Double inflation via non-minimally coupled spectator

The Universe is believed to have experienced a period of nearly exponential expansion, dubbed inflation, in its very early stage. This inflationary period not only helps address the flatness and horizon problems but also generically provides a mechanism to generate primordial density fluctuations necessary for structure formation. A simple possibility is that a single scalar field, called inflaton, is responsible for both driving the inflationary expansion and also generating the primordial fluctuations. Since the nature of the primordial fluctuations depends on models of the inflaton, cosmological observations such as the cosmic microwave background (CMB) can probe these models, and indeed, data from the Planck satellite in combination with other observations, especially CMB B-mode polarization experiments such as BICEP/Keck, provide constraints on these models [1, 2].

Though having only a single field responsible for inflation is attractive, the simplest inflationary potentials are now severely constrained by cosmological data and many have been ruled out. Therefore multi-field models have also been extensively discussed in the literature. These include cases where the inflaton is only responsible for the inflationary expansion while another scalar field, whose energy density is negligible during inflation, and irrelevant to the inflationary dynamics (referred to as a spectator field), is responsible for generating primordial fluctuations. Examples of such models include the curvaton model [3, 4, 5] and modulated reheating scenarios [6, 7]. In high energy theories such as string theory, the existence of multiple scalar fields is generic, which also motivates a multi-field inflation model. In addition, when a spectator field is responsible for primordial fluctuations, the predictions for primordial power spectra are modified and, in some cases, even if the model is excluded by the data as a single-field inflation model, it can be resurrected in the framework of spectator field models; see, e.g., [8, 9, 10, 11, 12, 13, 14] for the curvaton model and [15] for the modulated reheating scenario.

Another extension of a single-field inflation model is to introduce a non-minimal coupling to gravity. A well-known example is the Higgs inflation [16, 17] where the inflaton is identified with the Standard Model Higgs field with non-minimal coupling to gravity.¹¹1 There have been early works along this line. See [18, 19, 20, 21] for such early studies. The potential for the Higgs field is quartic during inflation in the pure Standard Model, and therefore the pure Standard Model Higgs inflation is excluded by observational data. However, by introducing a non-minimal coupling, the model becomes consistent with cosmological observations in terms of the spectral index and the tensor-to-scalar ratio (the suppression of the tensor-to-scalar ratio has already been discussed in [21]). Inflationary models with non-minimal coupling have become an extensively studied research area as they generally predict modified inflationary observables; see, e.g., [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. In addition, non-minimal couplings generically arise radiatively in quantum field theory in curved space-time [35, 36] even if they do not exist at tree level. Therefore inflation models with non-minimal coupling to gravity are well motivated theoretically.

Given that both multi-field inflationary models and non-minimal couplings are well motivated, it is tempting to consider multi-field models with non-minimal couplings; see, e.g., [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] for works along this line. In this paper, we consider a multi-field model in which there exist two scalar fields: a minimally coupled field $\Phi$ and non-minimally coupled one $S$, with $\Phi$ being the inflaton and $S$ a spectator initially. As will be shown in this paper, due to the spectator non-minimal coupling, its amplitude can grow during the inflationary phase driven by the $\Phi$ field until eventually the $S$ field acquires super-Planckian field values, which, in turn, drives a second inflationary phase after the end of the first inflationary epoch driven by the $\Phi$ field.²²2 Even in minimally coupled models, when the spectator field is assumed to have a super-Planckian amplitude, the second inflationary phase can be driven by the spectator field (these kinds of models are sometimes called “inflating curvaton models”) [8, 9, 11, 48, 49].

We also discuss primordial fluctuations in such models. Since isocurvature fluctuations generated by the spectator field can source the adiabatic ones, the existence of the spectator non-minimal coupling can also modify primordial power spectra from its minimally coupled counterpart. In particular, the predictions for the spectral index $n_{s}$ and the tensor-to-scalar ratio $r$ are modified. This issue has already been discussed for single-field non-minimally coupled models (see, e.g., [22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 34]). We show in this paper that the spectator non-minimal coupling can also relax the constraints on $n_{s}$ and $r$ so that even inflation models that are excluded in the single-field minimally coupled framework, such as the quadratic chaotic inflation, can become viable due to the existence of the spectator non-minimal coupling.

