a-cosmology-group/acg#135

I hope they keep the horn as a historic monument, a reminder that our science can be thrown off course by a misinterpreted observation.

In what follows, three references are made to posts in discussion 127 named Can the CMB temperature be less than 2.7 K ? Important for our theory is the following function, derived in the first reference.

$$ f(x) = \frac{2-e^{-x}(x^2+2x+2)}{x} \quad \mbox{where} \quad x = \frac{c}{H}.R $$

$x$ is a dimensionless variable, with $c=$ lightspeed, $H=$ Hubble parameter, $R=$ distance. All values of the function must be below a certain value. For e.g. the ΛCDM model we have found that $f(x) \lt 1/3$ . For other values of $x$ that part of the universe is a Black Hole, meaning that no light from there can enter our cosmic bubble. We cannot observe anything where $f(x) \gt 1/3$, according to ΛCDM. However, alternative cosmic models can be devised. The most extreme one is defined by calculating the maximum of the function $f(x)$. Done numerically:

f(x) := (2-exp(-x)*(x^2+2*x+2))/x;
diff(f(x),x); s := fsolve(%=0,{x});
assign(s);
w := (2-exp(-x)*(x^2+2*x+2))/x;

                        s := {x = 3.383634283}

                          w := 0.3883945571

This results in a condition $f(x) \lt 0.3883945571$, a bit less restrictive than ΛCDM, with far reaching consequences though. In a second reference we found for the total mass of our infinite SEU (Static Euclidean Universe) with $G=$ gravitational constant:

$$ M_U = \frac{c^3}{w.G.H} \quad \mbox{where} \quad w = 0.3883945571 $$

There is also the mass of our Observable universe, which is - easy to derive - related to the mass of the whole universe as

$$ M_O = \frac{W.w}{2} M_U = \frac{W}{2}\frac{c^3}{G.H} \quad \mbox{where} \quad W = 3.383634283 $$

And the radius of our Observable universe is

$$ R_O = W\frac{c}{H} $$

In a third reference we found for the gravitational self energy density:

$$ u_G = - GM^2/(8\pi r^4) $$

Integrating over our Observable universe:

$$ U_G = \int_{R_O}^\infty \left[ - \frac{G M_O^2}{8\pi r^4} 4\pi r^2 \right] dr = - \frac{G M_O^2}{2}\int_{R_O}^\infty \frac{dr}{r^2} = - \frac{G M_O^2}{2R_O} $$

The latter outcome is also known from the booklet Relativity Reexamined by Léon Brillouin: formula (7.13) times $c^2$. But now it goes wrong because, due to $E=mc^2$, energy has a mass. Therefore we must integrate the energy density according to the Variable Mass Theory; we cannot assume that it’s a constant. The total gravitational self energy of the observable universe herewith becomes

$$ U_G = - \frac{G M_O^2}{2} \int_{R_O}^\infty \frac{e^{-H/c.r}}{r^2}dr = - \frac{G M_O^2}{2R_O}.W.\mbox{VMT} $$

Computing the integral by hand is a bit difficult, but we let a Computer Algebra System (MAPLE) do the work.

$$ \mbox{VMT} = \int_W^\infty \frac{e^{-x}}{x^2}dx = \frac{e^{-W}}{W} - \mbox{Ei}(W) \quad \mbox{with} \quad \mbox{Ei}(x) = \int_1^\infty \frac{e^{-xt}}{t}dt $$

With a slight modification it is assumed again that

Uniform CMB energy + Gravitational self energy = 0

Where

$$ U_{CMB} = \frac{4}{3}\pi R_O^3.\frac{4\sigma}{c}\Theta^4 $$

And so

$$ \frac{4\sigma}{c}\Theta^4 = \frac{G M_O^2}{2R_O.4/3.\pi R_O^3} W.\mbox{VMT} = \frac{3.\mbox{VMT}}{32\pi.W} \frac{(H.c)^2}{G} $$

W := 3.383634283;
G := 6.67408*10^(-11);
Mpc := 3.08567758*10^22;
H := 73.4; # Riess
H := H*1000/Mpc;
c := 299792458;
VMT := int(exp(-x)/x^2,x=W..infinity);
u_G := (3*VMT)/(32*Pi*W)*(H*c)^2/G;
sigma := 5.670374419*10^(-8);
Theta := evalf((c/(4*sigma)*u_G)^(1/4));

                         Theta := 3.638225492 # Kelvin

No expanding space-time, no Big Bang, no Co(s)mic Inflation, no Primordial Relic Radiation, no fuzz parameters, no tweaking, no nothing of the kind. And yet a reasonable estimate of the Cosmic Microwave Background Θemperature.