calculating non-adiabatic pressure perturbations during multi-field inflation
DESCRIPTION
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 .TRANSCRIPT
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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
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Adiabatic evolution
δX
X=δY
Y
I Generalised form of fluid adiabaticityI Small changes in one component are rapidly
reflected in others
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Adiabatic evolution
δP
P=δρ
ρ
I Generalised form of fluid adiabaticityI Small changes in one component are rapidly
reflected in others
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Non-adiabatic Pressure
δP = (P /ρ)︸ ︷︷ ︸c2s
δρ + . . .
δPnad = δP − c2sδρ
Comoving entropy perturbation:
S =H
PδPnad
Gordon et al 2001, Malik & Wands 2005
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Non-adiabatic Pressure
δP = (P /ρ)︸ ︷︷ ︸c2s
δρ + . . .
δPnad = δP − c2sδρ
Comoving entropy perturbation:
S =H
PδPnad
Gordon et al 2001, Malik & Wands 2005
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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.
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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.
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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.
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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
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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
. . .
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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
. . .
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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)
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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)
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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
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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
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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
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Double Quadratic: δP, δPnad
0102030405060Nend −N
10−55
10−49
10−43
10−37
10−31
10−25
10−19
k3PδP /(2π2)
k3PδPnad/(2π2)
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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)
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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
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Hybrid Quartic: R,S, S
01020304050Nend −N
10−22
10−18
10−14
10−10
10−6
k3PR/(2π2)
k3PS/(2π2)
k3PS/(2π2)
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Hybrid Quartic: last 5 efolds
012345Nend −N
10−22
10−18
10−14
10−10k3PR/(2π2)
k3PS/(2π2)
k3PS/(2π2)
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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)
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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
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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)
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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.
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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
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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