Analysis of R^{p} inflationary model as p\geqslant 2

# Analysis of Rp inflationary model as p⩾2

Lei-Hua Liu Lei-Hua LiuInstitute for Theoretical Physics, Spinoza Institute and the Center for Extreme Matter and Emergent Phenomena (EMME), Utrecht University, Buys Ballot Building Princetonplein 5, 3584 CC Utrecht, the Netherlands
Tel.: +31 (30) 253 5907
22email: L.Liu1@uu.nl
###### Abstract

We study the inflationary model of Muller:1989rp () for using the result of Ref. Motohashi:2014tra (). After reproducing the observable quantities: the power spectral index , its corresponding running and the tensor to scalar ration in terms of e-folding number and , we show that inflation model is still alive as is from to . In this range, our calculation confirms that and agree with observations and is of order which needs more precise observational constraints. We find that, as the value of increases, all , and decrease. The precise interdependence between these observables is such that this class of models can in principle be tested by the next generation of dedicated satellite CMB probes.

###### Keywords:
Inflation, consistency relations, constraints
###### pacs:
98.80.-k, 98.80.Bp, 98.80.Es.

## 1 Introduction

Currently, model is the paradigm providing a consistent explanation for the acceleration expansion of universe, the formation of large scale structure, cold dark matter and even for the most mysterious dark energy. However, it still suffers from the horizon, flatness, homogeneity and so-called magnetic monopole problems. When supplementing with a scenario of inflation Starobinsky:1980te (); Guth:1980zm (); Linde:1981mu (), these problems can be solved elegantly. In this framework, we need an additional scalar field called inflaton to trigger the inflationary period. In order to relate to the standard model (SM), there are lots of SM particles can be produced from preheating process Kofman:1997yn (). Following Occam’s razor principle, if we embed the inflaton field in the a Higgs sector, one can rule out this model due to its large value of . However, it is still alive by introducing a non-minimal coupling between Ricci scalar and Higgs field  Bezrukov:2007ep ().

By introducing the non-minimal coupling, a notable alternative called modified gravity is proposed whose effective scalar fields are called scalarons – which can be generated by e.g. quantum effects. To be more precise, generic quantum fluctuations from matter around the Plank energy scale can generate higher derivative local gravitational operators in the effective action. One class of these operators are a function of the Ricci scalar denoted by . By introducing a Lagrange multiplier and transforming into the Einstein frame, these operators can play a role of inflaton. The most simple and successful model was proposed by Starobinsky Starobinsky:1980te () whose Lagrangian is , where is the reduced Plank mass and . Its predictions still agree very well with the current observational constraints. Ade:2015xua ().

The high accuracy of inflation model motivates us to propose various modified gravity models to mimic the evolution of universe model under the framework of gravity, i.e. by choosing suitable form for including non-linear terms, the accelerating expansion without cosmological constant can be reproduced Carroll:2003wy (); Capozziello:2002rd (); Sotiriou:2006qn (). These gravity models even can pass Solar sysmem tests Hu:2007nk (); Nojiri:2007cq (). Further, these models unify the inflation and cosmic acceleration Nojiri:2003ft (); Nojiri:2010wj (). Models such as , where and are functions of Ricci scalar, are still suffering from the singularity problem since the scalar mass and Ricci scalar are divergent in the very beginning of the Universe Starobinsky:2007hu (); Tsujikawa:2007xu (). This crucial problem can be solved by inserting the term into Appleby:2009uf (), similar idea was proposed for solving the finite time singularity Bamba:2008ut (); Nojiri:2008fk (). The dynamical part for inflation is the same, the deviation will appear in the reheat phase dominated by the kinetic term of scalaron Motohashi:2012tt (). Meanwhile, it enhances the tensor power spectrum Nishizawa:2014zra (). Inspired by the ultraviolet complete theory of quantum gravity, Ref. Huang:2013hsb () considers that a polynomial f(R) inflation model where , they find that is exponentially suppressed. Also nearly the Starobinsky inflationary model, Ref. Sebastiani:2013eqa () generalizes to the class of inflationary scalar potentials , in which can be included in where it is the polynomial of Ricci scalar to present various models of gravity. In the light of conformal transformation from Jordan frame to Einstein frame, one can reconstruct viable inflationary model Odintsov:2018ggm (); Kuiroukidis:2017dyz (); Oikonomou:2018npe (). Inspired by attractor Kallosh:2013hoa (); Kallosh:2015lwa (); Galante:2014ifa (), even one can reconstruct the gravity from attractor Miranda:2017juz ().

