  Knowledge of the spectrograph's instrumental profile (IP) provides important
information needed for wavelength calibration and for the use in scientific
analyses. This work develops new methods for IP reconstruction in high
resolution spectrographs equipped with Laser Frequency Comb calibration (LFC)
systems and assesses the impact that assumptions on IP shape have on achieving
accurate spectroscopic measurements. Astronomical LFCs produce $\approx10000$
bright, unresolved emission lines with known wavelengths, making them excellent
probes of the IP. New methods based on Gaussian Process regression were
developed to extract detailed information on the IP shape from this data.
Applying them to HARPS, an extremely stable spectrograph installed on the ESO
3.6m telescope, we reconstructed its IP at 512 locations of the detector,
covering 60% of the total detector area. We found that the HARPS IP is
asymmetric and that it varies smoothly across the detector. Empirical IP models
provide wavelength accuracy better than 10 ms$^{-1}$ (5 ms$^{-1}$) with 92%
(64%) probability. In comparison, reaching the same accuracy has a probability
of only 29% (8%) when a Gaussian IP shape is assumed. Furthermore, the Gaussian
assumption is associated with intra-order and inter-order distortions in the
HARPS wavelength scale as large as 60ms$^{-1}$. The spatial distribution of
these distortions suggests they may be related to spectrograph optics and
therefore may generally appear in cross-dispersed echelle spectrographs when
Gaussian IPs are used. Methods presented here can be applied to other
instruments equipped with LFCs, such as ESPRESSO, but also ANDES and G-CLEF in
the future. The empirical IPs will be crucial for obtaining objective and
unbiased measurements of fundamental constants from high resolution spectra, as
well as measurements of the redshift drift, isotopic abundances, and other
science cases.
