# Dark matter and baryon asymmetry from the very dawn of the Universe

Eugeny Babichev, Dmitry Gorbunov, Sabir Ramazanov

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Eugeny Babichev, Dmitry Gorbunov, Sabir Ramazanov. Dark matter and baryon asymmetry from the very dawn of the Universe. Physical Review D, 2018, 97 (12), pp.123543. 10.1103/Phys-RevD.97.123543. hal-01801920

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# Dark matter and baryon asymmetry from the very dawn of the Universe

Eugeny Babichev,¹,² Dmitry Gorbunov,³,⁴ and Sabir Ramazanov⁵

¹Laboratoire de Physique Théorique, CNRS, Univ. Paris-Sud, Université Paris-Saclay,
91405 Orsay, France

²UPMC-CNRS, UMR7095, Institut d'Astrophysique de Paris, GReCO,
98bis boulevard Arago, F-75014 Paris, France

³Institute for Nuclear Research of the Russian Academy of Sciences,
60th October Anniversary prospect 7a, Moscow 117312, Russia

⁴Moscow Institute of Physics and Technology, Institutsky per. 9, Dolgoprudny 141700, Russia

⁵CEICO-Central European Institute for Cosmology and Fundamental Physics,
Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, 182 21 Prague 8, Czech Republic

<figure><img src="image_2.png" /></figure>

(Received 23 May 2018; published 29 June 2018)

We propose a universal mechanism of producing dark matter and baryon (lepton) charge at the stage of the quasi-de Sitter expansion of the Universe—inflation. The key ingredient of the mechanism is a linear coupling of the field, responsible for a generation of dark matter or baryon (lepton) charge, to a function of the inflaton. During inflation this induces almost constant force dragging the corresponding field to the nonzero value. This force explicitly breaks quantum numbers associated with dark matter/baryon abundance at later stages. As a particular realization of the mechanism we introduce a superheavy complex scalar field with the mass larger than the Hubble rate during the last e-folds of inflation. The global $U(1)$ symmetry is violated due to the linear coupling of the phase of the complex scalar to the inflaton. The symmetry breaking leads to the generation of a nonzero Noether charge. The latter is directly related to the dark matter abundance, or, alternatively, can be converted into baryon asymmetry, if the complex scalar carries the baryon charge.

DOI: 10.1103/PhysRevD.97.123543

## I. INTRODUCTION

The earliest yet hypothetical stage of the Universe evolution—inflation (almost exponential expansion)—has been suggested to solve simultaneously the so-called initial condition problems of the hot big bang theory: singularity, flatness, horizon problems and the Universe overclosure by heavy relics [1–5]. By itself, this idea is unfalsifiable, unless we observe the spatial curvature, which is predicted to be exponentially small. However, if inflation is driven by a scalar field called inflaton, quantum fluctuations of the latter evolve into classical field perturbations, which serve as primordial matter perturbations sourcing the large-scale structure formation and anisotropy of the cosmic microwave background in the late Universe [6]. In this manner, inflation can solve another problem of the hot big bang theory and becomes testable through observed parameters of the matter power spectrum, which are determined by the inflaton dynamics.

In this paper we put forward an idea that two other problems of the hot big bang theory—dark matter (DM) and baryon asymmetry (BA) of the Universe—can also be solved at the inflationary stage by the same source, inflaton. From the analysis of cosmological data we know that the baryon component must be asymmetric at least from the epoch of the

primordial nucleosynthesis (temperatures above 1 MeV), while the DM component must be present in the Universe well before the plasma temperature drops to eV scale. Usually, two different mechanisms are invoked to produce DM and generate the asymmetry [7–10] at different stages starting from the Universe reheating. We suggest that both processes can happen even earlier, at inflation, and moreover, they can be triggered by one and the same source (inflaton in our case), which is arguably natural given the order-of-magnitude coincidence between DM and baryon contribution to the energy density of the present Universe.

The very idea seems to be in gross contradiction with our expectations about the properties of the Universe after inflation. Indeed, the latter gets rid of all the rubbish from previous stages and makes a huge part of the Universe, formed by many Hubble-size patches, flat, causal and empty. However, the inflaton perturbations provide one example of the remnants from the inflationary epoch which do not only survive until today but also are extremely important to shape realistically the late Universe. Encouraged by this example we introduce the interactions between the inflaton and DM, as well as the inflaton and a field carrying baryon charge. The interactions are designed in such a way that the inflaton plays a role of external source of DM and baryons. Being almost
constant during inflation, the source provides a nontrivial attractor for the DM field and the field carrying baryon charge. The latter property is welcome and inherent in the inflaton nature, which washes out any traces of the initial conditions at the beginning of inflation. The attractor solutions are governed by the chosen coupling constants and can explain both DM and BA at this earliest stage of the Universe evolution.

As for the concrete realization of the idea, we consider a massive complex scalar field charged under a global $U(1)$-symmetry group. We will be mainly (but not only) interested in the superheavy field with the mass $M$ larger than the Hubble rate $H$ during inflation, $M \gtrsim H$. Were the $U(1)$ symmetry exact at all the times, any preexisting $U(1)$ charge would decay exponentially fast in the course of the inflationary expansion, and no charge would be left at the hot epoch (Sec. II). The problem can be circumvented by breaking the $U(1)$ symmetry (Sec. III). To achieve this, we couple the phase $\varphi$ of the complex scalar field directly to the inflaton, i.e.,

$$ \sim \varphi \cdot T_{\text{infl}}, $$

where $T_{\text{infl}}$ is the stress-energy tensor of the inflaton. This coupling generates the nonzero Noether charge, and keeps it up while inflation proceeds.

We study cosmological perturbations in this model in Sec. IV. The adiabatic mode of the complex scalar perturbations is nearly constant beyond the horizon during inflation. In the case of the superheavy field, $M \gtrsim H$ (which we focus on in the bulk of the paper), the isocurvature mode is decaying exponentially fast. On the contrary, for masses $M \ll H$ (considered in the Appendix), large isocurvature perturbations are generated, unless the scale of inflation is very low or the complex field is ultralight. One can avoid these constraints by the price of introducing the non-minimal coupling to gravity. The latter translates into the large effective mass of the complex field during inflation and allows one to suppress the isocurvature mode. We consider this option in Sec. V.

So far, the nature of the complex scalar has been irrelevant for our discussion. Its physical consequences may be twofold after inflation. On one hand, one may assume that the nonzero baryon charge is attributed to the complex scalar. Then its Noether charge can be transmitted to the observed BA à la the Affleck-Dine mechanism [11]. On the other hand, the complex scalar can be viewed as DM. In the case of superheavy masses, $M \gtrsim H$, the energy density of DM is directly related to the Noether charge density, which redshifts as $1/a^3$. Each of the components of the complex field experiences coherent oscillations around the minimum, and DM behaves cosmologically as the pressureless perfect fluid.

In the rest of the paper we refer to the complex field as DM, unless the opposite is stated, although in most cases

the discussion remains essentially the same for both DM and the field carrying baryon charge. Indeed, we are primarily interested in the dynamics of the complex scalar during inflation and shortly afterwards. Then, for the inflaton it is fairly irrelevant, whether the complex scalar is attributed to DM or it is used to generate BA. There is a quantitative difference, however, because producing the right amount of DM and BA requires different coupling constants to the inflaton. We will get back to this point in Sec. III C, when making the relevant estimates.

The paper is organized as follows. In Sec. II as a warm up we briefly consider the case of a noninteracting complex scalar field at inflation (with more details given in the Appendix). In Sec. III, we discuss the production of the complex scalar through the $U(1)$-symmetry breaking coupling of its phase to the inflaton. We study linear adiabatic and isocurvature perturbations of the complex scalar in Sec. IV. In Sec. V, we suggest a way to avoid large isocurvature perturbations in the case of a light field. We discuss obtained results in Sec. VI.

