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UFIFT-QG-14-06

Excitation of Photons by Inflationary Gravitons

C. L. Wang∗ and R. P. Woodard†

Department of Physics, University of Florida, Gainesville, FL 32611

ABSTRACT

We use a recent result for the graviton contribution to the one loop vacuumpolarization to solve the effective field equations for dynamical photons onde Sitter background. Our results show that the electric field experiences asecular enhancement proportional to the number of inflationary e-foldings.We discuss the minimum this establishes for primordial inflation to seedcosmic magnetic fields.

PACS numbers: 04.62.+v, 98.80.Cq, 04.60.-m

∗ e-mail: clwang@ufl.edu† e-mail: woodard@phys.ufl.edu

1 Introduction

Because photons have zero mass it does not take much to affect the longwavelength modes. This has lead many to suspect that the explosive expan-sion of spacetime during primordial inflation might help to explain cosmicmagnetic fields [1]. However, the Maxwell Lagrangian is conformally invari-ant, which means that free photons cannot locally sense the expansion ofspacetime. The search for an inflationary connection has prompted investi-gations of explicit conformal breaking terms which might be present in theeffective action [2]. Quantum effects from the conformal anomaly have beenalso studied [3].

Conformal breaking from other particles can be communicated to pho-tons. No one knows the gravitational couplings of the charged partners ofthe Standard Model Higgs (which become the longitudinal polarizations ofthe W± at low energies) but it has been suggested that the inflationary pro-duction of minimally coupled Higgs scalars could endow the photon with amass during inflation [4, 5, 6], and that this might seed the ubiquous cosmicmagnetic fields of the current epoch [7, 8]. An explicit one loop computationof the massless charged scalar contribution to the vacuum polarization onde Sitter background [9, 10] has confirmed the photon mass conjecture [11],although more work needs to be done to connect this to cosmic magneticfields [12]. Similar one loop results pertain as well when the scalar has asmall mass [13, 14].

Because inflation produces more and more charged scalars as time pro-gresses (provided they are light and nearly minimally coupled) the effec-tive photon mass grows. The scalar mass remains small during this pro-cess [15, 16, 17] until a static, nonperturbative limit is eventually reached[18]. The vacuum energy drops while this occurs [19] and there are dramaticchanges in the electrodynamic forces exerted by point charges and currentdipoles [20].

The effects of charged, minimally coupled scalars are fascinating but de-pendent upon assumptions about the unknown conformal coupling of theHiggs. Gravitons also break conformal invariance so they too can commu-nicate the violence of primordial inflation to the photon sector [21, 22, 23].Graviton effects are weaker because they are mediated through derivativeinteractions, but they are universal. Hence they serve to establish the mini-mum level at which primordial inflation must affect electromagnetism. Thepurpose of this paper is to complete the derivation of these minimum effects.

1

Our technique is based on a recent dimensionally regulated and fullyrenormalized computation of the one loop graviton contribution to the vac-uum polarization i[µΠν ](x; x′) on de Sitter background [24]. We use this toquantum correct Maxwell’s equation,

∂ν[√−g gνρgµσFρσ(x)

]

+∫

d4x′[

µΠν]

(x; x′)Aν(x′) = Jν(x) , (1)

where gµν is the de Sitter metric, Fρσ ≡ ∂ρAσ−∂σAρ is the usual field strengthtensor and Jµ(x) is the current density. With Jµ(x) 6= 0 one can study howinflationary gravitons alter the electrodynamic response to standard sources,as has recently been done for a point charge and for a point magnetic dipolewith the following results [25]:

• An observer co-moving with respect to the sources (hence at an expo-nentially increasing physical distance) perceives the magnitude of thepoint charge to increase linearly with co-moving time and logarithmi-cally with the co-moving position;

• The co-moving observer reports only a negative logarithmic spatial vari-ation in the one loop field of the magnetic dipole;

• An observer at fixed invariant distance from the sources perceives nosecular change of the point charge; and

• The static observer reports a secular enhancement of the magneticdipole moment.

For our study we set Jµ(x) = 0 and work out the one loop corrections todynamical photons.

This paper has four sections of which the first is this Introduction. Insection 2 we use the vacuum polarization [24, 25] to derive an equation forthe one loop correction to spatial plane wave photon mode functions. Thisequation is solved in section 3. In section 4 we discuss the minimum ourresult sets for inflation to seed cosmic magnetic fields.

