# Phenomenology of self-interacting dark matter in a matter-dominated universe

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## Abstract

We study production of self-interacting dark matter (DM) during an early matter-dominated phase. As a benchmark scenario, we consider a model where the DM consists of singlet scalar particles coupled to the visible Standard Model (SM) sector via the Higgs portal. We consider scenarios where the initial DM abundance is set by either the usual thermal *freeze-out* or an alternative *freeze-in* mechanism, where DM was never in thermal equilibrium with the SM sector. For the first time, we take the effect of self-interactions within the hidden sector into account in determining the DM abundance, reminiscent to the Strongly Interacting Massive Particle (SIMP) scenario. In all cases, the number density of DM may change considerably compared to the standard radiation-dominated case, having important observational and experimental ramifications.

## 1 Introduction

The existence of dark matter (DM) seems indisputable. From the Cosmic Microwave Background radiation (CMB), large scale structure of the Universe and different physics at galactic scales, one can infer that there must be a long-lived, dynamically non-hot, non-baryonic matter component, whose abundance exceeds the amount of ordinary ‘baryonic’ matter roughly by a factor of five [1, 2, 3, 4] and which has been there from the hot Big Bang era until the present day. However, the non-gravitational nature of the DM component remains a mystery.

For a long time, Weakly Interacting Massive Particles (WIMPs) have been among the best-motivated DM candidates. The increasingly strong observational constraints on DM (see e.g. Ref. [5]) are, however, not only puzzling as such but are now forcing one to ask: is the standard WIMP paradigm just waning, or is it already dead? If so, what alternative explanations for the production and properties of DM do we have?

A simple alternative for the standard WIMPs is provided by relaxing the usual assumption that DM is a thermal relic, produced by the *freeze-out* (FO) mechanism in the early Universe, and assuming that it never entered in thermal equilibrium with the particles within the Standard Model of particle physics (SM). If that was the case, then the present DM abundance could have been produced by the so-called *freeze-in* (FI) mechanism, where the abundance results from decays and annihilations of SM particles into DM [6, 7, 8, 9, 10]. Assuming that DM never entered into thermal equilibrium with the particles in the visible SM sector typically amounts to choosing a very small coupling between the two sectors. A good thing about this is that then these so-called Feebly Interacting Massive Particles (FIMPs) easily evade the increasingly stringent observational constraints, yet an obvious hindrance is that this also makes the scenario inherently very difficult to test. For a recent review of FIMP DM models and observational constraints presented in the literature, see Ref. [11].

Another way to evade the experimental constraints is to consider non-standard cosmological histories [12]. We know that the Universe was effectively radiation-dominated (RD) at the time of Big Bang Nucleosynthesis (BBN) and one usually assumes that this was the case also at the time the DM component was produced, was it at the time of electroweak cross-over or at higher energy scales. However, there are no obvious reasons for limiting the DM studies on such cosmological expansion histories,^{1} as alternatives not only can lead to interesting observational ramifications but are also well-motivated. For example, an early matter-dominated (MD) phase can be caused by late-time reheating [18], massive meta-stable particles governing the energy density of the Universe (see Refs. [19, 20, 21] for recent works), moduli fields [22, 23, 24], and so on. The effect on the resulting DM yield can then be outstanding, as recently studied in detail in e.g. Refs. [19, 20, 21, 25, 26, 27, 28, 29, 30, 31, 32].

Indeed, when the expansion rate of the Universe differs from the usual RD case, it tends to effectively dilute the DM abundance when the era of non-standard expansion ends and the visible sector gets reheated (see also Refs. [27, 31] for DM production in fast-expanding universes and Refs. [26, 30] for co-decaying DM). This means, for example, that when the expansion was faster than in the RD case and the DM particles were initially in thermal equilibrium with the visible sector, they generically have to undergo freeze-out *earlier* than in the usual RD case, thus resulting in larger DM abundance to match the observed one. In case the DM particles interacted so feebly that they were never part of the equilibrium heat bath, the coupling between DM and the visible sector typically has to be orders of magnitude larger than in the usual freeze-in case to compensate the larger expansion rate. Production of DM during a non-standard expansion phase may thus result to important experimental and observational ramifications. Studying the effect non-standard cosmological histories have on different particle physics scenarios is thus not only of academic interest and also not limited to the final DM abundance, as different possibilities to test for example an early MD phase include formation of ultracompact substructures such as microhalos [33] or primordial black holes [34, 35, 36], as well as cosmological phase transitions with observational gravitational wave signatures [37] (see also Ref. [38]).

