Cosmic Acceleration as an Optical Illusion

Transcription

1 TUW-5-4 Cosmic Acceleration as an Optical Illusion arxiv: v2 [gr-qc] 24 Aug 205 Harald Skarke * Institut für Theoretische Physik, Technische Universität Wien Wiedner Hauptstraße 8 0, 040 Wien, Austria ABSTRACT We consider light propagation in an inhomogeneous irrotational dust universe with vanishing cosmological constant, with initial conditions as in standard linear perturbation theory. A nonperturbative approach to the dynamics of such a universe is combined with a distance formula based on the Sachs optical equations. Then a numerical study implies a redshift-distance relation that roughly agrees with observations. Interpreted in the standard homogeneous setup, our results would appear to imply the currently accepted values for the Hubble rate and the deceleration parameter; furthermore there is consistency with density perturbations at last scattering. The determination of these three quantities relies only on a single parameter related to a cutoff scale. Discrepancies with the existing literature are mainly due to effects beyond second order in perturbation theory. * [email protected]

2 Introduction The fact that cosmological observations do not conform to the predictions of Friedmann- Lemaitre-Robertson-Walker (FLRW) models with a vanishing cosmological constant Λ is usually interpreted as an indication that Λ differs from zero. Clearly our actual universe deviates from the idealized FLRW cases by hosting inhomogeneities, and there have been many suggestions that the latter might have effects which would explain the data without requiring Λ; see e.g. Ref. [] for an early proposal of this kind. The main challenge for any such claim is to explain why we perceive an accelerated expansion. Basically there are two possible routes as well as combinations of them. On the one hand the inhomogeneities might have an impact on the actual expansion of the universe (suitably defined in terms of the evolution of volumes of spatial regions). On the other hand there is the possibility that they affect light propagation in a subtle way which modifies the usual distance-redshift relations. In the present work we are mainly concerned with the second scenario, which relies on the obvious yet important insight that almost every single piece of evidence on the evolution of the cosmos relies on the observation of photons with telescopes or other devices; Ref. [2] provides a particularly forceful presentation of this point. There is an extensive amount of literature on light propagation in the presence of inhomogeneities; see e.g. Refs. [3, 4, 5, 6, 7, 8, 9, 0] for a small subset. Typical ingredients include the use of the Sachs optical equations [] from which a formula for the angular diameter distance d A can be derived, and approximations of the Dyer-Roeder type [2]. A somewhat different approach is pursued in Refs. [3, 4, 5, 6] and related papers, where a tailor-made coordinate system [7] is used. The present work will take the Sachs optical equations as a starting point, but will use them to analyse the evolution of the structure distance (cf. Weinberg [8]) d S = ( + z)d A. The result, a second order ordinary differential equation, looks more complicated at first sight than the corresponding formula for d A, but it turns out that the two nontrivial coefficients have very simple interpretations: one of them is a local (and directed) expansion rate that agrees with the standard Hubble rate in the homogeneous case, and the other one is a quantity that vanishes in a spatially flat homogeneous geometry. These expressions (more precisely: their suitably defined expectation values) are then computed non-perturbatively in the framework of a recently introduced statistical model [9] whose only assumptions are an irrotational dust

3 approximation for the matter content and initial conditions consistent with linear perturbation theory with only Gaussian fluctuations. With the help of some approximations (but not of the Dyer-Roeder type) and the use of a computer program we find that in such a universe with Λ = 0 there is a time t o with the following properties. An observer at t o will see redshift-distance pairs which, if interpreted with formulas that ignore the inhomogeneities, would indicate H(t o )t o, a deceleration parameter q(t o ) 0.5, and density perturbations at a redshift of z 090 from t o that agree with those assumed for dark matter at last scattering. In other words, such an observer sees what present day cosmologists see, despite living in a universe in which the cosmological constant vanishes. In the next section we derive a differential equation for the structure distance and discuss the meaning of its coefficients; furthermore we elucidate the relationship between local expansion data along a lightlike geodesic and the inferences that a cosmologist who ignores the inhomogeneities would make. In Sec. 3 the coefficients are computed explicitly for the cases of homogeneous and irrotational dust universes. Sec. 4 contains a brief summary of the methods of Ref. [9] for a non-perturbative statistical treatment of an irrotational dust universe with initial conditions from linear perturbation theory. In Sec. 5 the photon path average is introduced: this is the concept that we use to estimate the overall effect of the changing environments that a photon experiences on the way from its source to an observer. Sec. 6 contains calculations up to second order in perturbation theory (we will see that they do not suffice to produce the relevant effects). In Sec. 7 we present the results of a numerical computation that transcends perturbation theory: we find quantities that are in rough agreement with today s observations even though we assume Λ = 0. In the final section we briefly reiterate our findings, summarize the approximations that were made in deriving them, and discuss some discrepancies with existing literature as well as possible directions of future work. 2 Sachs equations and distance formulas Let us start with a brief summary of the homogeneous case in order to provide some reference points for our subsequent generalization. A homogeneous universe is usually described with the help of a time-dependent scale factor a(t) in terms of which the Hubble expansion rate is 2

4 defined as and the deceleration parameter as q = ä a ȧ 2 H(t) = ȧ(t) a(t), () = d dt ( ). (2) H The redshift z of a photon emitted at time t and observed at time t o, with both the source and the observer at rest with respect to a comoving frame, is given by which implies + z = a(t o) a(t), (3) H(t) = d ln( + z). (4) dt In the case of vanishing spatial curvature several distance formulas can be summarized as d = ( + z) λ z 0 H(z ) dz, (5) where we have to take λ = for the angular diameter distance d A, and λ = for the luminosity distance d L. The resulting identity d L /d A = ( + z) 2 actually holds in any pseudo-riemannian geometry; this is known as Etherington s theorem [20]. The simplest version of Eq. (5) occurs if we take d to be the geometric mean of d A and d L, d S = ( + z)d A = ( + z) d L, (6) for which there exists a variety of names in the literature; we will follow Weinberg [8] who calls d S the structure distance. Then λ = 0, and Eq. (5) implies and, with Eq. (4), H = dz dd S (7) dd S = ( + z)dt. (8) In the following we consider an arbitrary spacetime geometry. We want to analyse a lightlike geodesic corresponding to the path of a photon emitted at x µ e and observed at x µ o. With an affine parameter s and a corresponding tangent vector k µ = dx µ /ds the redshift z is determined in general by the formula + z = (u k) e (u k) o, (9) 3

