Nonminimally Assisted Inflation:
A General Analysis

The latest joint analysis of the Planck results [1] and BICEP/Keck (BK) data [2] has put a strong constraint on the spectral index, $0.958\leq n_{s}\leq 0.975$ (95% C.L.), and a stringent upper bound on the tensor-to-scalar ratio, $r\leq 0.036$ (95% C.L.). As a consequence, a plethora of single-field inflationary models have been ruled out. Notably, the chaotic inflation model [3] with a power-law potential $V(\phi)\sim\phi^{p}$, the power-law inflation model [4] with an exponential potential $V(\phi)\sim\exp(-\lambda\phi)$, the hybrid inflation model [5] with an effectively single-field potential $V(\phi)\sim 1+(\phi/\mu)^{2}$, and the loop inflation (also known as spontaneously broken SUSY) model [6] with $V(\phi)\sim 1+\lambda\log(\phi/M_{\rm P})$ are ruled out as either the tensor-to-scalar ratio $r$ or the spectral index $n_{s}$ is too large to be allowed by the latest bounds.

To save single-field inflationary models, many mechanisms have been proposed. For instance, an introduction of the so-called nonminimal coupling of the inflaton field to gravity of the form $\xi\phi^{2}R$ [7, 8, 9, 10, 11, 12], where $R$ is the Ricci scalar, is known to reduce the tensor-to-scalar ratio¹¹1 See also, e.g., Refs. [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] for the analysis of the effects of the nonminimal coupling to gravity in single-field inflation models for observables such as the spectral index and tensor-to-scalar ratio. . The nonminimally-coupled models predict similar tensor-to-scalar ratio and spectral index values as Starobinsky’s $R^{2}$ model [27] as they are approximately equivalent to each other under the slow-roll assumption for a particular combination of the inflaton potential and the nonminimal coupling function²²2 See, e.g., Refs. [28, 29, 30, 31] for the $R^{2}$-type models in the Palatini formulation. . Alternatively, one may modify the inflaton kinetic term. The $\alpha$-attractor model [32, 33] is a well-known example where the kinetic term has a pole. All of the aforementioned models, the nonminimally-coupled model, Starobinsky’s $R^{2}$ model, and the $\alpha$-attractor model, have one thing in common: The potential in the Einstein frame which can be achieved via the Weyl rescaling features a plateau region in the large-field limit where the inflaton slowly rolls during inflation.

In Ref. [34], an alternative approach has been proposed. Instead of directly modifying the inflaton sector in such a way that the Einstein-frame potential becomes flat, an extra scalar field $s$, dubbed an assistant field, is introduced. The assistant field does not directly interact with the inflaton field $\phi$, while it talks to gravity through the nonminimal coupling $\sim s^{m}R$ with $m>0$. The assistant field is further assumed to be effectively massless. Therefore, the potential in the original Jordan frame is given only by the original $\phi$ field. Even though the additional assistant field $s$ has no direct interaction with the original inflaton field $\phi$, due to its nonminimal coupling to the Ricci scalar, the potential in the Einstein frame and the dynamics of inflation become nontrivial. It was shown in Ref. [34] that the chaotic inflation model, when assisted by the assistant field, may be revived and become compatible with the latest observational constraints.

In this work, we generalise the analysis of Ref. [34] to a general inflationary model for the original inflaton field $\phi$. We remain agnostic as to the form of the $\phi$-field potential. We follow the same nomenclature as Ref. [34] and call the extra scalar field $s$ the assistant field. The characteristics of the assistant field are: (i) it does not directly couple to the original inflaton field $\phi$, (ii) it nonminimally couples to gravity through $s^{m}R$, and (iii) it is effectively massless. Including the assistant field $s$ and its explicit nonminimal coupling term, we perform a general analytical study about inflationary observables such as the spectral index and the tensor-to-scalar ratio. Our analytical formulae for the spectral index and the tensor-to-scalar ratio can readily be applied to a vast range of inflationary models, and one may easily see whether an otherwise ruled-out model can become revived with the help of the assistant field. We also discuss the non-Gaussianity and running of the spectral index that may become sizeable for multifield models, providing the corresponding analytical formula. To demonstrate how the assistant field affects observables in concrete models, we consider as an example three models for the original $\phi$ field, namely the loop inflation model, the power-law inflation model, and the hybrid inflation model, and show how the presence of the assistant field may bring the models to the observationally-favoured region.

