# Starobinsky-like two-field inflation

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## Abstract

We consider an extension of the Starobinsky model, whose parameters are functions of an extra scalar field. Our motivation is to test the robustness (or sensitivity) of the Starobinsky inflation against mixing scalaron with another (matter) scalar field. We find that the extended Starobinsky model is (classically) equivalent to the two-field inflation, with the scalar potential having a flat direction. For the sake of fully explicit calculations, we perform a numerical scan of the parameter space. Our findings support the viability of the Starobinsky-like two-field inflation for a certain range of its parameters, which is characterized by the scalar index \(n_s=0.96\pm 0.01\), the tensor-to-scalar ratio \(r<0.06\), and a small running of the scalar index at \(|\alpha _s|<0.05\).

### Keywords

Scalar Field Cosmic Microwave Background Scalar Potential Higgs Field Inflationary Model## 1 Introduction

Cosmological inflation in the early Universe is practically well established both theoretically and experimentally. It gives the universal solution to many problems of the Standard Cosmology, because it predicts homogeneity of our Universe at large scales, its spatial flatness, its large size and entropy, as well as the almost scale-invariant spectrum of cosmological perturbations, in remarkable agreement with the COBE, WMAP, PLANCK, and BICEP2 measurements of the cosmic microwave background (CMB) radiation spectrum. Inflation is also thought of as the amplifier of microscopic quantum field fluctuations in vacuum, and it is the only known mechanism for seeds of the macroscopic structure formation.

The standard mechanism of inflation in field theory uses a scalar field (called inflaton), whose potential energy drives inflation. The inflaton scalar potential should be flat enough to meet the slow-roll conditions during the inflationary stage. Physical nature and fundamental origin of inflaton and its interactions to the standard model (SM) elementary particles are unknown.

Starobinsky inflation [1, 2, 3, 4, 5] offers the gravitational origin of inflaton by identifying it with the spin-0 part of spacetime metric. In the Higgs inflation [6, 7, 8] inflaton is identified with the Higgs field of the SM. Both those single-field inflationary models offer the very economic and viable descriptions of chaotic inflation together with the clear origin of the inflaton field either from gravitational theory or from particle theory, respectively. As regards slow-roll inflation, the predictions of the Starobinsky and Higgs inflationary models are essentially *the same* (see below in this section).

*R*, where we have used the natural units with the reduced Planck mass \(M_\mathrm{Pl}=1\) and the spacetime signature \((+,-,-,-)\). Slow-roll inflation takes place in the high-curvature regime (\(M\ll H\ll 1\) and \(|\dot{H}|\ll H^2\)), where the Hubble function

*H*(

*t*) has been introduced. Then the Starobinsky inflationary solution (attractor!) takes the simple form

*M*whose value is fixed by the observational cosmic microwave background (CMB) data as \(M=(3.0 \times 10^{-6})({{50}\over {N_e}})\) where \(N_e\) is the e-foldings number. The predictions of the Starobinsky model (1.1) for the spectral indices \(n_s\approx 1-2/N_e\approx 0.964\), \(r\approx 12/N^2_e\approx 0.004\) and low non-Gaussianity are in agreement with the WMAP and PLANCK 2013 data (\(r<0.13\) and \(r<0.11\), respectively, at 95 % CL) [9], though they are in disagreement with the BICEP2 measurements (\(r=0.2+0.07,-0.05\)) [10]. The enhancement of the tensor-to-scalar-ratio

*r*of the Starobinsky model to the higher values can be achieved via modification of the simplest

*Ansatz*(1.1) by (matter) quantum corrections (beyond one loop) [11, 12]. However, the Planck 2015 data [13] excludes a significant enhancement of

*r*beyond \(r=0.08\). Therefore, the Starobinsky model (1.1) still perfectly fits the current observational data.

It raises the natural question on the theory side about the robustness of the simplest Starobinsky model (1.1) against mixing scalaron with other (matter) scalars. Though the current observational data favors a single-field inflation, it is very unlikely that any single-field inflationary model is capable to provide the ultimate description of inflation. As regards a more fundamental description of inflation in supergravity and string theories, multi-field inflation is a must; see e.g., Refs. [14, 15, 16]. The direct observational evidence for multi-field inflation would be a detection of primordial isocurvature perturbations beyond the adiabatic spectrum (see Appendix A for details).

In this paper, we study the two-field extensions of the Starobinsky model by non-minimal couplings, motivated by generic supergravity extensions of Eq. (1.1) in Ref. [17].

*R*; or in the large-field \(\phi \rightarrow +\infty \)) limit [8]. The scaling invariance is not exact for finite values of

*R*, and its violation is exactly measured by the slow-roll parameters, in full correspondence to the observed (nearly conformal) spectrum of the CMB perturbations. The (approximate) flatness of the inflaton scalar potential implies the (approximate) shift symmetry of the inflaton field. It also implies the alternative physical interpretation of the inflaton field as the pseudo-Nambu–Goldstone boson associated with spontaneous breaking of the scale invariance [20, 21, 22].

