text:
Dark energy as a mirage
Teppo Mattsson1,2,1
1 Helsinki Institute of Physics, P.O. Box 64, FIN-00014 University of Helsinki, Finland
2 Department of Physical Sciences, P.O. Box 64, FIN-00014 University of Helsinki, Finland
11E-mail:
Motivated by the observed cosmic matter distribution, we present the following conjecture: due to the formation of voids and opaque structures, the average matter density on the path of the light from the well-observed objects changes from ΩM≃1 in the homogeneous early universe to ΩM≃0 in the clumpy late universe, so that the average expansion rate increases along our line of sight from EdS expansion Ht≃2/3 at high redshifts to free expansion Ht≃1 at low redshifts. To calculate the modified observable distance-redshift relations, we introduce a generalized Dyer-Roeder method that allows for two crucial physical properties of the universe: inhomogeneities in the expansion rate and the growth of the nonlinear structures. By treating the transition redshift to the void-dominated era as a free parameter, we find a phenomenological fit to the observations from the CMB anisotropy, the position of the baryon oscillation peak, the magnitude-redshift relations of type Ia supernovae, the local Hubble flow and the nucleosynthesis, resulting in a concordant model of the universe with 90% dark matter, 10% baryons, no dark energy, 15 Gyr as the age of the universe and a natural value for the transition redshift z0=0.35. Unlike a large local void, the model respects the cosmological principle, further offering an explanation for the late onset of the perceived acceleration as a consequence of the forming nonlinear structures. Additional tests, such as quantitative predictions for angular deviations due to an anisotropic void distribution and a theoretical derivation of the model, can vindicate or falsify the interpretation that light propagation in voids is responsible for the perceived acceleration.
Inhomogeneous Cosmological Models, Dark Energy, Cosmology, Gravitation
††preprint: HIP-2007-64/TH
The widely adopted framework of modern cosmology is the spatially homogeneous and isotropic Friedman-Robertson-Walker spacetime, with the growth of structure described as linear perturbations evolving on the smooth FRW background. Accordingly, the standard model of cosmology is built on the assumption that the effects of the evident nonlinear inhomogeneities on the detectable light average out222Apart from gravitational lenses that are occasionally taken into account as additional corrections. over cosmological distances. This assumption – although perhaps in concordance with Newtonian intuition – lacks a convincing demonstration within general relativity, not to mention a mathematically rigorous proof. In fact, the assumption was questioned already in the 60's by Zel'dovich [1] , Feynman333Apparently, R. P. Feynman was also one of the pioneers of this subject, as [3, 4] cite a colloquium by Feynman in 1964., Bertotti [2] , Gunn [3] , Kantowski [4] and in the 70's by Dyer & Roeder [5, 6] , who suggested that the clumpiness of matter in the real universe has consequences on the observable distance-redshift relations that the simplified FRW description fails to capture.
At the time of this pioneering research, the cosmological observations were too inaccurate to distinguish the predictions of the proposed inhomogeneous models from the simple and elegant FRW solutions. Consequently, the efforts towards a more thorough description of the effects of the nonlinear structures on the distance-redshift relations never became part of the mainstream research. Although the structure formation itself has since then developed into a major part of modern cosmology, the emphasis is usually given either to the general relativistic but perturbative description of structures on large scales, or to the Newtonian description of the small scale structure e.g. in the context of galaxy formation.
At present, the unparalleled precision of the recent cosmological observations has raised the debate about the effect of the nonlinear structures on the distance-redshift relations once again topical. Indeed, the observations indicate the existence of inhomogeneities that seem to be on large enough scales to have cosmological significance but too strong to be described within the linear theory — in particular the observed voids and the clustering of matter as filaments of opaque galaxies between the voids [7, 8, 9, 10, 11, 12] . The increase in the precision of the data naturally demands a corresponding improvement in the theoretical model; otherwise there is an increased risk of drawing incorrect conclusions from the observations.
In fact, the discovery that the standard FRW cosmology needs a fine-tuned dark energy component [13, 14, 15] in order to account for observations [16, 17, 18, 19] , suggests that the precision of the data may already have surpassed the precision of the employed theoretical model. This is also backed up by a mysterious coincidence, pointed out by Schwarz [20] and Wetterich [21] and elaborated by Räsänen [22, 23, 24, 25] : the perceived increase in the cosmic expansion, ascribed to dark energy in the standard cosmology, happens to take place during the same era when nonlinear structures start to form at cosmologically significant scales.
Whereas the fine-tuning of dark energy only indicates that something fundamental may be lacking from the standard picture, the coincidence problem suggests a causal connection between cosmic acceleration and structure formation. Perhaps the most natural explanation would be, that the standard FRW description breaks down when the nonlinearities become dominant in the late universe and the need for a tiny cosmological constant is a manifestation of this breakdown, not evidence for dark energy [22] . In recent years, more and more authors have addressed this issue and substantially increased the general understanding of the subject, but crucial open questions still remain [26] . In this work, we address the issue by considering a phenomenological model that aims to: 1) clarify the physical interpretation for how the structure formation can actually mimic dark energy, and 2) provide a quantitative fit for the observations.
The paper is organized as follows. In Sects. 2.1–2.3, we review three physically independent mechanisms for how the cosmic inhomogeneities can induce accelerated expansion along our line of sight: global acceleration due to spatial variations in the expansion rate, faster local expansion rate due to a large local void, and biased light propagation through voids that expand faster than the average; possibilities to describe the total effect of the inhomogeneities have been discussed in Sect. 2.4. After reviewing the original Dyer-Roeder method in Sect. 2.5, and motivating the need to modify it, we introduce a generalized Dyer-Roeder method in Sect. 3.2 to allow for the total effect of the inhomogeneities on the observable distance-redshift relations. The distance-redshift relations are then calculated for an inhomogeneous model, based on the conjecture that the average matter density on the path of the light from the observed objects changes from ΩM≃1 in the homogeneous early universe to ΩM≃0 in the clumpy late universe. The physical foundations of the conjecture are summarized in Sect. 3.1. In Sect. 3.3, we construct a simple description of the structure formation in terms of a function α(z) that measures the deviation of the matter density along our line of sight from the FRW matter density. A comparison of the model with cosmological observations is performed in Sect. 3.4, and a discussion of the results is given in Sect. 3.5. Finally, Sect. 4 contains our conclusions.
2 Inhomogeneous generalizations of the FRW cosmology
Formally, the cosmic inhomogeneities pose the following problem: we have a physical system, the universe, with >1080 degrees of freedom but we want a model with ≲10 degrees of freedom, represented by the cosmological parameters. For a mathematical description of the universe, we thus need a coarse graining map: 1080 dof↦10 dof, that preserves the relevant physical structure of the spacetime but removes its mathematical complexity. As our cosmological knowledge is firmly based on observing light, the relevant structure is the correct optical properties of the universe. That is, we want the observable distance-redshift relations to be unaltered by the coarse graining.
A common critique against acceleration from inhomogeneities is that linearly perturbed FRW solutions seem to describe well even highly nonlinear density contrasts, δρ/¯ρ∼1030, while still having insignificant backreaction [27] . This is because the relevant perturbed gravitational potential can satisfy Φ≪1 even for the high density contrasts δρ/¯ρ≫1. However, this argumentation overlooks the crucial part of realistic nonlinear structures, namely inhomogeneities in the expansion rate [23] . Even a relatively small spatial variation in the expansion rate, δθ/¯θ∼O(0.1), can invalidate the linear perturbation theory by rendering the uniformly expanding FRW background inapplicable. Sometimes the assumption δθ/¯θ≪1 is hidden in the requirement of small peculiar velocities [28] , which appears to restrict the spatial variations in the expansion rate to be small as well [29] . As we next discuss, O(0.1) spatial variations in the expansion rate can indeed induce accelerated expansion via various mechanisms.
2.1 Backreaction due to inhomogeneous expansion
A central question in constructing the coarse grained description of the universe is how to average relativistic gravitational systems within Einstein's theory. As pointed out by Shirokov and Fisher [30] and later made more popular in the observational cosmology programme by Ellis and collaborators [31, 32, 34, 33] , it seems physically more correct to first calculate the Einstein field G(g) for the exact metric g and only then average ⟨G(g)⟩, than to calculate the Einstein field for an averaged metric G(⟨g⟩) as in the standard FRW cosmology. The reason is that the Einstein field is more closely related to physical quantities whereas the metric corresponds to a gravitational tensor potential, whose derivatives determine the physics. In general relativity, the field G depends nonlinearly on the metric g, so its evaluation does not commute with averaging and the non-vanishing difference, ⟨G(g)⟩−G(⟨g⟩)≠0, gives rise to what is known as the nonlinear backreaction; see [22, 23, 24] . Hence the issue is not only a conceptual one, but in general, the two approaches yield identical predictions only in the absence of nonlinear inhomogeneities.
Even if performed in the above explained order, the averaging is not free of problems. Firstly, the only practicable way to average tensors seems to be the manifestly coordinate-dependent averaging of components: ⟨G⟩≡⟨Gμν⟩. As we would like the coordinate-invariance of general relativity to hold also for the coarse-grained or averaged quantities, only the inherently invariant rank 0 tensors or scalars appear to have well-defined averages. Secondly, the desire in cosmology to divide the universe into temporal and spatial parts leads to the issue of foliation dependence: in order to take spatial averages, we must artificially break the symmetry between time and space inherent in general relativity.
In 1999, Buchert proposed a way to circumvent, or at least alleviate, these problems [35] . By using the fact that a part of the Einstein tensor equation can be decomposed into scalar equations, he derived a set of spatially averaged equations, now known as the Buchert equations, describing the averaged dynamics of a general irrotational dust universe. Moreover, he foliated the spacetime by using the proper time of the dust as a global time coordinate; a physically well justified choice in a dust universe, also supported by a comparison with exact observables performed in Ref. [36] .
To see how the backreaction can affect the observations, consider the Buchert acceleration equation [35] :
3¨aa=−4πG⟨ρ⟩+Q , (1)
where a(t) is a generalized scale factor, ρ is the matter density and the difference between Eq. (1) and its homogeneous FRW counterpart is known as the dynamical backreaction
Q≡23(⟨θ2⟩−⟨θ⟩2)−2⟨ σ2⟩ , (2)
where σ2 is the shear scalar and θ is the expansion scalar. The backreaction (2) in the Buchert equations explicitly demonstrate that averages of inhomogeneous quantities do not evolve in time like the corresponding homogeneous quantities.
From Eq. (1) it is obvious that, regardless of decelerating locally everywhere, with large enough backreaction Q>4πG⟨ρ⟩, the average expansion can accelerate without an exotic fluid with negative pressure or a modification of gravity. Physically, the global acceleration is possible because the volume can become dominated by fast-expanding regions [24] . This is realized with large enough variance of the expansion rate, if the counterbalancing average shear is not too large. The variance becomes large when contracting (θ<0) and expanding (θ>0) regions coexist. This is exactly what one would expect in the late universe with structures forming via gravitational collapse [37, 38] , so it has been conjectured that the average acceleration could explain the cosmological observations [23] .
