Re: [Phys-l] cosmology question

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding Brian W's welcome clarifying explanation:

> I think I may have offended you with my unseemly lunchtime haste,
> David.

Not at all.  I was only bewildered by the output you relayed to the list in response to what I wrote.

> Moreover, I likely pasted your Friedmann equation into the Alpha 
> computational engine in incomplete form.  So this afternoon, I
> cut and pasted the equation again, and took a screen shot of the
> Alpha effort.
> This is how you could access this server:
> http://www.wolframalpha.com/
>
> About Alpha: I seem to recall that when a list-member inquired
> about the fourier transforms of some inverse transcendental
> expressions last year, it was Alpha who was the sole responder.

I must have missed that.

> Though it is certainly the case that Alpha can produce results
> from materials where I am speechless, it too, has its naivete,
> and here it tries out on your equation:
>
> _http://s880.photobucket.com/albums/ac6/betwys/DavidBowmans%20Universe/_
>
> *http://tinyurl.com/3ztue8g
> *

It seems that it has misinterpreted the equation.

> It seems to be making heavy weather of the   [Omega]_lambda notation, 
> and appears to be assuming some constructs that were unintended.
> But how much less did I understand!
> It would be pleasant to be able to follow the notes of the most
> mathematically adept, but this is unlikely - and I must  content
> myself with crutches of this kind.
>
> Sincerely
>
> Brian Whatcott

Thanks for the explanation.

But if you wanted to see an explicit solution of the Friedmann equation, I could have provided it, had you asked.

The Friedmann-Robertson-Walker cosmological model that includes dark energy has a Friedmann equation which is a separable DE and can be formally integrated to give the elapsed time in terms of the spatial scale factor in terms of an incomplete elliptic integral of the first kind.  This elliptic integral function can be inverted, in principle, (and numerically in practice) to give the scale factor in terms of the elapsed time, i.e. x(t).  In the special case for when the model is restricted to being spatially flat, (so 1 = [Omega]_m + [Omega]_lambda) then the formal function inversion can be done explicitly in terms of a hyperbolic sine function, giving x(t) explicitly as:

x(t) = ((1/[Omega]_lambda - 1)*sinh^2(3*sqrt([Omega]_lambda)*H_0*t/2))^(1/3).

This functional form for the time dependence of the cosmological scale factor has an inflection point in it.  At early times when the universe is younger the matter concentration is so large that its effect on spacetime dominates the behavior, and the universe's expansion slows down and decelerates under the attractive effects of all that mutually gravitating matter.  But after the inflection point the matter in the universe is so diluted with all that space that was created that its mutual gravitation no longer dominates over the dark energy.  In this later stage of the expansion the dark energy's repulsive effects dominates (because the dark energy's repulsive contribution does not dilute away as the universe expands) and the universe's expansion therefore accelerates with time after the inflection point.  We are currently at a point in the history of the universe after this inflection point.  Eventually, in the distant future, the hyperbolic sine function will asymptotically approach an exponential function, and the universe's expansion will be exponential in form, i.e. a constant Hubble parameter DeSitter universe.  When this happens the asymptotic constant Hubble parameter will be

H = H_0*sqrt([Omega]_lambda)

and the universe will then have a size-doubling time of

T_double = ln(2)/H.

This model has the current age of the universe, t_0, given explicitly as:

t_0 = 2*arctanh(sqrt([Omega]_lambda))/(3*H_0*sqrt([Omega]_lambda)).

It also has the age of the universe when the inflection point occurred as:

t_inflection = 2*arctanh(1/sqrt(3))/(3*H_0*sqrt([Omega]_lambda)).

Using the recently determined values of

H_0 = 70.3 km/s/Mpc = 1/(13.91 Gyr)

and

[Omega]_lambda = 0.728,

along with the requirement of being spatially flat, results in a value of 

t_0 = 13.78 Gyr

for the current age of the universe.  It also has the age of the universe when the inflection point occurred as

t_inflection = 7.156 Gyr

which therefore seems to have happened about 6.623 Gyr ago.  In the very distant future the asymptotic exponential size-doubling time for the universe appears to approach

T_double = 11.30 Gyr.

BTW, in the olden days, before the discovery of the dark energy driven accelerating expansion of space, the Friedmann equation was also explicitly integrable for the x(t) function for any value of the mean matter density [Omega]_m.  This is because integrating the Friedmannn equation is pretty straightforward when its RHS is a linear polynomial rather than a cubic one.  If [Omega]_m > 1 then the universe is a finite sized 3-sphere and its scale factor, x(t), has the time dependence of a cycloid.  After one rolling revolution of the cycloid function the universe experiences a big crunch singularity.  If [Omega]_m = 1 then the universe is spatially flat and its scale factor x(t) follows a power law function in time, with a power law exponent of 2/3.  If [Omega]_m < 1 then the universe is a hyperbolic space and its scale factor x(t) is a hyperbolic analytic continuation of a cycloid in time, (i.e. a 'cycloid' with a negative rolling radius and an imaginary rolling angle).  Such a function, like the power law case, expands forever and does not experience a big crunch.  But in all those cases, because they don't have any dark energy, they always have a decelerating expansion (and an accelerating recontraction for the collapsing case) because of the global gravitational attractiveness of the matter, and the x(t) function is always concave down with time when there are no repulsive effects from the lacking dark energy.

David Bowman