Cosmic Acceleration as an Optical Illusion

Transcription

1 Theoretische Physik TU Wien / Uni Bielefeld April 2016 / Bielefeld based on arxiv: , PRD89 (2014) (arxiv: ), PRD90 (2014) (arxiv: ).

2 Motivation and basics Motivation: Do we need Λ 0? Why is that so hard to find out? What can one study? Toy models vs realistic universe (here: only the latter) Volume evolution or light propagation Basic idea: Consider a large domain D in an irrotational dust universe, divide D into infinitesimal regions actually small in cosmic terms, large enough for the irrotational dust approximation, follow the evolution of each such region, add the contributions.

3 Related work Introduction Many proposals for alternatives to Λ (see e.g. Ellis et al for a review). Work on volume evolution: authors such as Kolb, Matarrese, (Notari,) Riotto; Räsänen; Li, (Seikel,) Schwarz; (see Buchert for a review), acceleration as a real effect in an irrotational dust universe, Buchert s formalism here my approach is different! : redshift-distance relation up to second order perturbation theory worked out by Ben-Dayan, Gasperini, Marozzi, Nugier, Veneziano; Umeh, Clarkson, Maartens.

4 Outline Introduction 1 Introduction 2 General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution 3 Sachs Equations and distance formulas Photon path average Results 4

5 Irrotational dust: general setup General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution Spacetime manifold (4) M = R + M Energy-momentum tensor (4) T = diag(ρ, 0, 0, 0) Metric ds 2 = dt 2 + g ij (t, x)dx i dx j in synchronous gauge, any geometric quantity refers to the 3d-metric g! Expansion tensor θ i j = 1 2 gik ġ kj Scalar expansion rate θ = θ i i = Shear σ i j = θ i j θ 3 δi j, g g hence θ i j θj i = 1 3 θ2 + 2σ 2 with σ 2 = 1 2 σi j σj i. Ricci tensor R i j = R 3 δi j + r i j (with r i j... traceless part of R i j ) obeys Ṙij = θ k i jk + θk j ik θ ij klg kl θ ij.

6 Irrotational dust: dynamics General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution Energy-momentum conservation ρ + θρ = 0 Einstein equations 1 3 θ2 σ R Λ = 8πG Nρ, 2θ i + 3σ j i j = 0, σ i j + θσ i j + r i j = 0. ( ) 1 a local (t, x) = ˆρ 3 ρ(t,x)... the local scale factor (with ˆρ... a fixed quantity of dimension mass, e.g. M ); a(t, x) = a local (t, x) is just the side length of a cube of mass ˆρ consisting of material of density ρ.

7 Mass-weighted average General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution ( Energy-momentum conservation d dt ρ(x, t) ) g(x, t) = 0 the mass content m D = D ρ(x, t) g(x, t) d 3 x of any domain D M is time independent, ṁ D = 0 X ṁw = Ẋ mw for the mass weighted average of a scalar X, X mw (t) = 1 m D D X(x, t)ρ(x, t) g(x, t) d 3 x, N.B. X = Ẋ would not hold for a volume average X vol which would therefore require Buchert s formalism! But V D = D g(x, t) d 3 x = m D ρ 1 mw = m Ḓ ρ a 3 mw, X vol = Xa 3 mw / a 3 mw with a = a local.

8 General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution Rescaled quantities and their evolution ρ + θρ = 0 and a = (ˆρ/ρ) 1 3 Rescaled quantities imply ρ(t, x) θ(t, x) = ρ(t, x) = 3ȧ(t, x) a(t, x). ˆρ = a 3 ρ, ˆσ j i = a 3 σj i, ˆR = a 2 R, ˆr j i = a 2 rj i obey the evolution equations ˆρ = 0, ˆσ i j = aˆr i j, ˆR = 2a 3ˆσ i jˆr j ( ˆr j i = a 3 5 k ˆσi 4 kˆr j ˆσk j ˆr k i δi j ˆσk l ˆr k l i, ) + a 2 Y ki j k, with Y k ij = 3 4 (σk i j + σ k j i ) 1 2 g ijσ k m m σij k.

9 Local Friedmann equation General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution The evolution equation for the local scale factor (the 0-0 Einstein equation) can be written as ȧ 2 = 1 3 ˆσ2 ina πg N ˆρ a ˆR in Λ a2 4 9 t t in θ( t)a 4 ( t) ˆσ 2 ( t) d t. with t in... some initial time (typically t in = 0).