To clarify the essential point, consider two disconnected decelerating regions U and O that have initially the same volumes Vi. Let the region U represent an underdensity that initially expands at the rate θi>0 and let O represent an overdensity that initially contracts at the rate −θi<0. Clearly, the volume-averaged expansion rate is initially zero: ⟨θ⟩i=(Viθi+Vi(−θi))/2Vi=0. Later, the region U has expanded to take up a volume VU>Vi which is still expanding, θU>0, though slower than initially, θU<θi, whereas the region O has shrunken to a small virialized structure with volume VO≪VU which is essentially static, θO≃0. In the process, the volume-averaged expansion rate has become positive: ⟨θ⟩=(VUθU+VOθO)/(VU+VO)≃θU>0, confirming that the global expansion has accelerated regardless of decelerating locally at each point. By taking the junction conditions between the different regions into account, the emergence of the acceleration has been verified within exact solutions of general relativity [38, 39] .
The current studies suggest that while the dynamical backreaction (2) can be significant, perhaps increasing the average expansion by a factor ∼1.3, it may not be large enough to account for the required total increase factor ∼1.5 alone [40, 41, 42, 43, 44, 45] ; however, it has been argued that the approximations in the current calculations may underestimate the backreaction [24] . In Sect. 2.3 we propose that, due to the formation of opaque structures, the volume-averages do not necessarily capture the total effect of the inhomogeneities but the cosmic expansion may increase more rapidly along our line of sight to the observed objects than the global volume-average.
2.2 A large local void
When looking at the large-scale structure of the universe, perhaps the most eye-catching feature is that the major part of the universe appears to be taken up by voids of size O(10)...O(100) Mpc [7, 8, 9, 10, 11] , nearly empty regions expanding faster than the whole universe on average. Even larger voids could exist: it has recently been suggested that the correlation between the cold spot in the CMB and the deficit of radiogalaxies in the same direction might be evidence for a void of size ∼300 Mpc [12] ; though see [46] . It could also be possible that we would happen to live inside such a void [47, 48, 49, 50] , also known as Hubble Bubble in this context.
Clearly, if we were sitting in a special spatial location, such as inside a large local void, the global spatial averages may fail to describe our cosmological observations. In that case, an option444Another option could be to consider a different averaging scale for each redshift [36] . is to give up the notion of a global scale factor and study an inhomogeneous metric; a particularly useful metric for describing voids is the spherically symmetric Lemaître-Tolman-Bondi metric, an exact solution of the Einstein equations that Lemaître discovered in 1933 [51] ; for a review, see [52] . During the recent years, the LTB solution has been actively exploited in studying voids and the effects of inhomogeneities on the cosmological observations [53, 54, 55, 48, 49, 47, 56, 57, 58, 59, 60, 61, 62, 63] .
A key notion in understanding the physical basis of the inhomogeneity induced perception of acceleration is that the cosmological observations are made along our past light cone. A way to illustrate this is to consider the directional derivative along the light cone
ddz=∂xi∂z∂∂xi+∂t∂z∂∂t≈H−10(∂∂r−∂∂t) , (3)
where the approximation in the last step is more accurate for small distances, r≪H−10, but the sign is correct even for larger r. Eq. (3) tells us that negative radial variation roughly corresponds to positive time variation and that their relative contributions to the total variation w.r.t. the observable redshift cannot be distinguished. This is natural since by looking at a source, we simultaneously look into the past and spatially further.
Within the LTB model, the argument can be made more precise by comparing the expression of the luminosity distance in a matter dominated local void [36]
dLTBL(z)=H−10(0)[z+(14−H′0(0)H20(0))z2+
+(−18+13H′0(0)H20(0)+2H′20(0)H40(0)−12H′′0(0)H30(0))z3+O(z4)] , (4)
where ′≡∂/∂r, with its expression in the homogeneous and flat ΛCDM model:
dΛCDML(z)=H−10[z+(14+34ΩΛ)z2+(−18−ΩΛ+98Ω2Λ)z3+O(z4)] . (5)
The comparison between Eqs. (2.2) and (5) confirms that negative radial variation of the expansion rate, H′0(0)<0, is needed to mimic positive cosmological constant, ΩΛ>0, or to induce accelerated expansion along our line of sight. In other words, for the expansion rate to increase towards us along the past light cone, the LTB expansion function H0(r) must decrease as r grows. This is exactly what an observer inside a void would see, and has been shown to account for various cosmological observations without local acceleration of the expansion [55, 58, 56, 60, 61, 62] .
However, the explanation of the perceived cosmic acceleration as an effect induced by a local void does not come for free. Firstly, to respect the cosmological principle, we should not sit in the center of a spherically symmetric void, like fitting a local void for the observations seem to require; e.g. for the LTB model to remain consistent with the observed CMB dipole without a fine-tuned peculiar velocity, we should reside within a few percent of the size of the void from its center [64] . In addition, even if the void were not perfectly spherical and even if we were not near the center of the void, the cosmological principle would still be, at least weakly, violated: there are less galaxies in a void and if born in a random galaxy, it is much more probably located in an environment with lots of galaxies than in a relatively empty void. Another issue is that the required size of the local void seems to be a few times bigger than the typical size of the observed large voids. Moreover, the existence of such a void has not been observationally confirmed, although, due to most of the matter in the universe being dark, it may not be easy to disprove it either. In any case, without a natural explanation for these issues, the scenario with a large local void – if taken literally – does not appear likely. Instead, if considered to describe observations averaged over the celestial sphere, the LTB solution may still have relevance as an effective description for light propagation through several smaller voids.
2.3 Selection effects due to opaque structures
Even if we were not located inside a large local void, the observed cosmic network of smaller voids can still significantly affect the observable distance-redshift relations. Firstly, as discussed in Sect. 2.1, the spatial variations in the expansion rate can increase the volume-averaged global expansion rate. In addition, a complementary aspect we want to point out in this work is that the opacity of the cosmic structures can bias the observations, such that the light we see has traveled mostly in freely expanding voids at late times, and thereby affect the observed distance-redshift relations, independently from the dynamical backreaction. Although it may be difficult to distinguish the observational consequences of the selection effects and the backreaction, they are in principle distinct mechanisms: one could have large opaque structures that induce selection effects but give negligible backreaction or, on the other hand, perfectly transparent but highly nonlinear structures that produce backreaction but no selection effects. Naturally, in the real universe, both mechanisms may have significant effects on the observations. Mathematically, the difference between the purely dynamical backreaction and the biased light propagation in voids can be seen as the difference between the volume-averaged expansion rate and the average expansion rate along our line of sight.
The selection effects in the observations of cosmic light can be justified by the fact that the emptier a region in space, the easier it is to see an object through it. The bias can be further categorized roughly as two different cases: 1) an opaque structure on the foreground simply screens the further object behind it, or 2) the further object is seen but, due to foreground contamination, not clearly enough that the object would contribute to the high quality data. The first case is perhaps relatively rare, since only the nearest galaxies, including the Milky Way, cover a notable fraction of the celestial sphere. Instead, the second case may be more relevant as, due to their long distance, the cosmological objects tend to be dim, which means that they can be harder to detect if there are bright foreground objects nearby on the sky. In addition to the brightness of the foreground, the contamination can arise from gravitational lensing. An important implication of the contamination is that the effective area screened by the foreground objects may be significantly larger than their physical cross-section. As an example, large type Ia supernova surveys typically cover only ∼10−4 part of the sky [65] . It thus appears conceivable that the well-observed supernovae may consist more of objects with less than average intervening material on their foreground.
In addition to the kind of bias discussed above, there may be other kind of selection effects, arising from the fact that the dimmer the object the harder it is to detect. It appears non-trivial how this would affect the observed distance-redshift relations, since on one hand it could favor objects with higher luminosity but on the other hand disfavor objects with high redshift. In any case, for the statistically most relevant brighter supernovae at lower redshifts, we would expect this effect to be less significant. Furthermore, if the type Ia supernovae would be perfect standard candles, the bias towards higher intrinsic luminosity should not exist at all. In addition to selection effects, there could also be other systematic effects related to observational techniques [66] . Here we simply assume that such effects do not play a significant role.
Unfortunately, due to the complexity of the cosmological structures, it appears extremely difficult to derive quantitative estimates for the significance of the selection effects. A possible way to obtain such estimates could be a computer simulation where one studies the difference of the average expansion rate along our line of sight relative to the volume-averaged global expansion rate. Altogether, we can currently neither vindicate nor rule out the importance of cosmic selection effects from opaque structures.
2.4 Total effect of the inhomogeneities
As we have discussed in Sects. 2.1–2.3, it appears very difficult to derive from theory a realistic inhomogeneous model that would take into account all the relevant effects of the inhomogeneities. Instead of dealing with these difficulties, we employ a phenomenological approach based on physically justified assumptions in Sect. 3. The model contains a transition redshift to the void-dominated era as a new free parameter and the applied mathematical framework is provided by a generalized Dyer-Roeder method, introduced in Sect. 3.2. In this approach, the observable or optical distance-redshift relations are determined by the average properties of the space along our line of sights to the objects, instead of global volume-averages. The aim of the model is to provide only a simple description of what the overall effect of the cosmic inhomogeneities on the observed distance-redshift relations may be. Although based on assumptions, in the end, one can use the naturalness of the required value for the new parameter to judge the viability of the model, e.g. compared to the standard FRW cosmology with Λ as the additional phenomenological parameter.
Perhaps the first to discuss light propagation in the clumpy universe was Zel'dovich in 1963 [1] . Before that, the idealized FRW geometry had been used as the basis in deriving theoretical predictions for the observable distance-redshift relations. Later in the 60's, Zel'dovich's idea was elaborated at least by Bertotti [2] , Gunn [3] and Kantowski [4] . However, it was the works by Dyer and Roeder in the 70's that made the idea more famous [5, 6] ; consequently, the distance-redshift relations allowing for the effects of clumping of matter on light are usually referred to as the Dyer-Roeder relations.
It seems that the interest towards the effects of clumpiness on light propagation suffered a hiccup after 1976, when Weinberg argued against the use of the Dyer-Roeder version of the distance-redshift relations in place of the conventional FRW ones [67] . He proposed that the effect of light propagating in the empty intergalactic space would, on average, be canceled by an equal but opposite effect due to gravitational lensing caused by the matter clumps. However, Weinberg's argument has later been challenged [68] . In addition, we want to point out that the original Dyer-Roeder relations do not take two crucial physical properties of the universe into account: the inhomogeneities in the expansion rate and the growth of the inhomogeneities. Indeed, in Sect. 3 we demonstrate that the evolving inhomogeneous expansion can induce much stronger corrections to the observable distance-redshift relations than the mere clumpiness of matter in the original formulation of Dyer and Roeder – and hence unlikely to get counterbalanced by the gravitational lensing.
2.5 The original Dyer-Roeder method
Before presenting the generalized version of the Dyer-Roeder method in Sect. 3.2, we introduce the original formulation here for reference and to justify the modifications. For more discussion on the method, see Refs. [5, 6, 69, 70, 71, 72] .
