# Inflationary Universe from Anomaly-FreeFR-Gravity

Authors: Aroonkumar Beesham; Kazuharu Bamba
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

By adding a three-dimensional manifold to an eleven-dimensional manifold in supergravity, we obtain the action ofFR-gravity and find that it is anomaly-free. We calculate the scale factor of the inflationary universe in this model and observe that it is related to the slow-roll parameters. The tensor-scalar ratio Rtensor−scalar is in good agreement with experimental data.


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## Body

## 1. Introduction

One of the best models for considering an inflationary universe is suggested by modified theories of gravity likefR gravity, whose Lagrangian is a function fR of the scalar curvature R in a 4-dimensional universe [1–5]. To date, many models in this subject have been proposed. However, one of the best inflationary models in fR gravity is the pioneering Starobinsky R+R2 model, which still remains viable now in contrast to many other models proposed later and produces the best fit to existing observational data. Primordial curvature perturbations and gravitational waves in this model were first calculated in [6]. On the other hand, in these theories, the special property is the existence of the effective cosmological constant in such a way that early-time inflation and late-time cosmic acceleration are naturally unified within a single theory [1].Some of thesefR gravity theories, especially a power-law mechanism and Yang-Mills fR gravity, give the best fit values compatible with cosmological parameters like the tensor-to-scalar ratio within the permitted ranges derived by the Planck and BICEP2 observations [2]. In some other versions of these theories, a dynamical scalar field (φ) is coupled to gravity fR,φ whose dynamics allow for exit from inflation and which gives rise to the correct amount of inflation in agreement with observational data [3]. Also, in some modified gravity theories, F is a generic function of the curvature scalar and the Gauss-Bonnet topological parameter. Cosmological dynamical results which are obtained by two effective masses (lengths) correspond to both of these parameters and work, respectively, at early and very early epochs of the evolution of the universe [4]. Furthermore, modified teleparallel gravity leads to second-order equations and solves the particle horizon problem in a spatially flat cosmic space by providing an initial exponential expansion without resorting to an inflaton particle [5]. For more reviews on the connections between inflation, the dark energy problem, and modified gravity theories, see, for instance, [7].Now, the question that arises is whetherFR-gravity could be anomaly-free or not? Also, it is required to consider the role of anomaly cancellation in the evolution of the universe during the inflationary epoch. Recently, the exact forms of two-dimensional and four-dimensional anomalies from higher derivative gravity and FR-gravity have been calculated [8]. On the other hand, the conformal-anomaly driven inflation in FR-gravity without using the scalar-tensor representation has been studied. It has been shown that in FR-gravity, the curvature perturbations with their high amplitude which are consistent with observations and are created during inflation [6, 9].Besides these papers, there are some interesting works which connect M-theory toFR-gravity and inflation. For example, the first attempt to get FR from strings/M-theory is done long ago in [10]. On the other hand, the study of inflation in FR with scalars was done in k-essence fRfR gravity inflation [11, 12]. Also, the reconstruction of fϕRfϕR and mimetic gravity from viable slow-roll inflation has been discussed in [13]. And finally, the relation between Geometric Inflation and Dark Energy with Axion FRFR-gravity has been argued in [14]. These investigations advise us that there may be a relation between generalized gravity theories, M-theory, and inflation. Each gravity theory may have some anomalies which could be removed in a theory with higher dimensions. For example, all anomalies in string theory have been removed in 11-dimensional M-theory. Thus, we can conclude that anomalies in M-theory may be removed in a theory with 11+n-dimensions. In fact, first, we should consider the origin of M-theory and process of its birth on 11-dimensional manifold in a 11+n-dimensional manifold and then study its reduction to FR-gravity in four-dimensional universe. On the other hand, to find the relation between inflation and M-theory and gravity, we have to assume that our universe is located on a manifold which interact and exchange energy and fields with other manifolds in a higher dimensional world. These interactions force to manifolds to become larger and inflate. Motivated by these ideas, we propose a theory which lives on an (11+3)-dimensional manifold with two 11-dimensional manifolds and one 3-dimensional manifold. We will show that our universe is a part of an 11-dimensional manifold which is connected with the other 11-dimensional manifold by an extra 3-dimensional manifold. Two 11-dimensional manifolds interact with each other via exchanging fields which move along the 3-dimensional manifold. These fields are the main cause for the appearance of FR-gravity in the four-dimensional universe.This paper consists of two main parts. In Section2, we show that by adding 3-dimensional manifold to an 11-dimensional one, FR-gravity emerges. In Section 3, we obtain the cosmological parameters like the tensor to scalar ratio and compare with experimental data.


