Temperature evolution in the Early Universe and freeze-in at stronger coupling

Dark matter freeze-in at stronger coupling is operative when the Standard Model (SM) bath temperature never exceeds the dark matter mass. An attractive feature of this scenario is that it can be probed by direct detection experiments as well as at the LHC. In this work, we show how the mechanism can be realized in a simple UV complete framework, emphasizing the role of the maximal temperature of the SM thermal bath. We demonstrate that the maximal temperature can coincide with the reheating temperature or be close to it such that dark matter production is always Boltzmann-suppressed. This possibility is realized, for example, if the inflaton decays primarily into feebly interacting right-handed neutrinos, which subsequently generate the SM thermal bath. In this case, the SM sector temperature remains constant over cosmological times prior to reheating.


Introduction
The dark matter (DM) puzzle remains one of the outstanding questions in modern physics.While thermal dark matter in the form of a weakly-interacting-massive-particle (WIMP) emerges naturally in many extensions of the Standard Model, the lack of observational evidence in support of WIMPs [1] motivates us to explore further options.In particular, dark matter can be non-thermal and produced in a variety of ways.One popular mechanism is known as "freeze-in", whereby DM is gradually produced by the SM thermal bath [2][3][4][5].Its conventional version requires a feeble coupling between the SM sector and dark matter, as well as a high enough reheating temperature.This scenario assumes a zero DM abundance before the thermal production starts, which appears quite unlikely in the world with gravity [6].The freeze-in mechanism is also notoriously difficult to test, if possible at all.
The recently proposed "freeze-in at stronger coupling" [7] addresses some of the problematic aspects of traditional freeze-in models.It requires a low reheating temperature such that DM production is Boltzmann-suppressed.In this case, the coupling between DM and the SM sector is allowed to be up to O(1) without affecting the non-thermal nature of dark matter.This makes DM potentially observable in direct detection experiments as well as at the LHC.
The low reheating temperature allows for dilution of gravitationally produced dark matter, which otherwise is problematic in most models [6].Specifically, all particles are produced by gravity or via gravity-induced interactions both during and after inflation.This is particularly problematic for scalar DM, whose field fluctuations build up during inflation [8] resulting in DM overproduction, unless a suppression or dilution mechanism is invoked.Even if DM production gets suppressed during inflation, e.g.via a large Hubble-induced dark matter mass, it gets produced during preheating all the more efficiently.Also, classical and quantum gravitational effects induce higher dimensional operators, which couple dark matter s to the inflaton ϕ [9], e.g.
where C is a Wilson coefficient.During the inflaton oscillation epoch, this operator produces dark matter very efficiently in traditional (large field) inflation models, normally leading to a totally dark Universe.
The resulting abundance can be made consistent with observations only if the Universe undergoes a very long period of inflaton matter-domination, which dilutes all relativistic relics, or if the Wilson coefficient happens to be tiny, which cannot be guaranteed in the absence of a UV complete quantum gravity framework.The constraint can be put in the form [6] ∆ NR ≳ 10 6 C 2 ϕ 8 0 where characterizes the duration of the matter-dominated expansion period.Here H end and H reh are the Hubble rates at the end of inflation and at reheating, respectively; ϕ 0 is the inflaton field value at the end of inflation and m s is the DM mass.Taking ϕ 0 ∼ M Pl and H end ∼ 10 14 GeV, one obtains a very strong bound ∆ NR ≳ 10 17 C 2 ms GeV , which implies a low reheating temperature ∆ NR ≫ 1 (unless dark matter is super-light, m s ≪ eV, or C 2 happens to be vanishingly small).Similar considerations apply to fermionic dark matter, although the constraints are weaker [10].
Freeze-in at stronger coupling operates in models with low reheating temperature T R , thereby avoiding the problem of gravitational DM overproduction.The dark relics generated during and immediately after inflation are diluted away such that one may assume their negligible abundance at the onset of thermal production.If dark matter is heavier than T R , its production is Boltzmannsuppressed and its coupling to the SM fields can be significant.On the other hand, the DM abundance becomes very sensitive to the thermal history of the Universe.In particular, it is produced most efficiently when the SM bath temperature reaches its maximum, which generally happens before reheating.
