Observational constraints on slow-roll inflation coupled to a Gauss-Bonnet term

# Observational constraints on slow-roll inflation coupled to a Gauss-Bonnet term

Seoktae Koh Department of Science Education, Jeju National University, Jeju, 690-756, Korea    Bum-Hoon Lee    Wonwoo Lee    Gansukh Tumurtushaa Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea
Department of Physics, Sogang University, Seoul 121-742, Korea
###### Abstract

We study slow-roll inflation with a Gauss-Bonnet term that is coupled to an inflaton field nonminimally. We investigate the inflationary solutions for a specific type of the nonminimal coupling to the Gauss-Bonnet term and inflaton potential both analytically and numerically. We also calculate the observable quantities such as the power spectra of the scalar and tensor modes, the spectral indices, the tensor-to-scalar ratio and the running spectral indices. Finally, we constrain our result with the observational data by Planck and BICEP2 experiment.

## I Introduction

Recent experiments and observations including Planck Ade:2013zuv (), LHC Chatrchyan:2012ufa (), and BICEP2 Ade:2014xna () confirmed that the inflation paradigm is believed to be successful for explaining the evolution of our Universe and generation of large scale structure formation. The cosmic microwave background (CMB) observations by Planck and WMAP imply that our Universe is Gaussian, adiabatic, and nearly scale invariant. Although there are some debates Ijjas:2013vea (), the Planck data seem to favor the inflationary model with the simple scalar field potential, especially the convex-type potential Ade:2013uln (). But recent BICEP2 combined with Planck data seems to favor the concave-type potential, especially the potential.

While Planck and WMAP provide the upper bound on the tensor-to-scalar ratio (), recent BICEP2 telescope Ade:2014xna () at the South Pole reported the detection of B-mode polarization signal Michael:2014 (),Raphael:2014 () which is generated by the tensor perturbation (gravitational wave modes) in an inflationary period. According to BICEP2, at with disfavored at . This tensor-to-scalar ratio value is larger than the upper bound by Planck + WMAP. It has been widely studied how to reconcile this discrepancy between two data, and one simple resolution, which was suggested in Ref. Ade:2014xna (), is to consider the running spectral index, .

Although inflation is believed to solve a lot of the outstanding problems of the standard big bang cosmology such as the horizon and flatness problem, there are still several unsolved problems in an inflation scenario, for example, the flat potential problem, initial singularity problem, and quantum gravity (trans-Planckian problem).

Especially, if we think over the very early Universe approaching the Planck scale, we could consider Einstein gravity with some corrections as the effective theory of the ultimate quantum gravity. For instance, the higher derivative terms of gravity with nontrivial gravitational self-interactions naturally appear in the low energy limits of string theories. The presence of curvature squared terms such as a Gauss-Bonnet (GB) combination does not have any ghost particles as well as any problem with the unitarity. Additionally, the order of the gravitational equation of motion, the second-order derivatives of the metric tensor, does not change if there is no nonminimal coupling to a Gauss-Bonnet term Callan:1985ia (). Fortunately, the theory with a nonminimally coupled Gauss-Bonnet term could provide the possibility of avoiding the initial singularity of the Universe Antoniadis:1992rq (). It may violate the energy condition thanks to the presence of the term in the singularity theorem Hawking:1969sw (). In this perspective, one could introduce the Einstein theory of gravity having a scalar field with a nonminimally coupled Gauss-Bonnet term as the effective theory added a quantum correction.

