Inflation and Cyclotron Motion

# Inflation and Cyclotron Motion

Jeff Greensite Physics and Astronomy Department, San Francisco State University,
San Francisco, CA 94132, USA
July 27, 2019
###### Abstract

We consider, in the context of a braneworld cosmology, the motion of the universe coupled to a four-form gauge field, with constant field strength, defined in higher dimensions. It is found, under rather general initial conditions, that in this situation there is a period of exponential inflation combined with cyclotron motion in the inflaton field space. The main effect of the cyclotron motion is that slow roll conditions on the inflaton potential, which are typically necessary for exponential inflation, can be evaded. There are Landau levels associated with the four-form gauge field, and these correspond to quantum excitations of the inflaton field satisfying unconventional dispersion relations.

## I Introduction

Braneworld cosmology is a concept that exists in many variations. There are versions in which the higher dimensions are compactified, as in the Arkani-Hamed, Dimopoulous, Dvali proposal ArkaniHamed:1998nn (), or large but warped, as in the Randall-Sundrum model Randall:1999vf () and string-motivated DBI inflation Alishahiha:2004eh (); HenryTye:2006uv (). There is also the intriguing Dvali-Gabadadze-Porrati (DGP) version where the extra dimension is large but nearly flat Dvali:2000hr (). Consideration of the four-dimensional effective theory in the DGP model has led to a very general class of four-dimensional galileon models Nicolis:2008in () with powers of derivative terms greater than two, for which there now exists an extensive literature (see, e.g., Burrage:2010cu (); Trodden:2011xh (); Deffayet:2009mn (); Neveu:2016gxp () and references therein).

In this article I would like to describe some interesting features of the following action, describing a brane with standard model particle content evolving in a flat higher-dimensional background, with a coupling of the brane to an external four-form gauge field in the bulk:

 S = 116πG∫d4x√−gR+SSM −∫d4x√−g(12gμν∂μφs∂νφs+V(φ)) +q04!∫d4x Aabcd[ϕ(x)]ϵαβγδ∂αϕa∂βϕb∂γϕc∂δϕd,

where is the action of standard model (and possibly beyond-standard-model) fields, and

 gμν = ∂μϕAηAB∂νϕB   ,   A,B=0,1,...,D (2)

is the induced metric of a three brane in a -dimensional Minkowski space. We adopt the convention that upper case Latin indices run from 0 to , indices run from to , and all other lower case Latin indices run from to . We also define

 ϕs=1σ2φs, (3)

where is a constant with dimensions of mass. The fields are a set of inflaton fields, with the inflaton potential, and is a potential which is totally antisymmetric in the indices. It can be thought of as a four-form gauge field in dimensions. The induced metric corresponds to dimensions, however.

The main novelty of this formulation is the interaction of the braneworld with an external four-form gauge field in the bulk, and it is the purpose of this article to describe some possible consequences in an inflationary scenario. Like the DGP model there are large flat extra dimensions, but unlike that model there is no Einstein-Hilbert action in the bulk. Unlike Galileon models in general there is no galilean invariance, and the external four-form gauge field singles out special directions in the bulk. Inflation, in the scenario suggested below, is driven by inflaton fields with an ordinary potential in the inflaton action, rather than by galileon fields.

Without the external gauge field, a model with an Einstein-Hilbert action and other fields on the brane seems to have been first considered long ago by Regge and Teitelboim Regge (). The first question to ask of a model of this type is whether the equations of motion are equivalent, at the classical level, to standard general relativity at . The answer is: not quite. Denote

 Eμν ≡ δS[A=0]δgμν (4) = 12√−g{−18πGGμν+Tμν}.

Where is the stress-energy tensor of the standard model and inflaton fields. Then the field equations resulting from variation of the at are

 ηAB∂μ(Eμν∂νϕB)=0. (5)

These equations are obviously satisfied by the Einstein field equations . Moreover, any solution of can be embedded locally in a ten-dimensional flat Minkowski space, although globally an embedding may require still higher dimensions Clarke (). But of course there may be also be solutions of (5) which are not solutions of the Einstein equations. A simple (and intriguing) example is pure gravity with a cosmological constant, in which case

 Eμν=12√−g{−18πGGμν−λgμν}. (6)

In this case the equations of motion are certainly solved by de Sitter space, for which . But flat Minkowski space is also a solution: just choose and constant. Then , , and the equations of motion boil down to , which is satisfied trivially.

