APPENDIX I

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When a very distant galaxy or quasar is observed there are several
different quantities measured or calculated or previously given. These might
include the following quantities (very few of which are usually mentioned in
articles).

Measured:

La - the apparent luminosity (corrected for redshift as needed)
(The flux received)

z - the cosmological or relativistic Doppler red shift

N(z)obs - Observed galaxy count out to redshift z
Given
or assumed:

LA - the absolute luminosity assumed for the object (galaxy or
quasar or supernova) (The total power emitted over all angles)

d - the linear diameter assumed for the galaxy

R(t) - the scale factor of the universe (depending on the model)

r - the present mass density of the universe

l - the cosmological constant

G - the gravitational constant
Calculated:

Dl- the luminosity distance of the object (LA/4pLa)1/2

te - the age of the universe when the light was emitted that we
see now

tr - the age when the light is received (Also t0 for
the present age)

De - the proper distance away the object was at time
te

Dr - the proper distance away the object is at tr or
t0

Dlight - The proper distance away from us now of a light packet
emitted from our galaxy at time te

Dlt - the light "travel distance", ctlb

ve - the velocity of recession (rate of change of proper
distance) when the light was emitted we see now

vr - the velocity of recession (rate of change of proper
distance) when the light is received (now)

tlb - the "look back" time, i.e., the time the light has been
traveling

H0 - the Hubble "constant" at the present time

q - the angular diameter of the galaxy (could
also be measured)

r - the mass density of the universe (could also
be assumed)

Dph - the particle horizon

Deh - the event horizon

dN/dz - the relative number of galaxies counted in a telescope field of
view as a function of distance increment expressed in terms of z.

n - the number of new galaxies coming into view each year

Wm - the mass density of the
universe divided by the critical mass density (Could be assumed)

Wl - the cosmological constant, l, divided by 3H2 (or assumed)

Wk - the space curvature term,
k, divided by -H2 (or assumed)

APPENDIX II
Einstein deSitter cosmological model equations

Greek letters correctly.
If this is the case, email Ed Barlow, and he will fax or mail you a
written copy.

For this model the scale factor varies as R(t)~t2/3, the density is
critical, space is flat and t0=2/3H0. In a scientific
paper, these features are usually described by saying k=0 (space is flat),
q0=1/2 (the universe is decelerating), Wm=1 (the universe has the critical mass density)
and Wl=0 (there is
no cosmological constant).
(The subscript 0 denotes values at the
present time for the quantities listed.) Hence:

R(t0)/R(t)=(t0/t)2/3 =1+z
-------------------------(1)

The proper comoving distance of a galaxy
from us now is:

Dr=3ct02/3(t01/3-te1/3)
----------------------------(2)

(te is the time when the
light was emitted from it that we see now and t0 is
tr.)
This can be written as:

Dr=(2c/H0)(1-1/(1+z)1/2)
------------------------(2a)

Suppose we think of the question the other
way: how far away from us now is a light packet emitted from our galaxy at time
te? We can see that Dlight=Dr and hence
equations (2) and (2a) are correct for Dlight.
The proper comoving
distance away the galaxy was when it emitted the light we see now is:

De=Dr/(1+z)=3cte
2/3(t01/3- te1/3)
----------------(3)
De=(2c/H0)(1-1/(1+z)1/2)/(1+z)
------------------(3a)

This is, of course, the proper distance away from us the light packet was
when it left the galaxy so we can use this equation to trace the distance away
from us a light packet was from very very distant galaxies all the way back to
nearly the big bang when te~0 and z approaches infinity.

For this model, the particle horizon is at a proper distance of:

Dph=3ct0
-------------------------------------------(4)

Note: This is not ct0 as so many books state. (Maybe
they use the light travel distance, ctlb?)

There is no event horizon for this model.-------(4a)

There is a
distance concept called the luminosity distance, Dl.
If the total
luminosity of a galaxy is LA over all angles and if the observed
luminosity per unit area is La, then the luminosity distance is
defined as:

Dl =(LA/4pLa)1/2
---------------------------------(5)

and the present proper comoving
distance for this galaxy is:

Dr=Dl/(1+z)
---------------------------------------(5a)

The mass density for this
model is the critical density:

r=3H02/8pG ----------------------------------------(6)

As the particle horizon recedes, new galaxies are coming into view each year.
The number of such galaxies per year, n, is:

n=N0/t0
---------------------------------------------(7)

If there are 1011 galaxies now and t0=15
billion years, then n=7.

The observed angular diameter of a galaxy of linear diameter d is:

q=d/De
=(dH0/2c)(1+z)/(1-(1+z)-1/2) ------------(8)

This can also be expressed in terms of the present distance of the
galaxy, Dr as:

q=(d/Dr)/(1-Dr/3ct0)2
-----------------------------(8a)

Interestingly, this angle decreases
with increasing distance at first as would be expected, but then reaches a
minimum at z=1.25 and Dr=ct0 and gradually
increases for very distant galaxies and large z's.

