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arXiv:1107.3739v1 [astro-ph.CO] 19 Jul 2011

HIP-2011-20/TH

ITP-UU-11/28

SPIN-11/21

TTK-11-25

Non-metric chaotic inflation

Kari Enqvist∗

Physics Department, University of Helsinki, and Helsinki Institute of Physics, FIN-00014 University of Helsinki.

Tomi Koivisto†

Institute for Theoretical Physics and Spinoza Institute,

Leuvenlaan 4, 3584 CE Utrecht, The Netherlands.

Gerasimos Rigopoulos‡

Institut f¨ ur Theoretische Teilchenphysik und Kosmologie,

RWTH Aachen University,

D–52056 Aachen, Germany

(Dated: July 20, 2011)

We consider inflation within the context of what is arguably the simplest non-metric extension of

Einstein gravity. There non-metricity is described by a single graviscalar field with a non-minimal

kinetic coupling to the inflaton field Ψ, parameterized by a single parameter γ. We discuss the

implications of non-metricity for chaotic inflation and find that it significantly alters the inflaton

dynamics for field values Ψ ? MP/γ, dramatically changing the qualitative behaviour in this regime.

For potentials with a positive slope non-metricity imposes an upper bound on the possible number

of e-folds. For chaotic inflation with a monomial potential, the spectral index and the tensor-to-

scalar ratio receive small corrections dependent on the non-metricity parameter. We also argue that

significant post-inflationary non-metricity may be generated.

I. INTRODUCTION

Even at the classical level, the theory of gravity has still two major open issues. First, what is the form of the action?

Here investigations have focused on extensions of Einstein gravity, such as f(R) gravities [1], and their implications for

dark energy or inflation [2, 3]. But there is also the second, and arguably even more profound question: what are the

true gravitational degrees of freedom? Conventionally, one just assumes metricity: that gravity can be described by

the metric alone, setting the connectionˆΓ to be the Levi-Civita connection withˆΓ = Γ[gµν] by hand. In the Palatini

variation one takes the metric and the connection to be independent degrees of freedom [4], and varying both results

in equations of motion that differ from the metric case except if the Lagrangian has exactly the Einstein-Hilbert

form with f(R) = R. In the general Palatini case, the solution to equation of motion for the connection is not the

Levi-Civita connection, and hence such theory is called non-metric. In non-metric gravity, the metric is no longer

covariantly conserved.

However, several issues arise in the Palatini approach [5], which may stem from the nontensorial property of the

connection. Instead, one could consider theories with two metrics: one for describing the geometry of the manifold,

and the other for generating the connection. Indeed, the Palatini case can be seen as a special case of a more general

class of conformal non-metric theories. In these C-theories [6, 7] one postulates a metric for the connection, ˆ gµνthat

is related to the gravitational metric by a conformal factor C with ˆ gµν= Cgµν, the Palatini-f(R) then corresponding

to the choice C(R) = f′(R). The connection is then the Levi-Civita connection of the connection-generating metric

withˆΓ = Γ[ˆ gµν]. Such a relation was originally considered by Weyl in his conformally invariant theory of gravity [8].

Surprisingly, the subtle refinement of the fundamental spacetime structure turns out to have concrete and interesting

consequences, for one allowing non-metric degrees of freedom to propagate in vacuum. This is in contrast both to the

∗Electronic address: kari.enqvist@helsinki.fi

†Electronic address: T.S.Koivisto@uu.nl

‡Electronic address: rigopoulos@physik.rwth-aachen.de

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result of the standard metric variation (where the connection is fixed to be metric a priori) and the metric-affine, i.e.

Palatini, variation (where the connection is constrained a posteriori), which however can be recovered as specific limits

of C-theories. Thus, they provide a unified approach to a variety of previously seemingly very different alternative

gravity theories but reveal also novel possibilities hidden already in the Einstein-Hilbert action itself.

