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I was discussing the fact that if there was no matter in the universe, just vacuum and radiation, can we say that anything called gravity wouldn't exist?

In that universe, the Friedman equations would still be useful, but is it related with gravity? It just describes the expansion and the geometry of the universe, but it is developed from general relativity and has the $G$ constant in it. So, is gravity a valid thing in that universe?

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The Friedman Equations would still exist. In other words, $G_{\mu\nu}=8\pi G T_{\mu\nu}$ would still apply, but the assumption of vacuum means that $T_{\mu\nu}$ would be 0 (we can immediately see that the gravitational constant G falls out due to multiplication by zero). If I claim gravity is the curvature of spacetime, the $G_{\mu\nu}$ on the left hand side would still be around. (Also I'm assuming that the cosmological constant $\Lambda$ remains on the left hand side of the EFE since it originates in the gravitational action.)

Also, you said "just vacuum and radiation". Radiation is a source of gravity with equation of state $p=\frac{1}{3}\rho$. A universe with radiation in it is not truly a vacuum (e.g. $T_{\mu\nu}\neq 0$ in this case).

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