What is the conformal mode of a metric? I have a problem in terminology. This article talks about the conformal mode of a physical metric.
I know what a conformal transformation is. But what is the conformal mode of a metric?
 A: The metric tensor $g_{\mu\nu}$ in 4 dimensions has 10 independent components. Each of these has the potential of being a dynamical degree of freedom, also know as a "mode."  There are many different ways of parameterizing these 10 components.  In general, of the 10 components, you can always pick one of them to correspond to the overall local scale of the metric, and this is what they are referring to as the conformal mode.  One way of doing this would be to construct the physical metric $g_{\mu\nu}$ from an auxiliary metric $\tilde{g}_{\mu\nu}$ that is related by
$$g_{\mu\nu} = \Omega^2(x) \tilde{g}_{\mu\nu},$$
and require that $\det \tilde{g}_{\mu\nu}=1$.  Now the conformal mode is encoded in the scalar field $\Omega$, and $\tilde{g}_{\mu\nu}$ clearly no longer has a conformal degree of freedom, since the overall scale of it is not allowed to change.  Put another way, the 10 independent components of $g_{\mu\nu}$ are encoded by the 9 independent components of $\tilde{g}_{\mu\nu}$ (10 components minus one constraint due to the determinant condition), plus the one independent component in $\Omega$.
The mimetic dark matter theory does something similar, except they use $\tilde{g}^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi$  for the conformal degree of freedom.  
