The Effects of Inhomogeneities on the Universe Today. Antonio Riotto INFN, Padova

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1 The Effects of Inhomogeneities on the Universe Today Antonio Riotto INFN, Padova Frascati, November the 19th 2004

2 Plan of the talk Short introduction to Inflation Short introduction to cosmological perturbations The influence of perturbations Dark Energy? Conclusions Rocky Kolb (Fermilab/Chicago) Sabino Matarrese (Padova) Alessio Notari (McGill) Enrico Barausse (Sissa,Trieste) hep-ph/ , astroph/ , in preparation

3 From Inflation to the observed Universe

4 Inflation

5 Inflation

6 N 60 our inflation volume our Hubble volume size H 1 I e N size H 1 0

7 N>>60 our inflation volume our Hubble volume

8 From quantum fluctuations to the Large Scale Structure

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10 Comoving curvature perturbation

11 Power spectrum on superhorizon scales

12 In our local Universe both short- and long-wavelength modes are present

13 Evolution of H(z) is a key quantity H(z) is the most fundamental dynamical cosmological variable [e.g., all evidence for dark energy comes from evolution of H(z)]. H(z) in the FLRW (homogeneous/isotropic) model: ( aa & ) ( 8 ) 2 2 G00 = κ T00 κ = πg N H = κ ρ Usual assumption for perturbed FLRW model: An inhomogeneous Universe follows the evolution of a FLRW model of the same average density. Issues: gravity is nonlinear gravity is a long-range force

14 Evolution of H(z) is a key quantity H(z) is the most fundamental dynamical cosmological variable [e.g., all evidence for dark energy comes from evolution of H(z)]. H(z) in the FLRW (homogeneous/isotropic) model: 2 ( 2 G = κ T κ = 8πG ) (FLRW) N (perturbed FLRW) ( aa & ) H = κ ρ r FLRW r Gµν ( x, t) = Gµν ( t) + δgµν ( x, t) FLRW r 2 r G t + δg x, t = κ T x, t 3 () ( ) ( ) ( ) ( 2 aa & = κ ρ κ δg ) 00 Can think of κ 2 δ G 00 as kinetic energy of the gravitational field.. (a/a) 2 is not κ 2 ρ /3 H 2. Usual soundbite: δρ/ρ small δ G 00 small δh small.

15 Second-order order perturbation theory Expand energy-momentum tensor and metric tensor to 2 nd -order: T = T + T + T (0) (1) (2) µν µν µν µν g = g + g + g (0) (1) (2) µν µν µν µν Einstein tensor to 2 nd -order: G = G + G + G + G (0) (1) (11) (2) µν µν µν µν µν (0) (0) T µν g µν & are homogeneous & isotropic In synchronous gauge: δ gij a τ = ij i j i j r 1 r! ψ + r 1 r! χ χ = = ( r) ( ) ( ( r) ( r) ( r) ( r) 2 δ χ ) j χi ij = = = i ( r ) i ( r) i ( r ( χ 0; 0, ) i χ i χij 0) Metric perturbations in terms of peculiar gravitational potential ϕ : 2 2 r κ 2 (0) ( ) ( (1) r (0) ϕ x = a ρ δρ ( x) ρ ) 2

16 Metric perturbations in a ϕ is the peculiar gravitational potential r κ ϕ( x) = related to δρ/ρ by Poisson equation (0) a ρ δρ (1) ρ ( x) (0) r 2 (1) r 5 r τ 2 r ψ ( x, τ) = ϕ( x) + ϕ( x) (2) r 50 2 r 5τ, r r 4 r 2 r ψ τ ϕ ϕ ϕ, k ϕ ϕ K k ( x, ) = ( x) + ( x) ( x) + ( x) ( x) 4 τ r 10 r r ( 2 ( )) 2, ik ϕ x ϕ ( x) ϕ, ik ( x) Matter-dominated Universe

17 Einstein Equations 2 nd order ( ) G + G + G + G = κ T + T + T (0) (1) (11) (2) 2 (0) (1) (2) µν µν µν µν µν µν µν 00 : 2 G = κ T G + G + G (0) 2 (1) (11) (2) (1) (11) (2) 3 a G 2 = κ ρ + 00 G + 00 G00 a In a perturbed universe (a /a 2 ) 2. = (a/a) 2 is not κ 2 ρ /3. a /a 2 does not even describe the physical Hubble flow description of Hubble flow comes from evolution of ρ. Augment 00 equation. with continuity equation: D µ T µ0 = 0 ρ = ρ /a = θρ θ D µ u µ u µ is the fluid 4-velocity Hubble flow described by θ. For FLRW, θ = 3H.

