  At linear order in cosmological perturbations, departures from the growth in
the cosmological standard model can be quantified in terms of two functions of
redshift $z$ and Fourier number $k$. Previous studies have performed principal
component forecasts for several choices of these two functions, based on
expected capabilities of upcoming large structure surveys. It is typically
found that there will be many well-constrained degrees of freedom. However, not
all and, probably most, of these degrees of freedom were physical if the
parametrization had allowed for an arbitrary $k$-dependence. In this paper, we
restrict the $k$-dependence to that allowed in local theories of gravity under
the quasi-static approximation, i.e. ratios of polynomials in $k$, and identify
the best constrained features in the ($z$,$k$)-dependence of the commonly
considered functions $\mu$ and $\gamma$ as measured by an LSST-like weak
lensing survey. We estimate the uncertainty in the measurements of the
eigenmodes of modified growth. We find that imposing the theoretical prior on
$k$-dependence reduces the number of degrees of freedom and the covariance
between parameters. On the other hand, imaging surveys like LSST are not as
sensitive to the $z$-dependence as they are to the $k$-dependence of the
modified growth functions. This trade off provides us with, more or less, the
same number of well-constrained eigenmodes (with respect to our prior) as found
before, but now these modes are physical.