The nature of gravity is of great relevance for models of inflation [9]. Since inflation occurs at very high curvature

scales where one naturally expects corrections to Einstein’s theory, the ambiguities both in the form of gravitational

Lagrangian and its degrees of freedom should be taken into account. The purpose of this paper is to explore the

implications of the simplest conformal non-metricity to inflationary physics. In particular, we analyze the corrections

that inevitably appear to large-field chaotic inflation. Our approach is different from considering inflation to be driven

by non-metricity, which has been pursued in Refs. [10, 11]. The Palatini variation has been applied to a non-minimally

coupled inflation [12, 13], however there taking only into account the shift of the kinetic term [14], whereas our purpose

is to consider the generic lagrangian (for details, see appendix A).

In the next section, we couple a matter scalar field into gravity in the presence of Weyl non-metricity and derive

the equations of motion for homogeneous fields. In the following section we then analyze the dynamics of slow-roll

chaotic inflation and derive constraints on the amount of non-metricity chaotic inflation can tolerate. The results are

discussed in section IV. The generic starting lagrangian and the general field equations are confined to the appendices.

II. SCALAR FIELD AND NON-METRIC GRAVITY

Let us now consider the possibility that the spacetime connectionˆΓ is conformally related to the Levi-Civita

connection Γ with

ˆΓα

βγ= Γα

βγ+

?

δα

(βδλ

γ)−1

2gβγgαλ

?

∇λlogC , (1)

where C = C(x) is the conformal factor. The scalar curvature is then1

R ≡ gµνˆRµν= R −D − 1

4C2

?4C✷C + (D − 6)(∂C)2?.(2)

The conformal factor C represents a scalar degree of freedom. If we define ϕ by

C ≡ e

2κ

√

(D−1)(D−2)ϕ, (3)

the Einstein-Hilbert action can be written simply as

SE−H=

1

2κ2

?

dDx√−gR =

?

dDx√−g

?R

2κ2− (∂ϕ)2

?

, (4)

so the conformal relation contributes, effectively, a massless scalar field to the vacuum. This is a remarkably simple

way of introducing non-metricity and allows to study its effect on the coupling of gravity and matter.

For this purpose, let us consider a scalar matter field Ψ with a mass m. With the canonical action for the matter

field, the theory is metric, though more general than Einstein’s theory [15]. In the presence of non-metricity, one has

to reconsider the coupling of the scalar matter fields to gravity. There are several possible lagrangians that all reduce

to a canonical scalar field theory when the non-metricity vanishes, i.e. when C is a constant. The corresponding action

can then be interpreted as the work done along the path of the scalar particle, but now the free falling trajectories

are affected by the non-metricity of the connection. Hence the matter fields are coupled not only to the metric but

also to the connection, or equivalently, to the graviscalar field ϕ.

In the appendix A it is shown that the new effects in our case are captured by just a single term, with an unknown

coefficient γ in the effective lagrangian which becomes

2L =

1

κ2R − (∂ϕ)2− (∂Ψ)2+ 2γκΨΨ,αϕ,α− V (Ψ) , (5)

1Here ✷ is the metric d’Alembertian and other unhatted quantities are also metric unless otherwise specified. The reduced Planck mass

is denoted by κ ≡√8πG.

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where κ = 1/MP. We thus obtain effectively a two-field system with field dependent kinetic term. The non-metric

effects are suppressed by the Planck scale MP. The most general lagrangian is considered in the Jordan and in the

Einstein frame in the appendix A.

Let us now study the implications of the non-metric graviscalar ϕ for chaotic inflation. The equations of motion

for the homogenous scalar Ψ and graviscalar ϕ fields in FRW universe are given by

¨ ϕ + 3H ˙ ϕ −

γκ

1 − (γκΨ)2˙Ψ2+

γ2κ2

1 − (γκΨ)2Ψ˙Ψ2+

2γκ

1 − (γκΨ)2V = 0,

V,Ψ

1 − (γκΨ)2= 0,

(6)

¨Ψ + 3H˙Ψ − (7)

where the˙Ψ2terms arise from the field dependent metric in the kinetic term of (5). We can bring the field metric

into a flat form by defining new fields u± and v through du− =

du+=

?γ2κ2Ψ2− 1dΨ (for the region Ψ > 1/γκ), and dv = dϕ − γκΨdΨ. Then the scalar degrees of freedom are

given by

?1 − γ2κ2Ψ2dΨ (for the region Ψ < 1/γκ),

u− = Ψ

?1 − γ2κ2Ψ2

2

+arcsin(γκΨ)

2γκ

(Ψ < 1/γκ), (8)

u+ =

π

4γκ+ Ψ

?γ2κ2Ψ2− 1

2

−

ln

?

