Cosener s House 7 th June 003 Inflationary density perturbations David Wands Institute of Cosmology and Gravitation University of Portsmouth
outline! some motivation! Primordial Density Perturbation (and conserved quantities on large scales)! constraints on single-field models! predictions from multi-field models! conclusions
Cosmological inflation: Starobinsky (1980) Guth (1981) period of accelerated expansion in the very early universe requires negative pressure e.g. self-interacting scalar field V(φ) speculative and uncertain physics φ just the kind of peculiar cosmological behaviour we observe today
Motivation: inflation in very early universe testable through primordial perturbation spectra! radiation/matter density perturbations! + gravitational waves (we hope) gravitational instability new observational data offers precision tests of cosmological parameters and the Primordial Density Perturbation
Primordial Density Perturbation e.g., epoch of primordial nucleosynthesis cosmic fluid consists of photons, γ, neutrinos, ν, baryons, B, cold dark matter, CDM, (+quintessence?)! total density perturbation, or curvature perturbation R δρ δρ/ρ! relative density perturbations, or isocurvature pertbns S i =δ(n i /n γ )/(n i /n γ )! large-angle CMB: ( T/T) lss [ R - S m ] / 5
Conserved cosmological perturbations Lyth & Wands in preparation time t t 1 A B For every quantity, x, that obeys a local conservation equation dx dn space = y( x), e. g. & ρ = 3Hρ where dn = Hdt is the locally-defined expansion along comoving worldlines there is a conserved perturbation where perturbation δ x = x A -x B is a evaluated on hypersurfaces separated by uniform expansion N= lna m ζ δn = x m δ x y(x)
examples: (i) total energy conservation: dρ = dn H 1 & ρ = 3( ρ + for perfect fluid / adiabatic perturbations, P=P(ρ) δρ R ζ ρ = conserved 3( ρ + P) (ii) energy conservation for non-interacting perfect fluids: 1 δρi H ρ& i = 3( ρi + Pi ) where Pi = Pi ( ρi ) ζ i = 3( ρ + P ) (iii) conserved particle/quantum numbers (e.g., B, B-L, ) ni H 1 δ n& i = 3ni ζ i = 3n P) i i i
Primordial Density Perturbation (II) epoch of primordial nucleosynthesis perturbed cosmic fluid consists of photons, ζ γ, neutrinos, ζ ν, baryons, ζ B, cold dark matter, ζ CDM, (+quintessence, ζ Q ) total density perturbation, or curvature perturbation relative density perturbations, or isocurvature perturbtns R = i & ρi ζ i & ρ ( ) S = ζ ζ i 3 i γ
where do these perturbations come from?
perturbations in an FRW universe: wave equation δ & φ + 3Hδφ& δφ = Characteristic timescales for comoving wavemode k oscillation period/wavelength a / k Hubble damping time-scale H -1 small-scales k > ah under-damped oscillator large-scales k < ah over-damped oscillator ( frozen-in ) 0 vacuum comoving Hubble length H -1 / a frozen-in oscillates comoving wavelength, k -1 conformal time inflation accelerated expansion (or contraction) radiation or matter era decelerated expansion
Wilkinson Microwave Anisotropy Probe February 003 coherent oscillations in photon-baryon plasma from primordial density perturbations on super-horizon scales
Vacuum fluctuations V(φ) Hawking 8, Starobinsky 8, Guth & Pi 8 small-scale/underdamped zero-point fluctuations large-scale/overdamped perturbations in growing mode linear evolution Gaussian random field δφ 4π k 3 δφ k k = ah 3 (π ) π fluctuations of any light fields (m<3h/) `frozen-in on large scales = H δφ k φ e ik η k *** assumes Bunch-Davies vacuum on small scales *** all modes start sub-planck length for k/a > M Pl Niemeyer; Brandenberger & Martin (000) effect likely to be small for H << M Pl Starobinsky; Niemeyer; Easther et al ; Kaloper et al (00)
Inflaton -> matter perturbations R = for adiabatic perturbations on super-horizon scales R & = 0 during inflation scalar field fluctuation, δφ scalar curvature on uniform-field hypersurfaces Hδσ & σ during matter+radiation era density perturbation, δρ time scalar curvature on uniform-density hypersurfaces R = Hδρ & ρ ζ ζ = ζ = ζ = ζ = cdm ( ) necessarily adiabatic primordial perturbations γ B ν σ
Inflaton scenario: high energy / not-so-slow roll 1. large field ( ϕ < M Pl ) e.g. chaotic inflation see, e.g., Lyth & Riotto Kinney, Melchiorri & Riotto (001) 0 <η < ε not-so-high energy / very slow roll. small field e.g. new or natural inflation η < 0 3. hybrid inflation e.g., susy or sugra models 0 < ε <η slow-roll solution for potential-dominated, over-damped evolution gives useful approximation to growing mode for { ε, η } << 1 & 16π V H V 8π V H M P Vφ H ε M m P φφ η =
can be distinguished by observations slow time-dependence during inflation -> weak scale-dependence of spectra n = 1 6ε + η tensor/scalar ratio suppressed at low energies/slow-roll T R = 16ε
WMAP constraints (I) Microwave background only (WMAPext) Peiris et al (003) r < 1.8 Harrison-Zel dovich spectral index n R 1 6ε + η n 1, r 0
WMAP constraints (II) Microwave background + df + Ly-alpha Peiris et al (003) r < 1.8 spectral index n R 1 6ε + η
WMAP constraints (III) Microwave background + df + Ly-alpha Peiris et al (003) dn R / d ln k = 0.055 + 0.08 0.09 n R =1.13± 0.08 r < 1.8
scale-dependent tilt? dn R d ln k ( 3ε η) ξ 8ε third slow-roll parameter ξ V V 4 M Pl, φ 64π V, φφφ involving four derivatives of the potential, not two the beginning of the end for slow-roll? inflaton effective mass is not constant dη ξ + εη dn slow-roll inflation could be just a passing phase!
