Historical Models of the Universe
    The Friedmann model, which we examined in some detail, is only one 
of the many possible interpretations of Einstein's descriptions of the 
universe. We look at some of the others, now only of historical 
interest. We start with the genralized equations of the cosmos, where 
no presumption is made about LAMBDA and k. These equations were 
developed by LeMaitre in 1931. LeMaitre was unaware of Friedmann's 
work of 1922. To honor the two for arriving at virtually the same 
equations, these generalized equations are called the Friedmann-
LeMaitre equations. 
    LeMaitre championed the Friedmann model because it had a definite 
beginning for the universe, the moment for R = 0. He could not stomach 
the deSitter and Einstein models with their infinite prior existence 
for the universe. The universe to LeMaitre had to have a real and 
definitive origin. He called the state of R = 0 the 'primoidal atom'. 

 |                                                   | 
 | 2der(R,t) = -4 * pi * gamma * rho0 / (3 * R^2))   |
+|            + LAMBDA * R / 3                       | 
 |                                                   |        | 
 | 1der(R,t)^2 / 2 = 4 * pi * gamma * rho0 / (3 * R) |
 |                   + LAMBDA * R^2 / 6 - k * c^2    |

    We also have the definitions of H and q

 | OF H AND q                       | 
 |                                  | 
 | H = 1der(R,t) / R                | 
 |                                  | 
 | q = -R * 2der(R,t) / 1der(R,t)^2 | 
 |   = -2der(R,t) / (R * H^2)       | 

    The Friedmann-LeMaitre equations can be thrown into a form 
involving LAMBDA, k, H and q. First for LAMBDA 

     2der(R,t) = -4 * pi * gamma * rho0 / (3 * R^2) 
                + LAMBDA * R / 3 
     2der(R,t) / (R * H^2) * (R * H^2) 
       = -4 * pi * gamma * rho0 / (3 * R2)  LAMBDA * R / 3 

     -q * R * H^2 = -4 * pi * gamma * rho0 / (3 * R^2)
                    + LAMBDA * R / 3        

   q * R * H^2 = 4 * pi * gamma * rho0 / (3 * R^2) 
                 + LAMBDA * R / 3 

    LAMBDA * R / 3 = 4 * pi * gamma * rho0 / (3 * R^2) - q * R * H^2 

     LAMBDA = 4 * pi * gamma * rho0 / R^3 - 3 * q * H^2 

Then for k 
    1der(R,t)^2 / 2 = 4 * pi * gamma * rho0 / (3 * R) 
                      + LAMBDA * R^2 / 6 - k * c^2 
                    = 4 * pi * gamma * rho0 / (3 * R) 
                      + (4 * pi * gamma * rho0 / R^3 
                      - 3 * q * H^2) * R^2 / 6 - k * c^2 
                    = 4 * pi * gamma * rho0 / (3 * R) 
                      + 2 * pi * gamma * rho0 / (3 * R) 
                      - q * H^2 * R^2 / 2 - k *c^2 
                    = 6 * pi * gamma * rho0 / (3 * R) 
                      - q * H^2 * R^2 / 2 - k * c^2  

    1der(R,t)^2 = 12 * pi * gamma * rho0 / (3 * R)
                  - q * H^2 * R^2 - 2 * k * c^2 
                = 4 * pi * gamma * rho0 / R 
                  - q * H^2 * R^2 - 2 * k * c^2 
     1der(R,t)^2 / R^2 = 4 * pi * gamma * rho0 / R^3 - q * H^2
                         - 2 * k * c^2 / R^2 
     H^2 = 4 * pi * gamma * rho0 / R^3 - q * H^2 - 2 * k * c^2 / R^2 

    2 * k * c^2 / R^2 + H^2 = 4 * pi * gamma * rho0 / R^3 - q * H^2 

     2 * k * c^2 / R^2 = 4 * pi * gamma * rho0 / R^3 - q * H^2 - H^2 
                       = 4 * pi * gamma * rho0 / R^3 + (-q -1) * H^2 
                       = 4 * pi * gamma * rho0 / R^3 - (q + 1) * H^2
 | EQUATIONS RELATING LAMBDA, k, H, AND q                           | 
 |                                                                  | 
 | LAMBDA = 4 * pi * gamma * rho0 / R^3 - 3 * q * H^2               | 
 |                                                                  | 
 | 2 * k * c*2 / R^2 =  4 * pi * gamma * rho0 / R^3 - (q + 1) * H^2 | 

    In principle all the quantities on the right of these two 
equations are directly observable and mensurable, thereby firmly 
fixing the values for k and LAMBDA.  

