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Cosmological perturbations through the big bang

David Wands∗

Institue of Cosmology and Gravitation, University of Portsmouth,

Mercantile House, Portsmouth, PO1 2EG, United Kingdom

Several scenarios have been proposed in which primordial perturbations could originate fromquantum vacuum fluctuations in a phase corresponding to a collapse phase (in an Einstein frame)preceding the Big Bang. I briefly review three models which could produce scale-invariant spectraduring collapse: (1) curvature perturbations during pressureless collapse, (2) axion field perturba-tions in a pre big bang scenario, and (3) tachyonic fields during multiple-field ekpyrotic collapse.In the separate universes picture one can derive generalised perturbation equations to describe theevolution of large scale perturbations through a semi-classical bounce, assuming a large-scale limit inwhich inhomogeneous perturbations can be described by locally homogeneous patches. For adiabaticperturbations there exists a conserved curvature perturbation on large scales, but isocurvature per-turbations can change the curvature perturbation through the non-adiabatic pressure perturbationon large scales. Different models for the origin of large scale structure lead to different observationalpredictions, including gravitational waves and non-Gaussianity.

to appear in Advanced Science Letters, special issue on Quantum Gravity, Cosmology and Black Holes

I. INTRODUCTION

How did the universe begin? The standard HotBig Bang model, based on four-dimension Friedmann-Robertson-Walker (FRW) cosmology, starts with an ini-tial singularity where our notion of spacetime describedby Einstein’s general theory of relativity breaks down.But we do not expect general relativity, or any classi-cal theory of spacetime, to hold right up to a Big Bangsingularity. Quantum fluctuations about a simple FRWmetric in general relativity, including first-order inhomo-geneities in the geometry in a semi-classical description,become large as the energy density, and thus the cos-mological expansion rate H , becomes comparable to thePlanck scale, 1019 GeV. In alternative models, such asmodels with large extra dimensions, the classical four-dimensional effective theory may break down at muchlower energies.

Cosmology can be studied without worrying aboutwhat came before the Big Bang as long as we have someprescription for the initial conditions, at whatever timewe choose to apply the rules of general relativity, or somemodel of four-dimensional semi-classical gravity. In thehomogeneous and isotropic FRW cosmology it may besufficient to specify an initial thermal temperature andevolve this forward to the present day. But the standardHot Big Bang model does not give a unique prescriptionfor the initial distribution of inhomogeneities - spatialvariations in the matter and geometry across the initialspatial hypersurface. Indeed there is no reason that theyshould necessarily be small perturbations, but observa-tions (notably of the cosmic microwave background) sug-gest they are.

∗Electronic address: david.wands@port.ac.uk

There are two logical possibilities for the origin of pri-mordial perturbations. Either they are produced afterthe Big Bang, or they originate before the Big Bang.

There is a simple model to generate an initial spectrumprimordial perturbations due to vacuum fluctuations dur-ing inflation driven by a slowly-rolling, self-interactingscalar field. The accelerated expansion leads to the vac-uum fluctuations on small scales (much smaller than theHubble length, H−1) being swept up to large (super-Hubble) scales where they become “frozen-in” by the cos-mological expansion. The amplitude of the fluctuationsat the Hubble scale is proportional to H and the slowlyvarying expansion rate leads to an almost scale invariantspectrum. The weak interactions required for a slowly-rolling field naturally lead to an almost Gaussian distri-bution for the primordial perturbations on large scales.After almost 30 years of theoretical development this in-flationary picture of the early universe has become thestandard model for the origin of structure1. There is nosingle agreed model for which fundamental field is respon-sible for driving inflation and/or generating structure,but there are numerous possibilities based on extensionsbeyond the standard model of particle physics2.

But it is also possible that the large scale structureof our Universe is inherited from vacuum fluctuationsduring an earlier non-inflationary phase, before the BigBang3. It is this possibility that I will discuss in thispaper. There are many similarities with the inflationarymodel for the origin of structure in that one can calcu-late a spectrum of perturbations on large, super-Hubblescales in the Hot Big Bang model assuming only vacuumfluctuations on small, sub-Hubble scales in a precedingphase, only one now assumes that the preceding phasewas one of accelerated contraction (in the Einstein framewhere general relativity applies). This requires an inter-mediate bounce from contraction to expansion and one ofthe unresolved problems is whether such a bounce really

2

occurs in this manner and if so whether the perturba-tions spectrum calculated in the collapse phase can berelated to perturbations in the standard Hot Big Bang. Iwill argue that under fairly general conditions the spec-trum on large scales of interest can be expected to bepreserved through a bounce, while acknowledging thatthere is as yet no entirely satisfactory physical model forthe bounce.

The existence of a cosmological phase before the BigBang leads to a radically different view regarding theinitial conditions for the universe4,5, and such modelshave been criticised6,7 for requiring a very large universe(relative to the Planck scale) before the big bang. In-deed, as I will discuss, in some cases the generation of ascale-invariant spectrum of primordial perturbations re-quires the unstable growth of perturbations during col-lapse. However given the uncertainty in what constitutesa “natural” initial state for the cosmos, I will consider thepossible observational consequences of a collapse era.

II. HOMOGENEOUS COLLAPSE

In this paper I will consider a four-dimensional back-ground cosmological model that is spatially homogeneousand isotropic and therefore described by the Friedmann-Robertson-Walker metric

ds2 = −dt2 + a2(t)γijdxidxj . (1)

where γij is the metric on a maximally symmetric 3-spacewith uniform curvature K. The Hubble expansion rate(or collapse rate) is H ≡ a/a.

Local energy conservation gives the continuity equa-tion for matter

ρ = −3H(ρ+ P ) , (2)

where ρ is the energy density and P the isotropic pres-sure. For a linear barotropic equation of state P = wρthis can be integrated to give ρ ∝ a−3(1+w).

In a collapsing universe, a < 0, in the presence of mat-ter with ρ + 3P > 0 (or w > −1/3) the energy densitygrows faster than the spatial curvature K/a2. Note thatin an expanding universe one requires ρ + 3P < 0 (orw < −1/3) for the energy density to grow relative to thespatial density, and this is the usual condition for infla-tion. For simplicity I will assume in the following thatspatial curvature is negligible, so that γij = δij . Moreproblematic in a collapsing universe is anisotropic shear8.In the simplest case of a Bianchi I universe the shear isproportional to a−6 (where in this case we can still thinkof a3 as the volume factor). Thus the anisotropic sheargrows relative to matter in a collapsing universe for anymatter with P < ρ.

Although a fluid description yields simple linearbarotropic equation of state (for matter w = 0, or ra-diation w = 1/3) I will be interested in microphysicaldescription of the matter where one can use a quantum

vacuum state to set the initial conditions for inhomoge-neous perturbations at early times. Thus I will considercanonical scalar fields ϕI with energy density and pres-sure

ρ = V (ϕI) +∑

I

1

2ϕ2I , (3)

P = −V (ϕI) +∑

I

1

2ϕ2I . (4)

where V (ϕI) is the potential energy. In particular ascalar field with an exponential potential, VI(ϕI) ∝exp(−λIκϕI) where κ2 = 8πG, provides a simple modelwith P = wρ where 1 + w = λ2/3. This is the basisof both power-law inflation9 for λ2 < 2 and ekpyroticcollapse10 with λ2 ≫ 2.

Canonical scalar fields also have a kinetic-dominatedcosmology if the potential energy can be neglected suchthat P = ρ =

∑

I ϕ2I/2, corresponding to a stiff equation

of state with w = 1. Indeed in a collapsing universewhere the energy density grows as the universe collapses,the kinetic energy eventually dominates over any finitepotential energy.

