Observational tests of inflation with a field derivative coupling to gravity

Observational tests of inflation with a field derivative coupling to gravity

Shinji Tsujikawa Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
July 11, 2019
Abstract

A field kinetic coupling with the Einstein tensor leads to a gravitationally enhanced friction during inflation, by which even steep potentials with theoretically natural model parameters can drive cosmic acceleration. In the presence of this non-minimal derivative coupling we place observational constraints on a number of representative inflationary models such as chaotic inflation, inflation with exponential potentials, natural inflation, and hybrid inflation. We show that most of the models can be made compatible with the current observational data mainly due to the suppressed tensor-to-scalar ratio.

pacs:
98.80.Cq, 95.30.Cq

I Introduction

Inflation has been the backbone of the high-energy cosmology over the past 3 decades infpapers (). The most simple source for inflation is a minimally coupled scalar field (“inflaton”) with a slowly varying potential newinf (); chaotic () The spectra of density perturbations generated from the quantum fluctuations of inflaton are consistent with the temperature anisotropies observed in the Cosmic Microwave Background (CMB) infper ().

From the amplitude of the observed CMB anisotropies COBE (); WMAP7 () the typical mass scale of inflation is known to be around  GeV Liddle (). This is much larger than the electroweak scale ( GeV), which suggests the requirement of new physics beyond the Standard Model of particle physics Riotto (). In other words, for the potential with  GeV, the coupling is constrained to be from the CMB normalization Liddle (), but this is much smaller than the coupling constant of the Higgs boson particlereview ().

There have been attempts to accommodate the Higgs field for inflation. One is to use a non-minimal field coupling with the Ricci scalar Bezrukov () (see also Refs. Maeda ()). If the self coupling can be as large as from the CMB normalization Salopek (). Although this scenario is attractive, it is plagued by the unitary-violation problem associated with graviton exchange in scalar scattering around the energy scale (where  GeV is the reduced Planck mass) Burgess (). Since is around the energy scale of inflation, some strong coupling effect can give rise to additional corrections to the inflaton potential.

Another attempt is to employ a field derivative coupling with the Einstein tensor , i.e. , where is a constant having a dimension of mass Germani1 () (see also Ref. Amendola () for the original work). In the regime where the Hubble parameter is larger than the field evolves more slowly relative to the case of standard inflation due to a gravitationally enhanced friction. Hence it is possible to reconcile steep potentials such as () with the CMB observations.

In Refs. Germani1 (); Germaniper (); Germani2 (); Watanabe (); Germani11 () it was shown that, for a slow-rolling scalar field satisfying the condition , the strong coupling scale of the derivative coupling theory is around in a homogeneous and isotropic cosmological background. Provided that and are below the Planck scale, the theory is in a weak coupling regime with suppressed quantum corrections.

The property of the high cut-off scale around is associated with the fact that whenever the non-minimal derivative coupling to gravity dominates over the canonical kinetic term the theory possesses an asymptotic local shift symmetry for Germani11 (). This symmetry is related to the Galilean symmetry in Minkowski space-time Nicolis (), but the difference is that the coordinate in the derivative coupling theory on curved backgrounds is linked to the covariantly constant Killing vectors Germani11 (). In the presence of a slowly varying inflaton potential such a local symmetry is only softly broken, so that the potential can be protected against quantum corrections during inflation. The field self-interaction of the form Nicolis (); covaga (), which satisfies the Galilean symmetry in the limit of Minkowski space-time, also leads to the slow evolution of along the inflaton potential Kamada () (see also Refs. Galileoninf ()).

A nice feature of the non-minimal derivative coupling with the Einstein tensor is that the mechanism of the gravitationally enhanced friction works for general steep potentials. For instance, let us consider the potential of natural inflation, , where characterizes the scale of the breaking of a global shift symmetry Freese (). In order for this potential to be consistent with the CMB observations, we require that is larger than in conventional slow-roll inflation Savage (). Then the global symmetry is broken above the quantum gravity scale, in which case quantum field theory may be invalid naturalpro (). In the presence of the non-minimal derivative coupling to gravity, however, the scale can be much smaller than because of the gravitationally enhanced friction Germani2 (); Watanabe ().

Another example is the exponential potential , whose dominance leads to the power-law expansion of the Universe (with the scale factor , where is cosmic time) earlyexp (). In higher-dimensional gravitational theories, exponential potentials often arise as the curvature of internal spaces related with the geometry of extra dimensions expmoti (). In such cases the constant is usually larger than the order of unity, so that it is difficult to realize sufficient amount of inflation. As we will see later, this problem can be circumvented by taking into account the non-minimal derivative coupling. Moreover, unlike the standard case, inflation comes to end with gravitational particle production.

