Deriving Closed, Matter only, Friedmann Solutions

[Widdekind]

Trying to put explicit units back into this source (eqs. 121-125) could conceivably yield:

 

[math]A \equiv \frac{4 \pi G \rho_0 a_0^3}{3 c^2} = \frac{H_0^2 a_0^3}{2 c^2}\frac{\rho_0}{\rho_c} \equiv \frac{H_0^2 a_0^3}{2 c^2} \Omega_m[/math]

 

[math]a = A (1 - cos(\eta) )[/math]

 

[math]c t = A (\eta - sin(\eta) )[/math]

If so, one would presumably set a = a₀ in the first equation, to solve for the conformal time [math]\eta (d \eta \equiv c dt / a)[/math], and then plug the same, back into the second equation, to calculate the age of the current (closed) cosmos, given the original choice of (matter-only) density.

 

EDIT -- No, that wouldn't work. There are three (3) unknowns, a₀, t₀, [math]\eta_0[/math]. A third equation is required. What about...

 

[math]H \equiv \frac{\dot{a}}{a} = \frac{sin(\eta) \dot{\eta}}{1 - cos(\eta)}[/math]

 

[math]= \frac{c}{A} \frac{sin(\eta)}{(1 - cos(\eta))^2}[/math]

using the definition of conformal time (above). Thus, at the present epoch,

 

[math]H_0 = \frac{c}{A} \frac{sin(\eta_0)}{(1 - cos(\eta_0))^2} \implies \frac{H_0 A}{c} = \frac{sin(\eta_0)}{(1 - cos(\eta_0))^2}[/math]

 

[math]a_0 = A (1 - cos(\eta_0))[/math]

 

[math]\therefore \frac{H_0 a_0^2}{A c} = sin(\eta_0)= \sqrt{1 - cos(\eta_0)^2} = \sqrt{1 - (1 - \frac{a_0}{A})^2} = \sqrt{\frac{2 a_0}{A} - \frac{a_0^2}{A^2}}[/math]

 

[math]\therefore \frac{H_0 a_0^2}{c} = sin(\eta_0)= \sqrt{2 a_0 A - a_0^2}[/math]

 

[math]\therefore \frac{H_0^2 a_0^4}{c^2} = 2 a_0 A - a_0^2[/math]

 

[math]\therefore \frac{H_0^2 a_0^4}{c^2} = \frac{H_0^2 a_0^4}{c^2} \Omega_m - a_0^2[/math]

 

[math]\therefore \frac{H_0^2 a_0^4}{c^2}(\Omega_m - 1) = a_0^2[/math]

 

[math]\therefore a_0^2 = \frac{c^2}{H_0^2 \; \delta \Omega_m}[/math]

 

[math]\therefore a_0^2 \equiv \frac{D_H^2}{\delta \Omega_m}[/math]

where we have defined the mass-density over-contrast [math]\delta \Omega_m[/math] in the usual way.

 

This relation, in turn, allows us to determine the current conformal time [math]\eta_0[/math], from:

 

[math]a_0 = A (1 - cos(\eta_0))[/math]

where

[math]A = D_H \frac{\Omega_m}{2 \; \delta \Omega_m^{3/2}} = a_0 \frac{\Omega_m}{2 \; \delta \Omega_m}[/math]

 

[math]cos(\eta_0) = 1 - \frac{a_0}{A} = 1 - \frac{2 \; \delta \Omega_m}{\Omega_m}[/math]

Note that the conformal time runs from [math]0 < \eta < 2 \pi[/math], with the cosmic scale-factor a peaking at [math]\eta = \pi[/math]. Thus, the maximum 'size' & age of the universe are:

 

[math]a_{max} = 2 A[/math]

 

[math]c t_{max} = 2 \pi A[/math]

so that the 'size' growth-ratio is:

 

[math]\frac{a_{max}}{a_0} = \frac{\delta \Omega_m}{\Omega_m}[/math]

If so, then, for [math]\delta \Omega_m \approx 0.02[/math] & [math]D_H \approx 14 Gly[/math], space will expand by a further factor of 50 (from [math]a_0 \approx 100 \; Gly \rightarrow 5 \; Tly = 5000 \; Gly[/math]); and, the cosmos will exist, for a total of 16 Tyr = 16,000 Gyr. Note, though, that [math]a_0[/math] is a Radius of Curvature, so that the actual linear distance, through space, from (hyper-)pode to anti-(hyper-)pode, is [math]\pi[/math] times further. If so, then the maximum age of the universe exactly corresponds to the maximum 'girth', of space, at the 'equator' of spacetime. Traveling at the speed-of-light, matter would barely have time to 'circum-navigate' the 'hyper-globe' of spacetime.

