In the last Blog, How Aeons turned into Zeons, we derived a very simple formula for the expansion dynamics of the cosmos, normalized to a 'natural cosmological timescale' unit of 1 zeit = 17.3 billion years.
(2.1) H^{2 }= 1 + 0.443 S^{3}
where H is the fractional expansion rate and S is the 'stretchfactor', i.e. by how much distances have 'stretched' since light we observe has been emitted. It is basically the redshift factor z plus 1. By convention, S have a present value of 1.0; hence the present Hubble value H_{0} = (1.44)^{0.5} = 1.2 zeit^{1}. But what is the relationship between H and cosmological time?
Cosmological Time
Because of the instantaneous "interest on interest" nature of cosmic expansion, it follows an exponential pattern (e^{Ht}),
provided that H remains constant over time. Presently H is still
decreasing, so cosmological time is approximated by the following
natural logarithmic equation:^{[1]}
(2.2) t = ln[(H+1)/(H1)]/3 zeit.
Let us put our present H_{0} = 1.2 zeit^{1} in here (using Google or some other calculator) and check that we get the 0.8 zeit that we expect.
t_{now} = ln[2.2/0.2]/3 ~ 0.8 zeit.
Suppose
we observe the redshift from a specific galaxy as z=1, which means that
the light has left it when the wavelength stretch factor was S = z+1 = 2. Plugging S = 2 into eq. 2.1, we find that H = 2.13 zeit^{1} when the light was emitted. The cosmic time was
t_{then} = \ln[(2.13+1)/(2.131)]/3 = 0.34 zeit after the start of the present expansion.^{[2]}
The light from that galaxy obviously took 0.8  0.34 = 0.46 zeit to reach us  it is the socalled 'lookback time'.
Now
that we have a feeling for the relationships, what is better than
viewing it on a graph? Here you see the parameters mentioned so far
(against time).
You can see our present time (0.8 zeit) clearly where the red and blue curves cross, i.e. where both S and a are unity. You can also see how H starts high and asymptotically approaches unity. The graphs were plotted using LightCone 7zeit, a variant of the standard LightCone 7 cosmological calculator.
I have 'sneaked in' the curve for the scale factor (a=1/S),
but I did not give a direct relationship of it against time yet. You
get it by solving eqs. 2.1 and 2.2 simultaneously, which is not a simple
exercise at all. Fortunately, mathematics comes to the rescue and we
have a nifty solution for a(t), i.e. the scale factor as a function of
time:
(2.3) a(t) = sinh^{2/3}(1.5t)/1.3
where
sinh is the hyperbolic sine function. The 'hypersine' is obviously
related to the natural logarithm ln(x) and the natural exponent (e^{x}). The factor 1.3 is just scaling to make a_{now} = 1 at t_{now}
= 0.8 zeit, as is the convention. The importance of a(t) is that its
curve shows how a decelerating expansion that gradually changed over to
an accelerating expansion. The inflection point can be visually judged
to lie between 0.4 and 0.5 zeit. It is actually at 0.44 zeit.
Proper Distance
Proper
distance in cosmology is like measuring distance on a hypothetical
frozen view of the cosmos, i.e. with expansion stopped. Obviously we
cannot stop expansion, so how do we find proper distance? There are
various methods, but once we have the present and long term Hubble
values (H_{0} and H_{∞}), we only need to measure the redshift of a source and then calculate its proper distance from where we are.
Because
of the nonlinear expansion, there is no precise analytical solution
for the proper distance (that I know of). We have to numerically
integrate in small steps; but fortunately the definite integral is
rather neat:
(2.4) D_{now }= ∫_{1}^{S}(dS/H) = ∫_{1}^{S}[dS/√(0.44S^{3}+1)]
We
have substituted H from eq. 2.1 in here, so that we have a distance in
terms of the observable S = z+1. For calculation, you can use your
favorite integrator, but a rather cool web based one is available at
http://www.numberempire.com/definiteintegralcalculator.php.
For
our previous sample galaxy at S=2, I entered 1/sqrt(0.443x^3+1) into
the Function box, with 1 in the "From" and 2 in the "To" boxes and then
clicked "Compute"  it returned the answer as 0.64, which in our cas has
the units lightzeit. So we receive the light when the proper distance
to the galaxy is D_{now} = 0.64 lzeit. Since we have used
stretch S = 2, the proper distance when the photons left the galaxy was
half of 0.64, i.e. 0.32 lzeit; so we say that D_{then} = 0.32
lzeit for this galaxy. Recall that the light left that galaxy at t =
0.34 zeit, i.e. 0.46 zeit ago.
Here is a graph of D_{now} and D_{then}.
The red D_{then} curve is also known as the cosmological
lightcone. To the left of 0.8 zeit it represents our past lightcone and
to the right of 0.8 zeit our future lightcone. Near t = 0.0 the universe
was very dense and objects we now observe were actually very, very
close to our space locality. Distances then grew much faster than the
progress that photons could make in our direction and ancient photons
were at first dragged away from us. At around t = 0.23 zeit, the
fractional expansion rate dropped enough to allow photons to make
progress and eventually reach our telescopes.
To summarize, we
have now looked at a further three rather easy equations that allow us
to calculate the most common values for the standard cosmological model.
Part 3 will deal with the Hubble radius and two cosmological horizons that we should take notice of.
=0=
Endnotes
[1] The function e^{Ht }gives
the factor by which cosmic distances increase in a time interval t,
provided that H remains constant over this time interval. In the far
future, H will be a constant, but in the early universe H changed
significantly over periods longer than a million years. Hence the more
complex general solution.
[2] To verify that the approximations are valid, I used the LightCone 7zeit calculator for a 'oneshot' calculation with S_{upper}=2 and S_{step}=0.
[3] The D_{then} lightcone is where the LightCone 7 calculator got its name from.
