .We see that the time-scale for thermalization differs by orders of magnitude in various cosmological scenarios.The time needed for thermalization decreases with an increasing dissipation rate.
Using the expression of t th (z th ) in Eq. (2.21), we get the thermalization temperature T th as . (2.24) This generalized expression of T th matches with the expression of [26] in k 1 → 0 limit.

Spectrum of high-energy decay products after thermalization
The decay products are thermalized at z th , corresponding to a temperature T th .The thermalized sector dominates the energy density, as well as the number density of radiation after the thermalization, i.e., for z > z th .The meta-stable field decay does not end at z th .It rather continuously supplies high-energy particles with energy E ≃ m ϕ /2 > T (z > z th ) (temperature of the particles in the thermal bath).These high-energy particles take a finite time to thermalize.We will now estimate the spectrum of these particles with the background thermal bath in the next section.
The phase-space distribution of these out-of-equilibrium particles depends on the magnitude of their three-momentum p and time t.In a collinear splitting process, the transverse momenta of the daughter particles are negligible.We consider the phase space distribution function in one dimension 1 .The relevant Boltzmann equation governing the splitting processes is [27,29] (2.25) The first term on the right-hand side of Eq. (2.25) represents the contribution from the decay of the meta-stable field.The second and third terms describe the injection of particles with momentum p from the splitting of higher momentum (> p) particles.The last term illustrates the depletion of particles with a momentum of p due to splitting into daughter particles of momentum k.We use a hard infrared cutoff to regularize the infrared divergences in the last two integrals.This will not affect the result as long as p ≫ T .We avoid the collinear divergences by demanding a minimum splitting angle θ min which is of O((T /m ϕ ) 3/4 ).The splitting rate of high-energy decay products gets suppressed due to the LPM effect.The LPM suppressed splitting rate can be written as [29,34] where g * denotes the number of degrees of freedom of these low-energy particles in the thermal bath.Solving Eq. (2.25) using Eq.(2.26) we get [27,29] The decay products of the meta-stable field are thermalized at z th .The thermalized sector dominates the energy density of radiation after thermalization.The temperature of the thermal bath after thermalization evolved as where g * (T ) is the number of relativistic degrees of freedom of the thermal bath at temperature T .From the Eq.(2.28), one can see that depending upon the value of k 1 , the evolution of temperature during reheating can be divided into three categories: Reheating concludes when the energy density of the meta-stable field and radiation becomes equal, i.e., ρ ϕ (z Reh ) = ρ R (z Reh ) and the corresponding z is given by . (2.29) The temperature of the thermal bath corresponding to z Reh is given as (2.30) The particles in the thermal bath dominate the energy density of the universe after reheating, and the bath temperature redshifts inversely with the scale factor (a −1 ).

Dark matter production during reheating
In the previous section, we have derived the phase-space distribution of the non-thermal2 particles for time-dependent dissipation of the meta-stable field.In this section, we consider the impact of these highly energetic particles on dark matter production through freeze-in during the reheating of the universe.We assume that the dark matter does not couple directly to the meta-stable field.Therefore, we are not considering dark matter production from the meta-stable field.We concentrate on dark matter production from the collision between the high-energy decay products of the meta-stable field and the particles in the thermal bath.There are four production mechanisms of dark matter during reheating • Non-thermal(I): Production of dark matter from high-energy particles before thermalization.
• Non-thermal(II): Dark matter production from the collision between two high-energy particles after thermalization.
• Non-thermal(III): Production of dark matter from the collision between a high-energy particle and a particle in the thermal bath.
• Thermal(IV): Dark matter production from the collision between two particles in thermal bath.
The total cross-section for dark matter production can be parametrized as [26,28] where √ s is the center of mass energy of the process, M is a heavy mass-scale relevant to the process, λ is a dimensionless coupling and c n = 32 . While parametrizing the cross section as expressed in Eq. (3.1), we have assumed m χ , m inc , √ s ≪ M , with m inc being the mass of the incoming particles.Cross-section with n > 0 can be due to the interaction of dark matter with visible sector particles via a mediator having derivative couplings [22,43,44,48,49,77], where M can be identified as the cutoff scale of the theory.

