The epoch of reionization connects early cosmology with the birth of the first galaxies. When Reionization swept through the large-scale structure (the cosmic web), shock waves from rapidly moving ionization fronts heated the intergalactic gas to tens of thousands of kelvin. The excess pressure drove the gas to expand via the Jeans instability mechanism—much like heated air in a piston chamber pushing against the walls. After the fronts passed, the gas cooled, but the 'memory' of the thermal shock was preserved as a characteristic scale at which pressure gradients smoothed out the density. This quantity—the pressure-smoothing scale—carries integrated information about the total heating and remains one of the few available windows into an era when the universe was opaque to direct observation.
The authors used three complementary approaches. First, an ensemble of high-resolution cosmological hydrodynamical simulations from the Sherwood-Relics project, covering various reionization scenarios with differing end times and heat inputs. Second, a simplified expanding-spheres model that reproduces the gas's hydrodynamical response—'bubbles' with a Gaussian density profile and a radial expansion velocity of about 10 km/s were injected at random points in the volume. Third, solutions from linear perturbation theory with a time-dependent sound speed, exhibiting damped acoustic oscillations. To extract the characteristic peak, the power spectrum of the peculiar velocity gradient η and the flux spectrum in the Lyman-alpha forest (where distortions from the Doppler effect encode the velocity field) in an expanding universe governed by Hubble's law were fitted with a smooth empirical function, and the peak position was determined numerically. The same procedure was then applied to the observed flux spectrum from Boera et al. (2019) over the redshift range 4.2–5.0.
The simulation analysis revealed a pronounced peak in the peculiar velocity power spectrum on scales of 0.1–1.0 s/km, corresponding to gas expanding at ~10 km/s over typical sizes of ~40 kpc. This 'acoustic bump' is a direct imprint of the hydrodynamical response to reionization and, thanks to the expansion of the universe, shifts to larger scales as redshift decreases—much like a sound wave stretching in an expanding medium. The peak in the quasar flux power spectrum, proportional to k⁷ Pₕ(k), turned out to be broader but located on the same scales. Applying the method to real data made it possible to directly measure the pressure-smoothing scale for the first time: at z=4.2 it is >33.78 kiloparsecs (1σ), at z=4.6 it is 32.36 ± 8.35 kpc, and at z=5.0 it is 31.27 ± 18.28 kpc. These values are in good agreement with indirect estimates and theoretical predictions of the Gnedin & Hui filtering scale.
This new method radically changes how we probe the thermal history. Previously, information about pressure smoothing was extracted indirectly—from the suppression of the entire small-scale flux spectrum, where it is difficult to disentangle contributions from thermal effects, the Jeans scale, and the properties of dark matter. Now we isolate a specific acoustic feature with a clear physical origin. This will allow it to be used as an independent calibrator in full Lyman-alpha forest analyses and will significantly narrow uncertainties in constraints on dark matter models, down to the masses of warm dark matter particles.
The near future lies with new spectroscopic surveys—such as WEAVE, DESI, and the GHOSTLy instrument—which will provide orders of magnitude more quasar spectra with high signal-to-noise ratios. Increasing the maximum wavenumber kmax to 0.8 s/km (the current limit is about 0.2 s/km) will make the peak more pronounced and allow differentiation between scenarios with different reionization end times. Moreover, the method can be extended to analyze phase correlations in close quasar pairs, boosting statistical significance.
The results will impact several fields: early-universe cosmology, the theory of first galaxy formation, intergalactic medium physics, and indirect searches for dark matter particles.
Planned next steps include refining the empirical peak model to reduce systematic errors and incorporating new observational data at z>5.
This work directly addresses unresolved questions in physics: the timeline of reionization, the contribution of different ionizing sources, and the nature of warm dark matter, which leaves a similar imprint on small scales.
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What the work does
The authors reconstructed the scale of pressure smoothing in the intergalactic gas using Lyman-alpha forest spectra and numerical simulations. They combined the analysis of the Sherwood-Relics hydrodynamic simulations, a simplified model of expanding spheres, and linear theory to isolate a characteristic peak in the velocity and flux power spectrum. The method was applied to observational data from Boera et al. (2019) at redshifts 4.2–5.0, yielding the first direct measurements of the smoothing scale.
Where its strength is
This work offers a direct observational method to extract integrated information about the thermal history of reionization from one of the few available windows—the Lyman-alpha forest. The detected acoustic bump in the peculiar velocity power spectrum serves as a robust imprint of the hydrodynamic response, weakly dependent on model details.
Where it could go next
Worth trying
Why this may matter
Direct measurement of the smoothing scale opens the way to independent calibration of the thermal history of reionization, which is critical for understanding the nature of the first sources and the physics of the intergalactic medium.
Nearby works in our archive
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as written by the authors arXiv:2607.19938v1
The epoch of reionization leaves detectable imprints in the thermal history of the Universe. In particular, the fast ionization fronts passing over cold gas in the cosmic web overpressurize the gas, leading to the well-established phenomenon of pressure smoothing due to the hydrodynamic response of the gas. Although the effect has been indirectly measured from the power spectra of the Lyman-$$\alpha$$ forest over the past decades, very few examples exist of directly measuring the typical scale associated with this physical process. This work identifies an acoustic feature in the power spectrum of the projected peculiar velocity gradient $$\eta$$. Using toy models and linear theory, this work shows that the acoustic feature is likely strongly associated with the pressure smoothing scale. A feature at the same scale is imprinted on the small-scale flux power spectrum of the Lyman-$$\alpha$$ forest at $$k=0.1-1\;\mathrm{s/km}$$. A methodology developed for the simulations is applied to the latest measurements of the flux power spectra at $$z=4.2-5.0$$ to provide the first direct measurements of the pressure smoothing scale at high redshifts, $$\lambda_p(z=4.2) > 33.78\;\mathrm{ckpc}\;(1\sigma)$$, $$\lambda_p(z=4.6) = 32.36\pm8.35\;\mathrm{ckpc}$$ and $$\lambda_p(z=5.0)=31.27\pm18.28\;\mathrm{ckpc}$$. This proof-of-concept study paves the way for future observational programs aimed at recovering the thermal history using small-scale observations of the Lyman-$$\alpha$$ forest.
The expanding-spheres model used by the authors reproduced the acoustic peak in simulations with surprising accuracy, despite its simplicity. It turns out that the post-reionization universe was filled with myriads of 'bubbles' the size of a small galaxy, expanding at the sound speed in hot gas — about 10 km/s. The combined effect of these bubbles creates the detectable 'ringing'.