Non-perturbative approach for scalar particle production in Higgs-$R^{2}$ inflation

Cosmic inflation is the cornerstone of modern cosmology, providing a framework to understand the early universe and predict its fundamental characteristics. It describes a period of accelerated expansion driven by a scalar field $\phi$ slow-rolling towards the minimum of its potential $V(\phi)$. This mechanism addresses classical problems of the standard cosmological model and has become a central paradigm in cosmology [1, 2]. During the inflationary epoch, significant particle production occurs, especially as the kinetic energy of the inflaton field approaches its potential energy. This particle production can arise from gravitational effects due to the universe’s expansion and the dynamics of the inflaton field, serving as a classical background for propagating quantum fields [3, 4]. Known as gravitational particle production, this is a fundamental mechanism in quantum field theory in curved spacetime, offering insights into the origins of dark matter and the preheating and reheating epochs following inflation [5]. Several scenarios have been proposed for vacuum particle production during the early universe, particularly during inflation due to the evolution of the inflaton field and the universe’s expansion. Typically, the inflaton field is coupled to matter fields (bosonic, fermionic, or gauge fields) by a coupling constant $g$. At a critical value $\phi=\phi_{*}$ of the inflaton field, resonance begins and particle production is instantaneous [6, 7]. However, matter fields need not be directly coupled with the inflaton or standard model particles. Massive particles can be produced purely through gravitational effects, either via direct couplings with the gravitational field or induced couplings through a conformal transformation. If the matter field’s energy density contribution is negligible, the inflationary dynamics remain unaffected, leading to no back-reaction effects on spacetime geometry. Such fields are called spectator fields, and this mechanism provides a natural explanation for the origin of dark matter [8, 9, 10, 11, 12, 13]. Beyond dark matter origins, gravitational particle production is linked to significant early universe phenomena such as preheating after inflation [14, 15, 16, 17], primordial gravitational waves [18, 19, 20] and primordial magnetogenesis [21, 22, 23, 24]. Additionally, it imprints significant observable effects on the Cosmic Microwave Background (CMB), particularly in the generation of primordial non-Gaussianities [25, 26, 27].

In this work, we study gravitational particle production during and immediately after inflation driven by the SM Higgs boson and the scalaron field in the Higgs-$R^{2}$ inflation model. This model of inflation extends the SM Higgs inflationary model [28] to address the problem of large coupling [29, 30]. Here, cosmic inflation is driven by both the Higgs field and the extra scalar degree of freedom from the quadratic $R$ term, known as the scalaron. Unlike other multi-field inflation models, the Higgs-$R^{2}$ model effectively reduces to a single-field model during inflation, with the Higgs field acting as an auxiliary field while the scalaron drives inflation [31]. At the end of inflation during the reheating phase, both fields and the Ricci scalar oscillate around the potential’s minimum. This oscillatory behaviour enhance the scalar particle production through the non-perturbative mechanisms as tachyonic inestabilities [32, 33].

We propose an extension to the Higgs-$R^{2}$ model by introducing a massive scalar field $\chi$ as a spectator field that it not contributing to the energy density during the inflationary epoch with negligible interactions with the SM but considering a direct coupling to Ricci scalar via a non-minimal coupling term $\xi_{\chi}R\,\chi^{2}$. Initially, there is no explicit coupling between the inflationary sector and the spectator field in the Jordan frame; however, gravitational interactions between the scalaron, the Higgs field and free scalars induce couplings when we change the conformal frame. This process results in gravitational particle production with significant implications for preheating after inflation and background dynamics. Induced coupling in the Einstein frame between free scalars and scalar sector of the action provides an excellent channel of scalaron decay into light scalars. However, in the non-perturbative approach the field mass $m_{\chi}$ does not necessarily have to be light and we can explore the possibility of scalar particle production for spectator field masses $m_{\chi}>M/2$ where $M\simeq 1.3\times 10^{-5}\,M_{p}$ is the scalaron mass. We take into account the oscillatory behavior of background dynamics after inflation as a classical background, which results in particle production due to the time-dependent background inducing a time-dependent frequency in the mode equations for the spectator field $\chi$. To achieve this, we use the Bogoliubov approach [3] to compute the comoving number density $n_{k}(t)$ of particles created by the background dynamics in Higgs-$R^{2}$ inflation by solving the mode equations for the quantized modes $\chi$.

