Equations for Cosmological Expansion

I am going to get to a discussion on accelerated expansion in a later post, but I want to get into the equations now for use in both that discussion and one on “decelerated expansion”.  Unfortunately, it’s just the nature of the beast that I have to jump the gun on myself a little when I do this.  It’s also the case that easily accessible explanations are somewhat disparate being split across many webpages.  So, what I intend to do is bring a number of threads together in one place.

Here, it is stated that:

The acceleration equation describes the evolution of the scale factor with time

 

Then, later, it is stated that:

The most important property of dark energy is that it has negative pressure (repulsive action) which is distributed relatively homogeneously in space.

From this it follows that:

Assuming a flat universe (which is consistent with measurements) and noting that a flat universe is always flat, ρ=ρ_c=3H²/8πG.  So:

To be clear, a here is the scale factor and ä is the second time derivative of that scale factor.  An a with a single dot is the first time derivative of the scale factor and is used to defined the Hubble parameter:

The scale factor relates the change in proper distance between two objects over time, using the equation x(t)=a(t)x_o, or a(t)=x(t)/x_o, where x_o is the distance at some reference time (notionally t_o).

Note that there is a more easily generated notation for first and second time derivatives, namely da/dt and d²a/dt².  The widespread use of d to mean distance and ä to mean d²a/dt², where a is usually acceleration (or d²x/dt²), does not help things.  It’s almost like the people involved want to unnecessarily complicate things.  Rewriting the equation above using this notation:

Also,

So

In other words, the Hubble parameter gives us the speed of recession of an object, say a galaxy, divided by the distance of that object.  The use of the scale factor is just a minor complication of this concept.  (The most recent equation there is just a restatement of Hubble’s Law.)

At one point in the past, cosmologists were so convinced that the expansion of the universe was decreasing, they created a “deceleration parameter”, q, where:

And

 We know that, if H=1/t, then dH/dt=-1/t²=-H².  From this, we can see that if dH/dt=-H².b, where b is a constant, then H=1/(t.b).

Using the acceleration equation above, and the definition of the Hubble parameter:

 Therefore:

 Which implies that:

This Hubble parameter equation is invalid where the equation of state parameter w=-1, but note that if w=-1, then dH/dt=0, meaning that H=H_ic, which is to say some initial constant value and the acceleration involved is enough to precisely balance the inherent deceleration as t increases.  Note that normally, I’d use the subscript _o, but in this instance H_o is the value of the Hubble parameter today, not at some initial time.

There are effectively three zones established around two points, w=-1 and w=-1/3.

If w<-1, then it is understood that there is “phantom energy” which will lead to an eventual Big Rip.  The value of dH/dt becomes positive meaning that value of H increases with time.  This is what would happen with the value of H, assuming a current value of H=71 at 13.77 billion years and w=-1.03:

 

Precisely when the Big Rip happens depends a little on the magnitude of the delta-t (dt) used.  The larger the delta used, the further away the Big Rip seems, but it appears to be in the order of about 300 billion years away as dt approaches a unit of Planck length.  The time left before a Big Rip halves, approximately, if we use an equation of state parameter value of w=-1.06 and extends out towards infinity as the equation of state parameter approaches w=-1.

If w=-1/3, then we’d have the simple equation H=1/t, which would look like this:

 

As w→-1 from above, that curve would become less steep, approaching a flat line at the opening value (set at 71).

 

For values of w>-1/3, the curve would become steeper, for example:

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Since H=(dx/dt)/x, if H=1/t, then H=c/r_H, where r_H is the Hubble radius (or Hubble length).  The implication of this is that there is a horizon receding away at the speed of light.  If, however, H=1/(2t), that horizon would be receding away at half the speed of light, and so on.

Note that while it is generally said that the radiation-dominated and matter-dominated eras have “decelerated expansion”, this is a little inaccurate.  Expansion is slower in those eras, but in reality for a given state of equation parameter w=-1, the rate of expansion is fixed and for higher values of the equation of state parameter the rate at which the Hubble parameter decreases is accelerated.

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So, in brief:

• w<-1 – H increases until (eventually) an asymptotic point is reached, and the universe ends in a Big Rip.

• w=-1 – H remains constant.

• -1<w<-1/3 – H would decrease relatively gradually, at a greater rate as w→-1/3.  Effectively, it would be as if time were progressing more slowly, and “future” values of H would be reached later than would have been the case than when w=-1/3.

• w=-1/3 – H∝1/t, so that the Hubble parameter is related to the age of the universe.  It would be equal to the inverse age of the universe if it were always the case that w=-1/3, otherwise it would be equal to the inverse of the universe plus some offset related to the initial time that w=-1/3 and the value of H at that time.

• w>-1/3 – H would decrease more rapidly, increasingly so as the value of w increased. Effectively, it would be as if time were progressing more rapidly, and “future” values of H would be reached earlier than would have been the case than when w=-1/3.