Perturbations of Single-field Inflation in Modified Gravity Theory

In this paper, we study the case of single field inflation within the framework of modified gravity theory where the gravity part has an arbitrary form $f(R)$. Via a conformal transformation, this case can be transformed into its Einstein frame where it looks like a two-field inflation model. However, due to the existence of the isocurvature modes in such a multi-degree-of-freedom (m.d.o.f.) system, the (curvature) perturbations are not equivalent in two frames, so in despite of its convenience, it is illegal to treat the perturbations in its Einstein frame as the"real"ones as we always do for pure $f(R)$ theory or single field with nonminimal coupling. Here by pulling the results of curvature perturbations back into its original Jordan frame, we show explicitly the power spectrum and spectral index of the perturbations in the Jordan frame, as well as how it differs from the Einstein frame. We also fit our results with the newest Planck data. Since there are large parameter space in these models, we show that it is easy to fit the data very well.


I. INTRODUCTION
As a cosmological paradigm of the early universe, inflation has achieved many reputations thanks to its capability in addressing several problems of Big Bang cosmology [1][2][3] such as horizon, flatness, unwanted relics and so on (see [4][5][6] for early works). Furthermore, inflation predicts a nearly scale-invariant primordial power spectrum, which was verified to high precision by Cosmic Microwave Background (CMB) observations [7][8][9]. Due to these salient features, more and more people has been paying attention to the study of inflation and more and more models has been explored. Observational data has also been developed to better and better precision in order to verify/falsify these models.
We are interested in a subclass of inflationary models, containing one scalar field in framework of modified gravity [10][11][12]. In very high energy regime, Einstein gravity cannot hold because of the quantization disaster. In order to have a complete model for the early universe, Einstein gravity has to be modified. Furthermore, the universe may also contain scalar field(s) as its matter part, which is responsible for phase transition. An interesting realization of this kind of model is the Asymptotic Safe Gravity scenario [13,14], where the gravity is modified due to the running of gravitational coupling constant, while scalar field acts as Higgs boson [15][16][17]. This model can also be motivated by supergravity and string theory which require scalar fields to behave like moduli fields. For example, when transformed into its Einstein frame, this model becomes a two-field model, which can exist in Pre-Big-Bang scenario [18], where one of the fields be-comes dilaton while the other is viewed as an axion. It also can be seen as a 4D effective action of low-energy heterotic M-theory [19] in Ekpyrotic cosmology [20].
In usual modified gravity models, it is convinient to transform them from its original Jordan frame to the Einstein frame, where they look like field theory models minimally coupling to gravity, which is easier for the analysis. The transformation is called conformal transformation, of which we change the scaling of the metric while keep the inner constructure unaltered. It has been proved that for pure f (R) modified gravity theories or single scalar field nonminimally coupled to gravity, the two frames are equivalent and one can directly use what he gets in Einstein frame as his final result. However, for the models we are going to discuss about, as has been addressed in [21], the two frames are no longer the same, and a difference is generated in the results in the two frames. As will be seen below, the difference is mainly due to the generation of isocurvature perturbations. Therefore, when the Einstein frame analysis is done, it is necessary to pull everything back to its Jordan frame to get the right results. In like manner, we also have to perform the constraint on observable in its Jordan frame with observational data.
Recently, PLANCK satellite has released the most stringent data to constrain inflation models [8,9]. In their analysis, some basic inflationary models were checked to see if they can satisfy the constraints, but many other models, including the ones we're talking about, were not. In this paper, we will also compare the cosmological observable of this model, such as power spectrum of curvature perturbation, in its Jordan frame, to the PLANCK data. Our numerical plot shows that it is very easy to make the theoretical results consistent with the observational data.
The present paper is organized as follows. Section II gives the background of our model. We transfer the model into Einstein frame and solve the equation of motion to get the inflationary solution in Einstein frame. In Section III, we study the perturbation theory of our model in detail in the Einstein frame and then pull back to the Jordan frame. One can see that the results of the two frames will be quite different. Section IV is devoted to the constraints on this model by use of PLANCK data. Finally, we conclude with a discussion in Section V. Note that, we will work with the reduced Planck mass, M p = 1/ √ 8πG, where G is the gravitational constant, and adopt the mostly-plus metric sign convention (−, +, +, +).