We can identify three scalar-field collapse scenariosbased on the form of the potential11,12:

• Non-stiff collapse with P < ρ: stable with respectto spatial curvature for P > −ρ/3 but unstablewith respect to anisotropic shear.

• Pre-Big Bang3 collapse P = ρ: stable with respectto spatial curvature and marginally stable with re-spect to anisotropic shear8.

• Ekpyrotic collapse10 with P ≫ ρ: stable with re-spect to spatial curvature and anisotropic shear.

The last two are models which have been inspiredby ideas from string theory and are intrinsically higher-dimensional models. Nonetheless most of the quantita-tive results have been developed for effective theories de-scribing scalar fields in four-dimensional spacetime. Bothtake as their starting point the notion that string theoryshould be a self-consistent theory without the singulari-ties found in general relativity.

The pre Big Bang model3,13,14 assumes that our ob-servable Universe began in a low energy, weak couplingstate well described by a low energy effective action ofstring theory. Although sometimes written in terms ofan expanding cosmology in the string frame where thedilaton field is non-minimally coupled to the spacetimecurvature, this solution can be conformally transformedto a collapsing cosmology described by general relativ-ity in the so-called Einstein frame15. As the dynamics isdominated by the kinetic energy it is independent of theform of the potential or even the number of fields. As theenergy density becomes large, and the dilaton becomeslarge, the low-energy and weak-coupling approximationsinevitably break down. This offers the possibility that

3

the general relativistic singularity is resolved, but thisgoes beyond the low-energy effective description16,17.

The ekpyrotic model10,18,20 was originally motivatedby cosmological solutions describing the motion of branesin a higher-dimensional spacetime. But again this is usu-ally described by an effective theory of scalar fields ina four-dimensional spacetime. In contrast to the preBig Bang, it incorporates a negative effective poten-tial which is unbounded from below, leading to approx-imately power-law collapse model with w ≫ 1. Thisultra-stiff fast-roll collapse driven by a steep, negativepotential is in many ways dual to quasi-de Sitter, slow-roll inflation driven by a flat, positive potential. Unlikethe pre Big Bang the model approaches the weak cou-pling during the collapse phase. In the original modelthe authors appealed to the higher-dimensional picture toresolve the apparent singularity in the four-dimensionaleffective theory21.

III. LINEAR PERTURBATIONS DURING

COLLAPSE

A. Free field perturbations in an FRW cosmology

Let us first consider the dynamics of free field pertur-bations in a FRW background with scale factor a andHubble rate, H ≡ a/a, where a dot denotes derivativeswith respect to cosmic time t.

Consider an inhomogeneous perturbation, ϕI →ϕI(t) + δϕI(t,x), of the Klein-Gordon equation for ascalar field in an unperturbed FRW universe:

δϕI + 3H ˙δϕI +(

m2I −∇2

)

δϕI = 0 , (5)

where the effective mass-squared of the field is m2I =

∂2V/∂ϕ2I , and ∇2 is the spatial Laplacian. Decomposing

an arbitrary field perturbation into eigenmodes of thespatial Laplacian (Fourier modes in flat space) ∇2δϕI =−(k2/a2)δϕI , where k is the comoving wavenumber,we find that small-scale fluctuations undergo under-damped oscillations on sub-Hubble scales (with comovingwavenumber k > aH), but on large super-Hubble scales,k < aH, the modes are overdamped (or “frozen-in”).

This is most clearly seen in terms of the rescaled fieldand conformal time

vI = aδϕI , adη = dt . (6)

The Klein-Gordon equation (5) becomes

v′′I +

(

k2 + a2m2I −

a′′

a

)

vI = 0 . (7)

We have a simple harmonic oscillator with time-dependent effective mass

µ2 = m2Ia

2 − a′′

a=

(

m2 − 1 − 3w

2H2

)

a2 . (8)

For k2/a2 ≫ |1 − 3w|H2 and k2/a2 ≫ |m2| we canneglect the effective mass and we have essentially freeoscillations. Normalising the initial amplitude of thesesmall-scale fluctuations to the zero-point fluctuations ofa free field in flat spacetime we have22

δϕI ≃e−ikt/a

a√

2k. (9)

During an accelerated expansion or collapse |a| = |aH |increases and modes that start on sub-Hubble scales(k2 > a2H2) are stretched up to super-Hubble scales(k2 < a2H2). For k2 ≪ |1 − 3w|a2H2 we can neglectthe spatial gradients in Eq. (5). Perturbations in lightfields (with mass-squared m2 ≪ |1 − 3w|H2) becomeover-damped (or “frozen-in”) and Eq. (9) evaluated whenk ≃ aH gives the power spectrum for scalar field fluctu-ations at “Hubble-exit”

PδϕI|k=aH ≡ 4πk3

(2π)3

∣

∣δϕ2I

∣

∣

k=aH≃(

H

2π

)2

. (10)

Heavy fields with m2 ≫ |1 − 3w|H2/4 remain under-damped and have essentially no perturbations on super-Hubble scales. But light fields become over-damped andcan be treated as essentially classical perturbations witha Gaussian distribution on super-Hubble scales. Then onlarge scales we have

δϕI ≃ C +D

∫

dt

a3(t). (11)

where C and D are constants of integration. In an ex-panding universe with w < 1 the integral converges andthe field fluctuations become frozen-in on large scales.However in a collapsing universe with w < 1, there is agrowing mode at late times. This is due to the instabilitywith respect to the kinetic energy of the field which (like

anisotropic shear) grows as ˙δϕ2 ∝ a−6 in a collapsing

universe.Vacuum fluctuations in massless fields during quasi-

de Sitter expansion (|H | ≪ H2) produces approximatelyconstant amplitude of scalar field fluctuations at Hubble-exit which then remain approximately constant on super-Hubble scales, thus producing an approximately scale-invariant spectrum. During an accelerated collapse H2

grows rapidly (H = −3(1+w)H2/2) and thus the typicalamplitude of fluctuations at Hubble-exit grows rapidlywith time. During pre-big bang or ekpyrotic collapsewith w ≥ 1 these perturbations are frozen-in and henceminimally coupled, massless fields acquire a steep bluespectrum. On the other hand if w < 1 the instabilitycauses perturbations to grow on super-Hubble scales andin the particular case of a pressureless collapse (w = 0)the super-Hubble growth exactly matches the growth ofperturbations at Hubble-exit leading to a scale-invariantspectrum of perturbations on super-Hubble scales23,24,25.

It is interesting to note that it is the presence of an in-stability that enables the collapse phase with growingH2

4

to produce a scale-invariant spectrum26. It is also possi-ble to produce a scale invariant spectrum if the field hasa tachyonic mass, m2 < 0, which again leads to super-Hubble modes that grow at precisely the same rate as thefluctuations at Hubble-exit27. Another means to producea scale-invariant spectrum is due to a non-minimal cou-pling, as in the case of pseudo-scalar axion fields in thepre big bang scenario28.

Thus far we have neglected the interactions of thefield perturbations, including the gravitational coupling,which is valid only for isocurvature field perturbations,whose energy-momentum is negligible. In the next sec-tion we will include the effect of linear metric perturba-tions.

B. Scalar field and metric perturbations, with

interactions

To track the evolution of more general perturbationswe need to include interactions between fields and, evenin the absence of explicit interactions, we need to includegravitational coupling via metric perturbations.

For an inhomogeneous matter distribution the Einsteinequations imply that we must also consider inhomoge-neous metric perturbations about the spatially flat FRWmetric. The perturbed FRW spacetime is described bythe line element29

ds2 = −(1 + 2A)dt2 + 2a∂iBdxidt

+a2 [(1 − 2ψ)δij + 2∂ijE + hij ]dxidxj , (12)

where ∂i denotes the spatial partial derivative ∂/∂xi. Wewill use lower case latin indices to run over the 3 spatialcoordinates.