In this paper we will consider DM production during such an early MD phase. We will study DM production by both the freeze-out and freeze-in mechanisms, taking for the first time into account the effect that non-vanishing DM self-interactions can have. Instead of performing an intensive full-parameter scan, in this paper we will perform an analytical study of the different representative cases previously mentioned, which allows us to capture the essence of each scenario. Results of an exhaustive scan over the full parameter space in the usual freeze-out and freeze-in cases are presented in a companion paper [39], where we also discuss the effect of other non-standard cosmological histories. However, as we will show, already with the best-motivated non-standard case, an early phase of matter-domination, the DM phenomenology is very rich when the effect of DM self-interactions is taken into account, which is one of the reasons why we devote a separate paper for the analysis of this scenario only. Another important difference to Ref. [39] is that in this paper we will we make the usual assumption that the eventual decay of the energy density component responsible for the early matter-domination is instantaneous, whereas in Ref. [39] the duration of decay is taken to be finite. In this way, the two studies complement each other.

As we will show, the observational limits on DM self-interactions do not only rule out part of the parameter space for the model we will consider in this paper, but taking the detailed effect of DM self-interactions into account is crucial for determination of the final DM abundance, reminiscent to the so-called Strongly Interacting Massive Particle (SIMP) or *cannibal* DM scenarios [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63]. We will also discuss other prospects for detection of DM including collider, direct, and indirect detection experiments.

The paper is organized as follows: In Sect. 2, we will present a simple benchmark model where the DM particle is a real singlet scalar odd under a discrete \(\mathbb {Z}_2\) symmetry, and discuss what are the requirements for having an early MD phase prior to BBN. In Sect. 3, we turn into the DM production, discussing production by the usual freeze-out mechanism in Sect. 3.1 and by the freeze-in mechanism in Sect. 3.2. In Sect. 4, we discuss the experimental and observational ramifications, and present not only what part of the parameter space is already ruled out but also what part of it can be probed in the near future. Finally, we conclude with an outlook in Sect. 5.

## 2 The model

*s*. The only interaction between this hidden singlet sector and the visible SM sector is via the Higgs portal coupling \(\lambda _{hs}|\varPhi |^2s^2\), where \(\varPhi \) is the SM Higgs field. The scalar potential is

*SU*(2) gauge doublet in the unitary gauge and \(v=246\) GeV is the vacuum expectation value of the SM Higgs field. A discrete \(\mathbb {Z}_2\) symmetry, under which the DM is odd and the whole SM is even, has been assumed to stabilize the singlet scalar and make it a possible DM candidate. We assume \(\lambda _s>0\) and \(\mu _s>0\), so that the minimum of the potential in the

*s*direction is at \(s=0\) and \(m_s^2 \equiv \mu _s^2 + \lambda _{hs}\,v^2/2\) is the physical mass of

*s*after the spontaneous symmetry breaking in the SM sector. This implies \(\lambda _{hs}<2\,m_s^2/v^2\).

### 2.1 An early matter-dominated period

*H*is the Hubble scale, \(M_\mathrm{P}\) is the reduced Planck mass, and \(\rho _\mathrm{M}\) is the energy density of the matter-like component that is assumed to dominate over the SM energy density \(\rho _\mathrm{SM}\) and the singlet scalar energy density \(\rho _s\). We also assume that the SM was in thermal equilibrium for the whole duration of the early MD phase, so that

^{2}and

*T*is the SM bath temperature.