In our previous work [7], we have resorted to the low energy effective description and, in particular, to the instant reheating approximation, which assumes that dark matter is mostly produced at T R and the preexisting abundance can be neglected.The goal of the current work is to study under what circumstances this assumption is adequate and, more generally, explore cosmological frameworks where freeze-in at stronger coupling can be realized.We find broad classes of models in which the SM bath temperature stays constant over cosmological times such that the maximal and reheating temperature are naturally close.These allow for a reliable calculation of freeze-in DM abundance, without the problem of initial conditions inherent in high T R models, and justify our previous results.

Evolution of the Standard Model sector temperature
Little is known as to how exactly the SM fields were produced after inflation.They could be generated through their direct couplings to the inflaton [11] or result from interactions with a secondary field, e.g.decay thereof.The SM sector temperature exhibits very different evolution in these cases.
Let us consider a general case of the SM particle production from decay of a field χ, which does not necessarily dominate the energy density of the Universe.The SM quanta are typically produced in the relativistic regime, so we will assume that the SM energy density ρ scales as radiation.The energy density of the χ field is ρ χ , whose scaling is a −n , while that of the Hubble rate H is a −m , with a being the scale factor.The values of n and m are, in general, unrelated and depend on further details.Denoting the decay width of χ by Γ χ , we arrive at the following system: The label 0 refers to the initial moment a 0 = 1 corresponding to the end of inflation and ρ(1) = 0. We assume here that the SM sector does not contribute significantly to the energy balance of the Universe and that n, m are constant in the regime of interest.
The solution for ρ(a) is easily found as follows: ρ + 4Hρ = 1 a 4 d dt (a 4 ρ) and dt = da/(aH), so that and our boundary condition ρ(1) = 0 requires at a ≫ 1 since n − m < 4 for all cases of interest.In practice, this approximation becomes sensible already at a = O(1).
The above scaling is different from the usual 1/a 4 law applicable when the SM radiation dominates.In fact, n − m can be of either sign or zero.The SM energy density can thus grow in time or stay constant before reheating.Since thermalization in the SM sector is fast, one can associate the SM thermal bath temperature with ρ 1/4 , i.e.
where g * is the number of the SM degrees of freedom.Therefore, the SM sector temperature is allowed to grow in time or stay constant before reheating.The latter is defined as the time when the SM energy density starts dominating the energy balance of the Universe and is associated with temperature T R .
If the SM fields are produced directly by the inflaton, n = 2m and ρ ∝ a −m .This corresponds to the a −3/2 and a −2 scaling for the ϕ 2 and ϕ 4 inflaton potentials, respectively.In this case, the SM temperature decreases before reheating and the maximal temperature T max would typically be much larger than the reheating temperature T R [11][12][13].One should keep in mind that thermalization is a complicated process such that the true maximal temperature is normally lower than that assumed in the instant thermalization approximation [14].This effect is, however, insignificant for our purposes.
If the SM sector is produced by decay of a field other than the inflaton, the temperature scaling can be a 0 or a +|k| meaning that the maximal and reheating temperatures may be close and even coincide, A natural candidate for such a subdominant field is the right-handed neutrino ν R .It can be produced via inflaton decay and subsequently decay into SM states.The above calculation (2.3) applied to the neutrino production for the ϕ 4 and ϕ2 local inflaton potentials, respectively.1 ρ ν is now the source for the SM radiation, so using ν R as χ in (2.1), one finds n − m = 0 (2.8) as long as ν R remains subdominant in the energy balance.As a result, the SM bath temperature stays constant.In what follows, we study this scenario in more detail.We note that the SM temperature can also increase before reheating.This happens if n−m < 0, which can be realized in models where the SM sector is produced via decay of a sub-subdominant component in the energy density of the Universe.Indeed, the above considerations show that n can be zero while m > 0.

Example: SM sector production via ν R decay
Consider the possibility that the inflaton decays primarily into feebly interacting right-handed neutrinos, ϕ → ν R ν R .The relevant terms in the Lagrangian are where the Yukawa coupling are small, y ϕ , y ν ≪ 1, and H and ℓ are the Higgs and lepton doublet fields, respectively.Here we focus on the parameter regime where the direct coupling between the inflaton and the SM states is tiny such that ϕ → hh is insignificant.This can be enforced by an approximate Z 4 symmetry of the type The reheating process proceeds in two stages, where the final state in the ν R decay can be Hℓ or lighter states, if the Higgs channel is forbidden kinematically.We assume that ν R is feebly coupled such that both decays take place at late times and the inverse processes can be neglected.We also take the effective inflaton mass to be far above the ν R mass, hence the energy density of ν R scales as radiation until late times.Given the feebleness of the right-handed neutrino interactions, it does not form a thermal bath.
