Constraining the intergalactic medium at $z\approx$ 9.1 using LOFAR
  Epoch of Reionization observations
  We derive constraints on the thermal and ionization states of the
intergalactic medium (IGM) at redshift $\approx$ 9.1 using new upper limits on
the 21-cm power spectrum measured by the LOFAR radio-telescope and a prior on
the ionized fraction at that redshift estimated from recent cosmic microwave
background (CMB) observations. We have used results from the reionization
simulation code GRIZZLY and a Bayesian inference framework to constrain the
parameters which describe the physical state of the IGM. We find that, if the
gas heating remains negligible, an IGM with ionized fraction $\gtrsim 0.13$ and
a distribution of the ionized regions with a characteristic size $\gtrsim 8
~h^{-1}$ comoving megaparsec (Mpc) and a full width at the half maximum (FWHM)
$\gtrsim 16 ~h^{-1}$ Mpc is ruled out. For an IGM with a uniform spin
temperature $T_{\rm S} \gtrsim 3$ K, no constraints on the ionized component
can be computed. If the large-scale fluctuations of the signal are driven by
spin temperature fluctuations, an IGM with a volume fraction $\lesssim 0.34$ of
heated regions with a temperature larger than CMB, average gas temperature
7-160 K and a distribution of the heated regions with characteristic size
3.5-70 $h^{-1}$ Mpc and FWHM of $\lesssim 110$ $h^{-1}$ Mpc is ruled out. These
constraints are within the 95 per cent credible intervals. With more stringent
future upper limits from LOFAR at multiple redshifts, the constraints will
become tighter and will exclude an increasingly large region of the parameter
space.
