Inﬂation, geometry and...

Inflation, geometry and stochasticity

Sébastien Renaux-PetelCNRS - Institut d’Astrophysique de Paris

IHP Trimestre Analytics, Inference, and Computation in Cosmology: Advanced methods

18th September 2018

Based on Pinol, RP, Tada, 1806.10126, and in preparation

1. Introduction

2. Geometrical destabilization of inflation

3. Ito vs Stratonovich in single-field slow-roll inflation

4. Multifield inflation, and ‘derivation’

5. Stochastic diffusion in curved field space

Outline

Renaux-Petel, IAP

Planck all sky map

l(l+1)C

l/2�

[µK

2]

Quantum fluctuations stretched to

cosmological scales

QFT and GR!

Cosmological inflation provides seeds of CMB anisotropies and LSS

Renaux-Petel, IAP

- conceptually not satisfactory

background + fluctuations

Classical Quantum

�

��V (�)

�

- breaks down for very light scalar fields

Stochastic formalism

Conventional formulation

Renaux-Petel, IAP

time

physical scale

H�1

�k =a

kBackground

super-Hubble modes

Classical stochastic effective theory for

coarse-grained fields

d�

dN= �V 0(�)

3H2+

H

2⇡⇠

Continuous flow of initially sub-Hubble (UV) modes joining the super-Hubble (IR) sector

Open quantum system

normalized Gaussian white noise


⇠

IR modes

Renaux-Petel, IAP

Starobinsky, 86

cf C. Burgess’s talk

- reproduce non-trivial results from QFT in curved spacetime

- recover conventional perturbation theory in suitable classical limit

Success

Open questions

- EFT: delineate regime of validity

- non-Gaussianity of noise, non-white, non-Markovian, phase-space analysis,

curved field-space and covariance

- Eternal inflation and multiverse- Primordial Black Holes production Scope

- Geometrical destabilization of inflation


Starobinsky, Woodard et al

Vennin, Starobinsky

1. Introduction





Outline

Renaux-Petel, IAP

Realistic inflationary models have fields which live in an internal space with curved geometry.

This effect applies during inflation, it easily overcomes the effect of the potential, and can destabilize inflationary trajectories.

Initially neighboring geodesics tend to fall away from each other in the presence of negative curvature.

Geometrical destabilization of inflation

V (�1,�2)

�1

�2

Basic mechanism

Effective single-field dynamics

Light inflaton+

Extra heavy fields

Simplest ‘realistic’ models (hope):

(valley with steep walls)

RP, Turzynski, 1510.01281, PRL Editors’ Highlight

Renaux-Petel, IAP

V (�1,�2)

�1

�2

More realistic:

Geometrical instability

Light inflaton+

Extra heavy fields+

Curved field space

Basic mechanism

m2e↵ � ✏H2RM2

Pl

RP, Turzynski, 1510.01281, PRL Editors’ Highlight

Like eta-problem but no symmetry to help

L = �1

2(@�)2

✓1 + 2

�2

M2

◆� V (�)� 1

2(@�)2 � 1

2m2

h�2

Single-field slow-roll

Sidetracked inflation

m2e↵ = m2

h � 4✏H2M2Pl

M2

Fate of the instability?

Toy-model:

Backreaction of fluctuations on effective background?

?

Renaux-Petel, IAP

- Universal bound on curvature scale

- Modified ranking of inflationary models

Second phase of inflation: sidetracked inflationPremature end of inflation

Inhomogeneities dominate

Fate of the instability?

Inhomogeneities are shut off

RP, Turzynski, Vennin 1706.01835

RP, Turzynski, 1510.01281 Garcia-Saenz, RP, Ronayne, 1804.11279

OR

- Rich phenomenology

- EFT with imaginary sound speed

Garcia-Saenz, RP 1805.12563

Renaux-Petel, IAP

1. Introduction





Outline

Renaux-Petel, IAP

Stochastic calculus

NiN

BrownianW(N) ∆Wi : Gaussian (0, ∆N2)

h⇠(N)⇠(N 0)i = �(N �N 0)

N�i � [Ni, Ni+1]

Langevin equation:

Continuous limit of discrete process:

X’s properties depend on choice of

dX

dN= h(X) + g(X) ⇠

�Xi = h(X(N⇤i ))�Ni + g (X(N⇤

i ))�Wi

N⇤i

Renaux-Petel, IAP

Ito StratonovichN⇤

i = Ni N⇤i =

1

2(Ni +Ni+1)

dX = h dN + g dW dX = h dN + g � dW

df(X) = fXdX +1

2fXXg2dN df(X) = fXdX

Ito’s lemma: standard chain rule

Stochastic calculus

Renaux-Petel, IAP


i = Ni N⇤i =

1

2(Ni +Ni+1)


df(X) = fXdX +1



Stochastic calculus

P (X,N): pdf of X at time NFokker-Planck equation for

drift diffusion

@P

@N= LFP · P = � @

@X(D(X)P ) +

1

2

@2

@X2

�g2(X)P

�

Renaux-Petel, IAP


i = Ni N⇤i =

1

2(Ni +Ni+1)


df(X) = fXdX +1



Stochastic calculus

P (X,N): pdf of X at time NFokker-Planck equation for

@P

@N= LFP · P = � @

@X(D(X)P ) +

1

2

@2

@X2

�g2(X)P

�

DI = h DS = h+1

2g@g

@Xnoise-induced

drift

Ito ‘to respect causality’

• Stratonovich: white noises are idealizations of colored noises

• Discrepancy exceeds the accuracy of stochastic formalism

Ito versus Stratonovich for inflation

• Many papers: (not good reason)

Vilenkin 1999, Fujita, Kawasaki, Tada 2014, Tokuda & Tanaka 2017, …

Mezhlumian & Starobinsky 1994UV

UVIR

Mezhlumian & Starobinsky 1994

Vennin & Starobinsky 2015

smooth splitting between UV and IR modes

• Us: new perpective with requirement of covariance Renaux-Petel, IAP

3H2(�)M2Pl = V (�)


d�

dN= �V 0(�)

3H2+

H

2⇡⇠

Test scalar fields (e.g. in de Sitter)H(�) unambiguous

Inflaton:Ito versus Stratonovich ambiguity

of multiplicative noises

number of e-folds of inflation realized starting from �

stochastic quantity, directly related to observable curvature perturbation

⇣ = �N = N � hN i

N (�) :

cf. V. Vennin’s talk

Renaux-Petel, IAP

L†FP(�)�N (!,�) = �i! �N (!,�)

1

M2Pl

L†FP,I = v

@2

@�2� v0

v

@

@�

1

M2Pl

L†FP,S = v

@2

@�2� v0

v

⇣1� v

2

⌘ @

@�

v ⌘ V

24⇡2M4Pl

�N (!,�) ⌘ hei!N (�)i

In regime of validity of stochastic inflation v ⌧ 1

No difference between Ito and Stratonovich, in classical or stochastic regimes

All information in characteristic function


versus

v2v′�′�v′�2

> 1

Strato

Ito

sad

cl

10-8 10-5 0.01 1010-9

10-5

0.1

1000.0

107

φ

�ζ

Strato

Ito

sad

cl

10-8 10-5 0.01 100

1000

2000

3000

4000

5000

φ

⟨N⟩

V = Λ4 (1 − φ2

μ2 ), Λ = 10−2MPl, μ = 20MPl

No Ito-Stratonovich difference even in highly stochastic case

Single-field slow-roll: example

v2v′�′�v′�2

> 1

Renaux-Petel, IAP

1. Introduction





Outline

Renaux-Petel, IAP

Multivariate calculus

dXa

dN= ha(X) + ga↵(X) ⇠↵

h⇠↵(N)⇠�(N 0)i = �↵��(N �N 0)

Langevin equations

Renaux-Petel, IAP

Multivariate calculus

dXa

dN= ha(X) + ga↵(X) ⇠↵

h⇠↵(N)⇠�(N 0)i = �↵��(N �N 0)

DaI = ha

DaS = ha +

1

2gb↵

@ga↵@Xb

drift vectordiffusion matrixDab = ga↵g

b↵

Langevin equations

Fokker-Planck equation

@P

@N= LFP(X

a) · P

with LFP(Xa) = � @

@XaDa +

1

2

@2

@Xa@XbDab

Renaux-Petel, IAP

L = �1

2GIJ@µ�

I@µ�J � V

d�I

dN= � GIJV,J

3H2(�I)+ ⌅I

h⌅I(N)⌅J(N 0)i =✓H

2⇡

◆2

GIJ �(N �N 0)

⌅I = gI↵ ⇠↵ gI↵ =H

2⇡eI↵

Multifield generalization?