The energy density components satisfy the following equations ) ) where ρ ϕ , ρ ν , ρ are the inflaton, ν R and SM energy densities, respectively; Γ ϕ and Γ ν are the decay rates of the corresponding energy densities. 2The initial values for ρ ν and ρ are zero, and , where "0" refers to the end of inflation.Here we assume for definiteness a non-relativistic inflaton, i.e.
As long as the decay widths are small and Γ ϕ ρ ϕ ≫ Γ ν ρ ν , the energy densities are hierarchical and we recover the scaling behaviour of the solution described earlier: (2.15) at a > O(1).Furthermore, even if Γ ν is time-dependent, the SM energy density can be computed via This expression makes it explicit that if ρ ν scales as H and Γ ν = const, the SM energy density remains constant.The latter can also increase in time if Γ ν ρ ν /H grows, which could be due to the increase of ρ ν /H or broadening of the neutrino width.At late times, the different energy densities start approaching each other and their hierarchical structure gets lost.In this case, the full numerical solution is necessary (Fig. 1).Below, we distinguish the cases of Γ ϕ and Γ ν being vastly different, and being of similar size.Γ ϕ ≫ Γ ν .The inflaton field decays into the right-handed neutrinos and drops out of the energy balance when H ∼ Γ ϕ .Until SM reheating, the dominant role is taken by relativistic ν R .In this period, the SM temperature decreases as T ∝ a −1/2 (Fig. 2).Reheating occurs when H ∼ Γ ν .Γ ϕ ≪ Γ ν .As long as Γ ϕ ρ ϕ ≫ Γ ν ρ ν , the energy components evolve as in the previous case.The transition occurs when Γ ϕ ρ ϕ ∼ Γ ν ρ ν .The neutrino energy density ρ ν starts decreasing faster (ρ ν ∝ ρ ϕ ∝ a −3 ), although it does not decay exponentially quickly due to continuous replenishment from the inflaton sector.The SM sector takes over the subleading role in the energy balance until H ∼ Γ ϕ , when both ν R and ϕ decay exponentially quickly.The SM temperature decreases in this period as T ∝ a −3/8 (Fig. 2), so again the maximal temperature exceeds the reheating temperature.Γ ϕ ∼ Γ ν .This parameter choice realizes the possibility T max ≃ T R .Since the inflaton and neutrinos decay at the same time, the SM sector takes over the energy balance immediately thereafter.As Fig. 1 (bottom right) demonstrates, the temperature evolution can be approximated by a constant followed by a power law decrease T ∝ a −1 .
We therefore conclude that, in the simple framework where the SM sector is produced by the right-handed neutrino decay, the maximal and reheating temperatures do not differ significantly if the inflaton and ν R decay widths are similar.Their approximate equality One should also note that certain thermal effects can play a role in flattening the T (a) dependence prior to reheating.In particular, if T is far above the neutrino mass M ν , which we keep as a free parameter, the neutrino decay into the SM final states is blocked kinematically.This can be accounted for by replacing Γ ν → Γ ν θ(ρ ⋆ − ρ) in the evolution equations.Here ρ ⋆ is proportional to ν and for values of ρ ⋆ ∼ T 4 R , the SM temperature stays constant till reheating even if Γ ϕ ≫ Γ ν .Finally, let us estimate the size of the couplings required for a low reheating temperature.The relation where m ϕ and M ν are the inflaton and right-handed neutrino masses, respectively.This is because both ϕ → ν R ν R and ν R → Hℓ decay widths scale as the initial particle mass times the coupling squared.The typical parameter values in Fig. 1 are T R ∼ 100 GeV and Γ i ∼ 10 −14 GeV, which for m ϕ ∼ 10 13 GeV implies y ϕ ∼ 10 −13 .Assuming M ν ∼ 1 TeV, the neutrino Yukawa coupling is then y ν ∼ 10 −8 , consistent with its non-thermalization [15].(For light ν R , its decay can be loop-suppressed.)The inflaton coupling appears particularly small, although it is not surprising in inflationary cosmology given the flatness of the inflaton potential.The smallness of the couplings in the neutrino sector is also not unexpected, hence one may find the above scenario plausible, although the parameter choice can only be motivated in a more fundamental theory.
