Critical tests of inflation, MPA, Garching 6th November 2012 Curvaton! a worked example of local-type non-Gaussianity" David Wands Institute of Cosmology and Gravitation, University of Portsmouth Q: what is the curvaton? A: a light weakly-coupled scalar field Polonyi / moduli problem: - supersymmetric theories have many light weakly-coupled scalar fields - after inflation they are expected to be displaced from their true vacuum - they begin oscillating about true minimum when Hubble rate drops below effective mass of the field, H < m - oscillating field has (time-averaged) equation of state, P = 0 - hence density grows relative to radiation / photons - leads to early matter domination, incompatible with standard cosmology ln (ρ) -4/3 ργ ∝ V SOLUTION: -1 ρχ ∝ V decay ln (V) V(χ) curvaton scenario: Linde & Mukhanov 1997; Enqvist & Sloth, Lyth & Wands, Moroi & Takahashi 2001 χ - light field, m<<H, during inflation acquires almost scale-invariant, Gaussian distribution of field fluctuations δχ on large scales 2 2 - quadratic energy density for free field, ρχ=m χ /2 - spectrum of initially isocurvature density perturbations 1 δρ 2 δχ ζ ≈ χ ≈ χ 3 ρ 3 χ χ - transferred to radiation when curvaton decays with some efficiency, 0<r<1, where r ≈ Ωχ,decay 2r δχ ζ = rζ ≈ χ 3 χ non-adiabac field fluctuaons δχ inflaton !δχ δχ 2 $ ζ ~ r# + +...& " χ χ 2 % 18/2/2008 David Wands 4 non-linear curvature perturbation Lyth, Malik & Sasaki (2005), ρ dρ' ζ = δN + but see also Lyth & Wands (2003), ∫ρ 3[ρ'+P(ρ')] Rigopoulos, Shellard & van Tent (2003) Langlois & Vernizzi (2005) where, metric 2 2 2δN i j density ds = a e γijdx dx , ρ = ρ +δρ Hence N = curvature perturbation on uniform-density slices ζ = δ ρ=ρ 1 !ρ $ 3(1+w)ζ = ln# & ⇒ ρ = ρ e 3(1+ w) "ρ %δN=0 = density perturbation on uniform-curvature ( δN=0 ) slices curvaton perturbation Sasaki, Valiviita & Wands (2006) 1 ⎛ ρ ⎞ ζ = ln ⎜ χ ⎟ χ 3 ⎜ ρ ⎟ ⎝ χ ⎠δN =0 2 where curvaton density 1 2 2 3 m ζ χ χ ρχ = χ ⇒ e = 2 χ 2 expand order by order δ χ 3ζ = 2 1 χ1 χ 2 2 δ χ ⎛ δ χ ⎞ 3 2 (χ ) 5 3 9 2 2 1 ⇒ ζ = − ζ ⇔ f = − ζ χ 2 + ζ χ1 = + ⎜ ⎟ 2 1 NL χ ⎝ χ ⎠ 2 4 for purely Gaussian δχ sudden-decay approximation: Hdecay=Γ ζ on uniform-total-density hypersurface before curvaton decay: 4(ζ r −ζ ) 3(ζ χ −ζ ) ρr + ρχ = ρre + ρχ e = constant 4(ζ r −ζ ) 3(ζ χ −ζ ) ⇒ (1− Ωχ )e + Ωχ e =1 Sasaki, Valiviita & Wands (2006) expand order by order (and assuming, for simplicity, ζr=0) (1−Ωχ ) +Ωχ =1 4(1−Ωχ )ζ1 = 3Ωχ (ζ χ1 −ζ1) 2 2 4(1−Ωχ )ζ2 −16(1−Ωχ ) ζ1 = 3Ωχ (ζ χ 2 −ζ2 ) + 9Ωχ (ζ χ1 −ζ1) 4(1−Ωχ )ζ3 +... simplest quadratic curvaton: V=m2χ2/2 • first-order perturbations # & 3Ωχ ζ = r ζ χ where r = % ( ≤1 4 −Ω $% χ '(decay • second-order perturbations " 3 % 5 5 5r 5 ζ = − 2 − r ζ 2 ⇒ f = − − ≥ − 2 #$2r &' 1 NL 4r 3 6 4 • third-order perturbations 2 3 " 9 1 2 % 3 25" r 10r r % 9 ζ3 = $− + +10r + 3r 'ζ1 ⇒ gNL = − $1− − − ' ≤ # r 2 & 6r # 18 9 3 & 2 • Predictions of the simplest quadratic curvaton model: • for r ≈ 1 5 9 f = − , g = NL 4 NL 2 • for r << 1 36 10 f >>1 , τ = f 2 >> g = − f NL NL 25 NL NL 3 NL single source obeys Suyama-Yamaguchi equality Sasaki, Valiviita & Wands (2006) non-linearity