Generally, the Gauss-Bonnet term in four dimensions is known as the topological term, so the dynamics is not influenced by the Gauss-Bonnet term. In order to consider the effect of the Gauss-Bonnet term on the spacetime as well as the field evolution, the Gauss-Bonnet term is required to be coupled to the matter field. Recently a number of papers with this motivation were studied Hwang:2000 (),Kawai:1999 () and discussed phenomenology in detail in Satoh:2008 (),Satoh:2008a (),Satoh:2010 (),Guo:2010jr () and Jiang:2013gza (). In Refs. Guo:2010jr (),Jiang:2013gza (), the authors studied the specific inflationary model with the Gauss-Bonnet term constrained by the WMAP data in Guo:2010jr () and by the Planck data in Jiang:2013gza (). They analytically derived the power spectra of the scalar and tensor perturbations. They employed a monomial potential and an inverse monomial Gauss-Bonnet coupling that satisfies . These choices of the potential and Gauss-Bonnet coupling provide the relatively large parameter values, , to be consistent with observations and showed that a positive (or negative) coupling leads to a reduction (or enhancement) of the tensor-to-scalar ratio.

In this work, we try to relax the condition and then constrain from the recent observations by Planck and BICEP2. We also calculate the spectral indices of the scalar and tensor perturbations, the tensor-to-scalar ratio and the running spectral index.

The outline of this paper is as follows: In Sec. II, we set up the basic framework with the Gauss-Bonnet term for this work. The -folding numbers are calculated and then give a constraint on the model parameter. In Sec. III, we briefly review the linear perturbations with the Gauss-Bonnet coupling term and then calculate the observable quantities such as the power spectra, the spectral indices, the tensor-to-scalar ratio and the running spectral indices. In Sec. IV, we examine the specific models consistent with our motivations. We compare our result with the observational data by the Planck data and recent BICEP2. Finally, we summarize our results in Sec. V.

## Ii Slow-roll inflation with the GB term

We consider an action with the Gauss-Bonnet term that is coupled to a scalar field

 S= ∫Md4x√−g[12κ2R−12gμν∂μϕ∂νϕ−V(ϕ)−12ξ(ϕ)R2GB], (1)

where is an inflaton field with a potential , is the Ricci scalar curvature of the spacetime , is the Gauss-Bonnet term, and . The Gauss-Bonnet coupling is required to be a function of a scalar field in order to give nontrivial effects on the background dynamics.

Varying the action (1) with respect to and yields the Einstein and field equation

 Rμν−12gμνR=κ2(∂μϕ∂νϕ−12gμν(gρσ∂ρϕ∂σϕ+2V)+TGBμν), (2) □ϕ−Vϕ−12TGB=0, (3)

where . and are the energy-momentum tensor and its trace for the Gauss-Bonnet term, respectively, which are given by

 TGBμν= 4(∂ρ∂σξRμρνσ−□ξRμν+2∂ρ∂(μξRρν)−12∂μ∂νξR) −2(2∂ρ∂σξRρσ−□ξR)gμν, (4) TGB= ξϕR2GB. (5)

In a spatially flat Friedmann-Robertson-Walker universe with a scale factor ,

 ds2=−dt2+a(t)2δijdxidxj, (6)

the background Einstein and field equations yield

 H2=κ23(12˙ϕ2+V+12˙ξH3), (7) ˙H=−κ22(˙ϕ2−4¨ξH2−4˙ξH(2˙H−H2)), (8) ¨ϕ+3H˙ϕ+V,ϕ+12ξ,ϕH2(˙H+H2)=0, (9)

where a dot represents a derivative with respect to the cosmic time , denotes the Hubble parameter, and . Since is a function of , implies . If is a constant, then Eqs. (7)–(9) are reduced to those for standard inflation without the Gauss-Bonnet coupling.

In this work, we consider slow-roll inflation with the inflaton potential and Gauss-Bonnet coupling satisfying the slow-roll approximations

 ˙ϕ2/2≪V,¨ϕ≪3H˙ϕ,4˙ξH≪1,¨ξ≪˙ξH. (10)

In addition to the usual slow-roll approximations, we introduce two more conditions related to the Gauss-Bonnet coupling.

To reflect these approximations, we introduce the slow-roll parameters,

 ϵ =−˙HH2,η=¨HH˙H,ζ=...HH2˙H, δ1 =4κ2˙ξH,δ2=¨ξ˙ξH,δ3=...ξ˙ξH2. (11)

We have checked the validity of the new slow-roll parameters during an accelerating phase numerically in Fig. 1.