*non-minimal*coupling of the Higgs field to the spacetime scalar curvature [6]. It also has the approximate (rigid) scale invariance and, actually,

*the same*scalar potential (1.3) during slow-roll inflation [8]. The Higgs inflation is based on the Lagrangian (in the Jordan frame) [6]

It is still possible that inflaton is neither Starobinsky scalaron nor Higgs field, but a mixture of them. This possibility leads to a *two-field* inflation also. Another motivation to study the Starobinsky-like two-field inflation comes from 4D, \(N=1\) supergravity with chiral matter superfields, where inflaton is automatically extended to a complex field as the leading bosonic field component of an \(N=1\) scalar supermultiplet. For example, as demonstrated in Ref. [17], a generic \(N=1\) supergravity extension of the simplest Starobinsky model (1.1) leads to the non-minimal couplings of the Higgs field to both *R* and \(R^2\) gravity terms. It is commonplace in string cosmology that the inflaton is mixed with other scalars (moduli), so that a stabilization of the latter is required for inflation.

Our paper is organized as follows. In Sect. 2 we define the new class of two-field inflationary models as a combination (and a generalization) of Eqs. (1.1) and (1.4), and rewrite them to the more standard (dual) form. Those inflationary models interpolate between the Starobinsky and Higgs (single-field) inflationary models, and they can accommodate a broader range of values for the tensor-to-scalar ratio. In Sect. 3 we set up the equations of motion, and classify our model against the other two-field inflationary models studied in the literature. In Sect. 4 we focus on the particular case by dropping the Higgs part of the scalar potential. In Sect. 5 we summarize our numerical findings in the special two-field model of the Starobinsky-like inflation. Section 6 is our Conclusion. The technical details as regards linear perturbations, their spectra, and their evolution are collected in Appendix A.

## 2 Starobinsky- and Higgs-inspired two-field inflation models

*two*generic functions \(f(\phi )\) and \(M(\phi )\) in place of the constant parameters \(M_\mathrm{Pl}\) and

*M*of the original Starobinsky model (1.1).

^{1}

Both functions enter the Lagrangian (2.1) via their squares, in order to avoid ghosts. It is worth mentioning that both non-minimal couplings are required by renormalization of the \((R+R^2)\) gravity coupled to matter. In other words, we just replaced the parameters of the Starobinsky \((R+R^2)\) gravity by functions of a (Higgs) scalar field \(\phi \).

Should the scalar field \(\phi \) be stabilized by its scalar potential \(V(\phi )\) to some vacuum expectation value \(\phi _0\), our model reduces to the standard Starobinsky model (Sect. 1). Should the \(M^2(\phi )\) be sent to infinity, the Higgs inflationary setup is recovered.

Thus, the model (2.1) describes all the quintessence models with a non-minimal coupling to *R* (like the Higgs inflation) and the \(R+R^2\) gravity model of Starobinsky (1.1) as the particular cases. A non-minimal coupling to the \(R^2\) term is our new feature when \(M(\phi )\) is truly field-dependent.

*A*(to avoid ghosts), i.e.

*A*as the new independent scalar field (instead of \(\chi \)) and doing the field redefinition

*canonical*kinetic term of the scalar field \(\rho \). We find

*flat direction*along

We would like to emphasize that the discovered existence of a flat direction is automatic in the class of models under consideration, and it does not require supersymmetry. In a generic solution, two scalar fields \(\phi \) and \(\rho \) are going to evolve toward the flat direction.

*M*.

The choice (2.16) is also motivated by renormalizability. Though each of the quantum field theories (2.1) and (2.13) is not renormalizable as a theory of quantum gravity, it still makes sense to demand the (limited) renormalizability of the quantized scalar sector in a classical (curved) gravitational background. Then the non-minimal couplings (2.16) naturally arise with the renormalization counterterms [26, 27]. The Higgs potential (2.2) also fits the limited renormalizability requirement.

The field theory (2.13) has two real scalars \((\rho ,\phi )\) minimally coupled to the Einstein gravity and having the scalar potential (2.14). The kinetic term of the \(\rho \) scalar is canonically normalized, whereas the canonical term of the \(\phi \) scalar has the \(\rho \)-dependent factor. In the next sections we study two-field inflation in those models.

## 3 Classification of our model against the literature

### 3.1 Equations of motion

*matrices*, \((I,J=(\rho ,\phi )=1,2)\),

### 3.2 Correspondence of our model to the literature

## 4 Special case with \(V_\mathrm{H}=0\)

A simple two-field inflationary model of the same type as defined in our Eq. (2.1), though with a mass term instead of the Higgs scalar potential and *without* non-minimal interactions to *R* or \(R^2\), was considered in Ref. [32]. It was found by numerical calculations in Ref. [32] that the Starobinsky inflation is *robust* against that extension for a certain range of the ratio of two scalar masses. The model of Ref. [32] is in good agreement with the Planck data [13]. Multi-field dynamics of Higgs inflation in the presence of non-minimal couplings was analyzed in Ref. [33] where it was found that it is also in very good agreement with the Planck measurements.