5 where u e and u o are the normalized tangent vectors to the worldlines of the source and the observer, respectively. If we assume that we have a distinguished timelike coordinate t such that both the source and the observer have worldlines with normalized tangent vectors / t, and that s is normalized so that ds = dt at the observer, we get + z = dt ds, i.e. d ds = ( + z) d dt (to be evaluated at the source, i.e. at t = t e ; the same holds for the following equations). We write d or use dots when we treat t as parametrizing the geodesic, and we denote the partial dt derivative by the spacetime coordinate t = x 0 as 0 or. t The Sachs optical equations [] (see [2] for a textbook derivation) are (0) dθ opt ds + θ opt 2 + σ opt 2 = 2 R αβk α k β, () dσ opt ds + 2θ optσ opt = 2 R αβµνε α k β ε µ k ν, (2) where θ opt and σ opt are the expansion rate and the shear of the null bundle, respectively. In general the terms expansion rate and shear refer to the change in the size and the shape of a bundle of geodesics. Since we will later apply the same notions to worldlines of dust particles, we indicate with the subscript that we are referring to the optical quantities. Furthermore ε = ε () + ε (2) where ε (), ε (2) are spacelike unit vectors orthogonal both to k and to the observer s worldline; because of these properties the right-hand side of the second equation remains the same if the Riemann tensor R αβµν is replaced by the Weyl tensor C αβµν, and corresponding effects are often referred to as Weyl focusing. The angular diameter distance d A is determined by which can be used to reformulate the Sachs equations as d ds ln d A = θ opt, (3) d 2 d A ds 2 = ( σ opt R αβk α k β )d A, (4) d ds (σ optd 2 A) = 2 R αβµνε α k β ε µ k ν d 2 A. (5) We now want to transform Eq. (4) into an equation for the structure distance d S = ( + z)d A as a function of time. By using Eq. (0) we find d S [ln( + z)] d S + id S = 0 (6) 4

6 with i = ( + z) 2 ( σ opt R αβk α k β ) d2 ln( + z). (7) dt2 As we will demonstrate in Sec. 3, the quantity i actually vanishes for spatially flat homogeneous universes. In that case Eq. (6) is solved by d S = to t e ( + z)dt = z 0 (8) [ln( + z)] dz. Even for i 0 the introduction of d S is useful because we can simplify Eq. (6) by treating d S as a function of d S, which results in d 2 d S dd 2 S with boundary conditions at d S = 0 given by = i ( + z) 2 d S (9) d S = 0, dd S dd S =. (20) There is no perfectly natural way of generalizing the concept of a Hubble rate to an inhomogeneous universe. Two operational definitions of a Hubble rate associated with a specific point on a geodesic can be made as generalizations of Eq. (7): H inf = dz dd S, H = dz dd S. (2) Both formulas reduce to the standard Hubble rate for the case of a homogeneous spatially flat universe. While H inf is essentially the quantity that is inferred from observations under the assumption of flat homogeneity, H is the expansion at the source in the direction of the photon emission: by virtue of Eq. (8) we have H = d ln( + z), (22) dt in perfect analogy with Eq. (4); also note that H is just the second coefficient in Eq. (6). With the help of Eqs. (9) and (20) we find H = dd ds S = + H inf dd S 0 i to ( + z) d 2 S dd S = t i ( + z) d S dt. (23) This means that the two definitions of H coincide at the observer, H (t o ) = H inf (t o ) = H o, and that for positive i observations tend to overestimate and for negative i to underestimate 5

7 expansion rates in previous epochs; in particular, for sufficiently large negative i we can perceive acceleration even if it does not take place. As we have seen, someone who ignores the nonvanishing of i (in other words, any cosmologist believing in the standard concordance model) would interpret H inf as the Hubble rate. Furthermore, from Eq. (8) such a person would (wrongly!) infer a time parameter t inf with In fact, H inf and t inf satisfy an analogue of Eqs. (4) and (22): dt inf = dd S + z = d S dt. (24) + z H inf = dz dd S = ( + z) dz dt inf = d dt inf ln( + z). (25) Let us also introduce the deceleration parameters q inf = d ( ), dt inf H inf q = d ( ). (26) dt H By using the chain rule, the definitions of the various quantities and Eq. (6) one can show that they are related via q inf = q + i d S( + z). (27) ż This demonstrates again that negative i can lead to the perception of acceleration even if it does not take place. We can summarize the results of this section in the following way. From the values of the pairs (d S, z) along a given lightlike geodesic, without taking into account the quantity i that encodes the effects of curvature and inhomogeneity, one would infer an expansion history along that geodesic in terms of quantities t inf, H inf and q inf. The actual expansion history along that specific geodesic is encoded by t, H and q. The two sets of quantities are related by Eqs. (23), (27) and H inf dd S = H dd S = dz, (28) H inf dt inf = H dt = d ln( + z). (29) In reality we have at most a single data point (d S, z) for any observed direction, and we require a statistical analysis. As we will see, even H and q (suitably averaged over photon paths) can become quite different from the corresponding results from volume averaging. d S 6

8 3 Homogeneous and irrotational dust universes While all of our results up to now are exact in an arbitrary pseudo-riemannian geometry with a distinguished timelike coordinate, we assume in the following that the metric can be written, in the synchronous gauge, as ds 2 = g αβ dx α dx β = dt 2 + g ij (t, x)dx i dx j ; (30) this is true for any homogeneous spacetime as well as for irrotational dust, where the dust particles have constant space coordinates x i. We want to express our quantities in terms of the spatial 3-geometry with the time-dependent metric g ij. To distinguish it from the spacetime geometry we adopt the convention that an expression with greek indices or at least one index of zero or a left superscript of (4) pertains to the 4-metric g αβ, whereas any other quantity, in particular the Ricci scalar R = Ri, i refers to g ij. The connection coefficients for the metric (30) vanish if two or three indices are 0, and the non-vanishing coefficients are Γ 0ij = 2 0g ij, Γ i0j = Γ ij0 = 2 0g ij, (4) Γ ijk = Γ ijk, (3) with the notation 0 for / x 0 = / t and more generally µ for / x µ, so that d dt = 0 + ẋ i i. (32) The expansion tensor θj i and the scalar expansion rate θ are defined by θj i = 2 gik 0 g kj, θ = θi i = 0 g, (33) g and the shear is the traceless part of the expansion tensor, σ i j = θ i j 3 θδi j, σ 2 = 2 σi jσ j i. (34) The Riemann tensor R αβγδ can be expressed in terms of the expansion tensor and the Riemann tensor R ijkl of the spatial metric g ij : R 0i0j = g ik 0 θ k j θ ik θ k j, (35) R 0ijk = θ ij k θ ik j, (36) (4) R ijkl = R ijkl θ il θ jk + θ ik θ jl, (37) with θ ij = g ik θ k j and with the vertical strokes denoting covariant spatial derivatives. 7