18 nd The Hubble flow 2 nd order Hubble flow described by θ : 3 a θ = θ + θ + θ + θ = + θ + θ + θ a 2 (0) (1) (11) (2) (1) (11) (2) Einstein equations relate a /a 2 in terms of ρ & G, G, G (1) (11) (2) Define a δθ : (1) (11) (2) (1) (11) (2) (1) δθ θ + θ + θ G00 + G00 + G00 G00 = 3H 3H 6aH 72aH Notational point: could define H = (κ 2 ρ /3) 1/2 or H = (a /a 2 ) 1/2 H (κ 2 ρ /3) 1/2

19 It is not a backreaction (1) (11) (2) (1) (11) (2) (1) δθ θ + θ + θ G00 + G00 + G00 G00 = 3H 3H 6aH 72aH New terms present due to inhomogeneities New terms modify expansion But expansion does not produce inhomogeneities (although expansion does modify the growth of inhomogeneities)

20 : Averaging volume element: dx 3 γ ( τ, x) 3 dx γ ( τ, x) ( τ, x) ( τ ) r r O O 3 r dx γ ( τ, x) 2 1 r r r τ r = + ( x) + L= ( x) + L= ( x) ( x) + L 2 6 (1) (1) 2 γ 1 γ τ, 1 3 ψ τ, 1 5ϕ ϕ r O (0) (0) O = O 3 ψ O + 3 ψ O O (1) (1) (1) (1) (1) (1) (2) = O ( τ ) = dx O 3 (2) 3 dx r ( τ, x) dx O 3 (1) O ( τ, x) dxo ( τ, x) 3 (2) (1) 3 1 dx r 3 dx r

21 nd The Hubble flow 2 nd order Averages involve integrals of the form L 1 V ( R) ( L) dv V ( R ) Assume Gaussian window function Fourier transform of window function W 2 (kr) r 2R 3/2 3 dv = 4πr e dr V R = 2π R 2 2 ( ) ( ) 2 2 r r W kr = V R e ik x dv 1 2 ( ) ( ) r R exp V( R) k r 2 = e ( ) 1 kr 0 0 kr kr

22 Mean of δθ Typical expected value of δθ in V(R) ensemble average Hubblesize volume our inflation volume our Hubble volume

23 Crucial point Must remember the statistical nature of the vacuum fluctuations Our Universe corresponds to a typical member of the ensemble of possible Universes

24 Mean of δθ Express ϕ and its derivatives in terms of Fourier integrals ϕ = 3 dk ( 2π ) 3 ϕ e r k r r ik x ϕ Gaussian random variable w/ zero mean N-point functions: 3 r r 3 ϕ ( ) ( ) rϕr = 2π δ k1+ k2 P ( ) k1 k ϕ k1 2 6 r r r r 3 ( ) ( ) 3 ϕ ( ) rϕrϕrϕr = 2π δ k1+ k2 δ k3 + k4 P ( k1) P ( k3) k1 k2 k3 k4 ϕ ϕ r r r r 3( ) 3 + δ k ( ) 1+ k3 δ k2 + k4 Pϕ ( k1) Pϕ ( k2) r r r r 3( ) 3 + δ k ( ) 1+ k4 δ k2 + k3 Pϕ ( k1) Pϕ ( k2) P ϕ (k) = ϕ k 2 is the power spectrum of ϕ Cosmological Poisson equation relates P ϕ (k) and 2 (k,a) 2 (k,a) is the power spectrum of δρ/ρ

25 2 (k,a) Power spectrum of δρ/ρ in terms of T 2 (k), the transfer function, and A 10 5, a dimensionless amplitude: k 2 ( ka, ) A T ( k) = ah Harrison Zel dovich spectrum For Harrison Zel dovich: 2 ( k) k ln 4 2 ( k) k k 0 Other spectra, 2 (k) k 3+n as k 0 (n = 1 for H Z)

26 Mean of δθ (present value) 10 5 δθ / 3H ( 1) B B Γ Ω h exp Ω 2Ω Ω 0 0 k MAX (h Mpc 1 ) scales as (1+z) 1

27 Variance of δθ What we really want is δθ in our Hubble volume our inflation volume our Hubble volume

28 Variance of δθ ( ) ( ) 2 2 Var L L L 2 (1) (11) (2) (1) (11) (2) (1) δθ θ + θ + θ G00 + G00 + G00 G00 = 3H 3H 6aH 72aH τ = ah ϕ ϕ ϕ ϕ ϕ , i τ 2 2 τ, ij + ϕ ϕ + ϕ ϕ, i ϕ ϕ + ϕ ϕ, ij