A good starting point for quantitative considerations of light propagation in the inhomogeneous universe is the exact Sachs optical scalar equations [73] . They correspond to the scalar part of the Einstein tensor equation on null geodesics. The first one reads as
∂~θ∂λ+12~θ2+~σ2+Rμνkμkν=0 , (6)
where ~θ≡V−1∂V/∂λ is the expansion of the bundle of light rays taking up a volume V; λ is the affine parameter along the null geodesics xμ(λ) that gives the energy of the photons as measured by an observer with four-velocity uμ via E(λ)=−gμνuμ∂xν/∂λ; Rμν is the Ricci curvature tensor, whereas ~σ2 represents optical shear and kμ≡∂xμ/∂λ is the wave-vector of the light rays. The other Sachs equations are trivially satisfied when the optical shear is neglected as in the Dyer-Roeder approach and which is also an in-built assumption of the standard FRW description.
To rewrite Eq. (6) in terms of observables, we use ~θ=A−1∂A/∂λ where A is the cross-sectional area of the light bundle and collect the derivatives into one term as
∂~θ∂λ+12~θ2=2√A∂2√A∂λ2 , (7)
to obtain
2dA∂2∂λ2dA+8πGTμνkμkν=0 , (8)
where the Einstein equation has been employed in the form Rμν=8πGTμν+gμνR/2 with the facts that kμ is null, gμνkμkν=0, and that the angular diameter distance is proportional to the square root of the cross sectional area of the light bundle, dA∝√A.
By using ideal fluid to describe the energy content of the universe, we have the energy tensor Tμν=(ρ+p)uμuν+pgμν where uμ=δμ0 in the comoving coordinates and ρ=ρm+ρr stands for the energy density of matter plus radiation and p=pr is the radiation pressure. Hence, with the help of gμνuμkν=−E(λ), Eq. (8) yields
∂2∂λ2dA(λ)+4πG(ρ+p)E20(1+z)2dA(λ)=0 , (9)
where we have used the definition of the redshift, E(λ)/E0≡1+z, with E0 as the observed energy of the photons today. We still want the affine curve parameter λ in terms of the non-affine but observable redshift z:
∂∂λ=∂t∂λ∂z∂t∂∂z=E(λ)[−(1+z)H(z)]∂∂z=−E0(1+z)2H(z)∂∂z , (10)
where H(z) is the Hubble expansion along the path of the light and it has been assumed that the relation between the auxiliary scale factor and the redshift, a(0)/a(z)=1+z, is a good approximation and that the proper time t is independent of the spatial location (which is only possible if the vorticity of matter vanishes). Finally, by using Eq. (10) we can write Eq. (9) as
H(z)∂∂z[(1+z)2H(z)∂∂zdA(z)]+4πG[ρ(z)+p(z)]dA(z)=0 , (11)
where the dependence on the energy of the photons E0 canceled out.
By using the general relation between the angular and luminosity distances,
dL(z)=(1+z)2dA(z) , (12)
which holds for geodesic light in any spacetime as proved by Etherington in 1933 [74, 75, 52] , Eq. (11) can be transformed into the corresponding equation for the luminosity distance:
(1+z)2H(z)∂∂z[(1+z)2H(z)∂∂zdL(z)(1+z)2]+4πG[ρ(z)+p(z)]dL(z)=0 . (13)
The essential point Dyer and Roeder made was that, due to Eqs. (11) and (13) determining the optical properties of the universe, the matter density ρm should be interpreted as the average density of the intergalactic regions where the observed light has traveled. To describe this, they introduced a constant smoothness parameter α, representing the fraction of the matter density along our line of sight ρm to the volume-averaged matter density ¯ρm of the whole universe in the FRW description. That is, ρm≡α¯ρm in Eqs. (11) and (13), with the value α=0 corresponding to a late universe where all the matter is concentrated into opaque clumps and the value α=1 to the smooth early universe. Clearly, α can then be something between these two extremes.
Eqs. (11) and (13) are second order differential equations for dA(z) and dL(z) with the initial conditions: dA(0)=0, dA′(0)=H−10 and dL(0)=0, dL′(0)=H−10, where ′≡∂/∂z. Thus, we can easily write the solution to Eq. (13) as a power series in z:
dDRL(z)=H−10[z+14z2+(−18+1−α4)z3+O(z4)] , (14)
where it has been assumed that the Hubble expansion has the flat FRW form with pressureless matter as the only source, H(z)=H0(1+z)3/2, since the radiation is negligible at small redshifts where the solution (14) would only hold anyway.
By comparing the Dyer-Roeder luminosity distance (14) with its counterpart in the homogeneous ΛCDM model (5), we see that there is no way α could mimic Λ, as it does not even appear in the all-important second-order term. In fact, the effect of clumping via α<1 seems to effectively increase the deceleration of the expansion in concordance with the original results of Dyer and Roeder [5, 6] .
However, on physical grounds, one could expect just the opposite result: due to the local gravitational attraction of matter, a shearless and irrotational region of space with low mass density should decelerate less than a denser one. This can also be seen from the Raychaudhuri equation, an exact scalar part of the Einstein equation [52] :
−1V1/3d2dτ2V1/3=4πG3ρm+13(σ2−ω2) , (15)
where σ2 and ω2 represent the shear and vorticity of matter respectively, τ is the proper time of the matter particles and V is a local volume element, making V1/3 a generalized scale factor. As expected, in a region far enough from the high-density filaments so that σ2−ω2≪4πGρm, the deceleration or the left hand side of Eq. (15) is the smaller the lower the local matter density ρm.
We argue that these physically reasonable conclusions do not show up in the solution (14) due to the incomplete description of the inhomogeneities in the original Dyer-Roeder method. Firstly, matter density ρm is not the only quantity with inhomogeneous distribution, but the local Hubble expansion along our line of sight H(z) can differ from the FRW value as well. In fact, the effect of the expansion rate on the distance-redshift relations is more dominant than the effect of the matter extent, because the expansion is mainly responsible for the redshift. This has been demonstrated within the LTB model in Ref. [56] and is manifest already in the distance-redshift relations of the FRW models:
¯dL(z)=(1+z)H−10√|1−Ω0|Sk[√|1−Ω0|H0∫z0d~zH(~z)] , (16)
with only a relatively weak dependence on the overall density Ω0 via
Sk(x)=⎧⎪⎨⎪⎩sinh(x)$if$Ω0<1x$if$Ω0=1sin(x)$if$Ω0>1, (17)
whereas the expansion integrated over the past light cone, ∫z0H−1(~z)d~z, yields the main contribution to the relation (16).
Altogether, we propose that in order to make the method physically more correct, the Hubble expansion H(z) in Eqs. (11) and (13) should be interpreted as the average expansion rate of the regions where the detectable light travels. For this, we introduce another parameter β, the ratio of the expansion rate along our line of sight H(z) to the FRW value ¯H(z), so that H(z)≡β¯H(z) in Eqs. (11) and (13). Another crucial ingredient missing from the original formulation is that structures evolve in the universe; that is, both of the parameters should depend on the redshift: α(z) and β(z). The redshift-dependence of α was in fact already proposed by Linder in 1988 [69] , but perhaps because the corrections in the original Dyer-Roeder method are so weak (see Eq. (14)), the redshift dependence of α has won very little attention in the literature [76] . Instead, as we demonstrate in Sect. 3.2, it is only after taking into account both the parameter β and its dependence on the redshift that the results start to go more hand in hand with intuition.
3 Optically altered universe due to nonlinear structures
In this section, we introduce the principal model of this work, where light propagation in the real clumpy universe is described phenomenologically by a generalized Dyer-Roeder method. Within this approach, the universe can still be homogeneous on large scales, but the nonlinear inhomogeneities on smaller scales render its optical properties different. As discussed in Sect. 2, the optical overall effect of the inhomogeneities can result from: 1) biased light propagation through voids that expand faster than the average, 2) global increase in the volume-averaged expansion due to spatial variations in the expansion rate, 3) our possible special spatial location in the universe, or any combination of the three. There can also be effects that are not reducible to any of these; e.g. a clock-rate difference between underdense and overdense regions discussed in Refs. [41, 40, 42, 43, 44, 45] .
We proceed by summarizing the physical foundations of the model qualitatively in Sect. 3.1 and leave the discussion of its quantitative properties to Sects. 3.2 – 3.5.
3.1 The physical foundations of the model
The assumptions that form the physical foundations of the model, can be summarized in three main points:
The older the universe, the clumpier it is and hence the emptier and larger the voids.
The light from the well-observed objects travels mostly in the voids.
The emptier and larger the voids, the faster they expand relative to the average.
The first point is the standard notion of structure formation: initially overdense regions accrete more material via gravitational attraction and thus become denser and denser at the expense of the initially underdense regions, which thereby become emptier. Some observational evidence for the emptiness of voids has been found in galaxy surveys that show there to be typically ≲1% galaxies near the center of the voids relative to the average galaxy density [7] . The second assumption was explained in Sect. 2.3: the emptier a region the easier it is to see an object through it. The point three is an integrated effect of Eq. (15): denser regions have more mass and consequently stronger gravitational attraction to slow down the local expansion.
The combination of these properties suggests a physical explanation for how the nonlinear structure formation can be perceived as accelerated expansion without dark energy: as the detectable light traverses emptier and emptier regions, the expansion rate along the path of the photons increases relative to the average, causing accelerated expansion along our line of sight without local acceleration. More specifically, we present the following conjecture: due to the formation of voids and opaque structures, the average matter density on the path of the light from the well-observed objects changes from ΩM≃1 in the homogeneous early universe to ΩM≃0 in the clumpy late universe, so that the average expansion rate increases along our line of sight from EdS expansion Ht≃2/3 at high redshifts to free expansion Ht≃1 at low redshifts.
Optically, the proposed phenomenon bears close resemblance to the local Hubble Bubble, discussed in Sect. 2.2; see in particular Eqs. (2.2), (5) and the paragraph thereafter. However, unlike in the models with a large void in the local matter distribution, it is our special location in time, not in space, that is here crucial for the effect, so the cosmological principle is not violated. On the contrary, since every observer in the late universe sees the history of the structure formation, the phenomenon should look essentially the same no matter at which point the observer sits in the universe555This, at least, seems to distinguish the light propagation in freely expanding voids as the cause for the acceleration from Wiltshire's proposal whereby it is crucial that the observer sits in a static or gravitationally bound location, such as inside a galaxy, to perceive the inhomogeneity induced acceleration [40] ..
3.2 The generalized Dyer-Roeder method
In Sect. 2.5, we reviewed a quantitative method to allow for the optical effects of the clumpiness of matter in the universe, following the reasoning of Dyer and Roeder [5, 6] . The basic idea of this method is to employ the average density along the paths from the objects to the observer in calculating the distance-redshift relations for those objects, instead of using an average density of the whole universe like in the FRW description. Albeit a promising starting point, we argued that the original method falls short of capturing two crucial physical phenomena: inhomogeneities in the expansion rate and the growth of the nonlinear structures. To take these properties into account, we present a generalized version of the original Dyer-Roeder method.
Following the argumentation of Sect. 2.5, we take into account both the inhomogeneities in the matter distribution and in the expansion rate via the redshift-dependent parameters α(z) and β(z). Continuing to neglect the vorticity of matter, we still have a coordinate system where the matter four-velocity is just uμ=δμ0. Thus, we can simply substitute ρm(z)→α(z)¯ρm(z) and H(z)→β(z)¯H(z) in Eqs. (11) and (13) to obtain
β(z)¯H(z)∂∂z[(1+z)2β(z)¯H(z)∂∂zdA(z)]+4πG[α(z)¯ρm(z)+43ρr(z)]dA(z)=0 (18)
(1+z)2β(z)¯H(z)∂∂z[(1+z)2β(z)¯H(z)∂∂zdL(z)(1+z)2]+4πG[α(z)¯ρm(z)+43ρr(z)]dL(z)=0 , (19)
where the symbols with an overbar represent the auxiliary FRW quantities and we assume that radiation does not cluster: ρr(z)=¯ρr(z). The initial conditions for the differential equations (18) and (19) are: dx(0)=0, dx′(0)=H−10, where x∈{A,L}.
To work out how the inhomogeneities can modify the luminosity distance in the generalized Dyer-Roeder method, consider the solution to Eq. (19) as a power series in z:
H0dL(z)=[z+(1−β′(0)2β(0))z2+(β′2(0)3β2(0)−β′(0)3β(0)−β′′(0)6β(0)−α(0)4β2(0))z3+O(z4)], (20)
where radiation has again been neglected. Here the nonlinear structures, encapsulated in the functions α(z) and β(z), have much more notable role than in the original Dyer-Roeder method in Eq. (14). In addition, as can be seen by comparing the solution (20) to the luminosity distance in the flat ΛCDM model (5), the effect of the inhomogeneities is here importantly in the correct direction, mimicking Λ with β′(0)<0.
