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# Calculating Non-adiabatic Pressure Perturbations during Multi-field Inflation

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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 Pressure Perturbations during Multi-field Inflation

1. 1. Calculating Non-adiabatic Pressure Perturbations during Multi-ﬁeld Inﬂation Ian Huston Astronomy Unit, Queen Mary, University of LondonIH, A Christopherson, arXiv:1111.6919 (PRD85 063507) Software available at http://pyﬂation.ianhuston.net
2. 2. Adiabatic evolution δX δY = ˙ X ˙ Y Generalised form of ﬂuid adiabaticity Small changes in one component are rapidly reﬂected in others
3. 3. Adiabatic evolution δP δρ = P˙ ρ ˙ Generalised form of ﬂuid adiabaticity Small changes in one component are rapidly reﬂected in others
4. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 15. Numerical Results Three different potentials Check adiabatic and non-adiabatic perturbations Compare S and S evolution Consider isocurvature at end of inﬂation
16. 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. 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. 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. 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. 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. 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. 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. 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. 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. 25. Outcomes and FutureDirections 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. 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. 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