Inflationary density perturbations

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1 Cosener s House 7 th June 003 Inflationary density perturbations David Wands Institute of Cosmology and Gravitation University of Portsmouth

2 outline! some motivation! Primordial Density Perturbation (and conserved quantities on large scales)! constraints on single-field models! predictions from multi-field models! conclusions

3 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

+ gravitational waves (we hope) gravitational instability new observational

4 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

5 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

6 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)

7 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

8 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

9 where do these perturbations come from?

10 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

11 Wilkinson Microwave Anisotropy Probe February 003 coherent oscillations in photon-baryon plasma from primordial density perturbations on super-horizon scales

12 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)

13 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

14 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 =

15 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

16 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

17 WMAP constraints (II) Microwave background + df + Ly-alpha Peiris et al (003) r < 1.8 spectral index n R 1 6ε + η

18 WMAP constraints (III) Microwave background + df + Ly-alpha Peiris et al (003) dn R / d ln k = n R =1.13± 0.08 r < 1.8

19 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!

20 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

22 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

23 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+)

24 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

25 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

S B / ζ pre-wmap 95% c.f. Peiris et al f iso = S cdm /ζ 0.

26 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

27 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

28 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

29 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

30 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!

Inflation and the origin of structure in the Universe

Phi in the Sky, Porto 0 th July 004 Inflation and the origin of structure in the Universe David Wands Institute of Cosmology and Gravitation University of Portsmouth outline! motivation! the Primordial