II. THE MODEL
We start with the action: with κ 2 = 8πG, where f (R) is an arbitrary function of the Ricci curvatureR, and L s = −∇ µ ψ∇ µ ψ/2 − V(ψ) is the lagrangian for the matter field ψ. For later convenience, we refer variables with tilde to those in Jordan frame while their correspondence without tilde are in Einstein frame. This is just a simple example of modified gravity accompanied with normal matter, but a big difference from pure modified gravity is that it contains more than one degree of freedom, where at large scales, the curvature perturbation is no longer a conserved variable but sourced by isotropic perturbations, thus the system will not be in an adiabatic state. The equation of motion of the system (1) can be written as: whereH is the Hubble parameter, F is defined as F ≡ ∂f /∂R, and ρ s and P s are energy density and pressure of the scalar part, respectively. Prime denotes derivative witht, the cosmic time in Jordan frame. One can also define the "effective" energy density and pressure for the whole system, which arẽ satisfying the Friedmann equations 3H 2 = κ 2ρ and −2H ′ = κ 2 (ρ +P ).
As is well known, the system (1) can be written into a Brans-Dicke form by simple field redefinition. Define ϕ ≡ F , U (ϕ(R)) = FR − f (R), the action (1) becomes: which looks like two scalars, but still with one of them nonminimally coupled to gravity. The nonminimal coupling can be removed by further transformation, however, by the price of modifying the scaling of space-time, namely conformal transformation. To do this, we need to define a new metric, which we call the metric in Einstein frame. The metric in Einstein frame connects to the original metric (the metric in Jordan frame) as: where Ω ≡ √ ϕ. Therefore by manipulation we have: where From the above one can see, via conformal transformation one can transform our model (1) into a minimal coupling two-field inflation model. The equation of motion for such two-field models contains two second-order differential equations, but no higher-order derivatives involving, thus is easier to be solved. That is a very important reason why people like to do the calculations in Einstein frame rather than directly in the original Jordan frame. For the system (8), we would like to first analyze the background dynamics, and define some slow-roll parameters for later use. From (8), The energy density and the pressure in Einstein frame are: where dot denotes derivative w.r.t. cosmic time in Einstein frame t. By varying the action w.r.t. φ and ψ, we can obtain the equations of motion for both fields: Finally, the Friedmann equations read: Hereafter we take the unit such that κ 2 = 1. The various slow-roll parameters can be defined as: which should be much smaller than unity during inflation, and satisfy the relation: ǫ φ + ǫ ψ = ǫ. Under slowroll approximation, the equation of motion (12) can be solved to give:

A. Calculation of the perturbations in Einstein frame
In this subsection, we first calculate the perturbations of our model in its Einstein frame, namely in form of a two-field inflation model (8). The analysis of two-field inflation model has been well-developed and the detailed calculation can be found in e.g. [12,21,23,24], and here to be more concise we will only summarize their results which is needed for our later study. First of all, the perturbed metric can in general be formulated as: while the field can also be perturbed as It is convenient to define variables of perturbation that are invariant under gauge transformation. The oftenused gauge invariant variables are: where Φ = Ψ is the Newtonian potential, R is the comoving curvature perturbation, and Q i is the gauge-invariant perturbation the i-th field. Moreover, δq defined according to the relation ∂ i δq = δT 0 i where T 0 i is the (0, i) component of the energy-momentum tensor of action (8). In spatial-flat gauge which will be applied in this paper, one chooses ζ = E = 0, and thus Q i identified with δφ i .
