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Linearization stability results and active measurements for the Einstein-scalar field equations

We study the Einstein equations coupled with the scalar field equations, $\hbox{Ein}(g)=T$, $T=T(g,ϕ)+F^1$, and $\square_gϕ^\ell-m^2ϕ^\ell= F^2$, where the sources $F=(F^1, F^2)$ correspond to perturbations of the physical fields which we control. Here $ϕ=(ϕ^\ell)_{\ell=1}^L$ and $(M,g)$ is a 4-dimensional globally hyperbolic Lorentzian manifold. The sources $F$ need to be such that the fields $(g,ϕ,F)$ satisfy the conservation law $\hbox{div}_g(T)=0$. If $(g_ε,ϕ_ε)$ solves the above equations, $\dot g=\partial_εg_ε|_{ε=0}$, $\dotϕ=ϕ_ε|_{ε=0}$, and $f=(f^1,f^2)= \partial_εF_ε|_{ε=0}$ solve the linearized Einstein equations and the linearized conservation law $$ \frac 12 \hat g^{pk}\hat \nabla_p f^1_{kj}+ \sum_{\ell=1}^L f^2_\ell \, \partial_j\hatϕ_\ell=0, $$ where $\hat g= g_ε|_{ε=0}$ and $\hat ϕ= ϕ_ε|_{ε=0}$. Then $(\hat g,\hat ϕ)$ and $f$ have the linearization stability property. Here ask the converse: If $\dot g$, $\dot ϕ$, and $f$ solve the linearized Einstein equations and the linearized conservation law, are there $F_ε=(F^1_ε,F^2_ε)$ and $(g_ε,ϕ_ε)$ depending on $ε\in [0,ε_0)$, $ε_0>0$, such that $(g_ε,ϕ_ε)$ solves the Einstein-scalar field equations and the conservation law. When $\hat g$ and $\hat ϕ$ vary enough and $L\geq 5$, we prove a microlocal version of this: When $Y\subset M$ is a 2-surface and $(y,η)\in N^*Y$, there is $f$ that is a conormal distibutions wrt. the surface $Y$ with a given principal symbol at $(y,η)$ such that $(\hat g,\hat ϕ)$ and $f$ have the linearization stability property.