2 Effective Mode Equation for Photons

The purpose of this section is to convert the quantum corrected Maxwellequation (1) into a simple relation for the one loop corrections to the mode

2

function of a plane wave photon on de Sitter background. We first specializeto plane wave photons at one loop order. Then we discuss the restrictionsimposed by cosmology, by effective field theory and by our lack of knowledgeabout the initial state.

2.1 Perturbative Formulation

We work on spatially flat sections of de Sitter in conformal coordinates,

ds2 = a2(η)(

−dη2 + d~x · d~x)

, (2)

where a(η) = − 1Hη

= eHt is the scale factor and H is the Hubble parameter.

Hence the metric can be written as gµν = a2ηµν , where ηµν is the Minkowskimetric. Because the Maxwell Lagrangian is conformally invariant, all thescale factors cancel in the leftmost term of (1) and we can express it as∂νF

νµ(x), where we raise and lower indices with the Minkowski metric, F νµ ≡ηνρηµσFρσ.

We follow [24] in employing a noncovariant representation for the vacuumpolarization [26],

i [µΠν ] (x; x′) = (ηµνηρσ−ηµσηνρ) ∂ρ∂′σF (x; x′)

+ (ηµνηρσ−ηµσηνρ) ∂ρ∂′σG(x; x′) , (3)

where ηµν ≡ ηµν + δµ0 δν0 is the purely spatial part of the Minkowski metric.

(For the transformation to a covariant representation see [27].) Substituting(3) into (1) with Jµ = 0 and partially integrating the primed derivatives onthe right hand side gives,

∂νFνµ(x) = −∂ν

∫

d4x′

iF (x; x′)F νµ(x′) + iG(x; x′)F νµ(x′)

. (4)

Here a barred index on any tensor means that its 0-components vanish, forexample, V µ ≡ ηµνVν = V µ − δµ0V

0.Some nonperturbative statements can be made. For example, so long

as the vacuum polarization is computed (as the one loop correction was[24]) using electromagnetic and gravitational gauge conditions which respecthomogeneity and isotropy then the structure functions F (x; x′) and G(x; x′)can depend upon the spatial coordinates ~x and ~x′ only through the Euclideannorm of their difference ∆~x ≡ ~x− ~x′. If spatial surface terms vanish (which

3

will be shown in the next sub-section) we can reflect external space derivativesonto the field strengths inside the integral of expression (4),

∂νFνµ(x) = −∂0

∫

d4x′ iF (x; x′)F 0µ

−∫

d4x′

iF (x; x′)∂′jF

jµ(x′) + iG(x; x′)∂′jF

jµ(x′)

. (5)

One simple consequence of (5) is the validity, to all orders, of the generalform for the classical field strengths of plane wave photon solutions with wavevector ~k and transverse polarization vector εi(~k, λ),

F 0iph(x) = −∂0u(η, k)× εiei

~k·~x , F ijph(x) = u(η, k)× i[kiεj−kjεi]ei

~k·~x . (6)

To see this, first substitute (6) into the µ = 0 component of equation (5) andact the space derivatives using ∂jF

µνph = ikjF

µνph ,

ikjFj0ph(x) = 0−

∫

d4x′ iF (x; x′)ikjFj0ph(x

′) + 0 . (7)

Exploiting transversality (kjεj = 0) reduces expression (7) to a tautology of

the form 0 = 0. Now substitute (6) into the µ = i component of (5) andagain exploit transversality,

−(

∂20+k2

)

u(η, k)× εiei~k·~x = ∂0

∫

d4x′ iF (x; x′)∂′0u(η

′, k)× εiei~k·~x′

+∫

d4x′ i[

F (x; x′) + G(x; x′)]

k2u(η′, k)× εiei~k·~x′

, (8)

= εiei~k·~x ×

∂0

∫

d4x′ iF (x; x′)∂′0u(η

′, k)e−i~k·∆~x

+k2∫

d4x′[

iF (x; x′)+iG(x; x′)]

u(η′, k)e−i~k·∆~x

. (9)

Dispensing with the now-redundant factors of εiei~k·~x results in effective mode

equation for u(η, k),

(

∂20+k2

)

u(η, k) = −∂0

∫

d4x′ iF (x; x′)∂′0u(η

′, k)e−i~k·∆~x

−k2∫

d4x′[

iF (x; x′)+iG(x; x′)]

u(η′, k)e−i~k·∆~x . (10)

4

Because the structure functions can only depend upon the norm of ∆~x itis possible to quite generally reduce the right hand side of the effective modeequation (10) to a double integral over η′ and r ≡ ‖~x−~x‖. However, we mustat this point face the fact that the structure functions can only be computedto some finite order in the quantum gravitational loop counting parameterκ2 ≡ 16πG,

