A simple $F({\cal R},\phi)$ deformation of Starobinsky inflationary model

We study a model including a real scalar field $\phi$ non-minimally coupled to $F({\cal R})$ gravity, which is conformally equivalent to an Einstein-Hilbert theory, involving two real scalar fields. We consider three special cases of the potential of the field $\phi$ in the $F({\cal R})$-frame: a vanishing potential, a mass term and a Higgs potential. All these lead to non-trivial two-field potentials in the Einstein-frame which in particular directions resemble the well-known Starobinsky model. We find, that all these cases can yield viable inflationary models in complete agreement with current observational data.


Introduction
The mechanism of cosmological inflation [1][2][3][4][5] was first introduced in 1980's in order to solve crucial problems of the Big Bang Cosmology, such as the horizon, the flatness and the Monopole problems. Inflation is a period of accelerated (quasi-de Sitter) expansion of the very early Universe, which elegantly allows for near large-scale homogeneity and spatial flatness of our Universe. An extra bonus of introducing inflation in Standard Cosmology is that it can explain the formation of large-scale structure, being the only known mechanism to do this. Quantum fluctuations during the inflationary epoch presumably seeded the perturbations which grew under gravitational instability into the structures we observe today [6]. Due to this, the inflationary mechanism has been intensively studied resulting to a better theoretical understanding of it. Also, in recent years, the interest in inflationary cosmology has grown considerably because of the great amount of data made available sourcing from various cosmological surveys. Despite its successful predictions the origin of inflation is not well understood, as yet. It is more like a phenomenological construction, whose origin should be sought in some fundamental theory, such as high-energy particle physics or gravity.
In particle physics, superstring theory and supergravity, multiple scalar fields are involved and some of them may play the role of inflaton. Furthermore, in curved spacetime and in the context of renormalization of scalar fields we have the arise of non-minimal couplings between scalar fields and the Ricci scalar [7,8]. Thus it is reasonable to search for inflationary models which include many fields non-minimally coupled to gravity, whose potential energy dominates the energy-momentum tensor and drives inflation [9][10][11][12][13][14][15]. For a model of inflation to be viable it should be in agreement with the recent observational constraints for the spectral index, the tensor-to-scalar ratio and non-gaussianity (for multi-field inflation). These quantinties are defined in the context of cosmological perturbations [6,[16][17][18][19][20][21][22][23] and their constraints obtained from observetions of cosmic microwave background (CMB), according to [24,25]  Among the single-field models of inflation, various classes of models can be in agreement with the aforementioned constraints. The first is the well studied non-minimally coupled Higgs inflation [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] which provides a particle origin to inflaton, but is also strongly connected with gravity. The second is the class of models of chaotic inflation [45][46][47] and its variants, and another is natural inflation models [48,49]. Also a well-known motivated model is the Starobinsky model of inflation [1], which remarkably was proposed almost four decades ago and furnishes a gravitational origin to inflaton. This model can be seen as the simplest inflationary model within the context of F(R) theories of gravity [50][51][52][53][54][55][56][57][58][59][60], as the only extension from Einstein Gravity is the addition to the Hilbert -Einstein action of an extra R 2 term where above M is a parameter with dimensions of mass,g µν is the metric tensor andR is the Ricci scalar. In this paper, the reduced mass Planck is dimensionless and equal to 1, and we use the (+, −, −, −) spacetime signature notation. This action is classically equivalent, through a conformal transformation, to the following scalar-tensor theory with a non-minimal coupling between the scalar field φ and gravity Using the slow-roll inflation formalism and the agreement with the observables the value of M is restricted to M 1.3 · 10 −5 . The great success of Starobinsky inflation model and its elegant physical interpretation of inflaton has led to an intensive study of inflationary models that are extensions or modifications of this model in the framework of F (R) theories or F (R, φ) theories studied in the metric formalism  (see for instance [86] for a beautiful discussion of inflation in the Jordan frame). The latter have the meaning of being F (R) theories in the presence of scalar fields, which are in general non-minimally coupled to gravity. An alternative variational principle leading to the equations of motion of General Relativity, which has been intensively studied for cosmological purposes [87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102], is the Palatini formalism [52,54,[103][104][105], in which the metric tensor g µν and the connection Γ λ µν are treated as independent variables. Motivated by the multi-field scope of particle physics and the viability of Starobinsky inflation, in this paper we study the robustness of Starobinsky inflation in the presence of a scalar field non-minimally coupled to gravity both in the R and the R 2 term. This paper is organized as follows. In Section 2 we present the general theoretical setup of the model at hand. In Section 3 we specialize our study and obtain numerical results for the observables for the case where the pre-existing scalar field is massless and its potential is zero. In Section 4 we do the same work with the addition of a mass term for the pre-existing scalar field. In Section 5 we identify φ with the SM Higgs boson at the electroweak scale. We conclude and discuss potential extensions and future study of this model in Section 6.

