# FLRW Coupling to Perfect Fluid

In order to obtain the Friedmann equations from the Lagrangian formalism, as far as I understand, one way is to minimally couple a scalar field $$\phi(t)$$ to the FLRW metric, i.e.

$$S=\int d^4 x \sqrt{-g} \bigg(R - \frac{1}{2} g_{\alpha\beta} \partial^\alpha \phi \partial^\beta \phi - V(\phi) \bigg)$$.

Then, one computes the energy-momentum tensor $$T^{\mu\nu}$$ of the scalar field and identifies the pressure and energy density in terms of $$d\phi/dt$$ and $$V$$, one finds that the equations of motion derived from the action above are the Friedmann equations.

My question is: can one write the coupling of FLRW gravity to a perfect fluid in the Lagrangian formalism without introducing a scalar field in this way, instead perhaps with a term like $$g_{\mu\nu}T^{\mu\nu}$$?

Any kind of feedback would be appreciated.

Inflation considers scalar fields(as you said): $$\mathcal{L} = -\frac{1}{2} g_{\alpha\beta}\partial^\alpha\phi\partial^\beta\phi - V(\phi) \:.$$
$$\Lambda CDM$$ considers perfect fluids: $$\mathcal{L} = -\rho \:.$$
• Hi, thanks for the input and the reference. I see that the $\Lambda CDM$ Lagrangian does indeed give me the Friedmann equations. But so does the one for scalar fields, since $\rho =\frac{1}{2} \dot{\phi}^2 + V ( \phi)$ and $p = \frac{1}{2} \dot{\phi}^2 - V ( \phi)$ (For instance in blau.itp.unibe.ch/newlecturesGR.pdf equation 35.147). Could you say that these Lagrangians are related? Is there a way to obtain the Friedmann equations with a coupling term that is $\mathcal{L} = g_{\mu\nu} T^{\mu\nu}$? – Azarashi Aug 31 '20 at 12:46