The article is organized as follows: section 2 provides a brief introduction to the Higgs-$R^{2}$ model and the background dynamics during and after inflation for a set of suitable initial conditions. Section 3 analyzes the couplings induced by gravitational effects in the Einstein frame, showing the decay ratio of scalaron into light free scalars. This result indicates that the perturbative approach is kinematically valid only for masses $m_{\chi}<M/2$ during the reheating epoch after inflation when $|R|<M$. In section 4 we derive the mode equation for the spectator field, which is necessary to investigate the behaviour of the time-dependent frequency during and after inflation. We adapt a numerical approach to solve the mode equation for modes that leave $5$ e-folds before the horizon crossing during inflation, which ensures that each mode is within the horizon at the start of the numerical simulation.

Section 4 focuses on massive particle production during preheating. Finally, section 5 discusses the results and conclusions. In the following, we will use natural units in wihch $\hbar=1$, $c=1$, $\kappa_{\mathrm{B}}=1$ and we use the mass Planck as $8\pi G=M_{p}^{-2}$. We also work in the context of a spatially flat, homogeneous and isotropic universe described by the FLRW metric

where $t$ is the cosmic time, $\tau$ the conformal time and the relation between the two is $\mathrm{d}\tau=\mathrm{d}t/a(t)$ with $a(t)$ is the scale factor.

In this section, we will give a brief review of the Higgs-$R^{2}$ model in both Einstein and Jordan frames. For further details, we recommend the main references [31, 34, 30, 29] and the references therein. Let us get started the discussion in the Jordan frame; the model is giving by the action

where $h$ is the Higgs field in the unitary gauge, the subindex $J$ denotes variables defined in the Jordan frame, $\xi_{h}$ is the constant coupling between Higgs field and the gravitational sector, $\xi_{s}$ is a mass parameter and $\lambda\sim 10^{-3}$ is the quartic self constant coupling of the Higgs field. This model contains a quadratic term in $R_{J}$ described by the constant $\xi_{s}$, wihch comes from the $R^{2}$ model [35] . Such constant is related to the scalaron mass as

where $M\simeq 1.3\times 10^{-5}M_{p}$ in order to be consistent with the value of the amplitude of primordial perturbations $A_{\zeta}$ [36]. The model (2) can be regarded as a UV extension of Higgs inflation, so it is interesting to make a double interpretation of the gravitational coupling in this conformal frame. In Higgs inflation, the constant coupling $\xi_{h}$ must fulfill the condition $\xi_{h}^{2}/\lambda\sim 10^{10}$ in order to be consistent with CMB observations [28, 37]. This causes that the cutoff scale of the model to be below the natural energy scale (Planck scale) during inflation, making the predictions of the model questionable; however, it has been shown that including the Ricci scalar cuadratic term modifies the value of the energy scale, rasinig it up to the Planck scale. To solve the strong copuling problem present in the scalar sector, the parameter $\xi_{s}$ has to obey the inequality [29]

This inequality allow us to make a perturbative treatment of the model. In order to describe the inflationary dynamics of the model, it is more convenient to bring the action (2) into the canonical Einstein-Hilbert form. So that we need to carry out a conformal transformation $\Omega^{2}(x)$ introducing an extra scalar degree of freedom $\phi$ called scalaron, which comes from the quadratic term in $R_{J}$. Thus, to go from Jordan frame to Eintein frame one has to perform a conformal transformation of the space-time metric $g_{\mu\nu}\to\Psi^{2}(x)g_{\mu\nu}$ with $\Psi(x)>0$, where the correct form of it is

where $\alpha=\sqrt{2/3}M_{p}^{-1}$. The conformal transformation (5) removes the quadratic term of the scalar curvature $R_{J}$ introducing the scalaron $\phi$. Then in the Einstein frame the Higgs-$R^{2}$ model of inflation takes the form of a two-sacalar fields model with non-canonical kinetic term

which can be wirtten in the general action for multi-field inflationary models

where in our case $\phi^{I}=(\phi\,,h)$ and $G_{IJ}$ is the non-flat field-space metric given by

We can see that the non-canonical kinetic term is associated with the geometry of the field space via the metric $G_{IJ}$ [38]. On the other hand, the potential in the Einstein frame has interaction terms between the Higgs field $h$ and the scalaron $\phi$, and takes the form

The potential $V(\phi^{I})$ depends on both the $\phi$ and $h$, and mixes terms which make more difficult analyze analytically the background trayectories. Nevertheless, during inflation we can make an effective approximation of the model such that inflation is driven by the single scalar field $\phi$ [31], and with the following interesting features

1. 1.