If the slow-roll approximations (10) are taken into account, the background equations, (7)-(9), reduce to

 H2≃κ23V, (12) ˙H≃−κ22(˙ϕ2+4˙ξH3), (13) 3H˙ϕ+V,ϕ+12ξ,ϕH4≃0, (14)

which allows us to obtain the number of -folds

 N(ϕ)=∫tetHdt≃∫ϕϕe3κ2V3V,ϕ+4κ4ξ,ϕV2dϕ≡∫ϕϕeκ2Qdϕ. (15)

where a is determined from the condition and

 Q≡V,ϕV+43κ4ξ,ϕV. (16)

If we choose and , the number of -folds are calculated assuming is negligible compared to as

 N=∫ϕiϕeκ2Q≃κ2ϕ2i2n2F1(1,1n,1+1n;−αϕ2ni), (17)

where is the hypergeometric function and . We plot the number of -folds with for (solid line) and (dashed line) in Fig. 2. The condition of requires for and for . Here, is the value when becomes nearly constant. We find that approaches as decreases to . Because the hypergeometric function is constant for , cannot become larger than unless increases. and are required to obtain in Fig. 2.

It is convenient to express the slow-roll parameters (11) in terms of the potential and Gauss-Bonnet coupling:

 ϵ= 12κ2VϕVQ, (18) η= −VϕϕQκ2Vϕ−1κ2Qϕ, (19) ζ= VϕϕϕQ2κ4Vϕ+VϕϕQ22κ4V+3VϕϕQϕQκ4Vϕ+VϕQϕQ2κ4V +1κ4Q2ϕ+1κ4QϕϕQ, (20) δ1= −4κ23ξϕVQ, (21) δ2= −ξϕϕQκ2ξϕ−VϕQ2κ2V−1κ2Qϕ, (22) δ3= ξϕϕϕQ2κ4ξϕ+3ξϕϕVϕQ22κ4ξϕV+3ξϕϕQϕQκ4ξϕ+VϕϕQ22κ4V +2VϕQϕQκ4V+1κ4Q2ϕ+1κ4QQϕϕ. (23)

Equation (14) becomes for and

 ˙ϕ≈−√αξ0κ6(1+αϕ4), (24)

and then we get the solution assuming -term is negligibly small from Fig. 2,

 ϕ(t)∼−√αξ0κ6t+const. (25)

We find that this slow-roll trajectory is the attractor solution in Fig. 3. The slow-roll trajectories of Eqs.(12)–(14) were proved to be the attractor solutions generally when the Gauss-Bonnet term is coupled to the scalar field in Ref. Guo:2010jr (). We compare the attractor behavior for three cases: standard chaotic inflation (upper), chaotic inflation with the monomial Gauss-Bonnet coupling (middle), and chaotic inflation with the inverse monomial Gauss-Bonnet coupling (below) (which was considered in Ref. Guo:2010jr ()) in Fig. 3.

## Iii Linear Perturbations and Power Spectra

We briefly review the linear perturbations with the Gauss-Bonnet coupling in this section.

The linearized metric in the comoving gauge in which takes the form

 ds2=a(τ)2[−dτ2+{(1−2R)δij+hij}dxidxj], (26)

where represents the curvature perturbation on the uniform field hypersurfaces and is the tensor perturbation that satisfies .

If we perform the Fourier transform of and ,

 R(τ,x)= 1zs∫d3k(2π)3/2vs(τ,k)eik⋅x, (27) hij(τ,x)= 2zt∑λ∫d3k(2π)3/2vλt(τ,k)ϵλ,ijeik⋅x, (28)

where is a polarization tensor, Sasaki-Mukhanov equations for and are derived from linearizing Eqs. (2)–(3)

 v′′s+(c2sk2−z′′szs)vs=0, (29) v′′t+(c2tk2−z′′tzt)vt=0, (30)
 zs ≡  ⎷a2(˙ϕ2+6˙ξH3Δ)H2(1−12Δ)2,Δ=4κ2˙ξH1−4κ2˙ξH, (31) zt ≡√a2κ2(1−4κ2˙ξH), (32)

and

 c2s≡ 1+2(˙H−κ2˙ξH(H2+4˙H)+κ2¨ξH2)Δ2κ2˙ϕ2+6κ2˙ξH3Δ, (33) c2t≡ 1−4κ2(¨ξ−˙ξH)1−4κ2˙ξH. (34)