### 2.2 Constraints on the scenario

*after*the matter-like component has decayed into SM particles, \(T_\mathrm{end}'\), must be larger than the BBN temperature \(T_\mathrm{BBN}=4\) MeV. Second, the temperature has to be smaller than either the final freeze-out temperature or smaller than \(m_h\) in the freeze-in case in order not to re-trigger the DM yield after the decay of the matter-like component. As shown in the end of Appendix A, this amounts to requiring

*before*the end of matter-domination and \(x_\mathrm{FO}\equiv m_s/T_\text {FO}\), with \(T_\text {FO}\) being the DM freeze-out temperature. In the following, we will take the above ratio \(T_\mathrm{end}/m_h\) to be a free parameter, so that together with \(H_\mathrm{EW}\) it constitutes the set of our cosmological parameters, characterizing the duration of the early MD phase. The total parameter space is thus five-dimensional, consisting of the particle physics parameters \(\lambda _s\), \(\lambda _{hs}\) and \(m_s\), in addition to the cosmological parameters \(H_\text {EW}/m_h\) and \(T_\text {end}/m_h\).

*s*particles are non-relativistic, \(x_\mathrm{FO}>3\), as otherwise the scenario is subject to relativistic corrections that we are not taking into account in the present paper. Fourth, as discussed above, in a MD Universe \(H_{\mathrm {EW}}/m_h\gg 10^{-16}\). Fifth, as discussed below Eq. (1), the portal coupling has to satisfy \(\lambda _{hs}<2\,m_s^2/v^2\). Finally, the portal coupling has a further constraint when requiring or avoiding the thermalization of the two sectors, for the case of freeze-out and freeze-in, respectively. Depending on the strength of the portal coupling \(\lambda _{hs}\), the singlet scalar particles may or may not have been part of the equilibrium in the SM sector at the time the initial DM density was produced. The threshold value for \(\lambda _{hs}\) above which the DM sector equilibrates with the SM is

Before concluding this section let us note that the fact that now \(H_\mathrm{EW}\gg H_\mathrm{EW}^\mathrm{rad}\) means that in the freeze-out case the value of the portal coupling required to produce the observed DM abundance must be smaller than in the usual RD case, as the DM has to decouple earlier from the thermal bath in order to retain the required abundance. However, the faster expansion rate also means that now the threshold value for thermalization, Eq. (7), can be orders of magnitude larger than the corresponding value \(\lambda _{hs}\simeq 10^{-7}\) in the usual RD case. This makes the freeze-in scenario particularly interesting, as it might lead to important experimental ramifications, as we will discuss in Sect. 4.

## 3 Dark matter production

We start by reviewing the DM production within this model, briefly discussing two fundamental mechanisms that account for it: the freeze-out and the freeze-in scenarios.

*s*, its number density evolution is described by the Boltzmann equation:

*i*, \(\left| \mathcal {M}\right| ^{2}\) is the squared transition amplitude averaged over both initial and final states, \(f_i\) is the phase space density, \(+\) applies to bosons and − to fermions and

*i*. In the following, we will solve the relevant Boltzmann equations analytically in the regions of interest where different processes dominate at a time. A full parameter scan is performed in the pure freeze-out and freeze-in cases in Ref. [39].

In the freeze-out mechanism, DM was initially in thermal equilibrium with the SM sector. As soon as the interactions between the DM and the SM particles were no longer able to keep up with the Hubble expansion, the system departed from thermal equilibrium and the comoving DM abundance became constant. We will study the case of the DM freeze-out in an early MD era in Sect. 3.1.1 and then consider how a so-called cannibalism phase affects the DM yield in Sect. 3.1.2.

In the freeze-in scenario, the DM was never in thermal equilibrium with the visible sector, due to the very feeble interactions between them. The particles produced by this mechanism are known as FIMPs and their initial number density is, in the simplest case, negligible. The DM abundance is produced by the SM particle decays and annihilations, lasting until the number density of the SM particles becomes Boltzmann-suppressed. At this point, the comoving number density of DM particles becomes constant and the comoving DM abundance is said to ‘freeze in’. The evolution of the initial *s* number density can be tracked by the Boltzmann Eq. (8) as well. We discuss the DM freeze-in in an early MD era without cannibalism in Sect. 3.2.1 and with it in Sect. 3.2.2.

### 3.1 The freeze-out case

To study the effects of MD and DM self-interactions in a simple yet accurate way, in this section we assume the mass hierarchy \(m_b<m_s < 50\) GeV, where \(m_b\) is the mass of the *b*-quark and the upper limit is chosen to avoid complications with the Higgs resonance in our analytical calculations. Therefore, in this subsection, we will consider DM produced only by \(b\bar{b}\) annihilations and present the more general analysis in Ref. [39] for the pure freeze-out case without cannibalism.