Thus, in order to find a most economical generating theory of inflation without suffering from the singularity problem, the so-called inflation was proposed Muller:1989rp (); Gottlober:1992rg (). inflation model could also give the correction to inflation Codello:2014sua (); Ben-Dayan:2014isa (); Rinaldi:2014gua (). Together with this framework, inflation can also be reproduced in higher dimensions through compactification Nakada:2017uka (); Ketov:2017aau (). From perspective of dark energy, the constraint for inflation model can also be given Geng:2015vsa (). All of these models give the value of that is little larger when compared to inflation model as requiring . Here, we consider the case of in and then we study the scalar spectral index , the tensor-to-scalar ratio and the running of scalar spectral index in order to compare with current observational constraints.

The organization of the paper is as follows. In Section 2, we present the inflation model and the slow-roll approximation is achieved. In Section 3, results are presented according to the consistency relation. Section 4, we give our main results and conclusion.

## 2 The model

Recently, Ref. Motohashi:2014tra () derives a consistency relation in inflationary model. Using their results, we study this model as . Firstly, we recap how to get this model from gravity. The effective action comes from the most economical generalization of inflation,

 S=∫d4x√−gM2p2f(R), (1)

where is the reduced Plank mass and and is a constant and .

Next we need to proceed to investigate inflation governed by action (1). This action is on-shell equivalent to

 S=∫d4x√−gM2p2[f(Φ)+ω2(R−Φ)], (2)

where , is a real scalar field (dubbed scalaron in Starobinsky:1980te ()) and is a Lagrange multiplier (constraint) field. Upon varying the acition (2) and solving the resulting equation we can obtain,

 f′(Φ)−ω2=0, (3)

where

 f′(Φ)=dfdΦ≡F(Φ), (4)

Inserting action (3) into action (2) which is on-shell equivalent to (1), we obtain:

 S=∫d4x√−gM2p2[f(Φ)+F(Φ)(R−Φ)]. (5)

Note that as a scalar field is non-minimally coupled to gravity via the . Next step is to transform the Jordan frame (action (5)) into Einstein frame by a conformal transformation , where is a some specific local function. Upon this transformation is executed, action (5) becomes

 S=∫d4x√−gEM2p2[Ω2F(RE−6gμνE∇Eμ∇EνΩΩ)−Ω4(F(Φ)Φ−f)]. (6)

By choosing an appropriate function for conformal transformation,

 Ω2=1F(Φ), (7)

and then partially integrating the second term in the bracket of action (6), and dropping the boundary term, the action becomes,

 S=∫d4x√−gE[M2P2RE−3M2PgμνE∇EμΩ∇EνΩΩ2−12Ω4(FΦ−f)]. (8)

The higher gravitational operator has disappeared, but a new dynamical scalar field appeared named scalaron field. Note that scalaron is of non-canonical kinetic form, it can be changed into the canonical term by a simple transformation to Einstein frame,

 ϕE=−MP2√6ln(Ω(Φ)), (9)

where the mapping between these two fields is chosen such that as . The field can have the opposite sign since the resulting potential would be of mirror symmetry around of the potential from (9). With this in mind, action (9) finally becomes,

 S=∫d4x√−gE[M2P2RE−12gμνE∂μϕE∂νϕE−VE(ϕE)], (10)

where denotes the Einstein frame potential,

 VE(ϕE)=M2P2FΦ−fF2. (11)

In light of Eqs.(4), (7) and (9), we can write down the formula for ,

 F(ϕE)=exp(√23ϕEMP)=1+(p−1)λΦp−1. (12)

This equation defines the mapping . From a theoretical perspective of Ref. Starobinsky:1980te (), gravity is an effective field theory and every effective field theory can be quantized, here these two dynamical quantized fields are inflaton and graviton . In the proceeding part, we will discuss the dynamics of classical fields (condensate state of quantum state from macroscopic perspective) and the (tree level) dynamics of first order quantum perturbations.

### 2.1 Background dynamics and Cosmological perturbations

Action (10) is usually considered as driving inflation and it provides the quantum field for supporting large expectation values for inflation. Assuming that the field is approximately homogeneous with respect to some space-like hypersurface, it can be decomposed into the inflaton part and a small perturbation as follows,

 ^ϕE(x)=ϕE0(t)+^φE(x),ϕE0(t)=⟨^ϕE(x)⟩≡Tr[^ρ(t)^ϕE(x)]. (13)

Similarly, the tensor metric can also be decomposed into two parts (in Einstein frame),

 ^gμν(x) = gbμν(t)+δ^gμν(x), (14) gbμν(t) = ⟨^gμν(x)⟩=diag(−1,a2E(t),a2E(t),a2E(t)), (15)

the form of is written in conformal time, so the scale factor should be a function of conformal time, . Alternatively, and probably better, already at this state one can fix the gauge to the traceless-transverse gauge. In this gauge tensor perturbations are given by the spatial part of the metric perturbation, and thus can be written in the form, , with and .