## II. WARM UP: NONINTERACTING COMPLEX SCALAR DURING INFLATION

We start with the action of the canonical complex scalar field,

$$ S_{\Psi} = \int d^4 x \sqrt{-g} \left[ \frac{1}{2} |\partial_{\mu} \Psi|^2 - V(|\Psi|) \right], \quad (1) $$

where $V(|\Psi|)$ is the potential. In the bulk of the paper we assume that the scalar field is minimally coupled to gravity,<fnref n="1" />

$$ S_{\text{EH}} = \frac{M_{\text{Pl}}^2}{16\pi} \int d^4 x \sqrt{-g} R, $$

where $M_{\text{Pl}}$ is the Planck mass related to the Newton constant $G = M_{\text{Pl}}^{-2}$. We further neglect self-interaction of the scalar field. This is necessary once we treat the scalar field as DM. On the other hand, when $\Psi$ is viewed as a field generating BA, such an assumption is allowed although not necessary. Nevertheless, in order to keep the discussion homogeneous, we take the potential $V$ to be quadratic for both DM and the field generating BA,

$$ V(|\Psi|) = \frac{M^2 |\Psi|^2}{2}. $$

In terms of amplitude $\lambda$ and phase $\varphi$ of the complex scalar $\Psi = \lambda e^{i\varphi}$, action (1) takes the form

$$ S_{\Psi} = \int d^4 x \sqrt{-g} \left[ \frac{1}{2} (\partial_{\mu} \lambda)^2 + \frac{1}{2} \lambda^2 (\partial_{\mu} \varphi)^2 - V(\lambda) \right]. \quad (2) $$

<footnote n="1">¹The case of the nonminimal coupling is considered in Sec. V.</footnote>
We consider the situation, where the field Ψ is a spectator at very early times, and the cosmological evolution is dominated by the slowly rolling canonical inflaton field φ. Thus, the Friedmann equation reads

$$H^2 \approx \frac{8\pi}{3M_{\text{Pl}}^2} U,$$

where U is the inflaton potential. The derivative of the Hubble rate is given by

$$\dot{H} = -4\pi G \dot{\phi}^2.$$

Our main goal in the present work is to study the production of DM (or BA) through the interaction with the inflaton introduced in the next section. Beforehand, it is instructive to review the case when no interaction is present. The model (1) possesses global U(1) symmetry. The associated Noether current conservation reads

$$\nabla_{\mu} J^{\mu} = \frac{1}{\sqrt{-g}} \partial_{\mu} (\sqrt{-g} J^{\mu}) = 0, \qquad (3)$$

where $J_{\mu} \equiv \frac{\partial S}{\partial \partial^{\mu} \varphi} = \lambda^2 \partial_{\mu} \varphi$ is the Noether current. The Noether charge density is given by

$$Q \equiv J_0 = \lambda^2 \dot{\varphi}. \qquad (4)$$

As follows from Eq. (3), it redshifts as $Q \propto \frac{1}{a^3}$ in the expanding Universe. So, during the inflationary expansion, the Noether charge density vanishes exponentially, and can be safely set to zero.

In the case $Q = 0$ there are two options. If the mass of field Ψ is larger than the Hubble rate during inflation, its components $\Psi_1$ and $\Psi_2$ oscillate about their minima located at zero values and decay exponentially fast. No DM is generated in that case. In the opposite case of sufficiently light masses, $M \ll H$, components of the field Ψ are in the slow roll regime during inflation. After inflation, when the Hubble parameter drops below the mass $M$, i.e., $H \lesssim M$, the complex scalar starts oscillating, and its further evolution is that of the pressureless perfect fluid. This scenario is standard, and we relegate further details to the Appendix. There we show, in particular, that the scenario with $M \ll H$ imposes severe constraints on the parameter space: mass $M$ itself and the scale of inflation. At least one of them should be very small in order to not overproduce isocurvature perturbations.

It should be also stressed that no BA can be produced in the scenario considered in the present section. Indeed, BA is associated with the charge Q, which is zero. On the other hand, introducing U(1)-symmetry breaking interaction allows for generating the nonzero Noether charge Q during inflation. We show this in the next section.

### III. GENERATING DARK MATTER DURING INFLATION

#### A. Quasi-de Sitter expansion of the Universe

In the present section, we discuss the complex scalar field coupled to the inflaton. This, see below, opens up the opportunity of generating superheavy DM with masses $M \gtrsim H$. Notably, the main contribution to the relic abundance of DM can be produced during the quasi-de Sitter expansion of the Universe. This is in contrast to the other mechanisms of the superheavy DM production, which operate during preheating (see, e.g., the recent works [12,13]).

As in the bulk of the paper, we keep the “DM reference” for the complex scalar Ψ. However, let us reiterate that the inflaton is blind towards its nature, i.e., whether it carries baryon charge or not. Therefore, the mechanism for the production of the relic energy density of DM and BA, discussed below, is universal. There is, however, a difference in the strength of the coupling to the inflaton. We get back to this point, when relevant—in the end of Sec. III C.

We assume that the complex scalar field given by the action (1) is coupled to the inflaton through an interaction term $S_{\text{int}}$, which explicitly breaks U(1) symmetry. The simplest way to organize the symmetry breaking is by coupling the phase of the complex scalar to external matter. Note that by coupling to the Standard Model species one risks producing the large fifth force—in conflict with the Solar System tests.<fnref n="2" /> To be on the safe side, we assume that the phase interacts only with the inflaton. For the sake of concreteness, we choose the linear coupling of the phase to the trace of the inflaton stress-energy tensor $T_{\text{infl}} \equiv -(\partial_{\mu} \varphi)^2 + 4U(\varphi)$,

$$S_{\text{int}} = \int d^4 x \sqrt{-g} \cdot \beta \cdot \varphi \cdot T_{\text{infl}}, \qquad (5)$$

where β is dimensionless constant. Such a choice of coupling is not unique. Instead of the trace $T_{\text{infl}}$, one could assume a fairly arbitrary function of the inflaton field. The important things are (i) it should remain almost constant at inflation, (ii) the coupling breaks U(1) symmetry, (iii) the coupling is linear in the dynamical field (phase φ in our case). This leads to the production of the nonzero Noether charge as we will see later.

Two remarks about our choice of effective interactions are in order here. First, the interaction (5) is not periodic with respect to φ. Therefore, strictly speaking, φ cannot be a phase. However, one can think of (5) as an approximation of a periodic interaction, e.g., $\sim \sin(\beta\varphi)$ for small φ; in this

<footnote n="2">²Normally one expects the heavy degrees of freedom (d.o.f.) to decouple in the limit $M \to \infty$. However, coupling to the matter fields through the phase is a special case, because the phase field is massless. Therefore, the decoupling in the large mass limit $M \to \infty$ may not take place.</footnote>
case $\varphi$ is the phase. Second, the phase $\varphi$ is not well defined in the limit $\lambda \to 0$. This issue is irrelevant for the early Universe dynamics of the field $\Psi$, since the amplitude $\lambda$ is large at those times. At later stages, however, the field $\lambda$ redshifts to a tiny value, and the issue comes into play. In what follows, we assume that the interaction (5) can be properly regularized in the small $\lambda$ limit, so that the strong coupling regime never takes place. Being interested only in the early Universe dynamics, we do not specify the form of the regularization.

With all the ingredients added up, the full action of our model reads

$$S = S_{\text{EH}} + S_{\Psi} + S_{\text{int}} + S_{\text{infl}}, \qquad (6)$$

where $S_\Psi$, $S_{\text{EH}}$ and $S_{\text{int}}$ are given by Eqs. (1), (2) and (5) correspondingly, while $S_{\text{infl}}$ is the standard action for the inflaton.