F (x; x′) = 0 + κ2F(1)(x; x′) + κ4F(2)(x; x

′) + . . . , (11)

G(x; x′) = 0 + κ2G(1)(x; x′) + κ4G(2)(x; x

′) + . . . . (12)

At this time we have only the one loop results [24, 25],

iF(1) =−1

8π2

H2 [ln(a)+α]−[

ln(a)

3a2− β

a2

]

(

∂2+2Ha∂0)

+H

3a∂0

δ4(x−x′)

+a−1∂6

384π3

θ(∆η−∆x)

a′

(

ln[

1

4H2(∆η2−∆x2)

]

−1)

− H2

32π3

[

∂4

4+∂2∂2

0

]

×θ(∆η−∆x) ln[

1

4H2(∆η2−∆x2)

]

−[

∂4

4−∂2∂2

0

]

θ(∆η−∆x)

, (13)

iG(1) =H2

6π2

[

ln(a) +3

4γ]

δ4(x−x′)

+H2∂4

96π3

θ(∆η−∆x)(

ln[

1

4H2(∆η2−∆x2)

]

− 1)

. (14)

In these and subsequent expressions the coordinate separations are ∆η ≡η−η′ and ∆x ≡ ‖~x−~x′‖ and the flat space d’Alembertian is ∂2 ≡ ηµν∂µ∂ν =−∂2

0 +∇2.Because the structure functions are only known to a finite order in κ2

there is no alternative to making a similar expansion for the mode function,

u(η, k) = u0(η, k) + κ2u1(η, k) + κ4u2(η, k) + . . . (15)

Substituting expansions (11-12) and (15) into (10) and segregating terms ofthe same order in κ2 gives the tree order and one loop relations,(

∂20+k2

)

u0(η, k) = 0 , (16)(

∂20+k2

)

u1(η, k) = −∂0

∫

d4x′ iF(1)(x; x′)∂′

0u0(η′, k)e−i~k·∆~x

−k2∫

d4x′[

iF(1)(x; x′)+iG(1)(x; x

′)]

u0(η′, k)e−i~k·∆~x . (17)

By conformal invariance the tree order mode function is the same in de Sitterconformal coordinates as it is in flat space, u0(η, k) = e−ikη/

√2k.

5

2.2 Schwinger-Keldysh Formalism

The treatment (though not of course the explicit structure functions (13-14))we have given so far applies as well for traditional quantum field theory on flatspace (for example, see [28]). However, it is important to understand that theusual effective field equations describe matrix elements of the field operatorbetween states which are free in the asymptotic past and future. These in-outmatrix elements provide a correct description of scattering processes in flatspace, but they make little sense in cosmology because the universe beganwith a singularity at some finite time and no one knows how (or even if) itwill end. Persisting with the in-out effective field equations for inflationarycosmology would result in two embarrassments from the nonlocal source termon the right hand side of expression (17):

• Because the in-out structure functions do not vanish for x′µ outside thepast light-cone of xµ the right hand side of 17) would be dominated bycontributions from the far future when the inflated 3-volume is muchlarger;

• Because the in-out structure functions are complex they couple thereal and imaginary parts of the mode function u(η, k), making realfield strengths impossible.

The more meaningful effective field to study for cosmology is the trueexpectation value of the field operator in the presence of some state whichis released at a finite time. The appropriate field equations for studyingexpectation values are those of the Schwinger-Keldysh formalism [29, 30, 31,32, 33, 34, 35, 36, 37]. The associated one loop structure functions weregiven in expressions (13-14). Note that they are manifestly real, and thatthe factors of θ(∆η − ∆x) make each term vanish whenever the point x′µ

strays outside the past light-cone of xµ.1 These are important features of theSchwinger-Keldysh formalism which the in-out formalism lacks.

The constants α, β and γ which appear in expressions (13-14) representthe arbitrary finite parts of the three higher derivative counterterms whichwere needed to renormalize the vacuum polarization [24] because Einstein +Maxwell is not perturbatively renormalizable [38, 39]. (Appendix A discusses

1One consequence is that spatial integration by parts produces no surface terms in theSchwinger-Keldysh formalism. Partial integration in time can and does produce surfaceterms at the initial time.

