Spectral indices in active galactic nuclei as seen by Apertif and LOFAR

We present two new radio continuum images obtained with Apertif at 1.4 GHz. The images, produced with a direction-dependent calibration pipeline, cover 136 square degrees of the Lockman Hole and 24 square degrees of the ELAIS-N fields, with an average resolution of 17x12" and residual noise of 33 uJy/beam. With the improved depth of the images we found in total 63692 radio sources, many of which are detected for the first time at this frequency. With the addition of the previously published Apertif catalog for the Bootes field, we cross-match with the LOFAR deep-fields value-added catalogs at 150 MHz, resulting in a homogeneous sample of 10196 common sources with spectral index estimates, one of the largest to date. We analyze and discuss the correlations between spectral index, redshift, linear sources size, and radio luminosity, taking into account biases of flux-density-limited surveys. Our results suggest that the observed correlation between spectral index and redshift of active galactic nuclei can be attributed to the Malmquist bias reflecting an intrinsic relation between radio luminosity and the spectral index. We also find a correlation between spectral index and linear source size with more compact sources having steeper spectra.