## Thursday, July 26, 2012

### Big Rip in December 2012, first attempt

Wikipedia provides an equation which predicts how long it is until dark energy tears apart the universe, if it's dark energy of the "phantom" type. The neo-Mayan apocalypse is just five months away; what would have to be true, for the Big Rip to happen that soon?

I won't try to reproduce the equation here - just follow the link - but the basic things to note are that the left hand side has units of time, and says how long it is from now, "t_0", until the end, "t_rip"; on the right hand side, H_0 has units of inverse time, and all the other quantities are dimensionless. H_0 is the Hubble constant, which tells you how the speed with which the galaxies move away from us, increases with distance. Speed is distance over time, change of speed with distance is (distance over time) over distance, "distance" and "over distance" cancel out, and that's why H_0 is just a number times "1 over time".

But what is that number? Wikipedia gives us a bunch of estimates, I'll go with "72 km/s/Mpc" for the purposes of calculation. But for km and Mpc to cancel we need to match units... Actually, forget this, the calculation was already done: the "Hubble time" is 1/H_0 is 13.8 billion years.

So, let's go to that first equation. We have decided apriori that the end is this December, so t_rip - t_0 = 5 months = 0.4 years. That equals 1/H_0 x 2 x 1/[3.|1+w|.sqrt(1-Omega_m)], where w is the dark energy equation-of-state parameter and Omega_m is the matter density of the universe, including dark matter. Observationally, Omega_m is about 0.3, so we have:

0.4 = 13.8 x 10^9 x 2 x 1/[3.|1+w|.sqrt(0.7)]

0.4 x [3.|1+w|.sqrt(0.7)] = 13.8 x 10^9 x 2

|1+w| = 10^9 x (13.8 x 2) / (0.4 x 3 x sqrt(0.7))

(that denominator is 1.004, very close to 1)

So basically w approximately equals -2.7 x 10^10, which I think would be grossly incompatible with observation; implying that, if we want dark energy to plausibly end the universe in time for Christmas this year, we'll need a model more complicated than one with a constant equation of state.