Isocurvature perturbations in the Ekpyrotic universe

A. Notari (Scuola Normale Superiore; A. Riotto (INFN; Padova); Pisa)

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Isocurvature perturbations in the Ekpyrotic universe

A. Notari (Scuola Normale Superiore, Pisa), A. Riotto (INFN, Padova)

High Energy Physics - Theory · Submitted 2 May 2002 · 37 references

Pith's verdict: A modest but correct application of the Gordon et al. rotation formalism to the Ekpyrotic two-field system, with the headline claim conditional on a tuning and an unresolved bounce matching.

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arXiv IDhep-th/0205019 (version 2)

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CitationA. Notari (Scuola Normale Superiore, A. Riotto (INFN, Padova), Pisa). "Isocurvature perturbations in the Ekpyrotic universe." arXiv:hep-th/0205019 (2002). https://pith.science/paper/NDDT5UBU

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Notari and Riotto's real contribution is narrower than the abstract sounds, and more useful than it first appears. They take the Gordon–Wands–Bassett–Maartens rotation-in-field-space machinery and apply it to the Ekpyrotic system where the brane modulus φ and the dilaton χ have a kinetic coupling e^{-αχ}. The payoff is Eq. (3.31), where the gravitational potential is sourced only by the entropy field δs, and Eq. (3.38), the entropy equation with new α-dependent terms in Γ_α. That is a clean, correct piece of work, and they are upfront that the formalism is borrowed and that (3.38) reduces to [24] when α = 0. The derivation is explicit enough to check line by line, which I did on the key steps, and it holds up. The paper is also honest about what it does not know: the bounce matching is unresolved and they say so in Section 4 rather than papering over it. Soft spots, in proportion. The two-field truncation is the main one. Section 3 freezes β by asserting its mass squared is large compared with the kinetic term, but no V(β) is written down and the inequality is never checked against the Ekpyrotic background. The stress-test note is right that decoupling is assumed rather than shown. It is less damaging than it sounds, though: the kinetic term depends on β − χ, so what matters is that one combination is heavy, and the authors note the analysis goes through equally with χ frozen and β free. A third light field would add isocurvature, not remove it. So the qualitative claim survives; what is at risk is the specific cross-correlation coefficient in (3.43). Second, the striking claim that curvature is entirely seeded by isocurvature with a flat spectrum needs q ≈ 1/2, a tuning, plus a matching prescription across the bounce that nobody has. That is acknowledged, but it means the headline result is conditional on physics the paper cannot do yet. Who should read it: anyone working on Ekpyrotic perturbations, curvaton-type isocurvature mechanisms, or two-field perturbation theory with non-trivial kinetic terms. It is a standard reference for the point that isocurvature is unavoidable here. Recommendation: send it out. It is a correct, clearly written application with a real if modest result, and the limitations it admits are the field's limitations, not the paper's.

Referee report

Significance

The paper is a useful and explicit two-field treatment of the Ekpyrotic model at a time when the literature was almost entirely focused on the adiabatic spectrum. The derivation is checkable: I verified Eq. (3.25) by direct differentiation of the rotation angle, verified that Eq. (3.31) follows from the sum of (3.27) and (3.29) with (3.28) used to eliminate V_σδσ, and verified that the coefficient 2q²/(1-q)² in Eq. (3.45) is reproduced by C² ≃ 2qM_p⁴ together with the stated scaling solution. The α-dependent terms in Γ_α are new relative to the standard two-field result of Ref. [24]. The qualitative claim that isocurvature modes are generated whenever the five-brane moves is robust; the quantitative claims about cross-correlation and about curvature being entirely seeded by entropy are less so.

Major objections (3)

