Observables from Multi-field Inflationin Dyson Brownian Motion Potentials

Robert Rosati

November 30th, 2016

Outline

Motivation

InflationBasic CosmologyMulti-field inflationObservablesCurrent experimental bounds

Random potentialsRandom matricesDBM potentialsOur Model

Future DirectionsDBM ProblemsDias et al. method

Motivation

I Observations consistent with a single scalar field.

I However, can multiple light fields with a generic potentialproduce similar observational results? Attractor?

I Multifield inflation isn’t immediately as predictive assingle-field inflation.

Motivation

I Many fieldsI Expect moduli from type IIB, F-theory, and

supergravity compactifications.I Having many inflationary fields can help avoid

large-field inflation.

I Random matrix techniquesI Landscape of string vacua large-dimensional, likely

incredibly complicatedI DBM potentials only defined locally –

computationally efficient

What is inflation?

I Homogeneity, isotropy, and flatness give ds2 = dt2 − a(t)2dx2

I For a perfect fluid, stress-energy conservation gives

ρ+ 3H(p + ρ) = 0, where H ≡ a

a

I We can get a shrinking comoving Hubble sphere if

d

dt(aH)−1 = − a

a2< 0

=1

a(1− ε), where ε ≡ −H

H2

I Want inflation to last sufficiently long. Require η ≡ εHε small.

I We can write ε = 32(1 + p

ρ ) =⇒ inflate when we violate SEC

Multi-field inflation

I Minimally couple several scalar fields to gravity

S =

∫d4x√−g[M2

Pl

2R − 1

2δIJg

µν∂µφI∂νφ

J − V (φ)

]I Get φI + 3HφI + V,I = 0 =⇒ ρ, p = 1

2

(φ2 ± V (φ)

)I p/ρ→ −1 if 1

2 φ2 � V (φ)

I If additionally φI small =⇒ slow-roll approximation.

Quantum Perturbations

I φI (t, x) = φI (t) + Q I (t, x)

I Expand inflationary action, get equation of motion:

Q I + 3HQ I +

[k2

a2δIJ + M I

J

]QJ = 0

MIJ ≡ V,IJ − a−3∂t

[a3

HφI φJ

]

Adiabatic and Entropic modes

I Define an orthonormal basis {e(1)I , . . . , e(N)I } from

φI(n) ≡ ∂(n−1)t φI

Qσ ≡ e(1)I Q I , R ≡ H√

ΣI (φI )2Qσ

Qs ≡ e(2)I Q I

I In multifield inflation, adiabatic modes can be sourced by

entropic modes. Define η⊥ ≡−V,I e IsHσ

I Equation of motion for Qσ couples to Qs , giving superhorizonevolution of R

=⇒ R ≈ 2H2√

ΣI (φI )2η⊥Qs , k � aH

δN-formalism

I If all modes are adiabatic, then ζ ∼ R outsize horizon

I ζ(tc , x) ' N(tc , t∗, x)− N(tc , t∗), where N ≡∫ c∗ H dt

I δN(tc , t∗, x) = N,I δφI∗ + 1

2N,IJδφI∗δφ

J∗

I Write power spectrum in terms of δφI spectrum (≡ Q I )

〈ζk1ζk2〉 ≡ (2π)3δ(3)(k1 + k2)2π2

k31Pζ(k1), and Pδφ∗ ≡

H2∗

4π2

Pζ = ΣIN2,IPδφ∗

I Get spectral index as

nζ − 1 ≡d lnPζd ln k

= −2ε+2N,IN,IH N,JN,J

δN-formalism

I If all modes are adiabatic, then ζ ∼ R outsize horizon

I ζ(tc , x) ' N(tc , t∗, x)− N(tc , t∗), where N ≡∫ c∗ H dt

I δN(tc , t∗, x) = N,I δφI∗ + 1

2N,IJδφI∗δφ

J∗

I Write power spectrum in terms of δφI spectrum (≡ Q I )

〈ζk1ζk2〉 ≡ (2π)3δ(3)(k1 + k2)2π2

k31Pζ(k1), and Pδφ∗ ≡

H2∗

4π2

Pζ = ΣIN2,IPδφ∗

I Get spectral index as

nζ − 1 ≡d lnPζd ln k

= −2ε+2N,IN,IH N,JN,J

Observables

I (*) Ne ≡∫d ln a =

∫H dt

I (*) nsI Spectral running αs ≡ d ln ns

d ln k

I nt , r ≡ AtAs

I fNL (related to n-point functions of fields, n > 2)

Random Matrices

I Gaussian Orthogonal Ensemble: vab symmetric, i.i.d. w/variance 〈v2ab〉 = (1 + δab)σ2.

I Also require P(M) = P(UDU−1) = P(D)

I Eigenvalue distribution

P({λ1, . . . , λn}) = CN

N∏j=1

e−λ2j2

∏j 6=i

|λi − λj |

= CNe

−βH, β ≡ 1

H ≡∑j

λ2j2−∑j 6=i

log |λi − λj |

=⇒ P(λj) =1

πNσ2

√2Nσ2 − λ2j

Random Matrices

I Gaussian Orthogonal Ensemble: vab symmetric, i.i.d. w/variance 〈v2ab〉 = (1 + δab)σ2.