Theoretical setup
Our starting point is an inflationary model including a real scalar field φ non-minimally coupled to F (R) gravity. This model, is described by the action, in the F (R)-frame, For this theory we can obtain a better physical understanding by working in the Einstein frame. It can easily be seen that this theory is classically equivalent to as the equations of motion for the field ψ, which plays the role of a Lagrange multiplier, yield The corresponding equations of motion for Φ are then Thus the action (2.2) can be written in the following form in the Jordan frame In order to pass to the Einstein frame we perform a conformal transformation of the metric tensorg µν = g µν /2ψ [7,8]. Under this transformation the Ricci scalar transforms as The action, after eliminating a total derivative, is given by (2.7) Finally, using the field redefinition 2ψ = e 2 3 ρ , which leads to a canonical kinetic term for the field ρ , we obtain the following form for the action in the Einstein frame: In this the potential V (φ, ρ) is given by An interesting model belonging to this class of theories is that proposed in [72] where In this f (φ) and M 2 (φ) are two generic functions, which should be positive defined in order to avoid ghosts. In this model one can easily find, using equation (2.9), that the potential V (φ, ρ) is given by the following expression which however is in disagreement with the result derived in [72] 1 . In the following we shall study the cosmological prediction of this model with the potential as given by (2.11). The equations of motion for the two scalar fields (ρ, φ) in a spatially flat Friedman-Robertson-Walker spacetime assuming that the fields are only time-dependent, based on the observed homogeneity and isotropy of our Universe at large scales, following from the action (2.8) are given bÿ where we denote˙≡ d/dt, V ,I = ∂V /∂φ I with I = φ, ρ and H =α/α is the Hubble rate. The Einstein equations for the action (2.8) lead to the following Friedmann equations In order to find the time-evolution of the fields ρ, φ and the scale factor we need just to solve the equations of motion (2.13), (2.14) and (2.15). The equation (2.16) is not independent but it is related to the other three equations of motion. It is worth noting that the action (2.8) can be seen as a special case of the well-studied generalized non-sigma model of multifield inflation (2.17) In the above expression, the Latin indices account for the number of the fields of the theory and G IJ is the metric tensor of the curved field space manifold. Then, in correspondence with the theory of Gravity, we have the following definition for the covariant derivative: with A I is an arbitrary vector and Γ K IJ being the corresponding Christoffel symbols in the curved field space, calculated by the expression In this generalized model the slow-roll parameters are defined [79] as and should obey the ordinary (from one-field inflation models) slow-roll conditions 1 and η 1 .
Also for this case of inflationary models, there have been derived some well known expressions for the spectral index and the tensor-to-scalar ratio that we measure today from the CMB spectrum. The definition of n S and r happens in the context of cosmological perturbations. In the framework of this theory we have quantum fluctuations of the fields sourcing perturbations in the metric and vice versa. Let us briefly mention, using the description used in [10][11][12], the basic concepts for the understanding of the quantities needed for the calculation of n S and r in the concept of multi-field inflation. For a more complete view of this theory we recommend the reader to see these articles and the references therein [9][10][11][12][13][14][15][18][19][20][21][22][23].
The results that we will quote in the following are obtained by keeping only first order terms in the expansion of the spacetime-dependent fields, ϕ(x µ ), around the time-dependent background fields, φ(t), and of the spacetime metric around the spatially flat FRW background. A useful quantity for the simplification of the analysis of the multi-field cosmological perturbations [12,18] is the length of the velocity vector for the background fields given bẏ from which one can define the adiabatic and isocurvature directions in the curved field space via the unit vectorsσ where the turn-rate vector is given by ω I ≡σ I +Γ I JKσ JφK and ω = |ω I | [14]. It can be proved [9,12,19,22] that the spectral index at time t * when the perturbations of pivot scale during the inflationary era crossed outside the Hubble volume for the first time, can be determined via the power spectrum of the curvature perturbations from the expression n S (t * ) ≡ 1 + ∂ ln P R ∂ ln k t=t * = 1 − 6 (t * ) + 2η σσ (t * ) . (2.25) In the above equation (t) is the slow-roll parameter defined by (2.20) and η σσ(t) is a slow-roll parameter defined by [12] η σσ ≡σ where M I J is a mass-squared matrix appearing in the equation of motion of the gaugeinvariant, with respect to spacetime gauge transformations up to first order in the perturbations, Mukhanov-Sasaki variables [6,19,20] for the perturbations, Q I . Q I are constructed from a linear combination of field fluctuations and metric perturbations. M I J is given by [9][10][11][12] M The second term in (2.27) with the Riemann tensor in the curved field space vanishes because it is the contraction of a symmetric with an anti-symmetric tensor. At late times and in the long-wavelength, the transfer of power from isocurvature to adiabatic modes affects the spectral index, which becomes where cos(∆) ≡ T RS /(1 + T 2 RS ) 1/2 and [10-13, 20] are the transfer functions which relate the gauge-invariant perturbations at the time of pivot scale, t * , to their values at some later time t. In equation (2.29) the function β(t) is given by the following expression As it regards the tensor-to-scalar ratio, r, due to the evolution of the tensor pertubations just as in single-field inflationary models [10,11,19], it can be determined from the ratio between the power spectrum of the tensor perturbations and that of the curvature perturbations, resulting to (2.32) From the above expression it is obvious that if the maximum value r max = 16 (t * ) is in agreement with the observed constraint (1.1), there is no need of calculation of the transfer functions, as far as r is concerned.