Diversity of Cosmological Models
    The Einstein relativity theory can make rather specific 
predictions about so many aspects of the world around us and then turn 
uncertain in describing uniquely the world itself! This shortcoming 
comes from the uncertainty in the parameters needed to apply Einstein 
physics to the cosmos as a whole.
    On this cause there derived from it not merely several knids of 
universe but very different ones. Universes with no mass, with no 
gravity, with diffuse matter evenly spread thruout, with Euclid 
geometry, with momEuclid geometry, with the lambda force, with no 
lambda force, &c, &c. How can this be when in so many other situations 
the Einstein system is pretty much right on the money? No one knows. 
     There are hundreds of distinct models of the universe but most 
are mathematical manipulations having little to do with the observable 
world. Others probably deserve renewed study. The models described 
here include the famous ones and a bunch of lesser fame. We give a few 
paragraphs for each because the models are highly evolved in their 
maths and physics. However for those in the Friedmann-LeMaitre family 
we do give some of details. The models are laid out in alpha order 
over the next several sections. 
    The models are named for their inventors or promotors. cosmology 
was -- and is -- a close-knit disciline. A model promoted by a one nat 
have  features contributed by others. 
    Yhe models are discussed roughly in time dequence. Some decades 
had several current models. 

Einstein (Original) Universe 
    Einstein -- and most scientists -- believed in the mid 1910s that 
the universe was static and enduring. Those who suggested that the 
spiral nebulae were whole other Milky Ways assumed that these other 
systems were sitting at rest quietly beyond the Milky Way. Others 
questioned the Milky Way as merely one of the galaxies, an 'island 
universe'. They asserted that all of space is filled with stars like 
within the Milky Way. 
    In this millieu Einstein in 1917 set H = 0, R = 1. This made the 
model conform to the believed real world. For simplicity's sake 
Einstein allowed a uniform distribution of mass thruout the universe.  

     LAMBDA = 4 * pi * gamma * rho0 / R^3 - 3 * q * H^2 
            = 4 * pi * gamma * rho0 / 1 - 3 * q *H^2 
            = 4 * pi * gamma * rho0 - 0 
            = 4 * pi * gamma * rho0 

     2 * k * c^2 / R^2 = 4 * pi * gamma * rho0 / R^3 - (q + 1) * H^2 
                       = 4 * pi * gamma * rho0 / 1 - (q + 1) * H^2 
                      = 4 * pi * gamma * rho0 - (q + 1) * 0 
                      = 4 * pi * gamma * rho0 - 0 
                      = 4 * pi * gamma * rho0 
                      = LAMBDA 

    LAMBDA is equal to 4*pi*gamma*rho0, which is essentialy the 
collapsive force due to the universe's self-gravitation. LAMBDA is 
itself a 'force', exactly equal in magnitude, but opposite in 
direction, to the gravity force. LAMBDA exactly counteracts the 
collapse and maintains a static cosmos. 
    Moreover, Einstein believed strongly that the universe must have 
spherical geometry with k = +1. Einstein invented a cosmos of matter 
with no motion, matching the prevalent notions of the cosmos. 
    This universe is dynamicly unstable. If rho0 changes by even the 
slightest amount, the equation unbalances and the left or right side 
dominates. The universe will begin either a runaway collapse or 
runaway expansion. 
    A peculiar feature of the Einstein model is its frame of absolute 
reference! The matter all at rest within this universe constitutes a 
coordinate system against which absolutr -- not just relative -- 
motion can be assessed. This feature gave Einstein all levels of fits. 

 |                                | 
 | LAMBDA = 4 * pi * gamma * rho0 | 
 |                                | 
 | k = +1                         | 
 |                                | 
 | q = not applicable             | 

deSitter Universe 
    deSitter in 1917 held that the universe is actually empty. Only 
the Milky Way -- with the spiral nebulae within it -- occupried the 
whole of infinite space. He also believed in a flat cosmic geometry. 
These premises make k = 0 and rho = 0. deSitter objected to the innate 
instability of the Einstein model. It could not prevail today, butt 
already would have collapsed or dissipated. According to deSitter 
     LAMBDA = 4 * pi * gamma * rho0 / R^3 - 3 * q * H^2 
            = 4 * pi * gamma * 0 / R^3 - 3 * q * H^2 
            = 0 - 3 * q * H^2 
            = -3 * q * H^2 
     2 * k * c^2 / R^2 = 4 * pi * gamma * rho0 / R^3 - (q + 1) *H^2 
                       = 4 * pi * gamma * 0 / R^3 - (q + 1) * H^2 
                       = 0 - (q + 1) * H^2 
                       = -(q + 1) * H^2 