The rest of the paper is organised as follows. In Sec. 2, we start with a generic action for a multifield inflation model with a nonminimal coupling to gravity and perform the Weyl rescaling to the Einstein frame, setting the notations. Defining properties for the assistant field are introduced, and we set our model with a general potential for the original $\phi$ field. Section 3 presents the general analytical study for the spectral index, the tensor-to-scalar ratio, and the non-Gaussianity. We show how the inflationary observables are modified in comparison with the original predictions due to the presence of the assistant field. As an application of our general analytical formulae, three examples, the loop inflation model, power-law inflation, and hybrid inflation, are discussed in Sec. 4. We show how these models become compatible with the latest observational data. We conclude in Sec. 5. Appendix A summarises the computation and general behaviour of the running of the spectral index.

Let us consider the following generic action for multifield inflation:

$\displaystyle S=\int d^{4}x\,\sqrt{-g_{\rm J}}\left[f(\varphi^{i})g_{\rm J}^{\mu\nu}R_{{\rm J}\mu\nu}(\Gamma_{\rm J})-\frac{1}{2}G_{{\rm J}ij}g_{\rm J}^{\mu\nu}\partial_{\mu}\varphi^{i}\partial_{\nu}\varphi^{j}-V_{\rm J}(\varphi^{i})\right]\,,$

(2.1)

where the subscript J indicates that the action is written in the Jordan frame, $f(\varphi^{i})$ is a function of fields that represents the nonminimal coupling, $G_{{\rm J}ij}$ denotes the kinetic mixing between fields, and $V_{\rm J}(\varphi^{i})$ is the Jordan-frame potential³³3 Multifield inflation models with the nonminimal coupling to gravity have been discussed in a certain range of contexts. See, e.g., Refs. [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52] for such works. . Here, the Ricci tensor $R_{{\rm J}\mu\nu}$ is given as a function of the connection $\Gamma_{\rm J}$ to capture not only the standard metric formulation, but also the Palatini formulation. In the Palatini formulation, the metric and the connection are a priori independent to each other, while in the metric formulation, the connection is related to the metric, becoming the Levi-Civita connection. The connection is assumed to be torsion free.

Through the Weyl rescaling, $g_{{\rm J}\mu\nu}\rightarrow g_{{\rm E}\mu\nu}=\Omega^{2}g_{{\rm J}\mu\nu}$, the Jordan-frame action (2.1) can be brought to the Einstein frame, denoted by the subscript E. The conformal factor $\Omega^{2}$ is chosen to be $2f/M_{\rm P}^{2}$ with $M_{\rm P}$ being the reduced Planck mass. After the Weyl rescaling, we obtain the Einstein-frame action as follows:

$\displaystyle S=\int d^{4}x\,\sqrt{-g_{\rm E}}\left[\frac{M_{\rm P}^{2}}{2}g_{\rm E}^{\mu\nu}R_{{\rm E}\mu\nu}(\Gamma_{\rm E})-\frac{1}{2}G_{{\rm E}ij}g_{\rm E}^{\mu\nu}\partial_{\mu}\varphi^{i}\partial_{\nu}\varphi^{j}-V_{\rm E}\right]\,,$

(2.2)

where $V_{\rm E}$ is the Einstein-frame potential given by

$\displaystyle V_{\rm E}=\frac{V_{\rm J}}{\Omega^{4}}=\frac{M_{\rm P}^{4}V_{\rm J}}{4f^{2}}\,,$

(2.3)

and

$\displaystyle G_{{\rm E}ij}=\frac{M_{\rm P}^{2}}{2f}\left(G_{{\rm J}ij}+\kappa\frac{3f_{,i}f_{,j}}{f}\right)\,,$

(2.4)

with $f_{,i}\equiv\partial f/\partial\varphi^{i}$, represents the field-space metric. The parameter $\kappa$ parametrises which framework we work in; $\kappa=0$ denotes the Palatini formulation while $\kappa=1$ corresponds to the metric formulation. Hereinafter, we omit the subscript E for brevity.