6

the noncovariant counterterm resulting from the use of a de Sitter breakinggauge to compute the vacuum polarization [24].) No physical principle canfix these constants because those counterterms cannot actually be present infundamental theory. They are the price we must pay for using Einstein +Maxwell as a low energy effective field theory. In contrast, the logarithms ofthe scale factor with which the three constants are paired,

ln(a) + α ,1

3ln(a)− β , ln(a) +

3

4γ , (18)

represent unique and reliable predictions of the theory which must persistin whatever is the correct ultraviolet completion of Einstein + Maxwell. Atlate times these logarithms dwarf the unknown constants, which means thatwe can make reliable predictions in the late time regime. We are of coursemaking the usual assumption of low energy effective field theory that α, βand γ are of order one.

Another limit to the generality of our formalism is that the structurefunctions (13-14) were computed without correcting the free vacuum state.For in-out matrix elements we typically do not worry about correcting thestates because infinite time evolution is supposed to accomplish this in theweak operator sense. However, when the universe is released at a finitetime one must include at least the perturbative corrections to the initialstate. In the Schwinger-Keldysh formalism these corrections show up as newinteraction vertices on the initial value surface [40]. Unlike the finite partsof the higher derivative counterterms, it is perfectly possible to work thesecorrections out [41, 42, 43, 44, 45, 46]. However, there is no point to doingso because they give rise to surface terms which fall off like powers of theinflationary scale factor. We shall assume that these corrections simply serveto cancel the surface terms which would arise, at various points, from partialintegrations.

3 Solving the Equation

The purpose of this section is to solve equation (17) for u1(η, k) in the latetime limit for which reliable predictions can be made. First note that (17)can be expressed in terms of seven master integrals,

(∂20+k2)u1(η, k) = ik∂0

[

I1+I2+I3+I4+I5+I6+I7]

7

+k2[

−I1+1

3I2−I3+

1

3I4−I5−I6−I7

]

, (19)

where the various integrals are,

I1(η, k)≡−a−1(∂20+k2)3

384π3

∫

d4x′Θ

a′

[

ln

[

H2

4

(

∆η2−∆x2)

]

−1

]

u0(η′, k)e−i~k·∆~x,(20)

I2(η, k)≡−H2(∂20+k2)2

128π3

∫

d4x′Θ

[

ln

[

H2

4

(

∆η2−∆x2)

]

−1

]

u0(η′, k)e−i~k·∆~x,(21)

I3(η, k)≡H2(∂2

0+k2)∂20

32π3

∫

d4x′Θ

[

ln

[

H2

4

(

∆η2−∆x2)

]

+1

]

u0(η′, k)e−i~k·∆~x, (22)

I4(η, k)≡−H2 ln(a)

8π2

∫

d4x′δ4(x−x′)u0(η′, k)e−i~k·∆~x, (23)

I5(η, k)≡−a−2 ln(a)(∂20+k2)

24π2

∫

d4x′δ4(x−x′)u0(η′, k)e−i~k·∆~x, (24)

I6(η, k)≡Ha−1 ln(a)∂0

12π2

∫

d4x′δ4(x−x′)u0(η′, k)e−i~k·∆~x, (25)

I7(η, k)≡−Ha−1∂024π2

∫

d4x′δ4(x−x′)u0(η′, k)e−i~k·∆~x, (26)

To save space, we have defined the causality-enforcing θ-function as Θ ≡θ(∆η − ∆x). Of course the delta function terms (23-26) are trivial, andmost of the nonlocal contributions can be inferred from previous work [47].Technical details can be found in Appendix B but the results are,

I1(η, k) =H2u0(η, k)

48π2a

[1+2ik∆ηi+e2ik∆ηi

H2∆η2i

]

+[1+e2ik∆ηi

H∆ηi

]

−4ik

Hln(H∆ηi)−

2ik

H

∫ 1

0

dt

t

[

e2ik∆ηit−1]

, (27)

I2(η, k) =H2u0(η, k)

16π2

−2 ln(H∆ηi)−∫ 1

0

dt

t

[

e2ik∆ηit−1]

, (28)

I3(η, k) =H2u0(η, k)

16π2

[

6−4ik∆ηi+2e2ik∆ηi]

ln(H∆ηi)+e2ik∆ηi+7−2ik∆ηi

+∫ 1

0

dt

t

[

(3−2ik∆ηi)(e2ik∆ηit−1)+e2ik∆ηi(e−i2ik∆ηit−1)

]

, (29)

I4(η, k) = −H2 ln(a)

8π2× u0(η, k) , (30)

8

I5(η, k) = 0 , (31)

I6(η, k) = −ikH ln(a)

12π2a× u0(η, k) , (32)