The metric perturbations have been split into scalarand tensor parts according to their transformation prop-erties on the spatial hypersurfaces. The field equationsfor the scalar and tensor parts then decouple to linearorder. Vector metric perturbations are related to thedivergence-free part of the momentum, which vanishesidentically at linear order for minimally coupled scalarfields. However vector perturbations have been studied,for example, in the pre big bang model as possible sourceof primordial magnetic fields due to the non-minimal cou-pling of the dilaton field30.

The tensor perturbations, hij , are transverse (∂ihij =0) and trace-free (δijhij = 0). They are automaticallyindependent of coordinate gauge transformations. Thesedescribe gravitational waves as they are the free part ofthe gravitational field and evolve independently of linearmatter perturbations.

We can decompose arbitrary tensor perturbationsinto eigenmodes of the spatial Laplacian, ∇2eij =

−(k2/a2)e(+,×)ij , with two possible polarisation states, +

and ×, comoving wavenumber k, and scalar amplitudeh(t):

hij = h(t)e(+,×)ij (x) . (13)

The Einstein equations yield a wave equation for the am-plitude of the tensor metric perturbations

h+ 3Hh+k2

a2h = 0 , (14)

This is the same as the wave equation (5) for a masslessscalar field in an unperturbed FRW metric. Thus initialvacuum fluctuations on sub-Hubble scales give rise to apower spectrum for tensor metric fluctuations at Hubble-exit31 proportional to that given in Eq. (10)

Ph|k=ah = 64πG

(

H

2π

)2

. (15)

Note that as the Hubble rate approaches the Planck scale,H2 → G−1, the power in metric perturbations becomesof order unity, signalling the expected breakdown of thesemi-classical description.

The four scalar metric perturbations A, ∂iB, ψδij and∂ijE are constructed from 3-scalars, their derivatives,and the background spatial metric. In particular theintrinsic Ricci scalar curvature of constant time hyper-surfaces is given by

(3)R =4

a2∇2ψ . (16)

First-order perturbations of a canonical scalar field in afirst-order perturbed FRW universe obey the wave equa-tion31

δϕI + 3H ˙δϕI +k2

a2δϕI +

∑

J

VIJδϕJ

= −2VIA+ ϕI

[

A+ 3ψ +k2

a2(a2E − aB)

]

. (17)

where the mass-matrix VIJ ≡ ∂2V/∂ϕI∂ϕJ . The Ein-stein equations relate the scalar metric perturbationsto matter perturbations via the energy and momentumconstraints29

− 4πGδρ = 3H(

ψ +HA)

+k2

a2

[

ψ +H(a2E − aB)]

, (18)

−4πGδq = ψ +HA , (19)

where the energy and pressure perturbations and mo-mentum for n scalar fields are given by31

δρ =∑

I

[

ϕI

(

˙δϕI − ϕIA)

+ VIδϕI

]

, (20)

δP =∑

I

[

ϕI

(

˙δϕI − ϕIA)

− VIδϕI

]

, (21)

δq,i = −∑

I

ϕIδϕI,i , (22)

where VI ≡ ∂V/∂ϕI .

5

We can construct a variety of gauge-invariant combina-tions of the scalar metric perturbations. The longitudinalgauge corresponds to a specific gauge-transformation toa (zero-shear) frame such that E = B = 0, leaving thegauge-invariant variables

Φ ≡ A− d

dt

[

a2(E −B/a)]

, (23)

Ψ ≡ ψ + a2H(E −B/a) . (24)

Another variable commonly used to describe scalarperturbations during inflation is the field perturbationin the spatially flat gauge (where ψ = 0). This has thegauge-invariant definition32,33:

δϕIψ ≡ δϕI +ϕ

Hψ . (25)

It is possible to use the Einstein equations to elimi-nate the metric perturbations from the perturbed Klein-Gordon equation (17), and write a wave equation solelyin terms of the field perturbations in the spatially flatgauge34

δϕIψ + 3H ˙δϕIψ +k2

a2δϕIψ

+∑

J

[

VIJ − 8πG

a3

d

dt

(

a3

HϕI ϕJ

)]

δϕJψ = 0 . (26)

This generalises the single free-field Klein-Gordon equa-tion (5) to multiple, interacting fields.

For any light fields (whose masses are small comparedto the Hubble scale) the amplitude of perturbations atHubble-exit is approximately given by Eq. (10), howeverthe evolution on large, super-Hubble scales now dependson the fields interactions.

It is often useful to identify the “adiabatic” field per-turbation which is a perturbation forwards or backwardsalong the background trajectory in field space35 (see vanTent36 for the generalisation to non-canonical fields)

δσ =∑

I

ϕIδϕIσ2

, (27)

where

σ2 ≡∑

I

ϕ2I . (28)

The total energy, pressure and momentum perturbationsfor multiple fields in Eqs. (20–22) can be written as31

δρ = σ(

˙δσ − σA)

+ Vσδσ + Vsδs , (29)

δP = σ(

˙δσ − σA)

− Vσδσ , (30)

δq,i = −σδσ,i , (31)

where Vσ ≡ (∂V/∂ϕI)ϕI/σ. The only effect of isocur-vature field perturbations, orthogonal to the background

trajectory, is through a non-adiabatic pressure perturba-tion

Pnad,s = −2δsV

= 2

(

Vσδσ −∑

I

VIδϕI

)

. (32)

If the potential gradients vanish orthogonal to thebackground trajectory, δsV = 0 (for isocurvature fieldsat a local extremum of their potential), then the adia-batic and isocurvature field perturbations decouple. Theisocurvature perturbations obey the Klein-Gordon equa-tion (5) while the adiabatic field perturbations (on spa-tially flat hypersurfaces) obey the Klein-Gordon equationfor a single field in a perturbed FRW cosmology

δσψ +3H ˙δσψ +

[

k2

a2+ Vσσ − 8πG

a3

d

dt

(

a3

Hσ2

)]

δσψ = 0 .

(33)where the final term on the left-hand-side describes thegravitational back-reaction due to metric perturbations.This is compactly written in terms of conformal time andv ≡ aδσψ and z ≡ aσ/H to give

v′′ +

(

k2 − z′′

z

)

v = 0 . (34)

Analogous to Eq. (7) this leads to oscillating solutions onsmall scales, while on large scales where we neglect thespatial gradients we obtain

δσ ≃ Cσ

H+Dσ

H

∫

H2dt

a3σ2. (35)

In particular we see that the comoving curvature pertur-bation

R ≡ ψ +H

σδσ =

v

z, (36)

has a constant mode on large scales. During slow-rollinflation the second mode decays rapidly and is usuallyneglected, but it may become a growing mode during anaccelerated collapse.

IV. THREE WAYS TO SCALE-INVARIANT

SPECTRA

A. Collapse with P ≪ ρ

If we consider a power-law collapse, a ∝ (−t)p wherep = 2/3(1 +w), driven by a scalar field with exponential

potential then we have H = −4πGσ2 = −(1/p)H2 andhence z′′/z = a′′/a and Eq. (34) for the adiabatic fieldreduces to Eq. (7) for the isocurvature fields with

z′′

z=a′′

a=ν2 − (1/4)

η2. (37)

6

where

ν =3

2+

1

p− 1. (38)

The general solution is given in terms of Hankel func-tions of order |ν|

v =√

|kη|[

V+H(1)|ν| (|kη|) + V−H

(2)|ν| (|kη|)

]

. (39)

Normalising to the quantum vacuum on sub-Hubblescales at η → −∞ gives a spectrum of field perturba-tions on super-Hubble scales25 as η → 0

Pδσ =

(

2|ν|Γ(|ν|)(ν − 1/2)23/2Γ(3/2)

)2(H

2π

)2

|kη|3−2|ν|.