parameter see also Malik & Lyth (2006) c.f. exact (numerical) calculation rd 3 order non-linearity for curvaton Sasaki, Valiviita & Wands (astro-ph/0607627) full pdf for ζ from δN Sasaki, Valiviita & Wands (2006) probability distribution for ζ fNL bounds on curvaton parameters Fonseca & Wands (2011) see also Nakayama et al (2010) David Wands 13 fNL + rGW bounds on curvaton parameters Fonseca & Wands (2011) see also Nakayama et al (2010) Stochas(c VEV for curvaton: χ ~ H2 / m 4 => rGW > ( fNL / 518 ) Huang (2008) David Wands 14 Enqvist & Nurmi (2005) self-interacting curvaton Huang (2008) Enqvist et al (2009) n … 1 2 2 4 ! χ $ V (χ) = m χ + m # & 2 " f % assume curvaton is Gaussian at Hubble exit during inflation, but allow for non-linear evolution on large scales before oscillating about quadratic minimum 1 2 χosc = g(χ* ) ⇒ χosc = g + g'δχ + g''(δχ ) +... 2 nG from self-interacting curvaton • second-order perturbations 5 ! g''g $ 5 5r fNL = # +1&− − 4r " g'2 % 3 6 • third-order perturbations 2 9 ! g'''g g''g$ 9 ! g''g $ 1 ! g''g $ 2 gNL = # + 3 &− # +1&− #9 −1&+10r + 3r 4r2 " g'3 g'2 % r " g'2 % 2 " g'2 % • Predictions of the simplest quadratic curvaton model: • for r ≈ 1 5 " g''g % 9 " g'''g2 1 g''g % fNL = $ −1' , gNL = $ − + 2' 4 g'2 4 g'3 3 g'2 # & # & • for r << 1 2 5 ! g''g $ tree 36 2 9 ! g'''g g''g$ fNL = # +1& , τ NL = fNL ~ gNL = # + 3 & 4r " g'2 % 25 4r2 " g'3 g'2 % 2 for gNL >> fNL => loop corrections to τNL violating SY equality… an aside: loop correc(ons to SY equality 2 36 2 ' 81g ! k $* loop(k) f loop 1 NLP ln τ NL = ( NL ) ) + 2 # &, 25 ( 25 fNL " kIR %+ 2 for |gNL| >> fNL loop correc(ons violate SY equality (does not hold on all scales) -> generalised equality: Tasinato, Byrnes, Nurmi & Wands (2012) David Wands 17 Byrnes, Choi & Hall 2009 Khoury & Piazza 2009 scale-dependence of fNL? Sefusa, Liguori, Yadav, Jackson & Pajer 2009 Byrnes, Nurmi, Tasinato & Wands (2009); Byrnes, Gerstenlauer, Nurmi, Tasinato & Wands (2010) Ø power spectrum P (k) Nʹ2 P ζ = [ δϕ ]k=aH ⇒ scale-dependence d ln P d ln Nʹ2 n −1 ≡ ζ = H −1 − 2ε ζ d lnk dt Ø bispectrum 5 Nʹʹ f (k) ⎡ ⎤ NL = ⎢ 2 ⎥ 6 ⎣ Nʹ ⎦k=aH ⇒ scale-dependence d ln f NL Nʹ ⎛ V ʹʹʹ ⎞ n fNL ≡ = ⎜ 2ε (4ε − 3η)+ 2 ⎟ d lnk Nʹʹ ⎝ 3H ⎠ Ø e.g., self-int. curvaton Nʹ ⎛ V ʹʹʹ ⎞ n fNL = ⎜ 2 ⎟ Nʹʹ ⎝ 3H ⎠ scale-dependence probes self-interaction, not probed by power spectrum could be observable for curvaton models where gNL ∼ τNL (Byrnes et al 2011) scale-dependent fNL from curvaton + inflaton Byrnes, Nurmi, Tasinato & Wands (2009) 3 ζ (x) = ζ (x) +ζ (x) + f ζ 2 (x) ϕ χ 5 χχ χ Ø power spectrum Pζ (k) = Pζϕ (k) + Pζχ (k) Pζϕ (k) Pζχ (k) ln k Ø bispectrum Bζ (k) = Bζχ (k) Bζ (k) → f NL (k) ≈ 2 (Pζ (k)) Ø but gNL small ln k Conclusions: • Curvaton provides a simple (natural) model for large primordial non-Gaussianity • Quadratic (non-interacting) curvaton well-described by simplest local fNL • Many variants • self-interactions, inflaton+curvaton, multi-curvaton… offer wider range of observables • nfNL • large gNL • inhomogeneous non-Gaussianity • More precise data allows us to study more detailed models of inflation and origin of cosmological perturbations .