     2 * 0 * c^2 / R^2 = -(q + 1) * H^2 

     0 = -(q + 1) * H^2   
which is satisfied only by q+1 = 0 

     q + 1 = 0 

     q = -1 


     LAMBDA = -3 * q * H^2 
            = -3 * (-1) * H^2 
            = 3 * H^2 
    Here we have a situation where there is motion with no matter! Raw 
empty space just continuously outswells and dissipates and expands on 
its own! The LAMBDA repulsion force has no gravity force to check it 
and the universe simply grows without limit. At all times the geometry 
is flat, like a Euclid plane. 
    deSitter postulated that this continuous expansion could be 
monitored if there were visible specks, cosmic confetti, sprinkled in 
space. He, thus, is the first person to actually formulate the notion 
of the expanding universe. He could not then appreciate that the 
universe IS filled with cosmic confetti! The luminous specks are the 
    In the early 1920s some astronomers argued for the extragalactic 
nature of the spiral nebulae BECAUSE they exhibited redshifts 
indicative of the deSitter expansion! 

 |                   | 
 | LAMBDA = 3 * H^2  | 
 |                   | 
 | k = 0             | 
 |                   | 
 | q = -1            | 

Einstein-deSitter Universe 
    This model has LAMBDA = 0 and k = 0 and was developed by Einstein 
& deSitter in 1932. It is a sort of compromise between their earlier 
individual schemes. In this, now the 'standard' model, Einstein did 
away with his cosmological constant or lambda force, being that the 
universe was now observed to expand. later cosmologists and 
astronomers noted that Einstein's invention of lambda force in his 
first model was the biggest mistake he ever made in science. 

     2 * k * c^2 / R^2 = 4 * pi * gamma * rho0 / R^3 - (q + 1) * H^2 
     2 * 0 * c^2 / R^2 = 4 * pi * gamma * rho0 / R^3 - (q + 1) * H^2 
     0 = 4 * pi * gamma * rho0 / R^3 - (q + 1) * H^2 

     (q + 1) * H^2 = 4 * pi * gamma * rho0 / R^3 

     q + 1 = 4 * pi * gamma * rho0 / (R^3 * H^2) 

     LAMBDA = 4 * pi * gamma * rho0 / R^3 - 3 * q * H^2  

     0 = 4 * pi * gamma * rho0 / R^3 - 3 * q * H^2       

     3 * q * H^2 = 4 * pi * gamma * rho0 / R^3 
     3 * q = 4 * pi * gamma * rho0 / (R^3 * H^2) 
           = q + 1 
     2 * q = 1 
     q = 1/2 

Which is exactly the result we obtained from the detailed analysis; 
this is the deceleration parameter. Note that in the Einstein-deSitter 
universe rho is a derived quantity to satisfy k = 0 and q = 1/2. Thus 
     4 * pi * gamma * rho0 / (R^3 * H^2) = q + 1 

     4 * pi * gamma * rho0 / (3 * H^2) = 3/2 

     rho0 / R^3 = rho 
                = (3/2) * H^2 / (4 * pi * gamma) 
                = 3 * H^2 / (8 * pi * gamma) 

Which is exactly what we obtained from the detailed analysis. This is 
the closure or critical density. 
 |                            | 
 | LAMBDA = 0                 | 
 |                            | 
 | k = 0                      | 
 |                            | 
 | q = 1/2                    | 

