Determination of small-scale magnetic fields on Sun-like stars in the near-infrared using CRIRES$^+$

We aim to characterise the small-scale magnetic fields for a sample of 16 Sun-like stars and investigate the capabilities of the newly upgraded near-infrared (NIR) instrument CRIRES$^+$ at the VLT in the context of small-scale magnetic field studies. Our targets also had their magnetic fields studied in the optical, which allows us to compare magnetic field properties at different spatial scales on the stellar surface and to contrast small-scale magnetic field measurements at different wavelengths. We analyse the Zeeman broadening signature for six magnetically sensitive and insensitive \ion{Fe}{I} lines in the H-band to measure small-scale magnetic fields on the stellar surface. We use polarised radiative transfer modelling and NLTE departure coefficients in combination with MCMC to determine magnetic field characteristics together with non-magnetic stellar parameters. We use two different approaches to describe small-scale magnetic fields. The first is a two-component model with a single magnetic region and a free magnetic field strength. The second model contains multiple magnetic components with fixed magnetic field strengths. We find average magnetic field strengths ranging from $\sim 0.4$ kG down to $<0.1$ kG. The results align closely with other results from high resolution NIR spectrographs such as SPIRou. We find that the small-scale fields correlate with the large-scale fields and that the small-scale fields are at least 10 times stronger than the large-scale fields inferred with Zeeman Doppler imaging. The two- and multi-component models produce systematically different results as the strong fields from the multi-component model increase the obtained mean magnetic field strength. When comparing our results with the optical measurements of small-scale fields we find a systematic offset of 2--3 times stronger fields in the optical.