10 Linear perturbation theory: metric General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution Background: Einstein de Sitter, i.e. flat matter only, a EdS = const t 2 3. The relevant contributions all come from a single time-independent function C(x): g (LPT) ij (t, x) = a 2 EdS (t) ( with S ij (x) = i j C(x). δ ij a 2 EdS t 4 3 ) C(x)δ ij + t 2/3 S ij (x) Corresponds to Newtonian (longitudinal) gauge metric ds 2 = a 2 FLRW (t) ( (1 + 2Φ)dt 2 + (1 2Ψ)dx 2) with Φ = Ψ = C/3, a FLRW = a EdS.

11 General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution Linear perturbation theory: R, θ, r, σ, ρ Straightforward calculations result in first order expressions (with S ij = s ij δ ijs) R = 20 9 t 4 3 S, θ = 2t t 1 3 S, r i j = 5 9 t 4 3 δ ik s kj, σ i j = 1 3 t 1 3 δ ik s kj, 6πG N ρ = t t 4 3 S and initial conditions, at t in = 0, of lim t 0 a t 2/3 = (6πG N ˆρ) 1/3, ˆRin (x) = 20 9 (6πG N ˆρ) 2 3 S(x), (ˆσ in ) i j (x) = 0, (ˆr in) i j (x) = 5 9 (6πG N ˆρ) 2 3 δ ik s kj (x).

12 Dimensionless variables General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution The entries of the matrix S ij have dimensionality t 2/3 ; choose constant U such that Sij = S ij /U is dimensionless. Dimensionless quantities R in = 20 9 S, ( r in ) i j = 5 9 δik s kj. t = U 3 2 t, ā = (6πG N ˆρ) 1 3 Ua, R = (6πG N ˆρ) 2 3 U 1 ˆR, r = (6πG N ˆρ) 2 3 U 1ˆr, σ = (6πG N ˆρ) 1 U 3 2 ˆσ,

13 The probability distribution General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution C(x) = (a k + ib k )e i k x (2π) 3/2 d 3 k: Fourier modes a k = a k, b k = b k, Gaussian distribution: a k, a k independent unless k = k. S ij... symmetric Gaussian random matrix, invariant under orthogonal conjugation Using the theory of such matrices one finds p( S, δ, ϕ) e 1 10 S δ 2 δ 4 sin(3ϕ), where δ, ϕ parametrize the eigenvalues δ k of s ij via δ k = 2 3 δ cos(ϕ + 2πk 3 ); the normalization involves the value of ( 2 C) 2 : an integral requiring an ultraviolet cutoff (from now on: normalized quantities, written without bars).

14 Volume evolution Introduction General setup and mass-weighted average Rescaled quantities and their evolution Initial values from linear perturbation theory Volume evolution V D (t; S b, Λ) a 3 (t; S, δ, ϕ, Λ)e 1 10 (S S b) δ2 δ 4 sin(3ϕ) ds dδ dϕ can be computed numerically, using the evolution equations for a(t; S, δ, ϕ, Λ); implies volume scale factor a D (t) = V 1/3 D (t), Hubble rate H D (t) = ȧ D (t)/a D (t), deceleration parameter q D (t) = ä D a D /ȧ 2 D. For Λ = 0 these do not agree with what we observe (even though q D < 0 is possible for positive background curvature).

15 Flat homogeneous case Sachs Equations and distance formulas Photon path average Results Hubble rate H(t) = ȧ(t) a(t), deceleration parameter q = ä a = d ( 1 ) ȧ 2 dt H 1, redshift 1 + z = a(to) a(t) H(t) = d dt ln(1 + z), distance (assuming vanishing spatial curvature) d = (1 + z) λ z 0 1 H(z ) dz, with λ = 1... angular diameter distance d A, λ = luminosity distance d L, λ = 0... structure distance (Weinberg) d S, d L /d A = (1 + z) 2 actually holds generally (Etherington), then we define d S = (1 + z)d A = (1 + z) 1 d L, H = dz dd S, dd S = (1 + z)dt.

16 Redshift for general geometries Sachs Equations and distance formulas Photon path average Results x e µ, x o µ... loci of photon emission and observation, s... affine parameter for photon path (geodesic), k µ = dx µ /ds... tangent vector, u e, u o... normalized tangent vectors to the worldlines of the source and the observer: 1 + z = (u k)e (u k) o. If distinguished time coordinate t: u e = t e, u o = t o, s normalized to ds = dt at the observer, 1 + z = dt ds, i.e. d ds = (1 + z) d dt (with t = t e ). Notation: d dt or dot for t parametrizing the geodesic, 0 or t partial derivative by spacetime coordinate t = x 0.