Eddington Universe 
    From the one relativity theory come simultaneously and 
independently two completely different models of universe! Both enjoy 
infinite past life. One, by Einstein, has a uniform distridution of 
matter and the other, by deSitter, has no matter at all. One stays 
static (in the absence of disturbances) and the other continually 
swells ever outward. 
    Both had strong adhaerents in the late 1910s and early 1920s. In 
fact, both satisfied a very prevalent assumption. The universe had no 
specific beginning but lived for all past time. 
    Furthermore, the Einstein and the deSitter models were for a 
decade merely mathematical exercises with no appeal to astronomy. 
Largely this was due to the unawareness of the galaxies and of the 
Hubble expansion. These were not announced until the mid 1920s. 
    Eddington was the first regular astronomer to apply realtivity to 
astronomy and to elaborate a cosmological model. Previous workers were 
mathematicians or physicists. He in 1927 favored the infinite life in 
the past for the universe and believed in the expansion feature of 
deSitter. However, he felt that the universe had a indefinite prior 
life as an Einstein system and then suffered a disturbance. This may 
have been the condensation of gas clouds into galaxies. This triggered 
 runaway expansion, which persists to the present era. 

leMaitre Universe 
    leMaitre in 1931 cooked up a universe that began as a Friedmann 
universe with a closed and spherical geometry. When the universe 
reached the maximum scalefactor under this expansion mode it switched 
to a deSitter expansion, which we are in at the present epoch. He took 
this route becuase he knew only of the closed solution, k = +1, of the 
Friedmann model and could not stomach the ultimate destiny of it in a 
colossal collapse. 
    leMaitre was the first to assert an actual bigbang origin for the 
universe. Altho Friedmann postulated a zero volume origin, he never 
applied it to the realities of astronomy. leMaitre, on the other hand, 
insisted that the universe must have a definite creation, mainly from 
his profession as an abbe. leMaitre held that the universe really did 
begin as a 'primaeval atom' about the diameter of the Earth's orbit 
and as dense as the atomic nucleus. It by some superfission exploded 
into the universe we live in. 
    Later in 1950 leMatire added a 'dwell' phase, when R remains 
constant for some long while, between the initial Friedmann expansion 
and the later deSitter expansion. This accommodated new finds from 
geology and astrophysics which greatly lengthened the known age of the 
universe. The original model now had too short a lifespan and had to 
be somehow stretched out. 
   The dwell phase at constant R was to leMaitre the period of galaxy 
formation, a process requiring a billion years or so. Once fully 
developed, the galaxies could begin their present expansion along the 
deSitter part of his model. 

Tolman (Oscillating) Universe 
    In 1934 Tolman believed in a closed universe that will in time 
undergo a colossal collapse. Moew over he noted that the collapse is 
followed by the birth of a new universe, separate and distinct from 
the old. That is, the universe oscillates or cycles between bigbang 
and colossal collapse over and over again. 
    In the collapse all identity from the previous world is lost and 
the new one starts off with all-new physics. The values of the 
physical constants are in general unrelated to their old values. 
    However, the one thing that does percolate across the collapse is 
the entropy of the old universe. The new universe sets off with a base 
entropy equal to that of the old at collapse. Because entropy 
continuously increases during the life of the universe, each 
successive universe has greater and greater entropy. 
    The oscillating universe was abandoned when the standard bigbang 
model was generally accepted, which has but a single world expanding 
into infinite future time. Never the less the Tolman model is 
resurrected from time to time, as by Landsberg and by Wheeler in 1975. 

leMaitre-Tolman Universe 
    leMaitre in 1933 came up with the first nonhomogenous solution for 
the description of the universe. Previous models and most current ones 
assume the universe to be smooth and uniform thruout. Tolman continued 
leMaitre's work in 1934, but did not actually collaborate with him. 
   In this model there are cells of uniformity and smoothness in the 
universe within which the Friedmann scenarios operate.
    Each cell is isolated from the others and behaves like a 
miniuniverse with its own bigbang and Hubble expansion regime. The 
geometries of the cells depend on their individual densities and may 
be hyperbolic. planar, or spherical. Some cells are in expansion while 
others are contracting. 
    With this model leMaitre & Tolman were first to sugget a 
multiverse scehem for the cosmos. That we may live in one cell within 
a many-cellled world is revived from time to time, often in ignorance 
of leMaitre & Tolman. 