In order to test the viability of inflationary models with the non-minimal derivative coupling it is important to estimate the power spectra of density perturbations relevant to the CMB anisotropies. In Refs. Germaniper (); Watanabe () the authors computed the inflationary observables such as the scalar spectral index , the tensor-to-scalar ratio , and the nonlinear parameter of the equilateral scalar non-Gaussianities (see also Refs. KYY (); Gao (); DT11 ()). Since the scalar propagation speed is close to the speed of light during inflation, the scalar non-Gaussianities are suppressed to be small (). Hence and are the two main observables to distinguish between different inflaton potentials.

In this paper we shall place observational constraints on a number of representative inflationary models in the presence of the field derivative coupling with the Einstein tensor. We use the bounds derived from the joint data analysis of WMAP7 WMAP7 (), Baryon Acoustic Oscillations (BAO) BAO (), and the Hubble constant measurement (HST) HST (). Note that some constraints on Higgs inflation and natural inflation have been discussed in Refs. Germaniper (); Watanabe () without the CMB likelihood analysis. In Ref. Popa () the author carried out the cosmological Monte-Carlo simulation to test Higgs inflation with the field derivative coupling to gravity. Our analysis based on the recent observational data is general enough to cover a wide variety of models such as chaotic inflation chaotic (), inflation with exponential potentials earlyexp (), natural inflation Freese (), and hybrid inflation hybrid (). We show that the gravitationally enhanced friction mechanism can make most of the models compatible with the current observations.

Ii Background dynamics

We start with the following 4-dimensional action

(1)

where

(2)

Here is a determinant of the space-time metric , is the Ricci scalar, is the Einstein tensor, is a constant having a dimension of mass, and is the potential of a scalar field .

The action (1) belongs to a class of the most general scalar-tensor theories having second-order equations of motion (which is required to avoid the Ostrogradski instability) Horndeski (); Deffayet (); Char (); KYY (). The Lagrangian in such general Horndeski’s theories is the sum of the terms , , , and , where , () are functions of and , and Deffayet (); KYY (). The conditions for the avoidance of ghosts and Laplacian instabilities were derived in Refs. KYY (); DTjcap (). These conditions can be used to restrict the functional forms of , () to construct theoretically consistent models of inflation.

The non-minimal derivative coupling in Eq. (1) is recovered in the Horndeski’s Lagrangian by choosing the function after integration by parts. The sign in front of the term in Eq. (2) is chosen to avoid the appearance of ghosts in the scalar sector Germani1 (); Germani2 ().

In Ref. Germani11 () it was found that in a manifold having integrable (covariantly constant) Killing vectors the field Lagrangian in Eq. (1) is invariant under the (curved-space) Galilean transformation , where , , are constants and is a space-time coordinate. The existence of the Galilean symmetry has an advantage that the theory can be quantum mechanically under control Trod ().

Imposing the above Galilean symmetry in the curved background with integrable Killing vectors, Germani et al. Germani11 () showed that the second-order Lagrangians are restricted to take the forms or (plus a field derivative coupling with the double dual Riemann tensor). In the small derivative regime in which the condition is satisfied (e.g., during inflation), an approximate infinitesimal shift symmetry (where is an arbitrary function of space-time coordinates ) emerges for the Lagrangian , provided that the metric is shifted appropriately Germani11 (). The existence of such a gauge symmetry can allow the theory (1) to be protected against quantum corrections even up to the Planck scale. Note that the term does not possess such a general gauge shift symmetry.

The scale of unitarity violation for the theory (1) was estimated in Ref. Germani1 () in the context of Higgs inflation. In Standard Model we can consider the scattering via graviton exchange, where is one of the real scalar degrees of freedom for the Higgs doublet. We expand the metric in the form , where is the metric on the flat Friedmann-Lemaître-Robertson-Walker (FLRW) background ( is the scale factor with cosmic time ). We are interested in the high-friction regime in which the Hubble parameter (a dot represents a derivative with respect to ) is much larger than . In this regime the field is expanded as , where is a canonically normalized field perturbation. The first non-renormalizable operator associated with the interaction between gravitons and scalars is given by Germani1 ()

(3)

A power counting analysis gives the unitary bound . For the suppression of higher dimensional operators we require the condition . On using the relation this condition translates into , which is satisfied during inflation.