## 2. Emergence ofFR-Gravity on an (11+3)-Dimensional Manifold

In this section, we will assume that our universe has been constructed on a D3-brane. Thus, the action of gravity and matter in a 4-dimensional universe should be equal to the action of a D3-brane (see Figure1). This means that strings which live on a D3-brane produce gravity and different types of matter. Thus, each string (Xi) can be expanded in terms of curvatures (R), gauge fields (F), and scalars (ϕ) (see Figure 2). According to the Horava-Witten mechanism [15, 16], this D3-brane can be a part of a bigger 11-dimensional manifold on which the action of fields should be anomaly-free (see Figure 3). However, using the relation between strings and fields, we notice that the gauge variation of the action on this manifold is not zero, and some extra anomalies emerge. To remove these anomalies, we have to add an extra 11-dimensional manifold which is connected with the first 11-dimensional manifold by a 3-dimensional manifold. The extra fields which are produced by the interaction of the two manifolds produce FR-gravity (see Figure 4).Figure 1
4-dimensional universe on D3-brane.Figure 2
Different shapes of strings produce different types of fields.Figure 3
D3-brane is a part of 11-dimensional manifold.Figure 4
Two 11-dimensional manifolds interact via 3-dimensional manifold in 14-dimensional manifold.First, let us introduce the action of the D3-brane which is given by [17]:
(1)SD3=−TD3∫d4y−detγ¯ab+2πls2Fab,(2)γ¯ab=gμν∂aXμ∂bXν,(3)Fab=∂aAb−∂bAa,where Ab is the gauge field, Fab is the field strength, Xμ is the string, gμν is the metric, TD3 is the tension, and ls is the string length. Substituting X0=t and doing some mathematical calculations, the action of the D3-brane in equation (1) is given by [17]:
(4)SD3=−TD3∫d4y1+gij∂aXi∂aXj−4π2ls4FabFab,To construct our universe on a D3-brane, this action should be equal to the action of fields and gravity in a 4-dimensional universe. The action of matter and gravity is given by:(5)SGravity−Matter=−∫d4y−gR−gab∂aϕ∂bϕ−iψ¯γa∂aψ+1,where ϕ is the scalar field and ψ is the fermionic field. Putting equation (4) equal to equation (5) and doing some mathematical calculations, we obtain the relation between strings and matter fields:
(6)SGravity−Matter=SD3,(7)⇒Xi=∫dyi−1+4π2ls4FijFij+−ggijRij−gij∂iϕ∂jϕ−iψ¯γi∂iψ+12,where we have assumed TD3≃1 and 4π2ls4≃1. Using the above relation between matter fields and strings, we can reconsider the Horava-Witten mechanism. We will show that there are also other anomalies that have been ignored in previous considerations. Our goal is to show that by adding a 3-dimensional manifold to 11-dimensional spacetime in the Horava-Witten mechanism, all anomalies can be removed and an action without anomaly can be produced. This action is identical to the action of the modified gravity theory presented in [1–5, 7].At this stage, we introduce the Horava-Witten mechanism in 11-dimensional spacetime. In this theory, the bosonic part of the action in 11-dimensional supergravity (SUGRA) is given by [15, 16]:
(8)SBosonic−SUGRA=1κ¯2∫d11xg−12R−148GIJKLGIJKL+SCGG,(9)SCGG=−23456κ¯2∫M11d11xεI1I2⋯I11CI1I2I3GI4⋯I7GI8⋯I11,where εI1I2..Ik is the rank-k Levi-Civita pseudotensor, and CGG is used to denote the product term of the three-form field CI1I2I3 and four-form field GIJKL, which are directly related to the gauge field AI, field strength FIJ, and Ricci curvature RIJ [16]:
(10)GIJKL=−32κ2λ2εx11FIJFKL−RIJRKL+⋯,(11)δCABC=−κ262λ2δx11trεCFAB−εCRAB,(12)G11ABC=∂11CABC±23permutationsoftheindices11andABC+κ22λ2δx11ωABC,(13)δωABC=∂AtrεFBC+cyclicpermutationsofABC,(14)FIJ=∂IAJ−∂JAI,(15)RIJ=∂IΓJBB−∂JΓIBB+ΓJBAΓIAB−ΓIBAΓJAB,(16)ΓIJK=∂IgJK+∂KgIJ−∂JgIK,(17)GIJ=RIJ−12RgIJ,where ε and εC characterize infinitesimal gauge transformations [16]. Here, εx11 is 1 for x11>0 and −1 for x11<0 and also δx11=1/2∂εx11/∂x11 is the Dirac delta function. As usual [16], tr is 1/30th of the trace Tr in the adjoint representation for E8×E8. The ellipsis (⋯) denotes terms that are regular near x11=0 and hence vanish there [16]. It is clear from the above equation that G-fields can be produced by joining F-fields or R-fields. This means that G-fields are produced by joining two strings, each of which produces one gauge field or curvature (R) (see Figure 5).Figure 5
G-fields are produced by joining two strings.The gauge variation of the CGG term in the action yields the following equation [16]:
(18)δSCGG11=−23456κ¯2∫M11d11xεI1I2⋯I11δCI1I2I3GI4⋯I7GI8⋯I11≈−κ¯4128λ6∫M10Σn=15trFn−trRn+trFnR5−n,wheretrFn=trFI1I2..FI2n−1I2n=εI1I2..I2n−1I2nFI1I2..FI2n−1I2n or trRn=trRI1I2..RI2n−1I2n=εI1I2..I2n−1I2nRI1I2..RI2n−1I2n. The above terms cancel the anomaly of (SBosonic−SUGRA) in 11-dimensional manifolds [16]:
(19)δSCGG11=−δSBosonic−SUGRAanomaly.Thus,SCGG is necessary for anomaly cancellation. Our goal now is to find a good rationale for its inclusion. We also answer the issue of the relation between CGG terms in 11-dimensional supergravity and FR-gravity. In fact, we suggest a theory in which FR-gravity terms appear in the supergravity action without being added by hand. To this end, we choose a unified form for all particles and fields by using Nambu-Poisson brackets and properties of string fields (X). Solving equation (7) and assuming that changes of fields with respect to the coordinates be ingnorable, we obtain [18, 19]
(20)XI≈yI+yJFIJ−yJRIJ+yJ∂Iϕ∂Jϕ−iyJψ¯γJ∂Iψ+⋯⇒XI,XJ=ΣI,JεI′J′∂XI∂yI′∂XJ∂yJ′=FIJ−RIJ+∂Iϕ∂Jϕ−iψ¯γJ∂Iψ+⋯.where we introduced the scalar field ϕ. Also, AI is the gauge field and R is the curvature (R). In fact, the origin of all matter is the same and they are different shapes of strings. In the static state, all strings can be described by a unit vector (Xi). When these strings interact with each other or move, the initial symmetry is broken and fields and particles emerge. Using 4-dimensional instead of 2-dimensional brackets, we may derive the GG term GIJKLGIJKL in supergravity in terms of strings (X) [18, 19]:
(21)GIJKL=XI,XJ,XK,XL=εI′J′K′L′∂XI∂yI′∂XJ∂yJ′∂XK∂yK′∂XL∂yL′⇓∫d11xgGIJKLGIJKL=∫d11xgεI′J′K′L′∂XI∂yI′∂XJ∂yJ′∂XK∂yK′∂XL∂yL′εI′′J′′K′′L′′∂XI∂yI′′∂XJ∂yJ′′∂XK∂yK′′∂XL∂yL′′.Equation (21) helps us to extract the CGG terms from the GG terms in supergravity. To this aim, we will add a 3-dimensional manifold to the 11-dimensional manifold which connects it to other 11-dimensional manifold by applying the properties of strings (X) in the Nambu-Poisson brackets [19]:
(22)XI≈yI+yJFIJ−yJRIJ+yJ∂Iϕ∂Jϕ−iyJψ¯γJ∂Iψ+⋯⇓∂XI5∂yI5≈δyI5+⋯∂XI6∂yI6≈δyI6+⋯∂XI7∂yI7≈δyI7+⋯,(23)∫MN=3⟶∫yI5+yI6+yI7εI′5I′6I′7∂XI5∂yI′5∂XI6∂yI′6∂XI7∂yI′7=1+⋯,where ellipses (…) were used to represent higher-order derivatives. The integration is over a 3-dimensional manifold with coordinates (yI5,yI6,yI7), and consequently, the integration can be shown by ∫yI5+yI6+yI7=∫dyI5∫dyI6∫dyI7). This result shows that ignoring fluctuations of strings that leads to the production of fields, the result of integration over each 3-dimensional manifold tends to one. When we add one manifold to another, the integration will be the product of integration over each manifold.Extending the manifold over additional dimensions extends the integration volume element. By extending the 11-dimensional manifold in equation (21) with the 3-dimensional manifold of equation (23), we get [19]
(24)∫MN=3∫M11gGI1I2I3I4GI1I2I3I4=∫M11+yI5+yI6+yI7gεI′5I′6I′7GI1I2I3I4GI1I2I3I4∂XI5∂yI′5∂XI6∂yI′6∂XI7∂yI′7,where we notice that after making the identification [19]:
(25)CI5I6I7=εI′5I′6I′7∂XI5∂yI′5∂XI6∂yI′6∂XI7∂yI′7,we recover the CGG action in 11+3 dimensions [19]:
(26)SCGGN=11+3=∫M11+yI5+yI6+yI7gGI1I2I3I4GI1I2I3I4CI5I6I7.This equation has three interesting results: (1) the CGG term that appears in the supergravity action is a result of adding a 3-dimensional manifold to an 11-dimensional manifold. This manifold connects the first 11-dimensional manifold to the second one. Combining the 11-dimensional manifold with the 3-dimensional manifold yields 14-dimensional supergravity. The shape of the C-term is now clear in terms of string fields (XI). It is clear that C-fields are in fact G-fields such that one end of the strings is placed out of the 11-dimensional manifold and on the 3-dimensional manifold and for this reason, C-fields seem to have three ends only (see Figure 6).Figure 6
C-fields are produced from G-fields when one end of strings is placed on a 3-dimensional manifold and three ends are placed on an 11-dimensional manifold.To verify that the theory is correct, we should be able to get back the gauge variation of the CGG-action in equation (18) in terms of field strengths and curvature. To this end, using equations (20), (21), (23), and (25), we can calculate the gauge variation of C [19]:
(27)XI≈yI+yJFIJ−yJRIJ+yJ∂Iϕ∂Jϕ−iyJψ¯γJ∂Iψ+⋯⇓∂δAXI∂yI=δyI⇓∫MN=3+M11δACI5I6I7=∫MN=3+M11ΣI′5I′6I′7εI′5I′6I′7δA∂XI5∂yI′5∂XI6∂yI′6∂XI7∂yI′7=∫MN=3+M10ΣI′5I′6εI′5I′6∂XI5∂yI′5∂XI6∂yI′6=∫MN=3+M10FIJ−RIJ+∂Iϕ∂Jϕ−iψ¯γJ∂Iψ+⋯.=∫MN=3+M10FIJ−RIJ+∂Iϕ∂Jϕ−iψ¯γJ∂Iψ+⋯..where the ellipses (…) represent higher-order derivatives with respect to fields. Using equations (20), (21), (23), (25), and (27), we can calculate the gauge variation of the CGG action in the equation of (26):
