Question regarding the backreaction of a scalar field in curved spacetime

Question

Lets assume that I start with the following action:

$$ {\mathcal L}_1 = \frac{1}{2} \sqrt{-g} \left( \partial_\mu \phi \partial^\mu \phi - m^2 \phi \right) $$

where $g_{\mu \nu}$ is a FRW metric and $\phi$ a scalar field.

In addition assume that for 'whatever' reason there is a deviation of the above Lagrangian such that an arbitrary potential $V(\phi)$ is introduced so that $$ {\mathcal L}_2 = \frac{1}{2} \sqrt{-g} \left( \partial_\mu \phi \partial^\mu \phi - m^2 \phi + V(\phi)\right). $$

Surely, having already a matter field interact with gravity should produce a backreaction so that the initial metric $g_{\mu\nu} \rightarrow \tilde g_{\mu \nu}$ while the introduction of the interacting terms will further aid this process.

My question is how can I calculate the above backreaction. In particular, I know that I need to compute the eom's of the field and the Einstein equations and solve the system of differential equations, but I can't figure out whether I should calculate the eom's while taking into account the initial metric or not.

So for example, the eom's would be equal to: $$ \frac{\delta {\mathcal L}_2}{\delta \phi} \Bigg|_{g_{\mu \nu}} = 0 \quad or \quad \frac{\delta {\mathcal L}_2}{\delta \phi} \Bigg|_{\tilde g_{\mu \nu}} = 0 ??? $$

If the latter is true, than I should first compute the new metric $g'_{\mu \nu}$ via the Einstein equation $\delta G_{\mu \nu} \propto \delta T_{\mu \nu}$ and then plug it in above?

However, the Einstein equation contain $\phi$ as well making the problem a cyclic process. In my mind it is sort of a chicken vs egg kind of problem where one should assume that the introduction of matter at the initial geometry forces the latter to change. With that in mind I could argue that the eom with $g$ should have the correct physical interpretation. What are your thoughts? What am I missing?

Answer 1

The general coupled system of gravity (metric) and the scalar field can be described by the Lagrangian (expressed in $\hbar=c=1$ natural units often used in cosmology) $$\mathcal{L} = \sqrt{-g}\left[\frac{1}{8 \pi G} R + \frac{1}{2} (g^{\mu\nu}\partial_\mu\phi \partial_\nu \phi + m^2 \phi^2 + V(\phi)) \right]$$ where $R$ is the Ricci scalar of the metric and $G$ Newton's constant. By varying with respect to $g_{\mu\nu}$ one obtains the Einstein equations $$R^{\mu\nu} - \frac{1}{2} R g^{\mu\nu} = 8 \pi G T^{\mu\nu}$$ where $T^{\mu\nu}$ is obtained by the variation of the scalar part of the Lagrangian density $\mathcal{L}_{\phi}$ with respect to the metric
$$T^{\mu\nu} = \frac{-2}{\sqrt{-g}} \frac{\delta \mathcal{L}_\phi}{\delta g_{\mu\nu}} = \partial^\mu \phi \partial^\nu \phi - \frac{1}{2} g^{\mu\nu}\left[g^{\alpha\beta}\partial_\alpha\phi \partial_\beta \phi + m^2 \phi^2 + V(\phi)\right]$$ The equations of motion for $\phi$ are obtained by varying with respect to $\phi$ and yield $$\frac{1}{\sqrt{-g}}\partial_\nu \left( g^{\mu\nu} \partial_\mu \phi \right) + m^2\phi + V'(\phi) = 0$$

The two sets of equations above apply to the self-consistent dynamics of any metric and scalar-field configurations. Generally, the Einstein equations and the equations of motion for $\phi$ are a coupled set of equations that are solved simultaneously. In general, the equations simply have to be solved all at the same time with the same $\phi$ and $g_{\mu\nu}$ appearing in both.

For weak couplings, one can sometimes solve iteratively, such as e.g. using a zeroth-order metric field $g_{\mu\nu (0)}$, solving for the dynamics of $\phi$ in this metric, and feeding this into the Einstein equations to find a corrected metric $g_{\mu\nu(0)} + g_{\mu\nu(1)}$. Then you can plug this metric into the evolution equations for $\phi$ and so on. However, the quality of the approximation you get by a couple of iterations will depend a lot on the setup.

For example, when you assume that the configuration of $\phi$ is isotropic and homogeneous in the sense of FLRW cosmology and that the metric is a FLRW metric with an undetermined scale factor $a(t)$, the Einstein equations boil down to a set of Friedmann equations for the evolution of $a$ with a contribution of the scalar $T_{00}$ to energy density $\rho$ and $T_{ii}$ to pressure $p$. Additionally, the equation of motion for the scalar reduces to $$\ddot{\phi} + 3 \frac{\dot{a}}{a} \phi + m^2 \phi + V'(\phi) = 0$$ Again, you see that the evolution of $\phi$ feeds into the metric function $a$, and the evolution of $a$ feeds into the evolution of $\phi$. The equations cannot be generally considered as separate.

If, for instance, you assume that $\phi$ represents something like the dark energy field (quintessence) or an inflaton during an era when these fields dominate the dynamics of the universe, an iterative procedure is not a good idea, you should solve the full coupled dynamics. On the other hand, if you want to use it during an era where the scalar field has only a weak impact on the dynamics (such as the radiation- or matter-dominated era), using an iterative approach should work reasonably well.