#### 3.1.1 Freeze-out without cannibalism

*b*-quarks becomes smaller than the Hubble parameter. The DM abundance can then be calculated by taking into account the non-conservation of entropy (see Appendix A), yielding:

Figure 1 shows slices of the parameter space that give rise to the observed DM relic abundance. On the upper panel the cosmological parameters are fixed, \(H_\text {EW}/m_h=10^{-16}\) (black lines) and \(10^{-15}\) (blue lines), and \(T_\text {end}/m_h=10^{-6}\) (dashed lines) and \(10^{-4}\) (solid lines) while we scan over the relevant particle physics parameters (\(\lambda _{hs}\) and \(m_s\)). The upper left corner in red, corresponding to \(\lambda _{hs}>2\,m_s^2/v^2\), is excluded by the requirement discussed below Eq. (1). The figure shows that an increase in the dilution factor due to either an enhancement of the Hubble expansion rate \(H_\text {EW}\) or a decrease in the temperature \(T_\text {end}\) when the MD era ends has to be compensated with a higher DM abundance at the freeze-out. That, in turn, requires a smaller annihilation cross-section and hence a small \(\lambda _{hs}\). The dependence on the DM mass \(m_s\) is very mild.

#### 3.1.2 Freeze-out with cannibalism

*chemical*equilibrium with each other when \(\langle \sigma _{ss\rightarrow bb}v \rangle n_s/H=1\). However, the

*s*particles can maintain chemical equilibrium among themselves if number-changing interactions (namely, 4-to-2 annihilations with only DM particles both in the initial and final states, see Fig. 2) are still active. The condition for this so-called cannibalism is given by

*W*function. The DM abundance then becomes (see again Appendix A)

*kinetic*equilibrium at the time of DM freeze-out, so that temperature of the

*s*particle heat bath was the same as the SM temperature

*T*. The condition for this is \(\left. \langle \sigma _{sb\rightarrow sb}v \rangle n_b/H\right| _{x^\mathrm{c}_\mathrm{FO}} > 1\), where we have taken for simplicity \(\langle \sigma _{sb\rightarrow sb}v \rangle \simeq \langle \sigma _{ss\rightarrow b\bar{b}}v \rangle \) and \(n_b\) is the

*b*-quark number density.

Before closing this subsection, we present the results of an extensive scan over the parameter space for the DM freeze-out without (left column) and with (right column) cannibalism in Fig. 4. The blue regions produce the observed DM relic abundance, whereas the red regions correspond to the constraints discussed in Sect. 2.2. The plots generalize the results of Figs. 1 and 3. First, let us note that the usual RD scenario can be recovered by taking \(H_\text {EW}/m_h=H_\text {EW}^\text {rad}/m_h\simeq 1.76\times 10^{-16}\) and \(T_\text {end}/m_h=1\). This corresponds to \(\lambda _{hs}\simeq 10^{-1}\), in the case where DM mainly annihilates into *b*-quarks (\(m_b<m_s\lesssim 50\) GeV) and does not undergo a cannibalism phase. In the MD scenario the Higgs portal coupling \(\lambda _{hs}\) can reach much smaller values down to \(\mathcal {O}(10^{-4})\). Such small values naturally need large dilution factors, characterized by large expansion rates \(H_\text {EW}/m_h\) up to \(\mathcal {O}(10^{-13})\) and low temperatures for the end of the MD era, \(T_\text {end}/m_h\) down to \(\mathcal {O}(10^{-8})\). In the case with cannibalism, \(\lambda _{hs}\lesssim 10^{-2}\) while \(\lambda _s\gtrsim 10^{-2}\) due to the fact that the DM annihilation into SM particles must decouple earlier than the 4-to-2 annihilations. Finally, we note that in the scenario where freeze-out occurs during a standard RD phase, cannibalism would generically require non-perturbative values of \(\lambda _s\). As shown above, in the MD case the detailed effect of non-vanishing self-interactions can easily be taken into account, as the required values for \(\lambda _s\) can be much smaller. This result, along with its observational consequences that we will present in Sect. 4, are among the most important novelties of this work.

### 3.2 The freeze-in case

In this subsection we assume, for simplicity, the mass hierarchy \(m_s < m_h/2\), as we take the Higgs decay into two *s* to be the dominant production mechanism for DM. A more general analysis is again presented in Ref. [39] for the pure freeze-in case without cannibalism.