   $V(\phi^{I})$ shows two valleys and one ridge

2. 2.

   For $\xi_{h}$, $\xi_{s}$ possitive and enough large, the potential presents an attractor behavior

3. 3.

   If $\phi\gg M_{p}$ with arbitrary initial conditions, the fields $\phi$ and $h$ will always evolve towards one of the valleys and thus beginning the slow-roll evolution

4. 4.

   If $\phi<<M_{p}$, the potential is reduced to Higgs potential

   $\displaystyle V(\phi\ll M_{p}\,,h)\simeq V(h)=\dfrac{\lambda_{\mathrm{eff}}}{4}\,h^{4}\,,$

   (10)

   where $\lambda_{\mathrm{eff}}$ is the effective quartic self-coupling constant defined by

   $\displaystyle\lambda_{\mathrm{eff}}=\lambda+\dfrac{\xi_{h}^{2}}{\xi_{s}}\,.$

   (11)

   The condition $\lambda_{\mathrm{eff}}\lesssim 4\pi$ is neccessary for perturbative preheating

5. 5.

   If $h\ll M_{p}$, the potential is reduced to Starobinsky potential

   $\displaystyle V(\phi\,,h\ll M_{p})\simeq\dfrac{M_{p}^{4}}{4\xi_{s}}(1-e^{-\alpha\phi})^{2}\,.$

   (12)

We can see the positions of the valleys and the hill at $h=0$ in the figure 1. The inflationary dynamics ocurr along one of these valleys, showing a universal attractor behavior, so that inevitably the fields $\phi$ and $h$ will always fall into one of the valleys and then drive slow-roll inflation. To find the valleys, we can perform the analysis in both Einstein [31] and Jordan [39] frames. In the Einstein frame in the limit $\phi\gg M_{p}$ (it is required for enough long inflation) and $h\gg v$, where $v=246\,\mathrm{GeV}$, the valleys are found out by the condition

which implies that the Higgs field is completely determined by the inflaton field $\phi$

Because of the potential (9) is symmetric respect to transformation $h\to-h$, the two valleys are localizated by the condition (14). Thus, this shows that the Higgs field is an auxiliary field during inflation, so if we put the result (14) back into the Einstein potential (9), we give the effective potential along the valleys

Notice that the effective potential reduces to Higgs inflation if $\lambda\xi_{s}\ll\xi_{h}^{2}$ while one get $R^{2}$ inflation for $\lambda\xi_{s}\gg\xi_{h}^{2}$. In order to provide the correct amplitude of primordial scalar perturbations, the amplitude of (15) has to fulfill [40]

where $N_{*}\sim 50-60$ is the number of e-folds before the end of inflation, and $A_{\zeta}\sim 2\times 10^{-9}$ is the amplitude of the primordial power spectrum. For $\lambda=0.013$ and $\xi_{h}\,,\xi_{s}>0$ which satisfy (16), the inflationary dynamics is effectively the same as single-field slow-roll inflation. In this paper we will only analyze the standard parameter region $\xi_{h}\,,\xi_{s}\,,\lambda>0$. Specifically, if we have $\xi_{h}\gg 0$ the isocurvature effects of the model are absent and the bending is whithin the weak mixing case $\eta_{\perp}\to 0$ [31], and the isocurvature and adiabatic modes are not coupled so that they evolve independetly in the standard parameter region of Higgs-$R^{2}$ model. Therefore, the description of the dynamics inside the valleys is an effective single-field inflation.

In this section we study spectator fields present in the Higgs-$R^{2}$ model such as free scalar fields, initial assuming no direct coupling between the matter fields and the inflationary sector in the Jordan frame. In other words, we get started in the Jordan frame and after conformal transformation, the matter sector is gravitationally coupled to the inflatonary sector. In the Jordan frame our model is giving by

where $\psi_{i}$ are the different matter fields that are present during and after inflation. In this work we will consider a massive spectator scalar field $\chi$ that only interacts gravitationally. The action of such field is giving by

