Lindquist-Wheeler formulation of lattice universes

Rex G. Liu

Phys. Rev. D 92, 063529 – Published 22 September 2015

Abstract

This paper examines the properties of “lattice universes” wherein point masses are arranged in a regular lattice on spacelike hypersurfaces; open, flat, and closed universes are considered. The universes are modeled using the Lindquist-Wheeler (LW) approximation scheme, which approximates the space-time in each lattice cell by Schwarzschild geometry. Extending Lindquist and Wheeler’s work, we derive cosmological scale factors describing the evolution of all three types of universes, and we use these scale factors to show that the universes’ dynamics strongly resemble those of Friedmann-Lemaître-Robertson-Walker (FLRW) universes. In particular, we use the scale factors to make more salient the resemblance between Clifton and Ferreira’s Friedmann-like equations for the LW models and the actual Friedmann equations of FLRW space-times. Cosmological redshifts for such universes are then determined numerically, using a modification of Clifton and Ferreira’s approach; the redshifts are found to closely resemble their FLRW counterparts, though with certain differences attributable to the “lumpiness” in the underlying matter content. Most notably, the LW redshifts can differ from their FLRW counterparts by as much as 30%, even though they increase linearly with FLRW redshifts, and they exhibit a nonzero integrated Sachs-Wolfe effect, something which would not be possible in matter-dominated FLRW universes without a cosmological constant.

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Received 2 February 2015

DOI:https://doi.org/10.1103/PhysRevD.92.063529

Authors & Affiliations

Rex G. Liu^*

Trinity College, Cambridge CB2 1TQ, United Kingdom and DAMTP, CMS, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

^*R.Liu@damtp.cam.ac.uk

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Vol. 92, Iss. 6 — 15 September 2015

