Evidences of low-diffusion bubbles around Galactic pulsars

Recently, a few-degrees extended $\gamma$-ray halo in the direction of Geminga pulsar has been detected by HAWC, Milagro and Fermi-LAT. These observations can be interpreted with positrons ($e^+$) and electrons ($e^-$) accelerated by Geminga pulsar wind nebula (PWN), released in a Galactic environment with a low diffusion coefficient ($D_0$), and inverse Compton scattering (ICS) with the interstellar radiation fields. We inspect here how the morphology of the ICS $\gamma$-ray flux depends on the energy, the pulsar age and distance, and the strength and extension of the low-diffusion bubble. In particular we show that $\gamma$-ray experiments with a peak of sensitivity at TeV energies are the most promising ones to detect ICS halos. We perform a study of the sensitivity of HAWC, HESS and the future CTA experiment finding that, with efficiencies of the order of a few %, the first two experiments should have already detected a few tens of ICS halos while the latter will increase the number of detections by a factor of 4. We then consider a sample of sources associated to PWNe and detected in the HESS Galactic plane survey and in the second HAWC catalog. We use the information available in these catalogs for the $\gamma$-ray spatial morphology and flux of these sources to inspect the value of $D_0$ around them and the $e^{\pm}$ injection spectrum. All sources are detected as extended with a $\gamma$-ray emission extended about $15-80$ pc. Assuming that most of the $e^{\pm}$ accelerated by these sources have been released in the interstellar medium, the diffusion coefficient is $2-30 \cdot 10^{26}$ cm$^2$/s at 1 TeV, i.e. two orders of magnitude smaller than the value considered to be the average in the Galaxy. These observations imply that Galactic PWNe have low-diffusion bubbles with a size of at least 80 pc.