The basic equation of GR has a curvature constant Λ on the lefthand (geometric) side.(adsbygoogle = window.adsbygoogle || []).push({});

The Friedman equation is derived from the Einstein Field Equation by making a simplifying assumption of uniformity. As a spacetime curvature Λ can be written either in units of reciprocal area or reciprocal time-squared. So in the Friedman equation you might expect to see Λ ( with units of reciprocal time^{2}or length^{2}) again appearing on the lefthand side, and matter appearing on the right.

But that doesn't happen, so the question naturally comes up as to what the Friedman equation would look like if it were so. Well the Friedman is basically an equation governing the evolution of the expansion rate H(t). For simplicity I'll assume spatial flatness (measurements show largescale mean spatial curvature is nearly, if not precisely, zero). The Friedman actually tells us about the evolution of the SQUARE of the expansion rate, which in standard metric terms would be in units of seconds^{-2}. And Λ as reciprocal time^{2}can also be expressed in that unit.

H^{2}- Λ/3 = (8πG/3c^{2}) ρ

where ρ is the combined average energy density (ordinary and dark matter, electromagnetic radiation).

The righthand side goes to zero in the longterm future, due to expansion. So we can solve for the longterm Hubble expansion rate. Obviously it is (Λ/3)^{.5}and therefore the Hubble time is (Λ/3)^{-.5}

It turns out that the best-fit value of Λ, according to most recent Planck report, is

1.007 x 10^{-35}s^{-2}

If you use that value, the longterm Hubble time (Λ/3)^{-.5}comes out to 17.3 billion years.

If you then take the present-day Hubble time to be 14.4 billion years and evaluate the LHS of the Friedman equation for the present-day:

H^{2}- Λ/3 = (8πG/3c^{2}) ρ

at the mantissa level it is just 4.843 - 3.357 = 1.486

and you get 1.486 x 10^{-36}s^{-2}

This can then be multiplied by 3c^{2}/8πG to get the energy density corresponding to exact spatial flatness (energy equivalent of ordinary and dark matter etc).

It comes out to 0.2389 nanojoules per cubic meter. Google calculator gives this in the algebraically equivalent form 0.2389 nanopascal.

That is the critical density when what we are counting is just matter and energy that we actually know is energy, omitting a possibly fictitious "dark energy", which was already accounted for on the LHS by the curvature constant Lambda.

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# Form of Friedman eqn with Λ curvature constant

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