Milne Universe 
    Milne in 1935 developed a cosmology based on the absence of 
gravity! The universe is entirely kinematic with no forces at all! In 
this scheme LAMBDA = 0 and gamma = 0. 
     2 * k * c^2 / R^2 = 4 * pi * gamma * rho0 / R^3 - (q + 1) * H^2 
                       = 4 * pi * 0 * rho0 / R^3 - (q + 1) * H^2 
                       = 0 - (q + 1) * H^2  
                       = -(q + 1) * H^2 
                       = -(0 + 1) * H^2 
                       = -H^2 
which only k = -1 can satisfy 
     k = -1 
     LAMBDA = 4 * pi * gamma * rho0 / R^3 - 3 * q * H^2  
            = 4 * pi * 0 * rho0 / R^3 - 3 * q * H^2   
            = 0 - 3 * q * H^2)  
            = -3 * q * H^2 
     0 = -3 * q * H^2 

which only q = 0 can satisfy 

     q = 0 

    This universe, with mass but no gracity forcess, expands from a 
bigbang to infinite scalefactor at a constant H and no deceleration. 
 |                | 
 | LAMBDA = 0     | 
 |                | 
 | k = 0          | 
 |                | 
 | q = 0          | 

Dirac Universe 
    Dirac in 1937 hit on the concept that the basic parameters of 
nature are somehow interrelated. He claimed that the ratio of the 
Couloumb and Newton force in an atom and that of the size of an atom 
and the universe were so similar that they must actually be equal. 
    This, he concluded, meant that there was some inner connection 
between the cosmic and atomic worlds. One result of this reasoning was 
that the Newton constant varies with time. 

     Fc = qe * qp / (4 * pi * epsilon0 * r^2)  
     Fn = gamma * me * mp / r^2 

where Fc is the Coulomb force; Fn, Newton force; qe, me, electric 
charge and mass of electron; qp, mp, charge and mass of proton. 

     Fc / Fn = (qe * qp / (4 * pi * epsilon0 * r^2)) 
               / (gamma * me * mp / r^2) 
             = qe * qp / (4 * pi * epsilon0 * gamma * me * mp) 
             = (1.602E-19c) * (1.602E-19c) * (8.988E9n.m^2/c^2) 
               / (6.673E-11n.m^2/kg^2) * (9.109E-31kg) * (1.673E-28kg) 
             = 2.268E39 

     ru = c * T0 

     re = qe^2 / (4 * pi * epsilon0 * me * c^2) 
where ru is the raddius of the universe; re, radius of electron. 

    ru / re = (c * T0 / (qe^2 / (4 * pi * epsilon0 * me * c^2)) 
            = me * c^3 * T0 * 4 * pi * epsilon0 / qe^2 
            = (9.109E-31kg) * (2.998E8m/s)^3 * (5.870E16s) 
              / ((8.988E9n.m^2/c^2) * (1.602E-19c)^2) 
            = 6.246E39 

    Altho the two ratios are off by about 3:1, Dirac laid the 
discrepancy to inaccurate values for the ingredients. The peculiar 
slug 1/(4*pi*epsilon0) comes up over and over again in 
electromagnetics and has the consolidated value 8.988E9n.m^2/c^2. It 
is the Couloumb constant, parallel to the Newton constant. epsilon0 is 
the permittivity of space, 8.854E-12c^2/n.m^2. 
    Right away we see that T0 is far too low by today's estimation; 
it's based on Hubble's far too large H0. It was the best availabe at 
the time. There is also the issue of the 'radius' of an electron. 
Dirac treated the electron as a particle having a definible size. 
    At any rate by forcing the equality Dirac found that some physical 
factor had to vary with time. 

     Fc / Fn = ru / re 

     qe * qp / ( 4 * pi * epsilon0 * gamma * me * mp) 
      =  me * c^3 * T0 * 4 * pi * epsilon0 / qe^2 
The left side has all 'constants' while the right contains the 
variable T0. Clearly to Dirac certain of the 'constants' are actually 
functions of time. He chose gamma for he believed that all the others, 
coming from relativity and atomics, were somehow more basic and were 
truly constants. So 

     gamma = qe^2 * qe * qp / (16 * pi^2 * epsilon0^2 
             * c^3 * me^2 * mp * T0)) 
           = func(1/T0) 

    The Newton constant decreases with time! This has all sorts of 
consequences in the evolution of the universe. But most cosmologists 
dismiss Dirac's number playing as so much mumbo-jumbo. The results are 
mere coincidence with no deeper meaning. And gamma really is invariant 
over time. Never the less some astronomers today are intrigued by the 
close approximation of these (and other) ratios of constants. 