I7(η, k) =ikH

24π2a× u0(η, k) . (33)

Here and henceforth we define ∆ηi ≡ η − ηi = H−1(1 − 1a), where ηi is the

initial conformal time.A few comments are in order regarding the inverse factors of ∆ηi which

appear in expression (27) for I1(η, k). These factors diverge on the initialvalue surface and completely preclude any attempt to exactly solve the oneloop truncated effective field equations in their present form. That the prob-lem has nothing to do with de Sitter background is obvious from the fact thatthese very same divergences appear as well when the background is changedto flat space [28]. The problem arises instead because the vacuum polariza-tion was computed in free (Bunch-Davies) vacuum Ω0[A, h] [24]. Even in flatspace the true vacuum state wave functional Ω[A, h] requires perturbativecorrections [41, 42, 43, 44, 45, 46],

Ω[A, h] = Ω0[A, h]×

1 + κ∫

d3x1

∫

d3x2

∫

d3x3

×Ωµνρσ(~x1, ~x2, ~x1)hµν(ηi, ~x1)Aρ(ηi, ~x2)Aσ(ηi, ~x3) +O(κ2)

, (34)

where Ωµνρσ(~x1, ~x2, ~x3) is a C-number function which could be worked out— but has not been — the same way one computes corrections the simpleharmonic oscillator wave functions when the Hamiltonian contains an anhar-monic term. We stress that the problem derives from combining (D = 4)interactions with evolution from a finite time, and it cannot be solved by anyclever choice Gaussian initial state Ω0[A, h]. It can be avoided in flat spaceby taking the initial time to −∞ [28] but this is not an option for our de Sittercomputation owing to the factors of ln(a) and 1/a which are evident in ex-pressions (30) and (32-33).2 On the other hand, the initial value divergences

2The physical origin of this mathematical obstacle is inflationary particle productionwhich results in very high occupation numbers N(t, k) = [Ha(t)/2k]2 for graviton modeswhich have experienced first horizon crossing. The earlier one releases the initial state themore modes will have experienced first horizon crossing by any fixed late time.

9

i√2k Ii(η, k)

√2k ∂0Ii(η, k)

1 O( 1a) O(1)

2 O(1) O(1)

3 O(1) O(1)

4 −H2 ln(a)8π2 −H3a

8π2

5 0 0

6 O( ln(a)a

) ikH2 ln(a)12π2

7 O( 1a) O(1)

Table 1: Leading late time limiting forms for the integrals defined in expres-sions (20-26) and their first time derivatives. Only terms which show seculargrowth are given explicitly.

in expression (27) are all well behaved at late times,

1

H2∆η2i−→ 1 ,

1

H∆ηi−→ 1 , ln(H∆ηi) −→ 0 . (35)

The multiplicative factor of 1/a makes I1(η, k) go to zero at late times, so wecan avoid computing initial state corrections by simply working consistentlyin the late time regime, which is in any case necessary owing to the unknownfinite parts of the counterterms.

Table 1 gives the leading late time effect from each of the seven integralsand its first (conformal) time derivative. The dominant effect derives fromthe time derivative of I4(η, k),

(∂20+k2)u1(η, k) = −ikH3a

8π2× u0(η, k) +O

(

ln(a))

. (36)

Hence we find,

u1(η, k) =ikH ln(a)

8π2× u0(η, k) +O

(1

a

)

. (37)

From expression (6) follows that the one loop field strengths are,

κ2F 0i(1)(x) =

κ2H2

8π2

ln(a) +O(1)

× F 0i(0)(x) , (38)

10

κ2F ij(1)(x) =

κ2H2

8π2

ik ln(a)

Ha+O

(1

a

)

× F ij(0)(x) . (39)

4 Discussion

We have employed a previous computation of the one loop contribution tothe vacuum polarization from inflationary gravitons [24] to derive what hap-pens to photons during primordial inflation. Our results (38-39) for the fieldstrengths show that the electric field experiences a secular enhancement, rela-tive to its classical value. In contrast, the one loop correction to the magneticfield falls off with respect to its classical counterpart. Both results are con-sistent with the one loop photon wave function (37) relaxing to zero lessslowly (by one factor of ln(a)) than the classical mode function approachesa constant.

