Calculating Non-adiabatic Pressure Perturbations during Multi-field Inflation
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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.
![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](https://image.slidesharecdn.com/talk-120328101424-phpapp01/85/Calculating-Non-adiabatic-Pressure-Perturbations-during-Multi-field-Inflation-9-320.jpg)
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) ˙ ˙
- 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