9 We now want to specialize our analysis of photon paths to a metric of the type (30), with the assumption that both the source and the observer are comoving: x i e = const, x i o = const. Since Γ 0 ij = 2 0g ij, the 0-component of the geodesic equation is or, upon division by ( + z) 2 and application of Eq. (0), d 2 t ds ( 0g ij ) dxi dx j ds ds = 0 (38) d( + z) ( + z) 2 ds = 2 ( 0g ij )ẋ i ẋ j. (39) As ẋ µ is light-like and x 0 = t, the spatial part ẋ i must be a unit vector with respect to g ij, whereby the previous equation becomes Similarly we can transform the spatial component g ij ẋ i ẋ j =, (40) d dt ln( + z) = θ 3 σ ijẋ i ẋ j. (4) of the geodesic equation into d 2 x i ds 2 + dt 2θi j ds dx j ds + dx j dx k Γi jk ds ds = 0 (42) ẍ i + θ 3ẋi σ kl ẋ k ẋ l ẋ i + 2σ i jẋ j + Γ i jkẋ j ẋ k = 0. (43) Upon using this, together with (32), in the derivative of Eq. (4), we find d2 dt 2 ln(+z) = ( 0 +ẋ i i ) θ 3 +( 0σ ij +ẋ k k σ ij )ẋ i ẋ j 2σ ij ( θ 3ẋi σ kl ẋ k ẋ l ẋ i +2σ i kẋ k +Γ i klẋ k ẋ l )ẋ j. Note that up to now we have never used the Einstein equations ( ) R αβ (4) R Λ g αβ = 8πG N T αβ. (45) 2 Let us assume that the spatial part of the energy-momentum tensor is proportional to the metric, T ij = g ij T k k /3, and that T 0i = 0. This holds not only in the homogeneous case but also in the general irrotational dust case, where T ij = 0. Then Eq. (45) implies that the spacetime Ricci tensor R αβ must be of the same type, (4) R ij = g ij (4) R k k /3 and R 0i = 0, so that (44) R αβ k α k β = R 00 (k 0 ) g ijk i k j (4) Rk k = ( + z) 2 (R 00 + (4) R k 3 k); (46) 8

10 in the last step we have used k 0 = dx 0 /ds = + z and g ij k i k j = k µ k µ + (k 0 ) 2 = ( + z) 2. With the help of Eqs. (33) (37) this results in 2 ( + z) 2 R αβ k α k β = 3 0θ + R 6 σ2. (47) The traceless spatial part of the Einstein equations amounts to which implies 0 σ ij = θσ ij /3 + 2σ k i σ kj r ij, where 0 σ i j + θσ i j + r i j = 0, (48) r ij = R ij R 3 g ij (49) represents the traceless part of the spatial Ricci tensor. Using this after inserting Eqs. (44) and (47) into (7) we get i = ( + 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. (50) This result is still exact within the irrotational dust framework and also for any homogeneous cosmological model. In the latter case it reduces to i = R/6 = K/a 2 with K {, 0, } so that i/( + z) 2 = K/a 2 o is constant; thereby Eqs. (8), (9) lead to the well known distance formulas that involve sin or sinh functions for K 0. Let us also note that the equation (5) for the optical shear is determined by R αβµν ε α k β ε µ k ν = ( + 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 (5) for any metric of the type (30), as one can ascertain by using similar methods. This expression vanishes for any homogeneous model. 4 Mass-weighted average If we knew the spatial metric g ij in the vicinity of a given lightlike geodesic in an irrotational dust universe, we could now compute the redshift and the structure distance along that geodesic simply by solving Eqs. (4) and (6) with input from Eq. (50) (assuming we are also solving for 9

11 σ opt along the way). In practice we do not know the precise form of the metric and need to rely on a statistical model; in addition we have to make simplifications to keep the computations manageable. As we aim for results beyond perturbation theory, we choose the approach of Ref. [9] for our underlying statistical model. The present section is devoted to a brief summary of the relevant ideas and results. The central concept in this approach is the mass-weighted average [22] X mw (t) = X(x, t)ρ(x, t) g(x, t) d 3 x (52) m D D of a scalar quantity X, where D is a large domain (e.g. all of the visible universe), ρ(x, t) is the local mass density and m D = D ρ(x, t) g(x, t) d 3 x (53) is the mass content of D. For the case of an irrotational dust universe, energy conservation implies ( ρ(x, t) ) g(x, t) = 0 (54) t and therefore 0 X mw = 0 X mw. This makes it possible to evade the technical difficulties that arise with the more common volume average, where averaging and taking time derivatives do not commute. Nevertheless volume averages are easily computed within this approach as here a is the local scale factor defined as X vol = Xρ mw ρ mw = Xa3 mw a 3 mw ; (55) a(t, x) = ( ) ˆρ 3, (56) ρ(t, x) where ˆρ is an arbitrary fixed mass. Then the dust expansion rate can be expressed as and a set of rescaled quantities θ(t, x) = 0ρ(t, x) ρ(t, x) = 3 0a(t, x) a(t, x), (57) ˆρ = a 3 ρ, ˆσ j i = a 3 σj, i ˆR = a 2 R, ˆr j i = a 2 rj i (58) obeys the evolution equations 0 ˆρ = 0, 0ˆσ j i = aˆr j, i 0 ˆR = 2a 3ˆσ jˆr i j i, (59) 0ˆr j i = a ( ˆσi kˆr j k ˆσk j ˆr k i + ) 6 δi j ˆσ l k ˆr k l + a 2 Y ki j k, (60) 0