The enhancement we find seems to derive from the buffeting of photonsby inflationary gravitons. Even though the photon’s kinetic energy redshiftsto zero, its spin does not and this permits it to continue interacting withinflationary gravitons even at late times. The same ln(a) enhancement wasfound for massless fermions [48, 47, 50], and was explicitly tied to the spininteraction [49]. In contrast, massless, minimally coupled scalars neither ex-perience any significant effect from inflationary gravitons [51, 52], nor dothey induce a significant effect on inflationary gravitons [53, 54, 55]. Gravi-tons also have spin so it would be very interesting to see what they do tothemselves, as well as to the force of gravity.

An interesting technical detail concerns the comparison of our full oneloop computations with the result previously derived using the Hartree ap-proximation [24]. Both calculations give the same time dependence, confirm-ing the general reliability of the Hartree approximation for predicting thefunctional form. However, the sign of our exact computation differs fromthat of the Hartree result. This emphasizes the need for making exact com-putations, and has clear implications for gravitons [56].

One important consequence of our result is that quantum gravitationalperturbation theory must break down after an enormous number of e-foldingsln(a) ∼ 1/κ2H2, which is larger than 1010 in standard inflation. Note firstthat this eventual breakdown in no way invalidates the use of perturbationtheory at earlier times. The reason for working in the late time regime is thatwe have not included perturbative corrections to the initial state (although

11

this could be done) and that we do not know, and cannot know, the finiteparts α and β of the higher time derivative counterterms. However, as longas one makes the usual assumption that these finite parts are of order oneit will be seen that there is an enormous range of times for which the latetime limiting form (38) vastly dominates the unknown contributions, but isstill small compared with the classical result. For example, if it is agreed thatthe infrared logarithm ln(a) dominates when it has reached ln(a) >∼ 100, andthat it is still reliable for κ2H2 ln(a) <∼ 1/100, then we see that the regimeof validity for our result extends to a million e-foldings of inflation. Forsome perspective it should be recalled that there is no currently recognizedevidence for more than about 60 e-foldings of inflation.

Still, it is a fact that perturbation theory must eventually break downif inflation persists long enough, and we would dearly love to know whattakes place afterwards. The analogous problem for scalar potential modelson nondynamical de Sitter (whose loop corrections also involve factors ofln(a)) can be solved using Starobinsky’s stochastic formalism [57, 58]. Aproof has been given that this method reproduces the leading secular growthfactors at each order in perturbation theory [59, 60], and it has been extendedto include scalars coupled to fermions [61] and to electromagnetism [18].Detailed analysis of the nonperturbative resummation of this series of leadinglogarithms reveals that all three logical possibilities occur for the inducedvacuum energy:

• It can approach a small positive constant [58];

• It can approach a small negative constant [18]; or

• It can decrease without bound, resulting in a Big Rip singularity [61].

The stochastic formalism is also very useful in debunking techniques whichare sometimes proposed for evolving past the breakdown of perturbationtheory, for example, using order reduction to solve the one loop truncatedeffective field equations exactly [62],3 or employing variants of the renormal-ization group [63]. These techniques do give equations which can be evolvedpast the breakdown of perturbation theory, but the results are wrong [64].

3Order reduction is just a technique for preventing higher derivatives from introducingnew solutions, which problem does not even arise in our one loop solution. The problemcomes in solving the one loop truncated equations exactly. This is only valid if no equallystrong contributions occur in primitive contributions from higher loops. The stochasticformalism shows that they do.

12

Unfortunately, the presence of derivative interactions means that theproof of Starobinsky’s formalism which was given for scalar potential models[59, 60] does not apply to quantum gravity. One might speculate that thestochastic formalism is nonetheless correct, and it has been employed on thisbasis to estimate corrections to the inflationary power spectra [65]. Otherplausible approximations were earlier invoked to arrive at somewhat differentestimates [66, 67, 68]. And there has been recent work by Kitamoto and Ki-tazawa on secular growth corrections to gauge couplings [69, 70, 71, 72, 73].The key question is, does any specific formalism reproduce the correct seculargrowth factors from inflationary gravitons? Absent a proof, one can neverknow except by comparison with complete and fully renormalized resultssuch as the one (38) we have just derived. Indeed, expression (38) is onlythe second example of such a result so it effectively doubles what is reliablyknown about this fascinating phenomenon.

An appreciation of the importance of (38) comes by recalling what waslearned from the only other complete and fully renormalized graviton loopwhich shows a factor of ln(a), the 2005-6 computation of inflationary gravitoncorrections to massless fermions [48, 47, 50]. A diagram-by-diagram analysisof that result reached the following conclusions [49]:

• Naive application of the stochastic formalism does not correctly predictthe secular growth factor of ln(a);

• The factor of ln(a) derives entirely from diagrams which involve thefermion spin connection; and

• Because the ultraviolet sector of the fermion field contributes to thefactor of ln(a), one must leave the ultraviolet regularization on as wellfor the graviton.

