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Re: Old Stars



James McLean wrote:
Given my new understanding of this (Thanks!), a few questions come to mind:

Consider a cosmic background photon which was originally emitted someplace
which I'll call 'there', and is just now getting 'here'. Say this photon
comes from an event that was 10^10 years ago. How far apart were 'here'
and 'there' when the photon was emitted? How far apart are 'here' and
'there' now? If space has been expanding, it's not clear to me that the
answer to either question is 10^10 lightyears.

And Rick Tarara wrote:
Seems like the answer is tied up in general relativity and the
geometry of the Universe. Let me pose the same question a little
differently.
If about 12 billion years ago our friend 'Q' (from STNG) turned on a
flashlight for 1 second while standing at the 'center' of the Universe
(from previous Paul Camp writings I suspect there is no such
place--but--), where are the photons from that flashlight today? In a
Euclidian geometry the photons would be 12 billion lightyears away
(provided they didn't pass too near any large massive bodies and
provided the Universe itself isn't inside a black hole). But what
does GR say about these photons? Without knowing the mass and size of
the Universe, do we really know enough about it's geometry so as to be
able to answer the question?

Your both right. James is correct that photon propagation time does not give
proper separation in space, and Rick is correct that the energy density of the
universe crucially affects the geometry of the universe, and the geometry
affects the answer to the question about how far the photon source is from
the photon absorber given the photon propagation time. Rick is also correct
in suspecting that the universe has no "center".

Below are some answers to the questions that James and Rick pose above for a
model universe of a uniform density of pressureless dust which obeys GR
without a cosmological constant term. Let _rho_ = avg. mass density of the
universe and _rho_c =critical avg. mass density needed to just barely close
the universe. The answers are given for 4 cases: 1) _rho_ = 2 _rho_c,
2) _rho_ = _rho_c, 3) _rho_ = _rho_c/2, and 4) _rho_ = _rho_c/50. Case 1) is
a finite closed universe which will undergo a Big Crunch. In this case one
can imagine that the universe as the 3-d "surface" of a hypersphere in
4-dimensions whose radius expands from zero to a maximum value and then
shrinks back to zero size. (This is a 3-d version of the 2-d inflating balloon
model). Case 2 is the (theoretically preferred) borderline infinite flat
space model which expands forever. (It is sometimes imagined as a 3-d version
of an infinite flat sheet of rubber stretching uniformly in all directions.)
Cases 3) & 4) are examples of the infinite curved "hyperbolic" space model
which also expands forever (It is sometimes imagined as a sort of 3-d version
of a uniformly stretching rubber "saddle surface".) These last 2 cases can be
thought of as sort of an analytic continuation of the model in case 1) where
the central angle between 2 pts. on the "sphere" is imaginary--thus converting
all trig. functions to their hyperbolic analogs. Case 4) is typical of our
universe *only if* there is *no* dark matter. In all cases we will take the
current relative expansion rate of the universe H_0 = 1/(16.5 Gyr) =
= 59 km/s/Mpc. This value of H_0 (a.k.a the Hubble constant) is a typical
compromise among the competing oft-quoted values recently bandied about. In
the subsequent discussion we let: t = time coordinate since the BB, D_H(t) =
proper horizon distance, i.e. proper distance at time t to a photon source
which emitted its first photons at the t = 0 BB, t_0 = the current "age of the
universe", t_1 = 1.00 Gyr which is the assumed time since the BB when the
photons were emitted, D_1 = the proper distance "now" to a photon source whose
photons are just now reaching us if the photons were emitted at time t_1,
D_1' = the proper distance "then" (at emission time) between the photon source
and "here", f_1 = the ratio of the "scale size" of the universe "then" to the
"scale size" "now", and tau_1 = the time of propagation for the photons. In
addition we will consider a 2nd photon emission time of t_2 = 500 kyr. The
quantities D_2, D_2', f_2, and tau_2 are the corresponding quantities for the
photon emission time t_2 rather than t_1. Let R(t) = the "radius of the
universe" (cutting through the 4-th dimension to the center of the
hypersphere) at time t. Since Case 2) is flat we have R(t) = infinity for all
t > 0 in Case 2. For Cases 3) and 4) we take R(t) to be the effective
"hyperbolic radius" or "radius of hyperbolic curvature" of the universe.

Case 1: _rho_ = 2 _rho_c
t_0 = 9.42 Gyr R(t_0) = 16.50 Glyr D_H(t_0) = 25.92 Glyr
t_1 = 1.00 Gyr t_2 = 500 kyr = 0.0005 Gyr
tau_1 = 8.42 Gyr tau_2 = 9.42 Gyr
D_1 = 14.04 Glyr D_2 = 24.98 Glyr
D_1' = 3.48 Glyr D_2' = 0.04008 Glyr = 40.08 Mlyr
f_1 = 0.2482 f_2 = 0.001604
R(t_1) = 4.095 Glyr R(t_2) = 0.02647 Glyr = 26.47 Mlyr
D_H(t_1) = 2.95 Glyr D_H(t_2) = 0.00150 Glyr = 1.50 Mlyr

Case 2: _rho_ = _rho_c
t_0 = 11.00 Gyr R(t_0) = infinite D_H(t_0) = 33.00 Glyr
t_1 = 1.00 Gyr t_2 = 500 kyr = 0.0005 Gyr
tau_1 = 10.00 Gyr tau_2 = 11.00 Gyr
D_1 = 18.16 Glyr D_2 = 31.82 Glyr
D_1' = 3.67 Glyr D_2' = 0.04053 Glyr = 40.53 Mlyr
f_1 = 0.2022 f_2 = 0.001274
R(t_1) = infinite R(t_2) = infinite
D_H(t_1) = 3.00 Glyr D_H(t_2) = 0.0150 Glyr = 1.50 Mlyr

Case 3: _rho_ = _rho_c/2
t_0 = 12.43 Gyr R(t_0) = 23.33 Glyr D_H(t_0) = 41.13 Glyr
t_1 = 1.00 Gyr t_2 = 500 kyr = 0.0005 Gyr
tau_1 = 11.43 Gyr tau_2 = 12.43 Gyr
D_1 = 22.63 Glyr D_2 = 39.65 Glyr
D_1' = 3.75 Glyr D_2' = 0.04009 Glyr = 40.09 Mlyr
f_1 = 0.1656 f_2 = 0.001011
R(t_1) = 3.86 Glyr R(t_2) = 0.02359 Glyr = 23.59 Mlyr
D_H(t_1) = 3.06 Glyr D_H(t_2) = 0.00150 Glyr = 1.50 Mlyr

Case 4: _rho_ = _rho_c/50
t_0 = 15.94 Gyr R(t_0) = 16.67 Glyr D_H(t_0) = 88.14 Glyr
t_1 = 1.00 Gyr t_2 = 500 kyr = 0.0005 Gyr
tau_1 = 14.94 Gyr tau_2 = 15.94 Gyr
D_1 = 40.39 Glyr D_2 = 83.81 Glyr
D_1' = 3.22 Glyr D_2' = 0.02907 Glyr = 29.07 Mlyr
f_1 = 0.0796 f_2 = 0.0003469
R(t_1) = 1.33 Glyr R(t_2) = 0.00578 Glyr = 5.78 Mlyr
D_H(t_1) = 3.80 Glyr D_H(t_2) = 0.00150 Glyr = 1.50 Mlyr

David Bowman
dbowman@gtc.georgetown.ky.us