Our interests are the cases where the system is comprised of two fields, namely the original inflaton field $\phi$ and the assistant field $s$. The assistant field is effectively massless and does not directly couple to the $\phi$ field in the Jordan frame. These properties indicate that $G_{{\rm J}ij}=\delta_{ij}$, i.e., no kinetic mixing between the two scalar fields in the original Jordan frame, and that the Jordan-frame potential $V_{\rm J}$ becomes a function of only the $\phi$ field, i.e., $V_{\rm J}=V_{\rm J}(\phi)$. Furthermore, the assistant field nonminimally couples to gravity, i.e., $f=f(s)$. Upon imposing a $Z_{2}$ symmetry, a natural choice for the nonminimal coupling from the dimensional analysis viewpoint would be

$\displaystyle f(s)=\frac{M_{\rm P}^{2}}{2}\left[1+\xi_{2}\left(\frac{s}{M_{\rm P}}\right)^{2}\right]\,,$

(2.5)

where $\xi_{2}$ is a dimensionless coupling. Moreover, in order for the assistant field not to significantly alter, but to merely assist the inflationary dynamics, the nonminimal coupling term takes not too large values, i.e., $\xi_{2}(s/M_{\rm P})^{2}\ll 1$. We further note that the $\xi_{2}(s/M_{\rm P})^{2}\rightarrow 0$ limit corresponds to the original $\phi$-field inflation. In the current work, we focus on the quadratic nonminimal coupling of the form (2.5). However, our results can easily be extended to $f(s)=(M_{\rm P}^{2}/2)[1+\xi_{m}(s/M_{\rm P})^{m}]$ with $m>2$ in a straightforward manner.

The action in the Einstein frame (2.2) is therefore obtained as follows:

$\displaystyle S=\int d^{4}x\,\sqrt{-g}\,\left[\frac{M_{\rm P}^{2}}{2}g^{\mu\nu}R_{\mu\nu}(\Gamma)-\frac{1}{2}\mathcal{K}_{1}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}\mathcal{K}_{2}g^{\mu\nu}\partial_{\mu}s\partial_{\nu}s-V\right]\,,$

(2.6)

where

	$\displaystyle\mathcal{K}_{1}$	$\displaystyle=\frac{M_{\rm P}^{2}}{2f}\,,$		(2.7)
	$\displaystyle\mathcal{K}_{2}$	$\displaystyle=\frac{M_{\rm P}^{2}}{2f}+\kappa\frac{3M_{\rm P}^{2}}{2}\left(\frac{f_{,s}}{f}\right)^{2}\,,$		(2.8)

and the Einstein-frame potential is given by

$\displaystyle V=\frac{M_{\rm P}^{4}}{4f^{2}(s)}V_{\rm J}(\phi)\,.$

(2.9)

We note that the potential is product-separable, $V=K(s)V_{\rm J}(\phi)$, where $K(s)=M_{\rm P}^{4}/(4f^{2}(s))$. Upon canonically normalising the $s$ field through

$\displaystyle\left(\frac{\partial\sigma}{\partial s}\right)^{2}=\mathcal{K}_{2}\,,$

(2.10)

we may rewrite the action as

$\displaystyle S=\int d^{4}x\,\sqrt{-g}\,\left[\frac{M_{\rm P}^{2}}{2}g^{\mu\nu}R_{\mu\nu}(\Gamma)-\frac{1}{2}e^{2b}g^{\mu\nu}\partial_{\mu}\phi\partial_{\mu}\phi-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma-V\right]\,,$

(2.11)

where $b=b(\sigma(s))$ is defined via

$\displaystyle e^{2b}\equiv\mathcal{K}_{1}=\frac{M_{\rm P}^{2}}{2f}\,.$

(2.12)

We note that the action (2.11) has been explored in detail in Refs. [53, 54]. For brevity, in the following, we set $M_{\rm P}=1$.

Inflationary observables such as the scalar power spectrum $\mathcal{P}_{\zeta}$, the scalar spectral index $n_{s}$, the tensor-to-scalar ratio $r$, and the local-type shape-independent nonlinearity parameter $f_{\rm NL}^{\rm(local)}$ for the action (2.11) have been obtained in, e.g., Refs. [54, 34] by using the $\delta N$ formalism [55, 56, 57, 58, 59] under the slow-roll approximation. We do not repeat the derivation here and simply list the resultant expressions. The spectral index and the tensor-to-scalar ratio are given by