One should also note the claim that false indications of secular correctionscan come from neglecting diagrams [74].

It is not that approximate calculations are necessarily wrong but ratherthat we cannot currently judge their validity. When a technique is finallydevised for isolating the leading secular effects at each order and re-summingthem it will no doubt lead to simple approximations for quickly inferringjust the leading effects, without enduring the tedium of a complete and fullyrenormalized computation. We have reached this point with scalar potentialmodels through Starobinsky’s stochastic formalism. However, we are not

13

there yet for quantum gravity and we are not likely to get there, or be surethat we have even made any progress, without careful examination of exactresults such as expression (38). These comparisons sometimes reveal subtleproblems with plausible-sounding arguments such as the universal applica-bility of the stochastic formalism or the irrelevance of ultraviolet effects.

Finally, it is interesting to work out what our result says for the possi-bility of inflation seeding cosmic magnetic fields. We can use the 0-pointenergy to estimate the number of photons created by inflationary gravitons.Because the co-moving time t is related to the conformal time η by dt = adη,the physical Hamiltonian (which generates evolution in t) is 1/a times theconformal one. The physical 0-point energy in a single photon polarizationwave vector ~k is therefore,

1

2a

[

F 0iph × F 0i

ph

∗+

1

2F ijph × F ij

ph

∗]

=k2uu∗

2a, (40)

−→ k

2a×

1 +κ2H2 ln(a)

8π2+O(κ4)

. (41)

The occupation number N(η, k) is defined by equating the 0-point energy(41) to (N + 1

2)× k

a,

N =κ2H2 ln(a)

16π2+O(κ4) . (42)

The remainder of the computation is the same as the analysis [12] for themuch larger effect from inflationary scalars (if the Higgs is minimally coupledand still light at inflationary energy scales). Substituting our value (42)for the occupation number into equation (118) of that paper results in thefollowing estimate for the magnetic strength on scale ℓ0,

B2(ℓ0) ∼h2GH2

INI

8π3c4ℓ40, (43)

where HI is the inflationary Hubble parameter and NI is the value reachedby ln(a). Plugging in the numbers with NI ∼ 50 and ℓ0 ∼ 10 kpc gives,

B(ℓ0) ∼ 10−61 Gauss . (44)

This is far too small to have seeded today’s cosmic magnetic fields, but itdoes serve to establish the absolute minimum effect which must be presentfrom inflation.

14

5 Appendix A: de Sitter Breaking Gauges

Our analysis is based on an earlier computation [24] of the one loop vacuumpolarization from inflationary gravitons that was made using electromagneticand gravitational gauge fixing terms which respect homogeneity and isotropy,and also dilatation invariance, but not the three remaining symmetries of thede Sitter group. The special feature of these gauge fixing terms is that thephoton and graviton propagators consist of linear combinations of constanttensor factors times scalar propagators [75, 76]. This makes the computationtractable but it also means that renormalization can involve noninvariantcounterterms, and one such does occur [24].4 Similar noninvariant countert-erms arose from the graviton contribution to the one loop fermion self-energy[48] (see equations (74), (216) and (221))and the one loop self-mass-squaredof of a massless, minimally coupled scalar [51] (see equation (145) and Table8).

Those accustomed to modern techniques of renormalization in covariantgauges sometimes find the appearance of noninvariant counterterms to bedisconcerting. However, it is important to realize that they pose no problemof principle. The divergent part of the counterterm is of course fixed by theprimitive divergences it is to remove, and the finite part can be determinedto enforce physical symmetries. (In our case the focus on late times obviatesthe need for this as long as the finite part of the counterterm is assumedto be of order one.) The procedure is explained in older standard texts onquantum field theory, for example [77]. And it is important to recognize thatmany of the classic computations of quantum electrodynamics were in factperformed using noncovariant gauges [78].

Finally, it should be noted that an impressive amount of evidence hasbeen accumulated to support the consistency of this particular gauge. Thisevidence includes:

• An explicit check of the tree order gravitational Ward identity [79];

• An explicit check of the one graviton loop gravitational Ward identity[80];

4The vacuum polarization was recently computed in a manifestly de Sitter invariantgauge. Strangely enough, the noninvariant counterterm is still required owing the the time-ordering of interactions, the fact that gravitational interactions contain two derivatives,and the fact that the coincident graviton propagator diverges in de Sitter background [83].