These estimates also show that the direct decay ϕ → SM would be highly suppressed in this model.The inflaton coupling to the SM fields is generated at one loop, e.g. the coupling to the Higgs bilinear H † H is proportional to the loop factor × y ϕ y 2 ν M ν , where the last factor is required by the Z 4 symmetry [15] breaking.This makes the decay rate of ϕ → SM suppressed by about 56 orders of magnitude compared to that of ϕ → ν R ν R .More generally, the SM energy density generated via the cascade ϕ → ν R → SM scales as Γ ϕ Γ ν , whereas that from the direct decay ϕ → SM scales as Γ ϕ Γ 2 ν .Therefore, the latter can be made highly suppressed by decreasing Γ ν .
3 Dark matter freeze-in production at stronger coupling Let us consider freeze-in dark matter production in the class of models described in the previous section.The SM sector temperature has never been very high in this framework.The maximal and reheating temperatures are of similar size and possibly coincide.Before reheating, the Universe undergoes a long period of inflaton matter-dominated expansion which dilutes the relativistic relics generated right after inflation.Hence, one may focus entirely on thermal production of dark matter.We are interested in freeze-in at larger couplings [7], so we take the DM mass to be much larger than the SM temperature at any stage of the Universe evolution.On the other hand, the temperature profile is model-dependent and can be highly non-trivial in a multi-component Universe.We thus focus on a simplified case motivated in the previous section.Specifically, we assume that the relevant stage in the Universe evolution can be approximated by the period of constant SM temperature, followed by a power-law decrease T ∝ a −l as displayed in Fig. 1 (bottom right).We focus on the parameter regime in which the DM production via direct decay of the inflaton or the right-handed neutrino is highly suppressed or forbidden. 3onsider, for definiteness, real scalar dark matter s of mass m s which couples to the SM sector through the Higgs portal [16][17][18], where λ hs is the Higgs portal coupling.For T ≪ m s , dark matter is produced via scattering of energetic SM quanta from the "Boltzmann tail".The leading process is the Higgs and vector boson pair annihilation into pairs of s.For m s ≫ m h , all these modes can be accounted for by using four effective Higgs degrees of freedom, according to the Goldstone equivalence theorem.
The dark matter number density n satisfies the Boltzmann equation where h i represents 4 Higgs d.o.f. at high energies of order m s ≫ m h ; Γ h i h i →ss and Γ ss→h i h i are the DM production and annihilation rates per unit volume, respectively.Consider first the pure freeze-in regime, in which the annihilation term can be neglected.Since the initial state particles have energies far above the temperature, their distribution is given by the "Boltzmann tail".Then, the production rate per unit volume is [7] for 4 Higgs d.o.f.Integrating the Boltzmann equation, we get Here we use the scaling H = H 0 /a m , as before.Note that one cannot trade the integration variable a for T since T (a) can be constant.Using the explicit form of the reaction rate, we find Consider now two representative T (a) functions.

Constant T (a) = T max
This scaling applies when the SM sector is a sub-subdominant component in the energy density of the Universe.The integral is trivial and the total particle number is for a ≫ a 0 = 1.We see that the particle number (and the density) grows with a and is dominated by the late time contributions at largest a. 43.2 T (a) ∝ a −l During and after reheating, the SM temperature decreases.We may approximate it by a power law T (a) ∝ a −l , with l time-dependent but approximately constant during each period of interest.
Let us take where a max corresponds to the point where the temperature starts decreasing, and compute the particle number generated from a max on.Using the approximation for z 0 ≫ 1, obtained using integration by parts, we get This expression assumes 2m s /T max ≫ 1.In contrast to the previous case, the particle number is dominated by the early time contribution.This is due to the exponential suppression of particle production at lower temperatures.Both (3.6) and (3.9) can be rewritten in terms of H(a).The time dependence of the particle density is captured by , n(a) . (3.10)

Total particle number
Let us now combine the two contributions.Defining a max as the transition point from one scaling law to the other, we obtain the total DM particle number produced by the SM thermal bath, where a 3 max n(a max ) T =const is found from (3.6).DM production is most efficient in the vicinity of a = a max with the flat part of the temperature evolution giving the main contribution since Tmax 2ms ≪ 1.We note that the l-dependence of the result is very mild: l only affects the correction.