(40)Thus a power-law collapse gives rise to a power-law spec-trum for field fluctuations on super-Hubble scales withspectral tilt

∆nδσ ≡ d lnPδσd ln k

= 3 − 2|ν| . (41)

I have written these expressions in a way that makesclear that the spectral tilt is invariant under a change ofsign of ν → −ν, or equivalently25

p→ 1 − 2p

2 − 3p. (42)

In particular we see that a scale invariant spectrum offluctuations in the adiabatic field (∆nδσ = 0) may beproduced either from slow-roll inflation (w = −1 andν = 3/2) or a pressureless collapse23,25 (w = 0 andν = −3/2). This is because the general solution con-tains two modes and the transformation (42) swaps thegrowing and decaying modes at late times. In slow-rollinflation it is the constant mode outside the Hubble-scale which acquires a scale-invariant spectrum whereasin w = 0 collapse it is the time-dependent mode whichgrows rapidly Pδσ ∝ H2 outside the Hubble-scale. Thisis evidence of an instability of the background solutiondescribing pressureless collapse with ρ ∝ a−3 which isunstable to the growth of scalar field kinetic energy with˙δσ

2 ∝ a−6. This raises questions about how fine-tunedthe initial conditions would need to be to have a long-lasting, pressureless collapse phase. But if there is sucha phase, even if it is short-lived, then it can generatea scale-invariant spectrum of perturbations over somerange of scales.

Note that the spectrum of adiabatic field fluctuations,massless isocurvature field fluctuations, and gravitationalwaves24 all share the same scale-dependence in a power-law collapse. There is a simple relation between thepower of tensor to scalar metric perturbations duringpower-law collapse

r ≡ PhPR

=64πGσ2

H2=

16

p. (43)

This is small for slow-roll inflation, but is dangerouslylarge (r = 8) during pressureless collapse. Current ob-servational bounds37 require r < 0.3 at the time of last-scattering of the CMB. However whereas the tensor andscalar amplitudes are constant on large scales during con-ventional slow-roll inflation, both quantities are rapidlygrowing during pressureless collapse. Thus the final valuefor the tensor-to-scalar ratio will be model-dependent.In a simple bounce model38 it was found that the tensor-scalar ratio was small in only a small corner of parameterspace.

B. Pre big bang with P = ρ

The pre big bang scenario is based upon bosonic fieldsin the low-energy string effective action including thedilaton and other moduli fields3,13,14. Any finite po-tential becomes negligible as the energy density growsin a collapsing universe and hence the universe becomesdominated by the kinetic energy of the fields, leading topower-law collapse with w = 1 and p = 1/3. In this caseadiabatic and canonical isocurvature field perturbationshave a general solution given by Eq. (39) with Hankelfunctions of order ν = 0. This gives a strong blue spec-tral tilt ∆n = +3 in Eq. (41) for both the scalar and ten-sor metric perturbations during the pre big bang phase39,leaving essentially no perturbations on large scales.

Originally it was hoped that if the pre big bang couldprovide a homogeneous universe on large scales thencausal mechanisms such as cosmic strings or other topo-logical defects could source primordial perturbations.However subsequent observations37 have shown that anapproximately Gaussian distribution of adiabatic densityperturbations is required on super-Hubble scales by thetime of last scattering.

It turns out that some isocurvature fields will have verydifferent perturbation spectra if they are non-minimallycoupled to fields such as the dilaton which are rapidlyevolving during the pre big bang. In particular thepseudo-scalar axion in the four-dimensional effective ac-tion is coupled to the dilaton in an SL(2,R) invariantLagrangian14

L = −1

2(∇σ)2 − 1

2e2σ(∇χ)2 . (44)

The Klein-Gordon equation for isocurvature fluctuationsin the axion field is

δχ+ (3H + 2σ) δχ+k2

a2δχ = 0 . (45)

Analogous to Eq. (7) this can be written as

u′′ +

(

k2 − a′′

a

)

u = 0 , (46)

where u = aδχ and a ≡ eσa is the “scale factor” inthe conformal frame in which the axion (rather than the

7

dilaton) is minimally coupled. As a result the spectralindex for axion field perturbations turns out to be givenby28

∆nδχ = 3 − 2| cos ξ| , (47)

where ξ is an angle describing the rate at which the dila-ton rolls relative to other moduli fields. The invarianceof the spectra under cos ξ → − cos ξ corresponds to thepreviously noted invariance under ν → −ν which here co-incides with invariance under duality transformations ofthe string effective action14. Perturbations of the coupleddilaton-axion system can be shown to be invariant underSL(2,R) transformations of the background solutions28.

More generally there are many axion-type fields inthe four-dimensional effective theories with different cou-plings to the dilaton and/or other moduli fields. Forspecific parameters these may acquire scale-invariant, oralmost scale-invariant spectra14.

These isocurvature perturbations during the pre bigbang phase still need to be converted into adiabatic den-sity perturbations in the primordial era. This will happenif the axion field leads to a non-adiabatic pressure per-turbation, δPnad, and hence a perturbation in the localequation of state which changes the large-scale curvatureperturbation

R ≃ HδPnad

ρ+ P. (48)

In recent years a number of such mechanisms have beeninvestigated in the context of inflationary cosmology withmultiple fields.

In the curvaton scenario40,41, the axion survives fromthe pre big bang into the hot big bang phase, as a mas-sive, weakly coupled field. Although it’s initial energydensity is negligible, once it becomes non-relativistic itsenergy density grows relative to the radiation and caneventually come to contribute a significant fraction ofthe total energy density. The curvaton must decay be-fore primordial nucleosynthesis, but when it does so, anyperturbation in its energy density is transferred to theradiation density. Curvaton models have distinctive ob-servational signatures including the possibility of residualisocurvature modes42 or non-Gaussianity in the primor-dial density perturbation42,43.

Note that the pre big bang is only marginally stablewith respect to anisotropic shear8 in a collapsing uni-verse, and anisotropies grow during any collapse withP < ρ.

C. Ekpyrotic collapse with P ≫ ρ

The ekpyrotic scenario involves a collapse phase drivenby a steep and negative exponential potential in the four-dimensional effective action. This leads to an ultra-stiffequation of state w ≫ 1 and a rapidly increasing Hubblerate while the scale factor only slowly decreases. This

is in many ways the collapse equivalent of slow-roll in-flation where the scale factor rapidly increases while theHubble rate slowly decreases. The ekpyrotic collapse isthe stable attractor during collapse with respect to spa-tial curvature and shear, just as slow-roll inflation is thestable attractor during expansion.

However for w ≫ 1 and thus power-law collapse withp ≪ 1 the spectral index given in Eqs. (38) and (41)for scalar and tensor metric perturbations produced dur-ing collapse is steep and blue44 ∆n = +2. This isin contrast to the original ekpyrotic papers which cal-culated the spectrum of scalar metric perturbations inthe longitudinal gauge, where one finds a scale-invariantspectrum45. We will show in the next section that if thecollapse phase is connected to the hot big bang expansionby a non-singular bounce then we expect the comovingcurvature perturbation, and not the curvature perturba-tion in the longitudinal gauge to be conserved on largescales.

As in the pre big bang model, one requires instead aspectrum of almost scale-invariant perturbations in anisocurvature field to lead to an almost scale-invariantspectrum of primordial density perturbations. As in thethe pre big bang model this could be a pseudo-scalar ax-ion non-minimally coupled to the adiabatic field whichevolves during the ekpyrotic phase46. However in theekpyrotic phase the masses of the fields are not negli-gible compared with the Hubble rate and one can alsoconsider scale invariance due to a tachyonic mass of anisocurvature field.