DM production before thermalization
Since we assume that dark matter does not yield from the decay of the meta-stable field, the only production channel available for dark matter production before thermalization is γ(k 1 ) + γ(k 2 ) → χ(p 1 ) + χ(p 2 ).In this section, we consider dark matter production before thermalization from these high-energy decay products.The Boltzmann equation for the dark matter number density can be written as where σ(s) is the scattering cross-section for the process γ + γ → χ + χ. g χ and g R denotes the number of degrees of freedom (d.o.f) of χ and γ, respectively.Using the phase-space distribution function of the high-energy decay products given in Eq. (2.9), we get ⟨σv⟩ N T as The comoving number density of dark matter can be defined as Y χ (z) ≡ n χ (z) z 3 .We rewrite Eq. (3.2) in terms Y χ (z).We get As, the meta-stable field dominates the energy density of the universe during reheating, Hubble parameter H(z) can be expressed as Solving Eq. (3.4), we get the amount of dark matter produced from the high-energy decay products until thermalization

Dark matter production after thermalization
For z < z th , the entire contribution in dark matter number density comes from the highenergy particles.Whereas, for z > z th , there exists a thermal bath of particles with characteristics temperature T .We now focus on the dark matter produced during the period z th < z < z Reh .The relevant Boltzmann equation has the form where R(T ) denotes the production rate of dark matter.We can express the production rate as Where f psth R denotes the phase-space distribution of particles after thermalization, it can be either fR (p) given in Eq. (2.27) or f (eq) ξi where f (eq) ξi is the equilibrium phase space distribution, and it has the form f (eq) ξi = e (−E i /T ) , E i is the energy of ξ i .The Eq. (3.7) can be expressed in terms of the comoving number density of dark matter and the parameter z as The solution of the Eq.(3.9) gives the comoving number density of dark matter produced during the period z th < z < z Reh and is given by where Y (NT) χ (z th ) is given in Eq. (3.6).The task left for us is to calculate the production rate of dark matter for various production processes in this regime.As discussed earlier, there are three processes that contribute in dark matter production during the period z th < z < z Reh and is given by • collision between two high-energy particles.
• collision between a high-energy particle and a particle in thermal bath.
• a collision between two particles in thermal bath.

Non-thermal(II): Collision between two non-thermal particle with E 1 , E 2 > T
High-energy decay products are continuously produced from the decay of the meta-stable field until reheating.They take a finite time to attain a equilibrium with the thermal bath.Before their thermalization, these high-energy particles can collide with each other and yield a pair of dark matter particles if the production is kinematically available i.e., m χ < m ϕ /2.In this subsection, we calculate the number density of dark matter produced from the collision between two high energy particles with T < E 1 , E 2 ≤ m ϕ /2.This production channel is kinematically allowed as long as E 1 E 2 ≥ m 2 χ .We designate this production process as Non-thermal(II).The production rate of dark matter for the collision between two high-energy particles can be expressed as 6) . (3.11) Non-thermal(III): Collision between high energy particle with E 1 > T and a particle in the thermal bath Until now, we only considered dark matter production from non-thermal particles.We did not take into account the contribution of the thermal bath particles to dark matter production.But the collision of a high-energy particle and a particle in the thermal bath can produce a substantial amount of dark matter.In this subsection, we focus on calculating the amount of dark matter produced from the collision between a high energy particle with energy T < E 1 ≤ m ϕ 2 and a particle in the thermal bath with characteristic temperature T .This production channel is kinematically available as long as E 1 T ≳ m 2 χ .We label this production channel as Non-thermal(III).The production rate of dark matter for this channel can be written as One can estimate the minimum value of E 1 required for the pair production of dark matter and is given by where T min is the minimum temperature achieved by the thermal bath during reheating.
T min can be either T th or T Reh depending upon the value of k 1 .The comoving number density of non-thermal dark matter at a redshift z can be expressed as

