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every quantitative analysis of the primordial black hole ( pbh ) number and mass spectrum @xcite requires knowledge of the threshold parameter , @xmath2 , separating perturbations that form black holes from those that do not , and the resulting black hole mass , @xmath3 , as a function of distance from the threshold . in order to determine @xmath2 and @xmath3 for various initial conditions
, we performed one - dimensional , general relativistic simulations of the hydrodynamics of pbh formation in the radiation - dominated phase of the early universe .
three families of perturbation shapes were chosen to represent `` generic '' classes of initial data , reflecting the lack of information about the specific shape of primordial fluctuations .
the numerical technique is sketched in section [ numerics ] .
defined as the excess mass within the horizon sphere at the onset of the collapse , we find @xmath4 for all three perturbation shapes , indicating that the threshold value may indeed be universal ( section [ hydro ] ) . a numerical confirmation of the previously suggested power - law scaling of @xmath3 with @xmath5 @xcite , related to the well - known behavior of collapsing space  times at the critical point of black hole formation @xcite , is presented in section [ scaling ] . in this framework ,
the pbh mass spectrum is determined by the dimensionless coefficient , @xmath6 , the scaling exponent , @xmath7 , and the initial horizon mass , @xmath8 , such that @xmath9 we provide numerical results for @xmath6 and @xmath7 for the three perturbation families .
a more detailed description of the numerical technique and further results will be presented in a forthcoming publication @xcite .
the dynamics of collapsing density perturbations in the early universe are determined by the general relativistic equations of motion for a perfect fluid , the field equations , the first law of thermodynamics , and a radiation - dominated equation of state .
the assumption of spherical symmetry is well justified for large fluctuations in a gaussian distribution @xcite , reducing the problem to one spatial dimension .    for our simulations , we chose the formulation of the hydrodynamical equations by hernandez and misner @xcite as implemented by baumgarte et al .
@xcite . based on the original equations by misner and sharp @xcite ,
hernandez and misner proposed to exchange the schwarzschild time variable , @xmath10 , with the outgoing null coordinate , @xmath11 . in so doing ,
the hydrodynamical equations retain the lagrangian character of the misner  sharp equations but avoid crossing into the event horizon of a black hole once it has formed . covering the entire space 
time outside while asymptotically approaching the event horizon , the hernandez 
misner equations are perfectly suited to follow the evolution of a black hole for long times after its formation without encountering coordinate singularities .
this allowed us , in principle , to study the accretion of material onto newly formed pbhs for arbitrarily long times ( in contrast with earlier calculations @xcite ) . since the expanding outer regions of our simulated piece of the universe are most conveniently tracked in a comoving numerical reference frame , the lagrangian form of the hernandez 
misner equations is their second major asset .
it also provides a simple prescription for the outer boundary condition , which is defined to match the exact solution of the friedmann equations for a radiation dominated flat universe .
hence , the pressure follows the analytic solution @xmath12 where @xmath13 is the proper time of the outermost fluid element ( corresponding to the cosmological time @xmath10 in a friedmann  robertson  walker ( frw ) universe ) and @xmath14 and @xmath15 are the initial values for pressure and proper time .
we studied the spherically symmetric evolution of three families of curvature perturbations .
initial conditions were chosen to be perturbations of the energy density , @xmath16 , in unperturbed hubble flow specified at horizon crossing .
the first family of perturbations is described by a gaussian - shaped overdensity that asymptotically approaches the frw solution at large radii .
the other two families of initial conditions involve a spherical mexican hat function and a fourth order polynomial .
these functions are characterized by rarefaction regions outside of the horizon radius , @xmath17 , that identically compensate for the additional mass of the overdensities inside the horizon volume , so that the mass derived from the total integrated density profile is equal to that of an unperturbed frw solution . in our numerical experiments ,
the amplitude , @xmath18 , of the perturbations is used to tune the initial conditions to sub- or supercriticality with respect to black hole formation .
