Inflation in a conformally invariant twoscalarfield theory with an extra \(R^2\) term
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Abstract
We explore inflationary cosmology in a theory where there are two scalar fields which nonminimally couple to the Ricci scalar and an additional \(R^2\) term, which breaks the conformal invariance. Particularly, we investigate the slowroll inflation in the case of one dynamical scalar field and that of two dynamical scalar fields. It is explicitly demonstrated that the spectral index of the scalar mode of the density perturbations and the tensortoscalar ratio can be consistent with the observations obtaind by the recent Planck satellite. The graceful exit from the inflationary stage is achieved as in convenient \(R^2\) gravity. We also propose the generalization of the model under discussion with three scalar fields.
Keywords
Dark Energy Scalar Field Spectral Index Einstein Frame Jordan Frame1 Introduction
The natures on inflation [1, 2, 3, 4, 5] in the early universe have been revealed by the recent cosmological observations such as the Wilkinson Microwave Anisotropy Probe (WMAP) [6, 7], the Planck satellite [8, 9], and the BICEP2 experiment [10, 11] on the quite tiny anisotropy of the cosmic microwave background (CMB) radiation. Owing to the release of the recent observational data, in addition to seminal inflation with a single scalar field such as new inflation [3, 4], chaotic inflation [12], natural inflation [13], and powerlaw inflation with the exponential inflaton potential [14], novel models of singlefield inflation have been proposed in Refs. [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90]^{1} (for reviews on more various inflationary models, see, e.g., [124, 125, 126, 127, 128]).
In addition to inflationary models driven by a scalar field (i.e., the inflaton field) described above, there have been considered the socalled Starobinsky inflation [5, 129] originating from the higherorder curvature term such as an \(R^2\) term,^{2} where R is the Ricci scalar. This model is observationally supported by the Planck results. Such a theory can be interpreted as a kind of modified gravity theory including F(R) gravity to account for the latetime cosmic acceleration (for reviews on the dark energy problem and modified gravity theories, see, for instance, [130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140]). Various inflationary models in modified gravity theories corresponding to extensions of Starobinsky inflation have been explored in Refs. [141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153].
In this paper, we investigate inflation in a theory consisting of two scalar fields which nonminimally couple to the Ricci scalar and an additional \(R^2\) term.^{3} We consider the conformally invariant twoscalarfield theory in which the conformal invariance is broken by adding an \(R^2\) term. In particular, we explore the slowroll inflation in the cases of (i) one dynamical scalar field (namely, we set one of the two scalar fields constant) and (ii) two dynamical scalar fields. As a consequence, we analyze the spectral index of the scalar mode of the density perturbations and the tensortoscalar ratio and compare the theoretical results with the observational data obtained by the recent Planck satellite and the BICEP2 experiment. It is clearly shown that the spectral index and the tensortoscalar ratio can be compatible with the recent Planck results.
The motivation to propose our theory is to unify inflation in the early universe originating from the \(R^2\) term and the latetime cosmic acceleration, i.e., the dark energy dominated stage with dark matter. The \(R^2\) term is interpreted as the contribution from modified gravity, dark energy is described by one of the scalar fields, and dark matter is represented by the other scalar field. Furthermore, it seems that multiplefield inflation models can fit the Planck data better than singlefield inflation models. Inflationary models with multiscalar fields [155, 156, 157, 158, 159] including the socalled curvaton scenario [160, 161, 162, 163, 164] have been constructed and the cosmological perturbations in these models have also been investigated [165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177].
We use units of \(k_\mathrm {B} = c = \hbar = 1\) and express the gravitational constant \(8 \pi G_\mathrm {N}\) by \({\kappa }^2 \equiv 8\pi /{M_{\mathrm {Pl}}}^2\) with the Planck mass of \(M_{\mathrm {Pl}} = G_\mathrm {N}^{1/2} = 1.2 \times 10^{19}\) GeV.
The organization of the paper is as follows. In Sect. 2, we explain our model action and derive the gravitational field equation and the equations of motions for the scalar fields. We also examine the slowroll inflation in the case of one dynamical scalar field and study the dynamics of the system including the equilibrium points in detail. In Sect. 3, with the conformal transformation [178, 179], we explore inflationary cosmology in the Einstein frame. Especially, we study inflationary models in the case that the conformal scalar is the dynamical inflaton field and the other scalar fields are set constant. In Sect. 4, we investigate the slowroll inflation in the case of two dynamical scalar fields. Particularly, we consider the resultant spectral index and the tensortoscalar ratio in the Jordan frame (i.e., the original conformal frame). In Sect. 5, we explore the graceful exit from inflation, namely, the instability of the de Sitter solution in the present theory. As a demonstration, we concentrate on the case of one dynamical scalar field in the Einstein frame, because, as is described in Sect. 3, in this case the spectral index and the tensortoscalar ratio can be consistent with the recent Planck results. Finally, conclusions are described in Sect. 6.
2 Twoscalarfield model with breaking the conformal invariance
2.1 Model action and its transformation into the canonical form
The action in Eq. (2.1) without the \(R^2\) term has been proposed and studied in Refs. [180, 181, 182, 183, 184, 185, 186, 187, 188, 189].