What we want to calculate is the comoving curvature perturbation R, because it can be directly connected with the observables to test our model. However, what we can handle from the action (8) is the field perturbations Q i . Therefore, we need to connect between these two variables. One convenient technique is the so-called "(instantaneous) Adiabatic-Entropy decomposition" [23,24], where one can decompose the field perturbations in the field-space {φ,ψ} into adiabatic and entropy directions, which traces along with/orthogonal to the field trajectory. Perturbations along the two directions are often called adiabatic and entropy perturbations of field respectively, which is proportional to the curvature perturbation R and the isocurvature perturbation, S. Note that in single-field limit, there is only one field perturbation and the field-space is also one-dimensional, so the field perturbation can only goes along the field trajectory producing adiabatic perturbation, thus S → 0. In multi-field case however, isocurvature perturbations orthogonal to the trajectory can also exist, and as will be seen later, can act as a source of the adiabatic ones on large scales.
Following [12,21,23,24], we give the expression of adiabatic and entropy field perturbations (Q σ and Q s ) as: whereσ = φ2 + e 2bψ2 is the velocity of the background part of the adiabatic field σ. From the definition of R one can have in flat gauge. Similarly one can define the isocurvature perturbation as S = (H/σ)Q s . From action (8) as well as Eq. (23), one can finally obtain the equation of motion of Q σ and Q s after a long derivation [24]: where and where and the right hand side which is proportional to k 2 is negligible on large scales. By manipulating these equations, we can finally get a solution of Q σ and Q s as: where we have taken the slow-roll limit and assume that the correlation between Q σ and Q s is small. H 3/2 is the first kind Hankel function of order 3/2. In the k → ∞ limit, from the above solution, one obtains the Bunch-Davies vacuum solution: deep inside the horizon as usual, while in the k → 0 limit, one gets at the Hubble-exit time. Here τ ≡ a −1 (t)dt is the conformal time.
The inflationary observables are usually expressed in terms of power spectra and correlation functions. One could define the power spectrum for Q σ and Q s as: (32) where δ mn denotes that there are no correlations between Q σ and Q s until the Hubble-exit. This is actually reasonable, since inside the horizon, the correlations of the variables are determined by the communication relations of their production-annihilation operators, which should commute because they are independent degrees of freedom. Although they have coupling, it should be subdominant inside the horizon where |kτ | ≫ 1. Considering the relation between R, S and Q σ , Q s (Eq. (24) and the sentences thereafter), one can have and from the results (31), one has: at the Hubble-exit time. Here we denote the values of variables at Hubble-exit time by a star in the subscript. The spectral index of scalar spectrum at Hubble-exit time is then evaluated as One can also calculate the tensor perturbation of this model. In Einstein frame where the model behaves like two-field inflation model, the tensor power spectrum is the same as that of GR, which is One can also define the tensor-scalar ratio of the model, which is r ≡ P T /P R .

B. Perturbations at large scales
What we concerned more about is the values of perturbations that reenters the horizon. In single-field case, there is only curvature perturbation which is conserved at superhubble scales, so it is reasonable to take the Hubble-exit values of perturbation the same as those of the Hubble-reenter values, and thus the calculations in the above subsection is enough. However, in multi-field case, there is also isocurvature perturbation which will source the curvature one and the latter will evolve even after Hubble-exit, which we must take into account. We in this subsection give the formulation of perturbations at large scales as briefly as possible, while more detailed calculations can be found in the preceding works [24,25]. From the equations of motion of the field perturbation (25) and (27), one can get the varying of the curvature and isocurvature perturbations at large scales as: where By integration over time one can get the expressions for R and S at late time, which is (in matrix form): where The power spectra at horizon-reentering therefore can be expressed as: assuming that R and S are uncorrelated at Hubble-exit time. Define the rotation angle Θ such that sin Θ = T RS / 1 + T 2 RS , and applying the definition of spectral index in Eq. (35), one can furtherly have: where the definition of T RS has been used.