Not yet stochastic differential equations

and set of vielbeins eI↵eJ↵ = GIJ

with noise correlations

with

Renaux-Petel, IAP

Slow-roll, overdamped limit

Arbitrary choice of vielbeins:

no impact on Ito FP equation

matters for Stratonovich!

ga↵ ! ⌦ �↵ (X)ga�

• Vielbeins can a priori differ by arbitrary field-dependent rotations

Multifield generalization

Renaux-Petel, IAP

Arbitrary choice of vielbeins:

no impact on Ito FP equation

matters for Stratonovich!

• Field space covariance

Physical quantities should not depend on field redefinitions

ga↵ ! ⌦ �↵ (X)ga�

• Vielbeins can a priori differ by arbitrary field-dependent rotations

Covariance is respected only in Stratonovich interpretation!

(Ito lemma vs standard chain rule)


Renaux-Petel, IAP

Ps(�I) =

P (�I)pdet(GIJ)

should be a scalar under field redefinitions


Renaux-Petel, IAP

Ps(�I) =

P (�I)pdet(GIJ)

@Ps

@N= rI

✓V ,I

3H2Ps

◆+

1

2rI

✓H

2⇡rI

✓H

2⇡Ps

◆◆

+1

2rI

✓H

2⇡

◆2

eI↵(rJeJ↵)Ps

!



Stratonovich: manifestly covariant FP equation

Renaux-Petel, IAP

Ps(�I) =

P (�I)pdet(GIJ)

@Ps

@N= rI

✓V ,I

3H2Ps

◆+

1

2rI

✓H

2⇡rI

✓H

2⇡Ps

◆◆

+1

2rI

✓H

2⇡

◆2

eI↵(rJeJ↵)Ps

!



covariant derivatives


Renaux-Petel, IAP

Ps(�I) =

P (�I)pdet(GIJ)

@Ps

@N= rI

✓V ,I

3H2Ps

◆+

1

2rI

✓H

2⇡rI

✓H

2⇡Ps

◆◆

+1

2rI

✓H

2⇡

◆2

eI↵(rJeJ↵)Ps

!


covariant derivatives Spurious dependence on

the arbitrary choice of vielbeins, in curved or flat field space


but…


Renaux-Petel, IAP

Ps(�I) =

P (�I)pdet(GIJ)

@Ps

@N= rI

✓V ,I

3H2Ps

◆+

1

2rI

✓H

2⇡rI

✓H

2⇡Ps

◆◆

+1

2rI

✓H

2⇡

◆2

eI↵(rJeJ↵)Ps

!


New light on single-field slow-roll: field redefinition innocuous

only in Stratonovich



Renaux-Petel, IAP

• More rigorous approach:

• Heuristic approach:

�IR �UV+� =

classical equations of motion + time-dependence of coarse graining scale

Schwinger-Keldysh formalism (time-dependence) + Gaussian action for UV modes + phase space (effective Hamiltonian action)

integrate out UV modes at level of action

Derivation?

• Splitting

Renaux-Petel, IAP

W

DN⇡I = ⇡0I � �K

IJ�J0⇡K

3H2M2Pl = V (�I) +

1

2GIJ⇡I⇡J

First principles

Hamiltonian constraint

�I0 =GIJ⇡J

H+ ⇠IQ

DN⇡I = �3⇡I �VI

H+ ⇠P I

Langevin type equations in phase space

Correlation matrix of noises: power spectra of UV modes in IR background

Renaux-Petel, IAP

DN⇡I = ⇡0I � �K

IJ�J0⇡K

3H2M2Pl = V (�I) +

1

2GIJ⇡I⇡J

Correlation matrix of noises: power spectra of UV modes in IR background

First principles

Hamiltonian constraint

�I0 =GIJ⇡J

H+ ⇠IQ

DN⇡I = �3⇡I �VI

H+ ⇠P I

Langevin type equations in phase space

Complex!Non-Markovian

(noise depends on its past history)

Non-Gaussian and non-white noises (smooth splitting between UV and IR modes)

Renaux-Petel, IAP

Covariance

Even under simplifying assumptions of Markovian, Gaussian white noises, issue of covariance