the critical amplitude , @xmath19 , is strongly shape - dependent , varying between @xmath20 for mexican - hat - shaped perturbations and @xmath21 for the gaussian curve .
if , however , we define the control parameter @xmath1 as the additional mass inside @xmath17 in units of the horizon mass , we find strikingly similar values  @xmath22 ( mexican hat ) , @xmath23 ( gaussian curve ) , and @xmath24 ( polynomial )  for all three families of initial data in our study .
this number is considerably greater than the previously employed threshold @xmath25 following from analytical estimates @xcite . given the qualitative difference of the functional forms of the different perturbation families , the result that @xmath2 of the gaussian perturbation lies in between the critical values of the mass compensated functions is surprising .
our results suggest that @xmath4 ( with @xmath1 defined as above ) is a universal statement , i.e. , true for all perturbation shapes in the radiation - dominated regime .
this remains to be verified by means of additional experiments .
for a variety of matter models , it is well - known that the dynamics of near - critical collapse exhibit continuous or discrete self - similarity and power - law scaling of the black hole mass with the offset from the critical point \{eq .
( [ scale ] ) @xcite}. in particular , evans and coleman @xcite found self - similarity and mass scaling in numerical experiments of a collapsing radiation fluid .
they numerically determined the scaling exponent @xmath26 , followed by a linear perturbation analysis of the critical solution by koike et al .
@xcite that yielded @xmath27 . until recently , it was believed that entering the scaling regime requires a degree of fine - tuning of the initial data that is unnatural for any astrophysical application .
it was noted @xcite that fine - tuning to criticality occurs naturally in the case of pbhs forming from a steeply declining distribution of primordial density fluctuations , as generically predicted by inflationary scenarios . in the radiation - dominated cosmological epoch ,
the only difference with the fluid collapse studied by evans and coleman @xcite is the expanding , finite - density - background space 
time of a frw universe . assuming that self - similarity and mass scaling are consequences of an intermediate asymptotic solution that is independent of the asymptotic boundary conditions , eq .
( [ scale ] ) is applicable to pbh masses , allowing the derivation of a universal pbh initial mass function @xcite .
figure ( [ f7 ] ) presents numerical evidence for mass scaling according to eq .
( [ scale ] ) in black hole collapse in an asymptotic frw space  time .
all three perturbation families give rise to scaling solutions with a scaling exponent @xmath28 .
this work discusses numerical collapse simulations of three generic families of energy density perturbations , one with a finite total excess mass with respect to the unperturbed frw solution and two mass compensated ones . among various possible definitions for the collapse control parameter @xmath1 , the total excess gravitational mass of the perturbed space  time with respect to the unperturbed frw background enclosed in the initial horizon volume is the only one that gives rise to a similar threshold value for all three shape families , @xmath29 . whether this result is an indication for universality of @xmath2 in this specific definition needs to be verified with the help of additional simulations using a larger sample of initial perturbation shapes
.    the previously suggested @xcite scaling relation between @xmath3 and @xmath5 , based on the analogy with critical phenomena observed in near - critical black hole collapse in asymptotically non - expanding space 
times @xcite , is confirmed numerically for an asymptotic frw background . for the smallest black holes in this investigation ,
the scaling exponent is @xmath30 , which is identical to the non - expanding numerical and analytical results @xcite within our numerical accuracy .
the parameter @xmath6 of eq .
( [ scale ] ) , needed in addition to @xmath7 to evaluate the two - parameter pbh imf derived in @xcite , ranges from @xmath31 to @xmath32 .
first results of a numerical investigation of primordial black hole formation in the radiation dominated phase of the early universe are presented . 
 the simulations follow the gravitational collapse of three different families of high - amplitude density fluctuations imposed at the time of horizon crossing . 
 the threshold for black hole formation , @xmath0 , is found to be nearly identical for all perturbation families if the control parameter , @xmath1 , is chosen as the total excess mass within the initial horizon volume . 
 furthermore , we demonstrate that the scaling of black hole mass with distance from the formation threshold , known to occur in near - critical gravitational collapse , applies to primordial black hole formation .