What if Cosmic Structure ceased expanding?

[Widdekind]

What would happen, to the expansion of the universe, if Structure stopped stretching (as fast as the Voids)? Qualitatively, assuming a flat cosmology; and assuming that, in recent epochs (z<2) Structure has been expanding more slowly than Voids; so that by present epoch (z=0) Structure occupies only a quarter of the volume of space, whereas Voids occupy three quarters:

 

[math]\rho = \frac{M_S + M_V}{V_S + V_V} \approx \frac{M_{tot}}{V_{tot} \left( \frac{1}{4} \alpha^2 + \frac{3}{4} \alpha^3 \right) } = \frac{\rho_0}{\frac{1}{4} \alpha^2 + \frac{3}{4} \alpha^3} [/math]

where [math]\alpha \equiv a(t)/a_0[/math] is the normalized scale factor. So, from the first Friedmann equation:

 

[math]\frac{d\alpha}{\alpha} = \frac{dt}{T_0} \sqrt{ \left( \frac{\Omega_0 = 1}{\frac{1}{4} \alpha^2 + \frac{3}{4} \alpha^3} \right)}[/math]

 

[math]\frac{d\alpha}{\alpha} \sqrt{ \frac{1}{4} \alpha^2 + \frac{3}{4} \alpha^3 } = \frac{dt}{T_0} [/math]

 

[math]\frac{3 d\alpha}{2} \sqrt{ \frac{1 + 3 \alpha}{4} } = \frac{3 dt}{2 T_0} \equiv d\tau [/math]

 

[math]\Delta \left( \frac{4}{3} \left( \frac{1 + 3 \alpha}{4} \right)^{\frac{3}{2}} \right) = \Delta \tau[/math]

Such is the solution, for scale factor vs. lookback time, for slowly stretching Structure, from present epoch (z=0), back to the epoch when Structures occupied the same (relative) volume as Voids (z=2). At that epoch (z=2), the combined volumes of Structure & Voids was only 1/18^th that at present epoch. So, the cosmic average density was 18x that at present epoch. Thus, for earlier epochs (z>2), the above solution must be matched to a (re-scaled) standard solution, for flat matter-dominated expansion:

 

[math]\frac{d\alpha}{\alpha} = \frac{dt}{T_0} \sqrt{ \left( \frac{\Omega_2 = 18}{ \alpha^3} \right)}[/math]

 

[math]\Delta \left( \alpha^{\frac{3}{2}} \right) = \Delta \left( \tau \sqrt{18} \right)[/math]

Calculating the lookback time to (z=2), then adding the same to the second solution, and plotting, reveals the following final evolution for the (normalized) scale factor (x-axis) vs. lookback time (y-axis):

 

The scale factor initial grows with increasing time (decreasing lookback time) as (a~t^2/3), until (z=2), when Structures start to "stall", stretching slower. The slower stretching of Structure "holds everything up"; the scale factor grows much more slowly with increasing time thereafter, e.g. space only expands by a factor of 18x from (z=(2-0)) instead of 27x.

 

i was wondering if slowly expanding Structure could mimic the reported faster expanding Voids from current SNIa observations. However, slower stretching Structure effects the opposite of what was reported -- overall, spacetime stretches slower if Structure stalls out, not faster. So, slower stretching Structure does not match those reports. i offer the above equations for discussion -- are there any reasons why one could not make the density, in the RHS of the first Friedmann equation, any arbitrary function of the scale factor?

[Widdekind]

i suspect i may have made a mistake somewhere above. Please ponder the first Friedmann equation:

 

[math]\frac{1}{a} \frac{da}{dt} = H_0 \sqrt{\frac{\Omega}{\tilde{V}_V + \tilde{V}_S}}[/math]

If Structure stops expanding (as rapidly as Voids), then the volume occupied by Structure is less than it "should" be. That reduces the denominator, on the RHS. To compensate, on the LHS, the denominator (dt) must decrease, i.e. less time (dt) passes, for the same (percent) change in scale factor (da/a), i.e. the scale factor starts to "accelerate" its expansion, compared to what it "should" be doing. So, i offer, that Structure "stalling" out, could cause (from these crude calculations) the scale factor to, in turn, accelerate in its growth, so explaining the reported analyses of observations regarding SNIa.

 

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seeking simplicity, redefine "now" (z=0) so that the "present" epoch is the era when Structure (supposedly) started to slow its stretching, and "stall out" from the general expansion of the universe, afterwards occurring only in Voids. For sake of simplicity, assume a flat cosmology, wherein V & S each occupy half of all volume at "present" epoch; to "present" (z>0) the volumes occupied by V & S ~ (1/2)a³ + (1/2)a³; afterwards (z<0) the volumes increase as ~ (1/2)a³ + (1/2)a². The equations simplify considerably; and, do indeed, show that "Structure stall out" causes the scale factor growth to accelerate:

 

Edited December 3, 2012 by Widdekind

[Widdekind]

... seeking simplicity, redefine "now" (z=0) so that the "present" epoch is the era when Structure (supposedly) started to slow its stretching, and "stall out" from the general expansion of the universe, afterwards occurring only in Voids. For sake of simplicity, assume a flat cosmology, wherein V & S each occupy half of all volume at "present" epoch; to "present" (z>0) the volumes occupied by V & S ~ (1/2)a³ + (1/2)a³; afterwards (z<0) the volumes increase as ~ (1/2)a³ + (1/2)a². The equations simplify considerably; and, do indeed, show that "Structure stall out" causes the scale factor growth to accelerate:

 

 

One can imagine various mathematical forms modeling the slowing expansion, or even contraction, of regions of Structure. In the logical limit, the volume of regions of Structure vanishes:

 

[math]V_V + V_S = \frac{1}{2} a^3 + \frac{1}{2} a^3 \longrightarrow \frac{1}{2} a^3 + 0[/math]

In that limit, the scale factor begins growing nearly half-again as fast:

 

[math]\frac{d\alpha}{\alpha} \sqrt{ \frac{1}{2} \alpha^3 + \frac{1}{2} \alpha^3 } = \frac{dt}{T_0} [/math]

[math] \longrightarrow[/math]

[math]\frac{d\alpha}{\alpha} \sqrt{ \frac{1}{2} \alpha^3 + 0 } = \frac{dt}{T_0} [/math]

[math] \longrightarrow[/math]

[math]\frac{d\alpha}{\alpha} \sqrt{ \frac{1}{2}} \sqrt{ \alpha^3 } = \frac{dt}{T_0} [/math]

 

So, for a given increase in the scale factor, the shrinkage of structure reduces the LHS by the square-root of two; so, the RHS is reduced, i.e. less time passes during said expansion of the scale factor, i.e. expansion accelerates. That zero-size limit for Structure is (seemingly) the upper bound, to the effective acceleration of the (growth of the) scale factor, admissible, in this "Structure stall out" model.

 

Can not the density be treated as treated above? Could not the secession, of Structure, from the Hubble flow, cause and account for, the inferred acceleration of the expansion of the universe?