Phase-space pdf should be proper scalarP (�I ,⇡I , N)

Need to introduce arbitrary `square-roots' of the correlation matrix of the noises (gIQ↵, gP I↵)

covariance, but spurious ‘frame’ dependence

drops out but not covariant

Ito Stratonovich

Renaux-Petel, IAP

Stratonovich covariant Fokker-Planck equation:

@P

@N= �⇡I

HD�IP +

✓V,I

H+ 3⇡I

◆@⇡IP + 3nP

First line: neat Boltzmann-like reformulation

of deterministic evolution

Covariance

+1

2

�D�I

⇥gIQ↵·

⇤+ @⇡I [gP I↵·]

� �D�J (gJQ↵P ) + @⇡J (gP I↵P )

�

D�I = r�I + �KIJ⇡K@⇡J

phase-space covariant derivative

Second line: ’frame’ dependence…

Renaux-Petel, IAP

1. Introduction





Outline

Renaux-Petel, IAP

Stochastic diffusion in curved field space

Assuming inflationary stochastic anomaly is an artefact

And for test fields in dS space (simplicity)

@Ps

@N=

1

3H20

rI

�V ,IPs

�+

1

2

✓H0

2⇡

◆2

rIrIPs

Covariant equation for diffusion of light scalar fields in curved field space in overdamped limit.

Simplest multifield covariantization: no curvature invariants for effectively light fields

(like adding suitable noise-induced drift)

Renaux-Petel, IAP

(@�)2 + e2b(�)(@ )2

h�ni0 = � n

3H20

h�n�1V,�i

+1

2

✓H0

2⇡

◆2 ⇥n(n� 1)h�n�2i+ nh�n�1b,�i

⇤.

For sum-separable potentials, only depends on �

Stochastic diffusion in curved field space

Field space geometry

Easily solvable for polynomial V and b

Renaux-Petel, IAP

Case study: phi4 in hyperbolic geometry

Z2 symmetry of potential broken by geometry

b(�) = ��/MV (�) = ��4/4 R = �2/M2

Renaux-Petel, IAP


h�(N)i = � H20N

8⇡2M

1� �N2

12⇡2� �H2

0N3

768⇡4M2+ . . .

�

h�2(N)i = H20N

4⇡2

1 +

H20N

16⇡2M2� �N2

6⇡2� 7�H2

0N3

384⇡4M2+ . . .

�

Starting from �(�)

Double expansion in �

⇡2N2 H2

0

⇡2M2Nand

Z2 symmetry of potential broken by geometry

Benchmark for QFT computations

b(�) = ��/MV (�) = ��4/4 R = �2/M2

Renaux-Petel, IAP

h�ni0 = � n

3H20

h�n�1V,�i

+1

2

✓H0

2⇡

◆2 ⇥n(n� 1)h�n�2i+ nh�n�1b,�i

⇤.

Resummation at late times?


Renaux-Petel, IAP

Ve↵(�) = V (�)� 3H40/(8⇡

2) b(�)

Peq / e�8⇡2/(3H40 )Veff (�)

h�ni0 = � n

3H20

h�n�1V,�i

+1

2

✓H0

2⇡

◆2 ⇥n(n� 1)h�n�2i+ nh�n�1b,�i

⇤.

Resummation at late times?

Mapping to single-field model with effective potential:


e.g.Renaux-Petel, IAP

Conclusion

• New light on Ito vs Stratonovich dilemma with criterion of general covariance

• Manifestly covariant in Stratonovich only, at expense of frame-dependences: inflationary stochastic anomaly

• Single-field slow-roll: no practical difference, but Stratonovich is unambiguous

• Ambiguity in full phase space, or w. multiple fields

• Stochastic diffusion in curved field space: mapping to single-field models and benchmark model

Renaux-Petel, IAP

Open questions

• Fate of instability in geometrical destabilization of inflation, and general implications for inflation

• Scrutinize foundations of stochastic inflation: non-Gaussianity of noise, non-white, non-Markovian, phase-space analysis, curved field-space, covariance

• Inflationary stochastic anomaly: stochastic counterparts of notoriously difficult problem of maintaining general covariance in quantum theories

Renaux-Petel, IAP

Postdoc opportunity next Fall: ERC group at IAP

Thank you for your attention

Renaux-Petel, IAP

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