Here, a prime represents a derivative with respect to the conformal time .

and , where , can be written in terms of the slow-roll parameters Guo:2010jr ()Hwang:2005hb () using the definitions of the slow-roll parameters (11):

 zs=   ⎷a2κ22ϵ−δ1(1+2ϵ−δ2)+32δ1Δ(1−12Δ)2,Δ=δ11−δ1, (35) zt= √a2κ2(1−δ1), (36) c2s= 1−(4ϵ+δ1(1−4ϵ−δ2))Δ24ϵ−2δ1−2δ1(2ϵ−δ2)+3δ1Δ, (37) c2t= 1+δ1(1−δ2)1−δ1, (38)

where we have used the following relation from Eqs. (7)–(8):

 κ2˙ϕ2H2=2ϵ−δ1(1+2ϵ−δ2). (39)

If one keeps the leading order of the slow-roll parameters in using (35)–(36), Eqs. (29)–(30) become

 v′′A+(c2Ak2−ν2A−1/4τ2)vA=0, (40)

where the parameters are given by up to leading order in slow-roll parameters

 νs ≃32+ϵ+2ϵ(2ϵ+η)−δ1(δ2−ϵ)4ϵ−2δ1 (41) νt ≃32+ϵ. (42)

In deriving (40), we use the following relation:

 τ =−1aH11−ϵ. (43)

One can obtain the exact solutions for (40) assuming that the slow-roll parameters are constants,

 vA =√π|τ|2[cA1(k)H(1)νA(cAk|τ|)+cA2(k)H(2)νA(cAk|τ|)], (44)

where are the first and second kind Hankel functions. are the coefficients that are determined from the initial conditions and satisfy the normalization conditions

 |cA2|2−|cA1|2=1. (45)

If we adopt the Bunch-Davies vacuum for the initial fluctuation modes at by taking the positive mode frequency, the initial modes are given by

 vA=1√2cAkeicAk|τ|. (46)

These modes correspond to the choice of the coefficients

 cA1=ei(νA+12)π2,cA2=0, (47)

where we have used the asymptotic form of the Hankel functions in the limit ,

 H(1,2)νA(x)∼√2πxe±i(x−(νA+12)π2). (48)

Then the exact solution (44) becomes

 vA=√π|τ|2ei(νA+12)π2H(1)νA(cAk|τ|). (49)

The power spectra of the scalar and tensor modes are calculated with (49) on the large scales. Since the first kind Hankel function is approximated in the large scale limit () as

 H(1)νA∼21−e2iνAπ{1Γ(1+νA)(x2)νA−eiνAπΓ(1−νA)(x2)−νA}, (50)

where the second term is dominant, one can obtain the power spectra for the scalar and tensor modes on the large scales

 Ps =k32π2∣∣∣vszs∣∣∣2 ≃csc2νsππD2sΓ2(1−νs)1c3s|τ|2a2(csk|τ|2)3−2νs, (51) Pt =2k32π2∣∣∣2vtzt∣∣∣2 ≃8csc2νtππD2tΓ2(1−νt)1c3t|τ|2a2(ctk|τ|2)3−2νt, (52)

where the factor 2 of the tensor power spectrum comes from the two polarization states and we define

 zA ≡DAa2, D2s =2ϵ−δ1(1+2ϵ−δ2)+32δ1Δκ2(1−12Δ)2, D2t =1−δ1κ2.

