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This talk was given at the March 2012 UK Cosmology meeting at the University of Sussex. It describes work done in collaboration with Adam Christopherson published in Physical Review D and available of the arXiv at http://arxiv.org/abs/1111.6919 .
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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 [email protected]: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