  It has been realised only recently that TeV emission from blazars can
significantly heat the intergalactic medium (IGM) by pair-producing high-energy
electrons and positrons, which in turn excite vigorous plasma instabilities,
leading to a local dissipation of the pairs' kinetic energy. In this work, we
use cosmological hydrodynamical simulations to model the impact of this blazar
heating on the Lyman-alpha forest at redshifts z~2-3. We find that blazar
heating produces an inverted temperature-density relation in the IGM and
naturally resolves many of the problems present in previous simulations of the
forest that included photoheating alone. In particular, our simulations with
blazar heating simultaneously reproduce the observed effective optical depth
and temperature as a function of redshift, the observed probability
distribution functions of the transmitted flux, and the observed flux power
spectra, over the full redshift range 2<z<3 analysed here. Additionally, by
deblending the Lyman-alpha forest into a sum of thermally broadened individual
lines, we find superb agreement with the observed lower cutoff of the
line-width distribution and abundances of neutral hydrogen column densities.
Using the most recent constraints on the cosmic ultraviolet (UV) background,
this excellent agreement with observations does not require rescaling the
amplitude of the UV background; a procedure that was routinely used in the past
to match the observed level of transmitted flux. We also show that our
blazar-heated model matches the data better than standard simulations even when
such a rescaling is allowed. This concordance between Lyman-alpha data and
simulations, which are based on the most recent cosmological parameters,
suggests that the inclusion of blazar heating alleviates previous tensions on
constraints for sigma_8 derived from Lyman-alpha measurements and other
cosmological data. [abridged]