The spectral indices of the scalar and tensor modes and the tensor-to-scalar ratio are given by

 ns−1 ≡dlnPsdlnk =3−2νs≈−2ϵ−2ϵ(2ϵ+η)−δ1(δ2−ϵ)2ϵ−δ1, (53) nt ≡dlnPtdlnk=3−2νt≈−2ϵ, (54) r ≡PtPs≈8(2ϵ−δ1). (55)

We can also calculate the running spectral indices of the scalar and tensor modes

 dnsdlnk≈ −2ϵ(2ϵ+η)+(2ϵ(2ϵ+η)−δ1(δ2−ϵ))2(2ϵ−δ1)2 −2ϵ(8ϵ2+7ϵη+ζ)+δ1(ϵ2+ϵη+ϵδ2−δ3)2ϵ−δ1 (56) dntdlnk≈ −2(2ϵ2+ϵη). (57)

where we have used from (11)

 dϵdlnk =2ϵ2+ϵη, (58) dηdlnk =ϵη−η2+ζ, (59) dδ1dlnk =δ1(δ2−ϵ), (60) dδ2dlnk =ϵδ2−δ22+δ3. (61)

## Iv Models

In this section, we calculate the , and for the specific models using Eqs. (III)–(56) and then constrain our model predictions with the recent CMB observational data from Planck and BICEP2.

### iv.1 Exponential potential with an exponential Gauss-Bonnet coupling

Let us start with the exponential potential and exponential coupling to the GB term

 V(ϕ)=V0e−λϕ,ξ(ϕ)=ξ0e−λϕ, (62)

where and are constants. One can calculate the slow-roll parameters, (18)–(II), for the model given by (62)

 ϵ = 12λ2e−2λϕ(α+e2λϕ), (63) η = −λ2(3αe−2λϕ+1), (64) δ1 = −αλ2e−4λϕ(α+e2λϕ), (65) δ2 = −12λ2(7αe−2λϕ+3). (66)

Inflation ends at , although inflation does not stop naturally for this scenario, which gives the value of the field at the end of the inflation

 ϕe=−12λln(2−λ2αλ2), (67)

where . In this section we consider that the value of the field at the end of inflation is much smaller than that of the beginning, which means . Therefore, the number of -folds before the end of inflation is

 N≃∫ϕϕeκ2Qdϕ=−12λ2ln(α+e2λϕ). (68)

From (68), we obtain

 ϕ=12λln(e−2λ2N−α). (69)

After substituting the last result (69) into (III) and (55), the spectral index of the scalar modes and tensor-to-scalar ratio can be written as

 ns−1 = λ2(3αe−2λ2N−α−1),r=8λ2e−4λ2N(e−2λ2N−α)2. (70)

One, then, can write the relation between and as follows:

 r=−84α2e4λ2N−5αe2λ2N+1(ns−1). (71)

Before we compare our theoretical predictions with the observational data by Planck, one last thing that we need to check is the valid model parameter ranges for inflation to happen.

From (67)–(69) , we find that

 2−λ2αλ2>0,0<α+e2λϕ<1,ande−2λ2N>α. (72)

Since is always positive () and can be negative or positive, we can reach to the following results: if , , then . Or if , then . With these parameter ranges, we can freely choose the model parameters and that are valid for inflation to occur. Unfortunately, these parameter ranges of and are not favored by observational data.

### iv.2 Power-law potential and power-law Gauss-Bonnet coupling

We consider an inflationary model with the power-law potential and power-law coupling to the Gauss-Bonnet term characterized as follows:

 V(ϕ)=V0ϕn,ξ(ϕ)=ξ0ϕn. (73)

This class of potential has been widely studied as a simplest inflationary model and includes the simplest chaotic models, in which inflation starts from the large values of an inflaton field, .