Here and in the following, we take \(2\kappa ^2 = 1\). Regarding the property of the action in Eq. (2.1), we remark that if there does not exist an \(R^2\) term, this action is conformally invariant, while when the \(R^2\) term is added, the conformal invariance of this action is effectively broken.
We also note that our action may be considered to be invariant yet under the restricted conformal invariance [190].
2.2 Single dynamical scalar field model
If we take the number of efolds, whose value has to be large enough, such as \(N_e=50\)–60, to solve the socalled horizon and flatness problems and the values of \(n_\mathrm {s}\) and r suggested by the observations, it is apparently regarded that there are three equations, Eqs. (2.20), (2.23), and (2.24), for the three independent variables (\(\phi \), \(\alpha \), C). By solving these three equations, we can estimate viable values of our model parameters. However, \(\alpha \) and C are incorporated into all the equations in the form of \(x \equiv \alpha C\). Hence, it is necessary to analyze a system of two equations, for example, Eqs. (2.23) and (2.20), whereas r should depend on \(n_\mathrm {s}\). For this reason, we may try to modify the initial action in Eq. (2.1) in the following way. Let us suppose that s is an arbitrary numerical parameter. In this case, we have a system of three equations for three variables. We also remark that our potential is well known as a potential of “Spontaneous Symmetry Breaking Inflation (SSBI)” and a viable inflationary model can be constructed [127], although only for positive values of the parameter s.
According to the Planck 2015 results, \(n_{\mathrm {s}} = 0.968 \pm 0.006\, (68\,\%\,\mathrm {CL})\) [8, 9] and \(r < 0.11\, (95\,\%\,\mathrm {CL})\) [9]. These values are consistent with those obtained by the WMAP satellite [6, 7]. The BICEP2 experiment has suggested \(r=0.20^{+0.07}_{0.05}\, (68\,\%\,\mathrm {CL})\) [10], but recently the Bmode polarization of the CMB radiation is attributed to an effect of the dust, and not of primordial gravitational waves [11].
2.3 Equilibrium points
Next, we investigate equilibrium points in the system. We have a system consisting of two dynamical equations, Eqs. (2.11) and (2.12), with the constraint equation (2.13). To examine equilibrium points in this system, we need to rewrite the system of two second order differential equations as that of four first order differential equations. It is clear that this task is equivalent to an exploration of the shape of the potential, namely, to find its extreme values and study their natures. We execute the numerical analysis by using the graphics of the potential for several values of the parameters.
2.4 Recollapse and bounce solutions
Related to the bouncing solutions, we mention that the antigravity regime in the extended gravity theories with the Weyl invariance [180, 181, 182, 183, 184, 185, 198] including F(R) gravity [189] has been examined.
3 Inflationary cosmology
In this section, we reconsider the theory whose action is described by Eq. (2.1) and build an inflationary model in another way.
3.1 Conformal transformation
3.2 Inflationary model
To calculate the observables for inflationary models including the spectral index of curvature perturbations \(n_\mathrm {s}\) and the tensortoscalar ratio r, it is necessary for two scalar fields to be made constant. If the scalar field \(\lambda \) is a constant and one of the scalar fields \(\phi \) or u plays the role of the inflaton field, we obtain a similar theory to that described in the previous section.

Case 1: The relation \(K\equiv \left( s/12\right) \mathrm {e}^{\lambda }(\phi _0^2u_0^2)\gg 1\) is met. In this case, the first terms in the brackets \([\,\,]\) in Eqs. (3.16) and (3.17) may be neglected. Namely, the term with \(\mathrm {e}^{\lambda }\) is much smaller than any other terms. Accordingly, all the other terms have the multiplication factor \(\mathrm {e}^{2\lambda }\) incorporated in the same way, so that this overall factor can be removed from the other terms.

Case 2: The values of the first and second terms in the brackets \([\,\,]\) may be similar to each other, but the overall coefficient term \(s/\left( 6\alpha \right) \) is sufficiently small. As a consequence, the first terms in Eqs. (3.16) and (3.17) are suppressed so as to be much smaller than all the other terms in these equations.
3.2.1 Case 1
3.2.2 Case 2
4 Dynamical two scalar field model
In this section, we explore the theory proposed in Sect. 2, whose action is given by Eq. (2.1), and we consider how to realize inflation by using the two dynamical two scalar fields \(\phi \) and u.^{7}
The definitions of the slowroll parameters in Eqs. (2.21) and (2.22) are used to describe the slowroll inflation in the present theory. In addition, the spectral index \(n_\mathrm {s}\) of the curvature perturbations and the tensortoscalar ratio r are supposed to be represented as Eqs. (2.24) and (2.23), respectively. This assumption has been justified in Ref. [200].
As a result, in this simplest case (of function J), the Planck results cannot be realized. Nevertheless, the model with two dynamical scalar fields leads to significantly different results in comparison with the inflationary model with a single scalar field. There is another possibility for considering the more viable types of the function J, e.g., Eq. (2.29). In such a case, it is quite difficult to analyze the equations analytically (for example, \(N_e\) is described as an integral equation).
