Calculating Non-adiabatic Pressure  Perturbations during Multi-field             Inflation                Ian Huston        ...
Adiabatic evolution                 δX δY                    =                  ˙                  X   ˙                  ...
Adiabatic evolution                  δP    δρ                      =                   P˙    ρ                         ˙  ...
Non-adiabatic Pressure                 ˙ ˙          δP = (P /ρ) δρ + . . .                     c2                      s  ...
Non-adiabatic Pressure                 ˙ ˙          δP = (P /ρ) δρ + . . .                     c2                      s  ...
Motivations Many interesting effects when not purely adiabatic:     More interesting dynamics in larger phase space.     N...
Motivations Many interesting effects when not purely adiabatic:     More interesting dynamics in larger phase space.     N...
Motivations Many interesting effects when not purely adiabatic:     More interesting dynamics in larger phase space.     N...
Vorticity generation Vorticity can be sourced at second order from non-adiabatic pressure:        ω2ij − Hω2ij ∝ δρ,[j δPn...
ζ is not always conserved    ˙ = −H δPnad − Shear term    ζ           ρ+P   Need to prescribe reheating dynamics   Need to...
ζ is not always conserved    ˙ = −H δPnad − Shear term    ζ           ρ+P   Need to prescribe reheating dynamics   Need to...
Multi-field Inflation Two field systems:                    1 2            L=        ϕ + χ2 + V (ϕ, χ)                      ˙...
Multi-field Inflation Two field systems:                     1 2            L=         ϕ + χ2 + V (ϕ, χ)                     ...
Other decompositions Popular to rotate into “adiabatic” and “isocurvature” directions:                δσ = + cos θδϕ + sin...
Numerical Results   Three different potentials   Check adiabatic and non-adiabatic   perturbations   Compare S and S evolu...
Double Quadratic                         1       1               V (ϕ, χ) = m2 ϕ2 + m2 χ2                           ϕ     ...
Double Quadratic: δP, δPnad   10−19                                     k3 PδP /(2π 2 )   10−25                           ...
Double Quadratic: R, S, S    10−7    10−9   10−11   10−13                                        k3 PR /(2π 2 )        −15...
Hybrid Quartic                                   2                            χ2             ϕ2 2ϕ2 χ2       V (ϕ, χ) = Λ4...
Hybrid Quartic: R, S, S   10−6   10−10   10−14   10−18            k3 PR /(2π 2 )   10−22    k3 PS /(2π 2 )            k3 P...
Hybrid Quartic: last 5 efolds               k3 PR /(2π 2 )   10−10       k3 PS /(2π 2 )               k3 PS /(2π 2 )   10−...
Hybrid Quartic: end of inflation   10−8   10−10   10−12                            k3 PR /(2π 2 )   10−14                  ...
Product Exponential                                            2                V (ϕ, χ) = V0 ϕ2 e−λχ                     ...
Product exponential: δP, δPnad   10−26                                           k3 PδP /(2π 2 )        −28   10          ...
Outcomes and FutureDirections   Different evolution of δPnad and δs is clear (S vs S).   Scale dependence of S for these m...
Reproducibility    Download Pyflation at http://pyflation.ianhuston.net Code is also available as a git repository: $ git cl...
Summary  Non-adiabatic perturbations can change curvature  perturbations & source vorticity  Performed a non slow-roll cal...
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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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Transcript of "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
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