Alpher-Bethe-Gamow Universe 
    Until World War II cosmology was mostly geometric, with a crude 
attempt at phyiscal explanation by leMaitre. Following the war atomics 
emerged as a major division of physics. By accepting the Einstein-
deSitter or Friedmann-leMaitre model, physicists tried to work out the 
early mix of nuclides under the high densities and temperatures near 
the bigbang moment. 
    Alpher, Bethe, & Gamow in 1948 were the first to apply atomic 
physics to cosmology.  They derived the observed mix of hydrogen, 
helium, and heavies from the initial bigbang state. This replication 
of the observed elemental alloy from the bigbang added strong support 
for the bigbang theory as a real historical origin for the universe. 
    One result of their work was Gamow's 1948 prediction that there 
was a moment when the energy-dominated bigbang photosphere would 
dissolve into the present matter-dominated cosmos. We should be 
immersed in the residual photospheric radiation from that moment, 
whose temperature now should be about 5K. 
    There was no way for about 15 years to hunt for this radiation, 
given that electronics were still too weak and radioastronomy was too 
new in 1948. 
    A search for this relic background was started in 1964 by Dicke & 
Peebles but Penzias & Wilson found it first in 1965. Cosmologists 
generally regard this cosmic background radiation as almost a 
conclusinve proof of the bigbang theory. Even Hoyle threw up his hands 
and abdicated his competing steady-state theory. He quickly picked it 
up again with a steasy-state explanation of this radiation. 

    A-B-G, sometimes called the 'alpha-beta-gamma' team, did not coin 
the origin moment 'bigbang'. They merely descrined their model  
without a specific name. 
    The name 'big bang', as two words, was coined by Hoyle, who with
others proposed a rival cosmology, the steady state theory. Hoyle, 
contrary to legend, did not mean to ridicule Gamow's theory. He just 
wanted a short quick way to describe Gamow's scheme. Yes, the name
stands today, often as one word.
    In the early 1990s Sky and Telescope magazine put up a contest to
rename the bigbang model. It  asked astronomers for replacement terms.
S&T posted a judging panel that featured Carl Sagan, who just
presented his 'Cosmos' TV series with 'big bang' freely cited thruout
it. The project closed with no winner picked, S&T found that most
entries were silly, spiteful, hostile. And just about all astronomers
realized that Sagan will never adopt any new name for 'big bang'.

Bondi-Gold (Steady State) Universe 
    This, the steady-state system, is included here for completeness 
in the summary of other cosmological models. 
    Bondi & Gold in 1948 found a model extending the cosmological 
principle. In the other models the universe is the same at all 
positions at the same moment. But not at all moments. The universe 
evolves over time and R, rho, and H, for instance, are functions of 
time. Bondi & Gold asserted that the universe is the same in every 
place AND IN EVERY TIME. There is NO evolution. In this model LAMBDA = 
0, H = H0, rho0/R^3 = rho = rho0. This makes R^3 = 1. 
    The motivation for the steady-state model was to accommodate the 
observed Hubble expansion within a universe of infinite past life. 
Bondi & Gold had an abhorrence for the bigbang concept. 

     2 * k * c^2 / R^2 = 4 * pi * gamma * rho0 / R^3 - (q + 1) *H^2 
     2 * k0 * c^2 / R^2 = 4 * pi * gamma * rho0 / 1 -(q0 + 1) *  H0^2 

Because the right side is a constant, being made of all fixed 
quantities, the left side must also be a constant. It can not have a 
variable 1/R^2. To make the left side a constant, k0 = 0. 

     k0 = 0 

     2 * 0 * c^2 / R^2 = -(q0 + 1) * H0^2 

     0 = -(q0 + 1) * H0^2 

Because H0 is nonzero, only (q0+1) = 0 satisfies this equation 

     q0 + 1 - 0 

     q0 = -1 
     LAMBDA = 4 * pi * gamma * rho0 / R^3 - 3 * q * H^2  
            = 4 * pi * gamma * rho0 / 1 - 3 * (-1) * H0^2 
            = 4 * pi * gamma * rho0 / 1 - 3 * (-1) * H0^2 
            = 4 * pi * gamma * rho0 + 3 * H0^2      
            = 0 