The discussion of the unitary bound given above can be applied to the multi-field inflationary models in which one of the fields is not necessarily responsible for the cosmic acceleration. In single field models where only one field leads to inflation the unitary bound can be as close as the Planck scale in the regime where the condition is satisfied Watanabe (); Germani11 (). In this case the slow-roll evolution of the field suppresses the interaction (3) below the Planck scale.

In the following let us study the background dynamics for the theory described by the action (1). In the flat FLRW background the equations of motion following from the action (1) are

(4)
(5)

where . In order to solve the dynamical equations numerically, it is convenient to introduce the following dimensionless variables

(6)

Differentiating Eq. (4) with respect to and using Eq. (5) to eliminate , we obtain the second-order equation for the field . It then follows that

(7)
(8)
(9)

where

(10)

We are interested in slow-roll inflation in which cosmic acceleration is mainly driven by the potential energy . In this case Eqs. (4) and (5) reduce to

(11)
(12)

where

(13)

We define the following slow-roll parameters

(14)

For the validity of the slow-roll approximation we require that . Taking the time-derivative of Eq. (11) and using Eq. (12), we have

(15)

where

(16)

This shows that for and hence the evolution of the field slows down relative to that in standard slow-roll inflation.

The field value at the end of inflation is known by solving , i.e.

(17)

The number of e-foldings from the time during inflation to the time at the end of inflation is defined by . On using Eqs. (11) and (12), it follows that

(18)

If (i.e. ) during inflation, we can neglect the first term inside the bracket of Eq. (18) relative to the second one. This is not the case for inflation in which the transition from the regime to the regime occurs prior to the onset of reheating.

Figure 1: Evolution of the field versus for (i) and (ii) . The initial conditions at are chosen by using the slow-roll equations (11) and (12), i.e., (i) , , and (ii) , .

As an example, let us consider chaotic inflation chaotic () with the potential

(19)

where and are constants. In this case Eqs. (18) and (17) read

(20)
(21)

where

(22)

For the quadratic potential (i.e. and ) one has and

(23)

where . In the General Relativistic (GR) limit () this gives . In the high-friction limit () one has , which means that the field value is smaller than that in standard chaotic inflation.

In order to confirm the accuracy of the slow-roll approximation we solve the full equations of motion (7)-(9) numerically. In Fig. 1 the evolution of the field is plotted for the potential , i.e. and in Eqs. (8) and (9). We choose the initial conditions of and at by using the values derived under the slow-roll approximation.

The numerical simulations labeled as (i) and (ii) in Fig. 1 correspond to the parameters and , respectively. In the case (i) the numerical value of at the end of inflation () is , which means that the solution is in the high-friction regime during inflation. In this case drops below at the reheating stage. After inflation there is a transient period with in which the slow-roll condition is violated. In this regime some quantum corrections may come into play to the action (1). As long as such corrections are unimportant in the field equations (4) and (5), we find that the inflaton oscillation is not disturbed during the transient period (see Fig. 1). In the case (ii) we have at and hence the system enters the regime during inflation. In this case the qualitative behavior for the oscillation of inflaton at reheating is not much different from that in standard inflation.

We confirmed that the difference between the numerical and analytic values of the number of e-foldings acquired during inflation is usually less than a few percent. This shows that the slow-roll approximation employed in Eqs. (11) and (12) can be trustable.

Iii The spectra of density perturbations

The spectra of scalar and tensor perturbations generated in the theories given by the action (1) were derived in Refs. Germaniper (); KYY (); Watanabe (); DT11 (). Here we briefly review their formulas in order to apply them to concrete inflationary models in Sec. IV.

The perturbed metric about the flat FLRW background is given by permet ()

(24)

where , , and are scalar metric perturbations, and are tensor perturbations which are transverse and traceless. The spatial part of a gauge-transformation vector is fixed by gauging away a perturbation appearing as a form in the metric (24). We decompose the inflaton field into the background and inhomogeneous parts, as . In the following we choose the uniform-field gauge characterized by , which fixes the time-component of the vector .

Expanding the action (1) up to second order in perturbations and using the Hamiltonian and momentum constraints, we obtain the second-order action for scalar perturbations KYY (); DT11 ()

(25)

where

(26)
(27)

and

(28)

In order to avoid the appearance of scalar ghosts and Laplacian instabilities we require that and . Picking up the dominant contributions to and under the slow-roll approximation, we obtain

(29)
(30)

which mean that . The power spectrum of the curvature perturbation , which is evaluated at ( is a comoving wavenumber), is given by

(31)

where in the last approximate equality we used Eqs. (11), (13), (16), (29), and (30). The scalar spectral index is

(32)

where is defined in Eq. (16), and

(33)

In the high-friction limit () one has with , whereas in the GR limit ().