The look-back time, tr-te is given by:

t lb=t0(1-1/(1+z) 3/2)
-------------------------------(9)

A concept of recession velocity can be introduced as v=dD/dt
and so v0=dDr/dt and ve=dDe/dt.
(These velocities will be different in this model because the universe is
slowing down).

v0=2c(1-(1+z)-1/2)
---------------------------------(10)
ve=2c((1+z)1/2-1)
----------------------------------(10a)

Note: The recession velocity defined in this way can
clearly exceed c which is consistent with some texts, but many texts and
articles say that the velocity can never exceed c. For this definition of
velocity, velocities greater than c are possible for galaxies we can still
see.

The Hubble constant, H0, is defined as v=H0d, but for
this Einstein deSitter model, which v is it, ve or v0, and
which d is it, De or Dr? In terms of present
velocity, v0, the results are:

De=(v0/H0)(1-v0/2c)2
------------------------------(11)
Dr=v0/H0
(surprise!)--------------------------------(11a)

For a homogeneous, isotropic universe the galaxy count per increment
of z as function of z is:

dN/dz~(1-(1+z)-1/2)/(1+z)3/2
----------------------(12)

As a function of Dr, however it is
dN/dDr~Dr2 ----------(12a)

Appendix III
Equations for Models II through V

NOTE: If you don't have the "Symbol" font, you won't be able to view some of the
Greek letters correctly.
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written copy.

For models II through V of this paper, if the scale factor of the universe, R(t)
is known, the value of H(t) can be found as well as the trajectory in the
time-distance plane of receding galaxies (neglecting local motions) and of the
light packet reaching us now from these galaxies. The behavior of the mass
density over time and the impact of the cosmological constant can also be
determined. From general relativity an equation called the Friedmann equation
has been developed which permits R(t) to be determined. The equation is:

(da/dt)2=H02(Wm0/a+Wl0a2+Wk0)----------------(1)

Where:
a=R(t)/R(t0) (The scale factor
normalized to 1 at the present time.)

Wm0=present mass density/critical
mass density (8pGr0/3H02)
Wl0=the
cosmological constant term at the present time (l/3H02)
l is the cosmological constant itself
Wk0=the space curvature term at the present
time (-k/H02)
k is negative for
negatively curved space, 0 for flat space and positive for positively curved
space

From these definitions it can be seen that Wm0+Wl0+Wk0=1 a(t) can be found by step-by-step
integration of equation (1).

From this, H(t) can be found and the space-time trajectory of the light
package reaching us now can also be found. This is how the curves for FigureV
and VI were generated.

The values of Wm0, Wl0, and Wk0 for Models II through V are:

For Model II, 0, 0, 1

For Model III, 1, 0, 0

For Model IV, 0.3, 0, 0.7

For Model V, 0.3, 0.7, 0

Wm(a) - the mass density at any
time divided by the critical mass density is given by the equation:

Wm(a)=(Wm0/a)/(Wm0/a+Wl0a2+Wk0)---(2)

When this is evaluated for a given value of a, it can be determined as a
function of time since a(t) is known.

References:

J. V. Narlikar Introduction to Cosmology (Cambridge University
Press Cambridge, UK 1993)

Jeffrey Weeks The Shape of Space (Marcel Dekker, 1985)

Jean-Pierre Luminet et al. "Is Space Finite?". Sci. Am. April 1999.
90-97

P. J. E. Peebles. Principles of Physical Cosmology (Princeton
University Press, Princeton, NJ 1993) pp. 93-100 and other parts

Arjun Dey et al. "A Galaxy at z=5.34". Astr. Phys. J. 498. L93-L97
(1998 May 10)

A. G. Reiss et al. "Observational Evidence from Supernovae for an
Accelerating Universe and a Cosmological Constant". Astr. J. 116.
1009-1038

A.V. Filippenko and A. G. Reiss. "Results from the High-z Supernova Search
Team". Phy. Rpts. 307. 31-44

Virginia Trimble. "Closing in on Cosmology". Sky and Telescope. 97 no
2. 38 (1999)

Joshua Roth. "The Race to Map the Microwave Background" Sky and Telescope
98 no. 3 Sept. 1999 44

Craig J. Hogan et al. "Surveying Space-Time with Supernovae". Sci. Am.
Jan 1999. 46-51

Lawrence M. Krauss. "Cosmological Antigravity". Sci. Am. Jan 1999.
53-59

Martin A Bucher et al. "Inflation in a Low-Density Universe". Sci. Am.
Jan 1999. 62-69

J. D. Cohn. "Living with Lambda". Astro. Phy. & Space Sci. 259
213-234

Corey S. Powell "The Race to Discover how the Universe Will End" Discover
Vol 23 No 9 September 2002 49-55

Ron Cowen "Dark Doings" Science News Vol.165 No. 21 May 22, 2004.

Graphs
Fig I Model I Paths (moving
through space) and redshifts
Fig II Model II Paths
(steadily expanding space) and redshifts
Fig III Model III Einstein
deSitter standard model, t0=15
Fig IIIA Model III
Einstein deSitter standard model, t0=10
Fig IV Model III Distance
types and look-back times
Fig V Model IV Current mass
density 0.3 critical
Fig
VI Model V The accelerating universe
Fig VII Model III Apparent
galaxy size as a function of redshift
Fig VIII Redshift vs
magnitude test curves for new supernova data
Fig IX Galaxy count as a
function of redshift for two models

© 2000. Ed
Barlow -- All rights reserved.

URL this site: http://www.CosmologyModels.com