We present forecasts for constraints on cosmological models which can be obtained by forthcoming radio continuum surveys: the wide surveys with the LOw Frequency ARray (LOFAR), Australian Square Kilometre Array Pathfinder (ASKAP) and the Westerbork Observations of the Deep APERTIF Northern sky (WODAN). We use simulated catalogues appropriate to the planned surveys to predict measurements obtained with the source auto-correlation, the cross-correlation between radio sources and CMB maps (the Integrated Sachs-Wolfe effect), the cross-correlation of radio sources with foreground objects due to cosmic magnification, and a joint analysis together with the CMB power spectrum and supernovae. We show that near future radio surveys will bring complementary measurements to other experiments, probing different cosmological volumes, and having different systematics. Our results show that the unprecedented sky coverage of these surveys combined should provide the most significant measurement yet of the Integrated Sachs-Wolfe effect. In addition, we show that using the ISW effect will significantly tighten constraints on modified gravity parameters, while the best measurements of dark energy models will come from galaxy auto-correlation function analyses. Using the combination of EMU and WODAN to provide a full sky survey, it will be possible to measure the dark energy parameters with an uncertainty of {$sigma (w_0) = 0.05$, $sigma (w_a) = 0.12$} and the modified gravity parameters {$sigma (eta_0) = 0.10$, $sigma (mu_0) = 0.05$}, assuming Planck CMB+SN(current data) priors. Finally, we show that radio surveys would detect a primordial non-Gaussianity of $f_{rm NL}$ = 8 at 1-$sigma$ and we briefly discuss other promising probes.
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