The dimensionless parameter $\xi_{\chi}$ represents the non-minimal coupling to gravity sector and may take any value; however, studies based on electroweak vacuum stability [41] suggest that the value of non-minimal coupling constant should not exceed the value $10$. Moreover, there are two interest special cases: $\xi_{\chi}=0$ and $\xi_{\chi}=1/6$. The value $\xi_{\chi}=0$ is called minimal coupling case and the value $\xi_{\chi}=1/6$ is called conformal coupling. The first case corresponds to a free scalar field theory and the second one ensures the conformal invariance of the model if in addition $m_{\chi}=0$. Although there are no direct coupling between matter fields and the inflatonary sector in the first case, both scalaron $\phi$ and Higgs field $h$ gravitationally interact with every matter field is present during inflation and after inflation, and may occur gravitational particle production process. In the second one, free scalar $\chi$ is coupled to the Ricci scalar and the particle production may also ocurr as long as the matter fields are not conformally invariant, namely, if the scalar fied is massive $m_{\chi}\neq 0$. These kind of models which in the scalar fields interacts only gravitationally is specially interesting since it can explain the particle production process, with the gravitational field responsible for creating $\chi$ particles. This process of particle creation due to gravitational interactions has implications in the reheating process [17, 42, 43] and gravitational dark matter production [44, 45, 46, 12, 11, 9, 47].

What follows is to perform the conformal transformation (5) to the action (30) to change to the Einstein frame. Since $\chi$ is a spectator field, it remains energetically subdominant, $\rho_{\chi}\ll 3H^{2}M_{p}^{2}$, throughout the entire background evolution. Therefore, we can apply the same conformal transformation of the metric (5) as in the previous section. By doing so, we avoid considering back-reaction effects on the space-time geometry and the background dynamics. In the Einstein frame, we obtain the following action

where $\chi\to\tilde{\chi}=\Psi^{-1}\chi$ is the re-scaled scalar field and the Ricci scalar in the Einstein frame is given by $R=6(\dot{H}+2H^{2})$. We can perform another conformal transformation which involve the scalar field $\tilde{\chi}$ leading to another Einstein frame in order to remove the non-minimal coupling constant $\xi_{\chi}$. However, we are only interested in the situation where $\chi$ is an spectator field and both scalaron field $\phi$ and Higgs field are responsible of the dynamics during and after inflation. One can see that the conformal transformation (5) induces interaction terms between the scalaron $\phi$, the Higgs field $h$ and the free scalar $\tilde{\chi}$. We can identify the interaction terms from the following action

This action describes the interaction between the background fields $\phi$ and $h$ with the massive spectator field $\tilde{\chi}$. We can see that even with $\xi_{\chi}=0$ this interaction does not disappear, unless the field $\tilde{\chi}$ is conformal invariant ($m_{\chi}=0\,,\xi_{\chi}=1/6$). One can see this fact from the decay ratio of the scalaron into a pair of light scalar particles during reheating epoch [8], [48]

If $\xi_{\chi}=1/6$ and $m_{\chi}\to 0$, the decay ratio vanishes. In addition, if the mass of scalar field $\chi$ is $m_{\chi}>M/2$ where $M$ is scalaron mass term, the scalaron decay into a pair of $\chi$ particles is forbidden. Then, the perturbative production of $\chi$ particles is allowed only in the case of light scalars with mass $m_{\chi}<M/2$. However, it can also occur a scalar particle production by non perturbative effects by time-dependent background dynamics of $\phi$, $h$ and $a$ for a spectator field with mass $m_{\chi}>M/2$. In order to study this process it is necessary to find the equation of motion for the spectator field $\tilde{\chi}$ and track the mode evolution during and after inflation. Varying the action (31) respect to $\tilde{\chi}$ yields the following equation of motion

Notice that the background dynamics is explicit in the equation of the spectator field $\tilde{\chi}$ due to the conformal transformation. This induces a time-dependent effective mass term, $m_{\mathrm{eff}}^{2}(t),$ for the spectator field $\tilde{\chi}$. Consequently, in the Einstein frame, the model presents an interaction term with the dynamics of the classical background through the Ricci scalar, the scale factor $a$, and the dynamics of the fields $\phi$ and $h$. As a result, studying the solutions is generally highly complex from an analytical perspective; hence a numerical treatment of the background dynamics is necessary for a comprehensive analysis of mode evolution, specially after inflation when both fields $\phi$ and $h$ exhibit a strongly interacting anharmonic oscillatory behavior, as one can see in the figure 2. By numerically determining the background evolution of the scalaron, the Higgs field, and the scale factor, we can compute the Ricci scalar using $R=6(\dot{H}+2H^{2})$, ensuring that the dynamics are well-determined. However, during effective inflation along one of the potential’s valleys, we can approximate the background dynamics using the single-field equation of motion (21).

