The document discusses the effects of non-adiabatic pressure perturbations during multi-field inflation, highlighting their impact on curvature perturbations and vorticity generation. It outlines the motivations for exploring non-adiabatic dynamics and reviews calculations and numerical results related to different inflationary models. The study includes software tools for further analysis and emphasizes caution in making predictions involving isocurvature modes at the end of 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)
˙ ˙

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