    4 * pi * gamma * rho0 = -3 * H0^2 

    Recall that rho0 and H0 DO NOT vary over time; these are NOT 
merely the instant values of rho and H. The universe does expand at a 
constant rate H0 yet the density of matter within it remains the same 
rho0. How can rho0 remain constant when the space containing it 
outswells, tending to decrease rho0? Bondi & Gold postulate that new 
matter is spontaneously created to maintain the fixed rho0. 
    Any patch of the universe has a mix of 'old' matter carried apart 
by the expansion and 'new' matter formed to fill the regions vacated 
by the old. In this way the universe presents the same demeanor and 
guise not only everywhere but also everywhen. 
    The rate of this continuous creation, according to Bondi & Gold, 
is essentially the rate of increase of volume as it is filled with new 
matter of density rho0 

     V = R^3 

     1der(V,t) = 3 * R^2 * 1der(R,t) 
               = 3 * R^3 * (1der(R,t) / R) * R 
               = 3 * R^3 * H0 

     rho0 * 1der(V,t) / V = 1der(m,t) / V 
                          = 3 * R^3 * H0 * rho0 / V 
                          = 3 * H0 * rho0 

which with typical values for H0 and rho0 is 

     1der(m,t) / V = 3 * H0 * rho0 
                   = 3 * (1.6E-18/s) * (4E-28kg/m^3) 
                   = 19.2E-46kg/s.m^3 

This is an incomprehensibly tiny rate beyond all hope of detection. It 
is equivalent to about two Suns per cubic megalightyear per century! 
In practical terms it amounts to finding the two new stars created 
anywhere within the Milky Way and environs every 100 years! 

 |                     | 
 | LAMBDA = 0          | 
 |                     | 
 | k = 0               | 
 |                     | 
 | q = -1              | 

Godel Universe 
    In the usual cosmological model there is a clear segregation 
between the past and the future. The light from other parts of the 
univerwe came from long ago times in the past while that leaving us 
will reach pther places some long time in the future. This is the 
separation of past and future lightcones. A 'lightcone' is the locus 
 of lightrays (all EMR) converging onto or diverging outof a given 
point. The converging cone is the past lightcone; diverging, future. 
The cone analog comes from the representation of two dimensions of 
space and one of time in the x-y-ct system. In the real wolrd the 
'cone' is the three-dimensional locus in x-y-z-ct spacetime. 
    On a cylindrical surface, to cite a familiar geometry, two rays, 
forward and backward in time, at a point eventually intersect again 
elsewhere on the surface. A message on the forward ray could be passed 
to the backward ray at the intersection and modify its character. 
Godel first worked the details out in 1949: Events in the future can 
modulate events in the past! 
    A ray of light from us is sent into spacetime with a message. 
Normally only a future observer can receive this message and act on 
it. In Godel's model that ray could be captured by a 'future' observer 
who transfers the message to a new ray directed at us. We ourselves 
can ultimately get that message -- from our own past -- and therby be 
influenced by it. From the eye of the lightray, it goes into the 
future to reach the past! This sounds so bizarre, like some kind of 
time travel, that the Godel universe today is not widely supported. 

Alfven-Klein Universe 
    Alfven & Klein in 1965 were the first to invoke plasma physics 
into cosmology. Their model postulates that the world was from 
infinite past time filled with a rarified fluid of plasma. This 
consisted of protons, electrons, positrons (antielectrons), and 
negatrons (antiprotons). The fluid was so sparse that the particles 
were too far apart to interact. 
    At some moment a casual enhacement of local density collected the 
plasma into a contracting sphere under sel-fgravitation. As the 
material densified the particles began a violent reign of self 
annihilation. The resulting radiation was so intense that it heaved 
out the sphere to become the expanding univrse we now observe. This 
radiation is also the 2.7K relic radiation we see today. 
    The expansion carried away the remaining matter so that the world 
today is half particles and half antis. They are separated by distance 
in segregated sections of the universe. Only at the boundaries between 
the sections do they touch to produce violent explosive events, which
were discovered in the early 1960s.

Arp Universe 
    SInce the discovery of quasars in the early 1960s Arp has 
questioned their redshifts as caused by Hubble expansion. He claims 
that quasars are associated with regular, tho active, galaxies and are 
at the same distance away as them. The redshifts are produced by an 
as-yet incomprehensible process within the quasars. His evidence are 
alignments of quasars with galaxies, attachemnt of quasars to adjacent 
galaxies by bridges or threads, and foreground superposition of 
quasars on the image of a remoter galaxy. By Arp, the quasars are 
mixed in with galaxies and are not standing at the fartherst realms of 
the universe. 
       In 1990 he turned off to the bigbang models and started to 
develop a steady-state scheme, based on the Bondi-Gold model. In the 
mid 1990s it is still unfinished. The crux of the Arp system, as 
gleaned from partial publications of it, is that the quasars are the 
seats of continuous creation and that they disperse and turn into 
galaxies. In this way the mean density of the universe is held stable. 