For the system at hand one can easily infer, by comparing (2.8) and (2.17), identifies φ 1 by ρ and φ 2 by φ, that the metric tensor G IJ has the following form where the unit vectorsσ φ ,σ ρ appearing in the equations above arê (2.38) So far we have not specified the functions U (φ), f (φ) and M (φ) defining the function F (R, φ), (2.10), which defines the model given in (2.1). In order to proceed to predictions of the inflationary observables within the context of this type of models in the following chapters, and for different forms of the potential U (φ), we choose the functions M 2 (φ) and f (φ) to be the same as those employed in [72], In these α and β are some constants signalling deviations from Starobinsky model. In the next chapters we will be concentrated on the calculation of the spectral index n s and the max value of the tensor-to-scalar ratio in the pivot scale, while we will not make predictions for the non-gaussianity, which we expect to be negligible (as in the pure Starobinsky model) or very small and thus in agreement with the constraints [24,25].

The Case U (φ) = 0
In this section we review the special case, already studied in [72], when φ is a free massless field, in the F (R)-frame. We have U (φ) = 0 and thus the scalar potential V (φ, ρ) becomes  In order to calculate the scalar spectral index and the tensor-to-scalar ratio in the pivot scale, we first solve numerically the equations of motion (2.13) and (2.14) to find the time evolution of the fields φ and ρ. Our results are shown in Figure 2. In all graphs displayed the time t is normalized by M . From Figure 2 we can see that at the end of inflation the field ρ oscillates near in its approach to the minimum of the potential, whereas the field φ does not. In fact φ drops off rapidly. The oscillation of the field ρ can be seen after the time t = 605. Using the numerical results for φ and ρ, we then solve the equation (2.15), find the time dependence of the scale factor a(t) and plot the time evolution of the Hubble parameter and the number of e-foldings, as shown in Figure 4. From the right image of Figure 4 we can find the pivot scale used for the calculation of the spectral index and the tensor-to-scalar ratio. Furthermore, using the relations (2.35) and (2.36) we calculate the slow-roll parameters and η, respectively, and we present their behaviour as a function of time in Figure 5. From that figure we can see that the slow-roll parameters and η indeed obey the slow-roll conditions (2.22). Finally, using relations (2.25) and (2.32) we calculate the spectral index n s and the max value of the tensor-to-scalar ratio in the pivot scale. For N 50 − 60 we find that n s = 0.965 ± 0.004 and r < 0.005 .   It is worth pointing out, that trying different values for the initial conditions of the fields we observed that for bigger values of φ we have smaller duration of the inflationary period (number of e-foldings) but always in agreement with the observational constraints (1.1). In fact we have a wide range of allowed initial values for the field φ, for every initial value of ρ, to obtain at least 50 − 60 e-foldings of inflation. Therefore we do not have fine tuning for the initial conditions since a wide range of them leads to realistic inflation.   Now we make a simple but physical extension of the calculation we made in the previous section, by adding a mass term for the field φ in our action. The scalar potential V (φ, ρ) in this case takes the form This model for the case of α = β = 0 was studied in references [73,79] where it was found that it yields viable inflation without the need of fine-tunning. Specifically, in [79], using the δN formalism, the non-gaussianity was calculated and was found to be in agreement with the observed constraints [24,25]. Thus our expectation of feasible non-gaussianity for the case α = 0.01 and β = 0.001 is strengthened. In Figure 6 we see the profile of the potential for the choice of parameters α = 0.01, β = 0.001 and m = M = 1.3 · 10 −5 . From this graph we observe, in contrast to the potential considered in the previous section, that in this potential we have a unique Minkowski vacuum, at the point (φ min , ρ min ) = (0, 0). For the mass m we study the cases between m/M = 1 and m/M = 10 3 . Our results for the fields, the Hubble parameter and the number of e-foldings, as functions of time for three typical values of the ratio m/M are presented in Figure 7 and Figure 8. In all the cases we use the same initial conditions for the fields ρ = 7.4 and φ = 6. From these figures and our results we find that with increasing of the ratio m/M we have new effects that are absent in the massless case. First, we observe that we have the damped oscillation of the field φ, whose frequency is increased with increasing