A simple example is the case of two canonicalscalar fields, both with steep negative exponentialpotentials27,47,48,49

V (ϕ1, ϕ2) = −V1 exp(−λ1κϕ1)−V2 exp(−λ2κϕ2) . (49)

In slow-roll inflation it is known that a potential thatis a separable sum of exponentials leads to “assisted”inflation50 which is a power-law expansion with powerp =

∑

I 2/λ2I which is larger than the power pI = 2/λ2

I

that would be obtained for any of the fields on their own,“assisting” slow-roll. The same happens in ekpyrotic col-lapse with the potential (49), although the fact that p islarger than pI for a single field takes it further from theekpyrotic limit p→ 0.

The dynamics with multiple exponential potentialsis most easily understood via a fixed rotation in fieldspace51,52

φ =λ2ϕ1 + λ1ϕ2√

λ21 + λ2

2

, χ =λ1ϕ1 − λ1ϕ2√

λ21 + λ2

2

. (50)

The potential (49) is then given by

V (φ, χ) = U(χ) exp(−λκφ) , (51)

where

U(χ) = −U0

[

1 +λ2

2κ2(χ− χ0)

2 + . . .

]

. (52)

8

The “assisted” power-law solution corresponds to asolution where φ evolves along the extremum, χ =χ0 =constant. One can verify that perturbations in φdescribes adiabatic field perturbations (27) along this tra-jectory. Thus perturbations in χ are isocurvature pertur-bations described by Eq. (5) with a tachyonic mass

m2χ =

2κ2V

p= −9

2(w2 − 1)H2 . (53)

The time dependent effective mass term in Eq. (7) is then

µ2 = m2χa

2 − a′′

a=

(1/4)− ν2

η2, (54)

and the general solution is of the form given in Eq. (39)where the order

ν2 =9

4− 18(w + 1)

(3w + 1)2. (55)

Thus for w ≫ 1 we have ν ≃ (3/2)−(2/3w) and the spec-tral index for isocurvature field perturbations on super-Hubble scales which originate from vacuum fluctuationsis given by Eq. (41) with ∆nδχ → 0 as w → ∞. Notethat the effective mass of the isocurvature field dependsonly on the combined λ2 and not on the individual λI sowe do not require any cancellations between different pa-rameters to obtain a scale-invariant spectrum; this comesout automatically for a sufficiently steep potentials withcombined λ ≫ 1. On the other hand we have obtainedan isocurvature spectrum from vacuum fluctuations onlyat the expense of perturbing about an unstable solutionwith a tachyonic direction whose mass grows proportionalto the Hubble rate.

Different mechanisms have been studied which couldtransfer these isocurvature field perturbations to producean almost scale invariant spectrum of scalar metric per-turbations. This could occur at the bounce if the non-adiabatic field perturbations give rise to a significant non-adiabatic pressure perturbation during the bounce, or itcould occur during the collapse phase as the field rollsaway from the extremum χ = χ0 either due to terms inthe potential47,49 with ∂V/∂χ 6= 0, or simply due to thetachyonic instability itself53 which naturally causes anysmall initial deviation from χ = χ0 to grow. The stablelate time attractor for the potential (49) is an ekpyroticcollapse driven by a single field, φ1 or φ2, as in the orig-inal ekpyrotic model. Note however that isocurvaturefluctuations in the other field are not then close to scale-invariant.

It is worth noting that the simple model in Eq. (49)cannot produce a red tilt, ∆n < 0 favoured by currentobervations37. Thus one must introduce additional time-dependent terms either in U(χ) or breaking the exact ex-ponential potential for φ in Eq. (51). One might also hopeto include positive mass terms for χ to stabilise the χ0 atearly times54, e.g., a constant mass term which becomesdominates at early times but becomes negligible relative

to the growing tachyonic mass term at late times. Andeventually one must consider additional effects which willproduce a transition from collapse to expansion.

V. PERTURBATIONS THROUGH A BOUNCE

The calculations presented thus far are based on thedynamics of scalar fields in general relativity. This is a fa-miliar framework for theoretical cosmology and has beenthoroughly explored in the context of inflationary mod-els of the early universe and quintessence models of thelate universe. Thus the results are generally uncontro-versial, although differences in approach, notably choiceof gauge and conformal rescalings of the metric and/ornon-minimal coupling of fields to the spacetime curvaturelead to differences in the presentation of results.

However any collapse model must be connected to a ex-panding phase if it is to provide an explanation of initialconditions for the hot big bang cosmology, and in partic-ular the observed spectrum of almost Gaussian, almostscale-invariant, and almost adiabatic density perturba-tions before the last scattering of the cosmic microwavebackground37. This is not easy. In the context of spa-tially flat FRW models any bounce (H = 0, H > 0) re-quires violation of the null energy condition, ρ+ P < 0.This is not possible for any number of scalar fields witha canonical kinetic field for which ρ + P = σ2 ≥ 0 re-gardless of the potential energy. Instead one requiresghost fields with negative kinetic energy which generallyleads to instabilities55. The effective energy-momentumtensor of non-minimally coupled fields may violate thenull energy condition56 but we have calculated our met-ric perturbations during collapse in the Einstein frame,and wish to set initial conditions in the primordial ex-panding phase where general relativity is again assumedto be valid. Therefore we will require effective violationof the null energy condition in an Einstein frame. Prob-lems controlling the instabilities - usually by requiringsome UV-completion of a low energy effective field the-ory containing ghosts19,47,48 - leave models invoking abounce on much less secure foundation that other reali-sations of scalar fields in cosmology.

One might fear that no useful predictions can bemade in the absence of a detailed physical model for thebounce. However we can make some statements aboutthe primordial perturbations inherited from a precedingcollapse phase if we make apparently reasonable assump-tions about the bounce. A key assumption is that causal-ity which limits the physical scale over which a suddenbounce can alter the scale dependence of perturbations.

We will consider first the case of isocurvature field per-turbations which depend solely on the background evo-lution before considering the behaviour of scalar metricperturbations.

9

A. Isocurvature field perturbations

Let us consider the simplest case of a non-interactingscalar field perturbation obeying the Klein-Gordon equa-tion (5) in an FRW cosmology.

The effective mass, µ2 in Eq. (8), contains terms fromboth the physical mass, m2 and the expansion, a′′/a. Ifthe mass is bounded from below then all Fourier modeswith wavenumber k ≪ |am|min will follow the same evo-lution δϕI/δϕI(t0) = f(t) independent of wavenumberk. In this case the scale dependence of the spectrum ofperturbations (41) will not be changed. But if the spa-tial gradients grow larger than the physical mass, thenwe need to compare the gradients with the expansion.During collapse we related a′′/a to the comoving Hub-ble scale, a′′/a = −(3w − 1)a2H2. Clearly the Hubblelength, H−1, diverges at a bounce, but a′′ is finite andnon-zero at a simple bounce. In fact a′′/a will go throughzero at some point before a smooth bounce (where a(η) isanalytic) as it is negative during the collapse phase andpositive at the bounce, and then negative again during aradiation dominated expansion after the bounce.

Nonetheless if the bounce has a finite duration thenthere is always a finite scale over which the perturbationsevolution is significantly affected by the spatial gradients,and thus a long-wavelength regime in which the scaledependence is conserved.

In the long-wavelength limit we can model the bounceby a junction condition obtained by integrating theKlein-Gordon equation while imposing continuity of thefield and scale factor

[

v′IvI

]+

−

=

[

a′

a

]+

−

− a2M2I , (56)

where we can allow for a divergent mass-term throughthe bounce

M2I = lim

ǫ→0

∫ +ǫ

−ǫ

m2Idη . (57)

which could lead to a scale-independent change in theperturbations.