[Widdekind]

The 3D hyper-spherical volume, of a hyper-spherical Closed Cosmic spacetime, is (??):

 

[math]S_3 = \pi R^2 S_1 = \pi R^2 (2 \pi R) = 2 \pi^2 R^3[/math]

With [math]R^2 = a_0^2 = D_H^2 / \delta\Omega_m[/math], w.h.t.:

 

[math]S_3 = \frac{2 \pi^2 D_H^3}{\delta \Omega_m^{3/2}}[/math]

and [math]D_H = 4000 Mpc[/math], w.h.t.:

 

[math]S_3 = 15.5 \times 10^6 \; Gly^3[/math]

??? Note that S₂ = 107,000 Gly² (???).

[Widdekind]

If [math]a_0 = D_H / \delta \Omega_m^{1/2} \approx 100 \, Gly[/math], and if [math]H_0 = \dot{a_0} / a_0 \approx (14 Gyr)^{-1}[/math], then would that mean, that the current 'radial rate' of expansion, of Cosmic spacetime, through hyperspace, be [math]a_0 \times H_0 = c / \delta \Omega_m^{1/2} \approx 7 c[/math] (for the Closed cosmology considered) ??

Edited February 12, 2011 by Widdekind

[Widdekind]

The Proper Distance, from 'here & now' (coordinate origin, at time t = t₀), 'out & back' to some specified 'there & then' (co-moving coordinate, corresponding to some time t, or equivalently, some red-shift z), is (Carroll & Ostlie. Intro. Mod. Astrophys., p.1260):

 

[math]d(t) = R(t) \int_t^{t_0} \frac{c d\tau}{R(\tau)}[/math]

Now, can this equation not be re-written (??), as:

 

[math]d(t) |_{t=t_0} = R(t)|_{t=t_0} \int_R^{R_0} \frac{c \; dR}{R \frac{dR}{d\tau} } = R_0 \int_R^{R_0} \frac{c \; dR}{R^2 \, H }[/math]

If so, then, using [math]R/R_0 \equiv 1/(1+z)[/math], w.h.t.:

 

[math]dR = - \frac{R_0}{(1+z)^2} dz = - \frac{R^2 \, dz}{R_0}[/math]

s.t.:

 

[math]\therefore d_0 = -\int_z^0 \frac{c \, dz'}{H(z')}[/math]

Now, from the solutions of the GR equations, constrained by the Cosmological Principle, w.h.t.:

 

Rewritten in terms of red-shift z, instead of normalized scale-factor a = R/R₀ = 1/(1+z) (and ignoring Radiation & Curvature contributions), w.h.t.:

 

[math]d_0 = -\int_z^0 \frac{c \, dz'}{H_0 \sqrt{\Omega_M (1+z')^3 \; + \; \Omega_\Lambda} }[/math]

 

 

 

 

Flat, Matter-only, Critical Cosmology

 

Now, for a flat, matter-only (i.e., matter-critical) Cosmos, wherewithin [math]\Omega_M = 1[/math], w.h.t.:

 

[math]d_0 = -\int_z^0 \frac{c \, dz'}{H_0 \, (1+z')^{\frac{3}{2}}} = \frac{2 c}{H_0} (1+z')^{- \frac{1}{2}} |_{z'=z}^{0} = \frac{2 c}{H_0} \left( 1 - \frac{1}{\sqrt{1+z}} \right)[/math]

This seems to agree, w/ the afore-cited source, so can I conclude these equations are correct ??

 

 

 

 

Flat, Matter-and-Cosmological-Constant, Critical Cosmology

 

Numerical integration, of the fuller formula, including matter ([math]\Omega_M \approx 0.3[/math]) and 'dark energy' ([math]\Omega_\Lambda \approx 0.7[/math]) yields the following relative distances (in units of c / H₀ = 14 billion light-years), and fractional Visible Universe volumes, for a selection of survey red-shifts:

 

Survey    z        d             %VU        # VU galaxies (billions)
2MRS      0.03   0.0297962     1:1,364,782          67
HDF       4      1.67424       1:7.692948          600
HUDF      7      2.0145        1:4.416153          620
--       inf    3.30508          1

The actual existence of 'dark energy' seems to be more consistent, with the recently released 2MRS survey, as compared to the HDF & HUDF surveys.

 

 

 

'dark energy' was "turned on" at z=1 ??