The consequent constraint on the model is formulated in terms of the relic abundance of the s-particles, where g * is the number of the SM d.o.f. and n is the number density of the s-quanta.The observational constraint on the DM abundance is Y obs = 4.4 × 10 −10 GeV ms [19].Assuming that T max = T R , the Hubble rate at a = a max is determined by the SM temperature, H max ∝ T 2 max and l = 1.We thus get To obtain this expression, we have estimated Y at T = T max , after which the abundance remains approximately constant in the pure freeze-in regime.The correct DM relic abundance is reproduced for λ hs ∼ 10 −11 e ms/Tmax m s /GeV.The coupling can be as large as order one if m s ≫ T max .On the other hand, models with a high reheating temperature require λ hs ∼ 10 −11 [20,21].
In our previous work [7], we resorted to a low energy effective approach without specifying the full cosmological history.In particular, we assumed zero particle abundance before reheating, which amounts to computing particle production from the T ∝ a −l region only.Taking m = 3/2 corresponding to inflaton matter domination prior to reheating, we find that ignoring "prehistory" leads to underestimating the particle number by about a factor of 10.To reproduce the correct DM relic abundance, the effective approach required 2m s /T R ∼ 40 for typical parameter values.Since Y scales approximately as e −2ms/T R , the 10-fold increase in Y can be compensated by a 5% shift in T R .Hence, the full computation leads to a T R correction of order 5%, Equivalently, for a fixed T R , the DM mass m s increases by about 5%.Hence, the parameter space plot remains largely unchanged.This is shown in Fig. 3.The thin and thick colored lines correspond to the correct relic density computed within the effective approach of our previous work [7] and using the UV complete model presented here, respectively.At larger couplings, the annihilation effect becomes significant.The corresponding reaction rate per unit volume is given by [7] Γ(ss for m 2 s ≫ m 2 h and 4 Higgs d.o.f.Here v r is the relative particle velocity.As one increases the coupling further, the system thermalizes [7,22] and the resulting DM relic abundance is given by the purple line in Fig. 3. Fig. 4 illustrates the process of thermalization with our T (a) function.At low couplings, the DM particle number differs from its equilibrium value at all stages, apart from one point.For larger couplings, the DM density reaches the equilibrium value somewhat before a = a max , which is followed by DM freeze-out.For convenience, we redefine the scale factor in the plot as a/a max → a such that the SM temperature starts decreasing at a = 1 and T R = T (1).Dark matter particle number evolution.n eq represents the thermal particle density at the SM sector temperature T , which stays constant before a = 1 and decreases as 1/a after that.Left: pure freeze-in regime (λ hs = 10 −3 , m s = 602.5 GeV, T max = 30 GeV).Right: DM thermalization and freeze-out (λ hs = 0.195, m s = 651 GeV, T max = 30 GeV).

3.4
T max > T R case More generally, T max and T R can differ.For instance, if Γ ϕ ≫ Γ ν or Γ ϕ ≪ Γ ν , the maximal temperature exceeds the reheating temperature.In this case, the flat T = const stage, during which the inflaton field dominates, is followed by a period of a mild temperature decrease T ∝ a −l with l < 1.During this stage, the right-handed neutrinos take over the energy balance.As these decay, the SM bath comes to dominate and l becomes 1 signifying reheating.Thus, T R can be significantly lower than T max .Consider the pure freeze-in regime.It is clear from the above calculations that dark matter is mostly produced at the point a = a max .Hence, Eq. 3.11 applies to this case as well.However, the resulting DM abundance changes since the SM entropy gets produced until reheating, after which Y remains approximately constant.The result can be conveniently expressed as a correction factor to (3.13).Let us parametrize the scaling of the Hubble rate between a max and a R as H ∝ a −m ′ , such that H evolves with a according to (3.16) Since the DM particle number does not change between a max and a R , we obtain the following correction factor to the result (3.13): . (3.17) For example, in the case of radiation domination, m ′ = 2 and l = 1/2.Hence, the correction factor is (T R /T max ) 5 .Although this factor can be substantial, it does not change the parameter space picture drastically: taking T max = 10 T R results in about a 25% shift in m s .
Freeze-in production at stronger coupling implies that the maximal temperature is far below the DM mass.In this case, particle production is always Boltzmann-suppressed and of freezein type.This condition is typically violated if the inflaton decays directly into the SM states, implying a high T max .The dynamics of particle production would be more complicated in this case: depending on further details, DM may thermalize and freeze out before Boltzmann-suppressed freeze-in becomes operative.
