Constraints on the state of the IGM at $z\sim 8-10$ using redshifted 21-cm observations with LOFAR

The power spectra of the redshifted 21-cm signal from the Epoch of Reionization (EoR) contain information about the ionization and thermal states of the intergalactic medium (IGM), and depend on the properties of the EoR sources. Recently, Mertens et al 2025 has analysed 10 nights of LOFAR high-band data and estimated upper limits on the 21-cm power spectrum at redshifts 8.3, 9.1 and 10.1. Here we use these upper limit results to constrain the properties of the IGM at those redshifts. We focus on the properties of the ionized and heated regions where the temperature is larger than that of the CMB. We model the 21-cm power spectrum with the code GRIZZLY, and use a Bayesian inference framework to explore the source parameters for uniform priors on their ranges. The framework also provides information about the IGM properties in the form of derived parameters. In a model which includes a radio background in excess of the CMB, the 95 (68) per cent credible intervals of disfavoured models at redshift 9.1 for the chosen priors correspond to IGM states with averaged ionization and heated fraction below 0.46 ($\lesssim 0.05$), an average gas temperature below 44 K (4 K), and a characteristic size of the heated region $\lesssim 14 ~h^{-1} ~\mathrm{Mpc}$ ($\lesssim 3 ~h^{-1} ~\mathrm{Mpc}$). The 68 per cent credible interval suggests an excess radio background which is more than 100 per cent of the CMB at 1.42 GHz, while the 95 per cent credible interval of the radio background efficiency parameter spans the entire prior range. The behaviour of the credible intervals is similar at all redshifts. The models disfavoured by the LOFAR upper limits are extreme ones, as they are mainly driven by rare and large ionized or heated regions.