Varying action (6) with respect to $\varphi$, one gets in the homogeneous Universe

$$\frac{1}{a^3} \frac{d}{dt} (Qa^3) = \beta T_{\text{infl}}. \qquad (7)$$

Recall that $Q$ is defined by Eq. (4). In the slow roll regime, trace $T_{\text{infl}}$ is approximated by

$$T_{\text{infl}} \approx 4U.$$

Integrating Eq. (7) we obtain

$$Q(t) \approx \frac{4\beta}{a^3(t)} \int_{t_{\text{in}}}^{t} U(t') a^3(t') dt' + \frac{C}{a^3(t)}. \qquad (8)$$

Here $C$ is the constant of integration referring to the preinflationary charge, which should be specified by the initial conditions corresponding to the given class of inflationary models. For instance, in case of the chaotic inflation all the relevant physical quantities are of the Planckian order, while in case of the quantum Universe production all of them (except for the energy density) are zero. The related contribution [the last term in Eq. (8)] quickly redshifts away during inflation, and one can safely neglect it. On the other hand, the first term on the right-hand side of Eq. (8) does not disappear in the same manner, in spite of the factor $a^{-3}(t)$. To show it, note that in the inflationary Universe the scale factor $a$ grows almost exponentially, $a \propto e^{Ht}$, while $U$ and $H$ are slowly varying functions of time. Therefore, the integral in Eq. (8) is saturated on the upper boundary, and with the logarithmic accuracy one has

$$\int_{t_{\text{in}}}^{t} U(t') a^3(t') dt' \approx \frac{a^3(t) U(t)}{3H(t)}.$$

Finally, the estimate for the Noether charge density generated at some time $t$ during the quasi-de Sitter phase reads

$$Q(t) \sim 4\beta \frac{U(t)}{3H(t)} \propto H. \qquad (9)$$

For comparison, in the exact de Sitter space-time approximation (with constant $U$ and $H$) one would obtain $Q(t) = \frac{4\beta U}{3H}$. This explains the factor 3 in the denominator of Eq. (9). We see that the Noether charge density holds nearly constant, while its small variation with time is due to the nonzero slow roll parameters.

Next we vary action (6) with respect to amplitude $\lambda$,

$$\Box \lambda - \lambda \dot{\varphi}^2 + M^2 \lambda = 0. \qquad (10)$$

Expressing the time derivative of the field $\varphi$ from Eq. (4), we rewrite the latter equation as follows:

$$\ddot{\lambda} + 3H\dot{\lambda} - \frac{Q^2}{\lambda^3} + M^2\lambda = 0.$$

Hence, the field $\lambda$ resides in the effective potential

$$V_{\text{eff}} = \frac{M^2 \lambda^2}{2} + \frac{Q^2}{2\lambda^2}. \qquad (11)$$

The minimum of potential (11) is at $\bar{\lambda}$ given by

$$\bar{\lambda} = \sqrt{\frac{Q}{M}} \approx \sqrt{\frac{4\beta U(t)}{3MH(t)}}, \qquad (12)$$

where we made use of the estimate (9).

So far our discussion has been independent of the mass $M$ of the complex field. Since this point on, however, this aspect becomes crucial. If the field $\Psi$ is substantially light, $M \ll H$, its amplitude $\lambda$ enters the slow-roll regime at the onset of inflation akin to the noninteracting (free) light complex scalar. This model has known problems with isocurvature perturbations, as we mentioned in the Introduction and illustrate with calculations in the Appendix. However, if inflation is sufficiently long, the Hubble parameter may decrease significantly, so that the field becomes heavy (compared to the Hubble scale) by the end of inflation, $M > H$. One can expect such a scenario in chaotic inflation with a power-law inflaton potential. The conclusion about large isocurvature perturbations is not applicable in that scenario. In the next section, we show that they are exponentially suppressed compared to adiabatic ones. Thus, in the bulk of the paper, we mostly focus on the case of the superheavy DM with the mass

$$M \gtrsim H,$$
<figure><img src="image_3.png" /><figcaption>FIG. 1. Evolution of the field $\lambda$ is shown in the large mass regime $M \gtrsim H$, in the approximation of exact de Sitter space-time. We have chosen $M/H = 20$. The initial value for the derivative $\dot{\lambda}$ is set to zero, $\dot{\lambda}_i = 0$. The time range corresponds to 3 Hubble times. Independently of the initial value $\lambda_i$, the amplitude $\lambda$ relaxes to the minimum point $\bar{\lambda}$ (dashed line) of the effective potential (11).</figcaption></figure>

and hereafter by $H$ we assume somewhat loosely the value at $\sim 60$ e-folds before the end of inflation.

In Fig. 1, we show the evolution of the amplitude $\lambda$. We see that it relaxes to the minimum of the effective potential $\bar{\lambda}$ within a few Hubble times independently on its initial conditions. That is, the solution (12) is an attractor. We simplified the picture by considering the exact de Sitter space-time, in which case the solution (12) is just a constant. In reality, however, the field $\lambda$ does not get frozen out completely. Indeed, the parameters of the effective potential (11) are changing with time—although slowly. Therefore, the value of $\lambda$ corresponding to the minimum of the effective potential also shifts with time,

$$ \dot{\bar{\lambda}} \sim M_{\text{Pl}} \dot{H} \sqrt{\frac{\beta}{MH}}. \qquad (13) $$

The fact that $\dot{\bar{\lambda}}$ is nonzero is relevant for the evolution of the adiabatic perturbation mode discussed in Sec. IV.

While the field $\lambda$ takes a nearly constant value, the phase “velocity” also approaches constant $\dot{\varphi} \to M$. This follows from Eq. (10), where one sets time derivatives of the amplitude $\lambda$ to zero. We conclude that during the inflationary stage, the complex scalar spins fast in the configuration space with nearly constant amplitude and frequency; see Fig. 2.

## B. Preheating and later stages

After the end of inflation, the force associated with the inflaton decreases rapidly, and its further impact on the Noether charge evolution vanishes. Since this point on, the Noether charge density drops with the scale factor as $Q \sim 1/a^3$, i.e.,

<figure><img src="image_4.png" /><figcaption>FIG. 2. Evolution of the complex scalar with the quadratic potential is shown during inflation and in the radiation-dominated epoch. The superheavy DM with the mass $M \gtrsim H$ is considered. During inflation, the amplitude of the complex field resides in the minimum of the effective potential (11), $\lambda = \bar{\lambda}$, where $\bar{\lambda} = \sqrt{\frac{Q}{M}}$. The amplitude is nearly constant in the (quasi-)de Sitter space-time. Consequently, the trajectory of the complex field in the configuration space is almost circular. The uncertainties about the circle (captured by the thick line) reflect small deviations from the de Sitter space-time due to nonzero slow roll parameters. After the end of inflation, the trajectory is inspiralling with the radius decreasing as $1/a^{3/2}$. The latter reflects the drop of the amplitude $\lambda$ in the hot Universe.</figcaption></figure>

$$ Q(t > t_{\text{end}}) \approx Q(t = t_{\text{end}}) \frac{a_{\text{end}}^3}{a^3} \approx \frac{4\beta}{a^3(t)} \frac{a^3(t_{\text{end}})U(t_{\text{end}})}{3H(t_{\text{end}})}. \qquad (14) $$

Here $t_{\text{end}}$ designates the end of inflation, i.e., the moment, when slow roll parameters $\epsilon$ and $\eta$ become of the order unity, i.e., $\epsilon \sim 1$ and $\eta \sim 1$. That estimate holds, e.g., if we assume the immediate decay of the inflaton to the particle species. In reality, the evolution of the inflaton during the preheating stage may considerably alter this result. That is, the actual integral, which gives the contribution to the total Noether charge density, is given by

$$ \int_{t_{\text{in}}}^{t_{\text{end}}} \to \int_{t_{\text{in}}}^{t_{\text{end}}} + \int_{t_{\text{end}}}^{t_{\text{preheating}}}, \qquad (15) $$

where $t_{\text{preheating}}$ denotes the end of preheating. Generically, the contribution due to the second integral on the right-hand side is of the order of the first one (if preheating continues for a few Hubble times) or much larger than it (if preheating is long enough in terms of Hubble times). However, there are situations, which we are interested in, when the main contribution comes from the quasi-de Sitter expansion of the Universe. Apart from the general class of inflationary models with instant preheating, this occurs in a number of specific models. For example, consider the nonminimally coupled inflaton with the quartic potential [14–18],
$$S_{\text{infl}} = \int d^4 x \sqrt{-g} \left( \frac{1}{2} (\partial_{\mu} \phi)^2 + \frac{\xi}{2} R \phi^2 - \frac{h}{4} \phi^4 \right).$$

The role of the nonminimal coupling is to reduce the tensor-to-scalar ratio, which would otherwise violate the existing Planck bound [19]. The trace of the stress-energy tensor $T_{\text{infl}}$ rapidly decreases after the end of inflation following the restoration of the conformal symmetry in the cosmological background. Hence in this case, no DM is generated during the preheating epoch.

