Understanding Hubble's Law: Calculating Recession Velocity and Hubble's Constant

[thenewmans]

Can someone check my math? I want to be sure I understand Hubble’s Law. I know it’s not exact since there’s acceleration, inflation and flatness. But let’s leave all that out just to understand Hubble’s Law. The formulas are pretty simple. There are 2 parts one is for finding the recession velocity between objects and the other is Hubble’s Constant, which is not constant. It shrinks as the universe ages.

H = 1 / UniverseAge
V (km/s) = H (km/s/Mpc) * D (Mpc)

The tricky part is converting H to km/s/Mpc. So here’s a better version. You can use any time frame in place of years.

H(km/s/Mpc) = C(km/s) / C(Mpc/yr) * UniverseAge(yr)

For an object 1 billion light-years (307Mpc) away today:

H(0) = 300,000km/s / 307Mpc/Gyr * 13.7Gyr, H(0) = 71.4
v = 71 * 307, v = 22,000 km/s

That looks good. Here’s another one. For an object 1 kilometer away 1 second after the Big Bang started:

H = 300,000km/s / 9.72e-15Mpc/s * 1s, H(0) = 3.09e19
v = 3.09e19 * 3.24e-20, v = 1 km/s

Wow, funny how that works out. So 1 second after the Big Bang, an object 1 kilometer away is receding away at 1 kilometer per second. That has nothing to do with reality since that’s still in an inflation period. But I just want to know if that’s right according to Hubble’s Law.

 

[Wallace]

The statement that H = 1/age is only approximate. It happens to hold pretty well at this moment in the history of the Universe but didn't hold in the past and won't in the future. You certainly can't extrapolate this back to inflation unfortunately.

 

[thenewmans]

Thanks Wallace. You're right about that inflation business throwing a wrench in the works. But I'm glad to know I got Hubble's Law down.

 

[Wallace]

It's not just inflation, the function H(t) depends on a range of factors at all times in the history of the Universe. Only in some models or at certain times in some models does H(t) = 1/t, in general this is not true.