17 Sachs Equations Introduction Sachs Equations and distance formulas Photon path average Results dθ opt ds + θ opt 2 + σ opt 2 = 1 2 R αβk α k β, dσ opt ds + 2θ optσ opt = 1 2 R αβµνε α k β ε µ k ν, θ opt, σ opt... expansion rate and shear of the null bundle, ε = ε (1) + 1 ε (2) where ε (1), ε (2) are spacelike unit vectors with ε (i) k = ε (i) u o = 0, R αβµν (2 nd Eq.) can be replaced by the Weyl tensor C αβµν Weyl focusing.

18 Sachs Equations and distance formulas Photon path average Results Angular diameter and structure distance d A is determined by d ds ln d A = θ opt, the Sachs equations become d 2 d A ds 2 = ( σ opt R αβk α k β )d A, d ds (σ optd 2 A ) = 1 2 R αβµνε α k β ε µ k ν d 2 A. d A = (1 + z) 1 d S, d ds = (1 + z) d dt gives d S [ln(1 + z)] ḋs + id S = 0 with i = (1 + z) 2 ( σ opt R αβk α k β ) d 2 ln(1 + z). dt 2 d S = t o t e (1 + z)dt = z 1 0 [ln(1+z)] dz is a solution for i = 0 (holds e.g. for flat homogeneous universes!); useful because d 2 d S dd 2 S = i (1+z) 2 d S.

19 Sachs Equations and distance formulas Photon path average Results Inferred Hubble rate and deceleration Hubble rates H inf = dz dd S, H = dz dd S H H inf = 1 t o t i (1+z) d S dt, = d dt ln(1 + z), i > 0 (i < 0): observations over-(under-)estimate expansion rates in previous epochs, inferred time parameter dt inf = dd S 1+z = ḋs 1+z dt, H inf dt inf = H dt = d ln(1 + z), deceleration q inf = ( ) d 1 dt inf H inf 1, q = d dt q inf = q + i d S(1+z), ḋ S ż i < 0 can simulate acceleration. ( 1 H ) 1

20 Sachs Equations and distance formulas Photon path average Results Homogeneous and irrotational dust universes Assume Metric ds 2 = g αβ dx α dx β = dt 2 + g ij (t, x)dx i dx j, energy-momentum tensor with T ij = g ij T k k /3 and T 0i = 0. Then (using 1 + z = dt/ds and geodesic equations) one finds H = d dt ln(1 + z) = θ 3 + σ ijẋ i ẋ j, i = (1 + z) 2 σ opt 2 + R/6 σ 2 +( σ ij θ 2σ k i σ kj r ij + 2σ ij σ kl ẋ k ẋ l )ẋ i ẋ j +ẋ i i θ/3 + ẋ k ( k σ ij )ẋ i ẋ j 2σ ij Γ i klẋ k ẋ l ẋ j, R αβµν ε α k β ε µ k ν = (1 + z) 2 ( 2 3 θσ ij σ ik σ k j + 2r ij + ẋ l σ lm ẋ m σ ij ẋ l σ li ẋ m σ mj 4ẋ k σ i[j k] )ε i ε j.

21 Photon path average: basic ideas Sachs Equations and distance formulas Photon path average Results Idea: replace H and i by suitably defined averages; X pp (t) should be the average of X over all spatial positions x occupied by photons, directions v of propagation. Photon paths correspond to curves in x space. Flat homogeneus case: straight lines, if the shapes were not altered by inhomogeneities, then p(...) would give the distribution of the basic parameters with respect to the euclidean metric dl 2 = δ ij dx i dx j along such a curve; approximation: same distribution even in the general case.