After the inflaton decays, the complex scalar $\Psi$ becomes free modulo decays into the inflaton particles/interactions with Standard Model species mediated by the virtual inflaton. Given that $M \gtrsim H$, right after inflation the components of the complex field start oscillating around the origin with the amplitude decaying as $1/a^{3/2}$. Since this point on, the complex scalar mimics the pressureless perfect fluid: its energy density redshifts as $\rho_{\text{DM}}(t) = M^2\lambda^2(t) \propto 1/a^3(t)$. At the onset of oscillations, which start right after the end of inflation at the time $t = t_{\text{end}}$, the amplitude $\lambda$ is estimated by Eq. (12), and hence the DM energy density is given by

$$\rho_{\text{DM}}(t_{\text{end}}) \approx M^2 \bar{\lambda}^2 \approx MQ(t_{\text{end}}). \quad (16)$$

We see that it is fully determined by the Noether charge density generated and the mass of the scalar, and is independent of the preinflationary value of the field $\Psi$. Then it decreases following the behavior of the charge density $Q(t)$ as is written in Eq. (14). We require that the energy density (16) should match the abundance of DM observed today. Concretely,

$$\rho_{\text{DM}}(t_{\text{eq}}) \approx \frac{\beta M U(t_{\text{end}})}{H(t_{\text{end}})} \cdot \frac{a_{\text{end}}^3}{a_{\text{eq}}^3} \approx \rho_{\text{rad}}(t_{\text{eq}}),$$

where the subscript “eq” denotes the equality between the matter and the radiation. For the sake of simplicity, we assume that the radiation is produced immediately at the very end of inflation, so that $\rho_{\text{rad}}(t_{\text{end}}) \approx U(t_{\text{end}})$. Then one can write

$$\rho_{\text{rad}}(t_{\text{eq}}) \approx \rho_{\text{rad}}(t_{\text{end}}) \cdot \frac{a_{\text{end}}^4}{a_{\text{eq}}^4} \approx U(t_{\text{end}}) \cdot \frac{a_{\text{end}}^4}{a_{\text{eq}}^4}.$$

(In the first estimate here we neglected the change of the number of ultrarelativistic d.o.f. during the radiation-dominated epoch, which is straightforward to account for.) The estimate for the constant $\beta$ then takes the form

$$\beta \approx \frac{H(t_{\text{end}})}{M} \cdot \frac{a_{\text{end}}}{a_{\text{eq}}} \approx \frac{H(t_{\text{end}})}{M} \cdot \frac{T_{\text{eq}}}{T_{\text{end}}}.$$

Substituting the estimate for the Hubble rate $H(t_{\text{end}}) \sim \frac{T_{\text{end}}^2}{M_{\text{Pl}}}$, we obtain for the coupling constant $\beta$,

$$\beta \approx \frac{T_{\text{end}}}{M_{\text{Pl}}} \cdot \frac{T_{\text{eq}}}{M}. \quad (17)$$

Taking for instance $T_{\text{end}} \sim 10^{16}$ GeV (maximal possible temperature in the postinflationary Universe) and $M \sim H \sim 10^{-5} M_{\text{Pl}}$, one gets $\beta \sim 10^{-26}$. This is a rough estimate, which first assumes high scale inflation and then does not account for the details of the cosmological evolution during (p)reheating. However, the message is clear: production of superheavy complex field DM requires a tiny coupling to the inflaton. Indeed, we produce non-relativistic relics even before the radiation-dominated stage, where their relative contribution to the Universe energy density grows linearly with the scale factor. In the case of almost free relics, the growth must be compensated and only the strong suppression of the production rate can help. This is achieved by choosing a small value of the coupling constant. Since the coupling violates global $U(1)$ symmetry, it might be attributed to some nonperturbative quantum effect, so that the smallness may be natural.

For the coupling constant $\beta$ as small as $\beta \approx 10^{-26}$, one expects that the lifetime of DM exceeds the age of the Universe. This is the scenario assumed in the present work. Depending on the regularization of the interaction (5), however, the complex field $\Psi$ may turn out to be unstable with respect to the decay into the inflatons, so that no DM is left today. In that case, the modification of our scenario is required. For example, one may introduce interaction of the field $\Psi$ with some stable light particles $S$ (say, sterile neutrinos). If this interaction is stronger than the interaction with the inflaton, the field $\Psi$ predominantly decays into the particles $S$. Then, the Noether charge density produced during inflation is converted into the number density $n_S$ of particles $S$, i.e., $Q \approx n_S$. Note that the coupling constant $\beta$ required for this scenario is larger compared to that given in Eq. (17) by the ratio of the masses $M_{n_S}$. A detailed study of this scenario is out of the scope of the present work. In what follows, we assume that the field $\Psi$ is stable and constitutes most of DM in the Universe.

Let us make one remark in passing. In the limit, when the field $\lambda$ is initially frozen and $M \to \infty$, the model of the complex scalar effectively reduces to the following one:

$$S = \int d^4 x \sqrt{-g} \left[ \frac{\lambda^2}{2} (\partial_{\mu} \varphi)^2 - \frac{M^2 \lambda^2}{2} \right]; \quad (18)$$

see Fig. 3. Namely, the dynamics of the amplitude $\lambda$ freezes out, and the latter reduces to the Lagrange multiplier. Upon redefining the amplitude and the phase, i.e., $\lambda \to \lambda/M$ and $\varphi \to M\varphi$, the action (18) takes the form

$$S = \int d^4 x \sqrt{-g} \cdot \frac{\lambda^2}{2} \cdot [(\partial_{\mu} \varphi)^2 - 1]. \quad (19)$$

The stress-energy tensor of the model (19) is $T_{\mu\nu} = \lambda^2 \partial_{\mu}\varphi\partial_{\nu}\varphi$. It corresponds to the pressureless perfect fluid
<figure><img src="image_5.png" /><figcaption>FIG. 3. Evolution of the field $\lambda$ is shown in the large mass regime, i.e., $M \gg H$, during the radiation-dominated stage. We assume for simplicity that the latter follows immediately after inflation. The initial conditions at the beginning of the hot epoch (coinciding with the end of inflation, time $t_{end}$) are $\lambda(t_{end}) = \bar{\lambda}$ and $\dot{\lambda}(t_{end}) = 0$. The characteristic mass scale $M_0$ is chosen to be such that $\frac{M_0}{H(t_{end})} = 20$. For the masses $M \lesssim M_0$, the evolution of the amplitude exhibits clear oscillations about the average value. Instead, for $M \gg M_0$ oscillations are very small, and the change of the field $\lambda$ tracks the cosmological expansion, i.e., $\lambda \propto \frac{1}{a^{3/2}}$. In that case the evolution of the amplitude can be inferred from the simpler model (19), where $\lambda$ plays the role of the Lagrange multiplier.</figcaption></figure>

with $\lambda^2$ being the energy density and $\varphi$ the velocity potential. We see that at $M \to \infty$ the complex scalar exactly reproduces the pressureless perfect fluid. (Normally the pressureless perfect fluid is obtained from the dynamics of the scalar field upon averaging over its oscillations.) This holds true down to the times when caustics are formed and the fluid description breaks down. On the other hand, the model of the complex field always retains a nonsingular behavior.

The model (19) and its variations is nowadays widely discussed in the literature and goes under the name of the mimetic matter scenario [20,21].<fnref n="3" /> Originally, it has been designed as an alternative to the particle DM. According to our discussion above, the model (19) breaks down at the times of the caustic formation and must be completed. The superheavy complex scalar with initial conditions set by inflation may serve as a proper candidate for this completion [23].

