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 .

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)
˙ ˙

13. Multi-field Inflation
Two field systems:
1 2
L= ϕ + χ2 + V (ϕ, χ)
˙ ˙
2
Pressure perturbation
δP = ˙ ˙
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

17. Double Quadratic: δP, δPnad
10−19
k3 PδP /(2π 2 )
10−25 k3 PδPnad /(2π 2 )
10−31
10−37
10−43
10−49
10−55
60 50 40 30 20 10 0
Nend − N

18. Double Quadratic: R, S, S
10−7
10−9
10−11
10−13
k3 PR /(2π 2 )
−15
10 k3 PS /(2π 2 )
k3 PS /(2π 2 )
10−17
60 50 40 30 20 10 0
Nend − N

19. Hybrid Quartic
2
χ2 ϕ2 2ϕ2 χ2
V (ϕ, χ) = Λ4 1− 2 + + 2 2
v µ2 ϕc v
Parameters: v = 0.10MPL , ϕc = 0.01MPL , µ = 103 MPL
Normalisation: Λ = 2.36 × 10−4 MPL
Initial values: ϕ0 = 0.01MPL and χ0 = 1.63 × 10−9 MPL
At end of inflation nR = 0.932 (no running allowed)
Recent discussions: Kodama et al 2011, Avgoustidis et al 2012

20. Hybrid Quartic: R, S, S
10−6
10−10
10−14
10−18
k3 PR /(2π 2 )
10−22 k3 PS /(2π 2 )
k3 PS /(2π 2 )
50 40 30 20 10 0
Nend − N

21. Hybrid Quartic: last 5 efolds
k3 PR /(2π 2 )
10−10 k3 PS /(2π 2 )
k3 PS /(2π 2 )
10−14
10−18
10−22
5 4 3 2 1 0
Nend − N

22. Hybrid Quartic: end of inflation
10−8
10−10
10−12
k3 PR /(2π 2 )
10−14 k3 PS /(2π 2 )
k3 PS /(2π 2 )
10−16
10−3 10−2 10−1
k/Mpc−1

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

24. Product exponential: δP, δPnad
10−26
k3 PδP /(2π 2 )
−28
10 k3 PδPnad /(2π 2 )
10−30
10−32
10−34
10−36
10−38
10−40
60 50 40 30 20 10 0
Nend − N

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