The structure of this paper is the following. In the next section, we describe the setup for the multi-field inflation model with the non-minimal coupling we consider in this paper. Then in Section 3, we discuss the background evolution, particularly focusing on the growth of the spectator field during the initial inflationary phase driven by $\Phi$. We also argue that in our setup, a second inflationary phase is somewhat generic due to this growth originating from the existence of the spectator non-minimal coupling. In Section 4, we discuss primordial scalar and tensor power spectra in the model and how they are modified by changing the strength of the non-minimal coupling and the masses of the fields. We also investigate how the inflationary predictions are modified to relax the constraints on inflation models. The final section is devoted to conclusions and discussion.

We consider two scalar fields $\Phi$ and $S$ that play the role of inflaton and spectator, respectively, at the early stages of inflation. For simplicity, we assume that $\Phi$ is minimally coupled to gravity, while $S$ non-minimally at the leading order such that the action in the Jordan frame is given by

$$S_{\rm J}=\int\mathrm{d}^{4}x\sqrt{-g}\left(\frac{1}{2}M_{\rm pl}^{2}f(S)R-\frac{1}{2}\partial_{\mu}\Phi\,\partial^{\mu}\Phi-\frac{1}{2}\partial_{\mu}S\,\partial^{\mu}S-V_{\text{J}}(\Phi,S)\right),$$

(2.1)

where $M_{\rm pl}=1/\sqrt{8\pi G}=2.4\times 10^{18}\,\text{GeV}$ is the reduced Planck mass and $f(S)$ characterizes a non-minimal coupling between $S$ and $R$:

$$f(S)=1+\xi_{S}\left(\frac{S}{M_{\rm pl}}\right)^{2},$$

(2.2)

with $\xi_{S}$ being the non-minimal coupling constant, and $V_{\text{J}}$ is the Jordan frame potential. As a simplest non-trivial example, we assume that the potential takes the following form:

$$V_{\text{J}}(\Phi,S)=\frac{1}{2}m_{\Phi}^{2}\Phi^{2}+\frac{1}{2}m_{S}^{2}S^{2}.$$

(2.3)

To follow the evolution of the scalar fields and the cosmic expansion, we work in the Einstein frame. For a system with multiple scalar fields, one can write the action in the Einstein frame as³³3 For the formalism of a general multi-field inflation, see e.g., [50, 37, 39]. We follow the notation of [50].

$$S=\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}M_{\mathrm{pl}}^{2}R-\frac{1}{2}G_{IJ}\nabla_{\mu}\varphi^{I}\nabla^{\mu}\varphi^{J}-V(\varphi^{I})\right],$$

(2.4)

where the indices run for $I=\Phi,S$ with the abbreviation $\left(\varphi^{\Phi},\varphi^{S}\right)=\left(\Phi,S\right)$ for the field space. Here, $G_{IJ}$ is the field space metric given by

$$G_{\Phi\Phi}=f^{-1},\qquad G_{SS}=f^{-1}+\frac{3}{2}\frac{f_{S}f_{S}}{f^{2}},\qquad G_{\Phi S}=G_{S\Phi}=0,$$

(2.5)

with $f_{S}=\partial f/\partial S=2\xi_{S}S/M_{\mathrm{pl}}^{2}$, and $V(\varphi^{I})$ is the Einstein-frame potential related to the Jordan-frame counterpart as

$$V(\varphi^{I})=\frac{V_{\text{J}}(\varphi^{I})}{f^{2}}.$$

(2.6)

Let us first discuss the background evolution of the fields. We first focus on the case where the non-minimal coupling is small and therefore the field space metric component of the spectator can be approximated as $G_{SS}\simeq f^{-1}$. We can then introduce a canonically normalized spectator field⁴⁴4 The situation for the spectator is analogous to Higgs inflation in Palatini gravity [51, 52, 53, 54].