I Also require P(M) = P(UDU−1) = P(D)

I Eigenvalue distribution

P({λ1, . . . , λn}) = CN

N∏j=1

e−λ2j2

∏j 6=i

|λi − λj |

= CNe

−βH, β ≡ 1

H ≡∑j

λ2j2−∑j 6=i

log |λi − λj |

=⇒ P(λj) =1

πNσ2

√2Nσ2 − λ2j

Dyson Brownian Motion

I Dyson (1962) - This story is equivalent to particles movingunder Brownian Motion.

〈δvab〉 = −vab(s0)δs

Λh

〈(δvab)2〉 = (1 + δab)δs

Λhσ2

I 1304.3559 MMPW – Basic idea: identify the Hessian of ourpotential as a random matrix and evolve via DBM alonginflationary trajectory.

DBM Potentials (1304.3559)

I V = Λ4v

√Nf

(v0 + vaφ

a + 12vabφ

aφb)

, where φa ≡ φa/Λh

I Start with the Hessian out of equilibrium and relax to GOE

v0|p1 = v0|p0 + va|p0δφa

va|p1 = va|p0 + vab|p0δφb

vab|p1 = vab|p0 + δvab

I Points in field space separated by more than Λh should haveindependent GOE Hessians.

DBM potential evolution over 4Λh for Nf = 2

Eigenvalue distribution of Hessian

Effective masses during inflation

Our Model

I Few scenarios when inflation can occur – start near criticalpoints for now.

||va|| � 1

|min(Eig(vab))| � 1

I For now, assume gradient descent condition φI =−V,I3H

I Compute ns via δN-formalism

I Check entropic modes by computing η⊥ (identically zero withgradient-descent)

Freivogel ns (effective single-field model)

Our spectral index

Our spectral index, 24 runs, Nf = 20

Our spectral index, 10 runs, Nf = 50

Obstacles for DBM Potentials

I Hessians can’t be in GOE (Masoumi)I ∂iHjk = ∂jHik = ∂kHij

I Expect average number of minima 2−N of critical points,

but a random Hessian gives e−0.27N2.

I Morse theory gives Nmin − Nsaddle + Nmax = χ(M)

I Small field displacements (1608.00041 Freivogel et al.)

I Jumping Hessian could create non-Gaussianities.(1409.5135 Battefeld et al.)

Dias et al. (1604.05970) method

I Use techniques applicable to sum-separable potentials(Battefeld, Easther astro-ph/0610296) by rotating Hessian ateach timestep.

I Use transport method to propagate perturbations from onetime to the next.

I This allows us to compute the power spectrum directly =⇒full access to evolution of perturbations, entropic to adiabaticconversion

Dias et al. (1604.05970) method

I Use techniques applicable to sum-separable potentials(Battefeld, Easther astro-ph/0610296) by rotating Hessian ateach timestep.

I Use transport method to propagate perturbations from onetime to the next.

I This allows us to compute the power spectrum directly =⇒full access to evolution of perturbations, entropic to adiabaticconversion

Dias et al. (1604.05970) method

I Use techniques applicable to sum-separable potentials(Battefeld, Easther astro-ph/0610296) by rotating Hessian ateach timestep.

I Use transport method to propagate perturbations from onetime to the next.

I This allows us to compute the power spectrum directly =⇒full access to evolution of perturbations, entropic to adiabaticconversion

Dias et al. method

I Propagators from one patch to the next given by

δϕa∣∣pi+1

= Γab(pi+1, pi )δϕ

b|pi , with Γab ≡

∂φa|pi+1

∂φb|piI Full perturbation evolution given by path-ordered product of

propagators and orthogonal transformations

δ~φ∣∣pf

= OTpf

Γ(pf , pf−1)Opf . . .OTp1Γ(p1, p0)Op1 δ

~φ∣∣p0

≡ Γ(pf , p0) δ~φ∣∣p0

I ζ = Naδϕa

I Usual procedure to get ns , apply sum-separable potentialtechniques.

Dias et al. ns

Summary

I Preliminary results for ns close to Dias et al. , seem to be inmild tension with Freivogel et al.

I OutlookI Try to use Dias et al. method.I Drop slow-roll.I Improvements to potential.

Questions?

Backup Slides

Single-field Quantum Perturbations

I φ(τ, x) = φ(τ) + f (τ, x)/a(τ)

I Expand inflationary action to quadratic order, getMukhanov-Sasaki equation:

f ′′k +

(k2 − a′′

a

)fk = 0

I Get quantum statistics for the operatorf (τ, x) =

∫d3k

(2π)3/2(fk(τ)ak + f ∗k (τ)a†k)e ik·x

PR ≡1

2εMPl2Pδφ(k , τ) =

a−2

2εMPl2

d

d ln k〈|f 2|〉

≡ As

(k

k∗

)ns−1

Reheating

I After the end of inflation, the inflaton fields are veryenergetic, many tachyonic =⇒ particle production.

I To satisfy observational constraints (BBN), expect reheattemperature ≥ 5 MeV

I Multifield reheating model-dependent in general. Adiabaticmodes should survive, but entropic modes or non-gaussianitiescould decay (Leung et al. 1206.5196).

εV ≡M2

Pl

2V 2ΣIV

2,I

ηV ≡ M2Pl

ΣIJV,IV,IJV,JVΣIV

2,I