#### 3.2.1 Freeze-in without cannibalism

*s*-particles and \(n_h^\text {eq}\) is its equilibrium number density. These quantities are given by

*s*number density and

*a*is the scale factor, we get the comoving DM number density at infinity

^{3}

Figure 5 shows slices of the parameter space that give rise to the observed DM relic abundance. On the upper panel the cosmological parameters are fixed, \(H_\text {EW}/m_h=10^{-16}\) (black lines) and \(10^{-13}\) (blue lines), and \(T_\text {end}/m_h=10^{-5}\) (solid lines) and \(10^{-1}\) (dashed lines), while we scan over the relevant particle physics parameters (\(\lambda _{hs}\) and \(m_s\)). The upper left corner in red, corresponding to, \(\lambda _{hs}>2\,m_s^2/v^2\), is excluded. The figure shows again that an increase in the dilution factor due to either an enhancement of the Hubble expansion rate \(H_\text {EW}\) or a decrease in the temperature \(T_\text {end}\) when the MD era ends has to be compensated with a higher DM abundance at the freeze-out. This requires an increase in either \(m_s\) or the DM production via the Higgs decay (i.e. a bigger \(\lambda _{hs}\)). The thick dotted black line corresponds to the DM production in the usual RD scenario, characterized by \(T_\text {end}/m_h=1\) and \(H_\text {EW}/m_h=10^{-16}\). We note that, as expected, in the MD scenario the values for the required values for Higgs portal are always higher than in the RD case.

The same conclusion can be drawn from the lower panel of Fig. 5, where the particle physics parameters are fixed, \(m_s=0.1\) GeV (solid lines) and 10 GeV (dashed lines), and \(\lambda _{hs}=10^{-9}\) (blue lines) and \(10^{-5}\) (black lines), while we scan over the cosmological parameters. The left band corresponds to a scenario which is not MD (\(H_\text {EW}/m_h<10^{-16}\)). The lower left and the upper right corners correspond to scenarios where the resulting SM temperature after the MD era ends is either too small for successful BBN or so large that it re-triggers the DM yield, respectively. All three cases are excluded from our analysis. Observational constraints on the scenario will be discussed in Sect. 4.

As in the case of freeze-out, the result of Eq. (26) is the final DM abundance only if number-changing DM self-interactions do not become active and the *s* particles do not reach chemical equilibrium with themselves. This is the scenario we will now turn into.

#### 3.2.2 Freeze-in with cannibalism

*s*particles. In this case, the Boltzmann equation (8) is

For values of the portal coupling required by non-thermalization of the hidden sector with the SM sector \(\lambda _{hs}\lesssim (H_\mathrm{EW}/m_h)^{1/2}\), Eq. (7), the initial *s* particle number density in the hidden sector produced by Higgs decays is always smaller than the corresponding equilibrium number density. Thus, if the self-interactions are sufficiently strong (see below), the *s* particles can reach chemical equilibrium with themselves by first increasing their number density via 2-to-4 annihilations, and then undergo cannibalism when they become non-relativistic, as discussed in e.g. Refs. [46, 48, 54]. A possible caveat to this is the case where \(m_s\) is close to \(m_h\), as then the eventual *dark freeze-out* would occur before the yield from the SM sector has ended. In that case, the production mechanism is dubbed as *reannihilation* [74, 75]. Because in that case the *s* particles would not, in general, be in thermal equilibrium at the time of their freeze-out, finding the correct DM abundance requires solving the Boltzmann equation for the DM distribution function instead of number density, which is beyond the scope of this work. In this paper we therefore choose an approach where we solve the Boltzmann equation for DM number density but highlight the regime in our results where reannihilations could potentially alter our conclusions, and leave solving the Boltzmann equation for DM distribution function for future work. Because the freeze-in yield has ended by \(T\sim 0.1 m_h\) [75], we take this regime to be determined by \(m_s\gtrsim 10\) GeV. As we will show, this is only a small part of the observationally interesting parameter space, especially for DM self-interactions.

In the following, we will solve Eq. (27) in the limit where the self-interactions of *s* are large, to complement the usual freeze-in scenario discussed above. Note that the 2-to-2 scalar self-annihilations do not have a net effect on the final DM abundance and are therefore not included in Eq. (27).