Thermal(IV): Collision between two particles in thermal bath
After z > z th , there exists a thermal bath of particles.These particles can contribute significantly in dark matter production.In this subsection, we concentrate on determining the abundance of dark matter produced from the collision between two particles in thermal bath.We denote this process as Thermal(IV).The production rate of dark matter in this case is given as where f (eq) χi is the equilibrium phase space distribution and it has the form f ), E i is the energy of χ i .The function λ is given by (3.16) The comoving number density of thermal dark matter at a redshift z can be expressed as where the initial number density of thermal dark matter is

Dark matter relic density with non-instantaneous thermalization
In this section, we calculate the total dark matter relic density accumulated since the end of inflation until the present day.We begin this section by discussing the dark matter production after reheating is complete.The solution of the Boltzmann equation (Eq.(3.7)) gives [15,22,63] where R(T ) denotes the production rate of dark matter.It can be written as Integrating Eq. (4.1) using Eq.(4.2) gives, Dark matter production from the thermal bath particles occurs dominantly during reheating for n > −1.On the other hand, for n < −1 production dominantly happens after reheating.The present-day relic density of dark matter is given by [78], where ρ c , 3 denotes the critical density today and z 0 is present-day redshift of the universe.Thus, the total relic density of dark matter can be written as the sum of all the contributions • Ω χ, NT h 2 : Relic density of dark matter produced from the collision between two high-energy decay products before thermalization.
• Ω (2) χ, NT h 2 : Relic density of dark matter produced from the collision between two high-energy decay products after thermalization.
• Ω (3) χ, NT h 2 : Relic density of dark matter produced from the collision between a particle in thermal bath and a high-energy decay product after thermalization.
• Ω χ, Th h 2 : Relic density of dark matter produced from the collision between two particles in thermal bath after thermalization. 3The numerical value of ρc based upon present day Hubble expansion rate is given by, The relic density of dark matter produced from the scattering of two high-energy particles after thermalization has the expression for k 1 > 3/10.One can obtain the expression for k 1 < 3/10 by appropriately expressing z th (T th ) in terms of T Reh .Depending upon the value of n, the above expression can be approximated as The above expression of relic density for n < 1 can be re-expressed as The relic density of DM produced from the collision between a particle in thermal bath and a high-energy particle after thermalization can be expressed as where

We can approximate the expression of Ω
(3) χ, NT depending upon the value of n and the approximate expression is given as The relic density of DM for n < 1 has the form The relic density of dark matter produced from the collision between two particles in thermal bath has the form where