### C. Baryon asymmetry

While we mainly discussed DM so far, in an alternative scenario the generated Noether charge can explain BA of the Universe. For this scenario to be realized, one assigns

the nonzero baryon charge to the complex field $\Psi$. The rest of the scenario can be designed by analogy with the standard mechanisms of BA generation: the baryon charge should be transferred from $\Psi$ to the quark sector of the Standard Model. Assume for concreteness that the baryon charge of the field $\Psi$ equals unity, $B_{\Psi} = 1$. Then, one can write down the following interaction:

$$ \mathcal{L} = y \bar{n} S \Psi + \text{H.c.}, $$

where $y$ is the coupling constant, $n$ is the colorless fermionic operator formed by three Standard Model quarks (e.g., neutronlike fermion, $n = ddu$) and $S$ is another fermionic field, which is a singlet with respect to the Standard Model gauge group (e.g., sterile neutrino). Then BA in the quark sector is induced through the decays of the field $\Psi$. The resulting BA is given by

$$ \Delta_B \sim \frac{Q}{s}, $$

where $s$ is the entropy density.

Compared to the DM case, production of BA generically requires much larger coupling constant $\beta$. Let us show this explicitly. For this purpose, we again assume that the inflaton energy is converted immediately to radiation right after inflation. Then,

$$ \Delta_B \approx \frac{\beta U(t_{end}) T_{end}}{H(t_{end}) \rho_{rad}(t_{end})} \approx \frac{\beta T_{end}}{H(t_{end})} \approx \beta \cdot \sqrt{\frac{M_{Pl}}{H(t_{end})}}. $$

Here we use that $s = \frac{4\rho_{rad}}{3T}$ and $T \approx \sqrt{M_{Pl}H}$. The estimate for the coupling constant then follows from the observed BA of the Universe, $\Delta_B = 0.87 \times 10^{-10}$. We have

$$ \beta \approx 10^{-10} \sqrt{\frac{H(t_{end})}{M_{Pl}}}. \qquad (20) $$

Hence, for the high scale inflation with the Hubble rate $H \approx 10^{14}$ GeV, the coupling constant $\beta$ can be as large as $\beta \approx 10^{-12}$.

Some remarks are in order here. The estimate (20) does not depend on the lifetime of the field $\Psi$, since both $Q$ and $s$ drop similarly with the scale factor as the Universe enters the radiation-domination regime. On the other hand, this estimate changes if the energy density of the Universe scales differently than radiation after inflation. Say, for the matter-domination stage at preheating the relevant amount of BA requires bigger coupling by a ratio of the scale factors at reheating and at the end of inflation.

As we can see, scenarios for DM and BA generation involve coupling constants $\beta$ different by the factor $\sim 10^{-14}$. The appearance of this factor can be understood as follows. Assuming instant preheating the ratio of the coupling

<footnote n="3">³The original formulation of the mimetic matter scenario is different from Eq. (19). It stems from the particular conformal transformation of the metric. The equivalence with the model (19) has been first pointed out in Ref. [22].</footnote>
constants required for DM and BA generation is about the ratio of the proton mass (the lightest stable baryon) and the Hubble parameter at inflation (the smallest mass of the viable heavy DM), i.e., ~$m_p/H$~ ~$10^{-14}$.

## IV. PERTURBATIONS

Now let us discuss perturbations of the complex scalar field. We want to prove that only adiabatic fluctuations survive in the case of the superheavy DM, while isocurvature modes quickly decay behind the horizon. This is relevant in view of the strong Planck limits on the cold DM isocurvature perturbations [19]. On the other hand, for lighter DM, there is a nondecaying isocurvature mode, which can be large enough, unless the scale of inflation/the mass of DM are strongly constrained. This case is considered in the Appendix.

We choose to study the linear perturbations in the Newtonian gauge [24,25],

$$ds^2 = (1 + 2\Phi)dt^2 - a^2(1 - 2\Psi)\delta_{mn}dx^mdx^n,$$

so that

$$\delta g_{00} = 2\Phi, \quad \delta g_{ij} = 2a^2\Psi\delta_{ij}.$$

From $i \neq j$ components of the perturbed Einstein equations we have $\Psi = \Phi$, and the $0i$ component takes the form

$$\dot{\Phi} + H\Phi = 4\pi G(\dot{\Phi}\delta\Phi + \dot{\lambda}\delta\lambda + \lambda^2\dot{\varphi}\delta\varphi). \quad (21)$$

The linearized equations of motion for the amplitude $\lambda$ and the phase $\varphi$ are given by

$$\delta\ddot{\lambda} + 3H\delta\dot{\lambda} - \frac{1}{a^2}\partial_i\partial_i\delta\lambda - \delta\lambda\dot{\varphi}^2 - 2\lambda\dot{\varphi}\delta\dot{\varphi} + M^2\delta\lambda + 2M^2\lambda\Phi - 4\lambda\dot{\Phi} = 0 \quad (22)$$

and

$$\frac{1}{a^3} \frac{d}{dt} (a^3 \lambda^2 \delta \dot{\varphi}) - \frac{1}{a^2} \lambda^2 \partial_i \partial_i \delta \varphi - \frac{2}{a^3} \frac{d}{dt} (a^3 \lambda^2 \dot{\varphi}) \Phi - 4 \lambda^2 \dot{\varphi} \dot{\Phi} + \frac{1}{a^3} \frac{d}{dt} (a^3 \dot{\varphi} \delta \lambda^2) = \beta \delta T_{\text{infl}}, \quad (23)$$

respectively.

The analysis of perturbations parallels to that of inflation with multiple scalar fields. The adiabatic mode is given by [26]

$$\Phi = \Psi = C_1 \left( 1 - \frac{H}{a} \int_0^t a dt' \right) + C_2 \frac{H}{a} \quad (24)$$

and

$$\frac{\delta\Phi}{\dot{\Phi}} = \frac{\delta\lambda_{ad}}{\dot{\lambda}} = \frac{\delta\varphi_{ad}}{\dot{\varphi}} = \frac{1}{a} \left( C_1 \int_0^t a dt' - C_2 \right). \quad (25)$$

Here $C_1$ and $C_2$ are constants defined from the matching to the inflaton vacuum initial conditions. They correspond to the growing and decaying adiabatic modes, respectively. One can check that in the superhorizon regime the ansatz (24) and (25) passes the equations of motion (21)-(23). Note that the solution (24) and (25) is exact. The equalities (25) can be easily reformulated in terms of the complex scalar components $\Psi_1$ and $\Psi_2$,

$$\frac{\delta\Phi}{\dot{\Phi}} = \frac{\delta\Psi_{1,ad}}{\dot{\Psi}_1} = \frac{\delta\Psi_{2,ad}}{\dot{\Psi}_2}.$$

Notably, despite the nontrivial coupling to the inflaton via the phase field, the adiabatic condition has the same form as in the case of canonical noninteracting fields.

Let us switch to the case of isocurvature perturbations. For this purpose, one sets to zero gravitational potentials,

$$\Phi = \Psi = 0.$$

In the spectator approximation, which we assume here, the inflaton is the only source of metric fluctuations. Therefore, one may consistently set the isocurvature perturbation of the inflaton to zero, $\delta\phi_{iso} = 0$. Then the relevant system of equations is given by

$$\delta\ddot{\lambda}_{\text{iso}} + 3H\delta\dot{\lambda}_{\text{iso}} - \frac{1}{a^2}\partial_i\partial_i\delta\lambda_{\text{iso}} - \delta\lambda_{\text{iso}}\dot{\varphi}^2 - 2\lambda\dot{\varphi}\delta\dot{\varphi}_{\text{iso}} + M^2\delta\lambda_{\text{iso}} = 0 \quad (26)$$

and

$$\frac{1}{a^3} \frac{d}{dt} (a^3 \lambda^2 \delta \dot{\varphi})_{\text{iso}} - \frac{1}{a^2} \lambda^2 \partial_i \partial_i \delta \varphi_{\text{iso}} + \frac{1}{a^3} \frac{d}{dt} (a^3 \dot{\varphi} \delta \lambda_{\text{iso}}^2) = 0. \quad (27)$$

In the superhorizon regime, Eq. (27) takes the form

$$\frac{d}{dt} (a^3 \delta[\lambda^2 \dot{\varphi}]_{\text{iso}}) = 0.$$

Consequently, one has

$$\delta(\lambda^2 \dot{\varphi})_{\text{iso}} = \frac{C}{a^3}.$$

Hence, the isocurvature perturbation of the product $\lambda^2\dot{\varphi}$ decays fast behind the horizon, and one can write

$$\frac{\delta\dot{\varphi}_{\text{iso}}}{\dot{\varphi}} = -\frac{2\delta\lambda_{\text{iso}}}{\lambda}. \quad (28)$$

