Assuming a Robertson-Walker metric to describe homogeneous and isotropic cosmological models, Einstein equations with cosmological constant reduce to these 3 non-linear ordinary differential equations for a perfect fluid:

\begin{align} \dot{\rho} &= -3H(\rho + P) \tag{1} \\ \dot{H} &= -H^2-4\pi G(\rho + 3P)/3 + \lambda/3 \tag{2} \\ H^2 &= 8\pi G\rho/3-K/a^2+\lambda/3 \tag{3} \end{align}

Here, dot represent the derivative with respect to time and the function $H=H(t)= \dot{a}/a$ where $a = a(t)$ is the scale factor.

I know that (1) is the conservation of energy equation but i can't understand what (2) and (3) represent?


You can arrive at equations (2) and (3) by plugging the Robertson-Walker metric into the Einstein field equations. The universe is modelled as a perfect fluid so the energy-momentum tensor is $(T_{ab})=\text{Diag}(\rho,P,P,P)$ and the first equation comes from the $T_{00}$ (time) part of the field equations, the second from the spacial part once you have plugged in the metric.

Note that the three equations are not independent, though any choice of two of them are, the continuity equation (1) $\nabla^a T_ab =0 $ may be obtained directly from the field equations by applying $\nabla^a$ and applying the Bianchi identity and metricity property of the Levi-Civita connection.

Hope this helps :)

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  • $\begingroup$ Note that the question asks what those equations physically represent. The method of arriving at them was, from what I can infer, accepted and not in question. $\endgroup$ – Jim May 19 '15 at 18:28
  • $\begingroup$ Sorry, I thought it was implicit. The Friedmann equations are a manifestation of the Einstein field equations, which describe the relationship between the contents of spacetime and its geometry. Thus the equations (2),(3) are necessary and sufficient constraints on the metric in order that the universe obeys the laws of general relativity. ie. in order that gravity is taken account of. In this case I think the derivation is really all the physical meaning there is... The equations (2),(3) basically are the field equations! Thanks for pointing that out though :) $\endgroup$ – Aerinmund Fagelson May 19 '15 at 18:58
  • $\begingroup$ The friedmann equations have more physical meaning than that. For instance, (3) describes how the relative expansion of space is directly affected by the different energy densities that inhabit the universe. There are, of course, other interpretations as well. $\endgroup$ – Jim May 19 '15 at 19:03
  • $\begingroup$ Well, sure. It tells us that the expansion of space is affected by the different energy densities, because we are dealing with a model in which the universe must conform to the field equations, which take into account energy density among other things. But if you understand the field equations, and presumably one should know some GR before studying the Friedmann equations, then you already have the physical meaning. $\endgroup$ – Aerinmund Fagelson May 19 '15 at 19:12
  • $\begingroup$ I agree with that, but nevertheless, it couldn't hurt to write down the physical meaning of the friedmann equations specifically. Even though anyone that understands their origins should know it already. That is, after all, what the question asks for $\endgroup$ – Jim May 19 '15 at 19:17

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