  We present the first $\Lambda$CDM cosmological analysis performed on a galaxy
survey using marked power spectra. The marked power spectrum is the two-point
function of a marked field, where galaxies are weighted by a function that
depends on their local density. The presence of the mark leads these statistics
to contain higher-order information of the original galaxy field, making them a
good candidate to exploit the non-Gaussian information of a galaxy catalog. In
this work we make use of \simbig, a forward modeling framework for galaxy
clustering analyses, and perform simulation-based inference using normalizing
flows to infer the posterior distribution of the $\Lambda$CDM cosmological
parameters. We consider different mark configurations (ways to weight the
galaxy field) and deploy them in the \simbig~pipeline to analyze the
corresponding marked power spectra measured from a subset of the BOSS galaxy
sample. We analyze the redshift-space mark power spectra decomposed in $\ell =
0, 2, 4$ multipoles and include scales up to the non-linear regime. Among the
various mark configurations considered, the ones that give the most stringent
cosmological constraints produce posterior median and $68\%$ confidence limits
on the growth of structure parameters equal to
$\Omega_m=0.273^{+0.040}_{-0.030}$ and $\sigma_8=0.777^{+0.077}_{-0.071}$.
Compared to a perturbation theory analysis using the power spectrum of the same
dataset, the \simbig~marked power spectra constraints on $\sigma_8$ are up to
$1.2\times$ tighter, while no improvement is seen for the other cosmological
parameters.
