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 Pressure Perturbations during Multi-field Inflation
1. Calculating Non-adiabatic Pressure
Perturbations during Multi-field
Inflation
Ian Huston
Astronomy Unit, Queen Mary, University of London
IH, A Christopherson, arXiv:1111.6919 (PRD85 063507)
Software available at http://pyflation.ianhuston.net
2. Adiabatic evolution
δX δY
=
˙
X ˙
Y
Generalised form of fluid adiabaticity
Small changes in one component are rapidly
reflected in others
3. Adiabatic evolution
δP δρ
=
P˙ ρ
˙
Generalised form of fluid adiabaticity
Small changes in one component are rapidly
reflected in others
4. Non-adiabatic Pressure
˙ ˙
δP = (P /ρ) δρ + . . .
c2
s
δPnad = δP − c2δρ
s
Comoving entropy perturbation:
H
S= δP
P˙ nad
Gordon et al 2001, Malik & Wands 2005
5. Non-adiabatic Pressure
˙ ˙
δP = (P /ρ) δρ + . . .
c2
s
δPnad = δP − c2δρ
s
Comoving entropy perturbation:
H
S= δP
P˙ nad
Gordon et al 2001, Malik & Wands 2005
6. Motivations
Many interesting effects when not purely adiabatic:
More interesting dynamics in larger phase space.
Non-adiabatic perturbations can source vorticity.
Presence of non-adiabatic modes can affect
predictions of models through change in curvature
perturbations.
7. Motivations
Many interesting effects when not purely adiabatic:
More interesting dynamics in larger phase space.
Non-adiabatic perturbations can source vorticity.
Presence of non-adiabatic modes can affect
predictions of models through change in curvature
perturbations.
8. Motivations
Many interesting effects when not purely adiabatic:
More interesting dynamics in larger phase space.
Non-adiabatic perturbations can source vorticity.
Presence of non-adiabatic modes can affect
predictions of models through change in curvature
perturbations.
9. Vorticity generation
Vorticity can be sourced at second order from
non-adiabatic pressure:
ω2ij − Hω2ij ∝ δρ,[j δPnad,i]
˙
⇒ Vorticity can then source B-mode polarisation and/or
magnetic fields.
⇒ Possibly detectable in CMB.
Christopherson, Malik & Matravers 2009, 2011
10. ζ is not always conserved
˙ = −H δPnad − Shear term
ζ
ρ+P
Need to prescribe reheating dynamics
Need to follow evolution of ζ during radiation & matter
phases
Bardeen 1980
Garcia-Bellido & Wands 1996
Wands et al. 2000
Rigopoulos & Shellard 2003
...
11. ζ is not always conserved
˙ = −H δPnad − Shear term
ζ
ρ+P
Need to prescribe reheating dynamics
Need to follow evolution of ζ during radiation & matter
phases
Bardeen 1980
Garcia-Bellido & Wands 1996
Wands et al. 2000
Rigopoulos & Shellard 2003
...
12. Multi-field Inflation
Two field systems:
1 2
L= ϕ + χ2 + V (ϕ, χ)
˙ ˙
2
Energy density perturbation
δρ = ˙ ˙
ϕα δϕα − ϕ2 φ + V,α δϕα
˙α
α
where
Hφ = 4πG(ϕδϕ + χδχ)
˙ ˙
14. Other decompositions
Popular to rotate into “adiabatic” and “isocurvature”
directions:
δσ = + cos θδϕ + sin θδχ
δs = − sin θδϕ + cos θδχ
H
Can consider second entropy perturbation S = δs
σ˙
H
and compare with S = δPnad
P˙
Gordon et al 2001
Discussions in Saffin 2012, Mazumdar & Wang 2012
15. Numerical Results
Three different potentials
Check adiabatic and non-adiabatic
perturbations
Compare S and S evolution
Consider isocurvature at end of inflation
16. Double Quadratic
1 1
V (ϕ, χ) = m2 ϕ2 + m2 χ2
ϕ
2 2 χ
Parameters: mχ = 7mϕ
Normalisation: mϕ = 1.395 × 10−6 MPL
Initial values: ϕ0 = χ0 = 12MPL
At end of inflation nR = 0.937 (no running allowed)
Recent discussions: Lalak et al 2007, Avgoustidis et al 2012
23. Product Exponential
2
V (ϕ, χ) = V0 ϕ2 e−λχ
2
Parameter: λ = 0.05/MPL
Normalisation: V0 = 5.37 × 10−13 MPL
2
Initial values: ϕ0 = 18MPL and χ0 = 0.001MPL
At end of inflation nR = 0.794 (no running allowed)
Recent discussions: Byrnes et al 2008, Elliston et al 2011,
Dias & Seery 2012
25. Outcomes and Future
Directions
Different evolution of δPnad and δs is clear (S vs S).
Scale dependence of S for these models follows nR .
Need to be careful about making “predictions” when
large isocurvature fraction at end of inflation.
Follow isocurvature through reheating for multi-field
models to match requirements from CMB.
26. Reproducibility
Download Pyflation at http://pyflation.ianhuston.net
Code is also available as a git repository:
$ git clone git@bitbucket.org:ihuston/pyflation.git
Open Source (2-clause BSD license)
Documentation for each function
Can submit any changes to be added
Sign up for the ScienceCodeManifesto.org
27. Summary
Non-adiabatic perturbations can change curvature
perturbations & source vorticity
Performed a non slow-roll calculation of δPnad
Showed difference in evolution with δs
parametrisation, especially at late times
arXiv:1111.6919 now in Phys Rev D85, 063507
Download code from http://pyflation.ianhuston.net