The dynamical equation for the inflaton condensate in the background of an expanding universe is governed by,

 ¨ϕE0(t)+3HE˙ϕE0(t)+dVEdϕE0=0, (16)

where is the Hubble parameter in Einstein frame and we have neglected the backreaction from quantum fluctuations. Since inflaton drives inflation, the universe’s dynamics is governed by Friedmann (FLRW) equations,

 H2E ≡ (˙aEaE)2=13M2P⎛⎝˙ϕ2E02+VE(ϕE0)⎞⎠ (17) ˙HE = −˙ϕ2E02M2P, (18)

where is scale factor in Einstein frame and . The equation of motion (EOM) for the inflaton perturbation and graviton perturbation are governed by,

 (d2dt2+3HEddt+k2a2E+d2VEdϕ2E0)φ(t,k)=0 (19) (d2dt2+3HEddt+k2a2E)h(t,k)=0, (20)

where and are the Fourier modes of and and here . After adopting the zero curvature gauge which means that the spatial scalar graviton perturbations vanish, then we obtain the scalar and tensor spectra:

 Δ2s(k) = Δ2s∗(kk∗)ns(k)−1=k38π2ϵEM2P|φ(t,k)|2 Δ2t(k) = Δ2t∗(kk∗)nt(k)=2k3π2M2P|h(t,k)|2=16ϵEΔ2s, (21)

where or and it is a fiducial comoving momentum, and are the amplitude of scalar and tensor spectra evaluated at and and are the scalar and tensor spectral indices, respectively. These two spectra are obtained in slow roll approximation. Then by performing the canonical quantization and choosing the Bunch-Davies vacuum, we obtain identical expressions for and on the super-Hubble scales as in Ref. [37]. In order to characterize the amplitude of tensor perturbations, one defines the tensor-to scalar-ratio,

 r(k=k∗=0.05 Mpc−1)=Δ2t∗Δ2s∗. (22)

Being equipped with these observable quantities, we find their newest constraint from Ref. Ade:2015xua (),

 ns=0.9655±0.0062(68%CL,PlanckTT+lowP,α=0), (23)

Next we define the running of the spectral index,

 α≡[dns(k)dln(k)]k=k∗. (24)

Ref. Palanque-Delabrouille:2015pga () was able to show error-bars of and . These results should be confirmed in a future observation. Meanwhile, there is no direct measurement for tensor perturbations. Instead, the literature quotes upper bounds. For example, BICEP2/Keck and Planck Collaborations found Ade:2015tva () , more recently Array:2015xqh () BICEP2/Keck collaboration finds,

 r<0.09(95%CL,atk∗=0.05 Mpc−1)(BICEP2/Keck). (25)

Future observations, in particular space missions, such as CoRE and LiteBird, will significantly improve the upper bound on tensor perturbations. In the following section, we will consider observable predictions of inflationary model as , in which the corresponding value for is around as .

### 2.2 Cosmological perturbations in Rp inflationary model

Here, these observable quantities , and its running index are related to inflationary model in Einstein frame as it is well known that they are frame-independent Prokopec:2013zya (). In the light of Eqs. (1112) and definition of , the potential in Einstein frame can be explicitly written as,

 V(ϕE)=V0e−2√23ϕEMp(e√23ϕEMp−1)pp−1, (26)

where and it agrees with Motohashi:2014tra (). This potential recovers inflation as , for which that was first proposed by Ref. Barrow:1988xi (), is the energy scale which can be determined by the amplitude of the observed power spectrum for primordial perturbations and . In order to illustrate potential as , we also reproduce the plot showed in Ref. Motohashi:2014tra ().

Figure (1) shows three cases of potential for inflation when . For , this potential recovers the well-known inflation potential. When , the potential shows a deviation from inflation. As , there is a maximum value for . This scenario is quite different from inflation, there is a decay process for potential on the right of . However, this process is not physical since Ricci increases as increases, as can be seen from Eq. (12). As a consequence, the scalaron must shrink. Thus, we only need to consider the part that is left from for physical inflationary period.

In what follows we consider the scalar power spectral index , tensor to scalar ratio and its running index . Most inflationary models exhibit attractor behavior, which means that the physical parameters (tensor and scalar spectra) can be expressed in terms of the inflation amplitude individually, such as is a function of , this can be considered as so-called slow roll approximation. are small in slow roll approximation and we expand in these parameters. Thus, we want to apply this slow roll attractor to our mdoel. From the canonical quantization of scalar and tensor perturbations (20) and (21), we can see that and can be expressed in this attractor regime in terms of geometric slow roll parameters,

 ns=1−2ϵE−ηE,nt=−2ϵE, (27) ϵE=−˙HEH2E,ηE=˙ϵEHEϵE. (28)

The spectral index is a function of , meanwhile its running can also expressed in terms of geometrical slow roll parameters,