A criticism of Deser et at. Deser:1976qt () is that the embedding of a four-manifold is not unique. Some embeddings of a four-manifold may satisfy the equations of motion (5), and some may not. This fact does not necessarily rule out the embedding formulation of general relativity on experimental grounds; it could simply be that the alternative is selected by initial conditions on the .

When the four-form gauge field is included, there will in general be some deviation from the standard Einstein field equations. The equations of motion in this case are

 2ηAB∂μ(Eμν∂νϕB) −q04!FAabcdϵαβγδ∂αϕa∂βϕb∂γϕc∂δϕd=0, (7)

and

 ∂μ(√−ggμν∂νφs)−√−g∂V∂φs +q04!σ2Fsabcdϵαβγδ∂αϕa∂βϕb∂γϕc∂δϕd=0, (8)

where is the field strength

 Ffabcd=∂Aabcd∂ϕf−∂Afbcd∂ϕa+∂Afacd∂ϕb−∂Afabd∂ϕc+∂Afabc∂ϕd

corresponding to the four-form gauge field. These are supplemented by the usual equations of motions of the standard model fields.

In this article I would like to explore the cosmological consequences of these equations of motion in the simplest non-trivial case, namely, a constant field strength in a homogenous isotropic spacetime. For this purpose it will be sufficient to work in a five-dimensional embedding space, , with two inflaton fields , and ignoring, at the classical level, all standard model fields.

## Ii Inflation

It is well known that a four dimensional manifold described by a Friedman-Lemaitre metric can be embedded in five-dimensional space, and for simplicity we adopt the version with zero spatial curvature. We take the embedding to be Rosen (); LachiezeRey:2000my ()

 ϕ0 = 12{a(t)+∫tdt′da/dt′+a(t)r2} ϕ1 = a(t)rcos(θ) ϕ2 = a(t)rsin(θ)cos(χ) ϕ3 = a(t)rsin(θ)sin(χ) ϕ4 = 12{a(t)−∫tdt′da/dt′−a(t)r2}, (10)

and it is not hard to see that

 ds2 = ηABdϕAdϕB  ,  A,B=0,1,2,3,4 (11) = −dt2+a2(t)(dr2+r2(dθ2+sin2(θ)dχ2))

is the Friedman-Lemaitre metric. But let us also suppose that there is a four-form gauge field dependent on the coordinates , whose non-zero components are

 A5123[ϕ] = −12Bϕ6, A6123[ϕ] = 12Bϕ5. (12)

The four-form gauge field is antisymmetric under permutations of indices, but apart from (12) and components obtained from (12) by permutation, it is assumed that all other components vanish. This choice leads to a constant non-zero field strength , and we are interested in exploring the consequences for early-universe dynamics in a situation of this kind. In this context we also assume the simplest possible inflaton potential

 V[ϕ]=12m2φsφs . (13)

We begin with the usual simplifying assumptions of spatial homogeneity and isotropy, taking in particular

 ϕ5,6(x,y,z,t)=ϕ5,6(t), (14)

and for . In conjunction with (12), this has the consequence that

 FAabcdϵαβγδ∂αϕa∂βϕb∂γϕc∂δϕd=0. (15)

This is because two of the indices must be 5 and 6, so the expression necessarily includes at least one space derivative of , which vanishes according to (14). Then the equation of motion is satisfied by , which are the standard Einstein field equations. For a Friedman-Lemaitre metric, disregarding the other standard model fields, the Einstein equations are just the conventional expressions for the scale factor coupled to a pair of scalar fields:

 ˙a2a2 = 8πG3(12∂tφs∂tφs+12m2φsφs), ¨aa = 8πG3(−∂tφs∂tφs+12m2φsφs). (16)

The equations of motion for the , however, involve the field strength

 ∂2tφ5−qB∂tφ6+3˙aa∂tφ5+m2φ5 = 0, ∂2tφ6+qB∂tφ5+3˙aa∂tφ6+m2φ6 = 0, (17)

where . It is not hard to verify consistency of (16) and (17).