B. Adiabatic perturbations

We can derive a generalised equation for adiabaticperturbations by requiring that there exists a long-wavelength limit in which the evolution of the perturbeduniverse is the same as that of the FRW background57.The notion that long-wavelength perturbations can bemodelled as piecewise homogeneous universes is knownas the “separate universes” picture58,59,60. This is alsosometimes called the ultra-local approximation61. Oncethe background solution is specified, the evolution ofadiabatic perturbations on large scales is also implicitlyspecified since adiabatic perturbations are simply local

perturbations forwards or backwards along this back-ground solution. Consistency then requires that even ifa bounce solution invokes new physics, the same newphysics applies locally to long-wavelength perturbationsas applies to homogeneous FRW cosmology.

We consider a gravitational theory where homogeneousand isotropic spacetimes obey a Friedmann-type con-straint equation, determining the expansion rate of co-moving worldlines, θ, and an equation for its evolutionwith respect to the proper time, τ , along these worldlines,

θ2 = 3f , (58)

d

dτθ = −3

2g . (59)

For example, in loop quantum cosmology a modified ef-fective Friedmann equation (58) can be derived62 wheref = f(ρ) that leads to a cosmological bounce, and Car-dassian models63 where f(ρ) ∝ ρ+Cρn have been investi-gated. In both these examples local energy conservationalong comoving worldlines then fixes the form of g(ρ, p).

In general relativity we have f = 8πGρ and g =8πG(ρ+P ), where G is Newton’s constant. More gener-ally, one can always define an effective energy-momentumtensor such that the Einstein tensor Gµν = 8πGT eff

µν .From Eqs. (58) and (59) we can identify an effective den-sity and pressure:

ρeff ≡ f

8πG, peff ≡ g − f

8πG. (60)

Conservation of the Einstein tensor, ∇µGµν = 0, thenrequires conservation of the effective energy-momentumtensor, which implies

d

dτρeff = −θ(ρeff + peff) , (61)

or equivalently, from Eqs. (58) and (59),

d

dτf = −θg . (62)

However, in the following we will allow f and g to bearbitrary functions of energy, pressure or other variables.

In the linearly perturbed FRW cosmology (12) thereis a unit time-like vector field orthogonal to constant-ηspatial hypersurfaces64,

Nµ =1

a(1 −A,−Bi, ) , (63)

whose expansion rate is given by

θ = 3a′

a2(1 −A) − 3

aψ′ +

1

a∇2σ , (64)

where a prime denotes a derivative with respect to theconformal time η, and the anisotropic shear is

∇2σ = ∇2(E′ −B) . (65)

10

At zeroth-order the shear vanishes and the backgroundexpansion rate is θ0 = 3H/a, where H ≡ a′/a is theconformal Hubble parameter.

For the zeroth-order homogeneous (FRW) backgroundthe equations (58) and (59) can be written as

H2 =a2

3f0 , (66)

H2 −H′ =a2

2g0 . (67)

We then can apply Eqs. (58) and (59) where we takef = f0(η) + δf(η,x) and g = g0(η) + δg(η,x) and thelocal expansion rate is given, to first-order, by Eq. (64).Neglecting all spatial gradients, we can then write thefirst-order equations in terms of the lapse function A, itsderivative, the curvature perturbation ψ and its first andsecond derivatives,

− 3H(ψ′ + HA) =a2

2δf , (68)

ψ′′ −Hψ′ + HA′ + 2(H′ −H2)A =a2

2δg . (69)

Note that these equations are independent of two of thescalar metric perturbations, B and E in Eq. (12), whichdetermine the anisotropic shear (65), which vanishes inthis long-wavelength limit.

For adiabatic perturbations on large scales differentpatches of the inhomogeneous universe follow the sametrajectory in phase space, and the adiabatic perturba-tions correspond to a perturbation back or forward withrespect to this background trajectory60. In this case thehypersurfaces of uniform-θ and uniform-(dθ/dτ) coincide.To first-order this requires δg/g′0 = δf/f ′

0.More generally, we can write any perturbation δg as a

sum of its adiabatic and non-adiabatic parts,

δg =g′0f ′0

δf + δgnad , (70)

where δgnad is automatically gauge-invariant. Indeed,if we identify f with an effective density and g − fwith an effective pressure, then δgnad = 8πG[δpeff −(peff

0′/ρeff

0′)δρeff ] = 8πGδpeff

nad. If we assume f = f(ρ)in Eq. (58) and impose energy conservation, so thatdρ/dτ + θ(ρ + p) = 0 along comoving worldlines, thenwe would require from Eq. (59) that g = (df/dρ)(ρ + p)and then δgnad = (df/dρ)δpnad.

Using Eqs. (70), (66) and (67), we have from Eqs. (68)and (69) that

ψ′′ +3HH′ −H′′ −H3

H′ −H2ψ′

+HH′ −H3

H′ −H2A′ +

2H′2 −HH′′

H′ −H2A =

a2

2δgnad . (71)

Equation (71) includes the two gauge-dependent met-ric perturbations ψ and A. If we work in the longitudinal

gauge then we have Ψ = ψ = A in the absence of anyeffective anisotropic pressure29. (More generally one canuse the gauge freedom to work in a pseudo-longitudinalgauge57 which is constructed such that ψ = A.) We thenhave

Ψ′′ +4HH′ −H′′ − 2H3

H′ −H2Ψ′

+2H′2 −HH′′

H′ −H2Ψ =

a2

2δgnad . (72)

For adiabatic perturbations the right-hand-side vanishesand we have a homogeneous second-order evolution equa-tion for Ψ.

We can solve this equation by quadratures to find thegeneral solution58,65

Ψ = DHa2

+ C

[

−1 +Ha2

∫

a2 dη

]

, (73)

where C and D are constants of integration. Althoughthe differential equation (72) has a singular point whenH′ −H2 = 0, the solution (73) is clearly regular througha bounce.

If we use equations (60) to identify an effective densityand pressure on large scales, then one can show that ourgeneralised perturbation equation (72) can be written ina “general relativistic” form

Ψ′′ + 3(1 + c2effs )HΨ′

+[2H′ + (1 + 3c2effs )H2]Ψ = 4πGa2δpeffnad , (74)

where the effective adiabatic sound speed is

c2effs =peff′0

ρeff′0

=HH′ + H3 −H′′

3H(H′ −H2). (75)

Our results are consistent with previous work60,66

which pointed out that the curvature perturbation onuniform-density hypersurfaces60,67,

ζ ≡ −ψ − Hρ′δρ , (76)

is conserved for adiabatic perturbations on largescales assuming only local conservation of energy (seealso26,68,69). We can define a generalization

ζf ≡ −ψ −Hδf

f ′0

(77)

which is the gauge-invariant definition of the curvatureperturbation, −ψ, on uniform-expansion hypersurfaces,where δf = 0. Using Eqs. (66) and (68) we can write

ζf =1

H′ −H2

[

Hψ′ + (H2 −H′)ψ + H2A]

. (78)

In general relativity the uniform-density, uniform-expansion and comoving orthogonal hypersurfaces coin-cide in the long-wavelength limit and hence ζf = ζ. Using

11

Eqs. (68) and (69) for the evolution of perturbations onlarge scales we obtain

ζ′f =H

H′ −H2

a2

2δgnad . (79)

We see that ζf is constant in the large-scale limit formodified gravitational field equations, even allowing fornon-conservation of energy, if the perturbations are adi-abatic, i.e., δgnad = 0 in Eq. (70).

The growing mode solution (in an expanding universe)for the longitudinal gauge perturbation, Ψ+ ∝ C inEq. (73), corresponds to ζf = C, where C is a constantof integration. The decaying mode Ψ− = DH/a2, whichdominates during ekpyrotic collapse, does not contributeto the curvature perturbation ζf . In a simple cosmo-logical bounce model, assuming a specific ansatz for thebackground evolution, one can show57 that the growingmode of the curvature perturbation after the bounce doesnot receive a contribution from the growing mode in thecollapse phase.