Expressing the perturbation $\delta\dot{\varphi}$ from the latter and substituting into Eq. (26), we obtain
$$ \delta\ddot{\lambda}_{\text{iso}} + 3H\delta\dot{\lambda}_{\text{iso}} + M_{\text{eff}}^2\delta\lambda_{\text{iso}} = 0, $$

where

$$ M_{\text{eff}}^2 = M^2 + 3\dot{\phi}^2. \qquad (29) $$

As we assume the superheavy DM with the mass $M \gtrsim H$, one has $M_{\text{eff}} \gtrsim H$. Hence, the perturbation $\delta\lambda_{\text{iso}}$ relaxes to zero within a few Hubble times, independently of its initial value. At the same time, the adiabatic perturbation $\delta\lambda_{\text{ad}}$ remains constant behind the horizon,

$$ \delta\lambda_{\text{ad}} \sim \frac{\dot{\lambda}}{\dot{\phi}}\delta\phi \sim \sqrt{\frac{\beta H |\dot{H}|}{M}} \sim \sqrt{\frac{\epsilon\beta H^3}{M}}. $$

Here we use that $\dot{\phi} \sim \sqrt{M_{\text{Pl}}^2 |\dot{H}|}$, $\delta\phi \sim H$ and substitute Eq. (12). We conclude that for $M \gtrsim H$ perturbations are adiabatic with a very high accuracy.

On the contrary, for the light DM ($M \ll H$) discussed in the Appendix, the fate of perturbations is not that different from the case of the noninteracting complex field. Namely, isocurvature perturbations dominate over adiabatic ones, unless the scale of inflation is low and/or the mass $M$ is small. This severely constrains the scenario of light DM. Nevertheless, slightly extending the parameter space of the model allows one to suppress the isocurvature mode for essentially arbitrary $M$. We consider such a modification in the next section.

# V. NONMINIMAL COUPLING TO GRAVITY

## A. Inflationary stage

Having a superheavy complex field is not the only possibility to suppress isocurvature modes. Another option, which we consider in the present section, is to couple the amplitude of the complex scalar to the Ricci curvature. Similar ideas have been proposed in the context of axion models in Refs. [27,28]. We will see that in the presence of the nonminimal coupling, the mass of the field $\Psi$ remains essentially unconstrained. Such a freedom is not particularly relevant in the scenario, where the complex scalar carries baryon charge. On the other hand, it is important for the DM applications of our model. In particular, if the latter is ultralight, the composition of DM halos may have distinct features compared to the standard cold DM. Furthermore, for light DM, the coupling constant $\beta$ is allowed to be much larger than in the case of superheavy DM.

We modify the model by introducing the nonminimal coupling of $\Psi$ to gravity,

$$ S_{\text{nonmin}} = \int d^4 x \sqrt{-g} \left( \frac{\xi \lambda^2}{2} R \right). $$

During the inflationary expansion, the Ricci scalar $R = -12H^2 - 6\dot{H}$ holds nearly constant. Thus, the nonminimal coupling mimics the mass term of the complex field. The equation of motion (10) for the field $\lambda$ now takes the form,

$$ \ddot{\lambda} - \lambda \dot{\phi}^2 + 3H\dot{\lambda} + M_{\text{eff}}^2 \lambda = 0, $$

where the effective mass squared $M_{\text{eff}}^2$ is the sum of two contributions: one coming from the standard mass term and another due to the nonminimal coupling, i.e.,

$$ M_{\text{eff}}^2 \approx M^2 + 12\xi H^2 \qquad (30) $$

(we made use of the de Sitter approximation at this step). Note that the nonminimal coupling is irrelevant for the generation of the Noether charge density [as it does not break the $U(1)$-invariance]. That is, Eq. (4) holds true, and one can use it to express the time derivative $\dot{\phi}$. Repeating the steps from Sec. III A, we find the effective potential for the field $\lambda$,

$$ V_{\text{eff}} = \frac{M_{\text{eff}}^2 \lambda^2}{2} + \frac{Q^2}{2\lambda^2}. \qquad (31) $$

It coincides with the standard effective potential (11) modulo the replacement $M \to M_{\text{eff}}$. Thus, the minimum point of the new effective potential is given by Eq. (12), where one should just substitute $M_{\text{eff}}$ instead of $M$. If the effects of the nonminimal coupling are weak, so that $M_{\text{eff}} \approx M$, we return to the situation described in the previous sections. Here we are interested in the low mass limit, i.e.,

$$ M \ll \sqrt{\xi} H. $$

In that case, the nonminimal coupling dominates over the standard mass term, so that $M_{\text{eff}} \sim \sqrt{\xi} H$. Substituting the latter into Eq. (12), one gets for the minimum point of the new effective potential (31)

$$ \bar{\lambda}_{\text{nonmin}} \approx \frac{\sqrt{\beta} M_{\text{Pl}}}{\xi^{1/4}}. \qquad (32) $$

[The subscript “nonmin” is added to distinguish this minimum point from the one obtained in the case of superheavy DM; see Eq. (12).] During inflation, the amplitude $\lambda$ relaxes to $\bar{\lambda}$ within a few Hubble times, as in the case of the genuinely superheavy complex field. See Fig. 1.

The fate of adiabatic and isocurvature perturbations is also similar. The former obey Eqs. (24) and (25) in the Einstein frame, while the latter rapidly decay behind the horizon provided only that the parameter $\xi$ governing the nonminimal coupling is not small, i.e., $\xi \gtrsim 1$. Thus, with
the nonminimal coupling to gravity, the scenario with light DM may be viable as well.

## B. After inflation

We assume for the sake of simplicity that the radiation-dominated stage follows immediately after the end of the quasi-de Sitter expansion. Then, the Ricci scalar R abruptly drops to zero, $R \to 0$, and the mass term of the complex scalar takes the standard form. This leads to the abrupt change of the effective potential (31), so that we recover our original expression (11),

$$V_{\text{eff}} = -\frac{\xi R \lambda^2}{2} + \frac{Q^2}{2\lambda^2} \to \frac{M^2 \lambda^2}{2} + \frac{Q^2}{2\lambda^2}$$

(we assumed that $M^2 \ll |\xi R|$ during inflation). Now, the minimum of the effective potential is at $\bar{\lambda}$ given by Eq. (12). Note that $\bar{\lambda} \gg \bar{\lambda}_{\text{nonmin}}$. Consequently, the amplitude $\lambda$, which still resides at $\lambda = \bar{\lambda}_{\text{nonmin}}$ right after inflation, is away from the location of the true minimum $\bar{\lambda}$. Then, right after the end of inflation the amplitude $\lambda$ rolls towards $\bar{\lambda}$. This is reflected in the initial fast growth of $\lambda$ shown in Fig. 4.

Further evolution of the amplitude proceeds as follows. The complex scalar starts with a large kinetic energy due to rotations induced by the coupling to the inflaton, while the potential energy $\sim M^2 \lambda^2$ is yet negligible. At those times, the derivative of the amplitude decreases fast, $\dot{\lambda} \sim \frac{1}{a}$, and hence the growth of the amplitude quickly stabilizes.

<figure><img src="image_6.png" /><figcaption>FIG. 4. Evolution of the field $\lambda$ is shown during the radiation-dominated stage in the version of the model with the nonminimal coupling to gravity. We have chosen the coupling to the Ricci curvature $\xi = 1$. The small mass $M$ is assumed, i.e., $M/H(t_{\text{end}}) = 2 \times 10^{-3}$ and $Mt_{\text{end}} = 10^{-3}$. At the times $Mt \gtrsim 1$, the complex field exhibits damping oscillations around the minimum. We zoomed the nontrivial part of the evolution taking place at early times $Mt \ll 1$. Initially the amplitude $\lambda$ rolls fast; this reflects the instantaneous change of the structure of the effective potential (11) after inflation. Then the evolution of the field $\lambda$ slows down for the reasons described in the text. At the times $Mt \sim 1$ the oscillations start and the complex field enters the dustlike phase.</figcaption></figure>

Shortly, the kinetic energy, which redshifts as $\sim \frac{1}{a^6}$, drops below the potential energy, which is kept nearly constant, and the slow roll phase starts. At later times $t \sim H^{-1} \gtrsim M^{-1}$, the field $\Psi$ exhibits the standard oscillatory behavior, and its energy density $\rho_{\text{DM}} \sim M^2 \lambda^2 \sim 1/a^3$ mimics that of the pressureless perfect fluid.

