Population III Star Formation in an X-ray background: II. Protostellar
  Discs, Multiplicity and Mass Function of the Stars
  Disc fragmentation plays an important role in determining the number of
primordial stars (Pop III stars), their masses, and hence the initial mass
function. In this second paper of a series, we explore the effect of uniform
FUV H$_2$-photodissociating and X-ray radiation backgrounds on the formation of
Pop~III stars using a grid of high-resolution zoom-in simulations. We find
that, in an X-ray background, protostellar discs have lower surface density and
higher Toomre $Q$ parameter, so they are more stable. For this reason, X-ray
irradiated discs undergo fewer fragmentations and typically produce either
binary systems or low-multiplicity systems. In contrast, the cases with weak or
no X-ray irradiation produce systems with a typical multiplicity of $6 \pm 3$.
In addition, the most massive protostar in each system is smaller by roughly a
factor of two when the disc is irradiated by X-rays, due to lower accretion
rate. With these two effects combined, the initial mass function of fragments
becomes more top-heavy in a strong X-ray background and is well described by a
power-law with slope $1.53$ and high-mass cutoff of $61$ M$_\odot$. Without
X-rays, we find a slope $0.49$ and cutoff mass of $229$ M$_\odot$. Finally,
protostars migrate outward after their formation due to the accretion of
high-angular momentum gas from outside and the migration is more frequent and
significant in absence of X-ray irradiation.
