Deep uGMRT observations of the ELAIS-North 1 field: statistical properties of radio--infrared relations up to $z \sim$2

Comprehending the radio--infrared (IR) relations of the faint extragalactic radio sources is important for using radio emission as a tracer of star-formation in high redshift ($z$) star-forming galaxies (SFGs). Using deep uGMRT observations of the ELAIS-N1 field in the 0.3--0.5\,GHz range, we study the statistical properties of the radio--IR relations and the variation of the `$q$-parameter' up to $z=2$ after broadly classifying the faint sources as SFGs and AGN. We find the dust temperature ($\tdust$) to increase with $z$. This gives rise to $\qmir$, measured at $24\,\upmu$m, to increase with $z$ as the peak of IR emission shifts towards shorter wavelengths, resulting in the largest scatter among different measures of $q$-parameters. $\qfir$ measured at $70\,\upmu$m, and $q_{\rm TIR}$ using total-IR (TIR) emission are largely unaffected by $\tdust$. We observe strong, non-linear correlations between the radio luminosities at 0.4 and 1.4\,GHz with $70\,\upmu$m luminosity and TIR luminosity($\ltir$). To assess the possible role of the radio-continuum spectrum in making the relations non-linear, for the first time we study them at high $z$ using integrated radio luminosity ($\lrc$) in the range 0.1--2\,GHz. In SFGs, the $\lrc$--$\ltir$ relation remains non-linear with a slope of $1.07\pm0.02$, has a factor of 2 lower scatter compared to monochromatic radio luminosities, and $\qtirbol$ decreases with $z$ as $\qtirbol = (2.27 \pm 0.03)\,(1+z)^{-0.12 \pm 0.03}$. A redshift variation of $q$ is a natural consequence of non-linearity. We suggest that a redshift evolution of magnetic field strengths and/or cosmic ray acceleration efficiency in high-$z$ SFGs could give rise to non-linear radio--IR relations.