We emphasize that the solutions dA(z) and dL(z) to Eqs. (18) and (19) do not represent mathematically exact solutions of general relativity. Instead, their aim is to capture the most essential effects of the inhomogeneities on the observable properties of light that the simplified FRW description fails to describe. The crudest approximations in Eqs. (18) and (19) correspond to employing a different FRW result for each redshift. Even approximations of this kind have proven to be accurate in calculating the distance-redshift relations [36] . Nevertheless, in a more thorough treatment, e.g. the utilized FRW relation a(0)/a(z)=1+z should be replaced with a corrected inhomogeneous result; see [25] .
To calculate the observable distance-redshift relations dA(z) and dL(z) from Eqs. (18) and (19), we need the functions α(z) and β(z). An entirely phenomenological approach would be to determine them purely from the cosmological data analysis. In contrast, an ideal approach would be to derive α(z) and β(z) from the full Einstein equations starting with some initial perturbations in the early universe. In this work, we employ an analytic approach which falls somewhere between these two extremes: based on theoretical and observational reasoning, we deduce a form for the functions α(z) and β(z) containing a single phenomenological parameter that, given the approximations, would best correspond to the structure formation of the real universe; this is done in Sect. 3.3. In Sect. 3.4, we then fit the leftover parameter to various cosmological data.
3.3 A phenomenological description of the nonlinear structure formation
As boundary conditions given on a 3-dimensional hypersurface, the expansion rate H and the matter distribution ρm are independent (apart from some constraints). In theory, one could thus give them almost any profile e.g. along our past light cone; that is, choose α(z) and β(z) in Eqs. (18) and (19) arbitrarily. However, as suggested by the nearly uniform temperature of the CMB radiation, it is likely that the early universe has been very close to homogeneous. Thus, we only consider inhomogeneities that grow with time, which sets a constraint between the functions H(z) and ρm(z), or equivalently between α(z) and β(z), reducing the number of independent functions to one.
Some properties of α(z) and β(z) can be deduced without resorting to a specific model. Firstly, according to Eq. (15), initial underdensities evolve into regions that eventually begin to expand faster than the global average. As the detectable light is assumed to propagate mostly in these regions, we have α<1, hence implying that β>1. The requirement of positive mass density sets α≥0, while Eq. (15) implies the upper bound β≤3/2. For an explicit constraint between α(z) and β(z), a specific model is required.
The question for which we seek an answer is then: how much faster does a void with known matter density ρm<¯ρm expand than the FRW value ¯H, i.e. given 0≤α<1, what is 1<β≤3/2? We estimate this constraint by employing a different FRW result to each patch with different density and expansion rate. Given that we use the proper time of dust as our time coordinate, the constraint can be obtained by requiring a constant value for the age of the universe t0 between the different regions. This sets the matter density and expansion rate equal at t=0 for every patch, i.e. the condition that the universe was initially homogeneous. We do not, however, employ this method to model the dynamics of the universe as, e.g. due to vanishing vorticity and ignored mass flows, it cannot realistically describe the evolution of the structures at least when virialization becomes important.
As a starting point, we take the functions α(z) and β(z) to measure deviation from a flat FRW model with matter and radiation. Thus the quantities with an overbar in Eqs. (18)–(19) obey the relations
4πG¯ρm(z)=32¯H20¯Ωm(1+z)3 (21)
¯H(z)=¯H0(1+z)3/2√1+z¯Ωr , (22)
where ¯Ωm≡8πG¯ρm(0)/(3¯H20) and ¯Ωr≡8πG¯ρr(0)/(3¯H20) are the density parameters of the auxiliary FRW model (with the constraint ¯Ωm+¯Ωr=1) and ¯H0 would be interpreted as the average global expansion rate in the FRW description. However, if the dynamical backreaction (2) is significant, in the nonlinear late universe ¯ρm(z) and ¯H(z) are just auxiliary quantities with no obvious physical interpretation.
To estimate the constraint between α(z) and β(z), we first replace them by the auxiliary functions ~H0(z) and ~Ωm(z), defined through the equations:
α(z)=~H20(z)~Ωm(z)¯H20¯Ωm (23)
β(z)=~H0(z)¯H0 ⎷1+z~Ωm(z)+z(z+2)¯Ωr(1+z)(1+z¯Ωr) . (24)
With the definitions (23)–(24), the line-of-sight quantities ρm(z) and H(z) take the form
4πGρm(z)=4πGα(z)¯ρm(z)=32~H20(z)~Ωm(z)(1+z)3 (25)
H(z)=β(z)¯H(z)=~H0(z)(1+z)√1+z~Ωm(z)+z(z+2)¯Ωr , (26)
where, by construction, the deviation from homogeneity appears as the redshift dependence of ~H0(z) and ~Ωm(z) which in the limit z≫1 reduce to the constants ¯H0 and ¯Ωm. Apart from this limit, there is no straightforward physical interpretation for ~H0(z) and ~Ωm(z).
The next step is to obtain the constraint between ~H0(z) and ~Ωm(z) by requiring that the age of the universe is the same between the different FRW patches. The relation between ~H0(z) and ~Ωm(z) then determines a constraint between α(z) and β(z) through Eqs. (23) and (24). We are only applying the FRW results to voids, so Ωm≤1. In this case, the integration of the Friedman equation yields the age of the FRW dust universe:
t0=√1−Ωm−Ωmarsinh√1−ΩmΩmH0(1−Ωm)3/2 . (27)
We use the redshift to label the different patches on the present-day spatial hypersurface, so Ωm and H0 in Eq. (27) become functions of z. The requirement for the age in Eq. (27) to be equal between the different patches then yields H0 in terms of Ωm:
~H0(z)=t−10⎡⎢ ⎢ ⎢ ⎢⎣√1−~Ωm(z)−~Ωm(z)arsinh√1−~Ωm(z)~Ωm(z)(1−~Ωm(z))3/2⎤⎥ ⎥ ⎥ ⎥⎦≃t−10[1−13√~Ωm(z)] , (28)
where t0=constant is the age of the universe and in the last step we have employed an approximation which is accurate (|error| <1.5%) in the interval 0≤~Ωm(z)≤1.
Using Eqs. (23), (24) and (28), we can give ~Ωm(z) and ~H0(z) in terms of α(z),
~Ωm(z)=94(1−√1−89√¯Ωmα(z))2 (29)
~H0(z)=t−102(1+√1−89√¯Ωmα(z)) , (30)
so the line-of-sight quantities (25) and (26) become
4πGρm(z)=23t−20¯Ωmα(z)(1+z)3 (31)
H(z)=β(z)¯H(z)=t−102(1+z)(1+√1−89√¯Ωmα(z))×
√1+9z4(1−√1−89√¯Ωmα(z))2+z(z+2)(1−¯Ωm) . (32)
We impose two observationally motivated boundary values for the function α(z). Firstly, soon after decoupling the universe was still close to homogeneous, so the difference between the line-of-sight and globally averaged matter densities must have been negligible, corresponding to ρm→¯ρm or α(1100)≃1. At times earlier than z=1100, the distinction between the line-of-sight and volume averaged quantities does not make sense, since we cannot see beyond the opaque plasma wall formed by the last scattering surface. The second boundary value follows from the assumption that the regions where the detectable light from the well-observed objects travels today are nearly empty: ρm(0)≪¯ρm(0) or α(0)≃0. Altogether, we then have α(0)≃0 and α(1100)≃1.
With the endpoints α(0)≃0 and α(1100)≃1 of the function α(z) fixed, we still need its behavior between these points. Once initiated, the structures can only grow in a universe with no dark energy, so a logical starting point is that α(z) must be a monotonically increasing function of redshift. We choose the exact behavior of the function based on physical understanding about the structure formation.
A typical behavior of a small overdensity on a given scale L is that initially the excess decrease in its expansion rate is small, with linear proportionality to the initial conditions; but eventually, around a redshift z0(L), the gravity-driven cumulative increase in the clustering speed of the constituents collapses the system rapidly until it virializes due to pressure gradients and the conservation of angular momentum [77, 78] .
A function that encapsulates these qualitative features is the exponential, α(z)=α0+∑iαie−z/zi(Li), summed over different scales. For simplicity, we consider only a single redshift z0 when the voids start to dominate at the most significant scale. Taking into account the above deduced boundary values, we choose
~Ωm(z)=(1−e−z/z0)2 , (33)
Inserting Eq. (34) in Eq. (30), we also have
~H0(z)t0≡h(z)=13(2+e−z/z0) . (35)
In the end, we leave the transition redshift z0 as a free parameter. A more sophisticated numerical estimation of the function α(z) will be performed in a future work, in which even {zi(Li)}i is a derived, not free, set of parameters.
3.4 Comparing the model with cosmological observations
In this section, we aim to find out how the model with α(z) given by Eq. (34) fits the main cosmological data sets: the cosmic microwave background anisotropy, the position of the baryon acoustic oscillation peak inferred from the galaxy distribution, the magnitude-redshift relations of type Ia supernovae, the local measurements of the Hubble constant and the Big Bang nucleosynthesis. For a thorough and open-minded review of the current observational status, see Ref. [79] .
A good baseline for the studies is the model by Blanchard, Douspis, Rowan-Robinson and Sarkar [80, 81] , and especially its refined version by Hunt and Sarkar [82, 83, 84] . This is an FRW model with the following basic properties: no dark energy or Λ=0, a mixture of cold and hot dark matter ΩCDM=0.8 & ΩHDM=0.1, flat spatial geometry k=0, a low Hubble constant h=0.44 and a bump in the primordial perturbation spectrum at the scales k∼0.01...0.1 hMpc−1, which could arise e.g. from phase transitions during inflation [85] ; we shall call this the linear CHDM model. The model fits the CMB angular power spectra better than the standard ΛCDM model. It is also consistent with the nucleosynthesis constraints and with most of the features in the matter power spectrum of the galaxy distribution surveys. However, the model seems to fail in matching the observed magnitude-redshift relations of type Ia supernovae and the detected position of the baryon acoustic oscillation peak in the matter power spectrum and it is also in disagreement with the local measurements of the Hubble constant. For a more thorough description of the linear CHDM model, see [83] .
A candidate for hot dark matter would be three species of massive neutrinos with ∑imνi∼1.8 eV, yielding ΩHDM=Ων≃0.1. This is well below the present experimental upper bound of mν=2.3 eV [86] , but is expected to be detectable in the forthcoming KATRIN β-decay experiment [87] . Note that the sometimes quoted tighter upper bounds from cosmological observations depend heavily on the priors [88] . As neutrinos with mass higher than mν≳0.3 eV have been shown to cluster appreciably [89] , their existence should not alter the assumption that the detectable light travels in almost empty space today.
We consider a universe that is similar to the linear CHDM model at large redshifts, but instead of describing its optical properties with the standard FRW relations, we use the generalized Dyer-Roeder equations (18)–(19) to determine the observable distance-redshift relations. Accordingly, the growth of small deviations from homogeneity is assumed to follow linear perturbation theory, so apart from the changes in the associated distance measures, the perturbation spectra should be more or less unaltered. We call this the nonlinear CHDM model. We pose five observational tests for this model:
To obtain the same value for the angular diameter distance to the last scattering surface as in the linear CHDM model.
To match the position of the baryon oscillation peak in the matter power spectrum.
To fit the observed magnitude-redshift relations of the type Ia supernovae.
To explain the locally measured value of the Hubble constant.
To remain within the limits on the baryon density set by the BB nucleosynthesis.
By passing these tests successfully, the nonlinear CHDM model would explain the main cosmological data sets that support accelerated expansion. In Sects. 3.4.1–3.4.6, we discuss how the model survives these tests and the results are combined in Sect. 3.4.7.
3.4.1 The cosmic microwave background
For the values ¯ΩB=0.1, ¯h=0.44 and ¯Ω0=1, the fitting formula of Hu and Sugiyama [90] yields zdec=1101.35 as the decoupling redshift, for which we use the approximation zdec≃1100 throughout.
The fit of the linear CHDM model to the WMAP 3-year data has been thoroughly performed by Hunt and Sarkar [83] so we only need to consider how the nonlinearities modify this. As the nonlinear structures appear only in late times, it is essentially only the comoving angular diameter distance dc(z)≡(1+z)dA(z) and the late integrated Sachs-Wolfe effects that can change. However, as the model already fits the CMB data without the nonlinear structures, it is desirable that adding the nonlinearities would not alter the distance to the last scattering surface, dc(1100). Physically, this is possible because the negative spatial curvature and the faster Hubble expansion in the voids have the opposite effects on the comoving angular diameter distance dc(z), so with a suitable value for the new parameter z0, their effects can cancel.
By determining dc(z) numerically from Eq. (18), and ¯dc(z)=¯dL(z)/(1+z) from Eq. (16), we find that the value for z0 that sets dc(1100) and ¯dc(1100) identical, is z0=0.347. Values in the range z0=0.30...0.39 give less than 1% discrepancy between dc(1100) and ¯dc(1100). Although equal around z≈1100, the functions dc(z) and ¯dc(z) are not equal at lower redshifts.