22 Photon path average: definition Sachs Equations and distance formulas Photon path average Results From euclidean length to physical time: v i = dx i dl = ẋ i dt dl... tangent vector with δ ij v i v j = 1, g-norm g ij v i v j, use g ij ẋ i ẋ j = 1 dt = g ij v i v j dl. For every euclidean path segment dl average over the parameters of the model (... mw ), average over all directions v (... d 2 v), S 2 weight by the time dt = g ij v i v j dl spent in the segment S X g X pp = 2 ij v i v j d 2 v mw. S g 2 ij v i v j d 2 v mw If X depends on ẋ i explicitly, use ẋ i = v i. gij v i v j

23 Some approximations Sachs Equations and distance formulas Photon path average Results We want S X g 2 ij v i v j d 2 v mw for X = 1, X = H, X = i. Integrands: polynomial in v except for g ij v i v j. Diagonal basis, g ij = ḡ(δ ij + γ ij ), ḡ = g 11+g 22 +g 33 3 ( ) λ g ij v i v j = ḡ λ/2 (1 + λ 2 γ ijv i v j +...), omit anything quadratic or higher in γ ij, omit σ opt -term in the expression for i, then S 2 g ij v i v j d 2 v 4π ḡ, 1 4π ḡ S H 2 g ij v i v j d 2 v 5θ+4σ1 1 γ 11+4σ2 2γ 22+4σ3 3γ π ḡ 15, S i g 2 ij v i v j d 2 v ˆR in + 4 t 6a 2 9a 2 0 θ( t)a 2 σ 2 d t σ [(7σ1 1 θ + 7r (σ1 1 )2 )γ ].

24 Sachs Equations and distance formulas Photon path average Results Second order perturbation theory computations of H pp, i pp straightforward, but not all of the previous simplifications and approximations respect 2 nd order PT. Non-perturbative results with GNU octave: Euler method with logarithmic time steps, constant ˆr = ˆr in, three scenarios for modelling collapse: 1 collapse to half of maximal size, quantities retain values, 2 collapse to half of maximal size, then ignore, 3 ignore any region from the moment it starts collapsing (motivated by virial theorem, second one most realistic). ḡ mw = (g 11 + g 22 + g 33 )/3 mw close to its EdS equivalent t 2/3. Coding in plots: dashed lines for H, q, d S, solid lines for other quantities from scenarios 1/2/3.

25 Time evolution of Ht and i ḡ Sachs Equations and distance formulas Photon path average Results Ht i g time[units of Sec. 4] time[units of Sec. 4] Yellow line: result from volume averaging Strong deviations from homogeneous case: mainly from local anisotropy

26 Sachs Equations and distance formulas Photon path average Results Time evolution of distance and deceleration 10 1 structure distance[units of Sec. 4] deceleration time[units of Sec. 4] time[units of Sec. 4] t o chosen such that H o t o = 1 dotted black lines: ΛCDM

27 Distance in physical units Sachs Equations and distance formulas Photon path average Results 20 3 structure distance[gpc] distance divided by EdS value time[gyr] redshift z Black crosses: observed supernovae Better agreement for smaller redshift

28 Last scattering Introduction Sachs Equations and distance formulas Photon path average Results A combination of our programs and linear perturbation theory gives t ls (the time at which z = 1090). Distance to CMB: d S (t) = d S (0) + d (1) S t 1/3 + O(t 2/3 ) near t = 0 results in d S (t ls ) 20.7/20.9/19.8 Gpc for scenarios 1/2/3; overestimation by 50% (compared to Planck data): σ opt -term or measured vs computed d A? Density perturbations (total): ( ) ρ ρ = 1 ls 2 t 2 3 ls S = 1 2 ( ) Baryonic ones should satisfy ρ B /ρ B = 3 T /T (T... temperature), fits well with T /T 10 5, ρ/ρ 50 ρ B /ρ B (structure formation).

29 Summary of results Considering a universe that is matter dominated and obeys the Einstein equations, in its early stages was very close to being spatially flat and homogeneous, with only Gaussian perturbations, and has vanishing cosmological constant, Λ = 0, there is a time t o when observervations would suggest an inferred Hubble rate H inf such that H inf t o 1, an inferred deceleration parameter of q inf 0.5, and density perturbations at a redshift of 1090 that fit well with structure formation.

30 Approximations used Irrotational dust, statistical quantities expectation values, distribution as if photon paths were straight lines, simplified evolution for r ij, neglect of quadratic or higher terms in γ ij = ḡ 1 g ij δ ij, neglect of σ opt, i.e. Weyl focusing, numerical errors from discretization. The second and third approximation violate 2 nd order perturbation theory serious discrepancy with literature.

31 Outlook Introduction Work on all points in the approximation list, particular emphasis on those that violate 2 nd order PT Why do cancellations occur in 2 nd order PT? Better understanding of CMB Stochastic model: Joint distribution for (d S, z)?