If we set and in (17), then these equations are obviously the equations of motion of a charged particle moving, in the plane, under the influence of a magnetic field orthogonal to that plane; i.e. this is cyclotron motion. If we instead set , then these are the equations used in simple models of inflation. In models of that type it is normally important to impose slow roll conditions, which imply either a large initial value for the inflaton field, or else, unlike (13), a very flat potential (see, e.g., Chapter 8 in Peter:1208401 ()). For the simple potential (13) these slow roll conditions boil down to

 φsφs≫16πG, (18)

i.e. a large initial field.

The model we are discussing has a fairly large space of parameters and initial conditions but the time development is typically a spiral in the plane. What may be of interest is the fact that for it is possible to have a period of approximately exponential inflation, with a large number of e-foldings, even when the slow-roll condition (18) is strongly violated.111It should be noted, however, that there are other mechanisms for easing the slow roll conditions in the context of a braneworld cosmology, cf. Maartens:1999hf (). A single example should suffice. Working in Planck units, we choose parameters and initial conditions

 qB = 0.2  ,  m2=2×10−4, φ5(0) = 0   ,   φ6(0)=10−2, (∂tφ5)t=0 = 0   ,   (∂tφ6)t=0=0. (19)

The resulting spiral evolution in the plane is shown in Fig. 1, with and vs. cosmic time shown in Figs. 1 and 1 respectively. The expansion is very nearly a simple exponential up to in Planckian units, which is evident in the rather flat curves on the log-log plots, and the fact that

 ¨aa≈(˙aa)2 (20)

in this period. Expansion continues after this period, however, resulting in a total of about 100 e-foldings by .

The potential is responsible for a force towards the origin of the plane, while the “Lorentz force” due to the four form gauge field is directed away from the origin. Eventually these forces balance to produce a circular motion, spiraling towards the center. To see this, we plot the initial stage of the evolution in Fig. 2. In the absence of the gauge field, the system simply falls to the center, oscillating around the axis, and, because slow roll conditions are not satisfied, there is no inflationary period. The Lorentz force, however, deflects the initial fall to the center into an arc, and this interplay between the central potential, the Lorentz force, and gravitational friction continues until the inward and outward forces sum to a centripetal force for (roughly) circular motion, with gravitational friction causing a gradual spiral to the origin. The trajectory resulting from a quite different set of parameters is shown in Fig. 3. While this last example does not lead to many e-foldings, it does very clearly display the initial interplay of forces, leading to an eventual spiral towards the origin.

## Iii Landau Levels

After inflation, the constant field strength of the four-form gauge field still has an effect at the quantum level, in the form of Landau excitation levels of the quantized fields. We will see that these excitations satisfy a rather unusual dispersion relation.

We consider the post-inflationary period at some time where is negligible, . With given by the embedding (10), and as in (12), we have

 q04!Aabcd[ϕ(x)]ϵαβγδ∂αϕa∂βϕb∂γϕc∂δϕd (21) = qAs123ϵ0ijk∂tφs∂iϕ1∂jϕ2∂kϕ3 = qAs∂tφs(R3r2sinθ),

where . The factor of can be absorbed into a coordinate redefinition, and we then consider quantizing the action

 Sφ = ∫d4x(12∂tφs∂tφs−12∇φs⋅∇φs (22) −12m2φsφs+qAs(φ)∂tφs),

where again the index . The corresponding Hamiltonian is

 H = 12∫d3x{(ps−qAs)2+(∇φs)2+m2φsφs}, (23)

and have standard quantization conditions. Define

 ωk = √k2+14q2B2+m2 φs(x) = ∫d3k(2π)31√2ωk(as(k)eik⋅x+a†s(k)e−ik⋅x) ps(x) = ∫d3k(2π)3√2ωk12i(as(k)eik⋅x−a†s(k)e−ik⋅x), (24)

with the usual commutation relations

 [as(k1),a†r(k2)]=(2π)3δ3(k1−k2)δrs (25)

Then

 H = ∫d3k(2π)3{ωk(a†s(k)as(k)+δ3(0)) (26) +i12qB(a†5(k)a6(k)−a†6(k)a5(k))}.