Varying the Hubble constant does not change the limits on z0 as ¯H0 has essentially the same effect on both the "background" value ¯dc(1100) and the value dc(1100) which takes into account the nonlinearities. To conserve the successful angular power spectrum found in Ref. [83] , we fix the FRW Hubble constant to the value ¯H0=44 kms−1Mpc−1.
Within the linear perturbation theory, a flat matter dominated FRW model does not have a late integrated Sachs-Wolfe (LISW) effect, because the perturbed gravitational potentials do not evolve in time, ˙Φ=0. However, this does not necessarily hold in a universe with nonlinear structures if the CMB light has propagated mostly in voids. Indeed, with ˙Φ potentially non-zero along our line of sight, one would expect a non-trivial LISW effect from nonlinear inhomogeneities also in a matter dominated universe with initially flat FRW geometry. On the other hand, as the LISW affects the angular power spectrum only at low multipoles l, its contribution to the goodness of the overall fit could be insignificant due to the large cosmic variance ∝(2l+1)−1. Moreover, it might in fact be that the effects of the local structures dominate the angular power at the lowest multipoles [91, 92] . It is also possible that, even if the selection effects were significant in the supernova observations, they may be less important in the CMB observations. Altogether, along with other more involved issues, we postpone a quantitative discussion of the LISW to a future work.
Apart from the potential modifications due to the LISW effect at small l, the angular power spectrum of the nonlinear CHDM model with z0=0.347 is plotted in Fig. 1.
Figure 1: Binned WMAP data points in the temperature power spectrum with theoretical predictions from the best fit CHDM and ΛCDM models; figure from Hunt and Sarkar [83] .
3.4.2 Galaxy distribution
As long as the nonlinear inhomogeneities only affect light, the evolution of the linear perturbations remains the same as in the linear CHDM model. In the case of the CMB power spectrum this seems obvious, since the relevant evolution of the perturbations happened at z<1100 when the nonlinearities were negligible. However, as the galaxy distribution also reflects the dynamics of the late universe, it is not clear that the nonlinear inhomogeneities would not play a dynamical role here. We nevertheless neglect the potential effects of the nonlinearities on the evolution of the perturbations, but remind that, in a more realistic treatment, these effects should be taken into account in the matter power spectrum.
As shown in Refs. [81, 83] , the linear CHDM model fits the matter power spectrum apart from the position of the baryon acoustic oscillation peak. As in the CMB angular power spectrum, the main modification caused by the nonlinearities is the change in the distance-redshift relations. This opens up a possibility that the nonlinear CHDM model would fit the position of the baryon acoustic oscillation peak. To study this, we use the data from the Sloan Digital Sky Survey provided in Ref. [18] .
The detected position of the baryon acoustic oscillation peak in the matter power spectrum constrains the distance to the redshift z=0.35, corresponding to the acoustic oscillation scale today. More specifically, due to the different scaling of radial and angular distances, the exact distance measure constrained by the data reads as [18] :
DV(z)≡[zd2c(z)H(z)]1/3 . (36)
As the actual observations are made in the redshift space and the distance-redshift relation depends on the cosmological model, the "measured" values quoted in [18] are prior-dependent. We take this into account by using the more conservative 2σ error limits. The value of the distance (36) to the redshift z=0.35 deduced from the SDSS data is
DV(0.35)=1370±150 Mpc , (37)
where the limits corresponding to 2σ errors have been projected from Fig. 7 of Ref. [18] . With H0=66 kms−1Mpc−1 fixed, we find that the constraint set by Eq. (37) on the transition redshift parameter in Eq. (34) is z0>0.25, whereas the best fit value from the CMB analysis of Sect. 3.4.1, z0=0.347, gives DV(0.35)=1480 Mpc. Fixing z0=0.347 requires H0=71 kms−1Mpc−1 or ¯H0=47 kms−1Mpc−1 to give the best fit value DV(0.35)=1370 Mpc.
In the case that the statistical properties of the primordial perturbations are the same at every spatial location in the universe, the baryon oscillation peak in the matter spectrum and the first peak of the CMB angular power spectrum should correspond to the same physical feature just seen at different spatial locations and at different epochs. The physical scale associated with these oscillations is set by the sound horizon at recombination. As the distance (37) measures how large this length scale appears in the galaxy distribution, it is possible to combine it with the information of the associated scale of the peak in the CMB spectrum to make an additional observational test. The quantitative measure employed in Ref. [18] for this is defined as:
R0.35≡DV(0.35)dc(1100) . (38)
It should be again noted that due to the prior-dependency of the distance-redshift relations, Eisenstein et. al. quote several "measured" values for R0.35. As dc(1100)=13190 Mpc is the best fit value to the CMB data in the CHDM model, Eq. (37) yields:
R0.35=0.104±0.011 , (39)
where the limits should roughly correspond to 2σ errors in the CHDM model. We find that the values of z0 corresponding to these limits are: z0=0.29...0.83, with z0=0.53 yielding the best fit value R0.35=0.104. As R0.35 is a ratio of distances, it is independent of H0.
Albeit the nonlinear CHDM model with α(z) given by Eq. (34) is not as concordant with the SDSS data as with the other data sets, the value z0=0.35 still falls within the experimental limits. Moreover, the results represent a huge improvement compared to the linear CHDM model, which yields the values ¯DV(0.35)=1760 Mpc and ¯R0.35=0.134 that are both way off the rather conservative limits of Eqs. (37) and (39).
3.4.3 Type Ia supernovae
We use the Riess et. al. gold sample of 182 type Ia supernovae [17] to study how the nonlinear CHDM model fits the SN observations. For this, we use the conventional χ2 test:
χ2≡182∑n=1(mobs(zn)−m(zn)σn)2 , (40)
where m(zn)≡5log10(dL(zn)/Mpc)+25 is the theoretical prediction for the magnitude and σn is the estimated error for the measured magnitude mobs(zn) of a source at the redshift zn.
Just like the flat ΛCDM model with ¯ΩΛ and ¯H0 as free parameters, the nonlinear CHDM model has two free parameters: the transition redshift z0 and, say, the observable Hubble constant H0. There is, however, a perfect degeneracy between H0 and the maximum intrinsic luminosity of the type Ia supernovae, LIa, so the uncertainty in the value of LIa reflects a similar uncertainty in the value of H0 constrained from this data. For this reason, the combination (H0,LIa) is sometimes marginalized in the data analysis but we choose to instead use the fixed value for LIa employed by Riess et. al. in Ref. [17] .
By letting both parameters vary, we find the best fit values H0=65.5 kms−1Mpc−1 and z0=0.39 that yield χ2=161.1. A little inspection in the parameter space shows that there is degeneracy between z0 and H0, such that one can obtain good two-parametric fits (Δχ2≲5) roughly in the range: z0=0.2...0.8. Keeping instead H0 fixed to the value 66 kms−1Mpc−1, we find that the best fit transition redshift is z0=0.34, yielding χ2=161.4. The increase Δχ2=2.7 corresponds to 2σ errors in a one-parametric fit and yields for the limits: z0=0.34±0.100.08. The absolute fit 161.4/182=0.89 is good and represents a tremendous improvement to the linear CHDM model where χ2=1098 for ¯h=0.44 and even the best fit value ¯h=0.54 gives χ2=283.4.
For comparison, the best fit parameters of the ΛCDM model ¯ΩΛ=0.67 and ¯h=0.63 yield χ2=158.8. However, for the concordance values ¯ΩΛ=0.73 and ¯h=0.7 the fit is much worse: χ2=305.2, but this can be fixed by changing the assumed value of LIa or equivalently changing ¯h. Indeed, we find that the value ¯h=0.642 gives the best fit for the concordance ¯ΩΛ=0.73 ΛCDM model with χ2=162.5. Overall, the conclusion from comparing the results between the nonlinear CHDM model and the ΛCDM model is that the goodness of their fits to the Riess et. al. data are essentially equal. In Fig. 2, we plot the residual magnitude-redshift relations for three different models: the linear CHDM model with ¯h=0.54; the ΛCDM model with ¯ΩΛ=0.67 and ¯h=0.63; and the nonlinear CHDM model with z0=0.34 and h=0.66.
Figure 2: 16 binned data points from the Riess et. al. gold sample in the residual magnitude-redshift diagram with theoretical predictions from three different models: the linear CHDM model with ¯h=0.54 (dashed black curve), the ΛCDM model with ¯ΩΛ=0.67 and ¯h=0.63 (dashed dotted blue curve), and the nonlinear CHDM model with z0=0.34 and h=0.66 (solid red curve). We have subtracted the ¯h=0.66 freely expanding Milne universe relation from the curves to produce the residual diagram, so only the h=0.66 nonlinear CHDM model intersects the origin.
3.4.4 The Hubble Key Project
The best fit values obtained from the different data sets of Sects. 3.4.1, 3.4.2 and 3.4.3 for the observable Hubble constant are all consistent with the low redshift measurements of the expansion rate that should specifically probe the observable Hubble constant; the value H0=66 kms−1Mpc−1 sits firmly within the limits of the final results from the Hubble Key Project: H0=72±8 kms−1Mpc−1 [93] or H0=62.3±5.2 kms−1Mpc−1 [94] . In addition, there are persistent indications that one gets systematically lower values for the Hubble constant by observing objects at higher redshifts [79] . For the standard FRW cosmology this feature is an anomaly, while in the nonlinear CHDM model it is a prediction: towards higher redshifts, the optically perceived expansion should be closer to the (lower) FRW expansion rate due to the fact that the earlier universe was more homogeneous.
3.4.5 Age of the high redshift quasar APM 08279+5255
The particularly well-observed high redshift quasar APM 08279+5255 [95, 96] at z=3.91 with estimated age 2...3 Gyr has been proposed as a litmus test for every cosmological model [97] . For several cosmological models, most notably the homogeneous standard ΛCDM model where tage(3.91)=1.7 Gyr and almost all the dark energy models, the observed quasar implies an age problem: the universe appears to be younger than its contents.
The considered nonlinear model offers a resolution to the age problem: if the observed light from the quasar propagates in voids that expand faster than the average, the light redshifts more rapidly than expected from the stretching of the global FRW scale factor, implying that the quasar is seen in an older universe than predicted by the age-redshift relation of the homogeneous FRW model. An order of magnitude estimate can be obtained by approximating that the space along our line of sight to the quasar has expanded freely. This increases the age at a given redshift z relative to the FRW value by a factor √1+z, yielding tage=3 Gyr for the age of the universe at z=3.91, consistent with the estimated age 2...3 Gyr of the quasar APM 08279+5255.
3.4.6 The Big Bang Nucleosynthesis
Finally, we comment on the nucleosynthesis. The overall result from deducing the abundances of the light elements from the spectra of stars and galaxies is that the baryon density falls within the range 0.017≤¯ΩB¯h2≤0.024 at 95% confidence level [98] . As long as the early universe was nearly homogeneous and the constants of nature do not vary, these limits are model-independent. Therefore, the observed value should not be affected by the nonlinear inhomogeneities in the late-time universe so that the value for the baryon density, ¯ΩB¯h2=0.1⋅0.442=0.0194, falls within the above quoted observational limits also in the nonlinear CHDM model.
3.4.7 A concordant nonlinear CHDM model with no dark energy
By combining the results from the analysis of the different data sets in Sects. 3.4.1–3.4.6, we obtain a concordant cosmological model with the following properties: the observable Hubble constant h=0.66, the transition redshift to the era when nonlinear structures become dominant z0=0.35, baryon proportion ¯ΩB=0.1, cold dark matter proportion ¯ΩCDM=0.8, hot dark matter proportion ¯ΩHDM=0.1 and no dark energy Λ=0. All the data sets are consistent with these values and, apart from the SDSS data, the concordance value for z0 is actually very close to the best fit value of each separate data set; see Table 1. Moreover, the concordance value for H0 yields t0=H−10=14.8 Gyr as the age of the universe, consistent with the lower bound determined from the astronomical estimates for the age of the oldest globular clusters: 11.2 Gyr (at 95% confidence level) [99] . The main properties of the best fit nonlinear CHDM model are summarized in Table 2.
BAO peak
Type Ia SNe
Concordance value
Transition redshift z0
0.35±0.040.05 0.53±0.300.24 0.34±0.100.08 0.35
Table 1: The constraints on the transition redshift from each data set and the concordance value.
Transition redshift
z0 0.35
Observable Hubble constant
h 0.66
FRW Hubble constant
¯h 0.44
Age of the universe
t0 14.8 Gyr
Baryon proportion
¯ΩB 10%
Cold dark matter proportion
¯ΩCDM 80%
Hot dark matter proportion
¯ΩHDM 10%
Table 2: Best fit parameters for the nonlinear CHDM model with no dark energy. Only one of the three parameters h, ¯h and t0 is independent. It is noteworthy that a phenomenologically similar model was found by Wiltshire et. al.