[§3, first paragraph] The reduction from three moduli (φ, χ, β) to two fields is asserted rather than derived. The volume modulus is frozen by stating that ∂²V/∂β² ≫ e^{β-χ}φ̇²/M_p², but no V(β) is written down anywhere in the paper and the inequality is never checked against the Ekpyrotic background. Because the kinetic term (2.1) depends only on the combination β-χ, the physically meaningful statement is that one direction in modulus space is heavy; the paper should show that this direction decouples adiabatically and estimate the corrections it induces in V_s and in the α-dependent terms of Γ_α. The remark that the analysis applies equally if χ is frozen and β is free mitigates the concern but does not remove it, since the two-field rotation (3.20)-(3.24) is set up for a specific light combination. This is load-bearing for the quantitative cross-correlation in Eq. (3.43) and for the α-dependent terms in Eq. (3.38), though not for the qualitative statement that isocurvature is generated.
[§3.2, final paragraph and Eqs. (3.41)-(3.43)] The claim that curvature perturbations are 'entirely sourced' by entropy perturbations with a flat spectrum is asserted rather than derived. Eq. (3.31) is never solved with the source term included, no spectrum for Φ is computed, and there is no demonstration that the particular integral S(t) in Eq. (3.41) dominates the homogeneous modes f±. This matters because the paper itself notes on p. 10 that the homogeneous A-mode already has a scale-invariant spectrum in the single-field limit, so the added value of the entropy-sourcing mechanism needs to be quantified, for example through the relative amplitude of the particular and homogeneous solutions and the cross-correlation coefficient C_Φδs in Eq. (3.43). Relatedly, the initial conditions for δs are never specified, so the 'flat spectrum' statement in Eq. (3.45) concerns only the k-dependence and not the amplitude; without an amplitude the word 'entirely' in the abstract cannot be supported.
[Abstract, sentence s1; §3.2] The word 'generic' in the abstract should be qualified. The mixing is a consequence of the specific kinetic term (2.1) taken from Refs. [19, 20] and of the assumption that the five-brane modulus is the only light modulus besides χ. The paper's own three regimes show that the amplitude of the entropy source is strongly model-dependent: it is suppressed in the first two regimes and only becomes important in the third. The claim should be stated as 'generic given the effective action (2.1)', with an explicit statement of the conditions under which the mixing is non-negligible, rather than as a property of the Ekpyrotic Universe independent of the compactification details.

Minor objections (9)

[Abstract and §4] There are several typographical errors in the abstract and conclusions: 'isorcuvature' and 'pertubations' in the abstract, and 'isocuvarture' in the first sentence of §4. These should be corrected.
[§3, below Eq. (3.4)] The text reads 'were Vx denotes the derivative ∂V/∂x'; this should read 'where V_x denotes'.
[References, [17]] Reference [17] is incomplete: the entry for Bandos et al., Phys. Rev. Lett. 78, is missing its page number.
[Eqs. (3.9)-(3.11), (3.27)-(3.31)] The Einstein equations are written with a factor 4πM_p², whereas Eq. (3.5) implies G = 1/M_p², so the factor should presumably be 4π/M_p². The same comment applies to Eq. (3.30) and to the source term -8πM_p²V_sδs in Eq. (3.31). Please check the normalization throughout.
[Eq. (3.19)] The displayed prefactor (2V̇ + 3H(χ̇² + φ̇²e^{-αχ}))/(χ̇² + φ̇²e^{-αχ}) appears in the numerator, whereas the corresponding factor in Eq. (3.33) sits in the denominator. Please check the displayed form of Eq. (3.19).
[Eq. (3.36)] The second term as printed, H V_sδs(2V_σ + 3Hσ̇)σ̇, is dimensionally inconsistent with the first term; the factor (2V_σ + 3Hσ̇)σ̇ presumably belongs in the denominator. Please check.
[§3.2, sentence 'This will be the case if, for instance, A ≪ C/M_p'] This stated criterion is not satisfied in the case of interest. Using the relations αA = 2(3q-1)/(1-q), βA/M_p = 2/(1-q), αM_p/β = 3q-1 and C² ≃ 2qM_p⁴, the choice q = 1/2 gives A = √2M_p and C/M_p = M_p, so A is not much less than C/M_p. The conclusion nevertheless survives, because the actual ratio of brane-modulus to dilaton kinetic energy is 2qM_p²/(A²(1-q)²), which equals 2 at q = 1/2; the correct condition is A < 2M_p for q = 1/2. Please replace the stated criterion with the correct one.
[Eqs. (3.38) and (3.44)] The passage from Eq. (3.38) to Eq. (3.44) is not shown. In the brane-modulus-dominated limit (sinθ ≈ 1), 3θ̇² plus the σ̇² terms of Γ_α combine to -(α²/4)φ'²e^{-αχ}, which is presumably the origin of the retained term, but the dropping of V_ss and of the V_σ and V_s terms in Γ_α should be justified explicitly, since the flat-spectrum condition q ≈ 1/2 depends on the precise coefficient in Eq. (3.45).
[§4] The discussion of instanton/five-brane replacement and of the Type IIA analogy is interesting but is not connected to the perturbation analysis. The authors should state explicitly what happens to δs through the transition, or shorten this material and refer to Ref. [37].