Introduce

 b1(k) = 1√2(a5(k)+ia6(k)) b2(k) = 1√2(a5(k)−ia6(k)). (27)

which again have the usual commutation relations

 [bi(k1),b†j(k2)]=(2π)3δ3(k1−k2)δij (28)

with indices . The Hamiltonian takes the form

 H = ∫d3k{ωk(b†i(k)bi(k)+δ3(0)) (29) +12qB(b†1(k)b1(k)−b†2(k)b2(k))},

and the corresponding spectrum is

 E = ∑k{√k2+14q2B2+m2(n1(k)+n2(k)) (30) +12qB(n1(k)−n2(k))}+E0,

where are occupation numbers, is the ground state energy, and the sum runs over momenta with non-zero occupation numbers. We also find, by standard manipulations, the conserved total momentum

 Pi=∑kki(n1(k)+n2(k)). (31)

Were it not for the term proportional to in (30), the spectrum would simply consist of two types of particles of mass

 M′=√14q2B2+m2. (32)

Instead, defining , it is seen that excitations which are eigenstates of both and (with momentum eigenvalues ) satisfy dispersion relations

 E1(k) = √k2+M2+m2+M,   and E2(k) = √k2+M2+m2−M, (33)

respectively, which is clearly at odds with the relativistic expression for a free particle. But of course these excitations are not free particles, and the Lagrangian (22) they derive from is not Lorentz invariant, or even (unlike Newtonian mechanics) boost invariant. It is the external four-form gauge field which singles out a preferred time direction (much as, e.g., an ordinary background magnetic field along the -axis would introduce a preferred spatial direction for objects sensitive to that field), and the only remaining space-time symmetries are rotation and time/space translation invariance. Therefore the breaking of both Lorentz and boost invariance, so far as these inflaton excitations are concerned, is not a surprise. The question is how this breaking might manifest itself.

## Iv Properties of Landau level excitations

### iv.1 Group velocity

To begin with, consider how a wavepacket corresponding to a single “heavy” Landau excitation of energy , or a “light” Landau excitation of energy , and momentum , will propagate in time. Let correspond to a particle eigenstate of energy and momentum respectively, with conventional normalization

 |k,j⟩ = √2ωkbj(k)|0⟩ |x,j⟩ = ∫d3k(2π)3e−ik⋅x|k,j⟩ , (34)

and we consider initial wavepackets of the form

 |ψj⟩t=0 = ∫d3k(2π)31√2ωk|f(k)|k,j⟩ ψj(x,t=0) = ⟨x,j|ψj⟩t=0 (35) = ∫d3k(2π)3f(k)eik⋅x .

Then at a later time

 ψj(x,t) = ⟨x,j|e−iHt|ψj⟩t=0 (36) = e−i(3−2j)Mt∫d3k(2π)3f(k)ei(k⋅x−ωkt) .

From this we conclude that wavepackets of both heavy and light Landau excitations (we might as well call them “landons”) propagate with a group velocity appropriate to a particle of mass (for ). On the other hand, at low momenta in the frame singled out by the external four-form gauge field,

 E1(k) ≈ k22M+2M+m22M E2(k) ≈ k22M+m22M , (37)

which means that the rest energy of the heavy landons is approximately , while that of the light landons is approximately .

### iv.2 Scattering in a gravitational field

Because of the mismatch between the inertial mass in the momentum-dependent term and the rest energy, we may expect an apparent violation of the principle of equivalence, if it would be possible to somehow observe the motion of these excitations in a gravitational field. This can be verified by calculating the differential scattering cross section of heavy and light landons in the weak gravitational field of a static massive object of mass .

