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tokens: = PBH+CDM). We define the PBH+CDM mass fraction as where M PBH and M CDM are the total masses of PBHs and CDM particles. For our halo, we assume the Navarro–Frenk–White (NFW) form (Navarro et al. 1996 ) with scale density ρ 0 and scale length r s . The relaxation time is proportional to N /ln ( N ), where N is the number of particles. Assuming 100 M ⊙ PBHs and 10 7 M ⊙ halo, the particle relaxation will take between 1 and 12 Gyr depending on f m considered between 1 and 0.01. In comparison, the relaxation time for 10 8 (10 9 M ⊙ ) haloes are 10 (100) times longer than for 10 7 M ⊙ and longer than the age of the Universe, which is why we focus here on a 10 7 M ⊙ dwarf galaxy, starting at redshift z = 2 in the simulations. Given the halo mass and redshift, the halo concentration c 200 can be estimated from cosmological N -body simulations (Prada et al. 2012 ). Our halo is composed of DM and PBH particles with a total mass of 10 7 M ⊙ . To generate our NFW haloes, we use the initial-condition generator, magi . Adoption of a distribution-function-based method ensures that the final realization of the halo is in dynamical equilibrium (Miki Umemura 2018 ). We perform our simulations with the high performance collisionless N -body code, gothic (Miki Umemura 2017 ; Miki 2019 ). This gravitational octree code runs entirely on GPU and is accelerated by the use of hierarchical time steps in which a group of particles has the same time step. We have performed the simulations using NVIDIA GeForce GTX 1080 Ti (CUDA 9.0) on Intel Xeon Silver 4114 (GCC 4.8). We evolve different PBH+CDM haloes composed of 25, 50, 75, and 100 M ⊙ PBHs over 11 Gyr by adopting the softening length of ε 0 =1.331 pc and the accuracy control parameter of 2 −7 . All runs were made with CDM particles of 1 M ⊙ . We explored also PBH+CDM haloes with different mass fractions f m = [0.01, 0.1, 0.5]. In our simulations, we assume that the CDM and PBH components of PBH+CDM haloes initially follow NFW profiles with the same concentration. We tested two different scale lengths for the profile of the PBH component: and . The second corresponds to a scenario where the density of PBHs is enhanced in the central region. Indeed, we suppose that mass segregation of PBHs would occur, and this increases the density of PBHs at the centre. This scenario should also enhance the formation of cores due to dynamical heating of the CDM by PBHs. It will also accelerate the two-body relaxation. 3 RESULTS 3.1 Evidences for core formation from gravitational heating by PBHs First, we assess the impact on the PBH+CDM halo density profiles in our simulations of the two numerical parameters, which are the accuracy control parameter and the softening length ε. We tested 2 −6 , 2 −7 , 2 −8 , and 2 −9 for the accuracy control parameter in a PBH+CDM halo with f m = 0.1 and m PBH = 100 M ⊙ . The left-hand panel of the Fig. 1 shows that PBH+CDM density profiles hold the initial distribution for all the different values