# Perfect cosmological principle in co-moving co-ordinates

1. Oct 10, 2012

### johne1618

To quote cosmologist Ned Wright:

If the Universe is the same at all times, it is argued that the value of the Hubble parameter must be a constant $H_0$, so

$H = \frac{\dot{a}}{a} = H_0$

$a(t) = e^{H_0 (t - t_0)}$

where $t_0$ is the present age of the Universe.

This model is an expanding model that does not change its appearance over a time interval from $-\infty$ to $+\infty$.

However this model has been discredited as there is good evidence of a Big Bang Universe with a finite age.

But perhaps one could save the Perfect Cosmological Principle by saying the Universe should look the same at all times in co-moving cordinates.

Now we have

$H = \frac{c}{R_H}$

where $R_H$ is the Hubble radius.

Therefore if the Universe is to look the same at all times in co-moving co-ordinates then $R_H$ must be co-moving and so

$R_H = a(t) R_0$

where $R_0$ is constant.

Therefore we have

$\frac{\dot{a}}{a} = \frac{c}{a R_0}$

As $H_0 = c/R_0$ we have

$\frac{\dot{a}}{a} = \frac{H_0}{a}$

with solution

$a(t) = H_0 t$

This model is static in co-moving co-ordinates with a finite lifetime.

Last edited: Oct 10, 2012
2. Oct 11, 2012

### twofish-quant

Congratulations you've used reinvented the Milne model!!!!!

You can search the forums for lots of detail on the problems with Milne. The short answer is 1) it requires gravity to work differently than we think it does 2) it causes the early universe to expand rather slowly which causes problems with big bang nucleosynthesis and CMB 3) it doesn't fit with the most recent supernova measurements.

There is a variant of the Milne model called Milne-Dirac in which antimatter repels matter. In one recent thread, I was sort of defending it as "probably wrong, but not totally crazy."

3. Oct 11, 2012

### Chronos

Yes, twofish occasionally argues the less probable side of issues [he is old and enjoys stirring the pot now and then]. The odds favoring anti gravity from anti matter are not favorable. Gravity is almost surely mediated by an integer boson, and integer bosons appear to be indifferent to charge.

4. Oct 13, 2012

### johne1618

The co-moving perfect cosmological principle is also consistent with the Coasting Cosmology model with equation of state:

$p = -\frac{\rho c^2}{3}$

In any flat model without cosmological constant we have

$\rho \propto \frac{1}{R_H^2}$

In the CC model $R_H$ is constant in comoving co-ordinates and therefore the energy density $\rho$ is constant in co-moving co-ordinates.

This makes sense to me because if you factor out the expansion of space then you would expect the energy density to remain constant as energy is conserved locally.

Last edited: Oct 13, 2012
5. Oct 16, 2012

### clamtrox

This has actually been discussed quite a bit recently. Here's a paper you might find interesting: http://arxiv.org/abs/1109.5189

(and no, it's not correct :)