(28)δSCGGN=11+3=δ∫M11+MN=3gεI1I2I3I4I1′I2′I3′I4′I5I6I7εI~5I~6I~7∂XI5∂yI~5∂XI6∂yI~6∂XI7∂yI~7GI1I2I3I4GI1′I2′I3′I4′=δ∫M11+MN=3gεI1I2I3I4I1′I2′I3′I4′I5I6I7εI~5I~6I~7∂XI5∂yI~5∂XI6∂yI~6∂XI7∂yI~7,(29)0.5in×εI~1I~2I~3I~4∂XI1∂yI~1∂XI2∂yI~2∂XI3∂yI~3∂XI4∂yI~4εI~′1I~′2I~′3I~′4∂XI1′∂yI~′1∂XI2′∂yI~′2∂XI3′∂yI~′3∂XI4′∂yI~′4=∫M10+MN=3gεI1I2I3I4I1′I2′I3′I4′I5I6εI~5I~6∂XI5∂yI~5∂XI6∂yI~6,(30)0.5in×εI~1I~2I~3I~4∂XI1∂yI~1∂XI2∂yI~2∂XI3∂yI~3∂XI4∂yI~4εI~′1I~′2I~′3I~′4∂XI1′∂yI~′1∂XI2′∂yI~′2∂XI3′∂yI~′3∂XI4′∂yI~′4=∫M10+MN=3gεI1I2I3I4I1′I2′I3′I4′I5I6εI~4I~5∂XI4∂yI~4∂XI5∂yI~5εI~′4I~6∂XI4′∂yI~′4∂XI6∂yI~6,(31)0.5in×εI~1I~2∂XI1∂yI~1∂XI2∂yI~2εI~′1I~′2∂XI′1∂yI~′1∂XI′2∂yI~′2εI~3I~′3∂XI3∂I~I3∂XI′3∂yI~′3=∫M10+MN=3gΣn=15trFn−trRn+trFnR5−n+∫M11+MN=3gΣn=15Σm=05−nεJ1J2..J2n−1J2nRJ1J2..RJ2n−1J2nεI1I2..I2m−1I2m∂I1ϕ∂I2ϕ..∂I2m−1ϕ∂I2mϕ−im∫M11+MN=3gΣn=15Σm=05−nεJ1J2..J2n−1J2nRJ1J2..RJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ−im∫M11+MN=3gΣn=15Σm=0n−5Σk=05−n−mεJ1J2..J2n−1J2nRJ1J2..RJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ×εI1I2..I2k−1I2k∂I1ϕ∂I2ϕ..∂I2k−1ϕ∂I2kϕ+∫M11+MN=3gΣn=15Σm=05−nεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2m∂I1ϕ∂I2ϕ..∂I2m−1ϕ∂I2mϕ−im∫M11+MN=3gΣn=15Σm=05−nεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ−im∫M11+MN=3gΣn=15Σm=05−nΣk=05−n−mεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ×εI1I2..I2k−1I2k∂I1ϕ∂I2ϕ..∂I2k−1ϕ∂I2kϕ+⋯.The first term in equation (31) cancels the anomaly in equation (18). However, the other terms yield the action of modified gravity and matter. In fact, to remove the anomalies, we have to add two extra actions to equation (9), one is related to FR-gravity, and the second corresponds to matter. We can write
(32)SSUGRA=SGG+SCGG+SFR+SMatter,(33)SGG=1κ¯2∫d14xg−12R−148GIJKLGIJKL,(34)SCGG=−23456κ¯2∫M11d14xεI1I2⋯I11CI1I2I3GI4⋯I7GI8⋯I11,(35)SFR=∫M11+MN=3gFR,ϕ,ψ,(36)SMatter=∫M11+MN=3gΣn=16Σm=0n−6εJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2m∂I1ϕ∂I2ϕ..∂I2m−1ϕ∂I2mϕ−im∫M11+MN=3gΣn=16Σm=06−nεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ−im∫M11+MN=3gΣn=16Σm=06−nΣk=06−n−mεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ×εI1I2..I2k−1I2k∂I1ϕ∂I2ϕ..∂I2k−1ϕ∂I2kϕ+⋯,where FR,ϕ,ψ can be given by:
(37)FR,ϕ,ψ=Σn=16Σm=0n−6εJ1J2..J2n−1J2nRJ1J2..RJ2n−1J2nεI1I2..I2m−1I2m∂I1ϕ∂I2ϕ..∂I2m−1ϕ∂I2mϕ−imΣn=16Σm=06−nεJ1J2..J2n−1J2nRJ1J2..RJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ−imΣn=16Σm=06−nΣk=06−n−mεJ1J2..J2n−1J2nRJ1J2..RJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ×εI1I2..I2k−1I2k∂I1ϕ∂I2ϕ..∂I2k−1ϕ∂I2kϕ+⋯.ThisFR,ϕ,ψ-gravity is now free of any anomaly on the (11+3)-dimensional manifold. We can write
(38)δSSUGRA=δSGG+δSCGG+δSFR+δSMatter=0.The above equation shows that total action is free of any anomaly. This means that any variation in the supergravity action (δSGG) produces some extra terms as new anomalies. These anomalies could be cancelled by variations in the Chern-Simon action (δSCGG). Although, variations of these actions also produce some new anomalies, and these could be removed by variation of the f-gravity action (δSFR). Thus, the f‐gravity terms help to remove all anomalies.In fact, this type of gravity is produced by exchanging gravitons, scalars, and fermions between two 11-dimensional manifolds via one 3-dimensional manifold. The order of curvatures depends on the number of dimensions of the manifolds and also on the number of scalars or fermions which are attached to the curvatures. This result is in good agreement with previous predictions in [8, 9].