29 Variance of δθ sample terms dk ( ) ( ) Var ϕ = ah kaw, kr 1 4 k dk 2 dk 2 k1 k2 ϕ ah 3 ( k1 a) 3 ( k2 a) k 0 1 k 0 2 k2 k1 Var ϕ =,, ( ) ( ) 2W k R W k R W 2 (kr) acts as a filter function regulating ultraviolet but Var[ ϕ 2 ϕ ] singular in the infrared if n 1! as is Var[ ϕ 1 2 ϕ 1 ] traces to the infrared behavior of Var[ ϕ 1 ]

30 Variance of δθ infrared nature dk ( ) ( ) 1 ah kaw kr ϕ 1 4 k 0 Var ϕ =, = Var dk ϕ = ah ( kaw, ) ( kr) ϕ = k 1 0 Interested in R=R H For k < k H : W(kR H ) 1 and T 2 (k) 1,so 2 (k,a)w 2 (kr) k 3+n kh kh dk 3 n 2 dk + 3 n k ϕ + k 1 IR 5 k 1 IR k 0 0 Var ϕ Var kh kh 2 dk1 3 n dk n 2 k k2 = IR 1 k IR k 0 2 Var ϕ ϕ Var ϕ Var ϕ IR

31 Variance of δθ How to make sense of infrared singularity? 1. variance is infrared finite bluer than Harrison-Zel dovich 2. somehow infrared singular terms cancel 3. physical cutoff at Hubble radius: k MIN = k H

32 Variance of δθ (Var[δθ ]) 1/2 /3H cutoff at k=k H R (h 1 Mpc) k 4 scales as (1+z) 2 ; k 2 scales as (1+z) 1

33 Variance of δθ How to make sense of infrared singularity? 1. variance is infrared finite (δρ/ρ bluer than n = 1 as k 0) 2. somehow infrared-singular terms cancel 3. physical cutoff at Hubble radius: k MIN = k H 4. use cutoff determined by duration of inflation (sensitive to unknown nature of perturbations on scales greater than the Hubble radius)

34 Variance of δθ What we really want is δθ in our Hubble volume our inflation volume infrared cutoff at k = k MIN k MIN related to # of e-folds of inflation ln kh = N TOTAL N kmin H our Hubble volume

35 Variance of δθ How to make sense of infrared singularity? 1. variance is infrared finite bluer than Harrison-Zel dovich 2. somehow infrared singular terms cancel 3. introduce cutoff at Hubble radius: k MIN = k H 4. use cutoff determined by duration of inflation (sensitive to unknown nature of perturbations on scales greater than the Hubble radius) Are super Hubble perturbations physical? constant ϕ can be scaled out equations but can t get rid of ϕ 2 ϕ! could Var[δθ] be large enough to give δθ /3H O(1)?

36 Crucial technical points δρ 2 ϕ ρ on superhorizon scales density perturbs are perturbative ϕ 2 ϕ comes from e ϕ 2 ϕ may go beyond second-order order E.W. Kolb, S. Matarrese, A. Notari and A.R., astroph/

37 Entertaining conjecture

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43 Unperturbed Universe q 0 Ω 3 = i wi Ω i Acceleration implies a fluid with negative pressure Dark Energy

44 Distance-redshift relation in a perturbed Universe E. Barausse, S. Matarrese and A.R., In preparation

45 Distance-redshift relation in a perturbed Universe observer λ = 0 k µ ν = 0 k k µ = 0 µ k ν source λ = λ s photon momentum

46 Distance-redshift relation in a perturbed Universe energy momentum tensor Photon equations

47 expansion of the null congruence along the photon trajectory

48 Shear

49 Distance-redshift relation in a perturbed Universe Energy flux per unit surface measured by an observer with four-velocity

50 Distance-redshift relation in a perturbed Universe

51 Distance-redshift relation in a perturbed Universe

52 Distance-redshift relation in a perturbed Universe

53 Crucial aspects The Hubble parameter and the deceleration parameter are not deterministic Because of the statistical nature of vacuum fluctuations, the gravitational potential does not have well-defined values The theoretical predictions of the cosmological parameters come with a nonvanishing cosmological variance implying an intrinsic theoretical error

54 Variance of q Are super Hubble perturbations physical? Constant ϕ can be scaled out of equations...but can t get rid of ϕ 2 ϕ!

55 Variance of q What we really want is q in our Hubble volume our inflation volume infrared cutoff at k = k MIN k MIN related to # of e-folds of inflation ln kh = N TOTAL N kmin H our Hubble volume

56 for a flat spectrum, n=1

57 The Universe accelerates Dark Energy is present or It is matter-dominated and the value of deceleration parameter predicted by the theory at the unperturbed level comes with a large cosmological variance: negative values of q are allowed

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59 Deceleration parameter Cosmic Variance

60 Conclusions Inflation generates long wavelength perturbations Super-Hubble modes introduce intrinsic theoretical errors New effects to study No need of Dark Energy?