In this section, we numerically solve the equation of motion (34) along with the background equations (17), (18), (I) to obtain the mode functions $\chi$ for calculating the number density of particles produced, $n_{k}$. The equation of motion for $\tilde{\chi}$ considering a FLRW universe (I) can be written in terms of the cosmic time $t$ as

where $\Omega^{2}(t)$ is a background variables dependent and is given by

Rescaling the field $\tilde{\chi}\to a^{-3/2}(t)\,\tilde{\chi}(t,\mathbf{r})$ and using the background equations of motion (17), (20) the equation for the scalar field $\tilde{\chi}$ becomes a oscillator-like equation with time-dependent frequency

where $\omega^{2}(t)$ is given by

Particle production in the early universe is a process of quantum field theory in curved spacetime, where the gravitational field is not quantized, but the free scalar field $\chi$ is quantized. Therefore, the production of $\chi$ particles is a quantum effect [3, 4, 5]. We proceed by quantizing the spectator field in the Heisenberg picture so that the $\chi$ field and its conjugate momentum $\pi_{\chi}=\dot{\chi}$ are treated as field operators. In Fourier space the rescaled field can be written as

where $\mathbf{k}$ is the comoving wave number and $\hat{a}_{\mathbf{k}}$, $\hat{a}^{\dagger}_{\mathbf{k}}$ are the creation and annihilation operators respectively, which obey the standard commutation relations at equal times $[a_{\mathbf{k}}\,,a^{\dagger}_{\mathbf{k^{\prime}}}]=(2\pi)^{3}\delta^{(3)}(\mathbf{k}-\mathbf{k}^{\prime})$ and $[a_{\mathbf{k}}\,,a_{\mathbf{k^{\prime}}}]=[a^{\dagger}_{\mathbf{k}}\,,a^{\dagger}_{\mathbf{k^{\prime}}}]=0$. The field equation for Fourier mode $X_{k}(t)$ is given by

with

and the following effective mass term

The equation for $X_{k}$ describes an oscillator with time-dependent frequency $\omega_{k}^{2}(t)$, hence to track the evolution of the modes $X_{k}$ during and after inflation, it is necessary to analyze $\omega_{k}^{2}(t)$. One can see that during the quasi De Sitter phase during inflation when the effective single field inflation is valid, the effective mass is simplified because of $\dot{H}\to 0$. In this case one can see that the effective mass takes the form

We can see that the mass term is exponentially suppressed during inflation when $\phi\gg M_{p}$, making the mass of the spectator field much smaller than the Hubble parameter at the end of inflation, $m_{\chi}\ll H_{\mathrm{end}}$. However, after inflation, when the scalaron takes values $\phi\ll M_{p}$, the free scalar $\tilde{\chi}$ becomes superheavy, $m_{\chi}\gg H_{\mathrm{end}}$, due to the exponential dependence on the scalaron $\phi$. Then at the beginning of inflation, when $t\ll t_{\mathrm{end}}$ the Ricci scalar is $R\simeq 12H^{2}$ and the dominant terms in (41) are $k^{2}/a^{2}$, $H^{2}$ and $12\xi_{\chi}H^{2}$. However, towards the end of inflation, the term $\dot{H}$ cannot be neglected, as after inflation, $\dot{H}$ and the background oscillations dominate the dynamics. The term $V_{\phi}$ regulates the oscillations of the effective mass after inflation, but it is small compared to $H^{2}$ and $\dot{H}$. The contribution of each term depends on the moment at which it is considered. The mass term benefits from exponential growth when $\phi\ll M_{p}$, as we anticipated earlier. Therefore, after inflation, the dominant terms are the mass term and $\dot{H}$. Additionally, if the coupling constant is $\xi_{\chi}<1/6$, the effective mass could be negative during inflation. In such a case, the frequency $\omega_{k}^{2}(t)$ can become negative for super-horizon modes $k\ll aH$, leading to a exponential mode growth $\tilde{\chi}\sim e^{|\omega_{k}|t}$ during inflation due to tachyonic instability. This regime may result in a very efficient amplification of particle production [42, 33].