## 3. Inflationary Universe inFR,ϕ,ψ-Gravity

In this section, we construct a 4-dimensional universe on a (11+3)-dimensional manifold. We will show that interaction between two 11-dimensional manifolds in this system has a direct effect on the evolution of the universe during the inflationary epoch. We will obtain the scale factor of the universe in terms of the parameters of the system. Using the action in equation of (36) and the relation between G, C, F, and R-terms in equation (17), we can obtain the following field equations:
(39)Rij−12Rgij+12gijFR,ϕ,ψ−RijF′R,ϕ,ψ+∂i∂jF′R,ϕ,ψ−gij∂i∂iF′R,ϕ,ψ+12gijGIJKLGIJKL−RijGIJKLGIJKL′+∂i∂jGIJKLGIJKL′−gij∂i∂iGIJKLGIJKL′+εI′J′K′CI′J′K′12gijGIJKLGIJKL−RijGIJKLGIJKL′+∂i∂jGIJKLGIJKL′−gij∂i∂iGIJKLGIJKL′+GIJKLGIJKL12gijεI′J′K′CI′J′K′−RijεI′J′K′CI′J′K′′+∂i∂jεI′J′K′CI′J′K′′−gij∂i∂iεI′J′K′CI′J′K′′+gijΣn=16Σm=0n−6εJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2m∂I1ϕ∂I2ϕ..∂I2m−1ϕ∂I2mϕ−imΣn=16Σm=06−nεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ−imΣn=16Σm=06−nΣk=06−n−mεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ×εI1I2..I2k−1I2k∂I1ϕ∂I2ϕ..∂I2k−1ϕ∂I2kϕ+⋯=0,where′ denotes the derivative with respect to the curvature R and
(40)∂∂xI∂∂∂Iϕ−∂∂ϕ×Σn=16Σm=0n−6εJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2m∂I1ϕ∂I2ϕ..∂I2m−1ϕ∂I2mϕ−imΣn=16Σm=06−nεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ−imΣn=16Σm=06−nΣk=06−n−mεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ×εI1I2..I2k−1I2k∂I1ϕ∂I2ϕ..∂I2k−1ϕ∂I2kϕ+⋯,(41)∂∂xI∂∂∂Iϕ−∂∂ϕ×GIJKLGIJKL+εI′J′K′CI′J′K′GIJKLGIJKL+FR,ϕ,ψ=0,(42)∂∂xI∂∂∂Iψ−∂∂ψ×Σn=16Σm=0n−6εJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2m∂I1ϕ∂I2ϕ..∂I2m−1ϕ∂I2mϕ−imΣn=16Σm=06−nεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ−imΣn=16Σm=06−nΣk=06−n−mεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ×εI1I2..I2k−1I2k∂I1ϕ∂I2ϕ..∂I2k−1ϕ∂I2kϕ+⋯,(43)∂∂xI∂∂∂Iψ−∂∂ψ×GIJKLGIJKL+εI′J′K′CI′J′K′GIJKLGIJKL+FR,ϕ,ψ=0,(44)∂∂xI∂∂∂IAJ−∂∂AI×Σn=16Σm=0n−6εJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2m∂I1ϕ∂I2ϕ..∂I2m−1ϕ∂I2mϕ−imΣn=16Σm=06−nεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ−imΣn=16Σm=06−nΣk=06−n−mεJ1J2..J2n−1J2nFJ1J2..FJ2n−1J2nεI1I2..I2m−1I2mψ¯γI1∂I2ψ..ψ¯γ2Im−1∂I2mψ×εI1I2..I2k−1I2k∂I1ϕ∂I2ϕ..∂I2k−1ϕ∂I2kϕ+⋯,(45)∂∂xI∂∂∂IAJ−∂∂AI×GIJKLGIJKL+εI′J′K′CI′J′K′GIJKLGIJKL+FR,ϕ,ψ=0.For a 4-dimensional universe, we haveR=12H2+6Ḣ, where H=ȧ/a is the Hubble parameter and a is the scale factor of the universe. Assuming that all fields depend only on time, and using the relation between scalars, gauge fields, curvatures, and strings in equation (17), we can solve equations (39), (41), (43), and (45) simultaneously and obtain
(46),at≈e−∫dtZt,(47)Zt≈Σn=16βnts2n1−tts2n1+Σm=06−nαmln2mts1−tts1+Σk=012−2n−2mλke2γkkts1−t/ts,(48)ϕt≈ψ1/2≈Σn=16βnts2n1−tts2nΣm=06−nαmln−2mtts×Z1/4t,(49)AIt≈εIΣm=16αmts1−tts2mΣk=06−mλk′e−2γk′kt/ts×lnZt,where ts is time of collision between the two 11-dimensional manifolds. This time is much larger than the present age of the universe (ts⋘13.5Gyr). The above result shows the coincidence with the birth of the universe (t=0) on the 11-dimensional manifold. The scale factor grows with time, while scalars, fermions, and gauge fields decrease and tend to zero at (t=ts). This is because with the passage of time, the two 11-dimensional manifolds come close to each other and all fields which live on the 3-dimensional manifold between them dissolve into the 11-dimensional manifolds. By the disappearance of these fields, gravitons and FR-gravity emerge that lead to the expansion and inflation of the universe (see Figure 7).Figure 7