5 Graceful exit from inflation
In this section, we investigate the graceful exit from inflation, namely, the instability of the de Sitter solution at the inflationary stage for the present theory. Especially, we demonstrate the instability of the de Sitter solution in the case of one dynamical scalar field in the Einstein frame. In this case, as shown in Sect. 3.2.1, the spectral index and the tensortoscalar ratio can be compatible with the observations by the Planck satellite. We also examine the contribution of an \(R^2\) term in the Jordan frame to the instability of the de Sitter solution during inflation.
We note that even if the other scalar field becomes dynamical, the procedure to examine the instability of the de Sitter solution is basically the same as the one demonstrated above. We examine the perturbations of the Hubble parameter by using the gravitational field equation with solutions for the equation of motions in terms of two dynamical scalar fields. Qualitatively, the form of the solution for the perturbations will be changed, but in principle there may exist a solution representing the property that the de Sitter solution is unstable.
The contribution of the \(R^2\) term in the Jordan frame to the instability of the de Sitter solution is included in the scalar field \(\lambda \) and its dynamics through the auxiliary field \(\Phi \) in the Einstein frame. This fact can be seen from the action in Eq. (2.2), and Eq. (3.3) with \(\Lambda = \mathrm {e}^{\lambda }\). Accordingly, it is considered that the \(R^2\) term can be related to the graceful exit from inflation, i.e., the instability of the de Sitter solution to describe the slowroll inflation.
6 Conclusions
In the present paper, we have studied inflationary cosmology in a theory where there exist two scalar fields nonminimally coupled to the Ricci scalar and an additional \(R^2\) term. We have investigated the slowroll inflation in the case of one dynamical scalar field and that of two dynamical scalar fields. We have analyzed the spectral index \(n_\mathrm {s}\) of the scalar mode of the density perturbations and the tensortoscalar ratio r in comparison with the observations of the recent Planck and BICEP2 results.
For the case of a single dynamical scalar field in the Jordan frame, if the number of efolds during inflation is \(50 \le N_e \le 60\), we have \(n_\mathrm {s} \approx 0.96\) and \(r = \mathcal {O} (0.1)\). On the other hand, in the Einstein frame, for the case of one dynamical scalar field, we can obtain \(n_\mathrm {s} \approx 0.96\) and \(r < 0.11\). These are consistent with the Planck 2015 results. Furthermore, for the case of two dynamical scalar fields in the Jordan frame, when \(50 \le N_e \le 60\), we obtain \(n_\mathrm {s} \approx 0.96\) and \(r = \mathcal {O} (0.1)\). As a result, we have found that in the present theory, the spectral index and the tensortoscalar ratio can be compatible with the recent Planck analysis.
We have also shown that in the present theory the de Sitter solution representing the inflationary stage is unstable, and therefore the universe can successfully exit from inflation. The \(R^2\) term in the Jordan frame is considered to be related to the instability of the de Sitter solution, namely, the graceful exit from the inflationary stage.
As the further developments on the cosmological consistent scenario in the present theory, it is possible to unify inflation in the early universe realized by the \(R^2\) term and the latetime cosmic acceleration by the dark energy component of one of the scalar fields. In this theory, dark matter can also be explained by the other scalar field. To construct such a unified scenario is the significant purpose in this work. In addition, the consequences in multiplefield inflation models seem to be more in correspondence with the observational data obtained from the Planck satellite than single field inflation models.
It is remarked that the action in Eq. (2.1) may be generalized so that the conformal invariance can be broken by an arbitrary function of the F(R) term. Thanks to the additional term in the gravity sector, the novel contributions to cosmology come from the breaking of conformal invariance of the theory. Thus, we have more possibilities to realize the unified scenario of inflation, dark energy, and dark matter mentioned above.
Footnotes
 1.
Recently, there have also been studied inflationary models with two/multiple scalar fields (or a complex scalar field with two scalar degrees of freedom) such as hybrid inflation [91] and a kind of its extensions [92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123].
 2.
Note that the Starobinsky or \(R^2\) inflation in the case without matter is equivalent to nonminimal Higgs inflation considered in Ref. [90].
 3.
In Ref. [154], inflationary cosmology has been studied in a theory with two scalar fields nonminimally coupling to the Ricci scalar.
 4.
 5.
Obviously, \(\sqrt{g}=\Lambda ^2\sqrt{\bar{g}}\).
 6.
Recall the facts that \(\phi =\phi _0\ne 0\), \(u=u_0\ne 0\), and \(u_0 \ne \pm \phi _0\).
 7.
It should be noted that in the framework of this theory, inflation is realized only for noninteracting scalar fields [199], whereas in the most general case, inflation in this theory has not been realized yet, and it is not so clear how to analyze the tensortoscalar ratio.
 8.
Even in this simplest case, the equations cannot be reduced to the onefield equations because of the kinetic terms.
Notes
Acknowledgments
This work was partially supported by the JSPS GrantinAid for Young Scientists (B) # 25800136 (K.B.), MINECO (Spain) project FIS201015640 and FIS201344881 (S.D.O.), and the RFBR Grant 140200894A (P.V.T.).
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