$$\chi\equiv\frac{M_{\rm pl}}{\sqrt{\xi_{S}}}\,\,{\rm arcsinh}\left(\sqrt{\xi_{S}}\frac{S}{M_{\rm pl}}\right)$$

(3.1)

to aid the intuitive understanding of the dynamics. Though this rescaling introduces some kinetic couplings between the spectator and inflaton fields, in the double inflation regime which we are interested in, such couplings will be unimportant. This is due firstly because of slow-roll dynamics and secondly due to relative smallness of the term $\xi_{S}S^{2}$ necessary to achieve desirable duration of inflation as we shall discuss.

The equation of motion for the inflaton is then given by

$$\ddot{\Phi}+\left[3H-\frac{2\sqrt{\xi_{S}}}{M_{\mathrm{pl}}}\tanh\left(\sqrt{\xi_{S}}M_{\mathrm{pl}}^{-1}\chi\right)\dot{\chi}\right]\dot{\Phi}+\frac{m_{\Phi}^{2}\Phi}{\cosh^{2}\left(\sqrt{\xi_{S}}M_{\mathrm{pl}}^{-1}\chi\right)}=0.$$

(3.2)

The presence of the spectator acts to scale down the effective mass of the inflaton and to introduce an extra anti-friction term. For slow-roll dynamics and moderate values of the spectator amplitude, however, these changes are negligible and the dynamics is virtually that of the usual quadratic chaotic inflation. The equation of motion for the canonical spectator, on the other hand, is

$$\ddot{\chi}+3H\dot{\chi}-\frac{2\sqrt{\xi_{S}}}{M_{\mathrm{pl}}}\frac{\sinh(\sqrt{\xi_{S}}M_{\mathrm{pl}}^{-1}\chi)}{\cosh^{5}(\sqrt{\xi_{S}}M_{\mathrm{pl}}^{-1}\chi)}\left[m_{\Phi}^{2}\Phi^{2}-\frac{m_{S}^{2}M_{\mathrm{pl}}^{2}}{2\xi_{S}}\left(1-\sinh^{2}(\sqrt{\xi_{S}}M_{\mathrm{pl}}^{-1}\chi)\right)\right]=0,$$

(3.3)

where we have neglected the term proportional to $\dot{\Phi}^{2}$ as it is negligible during slow-roll.

We can then deduce the dynamics as follows. Initially, the inflaton field dominates and induces a force that amplifies the spectator field. Then, as the amplitude of $\Phi$ decreases past the threshold

$$\Phi_{\mathrm{turn}}=\frac{M_{\mathrm{pl}}m_{S}}{\sqrt{2\xi_{S}}m_{\Phi}},$$

(3.4)

the force reverses and the spectator is driven back toward zero as long as its amplitude has not exceeded the critical value

$$\chi_{\mathrm{runaway}}=\frac{M_{\mathrm{pl}}}{\sqrt{\xi_{S}}}\operatorname{arcsinh}(1)\qquad\mathrm{or}\qquad S_{\mathrm{runaway}}=\frac{M_{\mathrm{pl}}}{\sqrt{\xi_{S}}}.$$

(3.5)

If this value is exceeded, the spectator is amplified indefinitely.

When the spectator is amplified to a large value, it comes to dominate the energy density, driving the second stage of inflation while it rolls back to zero. Essentially the physics in the regime of interest is described by the effective potential

$$V_{\rm eff}=\frac{\xi_{S}m_{\Phi}^{2}\Phi^{2}+m_{S}^{2}M_{\rm pl}^{2}~{}{\rm sinh}^{2}(\sqrt{\xi_{S}}\chi/M_{\rm pl})}{2\xi_{S}~{}{\rm cosh}^{4}(\sqrt{\xi_{S}}\chi/M_{\rm pl})},$$

(3.6)

which is depicted in Fig. 1: When $\Phi$ is large (upper left direction), $V_{\text{eff}}$ is convex in the direction of $\chi$, causing it to be amplified. At smaller values of $\Phi$ (lower right direction), $V_{\text{eff}}$ becomes concave in the $\chi$ direction, causing it to roll back, and drives the second inflationary stage.