VI. CONCLUSIONS

It is an intriguing possibility that the large-scale struc-ture of our Universe today could originate from vacuumfluctuations in a preceding collapse phase. Cosmologi-cal models including a bounce from collapse to expan-sion are certainly speculative as the end-point of gravita-tional collapse remains one of the outstanding challengesfor quantum theories of gravity. We might hope thatgravitational collapse should be non-singular, but thereis no guarantee that our notions of a semi-classical space-time will be preserved. Loop quantum cosmology offersone framework in which using the symmetries of FRWcosmology provides non-singular solutions for the homo-geneous background70, but the dynamical evolution of aninhomogeneous universe is not known.

While the process of the bounce remains uncertain,I have argued that assuming our semi-classical frame-work still holds through the bounce, then the dynam-ics of a sudden bounce should affect field perturbationsabove some critical scale in a scale-independent way. Andthough the gravitational field equations controlling theevolution of metric perturbations through the bouncemay be unknown, we can infer the general form of equa-tions governing the behaviour of metric perturbations ina long-wavelength limit in which the inhomogeneous uni-verse can be described locally in terms of homogeneouspatches. This shows that the comoving curvature pertur-bation is constant on large scales for adiabatic perturba-tions. Non-adiabatic pressure perturbations can changethe comoving curvature on large scales by changing thelocal equation of state60 and this could imprint the scale-dependence of isocurvature field perturbations generatedduring collapse onto the comoving curvature in the hotbig bang phase.

The decrease of the comoving Hubble length |H |−1/aduring accelerated collapse, like inflation in an expand-ing universe, leads to sub-Hubble vacuum fluctuationsproducing a spectrum on perturbations on super-Hubblescales. However there is no unique limit in which one canobtain a scale invariant power spectrum. I have high-lighted three possibilities: (1) comoving curvature per-turbations acquire a scale-invariant spectrum in a pres-sureless collapse, (2) isocurvature perturbations in axionfields in the pre big bang scenario, or (3) isocurvatureperturbations in a two-field model of ekpyrotic collapse.In all cases we require a significant non-adiabatic pressureperturbation to lead to a change in the comoving curva-ture perturbations on super-Hubble scales either duringthe collapse phase or subsequently.

In (1) and (3) we require an instability of the back-ground solution so that field perturbations can growrapidly on super-Hubble scales to keep pace with thegrowing Hubble rate during collapse. However in (2) thismay be avoided because the axion is non-minimally cou-pled to the dilaton and so the amplitude of its vacuumfluctuations is controlled by the Hubble rate in a con-formally related axion frame28. The axion can acquire ascale-invariant spectrum if the axion frame is undergo-ing inflation, while in the Einstein frame the universe iscollapsing.

The rapid growth of the field perturbations is likelyto also lead to the growth of second- and higher-orderperturbations which could lead to a non-Gaussian dis-tribution of primordial density perturbations. Thishas recently been calculated in multi-field ekpyroticscenarios71,72,73, case(3), and the current observationallimits provide significant constraints on the allowed pa-rameter values. Future observational limits should beable to effectively rule out such models as small non-Gaussianity seems to be incompatible with these fast-rollpotentials.

In the pre big bang or ekpyrotic collapse the gravi-tational waves, like the comoving curvature perturba-tion, acquire a steep blue spectrum during collapse.Thus there are essentially no gravitational waves on largescales. A detection of gravitational waves on the Hubblescale at last scattering of the CMB would rule out thesemodels. On the other hand during a pressureless collapsephase the gravitational waves acquire a scale-invariantpower spectrum, like the comoving curvature perturba-tion, and unlike during slow-roll inflation, their relativeamplitude is not suppressed by slow-roll parameters. Thelarge relative amplitude of gravitational waves rules outthis model unless the bounce phase strongly boosts therelative scalar perturbation, which may be possible insome cases38.

Ultimately we should should be able to use observa-tional evidence to confirm or rule out these very differentmodels for the origin of large scale structure from beforethe Big Bang.

12

Acknowledgements

I am grateful to numerous collaborators for many dis-cussions and collaborations upon which much of the workpresented here is based, including Laura Allen, Bruce

Bassett, Antonio Cardoso, Ed Copeland, Richard East-her, Kazuya Koyama, Andrew Liddle, Jim Lidsey, DavidLyth, Karim Malik, Shuntaro Mizuno, Shinji Tsujikawa,Carlo Ungarelli and Filippo Vernizzi. DW is supportedby STFC.

[1] A. R. Liddle and D. H. Lyth, Cosmological inflation andlarge-scale structure, Cambridge University Press, Cam-bridge (2000)

[2] D. H. Lyth and A. Riotto, Phys. Rept. 314, 1 (1999)[arXiv:hep-ph/9807278].

[3] M. Gasperini and G. Veneziano, Astropart. Phys. 1, 317(1993) [arXiv:hep-th/9211021].

[4] A. Buonanno, T. Damour and G. Veneziano, Nucl. Phys.B 543, 275 (1999) [arXiv:hep-th/9806230].

[5] P. J. Steinhardt and N. Turok, Phys. Rev. D 65, 126003(2002) [arXiv:hep-th/0111098].

[6] A. Linde, in The Future of Theoretical Physics and Cos-mology, Edited G. W. Gibbons, E. P. S. Shellard andS. J. Rankin, Cambridge University Press, Cambridge(2003) [arXiv:hep-th/0205259].

[7] M. S. Turner and E. J. Weinberg, Phys. Rev. D 56, 4604(1997) [arXiv:hep-th/9705035].

[8] K. E. Kunze and R. Durrer, Class. Quant. Grav. 17, 2597(2000) [arXiv:gr-qc/9912081].

[9] F. Lucchin and S. Matarrese, Phys. Rev. D 32, 1316(1985).

[10] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok,Phys. Rev. D 64, 123522 (2001) [arXiv:hep-th/0103239].

[11] J. K. Erickson, D. H. Wesley, P. J. Steinhardtand N. Turok, Phys. Rev. D 69, 063514 (2004)[arXiv:hep-th/0312009].

[12] I. P. C. Heard and D. Wands, Class. Quant. Grav. 19,5435 (2002) [arXiv:gr-qc/0206085].

[13] M. Gasperini and G. Veneziano, Phys. Rept. 373, 1(2003) [arXiv:hep-th/0207130].

[14] J. E. Lidsey, D. Wands and E. J. Copeland, Phys. Rept.337, 343 (2000) [arXiv:hep-th/9909061].

[15] M. Gasperini and G. Veneziano, Mod. Phys. Lett. A 8,3701 (1993) [arXiv:hep-th/9309023].

[16] R. Brustein and R. Madden, Phys. Rev. D 57, 712 (1998)[arXiv:hep-th/9708046].

[17] C. Cartier, E. J. Copeland and R. Madden, JHEP 0001,035 (2000) [arXiv:hep-th/9910169].

[18] R. Kallosh, L. Kofman and A. D. Linde, Phys. Rev. D64, 123523 (2001) [arXiv:hep-th/0104073].

[19] R. Kallosh, J. U. Kang, A. Linde and V. Mukhanov,JCAP 0804, 018 (2008) [arXiv:0712.2040 [hep-th]].

[20] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok,“A brief comment on ’The pyrotechnic universe’,”arXiv:hep-th/0105212.

[21] J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Stein-hardt and N. Turok, Phys. Rev. D 65, 086007 (2002)[arXiv:hep-th/0108187].

[22] J. E. Lidsey, A. R. Liddle, E. W. Kolb, E. J. Copeland,T. Barreiro and M. Abney, Rev. Mod. Phys. 69, 373(1997) [arXiv:astro-ph/9508078].