C. Pulling back perturbations into Jordan frame
In the previous sections we have calculated the perturbations of the model (1), including the scalar and tensor perturbations, in its Einstein frame. This is what usually people do to analyse modified gravity theories, for it is much easier to deal with pure field theories in Einstein frame and one doesn't need to bother with the higher order curvature terms. For the case of pure modified gravity or single field nonminimal coupled with gravity, the physical quantities such as curvature perturbation are invariant in both frames, i.e.R = R [22], and one can directly use the Einstein frame results to compare with the observations. For our case, however, it is no longer the case. To see this, let's pull the results we get in Einstein frame back into the Jordan frame and see how different they are. From the conformal transformation (7), we have: where Ω = √ ϕ = √ F . It is useful to define the Jordan-frame-based slowvarying parameter: where one could find from Eq. (3) that first three of these parameters have the relation: while we have ǫ φ ≃ 3ω 2 from Eqs. (9) and (15). Furthermore, one could get that the slow-roll parameters in the two frames are related as (up to leading order): ǫ =ǫ +ω ≃ǫ ψ + 3ω 2 , η =ηǫ +zω ǫ +ω .
Furthermore, from perturbed metric (17), one can havẽ and the curvature perturbation in Jordan frame is: From this expression, one can see that the curvature perturbation is not conformal invariant any more. To see this more clearly, one can calculate the difference of R of two frames, which is: where we have used the background equations. Note also that δq is defined from the equation δT 0 i = ∂ i δq, so one has:δ respectively. Although the right hand side of Eq. (49) looks some complicated, it only contains terms that involves δφ and δψ. Actually in such a two-field system (or f (R)+single field in Jordan frame), there are only two degrees of freedom and δφ and δψ can become a complete set which can present everything. Therefore, after some straightforward calculation and making use of the inverse transformation of Eq. (23), we finally expressR − R in terms of δφ and δψ (or R and S, because of their one-to-one correspondance) as: where higher order terms are omitted and as it is perturbation on large scales, Eq. (37) is used.
From the above formula one can see that, the curvature perturbation R in the two frames differs by a quantity proportional to the isocurvature perturbation, S. That means some parts of isocurvature perturbations has now been transferred into the adiabatic ones during frame transformation. So before and after transformation, we are actually talking about different "adiabatic perturbations". Due to this reason, although mathematically frames can be transformed smoothly, the observables we are concerning are different, and this difference is physical rather than artificial. Contrarily, in solo-degree-offreedom (s.d.o.f.) system where the isocurvature perturbations do not appear, this difference will vanish andR and R will coincide, which gives the equivalence between two frames. Note that although we only show this point by taking a small example in this paper, in like manner, it can also be applicable for more complicated modified gravity models with m.d.o.f..
According to (52), the power spectrum ofR is: where P R , P S and C RS can be related to their values at Hubble-crossing via relations (41). It is convenient to assume that the coefficients A and B in Eq. (37) are nearly constants, so one simply have: T RS ≃ A B (e BN * − 1), T SS ≃ e BN * , where N * denotes the efolding number from Hubble-crossing time to the end of inflation. Taking this into account, one can straightforwardly get the final power spectrum of curvature perturbation in the original Jordan frame, namelyR, as: and from Eq. (42) one can get the spectral index in Jordan frame as: From Eqs. (54) we can see that, when C = 1,PR will coincide with P R , so C will behave as an estimator of the inequality between Einstein and Jordan frame in our model. From the expressions of n R in Eq. (56), one could get nontrivial constraints on slow-roll parameters, which will be different from what is done in Einstein frame. The difference comes from two sources, one is due to the frame transformation, and the other is caused by the deviation of C from 1, namely the inequality between two frames. Moreover, it is also straightforward to get the tensor spectrumP T in Jordan frame of our model from Eq. (36). After taking conformal tranformations (43) and keeping only leading order, the result will bẽ which coincides with P T . This is because since tensor degrees of freedom of perturbations are decoupled from the scalar degrees of freedom, and in such a system there is only one tensor degree of freedom, so for tensor the two frames are equivelant, and the tensor spectra in the two frames differ only by a conformal transformation. Eqs.