 

From the afore-derived formula, one can compute a 'mixed Cosmology', wherein 'dark energy' was "turned on", at z = 1, for unspecified reasons:

 

[math]d_0 = -\int_z^0 \frac{c \, dz'}{H_0 \sqrt{\Omega_M (1+z')^3 \; + \; \Omega_K (1+z')^2 \; + \; \Omega_\Lambda} }[/math]

 

[math] = - \left( \int_1^0 + \int_z^1 \right) \frac{c \, dz'}{H_0 \sqrt{\Omega_M (1+z')^3 \; + \; \Omega_K (1+z')^2 \; + \; \Omega_\Lambda} }[/math]

 

[math] = - \frac{c}{H_0} \left( \int_1^0 \frac{dz'}{ \sqrt{\Omega_M (1+z')^3 \; + \; \Omega_\Lambda} } + \int_z^1 \frac{dz'}{\sqrt{\Omega_M (1+z')^3 \; + \; \Omega_K (1+z')^2}} \right)[/math]

 

Survey    z        d             %VU        # VU galaxies (billions)
2MRS      0.03   0.0297962     1:1,031,997          51
HDF       4      1.484067      1:8.352197          650
HUDF      7      1.775787      1:4.875157          680
--       inf    3.011067          1

Thus, an unexplained 'dark energy was activated at about z=1' scenario is noticeably less compatible, with the comparison, between the 2MRS vs. HDF/HUDF sky surveys.

 

What would such a scenario -- which would, presumably, have actually altered the curvature of this Cosmic space-time fabric -- imprint upon the 'here & now' observed CMBR ?? Can the current CMBR observations categorically exclude any kind of (large) global Cosmic curvature, at all previous epochs ??

Edited June 24, 2011 by Widdekind

[pantheory]

Widdekind,

...Thus, an unexplained 'dark energy was activated at about z=1' scenario is noticeably less compatible, with the comparison, between the 2MRS vs. HDF/HUDF sky surveys....

I think you need to calculate backwards, the reason being that binned data of type 1a supernova indicate a change concerning apparent brightness of supernova at about z=.6 . At redshifts smaller than this, supernova appear to be dimmer than expected on an average (binned data). For redshifts greater than z=.6, supernova appear to be brighter than expected with increasing brightness as distance increases. So calculations, I believe, should adjust for a conclusion that dark energy (DE) was activated at redshifts z~.6, and continuing to the present time. As for me, I simply believe the Hubble formula for distances needs a wee tweak. I made changes myself to the Hubble formula that seem to compensate for the divergence. I think this is a much simpler explanation for variations of supernova data (about 11% max. adjustment is needed based upon my calcs) rather than the DE hypothesis implying cosmological reformulation.

 

You also may wish to consider that according to the DE proposal there was never a Hubble Constant as in your formulation, i.e. no constant rate of expansion. Before z=.6 the rate of expansion accordingly increases as you go forward in time to the present day. Before z=.6 the rate of expansion accordingly increases as you go backward in time, the minimum expansion rate being at about z=.6 according to this hypothesis.

Edited June 27, 2011 by pantheory

[Widdekind]

The Proper Distance, from 'here & now' (coordinate origin, at time t = t₀), 'out & back' to some specified 'there & then' (co-moving coordinate, corresponding to some time t, or equivalently, some red-shift z), is (Carroll & Ostlie. Intro. Mod. Astrophys., p.1260):

 

[math]d(t) = R(t) \int_t^{t_0} \frac{c \, d\tau}{R(\tau)}[/math]

 

"Conformal Time", in closed cosmology, is equivalent to "Hyper-Angle" subtended from "Hyper-Center of Space-Time"

 

In any given time-interval dt, photons will always travel a true metric distance, of c x dt. Thus, the "Conformal Time":

 

[math]\Delta \eta \equiv \int_{t_1}^{t_2} \frac{c \, dt'}{R(t')}[/math]

can be visualized, for a closed cosmology, as a "hyper-angle", subtended from the "hyper-center of space-time", [math]d \theta = c \, dt / R(t)[/math]:

 

In a closed cosmology, then, co-moving coordinates correspond to "hyper-angular distances", measured "around the circumference" of space-time. Then, by such visualization, it is immediately apparent, why the proper distance, between two contemporaneous points in space, is equal to [math]D_P(t) = R(t) \times \Delta \eta = R \, \Delta \theta[/math].

 

This can be verified, by noting, from the FRW metric, for a closed cosmology (k = +1), that:

 

[math]\frac{c \, dt'}{R(t')} = \frac{ dw }{ \sqrt{1 - w^2} } = ds[/math]

where w, dw, s, ds are all defined relative to the fixed, co-moving, reference-frame, which expands with 'stretching' space-time:

 

Thus, for closed cosmologies, the Conformal Time has an unambiguous geometrical interpretation, as the "hyper-angular" co-moving coordinate.

Edited July 26, 2011 by Widdekind