Let with

 g00 = −(1−2GMr)  ,  gii=(1+2GMr) gμν = 0   (μ≠ν) , (38)

be the metric corresponding to the massive object at the origin, at distances such that the gravitational field is weak. For our purposes it is sufficient to ignore this restriction on , unless we are interested in large angle scattering. We first need the interaction Hamiltonian to lowest order in . For this we consider the part of the total action containing , where

 Sφ = −∫d4√−g(12gμν∂μφs∂νφs+12m2φsφs) SA = ∫d4x qAsϵ0ijk∂tφs∂iϕ1∂jϕ2∂kϕ3 . (39)

Expanding to first order in we have

 Sφ = ∫d4x{12(1+4GMr)∂tφsφs−12(∇φs)⋅(∇φs) (40) −12(1+2GMr)m2φsφs} .

To compute to leading order in we use

 SA≈SA(h=0)+∫d4xδSAδgμνhμν . (41)

Now depends on the metric through the . As noted already, there is no unique mapping from the metric to the three-brane coordinates, but this turns out not to be a problem. Choose any mapping and observe that, acting on any functional of the metric,

 δδ(∂μϕA) = ∂gαβ∂(∂μϕA)δδgαβ (42) = 2ηAB∂αϕBδδgαμ ,

which can be inverted to give

 δδgμν=12gμα∂αϕAδδ(∂νϕA) . (43)

Applying this operator to in (39), we find

 δSA = ∫d4x(δSAδgμν)gαβ=ηαβhμν (44) = 32∫d4x 2GMrqAs∂tφs .

Altogether

 S′ = ∫d4x{12(1+4GMr)∂tφs∂tφs−12(∇φs)⋅(∇φs) −12(1+2GMr)m2φsφs+(1+3GMr)qAs∂tφs} .

We go to the Hamiltonian formulation, introducing canonical momenta conjugate to the

 ps=(1+4GMr)∂tφs+(1+3GMr)qAs∂tφs , (46)

 H = ∫d3x{12(1+4GMr)−1(ps−(1+3GMr)qAs)(ps−(1+3GMr)qAs) (47) +12(∇φs)⋅(∇φs)+12(1+2GMr)m2φsφs} = H0+∫d3x{−2GMr(ps−gAs)(ps−gAs)+GMrm2φsφs−3GMrqAs(ps−gAs)} .

Then the Hamiltonian density in the interaction picture, to first order in , is 222Note that in the interaction picture the operator identification must be used for the interaction Hamiltonian density.

 HI=−2GMr{∂tφs∂tφs−12m2φsφs+32M(φ5∂tφ6−φ6∂tφ5)} . (48)

Using interaction picture operators

 φ5(x) = ∫d3k(2π)31√2ωk1√2(b1(k)ei(k⋅x−E1(k)t)+b†1(k)e−i(k⋅x−E1(k)t)+b2(k)ei(k⋅x−E2(k)t)+b†2(k)e−i(k⋅x−E2(k)t)) φ6(x) = ∫d3k(2π)31√2ωk1√2i(b1(k)ei(k⋅x−E1(k)t)−b†1(k)e−i(k⋅x−E1(k)t)−b2(k)ei(k⋅x−E2(k)t)+b†2(k)e−i(k⋅x−E2(k)t)) , (49)

we can compute matrix elements

 ⟨p2,j|∫d4xHI|p1,j⟩ , (50)

and from there it is a standard exercise to calculate the differential cross sections for the heavy/light () Landau excitations in the specified gravitational field. The answer is

 (dσdΩ)gravtype j=(GM)2(E2j(p)−12m2∓32MEj(p))2p4sin4(θ/2) , (51)

where the minus sign is for type 1 and the plus sign for type 2 landons. The type-changing cross sections, in which an initial type 1 landon scatters into a type 2 final state or vice versa, both vanish. We note that for normal scalar fields, i.e. , eq. (51) agrees with the gravitational cross section previously obtained by Golowich et al. Golowich:1990gp ().