Many of our results also apply if the SM sector is produced by a sub-subdominant component in the energy density of the Universe. 5In this case, the SM sector temperature increases according to a power law until the energy balance changes and the evolution of T flattens, followed by its decrease.DM production is most efficient at the end of the flat T period, hence the above considerations remain largely valid.
In summary, the main results of our work are presented in Fig. 3.The direct detection experiments already probe freeze-in DM up to 3 TeV.As their sensitivity increases, they will cover further consistent freeze-in models with higher reheating temperature or lower couplings.The difference between the full model results with T R = T max and those using the instant reheating approximation is small in terms of the {λ hs , m s } parameter space.If T max is significantly above T R , the {λ hs , m s } relation receives a correction due to (3.17): as long as m s ≫ T max .Due to the exponential mass-dependence, this results in a modest shift of the {λ hs , m s } curve at fixed T R to smaller masses.For the same reason, model-dependent prethermalization DM production via scattering of the high energy quanta against the SM thermal bath [23,24] is not expected to affect our results significantly.
At low m s , the direct detection sensitivity is normally superseded by that of the LHC via the constraint on the invisible Higgs decay.This also applies to the Higgs portal model with low T R [25].Indirect detection constraints are weak in the Higgs portal models [18].Let us note that there exist other examples of freeze-in models, although with a more complicated dark sector, that can potentially be probed by a combination of experiments [26][27][28].For example, milli-charged DM with a light mediator is already constrained by direct DM detection [26].Some freeze-in models assuming early matter-domination can also be probed via displaced vertices at colliders [29].
Finally, we note that baryogenesis in our framework can be realized via low-temperature Affleck-Dine mechanism [30].It requires the existence of a scalar condensate, which can naturally be generated, for instance, in the pseudo-Goldstone direction in the scalar potential [31].Further constraints on specific realizations of our scenario can be obtained by assuming a particular inflationary potential, as detailed in [32].

Conclusion
Dark matter freeze-in at stronger coupling is a well motivated scenario, which addresses the problem of initial conditions in conventional freeze-in models.It assumes a low reheating temperature allowing for dilution of gravitationally produced relics and making DM production Boltzmannsuppressed.As a result, the DM coupling to the SM fields is allowed to be O(1) rendering it potentially observable in direct detection experiments [7].On the other hand, the DM relic abundance in this framework is sensitive to the thermal history of the Standard Model sector and, in particular, to the relation between the maximal and reheating temperatures.
In this work, we have studied a class of models, in which the maximal and reheating temperatures are in the same ballpark, and can even coincide.This is the case when the SM sector is produced via decay of a subdominant component in the energy density of the Universe.The role of this component can be played by feebly interacting right-handed neutrinos ν R .If these couple to the inflaton much stronger than the SM fields do, the inflaton decays predominantly into pairs of ν R , which subsequently produce the SM sector.In this case, the SM bath temperature stays constant over cosmological times prior to reheating, explaining the proximity of the maximal and reheating temperatures.
Since the temperature evolution is known in this framework, the dark matter abundance can be calculated reliably.We find that the resulting constraint on the coupling vs mass is very close to that computed in the instant reheating approximation [7] as long as T R ≃ T max .If the maximal and reheating temperatures differ, the coupling receives a rescaling factor (3.18).
An attractive feature of freeze-in models with stronger coupling is that they are currently being probed by direct DM detection experiments.As the search sensitivity improves [33,34], further models with higher T R or lower couplings will be explored.In addition, the LHC can place a complementary constraint via precise measurements of the Higgs invisible decay.All in all, dark matter freeze-in at stronger coupling provides us with an interesting alternative to traditional freeze-in or freeze-out models.

- 1 Figure 2 .
Figure 2. Temperature evolution for T max > T R .The parameters are as in Fig. 1 (upper panels) with the focus on the vicinity of the reheating region.

Figure 3 .
Figure 3. Parameter space of the Higgs portal DM model.Along the colored lines, the correct relic density is reproduced for a given T R = T max (in GeV).The thin lines correspond to the calculation presented in[7], while the thick lines are obtained in the UV complete model of the current work.The purple line corresponds to the thermal DM abundance.The shaded area is excluded by direct DM detection (LZ 2022)[1] and the dashed line shows the "neutrino fog".