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The amplitude of fluctuations in the Ly-a forest on small spatial scales issensitive to the temperature of the IGM and its spatial fluctuations. Thetemperature of the IGM and its spatial variations contain important informationabout hydrogen and helium reionization. We present a new measurement of thesmall-scale structure in the Ly-a forest from 40 high resolution, highsignal-to-noise, VLT spectra at z=2.2-4.2. We convolve each Ly-a forestspectrum with a suitably chosen wavelet filter, which allows us to extract theamount of small-scale structure in the forest as a function of position acrosseach spectrum. We compare these measurements with high resolution hydrodynamicsimulations of the Ly-a forest which track more than 2 billion particles. Thiscomparison 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-sigmaconfidence. The temperature at lower redshift is consistent with the fall-offexpected from adiabatic cooling ($T_0 \propto (1+z)^2$), after the peaktemperature is reached near z=3.4. At z=4.2 our results favor a temperature ofT_0 = 15-20,000 K. However, owing mostly to uncertainties in the meantransmitted flux at this redshift, a cooler IGM model with T_0 = 10,000 K isonly disfavored at the 2-sigma level here, although such cool IGM models arestrongly discrepant with the z ~ 3-3.4 measurement. We do not detect largespatial fluctuations in the IGM temperature at any redshift covered by our dataset. The simplest interpretation of our measurements is that HeII reionizationcompletes sometime near z ~ 3.4, although statistical uncertainties are stilllarge [Abridged].

A Measurement of Small Scale Structure in the 2.2 < z < 4.2 Lyman-alpha Forest