Calculating Critical Density Using FRW and Its Implications

[CaptainMarvel]

Using the FRW:

<br /> \left( \frac {\dot{a}} {a} \right)^2 = \frac {8 \pi G \rho} {3} - \frac {k c^2} {a^2}<br />

We define critical density by setting k = 0 and rearranging to get:

<br /> \rho_c = \frac {3 H^2} {8 \pi G}<br />

Where:

<br /> H = \left( \frac {\dot{a}} {a} \right)<br />

My question is does \rho include the density contribution for Cosmological Constant (dark energy) \Lambda or is this derivation only for a Universe with no cosmological constant?

How does one then actually measure the density of Universe?

I know that the density has been measured to be slightly less than the critical density, but I thought we are meant to live in a flat Universe? Is this due to the cosmological constant and how is this reconciled with \rho not being exactly \rho_c?

Finally, I am right in saying that a Universe with \rho_c will stop expanding after infinite time, one with \rho &gt; \rho_c will collapse back on itself and one with \rho &lt; \rho_c will expand forever?

Many thanks.

 

[nicksauce]

Yes, \rho includes contribution from the cosmological constant. In other words, we can write the density as a function of scale factor as
<br /> \rho = \rho_c\left(\Omega_Ma^{-3} + \Omega_Ra^{-4} + \Omega_{\Lambda}\right)<br />

Finally, I am right in saying that a Universe with \rho_c will stop expanding after infinite time, one with \rho &gt; \rho_c will collapse back on itself and one with \rho &lt; \rho_c will expand forever?

This is basically correct, although I don't think, "stop expanding after infinite time" is a well-defined notion.

How does one then actually measure the density of Universe?

Fit supernova data and/or CMB data to different models and see what works best.

I know that the density has been measured to be slightly less than the critical density, but I thought we are meant to live in a flat Universe? Is this due to the cosmological constant and how is this reconciled with \rho not being exactly \rho_c?

The measurement of \rho is within error of being less than, equal to, or greater than the critical density. People say we live in a "flat universe", because the measured value is very close to the critical density.