*s*self-interactions in Eq. (27) become active if

*s*particle abundance produced by Higgs decays, where \(\chi _s^\infty \) is given by Eq. (25), and we have invoked the principle of detailed balance. The scale factor \(a_\text {nrel}\) when the

*s*particles become non-relativistic can be solved from

*s*particle momenta are \(p\simeq m_h/2\). As discussed in Refs. [48, 76], it indeed suffices to evaluate Eq. (28) at \(a_\mathrm{nrel}\), which is the latest moment when the

*s*particles can reach chemical equilibrium with themselves.

*s*field’s quartic self-interaction strength can be solved from Eq. (28) to be

*s*particles thermalize with themselves and the sector exhibits a cannibal phase before the final freeze-out of DM density from the hidden sector heat bath. The time of the dark freeze-out of

*s*particles can be solved in the standard way from Eq. (27) as the time when the 4-to-2 interaction rate equals the Hubble expansion rate

*H*is given by Eq. (13) and

*T*can be inferred from entropy conservation, as after the thermalization within the hidden sector the two entropy densities are separately conserved. First, consider the times when the

*s*particles are still relativistic, whence

*s*particles became non-relativistic and their final freeze-out, the ratio \(\zeta \) is

*T*as a function of the hidden sector temperature

*W*function. The final DM abundance after the freeze-out then is

*s*direction) are excluded. Again, an increase in the dilution factor due to either an enhancement of the Hubble expansion rate \(H_\text {EW}\) or a decrease in the temperature \(T_\text {end}\) when the MD era ends has to be compensated with a higher DM abundance at the dark freeze-out. This requires a smaller 4-to-2 annihilation cross-section and hence a small \(\lambda _s\).

Figure 7 depicts the results of an extensive scan over the parameter space for the DM freeze-in without (left column) and with (right column) cannibalism. The blue regions produce the observed DM relic abundance, the red regions correspond to the constraints discussed in Sect. 2.2. Other observational constraints on the scenario will be discussed in Sect. 4.

## 4 Observational properties

Finally, we turn into observational prospects, discussing collider signatures, direct and indirect detection, as well as the observational consequences of DM self-interactions.

### 4.1 Collider signatures

In MD cosmologies, the interaction rates required to produce the observed DM abundance via freeze-in could lead to displaced signals at the LHC and future colliders [25]. However, as in our scenario DM is produced via the decay of the Higgs, we will have no exotic signals displaced from the primary vertex.

### 4.2 Direct and indirect detection signatures

The current limits from the analysis of gamma-rays coming from dwarf spheroidal galaxies with Fermi-LAT and DES [108, 109, 110] do not probe relevant parts of our parameter space. In the case of freeze-in, indirect detection signals can be expected in scenarios where the singlet scalar is a mediator and the hidden sector exhibits a richer structure, as recently studied in Ref. [111].

### 4.3 Dark matter self-interactions

In the present case, no cosmological signatures can be expected. Even though in the case where the singlet scalar never thermalizes with the SM sector the DM generically comprises an isocurvature mode in the CMB fluctuations [72, 73], the relative amount of such perturbations gets strongly diluted due to \(\rho _s \ll \rho _\mathrm{tot}\), leaving no observable imprints on the CMB.

### 4.4 Results

Figure 8 depicts the detection prospects for frozen-out and frozen-in DM, with and without cannibalism. The green regions are excluded by different observations discussed in the above subsections: DM direct detection, invisible Higgs decay or DM self-interactions. The blue regions give rise to the observed DM relic abundance, the light blue region being already in tension with observations. The black thick dashed line corresponds to the bounds that might be reached by next generation direct detection DM experiments. The constraints discussed in Sect. 2.2 are shown in red. Finally, the black dotted line shows the parameters yielding the correct DM abundance in the usual RD scenario.

In the MD scenario, DM direct detection already excludes an important region of the parameter space for the freeze-out case both with and without cannibalism. More interestingly, the next generation of DM direct detection experiments will be able to probe almost the whole region of parameter space compatible with the DM relic abundance, for the freeze-out scenario with \(m_s<m_h/2\).