	$\displaystyle n_{s}$	$\displaystyle=1-2\epsilon^{\sigma}_{*}-2\epsilon^{\phi}_{*}-\frac{4e^{-2X}}{u^{2}\alpha^{2}/\epsilon_{*}^{\sigma}+v^{2}/\epsilon_{*}^{\phi}}-\frac{1}{12}\frac{\eta_{*}^{b}+2\epsilon_{*}^{b}}{u^{2}\alpha^{2}/\epsilon_{*}^{\sigma}+v^{2}/\epsilon_{*}^{\phi}}\left(u\alpha\sqrt{\frac{\epsilon_{*}^{\phi}}{\epsilon_{*}^{\sigma}}}-v\sqrt{\frac{\epsilon_{*}^{\sigma}}{\epsilon_{*}^{\phi}}}\right)^{2}$
		$\displaystyle\quad+\frac{2}{u^{2}\alpha^{2}/\epsilon_{*}^{\sigma}+v^{2}/\epsilon_{*}^{\phi}}\left[u^{2}\alpha^{2}\frac{\eta_{*}^{\sigma\sigma}}{\epsilon_{*}^{\sigma}}+v^{2}\frac{\eta_{*}^{\phi\phi}}{\epsilon_{*}^{\phi}}+4uv\alpha+\frac{1}{2}s_{*}^{b}s_{*}^{\sigma}\sqrt{\epsilon_{*}^{b}\epsilon_{*}^{\sigma}}v\left(\frac{v}{\epsilon_{*}^{\phi}}-\frac{2u\alpha}{\epsilon_{*}^{\sigma}}\right)\right]\,,$		(3.1)
	$\displaystyle r$	$\displaystyle=\frac{16e^{-2X}}{u^{2}\alpha^{2}/\epsilon_{*}^{\sigma}+v^{2}/\epsilon_{*}^{\phi}}\,,$		(3.2)

where

$\displaystyle u\equiv\frac{\epsilon_{e}^{\sigma}}{\epsilon^{\sigma}_{e}+\epsilon^{\phi}_{e}}\,,\;\;v\equiv\frac{\epsilon_{e}^{\phi}}{\epsilon^{\sigma}_{e}+\epsilon^{\phi}_{e}}\,,\;\;X\equiv 2b_{e}-2b_{*}\,,\;\;\alpha\equiv e^{2b_{*}-2b_{e}}\left[1+\frac{\epsilon_{e}^{\phi}}{\epsilon_{e}^{\sigma}}\left(1-e^{2b_{e}-2b_{*}}\right)\right]\,.$

(3.3)

Here, $s^{\sigma}\equiv{\rm sgn}(V_{,\sigma})$ and $s^{b}\equiv{\rm sgn}(b_{,\sigma})$ denote the signs of the potential and $b$ derivatives with respect to the field $\sigma$, respectively. The subscript $*$ ($e$) indicates that the quantities are evaluated at the pivot scale (end of inflation). The slow-roll parameters are defined as

	$$\displaystyle\epsilon^{\sigma}\equiv\frac{1}{2}\left(\frac{V_{,\sigma}}{V}\right)^{2}=\frac{1}{2}\left(\frac{K_{,\sigma}}{K}\right)^{2}\,,\qquad\epsilon^{\phi}\equiv\frac{1}{2}\left(\frac{V_{,\phi}}{V}e^{-b}\right)^{2}=\frac{1}{2}\left(\frac{V_{{\rm J},\phi}}{V_{\rm J}}\right)^{2}e^{-2b}\,,$$
	$$\displaystyle\eta^{\sigma\sigma}\equiv\frac{V_{,\sigma\sigma}}{V}=\frac{K_{,\sigma\sigma}}{K}\,,\qquad\eta^{\phi\phi}\equiv\frac{V_{,\phi\phi}}{V}e^{-2b}=\frac{V_{{\rm J},\phi\phi}}{V_{\rm J}}e^{-2b}\,,$$
	$$\displaystyle\eta^{\phi\sigma}\equiv\frac{V_{,\phi\sigma}}{V}e^{-b}\,,\qquad\epsilon^{b}\equiv 8b_{,\sigma}^{2}\,,$$		(3.4)

and $\eta^{b}\equiv 16b_{,\sigma\sigma}$. The local-type, shape-independent, nonlinearity parameter is given by