However, for large enough modes $k$, namely

have positive frequency $\omega_{k}^{2}(t)>0$ even if $m_{\mathrm{eff}}^{2}<0$, which corresponds to sub-horizon regime $k\gg aH$. This occurs when the modes are well inside the horizon at the beginning of inflation. In the figure 3 we show the evolution of $\omega_{k}^{2}(t)$ for a non-minimal coupling $\xi=0$. Before $t_{\mathrm{cross}}$, the frequency is dominated by the decay of the physical mode $k/a\sim e^{-Ht}$. Once the mode crosses the horizon $(k\ll aH)$, $\omega_{k}^{2}(t)$ remains almost constant, and its value is negative since the coefficient of the dominant term $H^{2}$ is negative. For $t\gtrsim t_{\mathrm{end}}$, the frequency increases and oscillates after inflation as the background terms oscillate around the minimum of the potential (preheating). One can also observe growth due to the exponential dependence of the scalaron in the mass term toward the end of inflation.

To numerically solve equation (40) along with the background equations, we must specify suitable intial conditions that ensure the modes are within the horizon at early times. In the asymptotic limit $t\to 0$ and according to the behavior of $\omega_{k}^{2}(t)$ at the beginning of inflation, the dominant term is $k/a$. In this limit, the modes are sub-horizon $k\gg aH$, and the modes exhibit Minkowski-like behavior. This motivates choosing the Bunch-Davies vacuum as the initial condition [14]

As previously mentioned, for certain values of $\xi_{\chi}$, the modes experience tachyonic excitations during the time they are outside the horizon. Therefore, to ensure that each mode is within the horizon and to use $\eqref{BD vacuum}$ as the initial condition, we must track the evolution of the corresponding mode at times $t_{\mathrm{ini}}$ when the mode is inside the horizon. For concreteness, we assume that the Planck pivot scale $k_{*}$ leaves the horizon $N_{*}=60$ e-folds before the end of inflation and a total duration of inflation of $N_{\mathrm{tot}}=76.150$ e-folds. We also take the initial conditions to be the values of $X_{k}$ and the background variables 5 e-folds before the mode crosses the horizon of that mode. This guarantees that the initial conditions for the modes $X_{k}(t)$ and their derivatives $\dot{X}_{k}(t)$ can then be safely taken to be of Bunch-Davies type.

Once we have established the behavior of mode functions $X_{k}$ during and after inflation for some values of spectator mass, the non-minimal coupling $\xi_{\chi}$ and comoving wave number $k$, it is turn to analyze the non-perturbative particle production. The production of $\chi$ particles is due to the time dependence of the background variables, in particular the non-adiabatic behaviour of the frequency $\omega_{k}(t)$ that induces a mix among negative- and positive-frequency [3]. The adiabaticity condition is given by the adiabatic paramater $A_{k}$ defined as

The adiabatic evolution of $\omega_{k}(t)$ under the WKB approximation of the modes $X_{k}(t)$ guarantees that the positive-frequency mode at early times evolves into positive-frequency modes at late times. That is, for asymptotically early times $t\to 0$, the modes $X_{k}(t)$ evolve as [49]

This behavior usually happens at early times $t_{i}\to 0$ and at sufficiently late times $t_{f}\to+\infty$, but for intermediate times $t_{i}<t_{*}<t_{f}$ during the expansion of the universe the evolution of the modes is no longer adiabatic $|A_{k}|\gg 1$. During inflation, the positive-frequency solution corresponds to the Bunch-Davies vacuum (45), as described in the previous section. This evolution remains valid as long as the parameter $|A_{k}|\ll 1$ at all times. While this condition is met, no particle production occurs, as adiabatic evolution is ensured, and there is no mixing of negative- and positive-frequency modes. From figure 3, we can see that the evolution of $\omega_{k}(t)$ is smooth until the end of inflation. At that point, the condition $|A_{k}|\ll 1$ is strongly violated, leading to an enhancement in particle production and gravitational particle production is enhanced at the time $t_{*}$, which corresponds to the end of inflation. We show the non-adiabaticity evolution in the figure 4 parametrized by the adiabatic parameter $|A_{k}(t)|$. The mixture of positive and negative frequency modes at time $t_{\mathrm{end}}$ can be represented as the sum of the solutions $\exp(\pm i\int^{t}\omega_{k}(t^{\prime})\mathrm{d}t^{\prime})$ using the WKB approximation [14, 5]