γκΨ +

?γ2κ2Ψ2− 1

2γκ

?

(Ψ > 1/γκ), (9)

v = ϕ −1

2γκΨ2. (10)

In terms of the redefined fields the equations of motion now read

¨ u∓+ 3H ˙ u∓±

V,Ψ(u∓)

?|1 − γ2κ2Ψ2(u∓)|

= 0(11)

¨ v + 3H ˙ v = 0,

H2=1

3

(12)

?1

2˙ v2±1

2˙ u2

∓+ V (u∓)

?

.(13)

We immediately see that non-metricity qualitatively alters the field dynamics when γκu ? π/4 (Ψ ? 1/γκ). In

particular, for γκu ? π/4 (the + region) an attractive potential V translates into a repulsive force on the field and

vice versa. Taking for concreteness a monomial potential V (Ψ) = λκα−4Ψαwe see that the Klein-Gordon equation

(11) becomes

¨ u∓+ 3H ˙ u∓±αλ

κ3

1

γα−1

(γκΨ)α−1

?|1 − γ2κ2Ψ2(u∓)|

= 0(14)

where close to the point γκu = π/4 we have

γκΨ ≃ 1 + Sign

?

γκu −π

4

??

3

2√2

?2/3 ???γκu −π

4

???

2/3

(15)

For potentials V with a positive slope we see that a barrier forms at this point, separating the − from the + region.

The force exerted there is infinite and the field is driven away from Ψ = 1/γκ. In the + region the field then grows

without bound and is also a ghost. Even though solutions exist as long as its kinetic term doesn’t dominate, it is

difficult to see how this would lead to a sensible cosmology. On the other hand, the − region does lead to sensible

slow roll inflation. Slow roll breaks down close to the barrier and hence the total number of efolds in such a model is

bounded. Note that even though the force is infinite, the barrier has finite width and height and is thus in principle

crossable both classically and quantum mechanically. If the potential has a negative slope in the + region the force

is attractive (but again infinite at Ψ = 1/γκ). One can also see that the solution v = const is an attractor. This fixes

the graviscalar in terms of the inflaton to be

ϕ =1

2γκΨ2. (16)

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0.51.0 1.5 2.0

Γ?

0.5

1.0

1.5

Γu

0.2 0.4 0.60.8 1.01.21.4

Γu

?10

?5

5

10

FIG. 1: Left: The field u is plotted as a function of Ψ. Inversion of this graph gives Ψ(u). The dashed line shows the

approximation (15). Right: The third term of the field equation (14) (the negative of the “force” felt by the u field) for α = 2

(solid) and α = 4 (dashed). A repulsive barrier is formed at the point γu = π/4. (MP is set to unity in this figure.)

III.SLOW-ROLL CHAOTIC INFLATION

Let us now consider slow-roll chaotic inflation with a monomial potential V (Ψ) = λκα−4Ψα. The slow roll equations

of motion then read

3H ˙ u ≃ −λακα−4Ψ(u)α−1

?1 − γ2κ2Ψ(u)2,H2≃κ2

3λκα−4Ψ(u)α

(17)

while v = const. The inflaton, u starts to roll with some initial value u < π/8γκ. Using N as the time parameter,

and reverting from u back to Ψ for the moment, we obtain for the number of e-folds

dΨ

dN≃α

κ2

1

Ψ(1 − γ2κ2Ψ2), (18)

which is easily integrated to give

N ≃κ2

α

Ψ2

2

?

1 −γ2κ2

2

Ψ2

?

.(19)

The approximate equality holds after ignoring a constant of integration and focusing on field values relevant for

cosmological observations. Slow roll inflation can be supported in the region ˙ u2< 2V (Ψ), which implies that slow

roll breaks down at field values

Ψ2=

1 ∓

?