In this section, we discuss the evolution of fluctuations in our setup. First, we briefly summarize how we calculate the perturbations in the model. We use the formalism of [50, 37, 39] for decomposing the fluctuations into adiabatic and entropic components as follows. A scalar field $\varphi^{I}$ can be divided into the background part $\bar{\varphi}^{I}$ and its perturbation $Q^{I}$ as

$$\varphi^{I}=\bar{\varphi}^{I}+Q^{I}\,.$$

(4.1)

We can define a new basis $\{e_{n}^{I}\}~{}(n=\sigma,{s})$ such that the first vector points along the background trajectory, corresponding to the adiabatic mode whose basis vector can be written as

$$e_{\sigma}^{I}\equiv\frac{\dot{\varphi}^{I}}{\sqrt{G_{JK}\dot{\varphi}^{J}\dot{\varphi}^{K}}}=\frac{\dot{\varphi}^{I}}{\dot{\sigma}}\,,$$

(4.2)

where we have defined $\dot{\sigma}\equiv\sqrt{G_{IJ}\dot{\varphi}^{I}\dot{\varphi}^{J}}$. The remaining isocurvature direction is then perpendicular to the background trajectory. The perturbations $Q^{I}$ in the new basis are written as $Q^{I}=Q^{n}e_{n}^{I}$ with $n=\sigma,s$.

The equations of motion for the adiabatic and isocurvature modes $Q_{\sigma}$ and $Q_{s}$ are

	$\displaystyle\ddot{Q}_{\sigma}+3H\dot{Q}_{\sigma}+\left(\frac{k^{2}}{a^{2}}+\mu_{\sigma}^{2}\right)Q_{\sigma}$	$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}\Xi Q_{s}\Big{)}-\left(\frac{\dot{H}}{H}+\frac{V_{,\sigma}}{\dot{\sigma}}\right)\Xi Q_{s}$		(4.3)
and
	$\displaystyle\ddot{Q}_{s}+3H\dot{Q}_{s}+\left(\frac{k^{2}}{a^{2}}+\mu_{s}^{2}\right)Q_{s}$	$\displaystyle=-\Xi\left[\dot{Q}_{\sigma}+\left(\frac{\dot{H}}{H}-\frac{\ddot{\sigma}}{\dot{\sigma}}\right)Q_{\sigma}\right],$		(4.4)

respectively, where $\mu_{\sigma}$, $\mu_{s}$, and $\Xi$ are given by

	$\displaystyle\mu_{\sigma}^{2}$	$\displaystyle=$	$\displaystyle V_{;\sigma\sigma}-\frac{1}{a^{3}}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{a^{3}\dot{\sigma}^{2}}{H}\right)-\frac{V_{,{s}}^{2}}{\dot{\sigma}^{2}},$		(4.5)
	$\displaystyle\mu_{s}^{2}$	$\displaystyle=$	$\displaystyle V_{;{ss}}+\frac{1}{2}\dot{\sigma}^{2}\tilde{\mathcal{R}}-\frac{V_{,{s}}^{2}}{\dot{\sigma}^{2}},$		(4.6)
	$\displaystyle\Xi$	$\displaystyle=$	$\displaystyle-\frac{2V_{,{s}}}{\dot{\sigma}}\,,$		(4.7)

in which $V_{;\sigma\sigma}$ and $V_{;{ss}}$ are defined as projections of $\mathcal{D}_{I}\mathcal{D}_{J}V$ onto the appropriate directions and $\tilde{\mathcal{R}}$ is the Ricci curvature scalar of the field manifold defined by $G_{IJ}$. The comoving curvature perturbation in the flat gauge is given as

$$\mathcal{R}=\frac{H}{\dot{\sigma}}Q_{\sigma}\,.$$

(4.8)