The intrinsic tensor perturbation can be decomposed into two independent polarization modes, i.e. . In Fourier space we normalize the two modes, as (where ) and . Then the second-order action for tensor perturbations can be written as KYY (); DT11 ()

(34)

where

(35)
(36)

This shows that, unlike the scalar propagation speed squared, is slightly larger than 1 during inflation. Since Lorentz invariance is explicitly broken on the FLRW background, the superluminal mode does not necessarily imply a violation of causality. The tensor power spectrum is given by

(37)

which is evaluated at . The tensor spectral index is

(38)

The tensor-to-scalar ratio is

(39)

from which we obtain the consistency relation

(40)

This relation is the same as that in conventional inflation at leading order in slow-roll.

The running spectral indices and are second order in slow-roll parameters. They are set to be 0 in the CMB likelihood analysis. The consistency relation (40) reduces the inflationary observables to three, i.e., , , and . These observables are varied in the likelihood analysis with the pivot wavenumber  Mpc, by assuming the flat -cold-dark-matter model.

Since , the general formula for the equilateral non-Gaussianities of scalar perturbations DT11 () shows that the nonlinear parameter is of the order of Watanabe (). Hence the scalar non-Gaussianities do not provide additional constraints to those derived from the linear perturbations.

Iv Observational constraints

In the presence of the field derivative coupling to the Einstein tensor we place observational constraints on a number of models such as (i) chaotic inflation, (ii) inflation with exponential potentials, (iii) natural inflation, and (iv) hybrid inflation. Our analysis covers most of the representative inflaton potentials proposed in literature.

iv.1 Chaotic inflation

We start with chaotic inflation characterized by the potential

(41)

where, for , we use the notation as in the previous section. Under the slow-roll approximation the dimensionless field is related with the number of e-foldings as Eq. (20). From Eqs. (32) and (39) the scalar spectral index and the tensor-to-scalar ratio are given, respectively, by

(42)
(43)

where .

Figure 2: Observational constraints on chaotic inflation with the quadratic potential in the () plane with the three different numbers of e-foldings (). We evaluate the theoretical values of and in the range . The thin solid curves correspond to the 1 (inside) and 2 (outside) observational contours constrained by the joint data analysis of WMAP7, BAO, and HST. For larger values of the tensor-to-scalar ratio gets smaller, whereas the scalar spectral index increases.

In the limit that one has from Eqs. (20) and (21), in which case and are

(44)
(45)

These values correspond to those for standard chaotic inflation. If , then , for and , for . In another limit one has and

(46)
(47)

If , then , for and , for .

In the intermediate values of between we need to solve Eq. (20) for by using Eq. (21). When the field value can be expressed as Eq. (23), in which case and are known from Eqs. (42) and (43) for given values of and . In Fig. 2 we plot the theoretical values of and for as a function of (between ) with three different values of , together with the and observational contours constrained by the joint data analysis of WMAP7 WMAP7 (), BAO BAO (), and HST HST (). For these observables are close to the values estimated by Eqs. (44) and (45), whereas for they are close to those given by Eqs. (46) and (47). For larger , gets smaller whereas increases, so that the quadratic potential shows better compatibility with observations in the presence of the field derivative coupling to gravity.

Figure 3: Observational constraints on chaotic inflation with the quartic potential in the range with the three different values of . The and observational contours are the same as those in Fig. 2. While the standard case () is outside the bound, the field derivative coupling to gravity can make the quartic potential compatible with observations.

In Fig. 3 the theoretical values of and are plotted for as a function of between with . In the limit that the quartic potential is outside the observational contour for smaller than 70. In the presence of the field derivative coupling to gravity the model can be compatible with the current observations due to the suppressed tensor-to-scalar ratio and the larger spectral index. For the parameter is constrained to be

(48)

and (68 % CL). For the constraints are (95 % CL) and (68 % CL).

Using the scalar power spectrum (31), the CMB normalization by WMAP WMAP7 () corresponds to

(49)

where is the value of at . In the regime one has , so that Eq. (49) gives

(50)
(51)

In the case of the quartic potential it is possible to realize for .

iv.2 Exponential potentials

Let us proceed to the exponential potential

(52)

where and are constants. In this case one has and . Hence in standard slow-roll inflation we require the condition . Moreover this corresponds to the power-law inflation without the graceful exit earlyexp (). In the presence of the field derivative coupling to gravity, however, the slow-roll parameter can be smaller than 1 even for steep exponential potentials with . Inflation ends when grows to the order of unity, which is followed by reheating with gravitational particle production. This situation is analogous to that in braneworld inflation where the dominance of the density squared term () can lead to cosmic acceleration for steep exponential potentials Copeland ().