 α=−ηE(2ϵE+ξE),ξE≡˙ηEηEHE. (29)

Apparently, also depends on . Since current observations only constrain the upper limits of , we can denote by . Based on slow-roll approximation, one can also express the potential slow roll parameters in terms of the inflationary potential,

 ϵV = M2P2(V′EVE)2, ηV = M2PV′′EVE, ξ2V = M4PV′EV′′′EV2E, (30)

where . Together with and Friedmann equation (18), Eq. (30) implies that,

 ϵV=ϵE,ηE=−4ϵV−2ηV, (31)

and therefore

 ns=1−6ϵV+2ηV,nt=−2ϵV. (32)

Furthermore, one can show that,

 r = 16ϵE=16ϵV=−8nt (33) α = 16ϵVηV−24ϵ2V−2ξ2V. (34)

Eq. (33) is the one field consistency relation which can test the validity of single field inflation. Notice that , and are of first order in terms of slow roll parameters, while is of second order in slow roll parameters and thus it is tiny. Our later calculation will confirm our discussion. Finally, it is useful to define the number of e-foldings , which in slow-roll approximation reads,

 N≈NE=∫tetHEdtE≃1M2P∫ϕeEϕEVEVE,ϕdϕE. (35)

where denotes the time at the end of inflation () and . Since Jordan frame is the physical frame, it is worth in investigating how large is the difference for the number of e-folding between Einstein frame and Jordan frame. One can easily obtain the following relation between

 HJ=˙ΩΩ+HE. (36)

where , is Hubble parameter in Jordan frame. Using this result into Eq. (35) and in the light of (12), we can derive a relation between the e-folding number,

 dNE=dNJ+dF2F. (37)

Upon taking account of Eqs. (7) and (9) with (37), we get that the difference for e-folding numbers in two frames is . Since is small in slow roll, we conclude that to good approximation, .

In the light of Eqs. (30), (11), (12) and Eq. (35), we can reproduce the most important observable quantities for , and from Ref. Motohashi:2014tra (),

 ns−1=−8(2−p)[(2−p)E2k+p(Ek−1)]3[2(p−1)Ek−p]2,r=64E2k(2−p)23[2(p−1)Ek−p]2,α=−32p(2−p)2Ek(Ek−1)(2Ek−3p+4)9[2(p−1)Ek−p]4, (38)

where . Arming with these three observable quantities in terms of the e-folding number and , in what follows we discuss observational constraints on , and .

## 3 Results

As the main result of this paper, we show how , and vary with the e-folding number and . The range of e-folding number here is .

In figure 2, we show how the tensor-to-scalar ratio depends on as a function of and . When approaches , it recovers the inflation in which as . There is a deviation from inflation varying with which shows decreases as enhances. By requring , the case of in inflation can be ruled out. As for the allowed range for , its magnitude lies from to within the reasonable range of .

In figure 3, we impose the identical constraints as showing in figure 2. We show how the scalar spectra index as a function of its running varies with and . As increases, the absolute value of will decrease and this trend changes dramatically as grows above . Thus, we can see that the validity of inflation is quite sensitive to . On the other hand, when e-folding number is of valid range from to adopting the same constraints as in (23), the corresponding value of is within . Thus, in order to detect in our model, the measurement ought to be improved about one order of magnitude with respect to current observations, which will occur when the next generation of CMB space observatories will be launched in space (such as COrE). Nevertheless, together with a detection of , an observation of would play an important milestone in testing various models in a near future. For completeness, we also give the plot of . Due to lacking more accurate data of and , we only show that the possible observables range as expected. Another feature of figure 4 is that enhances as decreases.

To summarize, we have shown that the observable such as , and are strong sensitive with in inflationary model when . The model is ruled out as because of rather small values of as shown in Figure (2). Due to its enhanced absolute value of , this class of inflationary model can be tested by future dedicated space CMB missions.

## 4 Conclusion

In this work, we analyse the inflationary model Motohashi:2014tra (), where is slightly larger than two. The effective potential for scalaron exhibits a local maximum. We show that there is inflation when scalaron rolls toward to the smaller value from the maximum as shown in Figure 1. One can also get inflation as the scalaron rolls towards the larger value, but it is not viable since it leads to a shrank universe. We perform an anlysis of observables in the model and show that this model is valid as which is presenting in Figure 23 and 4. When , in particular, the scalar spectral index becomes smaller than what observations suggest which can be shown in Figure 2. Generally, as increases, , and decrease, the next generation of CMB observable may be challenging.

## 5 Acknowledgments

I am grateful for inspiring discussions with Prof. T. Prokopec. This work is in part supported by the D-ITP consortium, a program of the Netherlands Organization for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). L.H Liu is funded by the Chinese Scholarschip Council (CSC).

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