Parameter space for cosmological models
In this section, we briefly discuss the microscopic models in which we can have different kinds of temperature evolution depending upon the values of k 1 and ω.Following ref. [37,63], for a rotating scalar field ϕ one can express the dissipation rate as Γ ϕ ∝ φ−2 for m ϕ ≫ T .φ redshifts as a −3/2 for a ϕ 2 potential gives k 1 = 3, ω = 0.For a scalar field ϕ oscillating near the minimum of the ϕ 2 potential, the average dissipation rate for gives k 1 = 3/2, ω = 0.If the scalar field oscillates around the minimum of the ϕ 2 potential that gives k 1 = 0, ω = 0.
Case A : Increasing temperature during reheating In Fig. (5.1), we exhibit the contours of observed dark matter relic density i.e., Ω χ h 2 = 0.12 inferred by PLANCK Collaboration [79] in T Reh − m χ plane for n = 0 (left) and n = 2 (right).In the regime m χ < T Reh , the blue-shaded region demonstrates the dominance of thermal contribution (thermal -IV) to the relic density of DM over other contributions.
The relic density of DM produced from the process thermal(IV) scales as Ω χ ∼ m χ T n+1 Reh .As a result, to produce the observed DM, one needs to reduce T Reh with increasing DM mass.The constant relic density contour falls off with a slope of 1/(n + 1) until the point where DM mass equals the reheating temperature i.e.T Reh = m χ .In the present scenario, T Reh being the maximum temperature attained by the thermal bath during the reheating, thermal production of DM is kinematically forbidden when m χ > T Reh .Therefore, one should expect a sharp increase in reheating temperature along the line m χ = T Reh .However, due to the tail of the modified Bessel function K 1 in Eq. (3.15), one gets a smooth increase in T Reh for m χ > T Reh .
In this regime where m χ > T Reh , although the pair production of DM from the scattering of two thermal bath particles is kinematically disallowed, DM can still be produced from the collision of a bath particle with a high-energy particle.For n = 2, the relic density for the above non-thermal process scales as Reh .So as one moves towards higher dark matter mass one needs to decrease T Reh .In contrast for n = 0, the relic density scales as Reh .In this case, the relic density of DM decreases with increasing DM mass for a fixed T Reh .As a result, T Reh must increase to attain the observed relic density.These behaviors are visible in the orange-colored region in the figure.From Eq. (4.9), one can see, as m 2 χ approaches m ϕ 2 T min , DM number density drops very close to zero.Therefore, one needs a higher reheating temperature T Reh to obtain the observed relic density, illustrated by the sharp increase in T Reh .
In the regime where m 2 χ > m ϕ 2 T min , the exhaustive contribution in DM relic density comes from purely non-thermal processes, i.e., from the scattering between two high-energy particles either before or after thermalization.The green-shaded region in fig.(5.1) delineates this non-thermal domination of the DM relic.In this region, non-thermal (II) contributes significantly to the relic density compared to non-thermal (I).From Eq. (4.7), one can see that the relic density of DM produced through non-thermal(II) scales as Ω χ h 2 ∼ m χ T 4

Reh
for n = 2.As one goes to a larger DM mass m χ , one needs to reduce the reheating temperature.However for n = 0, the relic density of the dark matter has the expression Reh .In a similar fashion as in the case of non-thermal (III), DM number density reduces with increasing dark matter mass, therefore, T Reh must increase for obtaining the observed relic density.From Eq. (4.6), it is visible that as the DM mass moves very close to m ϕ /2, due to the cancellation between m ϕ /2 and m χ , DM number C. The temperature remains constant for case B, which implies T Reh (T th ), can be considered as the minimum temperature of the bath during reheating.Only for Case C, there exists a small region when DM mass is approximately close to m ϕ /2, i.e., m χ ∼ m ϕ /2, the relic density is dominated by the DM produced from the scattering of high-energy particles before thermalization.As can be seen from the table (2.1) that the high-energy decay products take longer time to thermalize for k 1 = 0 as compared to non-zero k 1 .So, these high-energy particles which are not yet thermalized get sufficient amount of time to produce the observe dark matter relic.From the expression of T Reh given in Eq. (2.30), one can clearly see that T Reh depends on z Reh , which in turn depends on Γ 0 and k 1 for fixed H end .For a fixed Γ 0 and DM mass to attain observed dark matter relic, one needs different M for case B and case C compared to case A.
6 Comparison between various contributions in dark matter relic χ, NT + Ω χ, NT + Ω In the blue-colored region, f th > 1 indicating the dominance of thermal relic of DM over non-thermal relic, i.e., DM relic produced from the scattering of two high-energy particles before and after thermalization, and the scattering of a high-energy particle with a thermal bath particle after thermalization, and in the rest of the parameter space, f th < 1 connotes the non-thermal domination into DM relic.In this parameter space, there exists a subset of region where non-thermal contribution before thermalization dominates over non-thermal contributions after thermalization, measured by χ, NT + Ω In the pink-colored region f nt > 1.From figure (6.1), we observed that non-thermal contribution to DM relic is always dominant for n ≳ 3.5.For a higher DM mass, we even noticed non-thermal dominance when k 1 < 1 for all values of n.This is because the thermal production of DM falls exponentially fast when m χ becomes larger than T max (maximum temperature during reheating).DM produced before the thermalization of the decay products contributes significantly when n > 2 with k 1 < 0.3.These numerical results have also been verified with analytical results.