For the model with the choice of (73), the slow-roll parameters can be calculated using (18)–(II) as

 ϵ≃ n22κ2(1+αϕ2n)ϕ−2, (74) η≃ −nκ2[n−2+(3n−2)αϕ2n]ϕ−2, (75) ζ≃ n22κ4[16−14n+3n2+4(n−1)(7n−8)αϕ2n +(3n−2)(11n−8)α2ϕ4n]ϕ−4, (76) δ1≃ −n2κ2αϕ2n(1+αϕ2n)ϕ−2, (77) δ2≃ −n2κ2(3n−4+(7n−4)αϕ2n)ϕ−2, (78) δ3≃ n2κ4[8−10n+3n2+4(n−1)(5n−4)αϕ2n +(3n−2)(7n−4)α2ϕ4n]ϕ−4. (79)

The number of -folds before the end of inflation for the choices of (73) is given in (17) by

 N≃κ2ϕ22n2F1(1;1n;1+1n;−αϕ2n).

It turns out that for ; then we can reproduce the standard chaotic inflation results, . Here, we assume the term of to be much smaller than 1, so that we could expand the hypergeometric function up to the leading order in ,

 2F1(1;1n;1+1n;−αϕ2n)≈1−αϕ2nn+1+O(α2). (80)

Then the number of -folds becomes

 N≃κ2ϕ22n(1−αϕ2nn+1)+O(α2). (81)

As we described in Sec. II, for and for to have enough -folding, . This implies that can be treated as a small parameter.

We also expand to the leading order in , which is a dimensionless parameter, ,

 ϕ=ϕ(0)+~αϕ(1)+O(~α2). (82)

Substituting (82) into (81), we obtain

 ϕ≃√2nNκ2[1+α(2nN)n2(n+1)κ2n]. (83)

With (83), one can rewrite (74)–(78) as follows:

 ϵ≃ n4N+n2(2nN)nα4(1+n)Nκ2n, (84) η≃ 2−n2N−3n2(2nN)nα2(1+n)Nκ2n, (85) ζ≃ (n−2)(3n−8)8N2+n2(14n−19)(2nN)nα4(1+n)N2κ2n, (86) δ1≃ −n(2nN)nα2Nκ2n, (87) δ2≃ 4−3n4N−7n2(2nN)nα4(1+n)Nκ2n, (88) δ3≃ (n−2)(3n−4)4N2+n2(10n−11)(2nN)nα2(1+n)N2κ2n. (89)

Substituting (84)–(89) into (III)–(57), we obtain , and , respectively, as follows:

 ns−1≃ −n+22N+n(3n+2)(2nN)nα2(1+n)Nκ2n, (90) nt≃ −n2N−n2(2nN)nα2(1+n)Nκ2n, (91) r≃ 4nN+4n(2n+1)(2nN)nα(1+n)Nκ2n, (92) dnsdlnk≃ −n+22N2−n(n−1)(3n+2)(2nN)nα2(1+n)N2κ2n, (93) dntdlnk≃ −n2N2+n2(n−1)(2nN)nα2(1+n)N2κ2n. (94)

Figures 46 show the - contour plot of the models that are given by (73) with , , and for the different values of and in comparison with the observational data. The red contour comes from the Planck data and the BICEP2 data set are included in the blue contour. The Planck and WMAP data constrain on as , but BICEP2 claims that . There seems to be some discrepancy between Planck and BICEP2. One way out of this discrepancy might be to take into account the running spectral index of the scalar modes Ade:2014xna ().

Black, brown, and gray dashed lines represent the theoretical predictions for (black), (brown), and (gray), respectively, and the pairs of red and blue dots represent and , respectively, in Figs. 46.

Without the Gauss-Bonnet term ), Planck data say that the model lies well outside of the joint 99.7% CL (confidence level) region in the plane (Fig. 6) and the model lie outside of the 95% CL region for (Fig. 5). On the contrary, the inflationary models with lies within the CL regions (Fig. 4). If we consider the combination of BICEP2 and Planck, even for reside within the CL regions, but model might be ruled out.

Both and are suppressed if and has negative values, but, for , those are enhanced. These results are completely opposite compared to Ref. Guo:2010jr (), in which is enhanced for negative and reduced for positive for with . Because becomes suppressed as decreases, Planck data alone favor the model, but the BICEP2 + Planck favors . Even for , BICEP2 with Planck seems to rule out at CL (Fig. 4). For with (Fig. 5), negative