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Images

Figure 1
Plots of $a$ vs $t$ for open, flat, and closed dust-filled universes.
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Figure 2
A test particle comoving with the boundary between two cells must be simultaneously falling toward both central masses. Thus the boundary itself must expand and contract accordingly.
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Figure 3
Hypersurfaces of constant Schwarzschild time $t$ intercept each other rather than mesh together at the cell boundaries. The normals from different cells point in different directions at the boundary. Therefore, Schwarzschild $t$ cannot serve as a global cosmological time coordinate for our model.
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Figure 4
Hypersurfaces of constant $τ_{CF}$ and $τ_{LW}$ mesh at the boundaries, and the normals from the two cells coincide. Therefore either can serve as cosmological time.
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Figure 5
A qualitative plot of various comoving test-particle trajectories (dashed line) inside a closed Schwarzschild cell and of various constant $τ_{CF}$ surfaces (dotted line); the plot is of Schwarzschild coordinates $r$ vs $t$ . The cell boundary’s trajectory has been drawn with a solid line. The $r$ axis passes through the boundary’s moment of maximum expansion, which happens when $τ_{CF} = τ_{\max}$ . Since the radial velocity must change sign abruptly when the boundary starts closing, there is a noticeable discontinuity in the gradients of the test particle graphs at the $r$ axis. The constant $τ_{CF}$ surfaces for $τ_{CF}$ just greater than and less than $τ_{\max}$ are not identical. Between the two surfaces is a “gap” region, shaded in the figure, where space-time points do not have a well-defined cosmological time $τ_{CF}$ .
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Figure 6
Rescaling the cell size by $ξ$ rescales the entire lattice universe by $ξ$ as well.
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Figure 7
A spherical cell of radius $a_{LW} \sin ψ$ and an infinitesimally smaller shell, indicated by the dashed line, have been embedded in a three-dimensional hypersphere of radius $a_{LW}$ , with one of the angular dimensions suppressed. The radial distance between the two shells, as measured along the 3-sphere, is $a_{LW} d ψ$ . Because of the curvature of the underlying 3-sphere, we have that $d (circumference) / d (radial distance) = 2 π \cos ψ$ .
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Figure 8
An analogous embedding in a flat hypersurface. In this case, the circumference of the boundary is $2 π a_{LW}$ ; the radial distance between the two shells is $d r$ ; and $d (circumference) / d (radial distance) = 2 π$ .
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Figure 9
An analogous embedding in a hyperbolic hypersurface. The circumference of the boundary is $2 π a_{LW} \sinh ψ$ ; the radial distance between the two shells is $a_{LW} d ψ$ ; and $d (circumference) / d (radial distance) = 2 π \cosh ψ$ .
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Figure 10
A photon travels from radius $r_{i}$ at cosmological time $τ_{i}$ in one cell to radius $r_{f}$ at time $τ_{f}$ in the next, as indicated by the long solid arrow. We assume that the photon always passes through the boundary in the manner illustrated here: that is, the point of crossing is a point of tangency between the boundaries of the two cells. A photon is emitted at $(τ_{i}, r_{i})$ by a source traveling along a geodesic given by (10) and observed by an observer in another cell. Although in a different cell from the source, the observer is still “comoving” with the source; that is, for any $τ$ , the observer is always at the same radius as the source in their respective cells, and this requires the observer to travel along a geodesic with the same $E$ as the source.
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Figure 11
A flat lattice universe with cubic cells approximated by spheres. The central masses are depicted by dots. A photon will on average travel through an equal number of gap and overlap regions. Because we are using spherical boundaries, at boundary crossing A, the photon will jump over a gap region, but at crossing B, it will pass through the overlap twice. The cubic boundaries are tangent to each other everywhere along the boundaries while the spherical boundaries are nowhere tangent to each other except where they intersect; though even there, the tangency is only in the entire ( $3 + 1$ )-dimensional space-time, not in any specific three-dimensional constant- $τ$ hypersurface. However, in the LW approximation, we shall assume that the spherical boundaries are everywhere tangent to each other, both in the entire ( $3 + 1$ )-dimensional space-time and in each three-dimensional constant- $τ$ hypersurface; this is because we are treating the spherical boundaries as if they were approximately identical to the cubic ones.
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Figure 12
On the left is an example of the original Schwarzschild block with one angular coordinate suppressed, and on the right is the Regge block to which it is mapped. $(τ, ρ, ψ)$ is a Minkowski coordinate system within the Regge block.
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Figure 13