As a result, the amplitude $\lambda$ does not change considerably by the beginning of oscillations and can be roughly estimated by Eq. (32). Hence, its value (and, thus, the relic energy density of DM) is attached to the Noether charge density generated by the end of inflation, as in the case of the superheavy complex scalar. The combination of constants $\beta, \xi$ and the mass $M$ can be constrained from the observed relic abundance of DM. Namely, at the matter-radiation equality, DM energy density is estimated by

$$\rho_{\text{DM}}(t_{\text{eq}}) \sim M^2 \lambda^2(t_{\text{eq}}) \sim M^2 \bar{\lambda}_{\text{nonmin}}^2 \cdot \frac{T_{\text{eq}}^3}{T_{H \sim M}^3} \sim \rho_{\text{rad}}(t_{\text{eq}}) \sim T_{\text{eq}}^4,$$

where $T_{\text{eq}}$ is the equilibrium temperature, $T_{\text{eq}} \sim 1$ eV and $T_{H \sim M}$ is temperature of the Universe at the onset of oscillations, $T_{H \sim M} \sim \sqrt{MM_{\text{Pl}}}$. We get an estimate of the coupling constant $\beta$,

$$\beta \sim \frac{\sqrt{\xi}}{\sqrt{M \cdot M_{\text{Pl}}} \cdot T_{\text{eq}}}.$$

For example, the widely discussed in literature ultralight scalar field DM of mass $M \simeq 10^{-21}$ eV requires coupling $\beta \sim 10^{-3}$. Thus, in the scenario with the nonminimal coupling to gravity, the constant $\beta$ is not necessarily very small. This is in contrast to the case of superheavy DM considered in the previous sections.

The field DM with such low masses may affect the standard picture of the halo formation. Generically, classical field DM obeys the Schröedinger-Poisson system of equations [29–32], while the standard cold DM obeys the Vlasov-Poisson system. The difference between the two scenarios is particularly strong in the ultralight case [33–35]. This has been used to set the lower limit on the axion mass: $m > 2 - 3 \times 10^{-21}$ eV [36,37]. The latter bound is applicable to our case as well.

One important remark is in order here. Having a relatively large coupling constant $\beta$ appears to be advantageous from the viewpoint of testability of the model. On the other hand, this may lead to thermalization of DM through the inflaton coupled to the Standard Model fields produced in the course of (p)reheating. Thermalized DM with the mass $M \simeq 10^{-21}$ eV behaves as hot DM and can constitute at best a few percent of the total DM in the Universe. Another issue is related to the stability of the produced component (DM and baryons) with respect to decay into light Standard Model particles through the higher order operators induced by the coupling to inflaton. This study is out of the scope of the present work.
## VI. DISCUSSION

In this work we suggested a mechanism to generate DM/BA from inflation. We focused on a particular model of a superheavy complex scalar field with global $U(1)$ symmetry explicitly broken by the direct coupling of its phase to the inflaton.

It is worth noting that were we only interested in DM generation during inflation, a simpler model with a real scalar field $\Phi$ would suffice. Indeed, once a linear coupling is introduced, $\sim \Phi \cdot T_{\text{infl}}$ [cf. Eq. (5)], the source term appears in the equation for the scalar field. The latter produces the nonzero expectation value of the field $\Phi$ during inflation. Upon the inflaton decay, the vacuum value again takes the zeroth value. The scalar $\Phi$ then oscillates about its minimum with the amplitude set by its expectation value during inflation.

Nevertheless, the model of the complex scalar field is more advantageous because it provides a link to BA as well. The latter can be generated using the same mechanism, which leads to DM production. In this case, the complex field $\Psi$ must carry nonzero baryon charge. After inflation, the field $\Psi$ eventually decays converting its Noether charge to the Standard Model quark sector. The resulting BA of the Universe is given by $\Delta_B \sim Q_s$. The mechanisms of the Noether charge transfer to BA are rather generic and not specific for our particular model.

In the present work, we did not discuss possible signatures of the mechanism proposed in the current and future data. The observational signatures depend on particular couplings of the complex scalar to Standard Model fields. Typically, the couplings to the inflaton make DM particles and baryons unstable, though with the lifetimes largely exceeding the age of the Universe. (See, however, the qualification in Sec. III B.) Hence, if we insist on the superheavy DM, these couplings may lead to a potentially observable flux of ultrahigh energy cosmic rays. See Refs. [38,39] for the state of art. On the other hand, the constraint on the mass $M$ can be significantly relaxed, upon introducing the nonminimal coupling to gravity (see the discussion in Sec. V). If the mass $M$ is in the ultralight range, the evolution of halos may be considerably affected compared to the cold DM predictions. This is a universal prediction for all the ultralight DM models. Whether our mechanism of DM production implies some specific signatures (e.g., due to the induced self-coupling from the exchange by the virtual inflaton), which can be tested with observations of the cosmic structures at small scales, remains to be clarified.

## ACKNOWLEDGMENTS

E. B. acknowledges support from “Projets de recherche conjoint de Centre National de la Recherche Scientifique/Russian Foundation for Basic Research” PRC CNRS/RFBR (2018–2020) No. 1985 “Gravité modifiée et trous noirs: signatures expérimentales et modèles consistants,” and from

the research program “Programme national de cosmologie et galaxies” of the CNRS/INSU, France. The investigation of the baryon asymmetry generation (D. G.) is supported by the RSF Grant No. 14-22-00161. The work of S. R. is supported by the funds from the European Regional Development Fund and the Czech Ministry of Education, Youth and Sports (MŠMT): Project No. CoGraDS-CZ.02.1.01/0.0/0.0/15_003/0000437.

## APPENDIX: LIGHT DM ($M \ll H$) MINIMALLY COUPLED TO GRAVITY

### 1. Noninteracting case

In this Appendix, we discuss DM composed of the complex scalar field $\Psi$ with the mass $M$ much smaller than the Hubble rate $H$ at inflation, $M \ll H$. We start with the noninteracting case. In this situation, the Noether charge density $Q = \lambda^2 \dot{\phi}$ equals zero, $Q = 0$. Hence, $\dot{\phi} = 0$, and the complex field evolves only in the radial direction. Its evolution is thus indistinguishable from that of the real scalar field. Namely, it slowly rolls during inflation. At the hot epoch, as the Hubble rate drops substantially, so that $H(t_{\text{osc}}) \sim M$, the standard dustlike evolution begins; $t_{\text{osc}}$ is the time, when the field $\Psi$ starts oscillating.

