Calculating Non-adiabatic Pressure Perturbations during Multi-field Inflation

Calculating Non-adiabatic PressurePerturbations during Multi-field

Inflation

Ian HustonAstronomy Unit, Queen Mary, University of London

IH, A Christopherson, arXiv:1111.6919 (PRD85 063507)Software available at http://pyflation.ianhuston.net

Adiabatic evolution

δX

X=δY

Y

I Generalised form of fluid adiabaticityI Small changes in one component are rapidly

reflected in others

Adiabatic evolution

δP

P=δρ

ρ

I Generalised form of fluid adiabaticityI Small changes in one component are rapidly

reflected in others

Non-adiabatic Pressure

δP = (P /ρ)︸ ︷︷ ︸c2s

δρ + . . .

δPnad = δP − c2sδρ

Comoving entropy perturbation:

S =H

PδPnad

Gordon et al 2001, Malik & Wands 2005

Non-adiabatic Pressure

δP = (P /ρ)︸ ︷︷ ︸c2s

δρ + . . .

δPnad = δP − c2sδρ

Comoving entropy perturbation:

S =H

PδPnad

Gordon et al 2001, Malik & Wands 2005

Motivations

Many interesting effects when not purely adiabatic:

I More interesting dynamics in larger phase space.

I Non-adiabatic perturbations can source vorticity.

I Presence of non-adiabatic modes can affectpredictions of models through change in curvatureperturbations.

Motivations

Many interesting effects when not purely adiabatic:

I More interesting dynamics in larger phase space.

I Non-adiabatic perturbations can source vorticity.

I Presence of non-adiabatic modes can affectpredictions of models through change in curvatureperturbations.

Motivations

Many interesting effects when not purely adiabatic:

I More interesting dynamics in larger phase space.

I Non-adiabatic perturbations can source vorticity.

I Presence of non-adiabatic modes can affectpredictions of models through change in curvatureperturbations.

Vorticity generationVorticity can be sourced at second order fromnon-adiabatic pressure:

ω2ij −Hω2ij ∝ δρ,[jδPnad,i]

⇒ Vorticity can then source B-mode polarisation and/ormagnetic fields.

⇒ Possibly detectable in CMB.

Christopherson, Malik & Matravers 2009, 2011

ζ is not always conserved

ζ = −H δPnad

ρ + P− Shear term

I Need to prescribe reheating dynamicsI Need to follow evolution of ζ during radiation & matter

phases

Bardeen 1980Garcia-Bellido & Wands 1996

Wands et al. 2000Rigopoulos & Shellard 2003

. . .

ζ is not always conserved

ζ = −H δPnad

ρ + P− Shear term

I Need to prescribe reheating dynamicsI Need to follow evolution of ζ during radiation & matter

phases

Bardeen 1980Garcia-Bellido & Wands 1996

Wands et al. 2000Rigopoulos & Shellard 2003

. . .

Multi-field InflationTwo field systems:

L =1

2

(ϕ2 + χ2

)+ V (ϕ, χ)

Energy density perturbation

δρ =∑

α

(ϕα ˙δϕα − ϕ2

αφ+ V,αδϕα

)

whereHφ = 4πG(ϕδϕ+ χδχ)

Multi-field InflationTwo field systems:

L =1

2

(ϕ2 + χ2

)+ V (ϕ, χ)

Pressure perturbation

δP =∑

α

(ϕα ˙δϕα − ϕ2

αφ−V,αδϕα)

whereHφ = 4πG(ϕδϕ+ χδχ)

Other decompositionsPopular to rotate into “adiabatic” and “isocurvature”directions:

δσ = + cos θδϕ+ sin θδχ

δs = − sin θδϕ+ cos θδχ

Can consider second entropy perturbation S =H

σδs

and compare with S =H

PδPnad

Gordon et al 2001Discussions in Saffin 2012, Mazumdar & Wang 2012

Numerical Results

I Three different potentials

I Check adiabatic and non-adiabaticperturbations

I Compare S and S evolution

I Consider isocurvature at end of inflation

Double Quadratic

V (ϕ, χ) =1

2m2ϕϕ

2 +1

2m2χχ

2

I Parameters: mχ = 7mϕ

I Normalisation: mϕ = 1.395× 10−6MPL

I Initial values: ϕ0 = χ0 = 12MPL

I At end of inflation nR = 0.937 (no running allowed)

Recent discussions: Lalak et al 2007, Avgoustidis et al 2012

Double Quadratic: δP, δPnad

0102030405060Nend −N

10−55

10−49

10−43

10−37

10−31

10−25

10−19

k3PδP /(2π2)

k3PδPnad/(2π2)

Double Quadratic: R,S, S

0102030405060Nend −N

10−17

10−15

10−13

10−11

10−9

10−7

k3PR/(2π2)

k3PS/(2π2)

k3PS/(2π2)

Hybrid Quartic

V (ϕ, χ) = Λ4

[(1− χ2

v2

)2

+ϕ2

µ2+

2ϕ2χ2

ϕ2cv

2

]

I Parameters: v = 0.10MPL, ϕc = 0.01MPL, µ = 103MPL

I Normalisation: Λ = 2.36× 10−4MPL

I Initial values: ϕ0 = 0.01MPL and χ0 = 1.63× 10−9MPL

I At end of inflation nR = 0.932 (no running allowed)

Recent discussions: Kodama et al 2011, Avgoustidis et al 2012

Hybrid Quartic: R,S, S

01020304050Nend −N

10−22

10−18

10−14

10−10

10−6

k3PR/(2π2)

k3PS/(2π2)

k3PS/(2π2)

Hybrid Quartic: last 5 efolds

012345Nend −N

10−22

10−18

10−14

10−10k3PR/(2π2)

k3PS/(2π2)

k3PS/(2π2)

Hybrid Quartic: end of inflation

10−3 10−2 10−1

k/Mpc−1

10−16

10−14

10−12

10−10

10−8

k3PR/(2π2)

k3PS/(2π2)

k3PS/(2π2)

Product Exponential

V (ϕ, χ) = V0ϕ2e−λχ

2

I Parameter: λ = 0.05/M2PL

I Normalisation: V0 = 5.37× 10−13M2PL

I Initial values: ϕ0 = 18MPL and χ0 = 0.001MPL

I At end of inflation nR = 0.794 (no running allowed)

Recent discussions: Byrnes et al 2008, Elliston et al 2011,Dias & Seery 2012

Product exponential: δP, δPnad

0102030405060Nend −N

10−40

10−38

10−36

10−34

10−32

10−30

10−28

10−26

k3PδP /(2π2)

k3PδPnad/(2π2)

Outcomes and FutureDirections

I Different evolution of δPnad and δs is clear (S vs S).

I Scale dependence of S for these models follows nR.

I Need to be careful about making “predictions” whenlarge isocurvature fraction at end of inflation.

I Follow isocurvature through reheating for multi-fieldmodels to match requirements from CMB.

Reproducibility

Download Pyflation at http://pyflation.ianhuston.net

Code is also available as a git repository:

$ git clone git@bitbucket.org:ihuston/pyflation.git

I Open Source (2-clause BSD license)I Documentation for each functionI Can submit any changes to be addedI Sign up for the ScienceCodeManifesto.org

Summary

I Non-adiabatic perturbations can change curvatureperturbations & source vorticity

I Performed a non slow-roll calculation of δPnad

I Showed difference in evolution with δsparametrisation, especially at late times

I arXiv:1111.6919 now in Phys Rev D85, 063507

I Download code from http://pyflation.ianhuston.net