Gravitational instability induces a simple correlation between the large- and small-scale fluctuations of the Lyα flux spectrum. However, nongravitational processes involved in structure formation and evolution will alter such a correlation. In this Letter, we explore how scatter in the temperature-density relation of the intergalactic medium reduces the gravitationally induced scale-scale correlation. By examining whether or not observations of the correlation are close to that predicted by pure gravity, we put constraints on the scatter in the ρ-T relation and in turn on any physical process that would lead to scatter, e.g., strong fluctuations in the UV background or radiative transfer effects. By applying this method to high-resolution Keck spectra of Q1422+231 and HS 1946+7658, we find the predicted correlation signal induced by gravity and the diminishing of this correlation signal at small scales. This suggests extra physics affects the small-scale structure of the forest, and we can constrain the scatter in the ρ-T relation to a conservative 20% upper limit. A crude model suggests, if there is any spatial correlation of temperature, the coherence length scale must be smaller than ～ 0.3 h~(-1) Mpc to be consistent with the Keck data.
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