IV. FITTING WITH THE PLANCK DATA
In the previous section, we derived the perturbation generated by a "modified gravity+single scalar" system, and after calculating in Einstein frame, we pull it back to its original Jordan frame, which can be used to fit the data. Actually, one can see that the results are controlled by a series of parameters, namelyH * ,ǫ * ,η * ,ω * ,z * , F * and C, while C contains information after Hubble-exits, namely A, B and N * . Therefore, even we take explicit forms of f (R) to break the degeneracy of some of the parameters, there are still large parameter space that can easily make our model consistent with the data. Here for illumination we just show two simple examples. In the first example f (R) is taken to be of the form [26] f (R) = ξR n , F = nξR n−1 = nξ[6H 2 (2 −ǫ)] n−1 , (58) andω * andz * can be simply expressed in terms ofǫ * ,η * asω and Eqs. (54) and (56) becomes One can check that it returns to the results of standard inflation when ξ = n = 1.
Another example is the well-known Starobinsky model [4]: andω * andz * can be simply expressed in terms ofǫ * ,η * asω * ≃ − and Eqs. (54) and (56) becomes Interestingly, we find that in this case, although expressed using slow-roll parameters in Jordan frame, the expressions ofñ R has the same form of that in Einstein frame, namely Eq. (35), except for the C term. This holds for any value of ξ, and only depend on the quadratic scaling of the second term in f (R).
In figure 1 we plot the TT power spectrum with one set of parameters for each example, and compare them to the Planck data points. From the plot we can see that, with proper parameter choice, we can have TT power spectra which can be very well consistent with the Planck data. Note that there are large parameter spaces in both the two cases mentioned above, it is very easy to obtain such a result from both cases. We present our parameter choice in both cases in the caption of the figure, and in our numerical calculation, we obtained that A s ≈ 2.13 × 10 −9 , n s ≈ 0.9627 for the first example, while A s ≈ 2.17 × 10 −9 , n s ≈ 0.9628 for the second example, both consistent with the Planck constraints: ln(10 10 A s ) ≈ 3.089 +0.024 −0.027 and n s ≈ 0.9603 ± 0.0073 at 1σ level.

V. CONCLUSION
In this paper we discussed about a well-motivated subclass of inflationary models, namely a scalar field in f (R) modified gravity, and calculate the perturbations generated from this model. As a system of m.d.o.f., it is convenient to transform it into Einstein frame so that it becomes a minimal-coupling two-field system, the perturbations of which can be calculated in a standard way. However, contrary to the cases of pure f (R) modified gravity or single field models nonminimally coupled to gravity in which the Jordan and Einstein frames are equivalent, in our model the two frames are different. Therefore, one should "pull-back" the Einstein-frame results via the conformal transformations in order to take the "real" results of perturbations in the Jordan frame. In the solodegree-of-freedom system, however the observable quantities such as power spectrum and spectral indexes are given the same via computations in Jordan and Einstein frame, and one can directly use the results he gets in Einstein frame as his final results. The main reason that causes the difference in the two frames of our model is that, as a m.d.o.f. system, isocurvature perturbations will be generated, and the isocurvature perturbations in Einstein frame will contribute to the adiabatic perturbations in Jordan frame, so the adiabatic part of the perturbations in the two frames are no longer the same. In other words, although mathematically it has no problem to do such a transformation, physically the quantity we are concerning has changed. So the difference between Einstein and Jordan frames are physical, not only from conformal transformations. One can check that when the isocurvature fraction of the perturbations approaches zero, adiabatic perturbations in the two frames will again coincide. Our results are consistent with that in Ref. [21], though the analysis are different. For more general systems of m.d.o.f., it is straightforward to prove that the conclusions are the same.
We also plot the TT spectrum according to the Jordan frame results we've got and compare it to the Planck data, with two simple examples. In the first one f (R) takes the form of R n , while the other is the famous Starobinsky model, f (R) = R + ξR 2 . From the plot one can see that, since in both examples the parameter space is quite large, it is very easy to have the theoretical plots of the spectrum consistent with the data points. We also explicitly gave the spectrum amplitude and spectral indexes of the two examples respectively, which is well within the constraints given by the Planck data. Thus it also indicates that the model itself is interesting and deserves further investigations in the future.