1 −2α2

2γ2κ2

3γ2

. (20)

We see that the necessary condition for the slow roll regime to exist at all is γ <

down for two field values: the minus sign in (20) gives a lower bound on the value of Ψ, which in the limit γ → 0

reproduces the standard result Ψ2

slow roll is no longer possible. This is because when Ψ ∼ 1/γκ, non-metricity introduces corrections which make

the effective potential felt by the inflaton field steeper, rendering slow roll unattainable. Hence, we conclude that

non-metricity introduces a bound to the possible number of e-folds a chaotic inflation monomial potential can give. A

similar result holds for any slow roll potential with a positive slope. Assuming γ ≪ 1, we find that for the monomial

potential

?3/2α2and that slow roll breaks

1≃ α2/6κ2. However, there is also an upper bound for the field value beyond which

N <

1

4αγ2,(21)

while requiring that N ? 60 restricts γ to be

γ ?

1

16√α.

(22)

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Deep in the slow roll region where u ≪ π/8γκ we can approximate

Ψ(u) ≃ u +1

6γ2κ2u3

(23)

whence the equation of motion reads

¨ u + 3H ˙ u + αλκα−4uα−1

?

1 +2 + α

6

γ2κ2u2

?

= 0. (24)

This corresponds to an effective potential

V (u) = λκα−4uα?

1 +α

6γ2κ2u2?

(25)

and

H2=κ2

3

?˙ v2

2

+˙ u2

2

+ V (u)

?

(26)

Thus, in the slow-roll region non-metricity appears as a subdominant correction in the effective potential felt by the

inflaton scalar. The number of e-folds as a function of U is calculated to be

N ≃κ2

2αu2

?

1 −1

6γ2κ2u2

?

(27)

from which u can be written as a function of N

κ2u2≃ 2αN

?

1 +α

3γ2N

?

(28)

From the above results we obtain for the spectral index at order γ2

n − 1 = −6ǫ + 2η = −2 + α

2N

−α(α − 2)

2

γ2. (29)

Interestingly, the correction at this order vanishes for α = 2. The tensor to scalar ratio is calculated to be

r ≃ 12.4ǫ ≃ 3.1α

?1

N+ αγ2

?

(30)

Let us immediately note that the calculations for the spectral index and the tensor to scalar ratio, equations (29) and

(30), ignore the isocurvature perturbation associated with the extra degree of freedom. Although a proper treatment

requires a linear perturbation analysis starting from the initial lagrangian (5), we can make the following estimates:

In the {u,v} basis inflation proceeds entirely along the u direction, which definines the adiabatic perturbation, while

the isocurvature field corresponds to v. Since the trajectory is straight there is no coupling between the two at the

linear level. Thus we expect the single field conclusions (29) and (30) to hold for the adiabatic perturbation related

to u, or equivalently Ψ, while the perturbations δv(k) ∼ H/k3/2would imply a perturbation for the graviscalar

δϕ(k) ≃ (1 + γκΨ∗)H∗

k3/2

(31)

where the subscript ∗ as usual indicates evaluation at horizon exit. We discuss the implications of this briefly in the

discussion.

IV. DISCUSSION

The simplest possible implementation of a non-metric theory of gravity introduces a new scalar degree of freedom

associated with the conformal part of the metric. In turn, this can be cast as a theory of two scalar fields with a

non-trivial metric in field space. We have examined the implications of such a non-metric theory for inflation and

found that for inflaton values Ψ ? 1/γκ such a seemingly innocuous modification has profound implications for the

way inflation proceeds. In particular, as can be seen from eq (11) the effect of non-metricity is to effectively turn an

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attractive inflaton potential into a repulsive one and viceversa. Thus, an inflationary potential with a positive slope

becomes unstable in the region Ψ ? 1/γκ leading to a cosmology that cannot represent the inflationary past of our

universe. On the contrary, for values Ψ < 1/γκ slow roll inflation can proceed with few modifications, mainly in the

form of small γ corrections for the spectral index and the scalar to tensor ratio. The main difference from metric

inflation is that now the non-metricity parameter γ sets un upper bound for the possible number of e-folds, see eq

(21). In this sense non-metricity can act as an IR regulator for inflation.

It is perhaps interesting to note that potentials which are repulsive can turn attractive in the Ψ ? 1/γκ region

which could then support a viable inflationary cosmology, although again with a point of infinite force for the field at

Ψ = 1/γκ.

