Exploring the Thermal State of the Low-Density Intergalactic Medium at z=3 with an Ultra-High Signal-to-Noise QSO Spectrum

At low densities the standard ionisation history of the intergalactic medium (IGM) predicts a decreasing temperature of the IGM with decreasing density once hydrogen (and helium) reionisation is complete. Heating the high-redshift, low-density IGM above the temperature expected from photo-heating is difficult, and previous claims of high/rising temperatures in low density regions of the Universe based on the probability density function (PDF) of the opacity in Lyman-$\alpha$ forest data at $2<z<4$ have been met with considerable scepticism, particularly since they appear to be in tension with other constraints on the temperature-density relation (TDR). We utilize here an ultra-high signal-to-noise spectrum of the QSO HE0940-1050 and a novel technique to study the low opacity part of the PDF. We show that there is indeed evidence (at 90% confidence level) that a significant volume fraction of the under-dense regions at $z \sim 3$ has temperatures as high or higher than those at densities comparable to the mean and above. We further demonstrate that this conclusion is nevertheless consistent with measurements of a slope of the TDR in over-dense regions that imply a decreasing temperature with decreasing density, as expected if photo-heating of ionised hydrogen is the dominant heating process. We briefly discuss implications of our findings for the need to invoke either spatial temperature fluctuations, as expected during helium reionization, or additional processes that heat a significant volume fraction of the low-density IGM.