On the other hand, the regions favored by freeze-in could be tangentially probed by next generation of direct detection experiments. A particularly interesting thing in this case is that the effect of non-vanishing self-interactions seems to be crucial in determining whether the scenario can be tested by the next-generation direct detection experiments or not, as shown in the two lower panels of Fig. 8. In the standard RD case the freeze-in scenario obviously does not have any such observational consequences, as the required values of \(\lambda _{hs}\) are in that case much smaller regardless of the value of \(\lambda _s\). Note that in Ref. [39] we obtained the opposite result, showing that FIMP DM cannot be tested by the next-generation direct detection experiments. This conclusion, however, is due to different assumptions for the decay of the matter-like component, as discussed in Sect. 1. However, observational constraints on DM self-interactions already rule out a corner of the parameter space corresponding to MeV-scale masses regardless of the prospects for direct detection. Finally, the region between the two dashed lines in the case of freeze-in with cannibalism corresponds to 0.1 cm\(^2\)/g\(~<\sigma /m_s<10\) cm\(^2\)/g, the zone where the small-scale structure tensions can be alleviated.

## 5 Conclusions

In cosmology, one typically assumes that at early times the Universe was radiation-dominated from the end of inflation. However, there are no indispensable reasons to assume that, and alternative cosmologies not only can lead to interesting observational ramifications but are also well-motivated. For example, an early period of matter domination is still a perfectly viable option.

In this context, we studied different dark matter production mechanisms during an early MD era. We focused first on the usual case where DM is produced by the freeze-out mechanism, corresponding to the WIMP paradigm. Then, the assumption of thermal equilibrium with the SM was relaxed allowing the DM to be produced via the freeze-in mechanism, corresponding to FIMP DM. For these two cases, we took for the first time into account the effects of sizable self-interactions within the hidden sector. Indeed, as we showed in the present context, DM self-interactions can be crucial for the determination of the final DM relic abundance and observational consequences.

When the expansion rate of the Universe differs from the usual radiation-dominated case, it tends to effectively dilute the DM abundance when the era of non-standard expansion ends and the visible sector gets reheated. This means that in case the expansion was faster than in the RD case and the DM particles were initially in thermal equilibrium with the visible sector, they generically have to undergo freeze-out earlier than in the usual RD case, thus resulting in larger DM abundance to match the observed one. In case the DM particles interacted so feebly that they never became part of the SM equilibrium heat bath, the coupling between DM and the visible sector typically has to be orders of magnitude larger than in the usual freeze-in case to compensate the larger expansion rate. As we showed, sizable self-interactions can further complicate this picture. Production of self-interacting DM during a non-standard expansion phase may thus result in important experimental and observational ramifications, as shown in Fig. 8.

In this paper we studied a benchmark scenario where the SM is extended with a real singlet scalar DM, odd under a \(\mathbb {Z}_2\) symmetry. It would be interesting to see what are the consequences in other models where, for example, the hidden sector has a richer structure (e.g. sterile neutrinos, gauge structure, etc.) or where the DM is not coupled to the SM via the Higgs portal but via some other portal, for example the \(Z'\) or a lepton portal [134, 135, 136].

## Footnotes

- 1.
A possible caveat to this is the viability of models for baryogenesis in such scenarios. However, some studies have shown that baryogenesiswith a low reheating temperature may be much less difficult than expected [13, 14, 15, 16]. Furthermore, there are some baryogenesis scenarios with MD cosmologies [17].

- 2.
In the following sections we will neglect, for simplicity, the evolution of \(g_*\) during the DM production. A detailed effect of this is addressed in Ref. [39], although the correction this imposes is relatively small.

- 3.

## Notes

### Acknowledgements

We thank X. Chu, M. Heikinheimo, M. Lewicki, and V. Vaskonen for correspondence and discussions. T.T. acknowledges Universidad Antonio Nariño and Universidade do Porto for hospitality. C.C. is supported by the Fundação para a Ciência e Tecnologia (FCT) Grant PD/BD/114453/2016, T.T. by the Simons foundation and the U.K. Science and Technology Facilities Council Grant ST/J001546/1, and N.B. partially by Spanish MINECO under Grant FPA2017-84543-P. This project has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant agreements 674896 and 690575; and from Universidad Antonio Nariño Grants 2017239 and 2018204.

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