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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