Let us close by briefly discussing the effects of perturbations. We argued that non-metric inflation would give

rise to a fluctuation of the graviscalar ϕ in addition to the standard adiabatic metric perturbation, see eq (31).

After the inflaton decays the graviscalar perturbation would presumably persist along with the adiabatic metric one

ζ(k) ≃ H∗/√ǫk3/2. This would give rise to a non-metric contribution to the perturbed part of the connection, see

equation (1), with (ˆΓ − Γ)/Γ ∼√ǫ. Thus, non-metric inflation seems to be generating deviations of the perturbed

post-inflationary connection from the metric case. Such deviations could be used to constrain non-metricity further

and we will return to this issue in future work.

Acknowledgements

KE is supported by the Academy of Finland grants 218322 and 131454. GR is supported by the Gottfried Wilhelm

Leibniz programme of the Deutsche Forschungsgemeinschaft.

Appendix A: General scalar field couplings

The most general lagrangian for a scalar field Ψ that is parity-invariant, first order in curvature and at most

quadratic in derivatives of Ψ can be specified by six functions Ai(Ψ), i = 1,...,6 as:

L = A1(Ψ)R + A2(Ψ)(∂Ψ)2+ A3(Ψ)Ψ,νgαβˆ∇νgαβ+ A4(Ψ)Ψ,µˆ∇νgµν+ A5(Ψ)gµνˆ∇µˆ∇νΨ + A6(Ψ).

For generality, we included also the nonminimal coupling to curvature, A1(Ψ). The Palatini variation of this lagrangian

was done by Burton and Mann in [16]. Here we consider the case that the lagrangian is quadratic in the field Ψ,

(A1)

A1(Ψ) =1

2

?1

κ2+ αΨ2

?

,A2(Ψ) = α2,A3(Ψ) = α3Ψ,A4(Ψ) = α4Ψ,A5(Ψ) = α5Ψ,A6(Ψ) = −1

2m2Ψ2.

(A2)

By using conformal relations like (1), (2) and partial integrations, we find that the theory can be equivalently written

in the simple form

2L =

?1

κ2+ αΨ2

??

R − κ2(∂ϕ)2?

− β (∂Ψ)2+ 2γκΨΨ,αϕ,α− m2Ψ2, (A3)

where β = (α5− α2)/2 and

γ =

1

2?(D − 1)(D − 2)[(D − 1)α + 2Dα3+ 2α4+ (D − 2)α5] .(A4)

When α5− α2> 0, the kinetic term of the field can be made canonical, β = 1, by rescaling the field. m is the mass.

Thus it is clear that there is essentially just one new parameter, γ, which is due to the presence of non-metricity. Its

effect is to introduce a kinetic mixing between the two scalar degrees of freedom. Additionally, a nonminimal coupling

α ?= 0 results now also in a derivative interaction.

Appendix B: Alternative descriptions

1.Einstein frame

The rescaling

˜ gµν= (1 + ακ2Ψ2)

2

D−2gµν

(B1)

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brings the lagrangian into the form

˜L =

1

2κ2˜R +˜ X −˜V ,(B2)

where the components of the kinetic matrix defined by˜ X = −1

2GIJ˜ gµνφI

,µφJ

,νare given by

Gϕϕ= 1,GϕΨ= GΨϕ= −

γκΨ

1 + ακ2Ψ2,GΨΨ=

1

1 + ακ2Ψ2

?

β + 4D − 1

D − 2ακ2Ψ2

?

. (B3)

The potential is

˜V (Ψ) = (1 + ακ2Ψ2)

2

D−2V (Ψ). (B4)

In this paper we however focus on the case α = 0 where the frames coincide.

2. Diagonal frame

To clarify the field content of our the case (5), let us introduce the linear combinations

φ±= Ψ ± ϕ.(B5)

The Lagrangian can then be written, in terms of the eigenfunctions

λ±=1

2±γκ

4

(φ++ φ−) , (B6)

as

2L = R − λ+(∂φ−)2− λ−(∂φ+)2− 2V

?φ++ φ−

2

?

.(B7)

So the diagonalized fields have a nonminimal interaction and both are nontrivially coupled kinetically. Below ”the

barrier” γκΨ < 1 there are no ghosts.

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