During inflation, the amplitude $\lambda$ acquires isocurvature perturbations, which remain frozen beyond the horizon at the value $\delta\lambda_{\text{iso}} \approx \frac{H}{2\pi}$ until the onset of oscillations. At the hot radiation-dominated stage, the DM isocurvature perturbation is defined as

$$S_{\text{DM}} = 3H \left( \frac{\delta\rho_{\text{DM}}}{\dot{\rho}_{\text{DM}}} - \frac{\delta\rho_{\gamma}}{\dot{\rho}_{\gamma}} \right) = 3H \frac{\delta\rho_{\text{DM,iso}}}{\dot{\rho}_{\text{DM}}}.$$

Here $\rho_{\text{DM}}$ and $\rho_{\gamma}$ are the DM and radiation energy density; $\delta\rho_{\text{DM}}$ and $\delta\rho_{\gamma}$ are their perturbations, respectively. In the second equality above we use the fact that the isocurvature perturbation of ultrarelativistic matter equals zero during the radiation-dominated stage (by definition). Using that $\dot{\rho}_{\text{DM}} + 3H\rho_{\text{DM}} = 0$ during oscillations, one gets for the power spectrum of DM isocurvature perturbations,

$$P_{S_{\text{DM}}} = \left\langle \left( \frac{\delta \rho_{\text{DM,iso}}}{\rho_{\text{DM}}} \right)^2 \right\rangle.$$

We then substitute $\rho_{\text{DM}} = M^2 \lambda^2$ and $\delta\rho_{\text{DM}} = 2M^2 \lambda \delta\lambda$ (these are the values obtained upon averaging over oscillations). The background value $\lambda$ and the perturbation $\delta\lambda_{\text{iso}}$ both redshift with time as $\lambda$, $\delta\lambda_{\text{iso}} \propto \frac{1}{a^{3/2}}$. Important is that their ratio remains constant. Hence, the power spectrum is also constant, and is defined by the value of $\lambda$ and the Hubble parameter during inflationary stage,

$$P_{S_{\text{DM}}} = \frac{H^2}{\pi^2 \lambda^2} \qquad (\text{A1})$$
The allowed amount of isocurvature perturbations uncorrelated with adiabatic fluctuations is limited by the Planck data as $\frac{\mathcal{P}_{\text{SDM}}}{\mathcal{P}_{\text{R}}} < 0.038$ [19], where $\mathcal{P}_{\text{R}} = \frac{H^2}{\pi \epsilon M_{\text{Pl}}^2}$ is the spectrum of adiabatic perturbations and $\epsilon$ is the slow roll parameter. Thus the amplitude $\lambda$ is bounded as

$$ \lambda \gtrsim 10 M_{\text{Pl}} \sqrt{\epsilon}. \qquad (\text{A2}) $$

On the other hand, the requirement that the right amount of DM is produced fixes the amplitude $\lambda$ at the onset of oscillations,

$$ M^2 \lambda^2 \cdot \frac{T_{\text{eq}}^3}{T_{\text{osc}}^3} \sim \rho_{\text{rad}}(t_{\text{eq}}), $$

where the radiation energy density is given by

$$ \rho_{\text{rad}}(t_{\text{eq}}) = \frac{g_{*}(T_{\text{eq}}) \pi^2 T_{\text{eq}}^4}{30}. $$

Here “eq” denotes the matter-radiation equality, $T_{\text{eq}}$ is the corresponding temperature, $T_{\text{eq}} \simeq 1$ eV, and $g_{*}$ is the number of ultrarelativistic d.o.f., $g_{*}(T_{\text{eq}}) \simeq 10$; $T_{\text{osc}}$ is the temperature of the Universe at the time, when the scalar field starts to oscillate. This occurs, when the Hubble rate is of the order of the mass $M$, i.e., $H(T_{\text{osc}}) \sim M$. Hence, $T_{\text{osc}} \simeq \frac{\sqrt{MM_{\text{Pl}}}}{g_{*}^{1/4}}$, where we make use of the relation between the Hubble rate and temperature at the radiation dominated stage, $H(T) \approx 1.66 \sqrt{g_{*}(T)} \frac{T^2}{M_{\text{Pl}}}$. We conclude with the following estimate for the field $\Psi$ amplitude at the onset of oscillations$^4$:

$$ \lambda \approx \frac{\pi}{\sqrt{30}} \frac{\sqrt{T_{\text{eq}} \cdot M_{\text{Pl}}}}{g_{*}^{3/8}} \left(\frac{M_{\text{Pl}}}{M}\right)^{1/4}. \qquad (\text{A3}) $$

Together with the constraint (A2), this severely limits the range of allowed masses $M$, i.e., $M \lesssim \frac{10^{-34}}{3\epsilon^2} \text{eV}$. In particular, for $M \gtrsim 10^{-21}$ eV, one gets $\epsilon \lesssim 2 \times 10^{-7}$. While these are the crude estimates, the message is clear: DM production is possible only for very light masses $M$ and at inflation with the low energy scale.

## 2. General case

Now let us switch to the case when the light complex scalar interacts with the inflaton through the phase. Again, as in the noninteracting case considered above, the amplitude $\lambda$ slowly rolls (or ends up slowly rolling) the effective potential (11); see Fig. 5. This fact alone sets the lower bound on the physically relevant values of the amplitude $\lambda$. Let us show this explicitly. In the slow roll regime, the equation for the amplitude simplifies to

$$ 3H\dot{\lambda} = \frac{Q^2}{\lambda^3} - M^2\lambda, \qquad (\text{A4}) $$

<figure><img src="image_7.png" /><figcaption>FIG. 5. Evolution of the field $\lambda$ is shown in the small mass regime $M \ll H$. We have chosen $\frac{M}{H} = 0.1$. The approximation of the de Sitter space-time is used. Independently of the initial value $\lambda_i$, the amplitude $\lambda$ is always in the slow roll regime during inflation (green and orange lines) or quickly ends up slowly rolling its effective potential (blue line).</figcaption></figure>

where the Noether charge density is still given by Eq. (4). If $\lambda \gg \bar{\lambda}$, the first term on the right-hand side is negligible, and the dynamics of the field $\lambda$ is determined by the mass term. In that case, the slow roll of the amplitude is guaranteed automatically, since we assume $M \ll H$. Therefore, we will focus on small field values $\lambda \ll \bar{\lambda}$ and neglect the second term on the right-hand side of Eq. (A4). In the exact de Sitter space-time approximation the solution of Eq. (A4) reads

$$ \lambda = \left( \frac{4Q^2}{3H} t + C \right)^{1/4}, \qquad (\text{A5}) $$

where $C$ is the integration constant. The slow roll approximation is valid provided that the following condition is fulfilled:

$$ \left| \frac{\ddot{\lambda}}{3H\dot{\lambda}} \right| \ll 1. $$

Once this condition is satisfied at some time $t$, it becomes progressively better later on. Substituting the solution (A5), we can rewrite the latter inequality as

$$ \frac{Q^2}{H^2} \ll \frac{4Q^2}{3H} t + C. $$

If $\frac{Q^2}{H} t \gg C$, one gets $Ht \gg 1$. Substituting this condition into the solution (A5), we obtain the desired bound

$$ \sqrt{\frac{Q}{H}} \ll \lambda. \qquad (\text{A6}) $$

One can check that the same inequality follows, if $C \gg \frac{Q^2}{H} t$. This lower bound translates into the upper bound on the phase velocity,

<footnote>$^4$We ignore the possible difference in the number of ultrarelativistic d.o.f. at the temperatures $T_{\text{osc}}$ and $T_{\text{eq}}$.</footnote>
$$ \dot{\phi} \ll H. \qquad (A7) $$

Following the inflaton decay, the complex scalar (now, a free field) continues the slow roll until the point during the radiation-dominated stage, when $H \sim M$. Since then, it experiences oscillations and cosmologically behaves as dust. The constant $\beta$ can be constrained from the observed relic abundance of DM today and the inequality (A6). The result reads

$$ \beta \ll \frac{T_{\text{eq}}}{\sqrt{M \cdot M_{\text{Pl}}}}. $$

Note that the scenario with light DM allows for much larger values of the constant $\beta$. So, for $M \sim 1$ eV, it can be as large as $\beta \sim 10^{-14}$. From the viewpoint of testability of the model, this may be considered as an advantage over the case of the

superheavy DM, where the coupling constant as small as $\beta \sim 10^{-26}$ is required.

Despite this advantage, the case of light DM is plagued by large isocurvature perturbations. Indeed, the latter for the field $\lambda$ are determined by the effective mass (29). Given the upper bound (A7) and our assumption $M \ll H$, one gets $M_{\text{eff}} \ll H$. Consequently, isocurvature perturbations of the field $\lambda$ are those of the massless scalar field, and can be estimated by $\delta\lambda_{\text{iso}} \approx \frac{H}{2\pi}$. It is then straightforward to show that the spectrum of isocurvature perturbations can be estimated by Eq. (A1). The rest of the estimates is also the same as in the noninteracting case considered in the first part of the Appendix. To conclude, the case of light DM is excluded by the current cosmological observations, unless the scale of inflation and/or the mass $M$ are very low.

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