The amplitude of fluctuations in the Ly-a forest on small spatial scales is sensitive to the temperature of the IGM and its spatial fluctuations. The temperature of the IGM and its spatial variations contain important information about hydrogen and helium reionization. We present a new measurement of the small-scale structure in the Ly-a forest from 40 high resolution, high signal-to-noise, VLT spectra at z=2.2-4.2. We convolve each Ly-a forest spectrum with a suitably chosen wavelet filter, which allows us to extract the amount of small-scale structure in the forest as a function of position across each spectrum. We compare these measurements with high resolution hydrodynamic simulations of the Ly-a forest which track more than 2 billion particles. This comparison suggests that the IGM temperature close to the cosmic mean density (T_0) peaks near z=3.4, at which point it is greater than 20,000 K at 2-sigma confidence. The temperature at lower redshift is consistent with the fall-off expected from adiabatic cooling ($T_0 propto (1+z)^2$), after the peak temperature is reached near z=3.4. At z=4.2 our results favor a temperature of T_0 = 15-20,000 K. However, owing mostly to uncertainties in the mean transmitted flux at this redshift, a cooler IGM model with T_0 = 10,000 K is only disfavored at the 2-sigma level here, although such cool IGM models are strongly discrepant with the z ~ 3-3.4 measurement. We do not detect large spatial fluctuations in the IGM temperature at any redshift covered by our data set. The simplest interpretation of our measurements is that HeII reionization completes sometime near z ~ 3.4, although statistical uncertainties are still large [Abridged].
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