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The paper

The paper claims

The Ekpyrotic universe cannot avoid making entropy perturbations, because the moving five-brane's kinetic energy is tied to the dilaton and the size of the extra dimension.

The reading

The Ekpyrotic scenario replaces cosmic inflation with a slow collision: a five-brane drifts down a fifth dimension and strikes the boundary brane we live on, and the resulting contraction is supposed to seed today's structure. This paper asks whether that scenario produces only the ordinary, everywhere-the-same (adiabatic) density perturbations, or whether it also produces isocurvature perturbations, the kind that leave the total density unchanged while shifting the balance between components. The authors point out that the brane modulus — the field recording how far the five-brane sits from us — enters the four-dimensional action with a kinetic term multiplied by $e^{(b- chi)/M_p}$, so its motion automatically drags the dilaton and the volume modulus along with it. Once several fields move, perturbations in them mix, and entropy modes are not optional. Using a rotation in field space, the authors split the physics cleanly into one field along the trajectory and one perpendicular to it, and show that the gravitational potential is sourced only by the perpendicular (entropy) field.

What carries the argument

The rotation in field space is the load-bearing device. Defining $dot{sigma} = costheta dot{chi} + sintheta dot{phi}, e^{-alpha chi/2}$ and $delta s = -sintheta, delta chi + costheta, deltaphi, e^{-alpha chi/2}$, with the angle fixed by the ratio of the background field velocities, separates the two-field system into a direction along the trajectory and a direction orthogonal to it. The adiabatic field then obeys the single-field equation $Ddot{sigma} + 3H dot{sigma} + V_sigma = 0$, while all the multi-field information is pushed into $delta s$ and into the source term proportional to $V_s delta s$ in the gravitational-potential equation. The angle itself evolves according to $dot{theta} = -V_s/dot{sigma} + (1/2)alpha sintheta dot{sigma}$, and that $alpha$-term is where the ekpyrotic kinetic coupling leaves its fingerprint on the entropy mass. A second piece of machinery is the relation $dot{mathcal{R}} = -3H (dot{p}/dot{rho}) S$, which converts the entropy perturbation directly into a change of the curvature perturbation on large scales, making the claim about entropy seeding curvature quantitative rather than rhetorical.

Extended reading notes

Core claim

The paper's claim is that isocurvature perturbations are a generic prediction of the Ekpyrotic Universe, not a special-case accident. The reason is visible already in the kinetic Lagrangian: the brane modulus appears as $e^{b- chi over M_p} partial_mu phi partial^mu phi$, so a moving five-brane forces nontrivial dynamics on the dilaton and volume modulus, and exact constant solutions for those fields do not exist. The authors then reduce to two fields (brane modulus and dilaton, with the volume modulus held at its minimum) and apply the adiabatic/entropy decomposition: an adiabatic field $sigma$ measuring path length along the classical trajectory, and an entropy field $s$ orthogonal to it. In these variables the equation for the gravitational potential becomes $Ddot{Phi} + (H - 2 Ddot{sigma}/Ddot{sigma}) Ddot{Phi} + (2Ddot{H} - 2H Ddot{sigma}/Ddot{sigma})Phi + (k^2/a^2)Phi = -8pi V_s delta s / M_p^2$, with the entropy perturbation as the only source on the right-hand side. The entropy field itself obeys a homogeneous second-order equation on large scales, decoupled from the metric, with new $alpha$-dependent terms in the mass-like coefficient. The authors also exhibit a concrete regime — brane-modulus potential negligible, dilaton potential $-M^4 e^{-beta chi/M_p}$, brane kinetic energy dominating — in which the entropy spectrum is nearly scale-invariant for $q simeq 1/2$, so that a flat spectrum of curvature perturbations can be generated entirely from entropy seeds with maximal cross-correlation.

Load-bearing premise

The derivation reduces a three-field system to two by assuming the volume modulus sits at the minimum of its potential with a mass far larger than the kinetic energy scale set by the moving brane; the authors state this assumption but do not demonstrate that it holds in the ekpyrotic background.