12 where Y k ij = 3 4 (σk i j + σ k j i) 2 g ijσ k m m σij k. (6) The initial values for these evolution equations can be found by comparison with linear perturbation theory: upon neglecting vector, tensor and decaying scalar modes the space metric g (LPT) ij (t, x) at early times can be expressed in terms of a single time-independent scalar Gaussian random function C(x) as g (LPT) ij (t, x) = a 2 EdS(t) ( δ ij a 2 EdS t 4 3 ) C(x)δ ij + t 2 3 i j C(x) ; (62) here a EdS = const t 2/3 is the standard EdS (Einstein-de Sitter, i.e. flat matter-only FLRW) scale factor. By comparing with section 5.3 of Ref. [8] one finds that this metric is equivalent to a Newtonian gauge metric with Φ = Ψ = C/3. It turns out that the initial conditions for our evolution equations are lim t 0 a t 2 3 = (6πG N ˆρ) /3, (63) ˆσ in (x) = 0, (64) ˆR in (x) = 20 9 (6πG N ˆρ) 2 3 S(x), (65) (ˆr in ) i j(x) = 5 9 (6πG N ˆρ) 2 3 δ ik s kj (x), (66) where S and s kj are the trace and traceless parts of the matrix i j C(x) = S ij (x) = s ij (x) + 3 δ ijs(x) (67) of second derivatives of the function C(x). In this setup it can be shown that ˆR(t) = ˆR in + 2a 4 (t) ˆσ 2 (t) and that the evolution equation of the local scale factor a(x, t) is t ( 0 a) 2 = 8 3 πg N ˆρ a 6 ˆR in + 3 Λ a2 4 9 t in θ( t)a 4 ( t) ˆσ 2 ( t) d t, (68) t t in θ( t)a 4 ( t) ˆσ 2 ( t) d t. (69) As long as one neglects the last term (a 2 Y ki j k) in Eq. (60), the evolution in a given region will depend only on the initial conditions within that region; furthermore, if one chooses a coordinate system in which the symmetric matrix S ij (x) is diagonal then r ij and σ ij will be diagonal in that system at any time t. In this way it suffices to work with the probability

13 distribution for the three eigenvalues of S ij. As shown in Ref. [9], the assumption that C(x) is a Gaussian random field suffices to compute this distribution explicitly in terms of a single dimensionful parameter which is related to the value of an integral that requires an ultraviolet cutoff. Then one can switch to dimensionless units by taking a specific value for this parameter. With the computationally convenient choice that was adopted in Ref. [9] and that will also be used here, one finds S 2 mw = 5, s ij s kl δ ik δ jl mw = 0/3. (70) If one also chooses ˆρ such that 6πG N ˆρ = in the corresponding units then the perturbative series for a starts as a(x, t) = t 2 S(x) t 4 S 2 (x) + 2s ij (x)s kl (x)δ ik δ jl 3 t , (7) 84 where we have neglected cubic and higher orders in perturbation theory as well as second order terms involving higher than second derivatives of C(x). 5 Photon path average Finally we want to connect the distance formula (6), which relies on the values of the quantities H = [ln( + z)] and i along a photon path, with the model of Ref. [9] as summarized above. We propose to do the following. We replace the right-hand sides of Eqs. (4) and (50) by suitable expectation values which we will denote by... pp, where the subscript stands for photon path. The idea is that X pp (t) should be the average of X over all spatial positions x occupied by a photon of a given type (e.g. supernova or CMB) at the time t. Then X = X pp + X with X pp = 0, and by the linearity of Eq. (6) the contribution of X cancels in the limit in which a photon probes the whole probability space within negligible time. Every photon path corresponds to a curve C in x space (the R 3 parametrized by the spatial coordinates x, x 2, x 3 ) that ends at x o. In the flat homogeneus case these curves are just straight lines. If the shapes of these curves were not altered by the presence of inhomogeneities, then our model would tell us how the basic parameters are distributed with respect to the euclidean metric dl 2 = δ ij dx i dx j along such a curve. We will make the simple approximation of assuming the same distribution even in the general case. As a next step we want to move 2

14 on to a description that is based on physical time rather than euclidean length. We denote by v i = dxi dl = ẋ i dt dl the tangent vector to C normalized to euclidean unit length, i.e. δ ij v i v j =. Upon taking the g-norm g ij v i v j of v and using Eq. (40) we find (72) dt = g ij v i v j dl, (73) which reflects the fact that the photon flight time is proportional to the traversed distance as measured with the physical metric g. For any path segment of length dl we average over the three basic parameters of the model (indicated by... mw ) and over all directions v, and weight by the time dt = g ij v i v j dl spent in such a segment. This results in X pp = S 2 X g ij v i v j d 2 v mw S 2 gij v i v j d 2 v mw, (74) where the integrations are taken over the unit sphere S 2 = {v : δ ij v i v j = } in tangent space; if X depends on ẋ i explicitly, we make use of which follows from Eqs. (72), (73). ẋ i = v i gij v i v j (75) Our aim is to compute X pp for the nontrivial coefficients in Eq. (6), i.e. for the cases X = [ln( + z)] and X = i. To this end we require integrals over S 2 of expressions that are polynomials in the v i except for the occurrence of factors of g ij v i v j. Since exact results would involve elliptic functions we work in a basis in which the metric is diagonal and write with Then g ij = ḡ(δ ij + γ ij ) (76) ḡ = g + g 22 + g 33, γ + γ 22 + γ 33 = 0. (77) 3 ( ) λ ( ) λ gij v i v j = ḡ( + γ ij v i v j ) = ḡ λ/2 ( + λ 2 γ ijv i v j +...) (78) on the sphere δ ij v i v j =. For each term in this expansion we require only integrals of polynomials in the v i, such as (v i ) 2n d 2 v = 4π/(2n + ), (v ) 2 (v 2 ) 2 d 2 v = 4π/5, (79) S 2 S 2 (v ) 4 (v 2 ) 2 d 2 v = 4π/35, (v ) 2 (v 2 ) 2 (v 3 ) 2 d 2 v = 4π/05. (80) S 2 S 2 3

