  A new and promising avenue was recently developed for analyzing large-scale
structure data with a model-independent approach, in which the linear power
spectrum shape is parametrized with a large number of freely varying wavebands
rather than by assuming specific cosmological models. We call this method
FreePower.
  Here we show, using a Fisher matrix approach, that precision of this method
for the case of the one-loop power spectrum is greatly improved with the
inclusion of the tree-level bispectrum. We also show that accuracy can be
similarly improved by employing perturbation theory kernels whose structure is
entirely determined by symmetries instead of evolution equations valid in
particular models (like in the usual Einstein-deSitter approximation). The main
result is that with the Euclid survey one can precisely measure the Hubble
function, distance and ($k$-independent) growth rate $f(z)$ in seven redshift
bins in the range $z\in [0.6,\, 2.0]$. The typical errors for the lowest
$z$bins are around 1\% (for $H$), 0.5--1\% (for $D$), and 1--3\% (for $f$). The
use of general perturbation theory allows us, for the first time, to study
constraints on the nonlinear kernels of cosmological perturbations, that is,
beyond the linear growth factor, showing that they can be probed at the
10--20\% level. We find that the combination of spectrum and bispectrum is
particularly effective in constraining the perturbation parameters, both at
linear and quadratic order.