The evolution of ${\cal R}$ can be calculated by solving the equations of motion (4.3) and (4.4) along with the background ones. The primordial (scalar) power spectrum ${\cal P_{R}}$ is computed by evaluating ${\cal R}$ at the end of inflation and given by

$\displaystyle{\cal P}_{\zeta}(k)$

$\displaystyle=$

$\displaystyle\frac{k^{3}}{2\pi^{2}}|{\cal R_{\mathbf{k}}}|^{2}\,,$

(4.9)

while its scale dependence is characterized by the spectral index

$$n_{s}-1\equiv\frac{d\log{\cal P_{\zeta}}}{d\log k}\,.$$

(4.10)

We also discuss primordial gravitational waves generated during inflation. Gravitational waves correspond to the tensor perturbations whose power spectrum can be calculated with the standard procedure (see, e.g., [55]) as

$${\cal P}_{T}(k)=\frac{k^{3}}{2\pi^{2}}|v_{k}|^{2}\,,$$

(4.11)

where $v_{k}$ is the Fourier component of $v_{ij}=M_{\rm pl}h_{ij}/(2a)$ in a plane wave expansion, with $h_{ij}$ being the tensor perturbation in the FRW metric. $v_{k}$ satisfies the following equation of motion:

$$\ddot{v_{k}}+H\dot{v_{k}}+\left(\frac{k^{2}}{a^{2}}-H^{2}-\frac{\ddot{a}}{a}\right)v_{k}=0.$$

(4.12)

To quantify the size of the tensor power spectrum, the tensor-to-scalar ratio $r$ is commonly adopted, which can be compared with observational constraints. $r$ is defined at some reference scale $k=k_{\ast}$ as

$$r\equiv\left.\frac{{\cal P}_{T}}{{\cal P}_{\zeta}}\right|_{k=k_{\ast}}\,.$$

(4.13)

In this paper, we have investigated the two-field inflation model where the field $S$ that initially plays the role of spectator comes to dominate and causes the second inflation due to the non-minimal coupling to gravity. We have first studied the evolution of the background quantities of the fields and have shown that the second inflationary phase is generically realized due to the existence of the spectator non-minimal coupling $\xi_{S}$. When $\xi_{S}$ is positive, $S$ field can grow during the first inflationary phase driven by $\Phi$ and once $S$ acquires a super-Planckian field value, it can generate the second inflation. However, when $\xi_{S}$ or the initial amplitude is too large a runaway behavior can result where the field keeps growing and the second inflation does not terminate. However, in a sizable region of parameter space, the $S$ field drives a finite amount of inflation and then returns to zero.

We also investigated the power spectra for both the scalar and tensor modes in the model. As discussed in Sec. 4, adiabatic fluctuations are more sourced by isocurvature ones as $\xi_{S}S^{2}$ (or $\xi_{S}$) increases. Due to this isocurvature sourcing augmented by the spectator non-minimal coupling, the scale-dependence of the scalar power spectrum is modified. Although the tensor power spectrum itself is not directly affected by this isocurvature sourcing, the overall amplitude can change through the requirement of the normalization condition imposed on the scalar power spectrum. But due to the change of the scalar mode, the tensor-to-scalar ratio $r$ can also be affected. We explicitly showed $n_{s}$ and $r$ are modified and can fit well within the current observational constraints. With this setup, the quadratic chaotic inflation model can become viable by assuming a spectator with non-minimal coupling to gravity.

Multi-field dynamics is common to high energy models and non-minimal coupling to gravity is a generic feature of quantum field theory in curved space-time. Therefore, it is necessary to consider the interplay of such features in the early Universe. As we have shown, even simple dynamics can have a significant impact on cosmological predictions and change the picture considerably from the single-field case.

This work is supported in part by the Communidad de Madrid “Atracción de Talento investigador” Grant No. 2017-T1/TIC-5305 and the MICINN (Spain) project PID2019-107394GB-I00 (SR) as well as by JSPS KAKENHI Grant Numbers 19H01899 (KO), 21H01107 (KO), 17H01131 (TT), and 19K03874 (TT) and MEXT KAKENHI Grant Number 19H05110 (TT).