	$\displaystyle-\frac{6}{5}f_{\rm NL}^{\rm(local)}$	$\displaystyle=\frac{2e^{-X}}{(u^{2}\alpha^{2}/\epsilon_{*}^{\sigma}+v^{2}/\epsilon_{*}^{\phi})^{2}}\Bigg{[}\left(1-\frac{\eta_{*}^{\sigma\sigma}}{2\epsilon_{*}^{\sigma}}\right)\frac{u^{3}\alpha^{3}}{\epsilon_{*}^{\sigma}}+\left(1-\frac{\eta_{*}^{\phi\phi}}{2\epsilon_{*}^{\phi}}\right)\frac{v^{3}}{\epsilon_{*}^{\phi}}$
		$\displaystyle\qquad+\frac{1}{2}s_{*}^{b}s_{*}^{\sigma}\frac{u^{2}v\alpha^{2}}{\epsilon_{*}^{\sigma}}\sqrt{\frac{\epsilon_{*}^{b}}{\epsilon_{*}^{\sigma}}}+\left(\frac{u\alpha}{\epsilon_{*}^{\sigma}}-\frac{v}{\epsilon_{*}^{\phi}}\right)^{2}e^{X}\mathcal{C}\Bigg{]}\,,$		(3.5)

where

$\displaystyle\mathcal{C}$

$\displaystyle\equiv\frac{\epsilon_{e}^{\sigma}\epsilon_{e}^{\phi}}{\epsilon_{e}^{2}}\left(\frac{\epsilon_{e}^{\sigma}\eta_{e}^{\phi\phi}+\epsilon_{e}^{\phi}\eta_{e}^{\sigma\sigma}}{\epsilon_{e}}-4\frac{\epsilon_{e}^{\phi}\epsilon_{e}^{\sigma}}{\epsilon_{e}}-\frac{1}{2}s_{e}^{\sigma}s_{e}^{b}\sqrt{\frac{\epsilon_{e}^{b}}{\epsilon_{e}^{\sigma}}}\frac{(\epsilon_{e}^{\phi})^{2}}{\epsilon_{e}}\right)\,.$

(3.6)

We leave detailed discussion on the running of the spectral index $\alpha_{s}$ in Appendix A.

The Einstein-frame potential for the assistant field is given by

$\displaystyle K(s)=\frac{1}{(1+\xi_{2}s^{2})^{2}}\,,$

(3.7)

as can be seen from Eqs. (2.5) and (2.9). Thus, it is straightforward to obtain the slow-roll parameters for the $s$ field or, equivalently, its canonically-normalised version $\sigma$. They are given by

$$\displaystyle\epsilon^{\sigma}=\epsilon^{b}=\frac{8\xi_{2}^{2}s^{2}}{1+\xi_{2}s^{2}(1+6\kappa\xi_{2})}\,,\quad\eta^{\sigma\sigma}=\frac{4\xi_{2}[-1+3\xi_{2}s^{2}+4\xi_{2}^{2}s^{4}(1+6\kappa\xi_{2})]}{[1+\xi_{2}s^{2}(1+6\kappa\xi_{2})]^{2}}\,,$$

(3.8)

where we have used Eq. (2.10). Note also that $\eta^{b}=-16\xi_{2}(1+\xi_{2}s^{2})/[1+\xi_{2}s^{2}(1+6\kappa\xi_{2})]^{2}$. One may regard the $s$-field value at the pivot scale, $s_{*}$, as an input parameter⁴⁴4 Quantum kick for the assistant field in the de Sitter phase can be estimated as $(\Delta\sigma)^{2}=\mathcal{K}_{2}(\Delta s)^{2}\simeq N^{2}H_{*}^{2}/(2\pi)^{2}\simeq N^{2}rA_{s}/8$, where $A_{s}\approx 2\times 10^{-9}$ is the magnitude of the scalar power spectrum at the pivot scale and $N$ is the number of $e$-folds. In the $\xi_{2}s^{2}\ll 1$ limit, we find $\Delta s\lesssim 4.2\times 10^{-4}$ ($1.8\times 10^{-4}$) for $r<0.2$ (0.035) with $N=60$. As far as a relatively larger value of $s$ is considered, such fluctuations can safely be neglected. . The $s$-field value at the end of inflation is then given by requiring the number of $e$-folds, $N$, to be $N_{e}$; in this work, we choose $N_{e}=60$ from the end of inflation. The number of $e$-folds can be solely given by the assistant field $s$,

$\displaystyle N=\int_{e}^{*}\frac{K}{K_{,\sigma}}\,d\sigma=\frac{1}{4\xi_{2}}\left[\ln\left(\frac{s_{e}}{s_{*}}\right)+3\kappa\xi_{2}\ln\left(\frac{1+\xi_{2}s_{e}^{2}}{1+\xi_{2}s_{*}^{2}}\right)\right]\,.$