the ratio m/M . Thus we may have a possible contribution of the field φ to the reheating as m increases. This effect can be also observed for the massless case we studied in the previous section for larger values of the constant α (larger than 1), but for these values the viability of the inflationary model breaks down. Second, we see that in the beginning we have a valid increase of the field ρ which first reaches a maximum value and then starts the slow-roll inflationary process. This maximum value is increased with increasing the ratio m/M . Specifically, in the case m/M = 1 the field ρ does not increase, in the case m/M = 10 2 increases till the maximum value 7.6 and in the case m/M = 10 3 till 8.4. Last, we notice that with increasing the ratio m/M the duration of the inflationary period becomes longer. From our results we find that the values for the observables in the case of including the mass term in the potential agree with the constraints (1.1), for values of m/M ranging from 1 to 10 3 , and thus also this case leads to viable inflation. For example for the case of m/M = 1 the spectral index and the tensor-to-scalar ratio are n s = 0.963 ± 0.004 and r < 0.005, while in the case of m/M = 10 3 , they are n s = 0.963 ± 0.003 and r < 0.005.

The Case
In the following, we shortly study the case where φ is identified as the Standard Model Higgs boson. The scalar potential V (φ, ρ) in this case takes the form We focus on the case where λ = m 2 h /2υ 2 0.13 is fixed by the measured Higgs vacuum expectation value υ 246 GeV and Higgs boson mass m h 125 GeV at the electroweak scale. The case β = 0 was exhaustively studied in [75] where it is also being studied the case of smaller values of λ that are suggested by the Standard Model RG flow, which drives the running coupling λ to very small values at high energy scales.
The potential (5.1) is positive defined and it contains a two-fold degenerated Minkowski minimum and a saddle-point, being a minimum for ρ and a maximum for φ, which respectively are given by From the above equations we observe that the places of the extrema of the potential and the values of the potential calculated there do not depend on the constant β and thus are the same with the places found for the case β = 0 studied within [75]. We study this model for different initial values for φ, φ = 6, 1, 0.6 and 10 −3 and ρ = 7.4. From Figure 9 we observe that we have the damped oscillation of the Higgs field φ, whose frequency is increased with increasing the initial condition. Thus we may have a possible contribution from the Higgs field to the reheating. For φ in 10 −3 we have no oscillations. The behaviour of the field ρ does not change in this range of initial values of φ. In all cases the spectral index and the tensor-to-scalar ratio are n s = 0.964 ± 0.004 and r < 0.005, leading thus to viable inflation.

Conclusion
Inspired by the success of the Starobinsky model and the multifield nature of theories of particle physics we study the inflationary model proposed in [72]. In the begining, we review the feauture of a general F (R, φ) theory of gravity, which is conformally equivalent to an Einstein-Hilbert theory including two scalar fields. Specializing to the case F (R, φ) = − 1 2 f (φ)R + 1 12M 2 (φ) R 2 and U (φ) = 0, as considered in [72], we found that it leads to a two-field potential differing from the one given in [72]. However this also yields a viable inflationary model, as concluded therein.
Moreover, we considered other cases, as well, by adding a mass term for the field φ in the action. We found that this case also yields a viable inflationary model, while the evolution of the fields depends mostly on the ratio m/M . More specifically, we found that with increasing the ratio m/M we have new effects that are absent in the massless case, such as the observation of damped oscillations of the field φ, whose frequency is increased with increasing the ratio m/M . The ratio m/M affects the duration of the inflationary period, too. Furthermore, in a Higgs-like potential, identifying the field φ with the Standard Model Higgs boson, we also found a viable inflation.
Within the goals of future work, in the context of this model, and in order to check its viability, would be the study of reheating and preheating after the end of inflation, but this lies beyond the scope of this paper. Regarding possible extensions of this model these mainly include considerations of other physical scalar potentials, in the place of the U (φ) studied here and/or the insertion of more complicated non-minimal couplings between gravity and the preexisting scalar field which are motivated by higher dimensional theories, such as non-minimal kinetic terms and d' Alembertian of Ricci scalar terms, or Supergravity theories.