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A meth trial scheduled this month for John Manis Richards, 40, of Grantsville, has been postponed, according to Calhoun prosecutor Matt Minney, although a status conference is still scheduled for August 29th.
Richards, who has a long criminal record, has hired and fired a number of lawyers, most of them court-appointed. He has now been appointed a new attorney from the Kanawha County public defender's office, with the court reportedly telling Richards he will no longer be entitled to another attorney.
His trial will likely be re-scheduled in late fall.
Richards is currently incarcerated in Central Regional Jail on larceny charges.
A Calhoun grand jury indicted John Manis Richards in January for attempting to operate a clandestine meth lab, one count; possession of pseudoephedrine with the intent to manufacture meth, two counts; conspiracy to operate a clandestine drug lab, one count.
Authorities arrested Richards last August after Sgt. Tom Yanero, Trooper First Class Jeff Hunt and then Grantsville Police Chief Ron Gordon obtained a search warrant for his south Grantsville residence.
At the time of the arrest, Sgt. Darrin Campbell of the WV State Police said a field test was positive for the presence of meth, while officers did a search of the property.
A search led by Trooper Hunt "observed a quantity of pseudoephedrine, along with plastic tubing, three bottles of Heet, three bottles of iodine, acetone and Coleman camp fuel."
Officer Hunt reported "the presence of a minor child, who is the natural child of the defendant's domestic partner, in the same vehicle that was used to transport the meth lab."
Since his arrest, Richards has distributed several video tapes which contain "educational information" regarding the manufacture and hazards of meth.
The tapes also appear to show drug deals being completed in Calhoun and Gilmer County, supposedly filmed with a hidden camera.
After Richards was charged, his claim-to-fame has been to establish a web-blog "The Lone Meth Ranger," saying he has been involved in a war against meth for several years, and has provided a chemical mix to some of the producers that "flops their batches," or neutralizes the meth.
Richards says he has been involved with several other undercover individuals who have been active in a Meth Eradication Team (MET). Richards could face 2-10 years in prison and a fine of $5,000-$25,000 over his attempt to operate a drug lab.
On the possession of a substance as a precursor to manufacture meth, Richards faces penalties of 2-10 years in prison and a fine up to $25,000.
While neighboring counties have hundreds of meth-related arrests on the books, Calhoun has had only a handful, and some of those are cases which originated elsewhere.
Richards claims his crusading has eradicated the meth scourge from the community, frightening producers into moving to other counties.
Hur Herald ©from Sunny Cal
The information on these pages, to the extent the law allows, remains the exclusive property of Bob Weaver and The Hur Herald. information cannot be not be used in any type of commercial endeavor, or used on a web site without the express permission of the owner. Hur Herald published printed editions 1996-1999, Online ©Hur Herald Publishing, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019