Now let us go to the low-momentum limit. For comparison, the differential cross section for a particle of mass in a potential

 V(r)=−λr , (52)

computed via the Born approximation in non-relativistic quantum mechanics is the familiar Rutherford result

 (dσdΩ)Ruth=14λ21m2v4sin4(θ/2) . (53)

For normal scalar particles (), using (51) with the approximations (37) in the low momentum limit, the gravitional cross section can be expressed

 (dσdΩ)gravnormal=14(GMm)21m2v4sin4(θ/2) , (54)

which, comparing to the Rutherford potential, corresponds to scattering from the potential

 V(r)=−GMmr . (55)

In other words, the gravitational mass and the inertial mass are the same. In contrast, for landons of types 1 and 2, eq. (51) becomes in the limit

 (dσdΩ)gravtype 1 = 14(GM2M)21M2v4sin4(θ/2) (dσdΩ)gravtype 2 = 14(GMm22M)21M2v4sin4(θ/2) . (56)

This is a result that we might have guessed. By comparison to the Rutherford cross-section, the gravitational masses of both types 1 and 2 landons are equal to their rest energies, which (for ) are and respectively, while the inertial mass, in accordance with its appearance in group velocity, is approximately in both cases.

The principle of equivalence, of course, asserts the identity of gravitational and inertial mass, which would seem to be badly violated for both heavy and light landons. Indeed, in the present scenario, if it were possible to drop a heavy and a light landon from the top of a tall building and observe how they propagate, the heavy landon would accelerate at , while the light landon would drift downwards (assuming ) only very slowly, with acceleration . These odd effects should be viewed as only an apparent violation of the equivalence principle, arising due to interaction with an external four-form gauge field that singles out a particular time direction. A rough analogy might be the retardation in the gravitational acceleration of a falling conducting ring in the presence of a constant magnetic field directed parallel to gravitational field. If we were unaware of the external field, this might also seem like a violation of the principle of equivalence, rather than a manifestation of Lenz’s Law. In the present situation, the external four-form gauge field makes a contribution to the landon rest energies which cannot be absorbed into the inertial masses, resulting in both an unusual dispersion relation, and a seeming violation of the equivalence principle.

### iv.3 Energy density in the early Universe

If the inflaton field couples only to gravity and the external four-form gauge field, as assumed from the beginning in (LABEL:S0), then observations of the sort just mentioned would be difficult carry out, and it may be more useful to look for signatures of the unconventional dispersion relations in the early universe, due to an unconventional equation of state. Since it requires an energy of at least to pair-create the heavy excitations, and assuming is in Planck units, then after inflation the number density of these objects is fixed. Assuming a dilute ideal gas, the equation of state is conventional:

 ρ=n(2M+m22M)+32P , (57)

where are energy and number density, respectively, and is pressure. The result follows from Boltzmann statistics, plus the fact that, in a non-relativistic regime where (37) applies, momentum degrees of freedom enter quadratically. Hence the equipartition theorem applies, and the result is no different than that of a monatomic ideal gas, with particles of rest energy . Heavy excitations would contribute to deceleration in the matter-dominated era, but their contribution cannot be easily distinguished from that of other types of matter.

The situation is more interesting with respect to light excitations. It is assumed that the rest energy is so small that the number of these excitations is not fixed in the hot environment of the early universe333At least, the number is not fixed if there are any interaction terms in the inflaton potential. If this is not the case and the number is fixed, then the analysis is the same as for an ideal gas with particle rest mass . Taking , result is , which, it will be seen, is the same as the grand canonical result derived below. and the chemical potential can be taken to be zero. In that situation, as with photons, it is necessary to carry out the analysis in a grand canonical ensemble. Following the usual analysis, the logarithm of the grand canonical partition function is

 logZ = −V∫d3k(2π)3ln(1−e−βE2(k)) (58) = βVP ,

with defined in (37). The energy density is

 ρ=∫d3k(2π)3E2(k)eβE2(k)−1 . (59)

We assume that in the early universe , and observe that

 ddkln(1−e−βE2(k)) = βeβE2(k)−1ddk(k2+m22M) (60) = βkM1eβE2(k)−1

Applying this identity we have

 ρ = 4π(2π)3∫∞0dk k2E2(k)Mβkddkln(1−e−βE2(k)) = 4π(2π)3MβkE2(k)ln(1−e−βE2(k))∣∣∞0 −4π(2π)3Mβ∫∞0dk(ddkkE2(k))ln(1−e−βE2(k))

The boundary terms go to zero linearly with as , and exponentially to zero like as . Carrying out the derivative inside the integral we have

 ρ = −4π(2π)332β∫∞0dk k2ln(1−e−βE2(k)) (62) −4π(2π)312β∫