15 From now on we simply omit any terms that are of quadratic or higher order in the γ ij. While this may look excessively crude, one can check that even in the extremal cases of one or two vanishing eigenvalues the error is at most around 5%. For the integral in the denominator of (74) this gives, upon using (77), S 2 gij v i v j d 2 v 4π ḡ. (8) According to Eq. (4), [ln( + z)] = θ/3 + σ ij ẋ i ẋ j. Since θ has no direction dependence, θ gij v S 3 i v j d 2 v = θ gij v 3 i v j d 2 v 4π ḡ θ 2 S 3. (82) 2 In evaluating the second term we use the fact that σ ij is diagonal in the same coordinate system in which g ij is: σ ij ẋ i ẋ j g ij v i v j = σk j g ki v i v j gij v i v j = ḡ ( 3 ) ( σi( i + γ ii )(v i ) 2 2 i= ) 3 γ ii (v i ) (83) Upon restricting this to terms linear in γ ij and using the formulas (79) and (77) we get S 2 σ ij ẋ i ẋ j g ij v i v j d 2 v 6 5 π ḡ(σ γ + σ 2 2γ 22 + σ 3 3γ 33 ). (84) Combining our results gives [ln( + z)] pp ḡ(5θ + 4σ γ + 4σ 2 2γ σ 3 3γ 33 ) mw 5 ḡ mw. (85) Next we turn our attention to i pp. Since no direction is singled out, the expressions in the second line of Eq. (50), which are all odd under ẋ i ẋ i, do not contribute after averaging. The optical shear σ opt is determined by Eq. (5). The behaviour for small t o t is easily found to be σ opt 6 (t o t)r αβµν ε α k β ε µ k ν, i.e. well-behaved. Under a 90 rotation ε () ε (2), ε (2) ε () the right-hand side of Eq. (5) changes sign, hence its photon path average vanishes and the behaviour of σ opt resembles a random walk around zero. We will neglect the corresponding term in i. According to Eq. (68), R = a 2 ˆRin + 2σ t 3 a 2 θ( t)a 2 σ 2 d t, (86) where ˆR in = lim t 0 a 2 R. The contribution of ( σ ij θ r ij )ẋ i ẋ j can be treated like that of σ ij ẋ i ẋ j before, resulting in 0 S 2 ( σ ij θ r ij )ẋ i ẋ j g ij v i v j d 2 v 6 5 π ḡ[(σ θ + r )γ +...]. (87) 4 i=

16 With slightly more work we also find and S 2 2σ k i σ kj ẋ i ẋ j g ij v i v j d 2 v 8 5 π ḡ[(σ ) 2 (5 + 4γ ) +...] (88) 2σ ij σ kl ẋ i ẋ j ẋ k ẋ l g ij v i v j d 2 v 6 S 05 π ḡ[(σ) 2 (7 + 8γ ) +...]. (89) 2 Putting the pieces together we obtain i g ij v 4π i v j d 2 v ( ˆRin ḡ S 6a + 4 ) t θ( t)a 2 σ 2 d t a σ [(7σ θ + 7r + 6(σ) 2 )γ +...]. (90) Our formulas rely explicitly on the spatial metric g ij in the diagonal basis. To obtain it from the quantities whose evolution is studied in Sec. 4 we use which implies 2 0 ln g = 2 g 0 g = θ = θ 3 + σ = (ln a) + σ (9) ( g (t) = const a 2 exp 2 t 0 ) σ( t)d t, (92) with analogous expressions for g 22 and g 33. Comparison with Eq. (62) shows that the constant must be the same in each case, and that setting it to corresponds to a normalization where a 2 = a 2 FLRW. 6 Perturbative results Before proceeding to the results of a non-perturbative numerical computation, let us first assume that we are still so close to the EdS case that in most regions perturbation theory provides a good approximation. We work with the dimensionless quantities described at the end of Sec. 4. Again our first goal is the photon path average of the right-hand side of Eq. (4). From Eq. (7) we find (to the same accuracy as there) ( θ(x, t) = 2t + S(x) 6 t 2 3S 2 (x) + 2s ij (x)s kl (x)δ ik δ jl t ). (93) The approximation (8) is valid at linear order, and with Eq. (92) and the fact that σ j i traceless we get I := gij v i v j d 2 v = 4πa + O(2); (94) S 2 5 is

17 O(n) means an expression of n th or higher order in perturbation theory. Since the perturbative expansions θ = θ (0) + θ () + θ (2) + O(3) and I = I (0) + I () + I (2) + O(3) have deterministic leading terms (i.e., θ (0) = θ (0) mw and I (0) = I (0) mw ) and first order terms whose expectation values vanish (i.e., θ () mw = 0 and I () mw = 0), we get θ pp = θi mw I mw = θ (0) + θ (2) + θ() I () I (0) mw + O(3); (95) note that I (2) has dropped out at quadratic order so that Eqs. (70), (7), (93) and (94) suffice for computing θ pp 2t 5 9 t 3 (96) to the same order as a and θ before. The approximation (84) implies σ ij ẋ i ẋ j pp 4 5 σ γ + σ 2 2γ 22 + σ 3 3γ 33 mw (97) at leading (second) order. This can be evaluated via t σ = a 3 a ˆr d t 3 5 t 3 ˆr 3 t 3 s (98) (here and in the following equation we only consider leading orders), 0 γ = g ḡ t t e2 0 σ d t 2 σd t t 2 3 s (99) 0 and Eq. (70); the result is σ ij ẋ i ẋ j pp 8 27 t 3. (00) Combining this with Eq. (96) we obtain H pp 2 3 t + 9 t 3, (0) where the approximation again neglects terms of cubic or higher order in perturbation theory as well as terms with higher derivatives of C(x). In order to compute i pp up to second order in perturbation theory we require the massweighted average of Eq. (90). We begin with ḡ ˆR in a 2 mw ˆR in a mw t 3 S( 6 t 2 3 S) mw = 0 27 S2 mw = 50 27, (02) 6