A plot of $z_{LW}$ against $z_{FLRW}$ for 15 trajectories in a flat universe with an initial cell size of $r_{b_{0}} = 3 \times 10^{4} R$ . The initial angle $θ_{n}$ of the trajectory is given in the legend. Each trajectory was traced across 50 cells. A linear regression has been performed for each graph, with the first five data points excluded so as to focus only on the linear regime. The regression equations and corresponding root-mean squares (RMS) of the residuals are listed above in order of decreasing $θ_{n}$ .
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Figure 14
A plot of redshifts $z$ against the radius of observation $r_{obs}$ for photons traveling in the flat universe with an initial cell size of $r_{b_{0}} = 3 \times 10^{4} R$ . The photons were traveling initially in the directions of $θ_{0} = π / 2$ and $θ_{29} = 311 π / 3000$ . Plots for both LW and FLRW redshifts are shown. All four series start at the same data point, and there is a clear jump in the LW graphs between the first and second data points.
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Figure 15
A plot of redshifts $z$ against the radius of observation $r_{obs}$ for a photon starting instead with a position and trajectory corresponding to the second data point of the $θ_{0} = π / 2$ trajectory in Fig. 14. Plots for both LW and FLRW redshifts are shown. This time, both graphs extend smoothly out from the zero-redshift data point.
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Figure 16
A plot of photon frequencies $ν_{LW}$ against the radius of observation $r_{obs}$ for a photon traveling initially in the direction of $θ_{0} = π / 2$ ; the initial cell size is $r_{b_{0}} = 3 \times 10^{4} R$ . Each graph represents the photon’s frequencies within a single cell; these are the frequencies that would be seen by a comoving observer if the photon intercepted the observer at $r_{obs}$ . The analytic frequencies are given by (70), and graphs for the first seven cells traversed are shown. Frequencies were also computed numerically by simulating the propagation of a photon across the first cell only; the corresponding graph is shown and completely overlaps with its analytic counterpart.
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Figure 17
The equivalent plot to Fig. 16 but for a photon traveling initially in the direction of $θ_{29} = 311 π / 3000$ . Once again, the numerical and analytic graphs for cell 1 overlap completely.
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Figure 18
The equivalent plot to Fig. 16 but showing the redshifts $z_{LW}$ of cell 1 only against the radius of observation $r_{obs}$ . The two graphs overlap each other completely.
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Figure 19
(a) A plot of $z_{LW}$ against $z_{FLRW}$ for a tangential and a nearly radial trajectory in the flat universe with initial cell size of $r_{b_{0}} = 3 \times 10^{4} R$ . This is the equivalent plot to that of Fig. 13 but with the photon starting at $10^{4} R$ instead. The regression lines now pass very close to the origin, and the initial curve is nearly absent; we also note that the gradients are nearly unity. Other trajectories, not shown, showed similar behavior. (b) A plot of $z_{LW} - z_{FLRW}$ against $z_{FLRW}$ for 15 trajectories in the same universe, including the two trajectories in (a). Each graph still shows an initial curve, but it is small and short lived, and linearity is soon established.
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Figure 20
A plot of $z_{LW}$ against $z_{FLRW}$ for the $θ_{0} = π / 2$ photon trajectory in the (a) $r_{b_{0}} = 10^{5} R$ universe, (b) $r_{b_{0}} = 10^{6} R$ universe, (c) $r_{b_{0}} = 10^{7} R$ universe, and (d) $r_{b_{0}} = 10^{8} R$ universe. $Δ r_{0}$ was $10^{- 5} R$ for (a) and (b), $10^{- 4} R$ for (c), and $10^{- 3} R$ for (d).
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Figure 21
(a) A plot of $z_{LW}$ against $z_{FLRW}$ for a tangential and a nearly radial trajectory starting at radius $10^{7} R$ in the $r_{b_{0}} = 10^{8} R$ flat universe. Each trajectory was traced across 15 cells. Compared to Fig. 19, the regression lines here pass even closer to the origin; the initial curve is even more suppressed; and the gradients are even closer to unity. Other trajectories, not shown, showed similar behavior. (b) A plot of $z_{LW} - z_{FLRW}$ against $z_{FLRW}$ for 15 trajectories in the same universe, including the two trajectories in (a). Compared to Fig. 19, the initial curve here is even smaller and more short lived.
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Figure 22
A plot of the gradients of $z_{LW}$ vs $z_{FLRW}$ graphs against $\cos (θ_{n})$ for flat universes of different initial cell size $r_{b_{0}}$ . A linear regression has been performed on each graph, and the regression equations are displayed in order of increasing $r_{b_{0}}$ .
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Figure 23
A plot of $z_{LW}$ against $z_{FLRW}$ for 12 trajectories in the $E_{b} = 1.1$ universe. For this simulation, $Δ r_{0}$ was $10^{- 3} R$ . The initial angle $θ_{n}$ of the trajectory is given in the legend. Each trajectory was traced across 15 cells. A linear regression has been performed for each graph. The regression equations and corresponding root-mean squares of the residuals are listed above in order of decreasing $θ_{n}$ .
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Figure 24
Plot of the gradients of $z_{LW}$ vs $z_{FLRW}$ graphs against $\cos (θ_{n})$ for open universes of different values of $E_{b}$ . Also shown is a plot for the flat universe, corresponding to $E_{b} = 1$ . A linear regression has been performed on each graph, and both the regression equation and the root-mean square of the residuals are displayed in order of increasing $E_{b}$ .