redpajama_set_name: RedPajamaCommonCrawl


Rasika Dugal elegant look in a white saree with black simple patch border teamed with a matching blouse.
For a TV show Regina Cassandra beautiful appearance in a Purvi Doshi embroidery skirt,teamed with a bell sleeves backless crop top,she looked good.
Neha dhupia elegant look in hand woven khadi saree by purvi doshi teamed with a embroidery quarter sleeves blouse.
Tapsee stylish look in a mirror work embroidery khadi out fit by purvi doshi,teamed with ruffled jeans.
For an event upasana elegant appearance in a mirror work embroidery lehenga by doshi teamed with a matching short kurti,she finished out the look with good earth jewellery.


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Alcohol's ESG activations around the world – May 2022 – The IARD Digest
What has the alcohol industry been up to this month on the Environment, Social & Governance fronts?
The IARD editorial team
Once a month, the drinks industry-funded International Alliance for Responsible Drinking, which covers alcohol-related health and policy research worldwide, reviews some of the alcohol category's most recent Environment, Social & Governance initiatives. Here's what's been going on in May.
Beam Suntory launched its third annual 'Together for Good' global event this month, towards its target of donating 1m employee volunteer hours to communities by 2030. The company's staff have been reforesting reclaimed mine land in Kentucky, cleaning waterways in Singapore and Chicago and supporting flood victims in Australia.
The brand owner's parent company, Suntory Group, joined the 'Taskforce on Nature-related Financial Disclosures' (TNFD) forum in Japan. The aim of the project is to develop and deliver a risk management and disclosure framework for organisations to report and act on evolving nature-related risks.
In the UK, Asahi has been focusing on diversity and inclusion through its 'Shine As You Are" action plan. The company has an aligned set of commitments across its global business, with targets such as an equal balance of women in its leadership teams by 2030.
In South Africa, Bacardi launched the latest round of its 'Shake Your Future' free bartender training programme for disadvantaged young adults. South Africa's youth unemployment rate is 66.5%, so the programme provides them with vital training and job opportunities while supporting on-premise outlets that were badly affected by the COVID pandemic and are struggling to find skilled staff. Since 2018, the initiative has trained young adults in France, Spain, and Italy, with 80% of graduates securing careers in the hospitality industry: Bacardi has also launched the programme in the UK for the first time.
Back to the UK and Carlsberg's Carlsberg Marston's Brewing Co (CMBC) published its first 'Sustainability Report' this month. The report showcases projects from the JV's first year, and its progress in achieving "zero carbon footprint, zero water waste, zero irresponsible drinking and a zero accidents culture", as part of its commitment to the brewer's wider 'Together Towards ZERO' sustainability programme.
Anheuser-Busch InBev has invited small companies to apply for its annual '100+ Accelerator' initiative. The programme, which is a collaboration with The Coca-Cola Co, Colgate-Palmolive Co and Unilever, helps start-ups find solutions to sustainability challenges by overhauling supply chains and scaling innovative solutions. One initiative from the 2021 cohort provided solar-powered streetlights to Ecuador's Yacupamba farmer community.
Diageo partnered with Ecospirits to pilot a closed-loop packaging solution, initially in South-East Asia. The project involves Smirnoff and Captain Morgan using reusable 4.5-litre glass containers, which will be returned by on-premise outlets once empty.
In the US, Molson Coors Beverage Co commenced a major overhaul of its brewery in Golden, Colorado. When completed in 2024, the facility will be one of the world's most efficient, saving 100m gallons of water a year, reducing beer waste by 25%, increasing CO2 capture by 30%, and reducing energy usage by 35%.
Moët Hennessy has debuted the first rosé wine from Provence to be marketed in a flat bottle made from 100% recycled ocean plastic waste. Rosé from Provence is typically bottled in thick-walled glass, which accounts for 40% of the energy consumed in production and distribution and makes sustainable distribution "impossible."
Pernod Ricard's Chivas Brothers division has lined up a US$110m investment plan to upgrade sustainable distillation technologies and increase capacity at two of its Scotch whisky distilleries. The expansion will accelerate Chivas' 2026 carbon-neutral distillation goal by installing new bio plants and Mechanical Vapour Recompression (MVR) fan technology for pot-still distillation.
In Brazil, Heineken has launched the 'Heineken Institute', which will work with vulnerable individuals in society who are part of their value chain and need support, including street vendors, waste collectors, and underprivileged young adults. The institute will enable Heineken to scale up and partner with NGOs, governments, and others.
Kirin is trialling label-less bottles with some major retailers in the Tokyo area. The 'Kirin Namacha Paper Sticker Labelless' project uses 100% recycled PET bottles, with a small product information sticker made of paper, instead of the usual full label. Kirin is promoting efforts to reduce the volume of plastic used in packaging and is trialling the new format for six-packs and cases.
Finally, in Australia, Kirin's Four Pillars Gin brand has been certified carbon-neutral, making it the first gin distillery in the country to achieve carbon neutrality across its operations.
For further details on The International Alliance for Responsible Drinking, click here.
How AR will revolutionise consumer marketing and ease supply chain pressures – Click here for a Just Drinks focus
Sweden Hot Drinks - Market Assessment and Forecasts to 2024
Mexico Hot Drinks - Market Assessment and Forecasts to 2024
United States of America (USA) Gin and Genever (Spirits) Market Size, Growth and Forecast Analytics to 2025
Syrups (Syrups and Spreads) Market in the United Arab Emirates - Outlook to 2024; Market Size, Growth and Forecast Analytics (updated with COVID-19 Impact)
United States of America (USA) Rum (Spirits) Market Size, Growth and Forecast Analytics to 2025
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Suntory Group
Low- and no-alcohol in high spirits as consumer trends continue to drive growth
Diageo looks to add value to its rum offer with Don Papa buy
Can Uganda become a coffee powerhouse?
Top technology investment trends to watch in 2023


What must employers do to protect you from drunk truck drivers? | The Law Offices of Miller & Bicklein, P.C.
What must employers do to protect you from drunk truck drivers?
On behalf of The Law Offices of Miller & Bicklein, P.C. posted in truck accidents on Monday, March 25, 2019.
If you have ever taken an extended drive, then you know just how taxing spending hours behind the wheel can be. For this reason, you may be inclined to be compassionate towards the truck driver that caused your accident in San Antonio. Yet those feelings of empathy can quickly turn to anger if you discover that the driver was engaged in reckless activity. Perhaps no activity is more reckless than drinking and driving, and you might certainly think that a truck driver would know better than to endanger themselves and others in such a way. By extension, you might also wonder what responsible company would allow such a driver behind the wheel of a massive vehicle like a semi-truck.
The fact that motor and freight carriers can be held responsible for the actions of their employees has been detailed on this blog in the past. Yet that was specifically in reference to a driver's actions while completing a route. What sort of responsibility does a company have to ensure that you and others never have the potential to encounter one of its employees drunk while behind the wheel of one of its vehicles?
Furthermore, an employer cannot allow a driver to drive if it has knowledge that the driver has consumed alcohol while on duty or within four hours of consuming alcohol prior to coming on duty.
Related Posts: Vicarious liability in truck accidents, How many hours can a trucker drive in one day?, Accident reconstructionists can determine liability, How can I safely drive with commercial trucks?


So I had a pretty eventful year overall. Here's a summary of everything. As you can see, I maxed out over 200 miles in August and the majority of my miles are running.
The half still remains my favorite distance.
Michigan Mile(Flint, MI) 8/23(completed) - a new PR!
I ran my first triathlon - the Stony Creek Off Road Sprint Tri - and it was a fantastic experience. It was much harder than I ever thought it would be, but it was off road so that made it more difficult.
I had one one PR and that was for my first one mile race ever.
All my other PR's are for 2011 and 2012. I guess I'm getting slower! I did, however, get two age group awards including a 10k AG award at Alpenfest in Gaylord which was the second year in a row I placed in my AG. I also placed 3rd in the Great Lakes Bay half marathon, the first time I've ever placed in this distance which was exciting before because it's usually fairly competitive at that distance.
I ran three trail races this year and loved them all! I also ran eight races outside of Southeast Michigan. I ran my first race in Indiana and added a third state to my marathons. This was also my first failed attempt to BQ. I won't be attempting a BQ in 2014.
I was fortunate enough to have five race weekends as vacations this year including three with my running buddies. These included Midland, Charlevoix, Gaylord, Glen Arbor and Indianapolis. These weekends are so much fun and the plan is to do more of them in 2014.
Gaylord Alpenfest 10k - AG win!
Hey - who's that guy to my right with no beard?
Of all 23 races I ran, I ran 17 races new to me for the year. I love trying out new races so I get a variety. Of course, I had some favorites that I ran again as well. You can read my "best of 2013" post HERE.
Overall, it was a really fun year of training and racing and I look forward to seeing what 2014 brings. Given my schedule is already fairly full through August, I know that I'll be having lots of fun sharing my races with all of you again.
Nice job in 2013! You sure did run a lot of races and I've enjoyed reading the recaps. There are plenty of races I'm interested in trying thanks to your recaps. I'm looking forward to hearing about your upcoming 2014 races too!
Thanks. It looks like you had a great year as well as a good one coming!


Brichard's chalinochromis originate from Lake Tanganyika where they inhabit the rocky outcrops.
Brichard's chalinochromis are quite a rare addition to the aquarium hobby and should only be cared for by keepers with some experience of keeping aggressive cichlids. Another unusual trat with these fish is the dramatic change of colouration when they mature from juveniles to adult fish. As juveniles they display barring all along their body but this disappears as they mature only leaving the black barring around the head hence their common name of the masked brichardi. Adult specimens average a length of round 3.5 inches but there are cases where they have grown slightly larger. They are extremely aggressive towards other tank mates if allowed to be, they should only be housed with other species of fish that can handle themselves, it is also advised to only house one male in the aquarium with a group of females.
The expected lifespan for Brichard's chalinochromis is 5-8 years if cared for correctly.
It is wise to give these fish a lot of space, despite their smaller size they do need to establish territories. They can be housed in a 30 gallon aquarium but giving them space in a 50 gallon aquarium could allow for more hiding places and larger areas for territories. Add rockwork to the rear of the aquarium and it is also advised to use sand for the substrate to make them feel at home. Brichard's chalinochromis prefer a medium water flow so place the outlet nozzles from the filter carefully and try to avoid still spots in the rockwork if possible. The temperature should be set at 26°C (79°F), and keep the pH range between 8.0-8.9. Regular water changes should be performed, at least 10% weekly and always keep a close eye on the fish to make sure that individual specimens are not being bullied or harassed.
Brichard's chalinochromis have a ravenous appetite and will accept most foods offered. They will readily accept quality commercial flake food or small cichlid pellets but this should be varied by offering live or frozen foods. If offering blood worms these should be fed sparingly or it could lead to digestive problems.
There is no reliable way of sexing these fish for amateur fishkeepers, they can be vented by expert keepers but even this can lead to errors at times.
Brichard's chalinochromis are secretive spawners and the females will often deposit their eggs in the rockwork without the keeper being aware. The females tend to guard the eggs while the male may hold a spot a distance away from the nest but he will constantly monitor the spawning site and occasionally swim over to make sure all is well. The parent fish are extremely defensive of the fry so it may be best to breed these fish in a separate breeding tank to avoid major conflicts with other tank mates. The fry are well defended for up to 4 weeks even from previous fry that have have been spawned from the same parent fish but after this time the fry will be released to wander around the tank and mix with other occupants of the tank.


These distinctive flag display cases are exceptionally beautiful!
These attractive cases are made of rich mahogany-colored hardwood and a thick glass for display. Each holds up to a 5' x 9-1/2' cotton flag. A die-cast, gold-plated U.S. Armed Forces medallion is centered on a bevel-cut double mat. The case is felt lined with an easy-open back panel to insert the flag and scratch-resistant bottom. Outside dimensions of the case are 26"L x 13"H x 3-1/2"D.


Prizes are a Camrbidge Audio ALVA TT Direct Drive Turntable with Bluetooth aptX HD and a $250 World Wide Stereo eGift Card.
The approximate retail value of all prizes is $1,949.99.
Open only to those who sign up at the online sweepstakes page and who are at least 13+ years old at the time of entry. The sweepstakes is open to the legal residents of the United States and is void where prohibited by law.
You may enter once daily and you must fill in the information requested.


