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*From*: David Bowman <David_Bowman@georgetowncollege.edu>*Date*: Sat, 10 Oct 2020 02:53:24 +0000

Regarding

One reasonable choice is the time when the universe cooled to

the point where it was no longer a plasma (free electrons and

protons) and became instead a bunch of neutral hydrogen atoms.

This event is observable in the microwave background radiation.

Before this, the universe was opaque to radiation, so nothing

before that will be "visible", at least not using optical (or

radio) telescopes.

If one takes the latest data obtained from the (now retired) Planck spacecraft & combines them with constraints from Baryon Acoustic Oscillations, CMB lensing effects and some other stuff we have the best fit (2018 final) experimental cosmological parameters <https://en.wikipedia.org/wiki/Planck_(spacecraft)> being:

H_0 = 67.66 km/s/Mpc = 1/(14.451555 Gly) (current Hubble parameter)

Omega_lambda = 0.6889 (dimensionless dark energy density in units of its critical value)

Omega_m = 0.3111 (dimensionless total, i.e. normal + dark, matter density in units of critical value)

Decoupling redshift z_* = 1089.8 (current mean redshift of CMB since it last scattered in a frame in which it is maximally isotropic, i.e. no dipole component)

k = 0 (spatially flat on largest length scales) This means 1 = Omega_M + Omega_lambda.

If we use the commonly accepted lambda-CDM cosmological model (i.e. dark energy acts like a simple cosmological constant combined with cold dark matter) and homogenize and simplify it so a flat (k=0) homogeneous isotropic Friedmann–Lemaître–Robertson–Walker metric describes the Universe's dynamics with a pressureless dust of matter one can find a closed form formula for the elapsed time of flight for radiation observed as a function of its redshift (assuming that redshift is 100% cosmological in origin). Also the current distance away from the observer for the source/scattering site of such radiation as a function of its redshift is also doable as a simple 1-dim definite integral (which could found in terms of hypergeometric functions, but is much simpler to just evaluate numerically). Applying these results to the CMB, whose redshift is 1089.8, we find that the time since the CMB was last scattered, at matter/radiation decoupling is 13.79102 Ga, and that surface of last CMB scattering at decoupling is currently 45.49457 Gly away from us and is receding from us at a speed of 3.148075*c (as per Hubble's law).

Other choices are possible, but let's not get into that.

One possible choice of dubious validity is to simply use the above simplified model and extrapolate it all the way back to z -> ∞. Doing so goes all the way back to the causally hard particle horizon reflecting the finite time available since the initial singularity. Doing this yields a slightly longer time than back to decoupling since that decoupling time happened far closer to the singularity in time than we currently are. Doing this extrapolation gives a total age for the universe of 13.79150 Ga, and yields a particle horizon distance from us of 47.06357 Gly away from us, and it is receding from us at 3.256644*c.

But this extra extrapolation all the way back to z -> ∞ is not really legitimate because the simplified model so extrapolated assumes the universe is a homogeneous pressureless dust of matter supplemented with a fixed cosmological constant/dark energy density. Unfortunately at the early stages of the universe's life it wasn't such a thing. Rather it was dominated by radiation, not matter, and that radiation exerted quite a significant pressure. This means a correct model needs to more accurately reflect the conditions that prevailed then. It also neglects even more esoteric effects from the early universe, e.g., acoustic fluctuations, field theoretic symmetry breaking phase transitions, inflation, and other complicating factors. All those effects don't change the order of magnitude of the *time interval* from creation to the CMB matter/radiation decoupling. But the simple model is only qualitatively correct and not quantitatively correct in that early era. In fact we can subtract our above age numbers 13.79150 Ga - 13.79102 Ga and get a time interval from creation to decoupling of 4.8x10^5 years. But supposedly this time interval was actually more like 3.8x10^5 years when the relevant complicating early universe effects actually *are* included in the calculation.

A similar subtraction for the difference between the location the CMB's last scattering surface and that of the particle horizon at creation vis 47.06357 Gly - 45.49457 Gly = 1.569 Gly is the prediction of the simple model for how much beyond the CMB last scattering surface is the hard particle horizon reflecting the creation singularity. But this is really extremely misleading. Neglected effects like inflation and other stuff have a much greater effect on the distances and rates of expansion in the initial phases of expansion than they do on the time scales involved. Because of this I put no confidence whatsoever in the supposed 47 Gly distance to the particle horizon. My guess is that that distance is really *much* farther away (if not actually infinitely far).

In case anyone is interested in the formulas for the above simplified (homogeneous, isotropic, pressureless, dust) model I include some of them below.

Let x = x(t) be the dimensionless linear scale size of the universe at time epoch t since creation in units of (i.e. divided by) its current scale size. Let t_0 be the current age of the universe (i.e. time ago when light was emitted which is just now received having an infinite redshift). (Thus x(t_0) = 1 by definition & x(0) = 0).

Then x(t) = A*(sinh(B*t))^(2/3)

Here A == (1/Omega_lambda - 1)^(1/3) and

B == (3/2)*H_0*sqrt(Omega_lambda)

t_0 = 2*arctanh(sqrt(Omega_lambda))/(3*H_0*sqrt(Omega_lambda))

Let Dt = Dt(z) = time interval over which a light beam is in flight from a source until it arrives here now having a redshift z.

Dt = B*ln((1 + 1/sqrt(Omega_lambda))*(z + 1)^(3/2)/(1 + sqrt(1 + (A*(z + 1))^3)))

Let D_0(z) be the current proper distance to a light source whose light we are now receiving and which has a redshift z.

H_0*D_0/c = Integral{1/(z + 1),1|dx/(x*sqrt(x + Omega_lambda*(x^4 - x)))}

Here the definite integral function has the structure: Integral{lowerlimit,upperlimit|integrand}.

David Bowman

**References**:**[Phys-L] cosmology***From:*Anthony Lapinski <alapinski@pds.org>

**Re: [Phys-L] cosmology***From:*John Denker <jsd@av8n.com>

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