18 where the approximations neglect contributions of third or higher order in perturbation theory; the linear term has dropped out upon averaging. All other expressions in Eq. (90) are explicitly of quadratic or higher order: with Eq. (98) we find σ t 3 (s ), (03) and Eq. (99) together with a 2 t 0 θa 2 σ 2 d t 6 t 2 3 (s ), (04) r = a 2ˆr 5 9 t 4 3 s (05) implies (σ θ + r )γ +... ( )t 4 3 t 2 3 (s ) = 9 t 2 3 (s ). (06) Combining all contributions and using Eq. (70) we arrive at ( i pp ( )0 3 ) t 2 3 = t 3. (07) 7 Non-perturbative results In this section we present the results of numerical computations performed with GNU octave [23]. We used the Euler method with logarithmic time steps to solve the evolution equations (59) and (69), but we assumed constant ˆr = ˆr in instead of using Eq. (60) (the last term in that equation cannot be described directly within the present model, and the other terms do not affect results very much, at least when volume evolutions are studied [9]). This was done for a large set of initial conditions, and the resulting values for a, σ, r and R were used to evaluate the formulas of Sec. 5, with an appropriate probability measure for each set of initital conditions. More algorithmic details can be found in the appendix of Ref. [9]. In regions that collapse, the treatment in terms of irrotational dust breaks down and it is necessary to give a prescription on how to proceed with them. We followed the standard assumption, as suggested by the virial theorem, that collapsing regions shrink to half of their maximal sizes; somewhat unrealistically we pretended that such regions contract according to the irrotational dust evolution equations until that size is reached. The collapsed regions themselves were then treated in two distinct ways: firstly, by just removing them from the 7

19 statistics, and secondly by keeping them and letting all quantities retain the values that they had in the last moment of collapse. The first approach makes more sense since it is doubtful whether many of the observed photons would have passed through a collapsed region, and also because the strong anisotropies that can occur during collapse should not persist in the virialized regions; nevertheless it is useful to have the other approach as well in order to get an idea of how strongly our results depend on details of modelling. We present our results mainly in the form of figures created by GNU octave [23]. The first three of them display the basic results of the averaging process, and the remaining figures show quantities that are derived from these by using formulas from Sec Figure : Time evolution of ḡ mw /t 2/3 Fig. displays the time evolution of ḡ mw = (g + g 22 + g 33 )/3 mw as computed 8

20 according to Eq. (92), divided by the EdS value of t 2/3. The blue line shows the result for the first scenario (collapsed regions removed from the statistics) and the green (highest) line gives the same curve for the other scenario. While the quotient remains extremely close to in the first case, even in the second case the deviation of the behaviour from the EdS values, as indicated by the straight red line, is not large (note the scale of the vertical axis) Figure 2: Time evolution of Ht Fig. 2 displays Ht over the time t for various versions of the Hubble rate H. As before, the blue and the green line correspond to the results of the non-perturbative computations; more precisely, the highest (green) line gives H pp t for the second and the second highest (blue) line for the first scenario, with H pp computed numerically via Eq. (85). The third line, in black, corresponds to the perturbative result (0), the fourth (yellow) line to Ht as computed via volume averaging, and the final red line shows the constant EdS value of H EdS t = 2/3. 9

21 The strong deviations from the homogeneous case are a consequence mainly of local anisotropy, by the following mechanism. Consider a region R characterized by some specific values of θ and σ ij and pick a frame {e, e 2, e 3 } in which σ ij is diagonal. Assume, without loss of generality, that σ > σ 22 and that originally R had the same diameters along the corresponding directions e, e 2. Even though the overall volume expansion of R is determined by θ, it will expand faster along e and more slowly along e 2, so that after a while R will have a larger extension in the e -direction than in the e 2 -direction. A photon traversing R along e will not only experience a stronger redshift per unit of time spent in R than one moving along e 2, but it will also spend more time in R. The corresponding weighting that favors directions with stronger expansion results in the effect that on average a photon traversing R experiences a higher redshift than the volume expansion of R would suggest Figure 3: Time evolution of i ḡ 20

22 In Fig. 3 the time evolution of i ḡ is displayed for our two non-perturbative scenarios; to be precise, i pp ḡ mw, i.e. the mass-weighted average of the right-hand side of Eq. (90) is shown. The sharply dropping green line corresponds to the second scenario, and the curved blue line to the first one. These results are contrasted with the perturbative result i ḡ 5/27 and the EdS value of i ḡ 0 as represented by the two horizontal lines (in black and red, respectively). Here the differences between the perturbative and non-perturbative results are not only enormous in magnitude but also change the direction of the effect. Once again the main contributions come from terms involving indicators of local anisotropy such as σ ij and r ij, as the form of the defining equation (50) suggests Figure 4: Structure distance over time Fig. 4 differs from the previous ones by relying not only on t e = t but also on t o, the present age of the universe expressed in the dimensionless units of Sec. 4. Here and elsewhere our 2

23 choice was simply to take t o as the time at which H t = (remember that H (t o ) = H inf (t o )). This is suggested by the fact that it seems to be a very good approximation in the case of the ΛCDM model and also close to lower bounds coming from ages of globular clusters; in a more general analysis one should probably also allow for values of H o t o somewhat above. For our first scenario we find t o.35 in this way. The three higher lines show various versions of the structure distance as functions of t = t e [0, t o ]: the highest line (in cyan) shows d S itself, the blue line below corresponds to d S, and the red line that starts late corresponds to an EdS universe with the same value of H o, which would have had a shorter lifetime up to now. The three lower lines correspond in an analogous way to the other scenario, where t o Figure 5: Structure distance over ln( + z) For producing Fig. 5, a plot of various versions of the structure distance over ln( + z), the result of Eq. (85) (as shown in Fig. 2) was integrated to get ln( + z) as a function of 22

24 t, and combined with the values for the structure distance as displayed in Fig. 4. The three lines that start with a higher slope correspond to the first and the other three lines correspond to the second scenario, with the colour coding the same as before. d (EdS) S This plot shows that < d S < d S, with differences of roughly the same size; i.e. the effect of a proper treatment of the second coefficient [ln( + z)] in Eq. (6) is of the same order of magnitude as that of a proper treatment of the third coefficient, the quantity i Figure 6: Deceleration over time Fig. 6 displays various versions of the deceleration parameter over the time t. The colour coding is the same as in the previous plots: the blue and green lines give q for the first and second scenario, respectively, and the lines in cyan and magenta the corresponding values of q inf ; in each case the lines end at our choice for t o. Again the straight red line represents the EdS scenario, where q /2, and the yellow line which shows only a slight downward slope 23