Starting this week (Wednesday March 13, 2019 or soon after) PPS will commence GPM IMERG data reprocessing for IMERG V06. PPS will start reprocessing with the GPM-era data and will make these products available as they are archived.
Thereafter, we will reprocess the TRMM-era data. The first available products for the GPM-era will start with June 2014. TRMM-era products will span June 2000 - May 2014. Earlier TRMM-era processing, 1998-01 to 2000-05, will proceed when all required input data have been provided.
Starting early the week of March 04, 2019 or shortly after ; PPS will be doing a limited RETRO-processing of Level 1 - Level 3 F16 SSMIS as Version V05B for the period of April 2015 through January 2018. Please see the table below for the specific products and dates affected.
If you have previously obtained any of the data listed below, please discard these and replace this data with the retro-processed data as listed.
Beginning on Monday February 04, 2019, the Level 2 and Level 3 GPM GPROF products will be produced with an updated monthly emissivity class file (see developers comments below). PPS will increment the product version for all Level 2 and Level 3 GPROF products starting with the first full orbit on data date February 1, 2019. The current version of these products will cease with data date Jan 31, 2019 and will remain unchanged.
The new production version numbers/ designations will be incremented as shown in the table below.
Effective on the date and time above, all the NRY GPROF data product verson numbers were changed by increasing the letter of the data product version. This letter change means that data products were changed but no backward processing is undertaken. Only the products forward of the date/time have been altered. So, for GMI GPROF the vesion changed from V05A to V05B. Other instrument GPROF retrievals had a similar change but appropriate to their starting data product version.
Heavy precipitation has been falling in areas of California that were recently devastated by deadly wildfires. This flooding rainfall has resulted in evacuations in burn scarred areas such as Butte County where the deadly Camp Fire hit this month. Flash floods, debris flows and mudslides are now predicted in areas where deadly wildfires stripped away vegetation. On a positive note these Pacific storms are expected to dampen wildfires and replenish the Sierra Nevada snowpack. This snowpack is an important source of water for California's streams and rivers.


text:
Q: Tensorflow error in import tf.nn.rnn_cell I am using Tensorflow 1.0.0 and Python 3.5.
When I try to do:
cell = tf.nn.rnn_cell.BasicRNNCell(state_size)

I get the following error:
AttributeError
  <ipython-input-25-41a20d8458a7> in <module>()

      1 # Forward pass
      2 print(tf.__version__)
  --->3 cell = tf.nn.rnn_cell.BasicRNNCell(state_size)
      4 states_series, current_state = tf.nn.dynamic_rnn(cell, inputs_series, initial_state = init_state)
  AttributeError: module 'tensorflow.python.ops.nn' has no attribute 'rnn_cell'

Can someone help me?

A: TensorFlow changes a lot of APIs before 1.0.
You'll need to replace tf.nn.rnn_cell.BasicLSTMCell by tf.contrib.rnn.BasicLSTMCell

A: I have same problem in tensorflow 2.1 and when I used this code:
rnn_cells = tf.nn.rnn_cell.MultiRNNCell(
            [lstm_cell(size_layer) for _ in range(num_layers)],
            state_is_tuple = False,
        )

I faced with this error:
AttributeError: module 'tensorflow_core._api.v2.nn' has no attribute 'rnn_cell'

Finally, I replaced tf.nn.rnn_cell.MultiRNNCell by tf.compat.v1.nn.rnn_cell.MultiRNNCell, and then it worked well.
Please, replace tf.nn.rnn_cell.BasicRNNCell(state_size) by tf.compat.v1.nn.rnn_cell.BasicRNNCell(state_size).

redpajama_set_name: RedPajamaStackExchange


Cut the turkey wings and put them in the pot to boil for 20 minutes on their own. Then put the carrots on the potato onion salt oil and rice. Water we put as sparse or thick as we want our soup. When it's all boiling, we turn off the fire. Put the wings in a bowl and clean them from the bones. Cut them into small bits and add them to the pot.
For egg-lemon, beat the egg white in a bowl first, and then pour the egg yolk and beat them together. Continue to beat and add a little lemon juice. Finally, with a ladle we take the juice from the soup and continue to strike it slowly into the egg-lemon. When all the ingredients are joined to the bowl, pour them into the saucepan stirring continuously the soup. Serve warm with a bit of pepper on top.


Punjab State Power Corporation Limited (PSPCL) has released a recruitment notification for 850 posts of Lineman. Interested candidates may check the vacancy details and apply online from 02-11-2018 to 24-11-2018.
Educational Qualification: Candidates should have passed ITI in Wireman or Electrician trade.
How to Apply: Interested and eligible candidates may apply online through PSPCL website – https://www.pspcl.in/ – from 02-11-2018 to 24-11-2018.


Construction on the Hejaz Railway connecting Damascus to Medina started in 1900 under the Ottoman Sultan Abdul Hamid II. The al-'Ula station was toward the end of the line. It was abandoned when the line stopped working in 1920. The station house, fortifications, windmill, and water tower are still on the site, as are cars of derelict trains.


Second night in Lanxess Arena, Cologne, Wednesday September 5th 2018.
Lifting the roof of the Lanxess Arena in Cologne tonight with a special shoutout to longtime friend from the days when everyone gave everyone another name: 'Bottle Of Milk right in front of you'. Here's the full set list from Cologne 2.
Written by Christian Beyermann for Reflections Of Darkness website : '...Around half past 8pm, the lighting was dimmed, the Noel Gallagher song 'It's a Beautiful World' started playing, and the giant screens came to life with a video projection of a mash-up of scenes from Charlie Chaplin's iconic film 'The Great Dictator', interlaced with pictures of real and would-be dictators (think Kim Jong-Un and Donald Trump), destroyed cities all over Europe in the aftermath of WWII, and demonstrating youths bearing "Refugees welcome" banners... U2 started playing 'The Blackout' from the 2017 'Songs of Experience' album from inside the barricage, of which the main upper part was slowly lifted towards the ceiling. What became immediately clear is that the sound engineers had done a tremendous job. Wherever you were inside the huge arena, you were truly immersed into the soundscape of the concert without losing a sense of direction where the music originated, and all instruments and Bono's distinct vocals came across in a perfectly clear way. ..
The song 'Sunday Bloody Sunday' saw Bono and his band members marching on the barricage towards the other stage. The stage transition was also marked by an abrupt change in looks, as Bono donned a black coat, a top-hat, and a ton of make-up to mark his metamorphosis into a Mephistopheles -like character. This was augmented by the screen projection which positioned a devilish mask onto Bono's face, while he was singing songs like 'Elevation' and 'Vertigo'.
"Danke für die geile Nacht!"
zootopian Saydee was at the show and was updating us live with video snippets via twitter, you can review the night through her tweets here.
If you were at the second show in Cologne, add your own photos and reviews below.
Again no Amercan Soul ...... one of THE rock & roll songs U2 ever made ...... And for the real fans the same setlist and no AR experience ?!! But again AMAZING performance (litlle bit better then 1st cologne) !!!!!!!!!!!!!!!!!!!!!!!!!!!! Keep on going guys .... not only the best band of Ireland ....OF THE WORLD !!!
Full voice is back!!!!! We are so thankful for leading us to higher levels during the our own eXPERIENCE trip of emotional intelligence, which everybody should discover. We thank You for your true Love, which is born out of things needed to do in a long way and finally leads to ONE of the deepest hurts, to meet "IRIS" again. Healing... We know there is no feeded lie. Maybe even McPhisto (Good to have You back here Man!) thinks sometimes:"Get all these cynics into their clinics. We truly love Bono.
The personal touch added to the songs by Vino's spoken word poetry, personal annecdotes, open commentary about latest episodes line Bono's I'm losing his voice and gaining it back (Mephisto has something to say about that), just enough activism and politics to make it meaningful without losing touch, production level of imagery and soundscapes... All made for an unforgettable night.
One more chance and yes, satisfied. VERTIGO is pure ADRENALINE. (look up my niece's Adrenaline, it's not rock but she rocks!) I think I might want to sing. I know I want to go to Dublin.
From the Glens of Antrim to Koln.
What a show guys you are like a fine maturing wine. Belfast and Dublin you are in for one hell of a show. I loved the reworking of Pride and especially New Years Day. You never sit on you hands and play it safe.
My mom and I will travel all the way from Brazil just to enjoy the show! It will be my mom´s first concert ever. Can´t wait!


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Tag:	#EndSARS campaign
Nine positive lessons from #EndSARS campaign
It's often said that "experience is the best teacher" and while this might not hold water in certain cases, it's certainly true in…
Lekki shooting: I felt like I was dying, says Omoni Oboli
Omoni Oboli, an actress and film director, has narrated her battle with trauma after the shooting of unarmed #EndSARS protesters by soldiers in…
H.E.R shows support for #EndSARS protest during SNL show
Gabriella Wilson, an American singer better known as H.E.R, is the latest celebrity to show support for the #EndSARS protest. The award-winning singer…
#EndSARS: 'Don't stop accepting donations' — Genevieve begs Feminist Coalition
Genevieve Nnaji, a Nollywood actress, has pleaded with the Feminist Coalition to reconsider its decision to stop receiving donations aimed at supporting #EndSARS…
Instagram apologises for incorrectly flagging #EndSARS posts as false
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Q: Why BufferedInputStream#reset() did not thrown a RuntimeException? BufferedInputStream#mark(int) function gets as an argument the limit of bytes that can be read, once read, the mark become invalidated.
In the OCP book mark(int) is described as:

...you can call mark(int) with a read-ahead limit value. You can then
  read as many bytes as you want up to the limit value.

So the code below is setting limit value to 1 byte, after reading that byte, mark should be invalidated and calling .reset() function should throw a RuntimeException, yet it is not happening. Is it the JVM which is somehow overriding the argument passed to mark function?
public class Main {

public static void main(String[] args) throws IOException {

    Path path = Paths.get("Java8_IOandNIO\\src\\main\\resources\\abcd.txt");

    File f = new File(path.toString());

    FileInputStream fis = new FileInputStream(f);
    BufferedInputStream bis = new BufferedInputStream(fis);

    bis.mark(1);

    System.out.println((char) bis.read());
    System.out.println((char) bis.read());
    System.out.println((char) bis.read());

    bis.reset();

    System.out.println("called reset");

    System.out.println((char) bis.read());
    System.out.println((char) bis.read());
    System.out.println((char) bis.read());
  }
}

The code each time is printing data from the sample file:
A
B
C
called reset
A
B
C

A: Well, the documentation (original contract from InputStream) states:

If the method mark has not been called since the stream was created, or the number of bytes read from the stream since mark was last called is larger than the argument to mark at that last call, then an IOException might be thrown.

(Emphasis mine)
This means that the limit is a recommendation. It is not mandatory that the mark will be invalidated after that number of bytes have been read.

A: Because:

*

*the OCP you quoted doesn't say anything about throwing a RuntimeException;

*it doesn't say you can't necessarily read more;

*the OCP isn't a normative reference;

*the real normative reference doesn't say so either; and

*the stream is buffered, so it can support a mark of up to its internal buffer size, which is currently 8192 bytes.