Editorial extensions

If this is right

Any ekpyrotic model in which the five-brane moves must include entropy perturbations in its perturbation analysis; a single-field treatment is incomplete by construction.
The adiabatic and isocurvature modes are predicted to be cross-correlated before the collision, with a correlation $\langle \Phi(k)\, \delta s^*(k')\rangle$ proportional to the particular integral sourced by $V_s\delta s$.
In the regime where the brane-modulus potential is negligible and the dilaton potential dominates, a nearly scale-invariant entropy spectrum arises for $q \simeq 1/2$, giving a flat spectrum of curvature perturbations seeded entirely by entropy.
The authors' own numerical remark is that in the simplest ekpyrotic limit the induced cross-correlation has a blue spectrum, $C_{\Phi\delta s} \propto \alpha\, p\, M_p\, k^2$, so the size of the effect depends strongly on which regime the universe actually occupied.
Isocurvature perturbations can survive into the radiation era only if some component retains a nonzero relative entropy perturbation, which requires the brane modulus and the dilaton to decay into radiation and cold dark matter at different epochs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the entropy channel is really generic, the ekpyrotic scenario's predictive content shifts: the question stops being 'what is the adiabatic spectral index?' and becomes 'what is the correlation between curvature and entropy, and does it survive reheating?' A future measurement of a nonzero adiabatic-isocurvature cross-correlation at large scales would be a distinctive signature, though the autho
The paper's two-field reduction is a controlled approximation only if the volume modulus is much heavier than the kinetic energy scale $e^{(b-chi)/M_p}\dot{\phi}^2/M_p^2$. A three-field treatment would not remove the entropy production but would change the entropy mass matrix and hence the spectral index of the seeded curvature; the qualitative claim is robust, the quantitative one is not.
The mechanism here is structurally the same as curvaton-style and modulated-reheating seeding in ordinary inflation: a light or subdominant field carries the perturbation and later converts it into curvature. That suggests the ekpyrotic version inherits the same model-building freedom, and the same problem of explaining why the conversion happens at a particular epoch.
Nothing in the derivation depends on the ekpyrotic potential's exponential form except the background solution; the source term $-8\pi V_s\delta s/M_p^2$ would apply to any contracting multi-field model with a similar kinetic mixing, so the result is testable against broader bouncing cosmologies, not only this one.

Original abstract

The Ekpyrotic scenario assumes that our visible Universe is a boundary brane in a five-dimensional bulk and that the hot Big Bang occurs when a nearly supersymmetric five-brane travelling along the fifth dimension collides with our visible brane. We show that the generation of isocurvature perturbations is a generic prediction of the Ekpyrotic Universe. This is due to the interactions in the kinetic terms between the brane modulus parametrizing the position of the five-brane in the bulk and the dilaton and volume moduli. We show how to separate explicitly the adiabatic and isorcuvature modes by performing a rotation in field space. Our results indicate that adiabatic and isocurvature pertubations might be cross-correlated and that curvature perturbations might be entirely seeded by isocurvature perturbations.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The Ekpyrotic scenario assumes that our visible Universe is a boundary brane in a five-dimensional bulk and that the hot Big Bang occurs when a nearly supersymmetric five-brane travelling along the fifth dimension collides with our visible brane. We show that the generation of isocurvature perturbations is a generic prediction of the Ekpyrotic Universe. This is due to the interactions in the kinetic terms between the brane modulus parametrizing the position of the five-brane in the bulk and the dilaton and volume moduli. We show how to separate explicitly the adiabatic and isorcuvature modes by performing a rotation in field space. Our results indicate that adiabatic and isocurvature pertubations might be cross-correlated and that curvature perturbations might be entirely seeded by isocurvature perturbations.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Ekpyrotic scenario assumes that our visible Universe is a boundary brane in a five-dimensional bulk and that the hot Big Bang occurs when a nearly supersymmetric five-brane travelling along the fifth dimension collides with our visible brane.	background 1.00	0.05	0.03	asserted 1.00
s1	We show that the generation of isocurvature perturbations is a generic prediction of the Ekpyrotic Universe.	result 0.92	0.94	0.85	asserted 1.00
s2	This is due to the interactions in the kinetic terms between the brane modulus parametrizing the position of the five-brane in the bulk and the dilaton and volume moduli.	interpretation 0.71	0.51	0.08	asserted 1.00
s3	We show how to separate explicitly the adiabatic and isorcuvature modes by performing a rotation in field space.	method 0.97	0.95	0.07	asserted 1.00
s4	Our results indicate that adiabatic and isocurvature pertubations might be cross-correlated and that curvature perturbations might be entirely seeded by isocurvature perturbations.	interpretation 0.88	0.87	0.37	hedged possibility 0.81