(3.9)

In the Palatini formulation, $\kappa=0$, and one may obtain an exact analytical solution for $s_{e}$ as

$\displaystyle s_{e}=s_{*}e^{4\xi_{2}N_{e}}\,.$

(3.10)

In the metric formulation, on the other hand, such a closed analytical form for the solution does not exist. However, an approximated form may be found. In this paper, we assume that the nonminimal coupling of the assistant field $s$ is small. Thus, we can take the small nonminimal coupling limit where $\xi_{2}s^{2}\ll 1$. In this limit, we may approximate the number of $e$-folds as

$\displaystyle N\approx\frac{1}{4\xi_{2}}\ln\left(\frac{s_{e}}{s_{*}}\right)+\frac{3}{4}\xi_{2}\left(s_{e}^{2}-s_{*}^{2}\right)\,.$

(3.11)

Requiring $N=N_{e}$ then gives

$\displaystyle s_{e}\approx s_{*}e^{4\xi_{2}N_{e}}\left[1-3\xi_{2}^{2}s_{*}^{2}\left(e^{8\xi_{2}N_{e}}-1\right)\right]\,.$

(3.12)

Figure 1 shows the comparison between the analytical solution for $s_{e}$ in the Palatini formulation (3.10), the approximated analytical solution for $s_{e}$ in the metric formulation (3.12), and the numerical solution for $s_{e}$ found by using $N=N_{e}$ with Eq. (3.9) in the metric formulation. We stress that the solution (3.10) in the Palatini case is exact. Deviations between the approximated solution and the exact numerical solution in the metric case start to appear when the assumption of the small nonminimal coupling breaks down; we shall use $\xi_{2}s_{e}^{2}\leq 0.1$ as the condition for the small nonminimal coupling in this paper. We also observe that the difference between the Palatini case and the metric case is negligible in the small nonminimal coupling limit; deviations become visible when the nonminimal coupling gets larger.

In order to discuss effects of the assistant field $s$ on the original $\phi$-field inflation model in a general way, we remain as agnostic as possible as to the form for the $\phi$-field potential, $V_{\rm J}(\phi)$. In the absence of the assistant field, the spectral index and the tensor-to-scalar ratio for the original $\phi$-field inflation model, denoted respectively by $n_{s}^{(0)}$ and $r^{(0)}$, are given by

$\displaystyle n_{s}^{(0)}=1-6\epsilon^{(0)}_{*}+2\eta^{(0)}_{*}\,,\quad r^{(0)}=16\epsilon^{(0)}_{*}\,,$

(3.13)

under the slow-roll approximation, where the slow-roll parameters are defined as

$\displaystyle\epsilon^{(0)}\equiv\frac{1}{2}\left(\frac{V_{{\rm J},\phi}}{V_{\rm J}}\right)^{2}\,,\quad\eta^{(0)}\equiv\frac{V_{{\rm J},\phi\phi}}{V_{\rm J}}\,.$

(3.14)

Comparing with the slow-roll parameters defined in Eq. (3.4), we see that

$\displaystyle\epsilon^{\phi}=\epsilon^{(0)}e^{-2b}\,,\quad\eta^{\phi\phi}=\eta^{(0)}e^{-2b}\,.$

(3.15)

Writing $\epsilon^{(0)}_{*}$ and $\eta^{(0)}_{*}$ in terms of $n^{(0)}_{s}$ and $r^{(0)}$, we get

$\displaystyle\epsilon^{\phi}_{*}=\frac{r^{(0)}}{16}e^{-2b_{*}}\,,\quad\eta^{\phi\phi}_{*}=\frac{1}{2}\left(n^{(0)}_{s}-1+\frac{3}{8}r^{(0)}\right)e^{-2b_{*}}\,.$

(3.16)

We may thus substitute Eq. (3.16) into the observables, Eqs. (3.1), (3.2), and (3.5), to get rid of the information on the $\phi$ field at the pivot scale. There still remains, however, dependence on the $\phi$ field at the end of inflation through the $\phi$-field slow-roll parameters $\epsilon^{\phi}_{e}$ and $\eta^{\phi\phi}_{e}$ or, equivalently, $\epsilon^{(0)}_{e}$ and $\eta^{(0)}_{e}$. In the current work, we consider two classes.