The transitional millisecond pulsar IGR J18245-2452 during its 2013 outburst at X-rays and soft gamma-rays

IGR~J18245--2452/PSR J1824--2452I is one of the rare transitional accreting millisecond X-ray pulsars, showing direct evidence of switches between states of rotation powered radio pulsations and accretion powered X-ray pulsations, dubbed transitional pulsars. IGR~J18245--2452 is the only transitional pulsar so far to have shown a full accretion episode, reaching an X-ray luminosity of $\sim10^{37}$~erg~s$^{-1}$ permitting its discovery with INTEGRAL in 2013. In this paper, we report on a detailed analysis of the data collected with the IBIS/ISGRI and the two JEM-X monitors on-board INTEGRAL at the time of the 2013 outburst. We make use of some complementary data obtained with the instruments on-board XMM-Newton and Swift in order to perform the averaged broad-band spectral analysis of the source in the energy range 0.4 -- 250~keV. We have found that this spectrum is the hardest among the accreting millisecond X-ray pulsars. We improved the ephemeris, now valid across its full outburst, and report the detection of pulsed emission up to $\sim60$ keV in both the ISGRI ($10.9 \sigma$) and Fermi/GBM ($5.9 \sigma$) bandpass. The alignment of the ISGRI and Fermi GBM 20 -- 60 keV pulse profiles are consistent at a $\sim25\ \mu$s level. We compared the pulse profiles obtained at soft X-rays with \xmm\ with the soft \gr-ray ones, and derived the pulsed fractions of the fundamental and first harmonic, as well as the time lag of the fundamental harmonic, up to $150\ \mu$s, as a function of energy. We report on a thermonuclear X-ray burst detected with \Integ, and using the properties of the previously type-I X-ray burst, we show that all these events are powered primarily by helium ignited at a depth of $y_{\rm ign} \approx 2.7\times10^8$ g cm${}^{-2}$. For such a helium burst the estimated recurrence time of $\Delta t_{\rm rec}\approx5.6$ d is in agreement with the observations.