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Figure 25
A plot of the $z_{LW}$ vs $z_{FLRW}$ gradients against $E_{b}$ for a selection of initial photon trajectories. Each graph approaches an asymptotic value as $E_{b}$ increases.
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Figure 26
A plot of $z_{LW}$ against $z_{FLRW}$ for two trajectories of photons, $θ_{0} = π / 2$ and $θ_{29} = 17 π / 150$ . Each trajectory’s graph is divided into a subgraph for when the trajectory is outgoing and a separate subgraph for when the same trajectory becomes ingoing; the two subgraphs meet at the redshift data point closest to maximum expansion of the universe. There is a slight bend in the blueshift regime of the ingoing $θ_{29} = 17 π / 150$ graph.
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Figure 27
A plot of $z_{LW}$ against $z_{FLRW}$ while photons are outgoing. Graphs for a selection of photon trajectories $θ_{n}$ are shown. A linear regression was performed on each graph, and the regression equation is listed in order of decreasing $θ_{n}$ .
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Figure 28
A plot of $z_{LW}$ against $z_{FLRW}$ while photons are ingoing. Graphs for the same trajectories as in Fig. 27 are shown. A linear regression was performed on each graph, and the regression equation is listed in order of decreasing $θ_{n}$ . Though in this case, the regression only includes redshifts measured at $r_{obs}$ equal to at least $10 R$ , the photon’s starting radius, so as to exclude points in the initial bend.
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Figure 29
A plot of $z_{LW}$ vs $z_{FLRW}$ gradients against $\cos (θ_{n})$ , where $θ_{n}$ is the initial direction of the photon. The gradients for when the photons are outgoing and for when they are ingoing are plotted separately, and all gradients are obtained by performing a linear regression on $z_{LW}$ vs $z_{FLRW}$ data. To avoid the initial bend in the $z_{LW}$ vs $z_{FLRW}$ graphs for ingoing photons, the ingoing gradients are obtained by regressing over only data points for which $r_{obs}$ is at least equal to the photon’s starting radius of $10 R$ .
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Figure 30
A plot of $R / r_{\max}$ vs ${\dot{ρ}}_{init}$ for various $r_{init}$ . Each graph has 100 data points. A quadratic regression has been performed on each graph, and the regression equations are ordered from $r_{init} = 8 R$ to $512 R$ . The regressions were performed without a linear term; regression was also attempted with a linear term present, but it was found to be many orders of magnitude smaller than the other two terms. The simulation’s block parameters were $R = 31$ , $Δ t = 10 Δ r$ , $Δ r = R / 10^{5}$ , and $Δ ϕ = 2 π / (3 \times 10^{7})$ .
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Figure 31
A plot of $(u^{ρ})^{2}$ against ${\dot{\tilde{ρ}}}^{2}$ for various $r_{\max} < 0$ , where $u^{ρ}$ is given by (58). Each plot consists of velocities computed at 100 different radii along the course of the test particle’s trajectory. A linear regression was performed on each graph, and the corresponding equations are ordered from $R / r_{\max} = - 1 / 11$ to $- 1 / 91$ . The graphs of the regressions completely overlap each other. The gradients only begin deviating from unity at the $10^{- 5}$ order of magnitude. The simulation’s block parameters were $r_{init} = 2 R$ , $R = 21$ , $Δ t = Δ r = R / 10^{5}$ , and $Δ ϕ = 2 π / (3 \times 10^{7})$ .
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Figure 32
A plot of $1 / \dot{ρ}$ vs $r / R$ comparing simulated particle velocities with $u^{ρ}$ . For the simulated particles’ graphs, $\dot{ρ} = \dot{\tilde{ρ}}$ , while for the $u^{ρ}$ graph, $\dot{ρ} = u^{ρ}$ . The simulation was run for various sizes $Δ r$ of the Regge block. For all simulations, the particle was propagated from $r = 2 R$ to $3 \times 10^{4} R$ . The simulation’s other block parameters were $R = 1$ , $Δ t = Δ r$ , and $Δ ϕ = 2 π / (3 \times 10^{7})$ .
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Figure 33
A plot of $R / r_{\max}$ vs ${\dot{ρ}}_{init}$ for $r_{init} = 3 \times 10^{4} R$ . A quadratic regression has been performed and the corresponding equation shown. The simulation’s block parameters were $R = 1$ , $Δ t = 10 Δ r$ , $Δ r = R / 10$ , and $Δ ϕ = 2 π / (3 \times 10^{7})$ .
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Figure 34
A plot of $\dot{\tilde{ρ}}$ vs $r / R$ comparing a simulated particle velocity with $u^{ρ}$ . The simulation began at $r_{init} = 3 \times 10^{4} R$ , but only data points from $r = 2 R$ to $100 R$ are shown. The simulated velocity is always greater than or equal to $u^{ρ}$ . The simulation’s block parameters were $R = 1$ , $Δ t = Δ r = R$ , and $Δ ϕ = 2 π / (3 \times 10^{7})$ .
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Figure 35
The Regge block interpolation intersects the Schwarzschild graph at $r = r_{i}$ and $r = r_{i + 1}$ , where $r_{i} < r_{i + 1}$ and where $r_{i}$ corresponds to the origin of the graph; it interpolates the radial distance for only those values of $r$ lying between these two points of intersection. Since the Schwarzschild distance function is convex, the interpolation will always underestimate the distance in this regime.
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