Primordial features, in particular oscillatory signals, imprinted in the primordial power spectrum of density
perturbations represent a clear window of opportunity for detecting new physics at high-energy scales. Future
spectroscopic and photometric measurements from the space mission will provide unique constraints on the primordial power spectrum, thanks to the redshift coverage and high-accuracy measurement of nonlinear scales, thus allowing us to investigate deviations from the standard power-law primordial power spectrum. We consider two models with primordial undamped oscillations superimposed on the matter power spectrum described by $1 + A X X X X one linearly spaced in $k$ space with $ lin k/k_*$ where $k_* = Mpc $ and the other logarithmically spaced
in $k$ space with $ log We note that A X $ is the amplitude of the primordial feature,
$ X $ is the dimensionless frequency, and $ X lin log $. We provide forecasts from spectroscopic and photometric primary probes on the standard cosmological parameters $ m,0 b,0 $, $h$, $n_ s $, and $ and the primordial feature parameters A _X$, $ and $ We focus on the uncertainties of the primordial feature amplitude A _X$ and on the capability of to detect primordial features at a given frequency. We also study a nonlinear density reconstruction method in order to retrieve the oscillatory signals in the primordial power spectrum, which are damped on small scales in the late-time Universe due to cosmic structure formation. Finally, we also include the expected measurements from galaxy-clustering bispectrum and from observations of the cosmic microwave background (CMB). We forecast uncertainties in estimated values of the cosmological parameters with a Fisher matrix method applied to spectroscopic galaxy clustering ( weak lensing (WL), photometric galaxy clustering ( the cross correlation (XC) between and WL, the spectroscopic galaxy clustering bispectrum, the CMB temperature and $E$-mode polarisation, the temperature-polarisation cross correlation, and CMB weak lensing. We consider two sets of specifications for the probes (pessimistic and optimistic) and three different CMB experiment configurations, that is Planck Simons Observatory (SO), and CMB Stage-4 (CMB-S4).
We find the following percentage relative errors in the feature amplitude with primary probes: for the linear (logarithmic) feature model, with a fiducial value of A _X = 0.01$, $ = 10$, and $ = 0$: 21<!PCT!> (22<!PCT!>) in the pessimistic settings and 18<!PCT!> (18<!PCT!>) in the optimistic settings at a 68.3<!PCT!> confidence level (CL) using While the uncertainties on the feature amplitude are strongly dependent on the frequency value when single probes are considered, we find robust constraints on A X )$. Due to the inclusion of numerical
reconstruction, the bispectrum, SO-like CMB reduces the uncertainty on the primordial feature amplitude by 32<!PCT!>--48<!PCT!>, 50<!PCT!>--65<!PCT!>, and 15<!PCT!>--50<!PCT!> respectively. Combining all the sources of information explored expected from in combination with the future SO-like CMB experiment, we forecast A lin 0.001$ at a 68.3<!PCT!> CL and A log 0.001$ for rec + BS)+WL+ for both the optimistic and pessimistic settings over the frequency range