25 displays the values that one gets via volume averaging. Once again we see that the photon path prescription leads to strongly different results, with effects of roughly the same order coming from the more precise treatments of the two non-trivial coefficients in Eq. (6). While all the plots presented so far refer to time in terms of the mathematically convenient but observationally meaningless units of Sec. 4, the following plots use units in which the normalization is changed in such a way that t o = ; of course the distance measures are redefined accordingly Figure 7: Structure distance over time normalized to t o = Fig. 7 is identical to Fig. 4 except for the normalization which was modified via t t/t o, d d/t o. It shows that, with the correct scaling, the predictions of the two different scenarios actually differ less than it appeared originally. 24

26 Figure 8: Structure distance normalized to t o = over z Fig. 8 is a modified version of Fig. 5, the changes being the normalization to t o =, the narrower range of z-values, the use of z instead of ln( + z), and the inclusion of the ΛCDM scenario. Again the red (lowest) line shows the results for an EdS universe and the two highest lines (cyan and magenta) give the corresponding values for the observed structure distance d S. Among the remaining three lines, which are fairly close together, the blue and green ones display d S for the first and second scenario, respectively, while the pear-coloured one gives the redshift-distance relation for a ΛCDM universe with Ω Λ = Somewhat surprisingly, d S is closer to the ΛCDM values than d S here. Nevertheless the first scenario perfoms much better than the EdS case; actually it somewhat overestimates the deviation from EdS. The second scenario, in which collapsed regions are included with the values for [ln( + z)] and i that they had in the last moment of collapse, overestimates these deviations even more strongly. This 25

27 suggests that our models might be improved by introducing a smooth slowing of the collapse (as it happens in reality), with a corresponding smooth transition of [ln( + z)] and i to zero. What have we seen up to now? Considering a universe with Λ = 0 and with distributions of geometric quantities that follow directly from initial conditions based on a Gaussian distribution, and with photons that obey the Sachs optical equations, we have shown that the following facts hold: there is a time t o such that an observer at that time sees a redshift-distance relation remarkably similar to that predicted by the standard ΛCDM scenario, and if the observer analyses the data without taking into account the inhomogeneities, he will infer a Hubble rate H inf such that H inf t o = and a deceleration parameter q inf 0.5. A rough estimate is t o.35 in the dimensionless units of Sec. 4. Let us now use this result to make a statement on the time t ls of last scattering, from which the cosmic microwave background stems. Taking this as the time at which z = 090 and using our programs, we find t ls in our units (with t = 0 the instant at which the singularity would have occurred in a purely matter dominated universe). At this time linear perturbation theory is still perfectly valid so that we can compute density perturbations at last scattering with the help of formulas (7) and (70): ( ) ( ) ρ (a 3 ) = = ρ a 3 2 t 2 3 ls S = 2 ( ) (08) ls ls These are the density perturbations for the total matter, which are dominated by the ones for dark matter. According to Eq. (2.6.30) of Ref. [8], the density perturbations of baryonic matter satisfy ρ B /ρ B = 3 T/T, where T is temperature; using the commonly cited value of 0 5 for the relative temperature fluctuations in the CMB we find that the total density perturbations are roughly 50 times as large as those for the baryonic matter. This fits very well with the fact that dark matter decouples from photons (hence clumps gravitionally) earlier than baryons. Similar values for the ratios of the baryonic versus total density perturbations are required for structure formation; see e.g. Fig. of Ref. [24]. We can turn this argument around: from the density perturbations we see that the time of last scattering cannot have occurred significantly before the time t ls that corresponds to t o.35. But then it is clear that the inhomogeneities will have a significant impact on inferred Hubble and deceleration rates, so that the assumption that a homogeneous universe (with or without a cosmological constant) give correct predictions necessarily breaks down. Conversely, since we do not require a non-zero Λ to account for present observations the simplest assumption is to take Λ = 0. 26

28 8 Discussion and outlook Let us start our discussion with a brief reiteration of our assumptions and conclusions. 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, we found that there is a time t o such that observervations made at that time and interpreted with formulas appropriate to the homogeneous case, would suggest an inferred Hubble rate H inf such that H inf t o, an inferred deceleration parameter of q inf 0.5, and density perturbations at a redshift of 090 that fit well with values required at last scattering to lead to structure formation. In other words, an observer at time t o in such a universe sees essentially what present day cosmologists see, even though Λ vanishes. This is the consequence of a model that has only one parameter (the overall scale) which can be adjusted. Once this parameter has been fixed by any of the three quantities that were just mentioned (and thus t o identified with the present age of the universe), the prediction for either of the other two provides a highly nontrivial test. Our methods have peformed very well on both of them. In order to arrive at these results it is essential to consider the effects of inhomogeneities on light propagation (not just on the evolution of volumes), and to use a formalism that transcends perturbation theory. The main steps involve the derivation of the differential equation (6) for the structure distance d S = ( + z)d A, and the computation of the two non-trivial coefficients H = [ln( + z)] and i that occur in this equation. In the spatially flat homogeneous case H is just the usual Hubble rate and i = 0; otherwise each of these coefficients contributes significantly, with effects of roughly the same magnitude, to the deviations in the values of d S, 27