Figure 4: Observational constraints on inflation with the exponential potential in the regime for . The and observational contours are the same as those in Fig. 2. Even the steep exponential potentials with larger than 1 can be allowed from the current observational data.

We focus on the case in which the condition is satisfied during the whole stage of inflation. Using Eq. (17) the field value at the end of inflation can be estimated as

(53)

which implies that . From Eq. (18) it follows that

(54)

In the regime the scalar spectral index (32) and the tensor-to-scalar ratio (39) reduce to

(55)
(56)

which correspond to taking the limit in Eqs. (46) and (47).

In Fig. 4 we plot the theoretical values of and for as well as the 1 and 2 observational contours. This shows that even the steep exponential potentials with are compatible with the current observational data.

Using the WMAP normalization at in the regime , we obtain the constraint

(57)

For one has .

iv.3 Natural inflation

Natural inflation Freese () is described by the potential

(58)

where and are constants having the dimension of mass. In the absence of the field derivative coupling to gravity the above potential can be compatible with observational data only for Savage (). Then a global symmetry associated with the pseudo-Nambu-Goldstone-boson is broken above the quantum gravity scale, in which case standard quantum field theory may not be applicable. If the potential (58) originates from the string axion, the regime is not generally realized Banks (). This problem can be circumvented by taking into account the field derivative coupling to gravity111See Refs. axionlarge () for other attempts to realize in large-field axion models. Germani2 ().

In the following we focus on the case in which the condition is satisfied during the whole stage of inflation (). From Eq. (17) the end of inflation is characterized by

(59)

where and

(60)

The number of e-foldings is given by

(61)

where and

(62)

The scalar spectral index and the tensor-to-scalar ratio are

(63)
(64)

For given the field value is known from Eq. (59). Since is determined by Eq. (61) for given values of and , we can numerically evaluate and as functions of for several different numbers of e-foldings. In Fig. 5 we plot and in the range , together with the 1 and observational contours. For increasing , both and get larger.

Figure 5: Observational constraints on natural inflation with the potential in the range with for three different values of . The and observational contours are the same as those in Fig. 2. For larger , both and increase toward the values and .

In the limit that it is possible to estimate and analytically. From Eq. (59) one has and hence is close to , as . Using Eq. (61), it follows that . Then Eqs. (63) and (64) reduce to

(65)
(66)

These values correspond to in Eqs. (46) and (47). This comes from the fact that in the regime inflation occurs around the potential minimum at . As in the case of the quadratic potential with , natural inflation with is compatible with the current observations.

For the parameter is constrained to be

(67)

and (68 % CL). For the constraints are (95 % CL) and (68 % CL). In the regime the WMAP normalization at gives

(68)

If , then one has .

iv.4 Hybrid inflation

Finally we study hybrid inflation with the potential

(69)

where and are constants. Inflation ends at a bifurcation point given by due to the appearance of a tachyonic instability driven by another field . As in the case of the original hybrid inflation hybrid () we focus on the regime . Note that in another regime the situation is similar to that in chaotic inflation discussed in Sec. IV.1.

Using Eq. (18) the field value can be estimated as

(70)

where and are positive constants defined by

(71)

From Eqs. (32) and (39) we obtain

(72)
(73)

which mean that the scalar power spectrum is blued-tilted (). Compared to the standard hybrid inflation, the presence of the field derivative coupling to gravity () leads to close to 1 as well as the suppressed tensor-to-scalar ratio. In the limit that one has and , i.e. the Harrison-Zel’dovich (HZ) spectrum. The HZ spectrum is under the observational pressure WMAP7 (), but this property is subject to change depending on the assumptions about the reionization scenario reion (). The future high-precision observations will provide a more concrete answer about this issue.

In the regime and one has for . From the WMAP normalization it follows that

(74)

If , , , for example,  GeV. In this case Eqs. (72) and (73) give and , which is close to the HZ spectrum.

V Conclusions

In this paper we have studied observational constraints on a number of representative inflationary models with a field derivative coupling to the Einstein tensor, i.e. . Such a non-minimal derivative coupling has an asymptotic local shift symmetry for a slow-rolling scalar field satisfying the condition