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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References

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Generating Ekpyrotic Curvature Perturbations Before the Big Bang
hep-th 2007-02 conditional novelty 6.0 of 10

Entropic perturbations generated during an ekpyrotic contraction can be converted into scale-invariant curvature perturbations at a moduli-space bounce, yielding the observed amplitude and a spectral tilt in the range...
Adiabatic and entropy perturbations with interacting fluids and fields
astro-ph 2004-11 accept novelty 6.0 of 10

The paper derives a complete set of gauge-invariant equations for how adiabatic and entropy perturbations evolve in a universe of multiple interacting fluids and scalar fields, on all length scales.
Large-scale curvature and entropy perturbations for multiple interacting fluids
astro-ph 2002-11 accept novelty 6.0 of 10

A gauge-invariant definition of entropy perturbations that survives energy transfer shows that adiabatic perturbations stay adiabatic on large scales, and quantifies how a decaying curvaton converts isocurvature into ...
Decoherence out of fire: Purity loss in expanding and contracting universes
hep-th 2024-01 conditional novelty 5.0 of 10

Contracting ekpyrotic universes fully decohere the curvature perturbation before the bounce, whereas inflation recoheres for heavy isocurvature fields, so the cosmological background drives the result.
Non-Singular Bouncing Cosmology: Consistency of the Effective Description
hep-th 2015-12 conditional novelty 5.0 of 10

A ghost-condensate cosmological bounce is shown to be a trustworthy effective field theory: its strong-coupling scale exceeds the bounce energy density, and the apparent singularities in the cubic action at H = 0 and ...
Matter creation in a nonsingular bouncing cosmology
gr-qc 2014-06 conditional novelty 5.0 of 10

Gravitational particle production in the two-field matter bounce yields a reheating temperature of order 10^-7 Planck masses, sufficient to reheat the universe without additional couplings.
Anisotropy in a Nonsingular Bounce
gr-qc 2013-01 conditional novelty 5.0 of 10

In a Galileon-mediated non-singular bounce preceded by Ekpyrotic contraction, Bianchi-I anisotropies are damped during contraction and remain small through the bounce, requiring roughly seven or more Ekpyrotic e-folds...
Towards a Nonsingular Bouncing Cosmology
hep-th 2012-06 conditional novelty 5.0 of 10

A single scalar field model with Ekpyrotic contraction and a ghost-condensate bounce yields a nonsingular cosmology in which the curvature perturbation spectral index is preserved through the bounce.
On Perturbations of Quintom Bounce
hep-th 2007-11 conditional novelty 5.0 of 10

In a two-field Quintom bounce, short-wavelength cosmological perturbations transfer the growing contracting-phase mode into the dominant expanding-phase mode, whereas long-wavelength modes transfer only into the decay...
A smooth bouncing cosmology with scale invariant spectrum
hep-th 2007-02 conditional novelty 5.0 of 10

A smooth, singularity-free bouncing cosmology produces a nearly scale-invariant spectrum with a slightly blue tilt, no observable gravitational waves, and large local non-Gaussianity.
On the Construction of Nonsingular Pre-Big-Bang and Ekpyrotic Cosmologies and the Resulting Density Perturbations
hep-th 2002-07 conditional novelty 5.0 of 10

Higher-derivative string and quantum corrections smooth the cosmological bounce in Ekpyrotic models, and the resulting curvature perturbation has a blue spectral index of 3 rather than a scale-invariant one.
Scalar and Tensor Perturbations in DHOST Bounce Cosmology
gr-qc 2021-08 conditional novelty 4.0 of 10

In a DHOST bounce with thermal initial conditions, the scalar spectral index and amplitude can be matched to Planck data by choosing two parameters, while the tensor-to-scalar ratio comes out many orders of magnitude ...
Running of the scalar spectral index in bouncing cosmologies
astro-ph.CO 2015-07 conditional novelty 4.0 of 10