Conclusions
After inflation, a meta-stable field that dominates the energy density of the universe dissipates its energy density via the production of high-energy particles.In general, this dissipation rate can vary with time.In this article, we have studied the impact of this time-dependent dissipation in the production of radiation after the end of inflation.The decay products of this meta-stable field can take a finite amount of time to reach thermal equilibrium with each other.We have studied the thermalization of these high-energy decay products of this meta-stable field in this non-standard cosmological scenario.We find that the decay products take shorter time to thermalize for non-zero k 1 compared to the case with k 1 = 0.
During reheating, the meta-stable field produces a shower of high-energy particles.These particles take some amount of time to dissipate their energy into low-energy particles (soft) to reach in equilibrium with the thermal bath.The temperature of the thermal bath varies differently as compared to the case of a constant dissipation.This unusual variation of temperature can affect the dark matter abundance during this epoch.
In the standard picture, the particles in the thermal bath particles scatter with each other and produce dark matter.In addition to this, highly energetic particles can annihilate into pairs of dark matter.Scattering of these particles with thermal bath particles can also produce dark matter.We calculate the dark matter number density produced from these processes in various scenarios e.g., when the temperature remains constant, decreases, or increases during reheating.We have sketched the relic contours satisfying the observed dark matter abundance for n = 0 and n = 2, where n is the power of the center of mass energy in dark matter production cross-section.The production of dark matter from these highenergy particles is important when the maximum temperature of the universe is smaller than the DM mass.We observed that DM production from the bath particles dominates when m χ < T max .On the other side, when m χ > T max production of dark matter from the particles in the thermal bath gets kinematically suppressed.Dark matter still gets produced from the collision of a high-energy decay product and a particle in the thermal bath.This process gets kinematically blocked when m 2 χ > T min m ϕ /2.After the kinematical blocking of this process, the collision between two high-energy particles either before or after thermalization still can produce dark matter.We find that in the temperature decreasing scenario, dark matter produced from the high-energy particles before thermalization contributes significantly to dark matter relic density when m χ ∼ m ϕ /2.Finally, we compare the relic density of DM produced from various production channels and find that DM production from non-thermal processes is always dominant for n > 3.

B Reheating Temperature
In this subsection, we derive the expression for the reheating temperature.Reheating concludes when the energy density of the meta-stable field is equal to the energy density of radiation i.e., ρ ϕ (z Reh ) = ρ R (z Reh ).The energy density of the meta-stable field has the form ρ ϕ (z) ≈ ρ ϕ,end z −3(1+ω) = 3 M 2 pl H 2 end z −3(1+ω) .(B.1) The energy density of radiation can be approximated as . (B. 3) The temperature evolution of the universe after thermalization until reheating can be written as a function of redshift z and is given by, T (z) = 30 π 2 g * (T )

Figure 5 . 1 :
Figure 5.1: Different colored regions illustrate the parameter spaces in which various contributions to dark matter relic are dominant.Each dashed contour manifests the parameters that can elucidate the observed relic density of DM.We have displayed the specific value of M associated with each contour.We set k 1 = 3.0, ω = 0, H end = 10 12 GeV.We have fixed n = 0 and m ϕ = 3 × 10 13 GeV in the left panel, and n = 2 and m ϕ = 10 12 GeV in the right panel.

Figure 6 . 1 :
Figure 6.1:We compare various contributions to DM relic density, and we have shown explicitly the regions where one contribution dominates over the other in n − k 1 parameter space.The left, middle and right panels are for different values of DM mass.In the plots we have set ω = 0, Γ 0 = 10 −15 GeV, M = 10 −4 M pl .Blue colored region indicates that DM produced from the bath particles dominates the DM relic.In the non-thermal dominant region, pink-colored region illustrates that DM produced from the high-energy particles before thermalization dominates over the production after thermalization.

Table 2 . 2 :
Temperature evolution during reheating Condition Temperature T min T max Reheating condition i.e. ρ ϕ (z Reh ) = ρ R (z Reh ) gives Using the expression of z Reh we can write the reheating temperatureT Reh = 30 π 2 g * (T Reh )