15

• A detailed physical comparison between the noncovariant gauge resultfor the one (photon) loop contribution to a charged scalar self-mass-squared [15, 16] and the analogous covariant guage result [17]; and

• A computation of the linearized Weyl-Weyl correlator in both the non-covariant [81] and covariant gauges [82].

6 Appendix B: Integrals from Section 3

The purpose of this appendix is to summarize the necessary details for eval-uating the nonlocal terms (20-22), and show they indeed make no significantcontribution in the late time regime. The first step is to perform the angularintegrations using the formula,

∫

d3x′Θf(∆x)e−i~k·∆~x = 4π∫ ∆η

0drr2f(r)

sin(kr)

kr. (45)

Employing relation (45) in (20-22) allows us to write,

I1=−a−1(∂20+k2)3

96π2k

∫ η

ηi

dη′u(η′, k)

a′

∫ ∆η

0drr sin(kr)

ln

[

H2

4(∆η2−r2)

]

−1

, (46)

I2=−H2(∂20+k2)2

32π2k

∫ η

ηi

dη′u(η′, k)∫ ∆η

0drr sin(kr)

ln

[

H2

4(∆η2−r2)

]

−1

, (47)

I3=H2(∂2

0+k2)∂20

8π2k

∫ η

ηi

dη′u(η′, k)∫ ∆η

0drr sin(kr)

ln

[

H2

4(∆η2−r2)

]

+1

, (48)

where ηi ≡ −H−1 denotes the initial time. The next step is to perform thetwo independent radial integrations,

J1(∆η, k)≡∫ ∆η

0drr sin(kr) =

T (k∆η)

k2, (49)

J2(∆η, k)≡∫ ∆η

0drr sin(kr) ln

[

H2

4(∆η2−r2)

]

, (50)

=2 ln(H∆η)

k2T (k∆η) +

1

k2

∫ 1

0

dt

t

T [k∆η(1−2t)]−T (k∆η)

, (51)

where T (x) ≡ sin(x)−x cos(x) = 13x3 +O(x5). This allows us to express the

integrals I1−3(η, k) as,

I1(η, k)=−a−1(∂20+k2)3

96π2k

∫ η

ηi

dη′u(η′, k)

a′

J2(∆η, k)−J1(∆η, k)

, (52)

16

I2(η, k)=−H2(∂20+k2)2

32π2k

∫ η

ηi

dη′u(η′, k)

J2(∆η, k)−J1(∆η, k)

, (53)

I3(η, k)=H2∂2

0(∂20+k2)

8π2k

∫ η

ηi

dη′u(η′, k)

J2(∆η, k)+J1(∆η, k)

. (54)

Because the various integrands of (52-54) vanish like ∆η3 ln(∆η) at η′ = ηwe can pass one factor of the differential operator (∂2

0 + k2) through theintegration to act on J1(∆η, k) and J2(∆η, k),

I1(η, k) = −a−1(∂20+k2)2

48π2k

∫ η

ηi

dη′u(η′, k)

a′

2 sin(k∆η) ln(H∆η)

+∫ 1

0

dt

t

[

sin [k∆η(1−2t)]−sin(k∆η)]

, (55)

I2(η, k) = −H2(∂20+k2)

16π2k

∫ η

ηi

dη′u(η′, k)

2 sin(k∆η) ln(H∆η)

+∫ 1

0

dt

t

[

sin [k∆η(1−2t)]−sin(k∆η)]

, (56)

I3(η, k) =H2∂2

0

4π2k

∫ η

ηi

dη′u(η′, k)

2 sin(k∆η) ln(H∆η) + 2 sin(k∆η)

+∫ 1

0

dt

t

[

sin [k∆η(1−2t)]−sin(k∆η)]

. (57)

We can pass one more derivative through the integration using the identities,

(∂20+k2)

[

u(η′, k)f(∆η)]

= −(∂0−ik)× ∂′0

[

u(η′, k)f(∆η)]

, (58)

(∂20+k2)2

[u(η′, k)

a′f(∆η)

]

= −(∂20+k2)(∂0−ik)× ∂′

0

[u(η′, k)

a′f(∆η)

]

+H(∂0−ik)2 × ∂′0

[

u(η′, k)f(∆η)]

. (59)

The final results are (31-33).Acknowledgements

We are grateful for conversation and correspondence on this subject withS. Deser, K. E. Leonard, S. P. Miao and T. Prokopec. This work was partiallysupported by NSF grant PHY-1205591 and by the Institute for FundamentalTheory at the University of Florida.

17

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