*different case*with \(V_\mathrm{H}=0\) (or in the limit \(\lambda \rightarrow 0\)), when the functions \(f(\phi )\) and \(M(\phi )\) are given by Eq. (2.16), for simplicity. Then the scalar potential reads

## 5 Numerical results

*r*, we get \(r=0.056\pm 0.003\). The spectral scalar index running \(\alpha _s \equiv dn_s/d \ln k\) is \(|\alpha _s| <0.05\) in all our models.

Our numerical calculations in this section support the qualitative conclusion that the Starobinsky inflation is robust against the field dependence in the non-minimal functions \(f(\phi )\) and \(M(\phi )\), as long as the non-minimal coefficients \(\alpha \) and \(\beta \) are much less than 1. In other words, the Starobinsky inflation is *stable* against small deformations of the non-minimal couplings as long as those deformations are much less than of the order 1 (in Planck units). In the case of large deformations, inflation persists but is not viable.

We also found that at the end of inflation the scalaron field \(\rho \) oscillates near its minimum and thus contributes to (pre)heating, whereas the (matter) scalar field \(\phi \) does not, approaching a constant value. It can be already seen in Fig. 6a, b, but it is much better illustrated by our numerical findings in Fig. 7a, b. Actually, the field \(\phi \) starts oscillating and thus contributing to the reheating only when the parameter \(\alpha \) is much larger than 1, however, it does not lead to a viable inflation.

Finally, the numerical solutions to the perturbation equations for fluctuations \(\delta \rho \) and \(\delta \phi \) on the background specified by Fig. 6, in our primary example with the parameters \(\alpha =0.01\) and \(\beta =0.001\), are presented in Fig. 8.

## 6 Conclusion

We found that the Starobinsky inflation is robust against mixing scalaron with another (matter) scalar via non-minimal interactions of the latter with both *R* and \(R^2\) terms in the original (Jordan) frame, as long as the non-minimal field couplings are much smaller than one (in the Planck units). The non-minimal couplings were introduced by promoting the parameters of the original Starobinsky model to the (matter scalar) field-dependent functions, under the additional restriction of renormalizability of matter in the classical gravitational background.

We confirmed numerically that the inflationary trajectory in our two-field inflationary models remains close to the single-field attractor solution in the original Starobinsky model [1] under adding small non-minimal couplings to the *R* and \(R^2\) terms in Eq. (2.1). Our main statement is reflected in the title of our paper by calling our two-field inflationary models the *Starobinsky-like* ones. Though our numerical solutions to the dynamical equations (Sect. 5) were obtained by using some initial conditions for inflation, we found that the dependence of our solutions upon small changes in the initial conditions is weak and rather unimportant. It is related to the facts that (1) our numerical solutions also exhibit an attractor-type behavior (see e.g., Refs. [36, 37] for more), and (2) our scalar potentials do not have ridges that are generically present in multi-field inflation caused by non-minimal couplings and whose presence leads to strong dependence upon the initial conditions at the onset of inflation [38].

The two-field Starobinsky-like inflation becomes not viable when any of the non-minimal parameters is of the order one or larger. Our results are complementary to the findings of Ref. [32] where the robustness of the Starobinsky inflation was established in another two-field Starobinsky-like limit with \(\alpha =\beta =0\) and a non-vanishing mass term of the matter scalar.

The main difference of our two-field inflationary models against the single-field Starobinsky model is the presence of isocurvature perturbations. However, those perturbations turn out to be very small and (currently) undetectable. As was argued in Ref. [39], significant isocurvature perturbations in generic multi-field inflationary models with non-minimal couplings may account for the observed low power in the CMB angular power spectrum of temperature anisotropies at low multipoles [40]. However, in our models the isocurvature perturbations are not amplified enough to be the reason for that observation.

Though we did not investigate primordial non-Gaussianities in our Starobinsky-like two-field inflationary models, we expect them to be negligible, like the original (single-field) Starobinsky model.

The field-dependent couplings are quite natural from the viewpoint of *string theory* where all coupling constants are given by expectation values of scalar fields. As regards the physical meaning of our two scalars from the viewpoint of string theory, it is conceivable that scalaron is related to string theory dilaton, whereas another (matter) scalar is given by one of the moduli arising from superstring compactification. A detailed investigation of the possible connection of our models to string theory is beyond the scope of this paper.

## Footnotes

## Notes

### Acknowledgments

This work was supported by a Grant-in-Aid of the Japanese Society for Promotion of Science (JSPS) under No. 26400252, the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, the special TMU Fund for International Research, and the Competitiveness Enhancement Program of the Tomsk Polytechnic University in Russia. The authors are grateful to D. Kaiser and S. Vagnozzi for discussions and correspondence, and to the referee for careful reading of our submission and critical remarks.

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