Density of dark energy in Planck units

[marcus]

this is not to argue or persuade but just to share a calculation.
no answer needed unless you particularly like something about it.
several websites, eg. NedWright's Cosmology tutorial have formulas for things like the critical density and the estimate that
real density is very close to critical and that it is comprised of some 5 percent ordinary matter, 30 percent dark matter and 65 percent dark energy. Seems to be a fair amount of agreement about the rough percentages. And also that Hubble time is 14 billion years.

So it would be fairly easy for anybody to calculate from all that what the density of dark energy is---in whatever units---if anybody wanted to know. It is just a curiosity, what is it actually. How about calculating it in Planck units? Seem weird?

Well I did and I got that critical density is 1.77 E-123 Planck.

And (dark energy + dark matter + usual matter) is
(1.17 + 0.53 + 0.07)E-123 Planck. Or thereabouts---allowing for some variation in the percentages people cite.

If you want to check the reckoning
Planck density is c^7/hbar G^2
A day is 1603E45 Planck time.
So Hubble time 14E9 years is 8.19 E60 Planck time
Critical density is 3/8pi (hubbletime)^2 and that works out
fairly easily to 1.77E-123 and the rest is straightforward.

This 1.17E-123 agrees with what I've found on the web by way of estimates of the density of dark energy but expressed in other units like electronvolts etc.

Having dark energy density in some system of units let's one compare it to other energy densities. One rather nice comparison: it is one tenthousandth of the density of sunlight energy at this distace from the sun. Like...how much sunlight energy is there in a cubic kilometer in anyone instant? Well, the dark energy is one tenthousandth of that much energy.
Anybody here besides me like knowinging how much dark energy there is around?

 

[marcus]

finding the acceleration of expansion

occurred to me after posting the above that the numbers could go into John Baez's intuitive geometrical form of the 1916 Einstein equation and would give the rate that expansion is accelerating.

the Einstein equation can be written about V double dot, the second timederivative of a freefalling sperical blob of test particles,
I will write it V_ct ct. Using a subscript ct for derivative wrt ct.

quoting from Baez site:

V_ct ct / V = -4piG/c^4 [rho + P_x +P_y +P-z]

The pressure terms from ordinary and dark matter are negligible. rho is assumed 1.77E-123 of which 1.17E-123 is dark energy and that part has negative pressure -3 x 1.17E-123 Planck.

3 x 1.17 = 3.51

In evaluating RHS can ignore c and G.

RHS = -4pi [ + 1.77 - 3.51] E-123 = 22E-123 (planck area)^-1

Can make more intuitive by using a "planck minute" timescale of E45 Planck time---about a minute so call it a minute

The second derivative of volume, as fraction of the volume is then equal to 22E-33 per minute per minute.

In a minute, therefore, the rate of expansion of the volume increases by 22E-33 V per minute.

They seem to have found out about the accelerating expansion (which I just calculated back-of-envelope) because at a given redshift, like 1.5, some type Ia supernovas were DIMMER than
you'd calculate using the established 14 billion year Hubble time.
Heady stuff. Anybody with a firm grasp of the chain of reasoning want to clarify. A real revolution in cosmology.