Ekpyrotic cosmologies predict a negative running of the scalar spectral index while multi-field matter bounces predict a positive one, opposite in sign to the observationally favoured inflationary models.
Assisted Contraction
hep-th 2002-06 conditional novelty 4.0 of 10

In a contracting two-field universe with exponential potentials, a scale-invariant isocurvature spectrum arises from a tachyonic instability for slow or radiation-like contraction, while the curvature spectrum is scal...
A Critical Review of Classical Bouncing Cosmologies
astro-ph.CO 2014-06 accept novelty 3.0 of 10

Most bouncing cosmologies do not yet match current data or avoid known instabilities, but a few, notably the super-bounce and S-brane bounce, remain viable research directions.
Ekpyrotic and Cyclic Cosmology
astro-ph 2008-06 accept novelty 3.0 of 10

A contracting ekpyrotic phase before a brane collision can solve the standard cosmological puzzles and generate near-scale-invariant, strongly non-gaussian perturbations, with a blue gravitational-wave spectrum that d...
Bouncing Cosmologies: Progress and Problems
hep-th 2016-03 accept novelty 2.0 of 10

Bouncing cosmologies can reproduce inflation's near scale-invariant spectrum of primordial perturbations without a singularity or trans-Planckian modes, but every known implementation needs new physics and none is yet...
Cosmic Bounces and Cyclic Universes
hep-th 2011-06 accept novelty 2.0 of 10

Ekpyrotic and cyclic bounce models can replace inflation for the standard cosmological puzzles and make distinctive predictions, but the bounce itself remains the program's weakest link.
Local non-Gaussianity from inflation
astro-ph.CO 2010-04 accept novelty 2.0 of 10

Local primordial non-Gaussianity can be computed with the delta-N formalism and is predicted at potentially observable levels by curvaton, modulated reheating, and ekpyrotic models.
Ekpyrotic Non-Gaussianity -- A Review
hep-th 2010-01 accept novelty 2.0 of 10

Ekpyrotic models predict local non-Gaussianity ranging from f_NL of order minus forty (already in tension with data) to f_NL of order tens (testable by Planck), depending on when entropy perturbations convert to curvature.

How the literature uses it

32 arXiv papers cite this in 37 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 21 sentences without a confident label

Uses its method or tool 1 paper

Such an analysis can be conducted by treating the total source as a system consisting of two scalar fields, as in the references [ref] .

Cosmological Singularities in Brane Gravity · Inhomogeneous Perturbations · 0.85

Compares with it 1 paper

Whereas in models with a singular bounce such as the Pre-Big-Bang [ref] and the Ekpyrotic [ref] scenarios, the perturbations must be matched across a singular space-like surface, which leads to serious conceptual problems [ref] , in our case no such problems arise [Recently, a ghost condensation mechanism to smooth out the singularity in the Ekpyrotic scenario has been introduced [ref] , and it has been shown in this context, using the techniques developed in [ref] that making use of iso-curvature modes, the dominant spectrum in the contracting phase can be passed on to the dominant spectrum in the expanding phase.].

On Perturbations of Quintom Bounce · Discussion And Conclusions · 0.83

Cites it as background 12 papers

Regarding the first step of the entropic mechanism, two classes of the scalar field $χ$ have been studied in the literature: the entropic field $χ$ has a canonical kinetic term and an unstable potential [ref] , or the entropic field $χ$ has a non-minimal kinetic coupling to the ekpyrotic scalar field [ref] .

Fully viable DHOST bounce with extra scalar · Introduction · 0.96
Quantum perturbations can also be amplified during phases of cosmological contraction, in particular during an ekpyrotic phase of slow high-pressure contraction [ref] , or during a period of matter-dominated pressure-free contraction [ref] .

Converting entropy to curvature perturbations after a cosmic bounce · Introduction · 0.81
It provides a mechanism which inflation [ref] and many alternatives [ref] rely on to generate cosmological inhomogeneities seeding the Cosmic Microwave Background (CMB) [ref] and the Large Scale Structure (LSS) [ref] of the universe.

Decoherence out of fire: Purity loss in expanding and contracting universes · Introduction · 0.83

Discussion

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