digging deeper: additional fields may play an important role initial state ending inflation enhancing inflation new inflation hybrid inflation assisted inflation warm inflation brane-world inflation producing density perturbations may yield additional information non-gaussianity isocurvature (non-adiabatic) modes
Inflation -> primordial perturbations (II) scalar field fluctuations two fields (σ,χ) curvature of uniform-field slices R = * isocurvature S = * Hδσ & σ Hδχ & σ R S 1 = primordial 0 k<< ah density perturbations matter and radiation (m,γ) curvature of uniform-density slices R& = αhs, S& = βhs T T RS SS R S * * S = R = δn n inflation k = ah Hδρ & ρ δn model-dependent transfer functions m m Amendola, Gordon, Wands & Sasaki (00) Wands, Bartolo, Matarrese & Riotto (00) n γ γ
examples: field dynamics during inflation Polarski & Starobinsky; Sasaki & Stewart; Garcia-Bellido & Wands; Steinhardt & Mukhanov; Adams, Ross & Sarkar; Langlois (1996+) variable couplings during/after reheating Dvali, Gruzinov & Zaldariaga; Kofman (003) late-decaying scalar : the curvaton scenario Enqvist & Sloth; Lyth & Wands; Moroi & Takahashi (001+)
curvaton scenario: Lyth & Wands, Moroi & Takahashi, Enqvist & Sloth (00) assume negligible curvature perturbation during inflation R * = 0 light during inflation, hence acquires isocurvature spectrum late-decay, hence energy density non-negligible at decay large-scale density perturbation generated entirely by non-adiabatic modes after inflation R = T RS S * Ω T RS Ω χ, decay χ, decay S * δρ ρ χ χ ( δρ χ χ / ρ ) negligible gravitational waves 100%correlated residual isocurvature modes detectable non-gaussianity if Ω χ,decay <<1
primordial isocurvature perturbations from curvaton? Moroi & Takahashi; Lyth, Ungarelli & Wands 03 cdm, neutrinos, baryon asymmetry all created after curvaton decays ζ ( ζ γ = ζν = ζ = ζ ) Ωχ, decay ζ χ = cdm B S = 0 i cdm/baryon asymmetry created at high energies before curvaton decay ζ = ζ γ Ωχ, decayζ χ, ζ m = 0 S m = 3ζ 100% correlation between curvature and residual isocurvature mode naturally of same magnitude neutrino asymmetry (ξ<0.1) created at high energies before curvaton decay S υ 135 ξ = 7 π ζ
Observational constraints Gordon & Lewis, astro-ph/0148v using CMB + df + HST + BBN isocurvature / curvature ratio B = S B / ζ pre-wmap 95% c.f. Peiris et al f iso = S cdm /ζ 0.1 B and marginalised over correlation angle -> f iso < 0.33 post-wmap
isocurvature perturbations from curvaton (II) cdm/baryon asymmetry created by curvaton decay ζ = ζ = ζ γ Ωχ,decayζ χ, ζ m χ Lyth, Ungarelli & Wands 0 Gupta, Malik & Wands in preparation S m = ( ) χ, decay Ω ζ = 3 ζ 3 1 χ, decay χ 1 Ω Ω χ, decay curvature and isocuravture perturbations naturally of same magnitude relative magnitude related to non-gaussianity
non-gaussianity simplest kind of non-gaussianity: Komatsu & Spergel (001) Wang & Kamiokowski (000) recall that for curvaton corresponds to δρ ρ δ ρ Ω χ,decay δρ ρ δρ ρ significant constraints on f NL from WMAP f NL < 134 χ χ Ω δ1ρ + ρ χ,decay f NL δ1ρ ρ δχ δχ + χ χ 1ρ δχ 1 Ωχ,decay, f NL χ Ωχ,decay Lyth, Ungarelli & Wands 0 hence Ω χ,decay > 0.01 and 10-5 < δχ/χ < 10-3
observable parameters inflaton regime curvature & tensor perturbations n s & tensor/scalar ratio = n t curvaton regime curvature + isocurvature perturbations n s = n iso & isocurvature/curvature ratio intermediate regime n s, n iso, n corr, n t, tensor/scalar, iso/curvature, correlation Wands, Bartolo, Matarrese & Riotto, 0
Conclusions: 1. Observations of tilt of density perturbations (n 1) and gravitational waves (ε>0) can distinguish between slow-roll models. Isocurvature perturbations and/or non- Gaussianity may provide valuable info 3. Non-adiabatic perturbations in multi-field models are an additional source of curvature perturbations on large scales 4. Consistency relations remain an important test in multi-field models - can falsify slow-roll inflation 5. More precise data allows/requires us to study more detailed models!