Emergence of 4-dimensional universe on 11-dimensional manifold in11+3-dimensional space-time.We can test our model by calculating the magnitude of the slow-roll parameters and the tensor-to-scalar ratio (Rtensor−scalar) defined in [20] and compare with previous predictions:
(50)H=ȧa=−Ż⇒,(51)ε=−ḢH2=Z¨Ż2,(52)η=−H¨2HḢ=Z⃛2ŻZ¨,(53)Rtensor−scalar=16ε=16Z¨Ż2.Obviously, during the inflationary epoch, the age of the universe (t) is very smaller with respect to the time of collision between manifolds (ts) and, as a result, (ts−t−n≪1) where n is an integer. Using equation (53) and expanding the functions (eO and ln) by applying the Taylor series, we can derive the following results:
(54)0≪t≪ts⇒η≈Σβnts−t−2nln−2nts1−tts+Osmallerterms≪1,(55)ε≈Σαmts−t−2me−2γmmts1−t/ts+Osmallerterms≪1⇒Rtensor−scalar≪1.This equation indicates that the values of the slow-roll parameters and the tensor-to-scalar ratio are very much smaller than the ones which are consistent with the predictions of the experiments in ref [21]. Thus, this type of FR,ϕ,ψ-gravity which is produced due to the interaction of two 11-dimensional manifolds creates the correct values for the experimental parameters and can describe cosmological events.In Figures8 and 9, we have presented the scalar spectral index and the tensor to scalar ratio parameters in terms of time. It is clear that for the inflationary age of the universe and the colliding time (ts=33Gyr), the spectral index is less then or around 0.96 and the tensor to scalar ratio is less than 0.036, which is comparable with results of Planck [22].Figure 8
The scalar spectral index in terms of time (ts=33Gyr).Figure 9
Tensor to scalar ratio parameter in terms of time (ts=33Gyr).


## 4. Summary and Conclusion

Motivated by the successes of modified theories of gravity such asFR-gravity, in this paper, we have investigated how we could make FR-gravity anomaly-free. To achieve this, we considered the origin of the action of FR-gravity on an (11+3)-dimensional space-time. However, we found that this was not anomaly-free. So, we proposed a theory which lives on an (11+3)-dimensional manifold with two 11-dimensional manifolds and one 3-dimensional manifold. We showed that our universe is a part of one 11-dimensional manifold which is connected with the other 11-dimensional manifold by an extra 3-dimensional manifold. The two 11-dimensional manifolds interact with each other via exchanging fields which move along the 3-dimensional manifold. These fields are the main cause for the appearance of FR-gravity in the four-dimensional spacetime. This is because with the passage of time, the two 11-dimensional manifolds come close to each other and all fields which live on the 3-dimensional manifold between them dissolve into the 11-dimensional manifolds. As these fields disappear, gravitons and FR-gravity emerge that lead to the expansion and inflation of the universe. Hence, the interaction between the two 11-dimensional manifolds has a direct effect on the evolution of the universe during the inflationary era. We have calculated the scale factor of the inflationary universe and examined our model against WMAP and Planck experiments. We have obtained the slow-roll parameters and shown that the tensor-scalar ratio is much smaller than unity (Rtensor−scalar⋘1), which is consistent with experimental observations.



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