[23] F. Finelli and R. Brandenberger, Phys. Rev. D 65,103522 (2002) [arXiv:hep-th/0112249].

[24] A. A. Starobinsky, JETP Lett. 30, 682 (1979) [Pisma Zh.

Eksp. Teor. Fiz. 30, 719 (1979)].[25] D. Wands, Phys. Rev. D 60, 023507 (1999)

[arXiv:gr-qc/9809062].[26] P. Creminelli, A. Nicolis and M. Zaldarriaga, Phys. Rev.

D 71, 063505 (2005) [arXiv:hep-th/0411270].[27] F. Finelli, Phys. Lett. B 545, 1 (2002)

[arXiv:hep-th/0206112].[28] E. J. Copeland, R. Easther and D. Wands, Phys. Rev. D

56, 874 (1997) [arXiv:hep-th/9701082].[29] V. F. Mukhanov, H. A. Feldman and R. H. Branden-

berger, Phys. Rept. 215, 203 (1992).[30] D. Lemoine and M. Lemoine, Phys. Rev. D 52, 1955

(1995).[31] B. A. Bassett, S. Tsujikawa and D. Wands, Rev. Mod.

Phys. 78, 537 (2006) [arXiv:astro-ph/0507632].[32] V. F. Mukhanov, JETP Lett. 41, 493 (1985) [Pisma Zh.

Eksp. Teor. Fiz. 41, 402 (1985)].[33] M. Sasaki, Prog. Theor. Phys. 76, 1036 (1986).[34] M. Sasaki and E. D. Stewart, Prog. Theor. Phys. 95, 71

(1996) [arXiv:astro-ph/9507001].[35] C. Gordon, D. Wands, B. A. Bassett and

R. Maartens, Phys. Rev. D 63, 023506 (2001)[arXiv:astro-ph/0009131].

[36] B. van Tent, Class. Quant. Grav. 21, 349 (2004)[arXiv:astro-ph/0307048].

[37] J. Dunkley, E. Komatsu, M. R. Nolta, D. N. Spergel,D. Larson, G. Hinshaw, L. Page, C. L. Bennett,B. Gold, N. Jarosik, J. L. Weiland, M. Halpern,R. S. Hill, A. Kogut, M. Limon, S. S. Meyer,G. S. Tucker, E. Wollack, E. L. Wright, “Five-YearWilkinson Microwave Anisotropy Probe (WMAP) Ob-servations: Likelihoods and Parameters from the WMAPdata”, [arXiv:0803.0586 [astro-ph]].

[38] L. E. Allen and D. Wands, Phys. Rev. D 70, 063515(2004) [arXiv:astro-ph/0404441].

[39] R. Brustein, M. Gasperini, M. Giovannini,V. F. Mukhanov and G. Veneziano, Phys. Rev. D51, 6744 (1995) [arXiv:hep-th/9501066].

[40] K. Enqvist and M. S. Sloth, Nucl. Phys. B 626, 395(2002) [arXiv:hep-ph/0109214].

[41] D. H. Lyth and D. Wands, Phys. Lett. B 524, 5 (2002)[arXiv:hep-ph/0110002].

[42] D. H. Lyth, C. Ungarelli and D. Wands, Phys. Rev. D67, 023503 (2003) [arXiv:astro-ph/0208055].

[43] M. Sasaki, J. Valiviita and D. Wands, Phys. Rev. D 74,103003 (2006) [arXiv:astro-ph/0607627].

[44] D. H. Lyth, Phys. Lett. B 524, 1 (2002)[arXiv:hep-ph/0106153].

[45] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok,Phys. Rev. D 66, 046005 (2002) [arXiv:hep-th/0109050].

[46] F. Di Marco, F. Finelli and R. Brandenberger, Phys. Rev.D 67, 063512 (2003) [arXiv:astro-ph/0211276].

[47] E. I. Buchbinder, J. Khoury and B. A. Ovrut, Phys. Rev.D 76, 123503 (2007) [arXiv:hep-th/0702154].

13

[48] P. Creminelli and L. Senatore, JCAP 0711, 010 (2007)[arXiv:hep-th/0702165].

[49] J. L. Lehners, P. McFadden, N. Turok andP. J. Steinhardt, Phys. Rev. D 76, 103501 (2007)[arXiv:hep-th/0702153].

[50] A. R. Liddle, A. Mazumdar and F. E. Schunck, Phys.Rev. D 58, 061301 (1998) [arXiv:astro-ph/9804177].

[51] K. Koyama and D. Wands, JCAP 0704, 008 (2007)[arXiv:hep-th/0703040].

[52] K. A. Malik and D. Wands, Phys. Rev. D 59, 123501(1999) [arXiv:astro-ph/9812204].

[53] K. Koyama, S. Mizuno and D. Wands, Class. Quant.Grav. 24, 3919 (2007) [arXiv:0704.1152 [hep-th]].

[54] E. I. Buchbinder, J. Khoury and B. A. Ovrut, JHEP0711, 076 (2007) [arXiv:0706.3903 [hep-th]].

[55] S. M. Carroll, M. Hoffman and M. Trodden, Phys. Rev.D 68, 023509 (2003) [arXiv:astro-ph/0301273].

[56] S. M. Carroll, A. De Felice and M. Trodden, Phys. Rev.D 71, 023525 (2005) [arXiv:astro-ph/0408081].

[57] A. Cardoso and D. Wands, Phys. Rev. D 77, 123538(2008) [arXiv:0801.1667 [hep-th]].

[58] E. Bertschinger, Astrophys. J. 648, 797 (2006)[arXiv:astro-ph/0604485].

[59] D. S. Salopek and J. R. Bond, Phys. Rev. D 42, 3936(1990).

[60] D. Wands, K. A. Malik, D. H. Lyth andA. R. Liddle, Phys. Rev. D 62, 043527 (2000)[arXiv:astro-ph/0003278].

[61] J. K. Erickson, S. Gratton, P. J. Steinhardt and N. Turok,

Phys. Rev. D 75, 123507 (2007) [arXiv:hep-th/0607164].[62] P. Singh, K. Vandersloot and G. V. Vereshchagin, Phys.

Rev. D 74 (2006) 043510 [arXiv:gr-qc/0606032].[63] K. Freese and M. Lewis, Phys. Lett. B 540 (2002) 1

[arXiv:astro-ph/0201229].[64] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl.

78, 1 (1984).[65] D. Polarski and A. A. Starobinsky, Nucl. Phys. B 385

(1992) 623.[66] A. A. Starobinsky, S. Tsujikawa and J. Yokoyama, Nucl.

Phys. B 610 (2001) 383 [arXiv:astro-ph/0107555].[67] J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys.

Rev. D 28, 679 (1983).[68] D. H. Lyth, K. A. Malik and M. Sasaki, JCAP 0505, 004

(2005) [arXiv:astro-ph/0411220].[69] D. H. Lyth and D. Wands, Phys. Rev. D 68, 103515

(2003) [arXiv:astro-ph/0306498].[70] M. Bojowald, Phys. Rev. Lett. 86, 5227 (2001)

[arXiv:gr-qc/0102069].[71] E. I. Buchbinder, J. Khoury and B. A. Ovrut, Phys. Rev.

Lett. 100, 171302 (2008) [arXiv:0710.5172 [hep-th]].[72] K. Koyama, S. Mizuno, F. Vernizzi and D. Wands, JCAP

0711, 024 (2007) [arXiv:0708.4321 [hep-th]].[73] J. L. Lehners and P. J. Steinhardt, Phys. Rev. D 77,

063533 (2008) [arXiv:0712.3779 [hep-th]].[74] J. M. Bardeen, in Particle Physics and Cosmology,

Edited A. Zee, Gordon and Breach, London (1988).