Hoyle-Narlikar Universe 
    Hoyle & Narlikar in 1964 developed a steady-state universe of the 
smae geometry as Bondi & Gold, but derived from electromagnetics as 
well as gravity. They were the first to incorporate electromagnetics 
into cosmology, it being only newly discovered thru radioastronomy. 
Prior models ignored electromagnetics as insignificant or unknown. 
    In this model the cosmological redshift arises from the varying 
mass of the atoms causing the spectral lines. Mass increasess since 
the time the atom was created. The light is really enitted at the 
longer wavelength in remote past time and reamins fixed during the 
light's travel to us. There is no expansion redshift! 
    When in 1965 the cosmic background radiation was found by Penzias 
& Wilson, Hoyle and other steasy-state promotors offered a novel 
expalantion for it. The radiation is merely stellar and other regular 
radiation which has been thermalized over vast travel distances into a 
low grade 2.7K heatflux. The thermalizing agent is alleged to be thin 
long 'whiskers' of iron supposedly a copious product of supernovae. 

Brans-Dicke Universe 
    Brans & Dicke in 1961 looked to replace Einstein physics with an 
all-new system and to base a cosmology on it. In the Brans-Dicke world 
gravity is a derived function of the scalefactor and it does not enter 
into the equations of the universe. Under this scheme the motions of 
bodies in space should be quite different from that predicted by 
    They tested the model on Mercury's perihelion, one of the crown 
tests used to prove relativity. They could make their model satisfy 
the observed perihelion excess by claiming a oblate Sun. But in solar 
eclispes of the 70s careful attention to the profile of the Sun turned 
up no oblateness. When the model was tested on other solar system 
phaenomenoa and tweeked into agreement, it became indistinguishable 
from Einstein relativity. With the necessity and convenience of the 
Brans-Dicke univere eroded, It was abandoned by 1980. 
EdeWitt (Many-Worlds) Universe 
    By the 1970s cosmology had fused with quantum and atomic physics 
such that papers in the one field often appeared in journals of the 
others. The tools of quantum physics were applied to cosmology in many 
ways. One was that of deWitt in 1975. Quantum physics predicts the 
probablility of an event's many possible occurrences, like the spot on 
a screen where an electron will strike from a cathode ray tube. It can 
not tell which occurrence will take place for a given instance. 
    Yet the electron somehow chooses a one and only one spot to hit on 
each instance. How? deWitt and his school interpret this behavior as 
the coexistence simultaneously of many worlds intertwined in the one 
space and time. Each world allows for each occurrence of a event. When 
the electron leaves the tube it 'jumps onto' one of the many 
universes, one that controls one of the spots on the screen. It 
'rides' in that universe to the spot. On a second shot it may pick an 
other one of the universes and hit some other point. Thus we live in a 
multitude of worlds swopping in and out of our lifes at each occasion 
of a quantum event. 
    The mechanism by which these worlds coexist is the subject of 
rather byzantine quantum physics far beyond this paper. Since the 
1970s the many-worlds idea developed into that of multiverses nd is a 
large theme in a newer discipline of string cosmology. 

Guth (Inflationary) Universe 
    One of the problems unexplained by the standard bigbang model is 
the horizon problem. If the universe expanded according to the 
Einstein-deSitter plan even the minutest deviation from pure 
uniformity and smoothness would have unraveled everything into a 
clumpy blotchy universe. But the cosmos is, under the standard model, 
uniform and smooth.
    In order to keep it so there must have been no chance for blotches 
and clumps to develop. Guth in 1981 theorized that the instant after 
the genesis the universe swelled instantly from the singularity to a 
certain finite size with all uniformity preserved. From this moment 
the Einstein-deSitter (or other Friedmann) expansion commenced. With 
nothing now to introduce the clumps and blotches these never formed 
and the universe remains uniform and smooth. This model is also called 
the inflationary universe from this initial vast inflation before the 
expansion phase. 
    The Guth model is the first to be built from quantum physics. The 
earlier efforts at using quantum physics, as in the deWitt universe, 
were not thoroly worked out into free-standing cosmological models.