J-PLUS: Tomographic analysis of galaxy angular density and redshift fluctuations in Data Release 3. Constraints on photo-$z$ errors, linear bias, and peculiar velocities

The {\it Javalambre Photometric Local Universe Survey} (J-PLUS) is a {\it spectro-photometric} survey covering about 3,000~deg$^2$ in its third data release (DR3), and containing about 300,000 galaxies with high quality ({\it odds}$>0.8$) photometric redshifts (hereafter photo-$z$s). We use this galaxy sample to conduct a tomographic study of the counts and redshift angular fluctuations under Gaussian shells sampling the redshift range $z\in[0.05,0.25]$. We confront the angular power spectra of these observables measured under shells centered on 11 different redshifts with theoretical expectations derived from a linear Boltzmann code ({\tt ARFCAMB}). Overall we find that J-PLUS DR3 data are well reproduced by our linear, simplistic model. We obtain that counts (or density) angular fluctuations (hereafter ADF) are very sensitive to the linear galaxy bias $b_g(z)$, although weakly sensitive to radial peculiar velocities of the galaxy field, while suffering from systematics residuals for $z>0.15$. Angular redshift fluctuations (ARF), instead, show higher sensitivity to radial peculiar velocities and also higher sensitivity to the average uncertainty in photo-$z$s ($\sigma_{\rm Err}$), with no obvious impact from systematics. For $z<0.15$ both ADF and ARF agree on measuring a monotonically increasing linear bias varying from $b_g(z=0.05)\simeq 0.9\pm 0.06$ up to $b_g(z=0.15)\simeq 1.5\pm 0.05$, while, by first time, providing consistent measurements of $\sigma_{\rm Err}(z)\sim 0.014$ that are $\sim 40~\%$ higher than estimates from the photo-$z$ code {\tt LePhare}, ($\sigma_{\rm Err}^{\rm LePhare}=0.010$). As expected, this photo-$z$ uncertainty level prevents the detection of radial peculiar velocities in the modest volume sampled by J-PLUS DR3, although prospects for larger galaxy surveys of similar (and higher) photo-$z$ precision are promising.