29 H inf and q inf. The main source of discrepancies from FLRW universes is the local anisotropy, as encoded in the dust shear σ ij and the traceless part r ij of the Ricci tensor, and not so much the inhomogeneity which manifests itself by variations of the expansion rate θ and the spatial Ricci scalar R. While Eq. (6) is valid in an arbitrary geometry in which photons follow light-like geodesics, the subsequent computations required a number of approximations: The matter was modeled as irrotational dust. While this is an excellent approximation during expansion, it would not permit stable structures such as galaxies and clusters as the results of collapse. Our way of treating this problem, by simply assuming that collapse holds at half the maximum size, is certainly somewhat ambiguous. In particular, the differences between the two variants that we chose show that the results do depend on such details. As we argued in the discussion of Fig. 8, a smoother transition to the virialized state would probably improve our results even further. We have replaced statistical quantities by their expectation values in order to arrive at a description in which distance can be seen as a function of redshift, as in homogeneous models (cf. the first paragraph of Sec. 5). From the set of supernova data it is clear that this is a gross oversimplification. We assumed a distribution of photon paths in x space (the space in which our matter is at rest, which starts out as being almost perfectly euclidean) that was the same as if the photons moved along straight lines in that space. While exact evolution equations were used for the local scale factor a, the shear σ ij and the Ricci scalar R, the evolution of the traceless part r ij of the Ricci tensor was simplified by ignoring the right-hand side of Eq. (60). In our analysis of expressions that arise upon taking photon path averages, we have neglected terms of quadratic or higher order in γ ij (a scaled version of the traceless part of the metric g ij ). For reasons that we discussed after Eq. (85) we ignored Weyl focusing, i.e. the contribution of the optical shear σ opt. For the numerical treatment the time axis and the probability distribution for the background parameters were discretized. The resulting errors are, however, much smaller than those coming from the other approximations. 28

30 These points imply that we probably should not trust our results to hold to an accuracy of single percent level. Nevertheless it seems extremely unlikely that the combination of the resulting errors would be large enough to account for our results, and even more incredible that they would conspire to give two independent correct predictions. There are some discrepancies between the present work and parts of the literature that need to be addressed. It is sometimes claimed that redshift-distance relations in an inhomogeneous universe are mainly sensitive to a volume averaged geometry [4, 6, 7]. It is likely that the corresponding arguments do not properly account for the strong effects from dust shear that we found. In particular, the mechanism described after Fig. 2 relies on the fact that a generic photon will pass through many regions both with positive and with negative values of σ ij ẋ i ẋ j, which normally will not occur in simple toy models. The overall size of effects that we found is significantly larger than in what is perhaps the most advanced study up to now [4, 5]. But this is not a contradiction, since the conclusions of Refs. [4, 5] come from computations up to second order, whereas our methods go beyond perturbation theory. Indeed our calculations indicate that second order perturbation theory can be trusted up to roughly one third of the present age of the universe (cf. Figs. 2 and 3). Note, in particular, that the quantity i starts with the wrong sign for mimicking acceleration and retains this sign perturbatively, but makes a spectacular reversal non-perturbatively. Let us finally mention possible directions of future work. Clearly each of the aforementioned approximations deserves efforts on improvement; this spans the whole range from theory via refined algorithms to the use of more computer power. In particular, with respect to the second item one could try to formulate a stochastic model that would predict a joint distribution for the pair (d S, z). Our comparisons with known facts took place in the range of small to moderate redshifts (i.e. in the range of supernova data) and we have largely ignored other pillars of the ΛCDM model such as the cosmic microwave background (except for our considerations on density perturbations). Because of Etherington s theorem and the fact that every CMB photon experiences the same late time anisotropies as a supernova photon, we would expect consistency, but this should certainly be considered in greater detail. Acknowledgements: It is a pleasure to thank Anton Rebhan and Dominik Schwarz for helpful discussions, and Phil Bull for correspondence. 29

31 References [] D. J. Schwarz, astro-ph/ [2] R. Durrer, Phil. Trans. Roy. Soc. Lond. A 369, 502 (20) [J. Cosmol. 5, 6065 (20)] [arxiv: [astro-ph.co]]. [3] T. Mattsson, Gen. Rel. Grav. 42, 567 (200) [arxiv: [astro-ph]]. [4] S. Rasanen, JCAP 0902, 0 (2009) [arxiv: [astro-ph]]. [5] C. Clarkson, G. F. R. Ellis, A. Faltenbacher, R. Maartens, O. Umeh and J. P. Uzan, Mon. Not. Roy. Astron. Soc. 426, 2 (202) [arxiv: [astro-ph.co]]. [6] P. Bull and T. Clifton, Phys. Rev. D 85, 0352 (202) [arxiv: [astro-ph.co]]. [7] K. Bolejko and P. G. Ferreira, JCAP 205, 003 (202) [arxiv: [astro-ph.co]]. [8] O. Umeh, C. Clarkson and R. Maartens, Class. Quantum Grav. 3, (204) [arxiv: [astro-ph.co]]. [9] I. Ben-Dayan, R. Durrer, G. Marozzi and D. J. Schwarz, Phys. Rev. Lett. 2, 2230 (204) [arxiv: [astro-ph.co]]. [0] S. Bagheri and D. J. Schwarz, JCAP 40, no. 0, 073 (204) [arxiv: [astro-ph.co]]. [] R. K. Sachs, Proc. Roy. Soc. Lond. A 264, 309 (96). [2] C. C. Dyer and R. C. Roeder, Astrophys. J. 80, L3 (973). [3] I. Ben-Dayan, M. Gasperini, G. Marozzi, F. Nugier and G. Veneziano, JCAP 204, 036 (202) [arxiv: [astro-ph.co]]. [4] I. Ben-Dayan, M. Gasperini, G. Marozzi, F. Nugier and G. Veneziano, Phys. Rev. Lett. 0, no. 2, 0230 (203) [arxiv: [astro-ph.co]]. [5] I. Ben-Dayan, M. Gasperini, G. Marozzi, F. Nugier and G. Veneziano, JCAP 306, 002 (203) [arxiv: [astro-ph.co]]. [6] G. Fanizza, M. Gasperini, G. Marozzi and G. Veneziano, arxiv: [astro-ph.co]. [7] M. Gasperini, G. Marozzi, F. Nugier and G. Veneziano, JCAP 07, 008 (20) [arxiv:04.67 [astro-ph.co]]. [8] S. Weinberg, Cosmology, Oxford Univ. Pr. (2008). [9] H. Skarke, Phys. Rev. D 90, no. 6, (204) [arxiv: [astro-ph.co]]. [20] I. M. H. Etherington, Phil. Mag. 5, 76 (933), reprinted in Gen. Rel. Grav. 39, 055 (2007). 30

32 [2] N. Straumann, General Relativity, Second Edition, Springer (203). [22] H. Skarke, Phys. Rev. D 89, no. 4, (204) [arxiv: [astro-ph.co]]. [23] Octave community, GNU Octave 3.6.2, [24] S. Naoz and R. Barkana, Mon. Not. Roy. Astron. Soc. 362, 047 (2005) [astro-ph/050396]. 3