where $\alpha_{k}(t)$ and $\beta_{k}(t)$ are the time-dependent Bogolyubov coefficients. Comparing with the Bunch-Davies initial condition (47), one see thats $\alpha_{k}(t_{0})=1$ and $\beta_{k}(t_{0})=0$. The comoving particle number density $n_{\chi}(t)$ is related to the coefficient $\beta_{k}$ of the Bogolyubov transformations which relate the vacuum states at the asymptotic limits $t\to 0$ and $t\to\infty$, and coincides with $\beta_{k}(t)$ at late times. We can compute $\beta_{k}(t)$ by solving (40) and evaluating

The physical number density $n_{\chi}(t)$ in the late-time limit $t\to\infty$ is given by

where $n_{k}(t)$ is the comoving number density of particles with comoving wavenumber $k$.

In this study, we investigated the gravitational and non-perturbative production of free scalar particles, $\chi$, in Higgs-$R^{2}$ inflation. We began with a free spectator field, $\chi$, interacting solely through the gravitational term $\xi_{\chi}R$ in the Jordan frame. By applying a conformal transformation to transition into the Einstein frame, direct couplings between the $\chi$ field and the scalaron $\phi$ emerged. Within the context of inflationary dynamics in the Einstein frame, we derived the equation of motion for the $\chi$ field, taking into account the non-minimal coupling with R. We numerically solved this system of equations, imposing Bunch-Davies conditions on the $\chi$ field 5 e-folds before the modes crossed the horizon, to determine the mode functions of the $\chi$ field. From these solutions, we extracted the Bogolyubov coefficient $\beta_{k}(t)$ to calculate the comoving particle density and visualize their spectrum. We explored two distinct scenarios for the non-minimal coupling $\xi_{\chi}$, considering different mass ranges. For $\xi_{\chi}=1/6$ and light masses, we observed efficient particle production at the end of inflation due to the term $e^{-\alpha\phi}m_{\chi}^{2}$, which becomes dominant at that stage. The resulting power spectrum exhibited a blue-tilted behavior, indicating that particle production was dominated by modes near the Hubble radius $(aH)^{-1}$ at the end of inflation, while shorter wavelength modes were suppressed.

In the case of $\xi_{\chi}=1/6$ and for masses close to $m_{\chi}\sim M/2$, the spectrum tends to flatten out without exhibiting a peak in the UV region. This phenomenon is possibly due to the mass term that depends exponentially on the scalaron $\phi$, causing the effective mass to become superheavy at the end of inflation. In contrast, for $\xi_{\chi}=0$, the spectrum remained nearly scale-invariant for masses $m_{\chi}\gtrsim M/2$, with a blue tilt emerging for larger masses. For light masses and $\xi_{\chi}=0$, we find that the comoving number of particles created is significantly higher compared to the case of conformally coupled particles, which is due to the tachyonic growth of the modes $X_{k}(t)$. We have not attempted a detailed exploration of the parameter space of the Higgs-$R^{2}$ inflation model or the parameters of the scalar field $\chi$. Instead, we have taken the Higgs-$R^{2}$ inflation model, whose observables are consistent with Planck, and considered the possibility of non-perturbative particle production through the direct couplings that arise between the $\chi$ field and the inflationary sector when changing the conformal frame. We found that inflation driven by the Higgs field and the scalaron can produce scalar particles in the cases $\xi_{\chi}=0$ and $\xi_{\chi}=1/6$ and for a specific range of masses. This field could potentially contribute to the abundance of dark matter, for which its current abundance would need to be calculated.

There remains much to be done for future work, such as considering the production of spin-$1/2$ and spin-$1$ dark matter, the production of gauge bosons $W^{\pm}_{\mu}$, $Z_{\mu}$, as well as the implications on primordial non-Gaussianities in the CMB induced by particle production. It would also be interesting to study the isocurvature constraints for light scalar field masses in this model. We leave these intriguing topics for future studies.

We are very grateful to Dr. Marcos Garcia for his insightful comments and input, which made it possible to write the code for this article. We would also like to acknowledge Dr. Andrew Long for providing an example of numerical code. The work of F.P. has been sponsored by CONACHyT-Mexico through a doctoral scholarship. This work was also partially supported by CONACHyT-Mexico through the support CBF2023-2024-1937.