Paul R. Shapiro

Interstellar bubbles. II - Structure and evolution

The detailed structure of the interaction of a strong stellar wind with the interstellar medium is presented. First, an adiabatic similarity solution is given which is applicable at early times. Second, a similarity solution is derived which includes the effects of thermal conduction between the hot (about 1 million K) interior and the cold shell of swept-up interstellar matter. This solution is then modified to include the effects of radiative energy losses. The evolution of an interstellar bubble is calculated, including the radiative losses. The quantitative results for the outer-shell radius and velocity and the column density of highly ionized species such as O VI are within a factor 2 of the approximate results of Castor, McCray, and Weaver (1975). The effect of stellar motion on the structure of a bubble, the hydrodynamic stability of the outer shell, and the observable properties of the hot region and the outer shell are discussed.

Hydrogen molecules and the radiative cooling of pregalactic shocks

Detailed results for the hydrodynamical, thermal, ionization, and molecular formation history of postshock cooling flows behind steady state shocks in a primordial gas at redshifts z = 5, 10, and 20 are presented and analyzed for a wide range of shock velocities from 50 to 400 km/s. The nonequilibrium results indicate that, for a significant range of shock velocities, if the shock-heated gas can cool to 10,000 K within the age of the universe, then it quite commonly forms an H2 fraction in excess of 0.001 and cools at nearly constant pressure to less than 100 K. The presence of an external flux of ionizing and dissociating radiation can, for a range of fluxes similar to that expected from a background of quasars, actually increase the peak H2 concentration to values of order 10 to the -1.5 or higher; it also increases the cooling time to 100 K. 78 references.

Simulating Cosmic Reionization at Large Scales I: the Geometry of Reionization

We present the first large-scale radiative transfer simulat ions of cosmic reionization, in a simulation volume of (100 h 1 Mpc) 3 . This is more than a 2 orders of magnitude improvement over previous simulations. We achieve this by combining the results from extremely large, cosmological, N-body simulations with a new, fast and effici ent code for 3D radiative transfer, C 2 -Ray, which we have recently developed. These simulations allow us to do the first numerical studies of the large-scale structure of reionization w hich at the same time, and crucially, properly take account of the dwarf galaxy ionizing sources which are primarily responsible for reionization. In our realization, reionization starts around z � 21, and final overlap occurs by z � 11. The resulting electron-scattering optical depth is in goo d agreement with the firstyear WMAP polarization data. We show that reionization clearly proceeded in an inside-out fashion, with the high-density regions being ionized earli er, on average, than the voids. Ionization histories of smaller-size (5 to 10 comoving Mpc) subregions exabit a large scatter about the mean and do not describe the global reionization history well. This is true even when these subregions are at the mean density of the universe, which shows that small-box simulations of reionization have little predictive power for the evolut ion of the mean ionized fraction. The minimum reliable volume size for such predictions is � 30 Mpc. We derive the power-spectra of the neutral, ionized and total gas density fields and show t hat there is a significant boost of the density fluctuations in both the neutral and the ionized c omponents relative to the total at arcminute and larger scales. We find two populations of H II re gions according to their size, numerous, mid-sized (� 10 Mpc) regions and a few, rare, very large regions tens of Mpc in size. Thus, local overlap on fairly large scales of tens of Mp c is reached by z � 13, when our volume is only about 50% ionized, and well before the global overlap. We derive the statistical distributions of the ionized fraction and ionized gas densi ty at various scales and for the first time show that both distributions are clearly non-Gaussian. All these quantities are critical for predicting and interpreting the observational signals from reionization from a variety of observations like 21-cm emission, Ly-α emitter statistics, Gunn-Peterson optical depth and small-scale CMB secondary anisotropies due to patchy reionization.

Photoevaporation of cosmological minihaloes during reionization

Energy released by a small fraction of the baryons in the Universe, which condensed out while the intergalactic medium (IGM) was cold, dark and neutral, reheated and reionized it by redshift 6, exposing other baryons already condensed into dwarf-galaxy minihaloes to the glare of ionizing radiation. We present the first gas dynamical simulations of the photoevaporation of cosmological minihaloes overtaken by the ionization fronts which swept through the IGM during the reionization epoch in the currently favoured A cold dark matter (ACDM) universe, including the effects of radiative transfer. These simulations demonstrate the phenomenon of I-front trapping inside minihaloes, in which the weak, R-type fronts which travelled supersonically across the IGM decelerated when they encountered the dense, neutral gas inside minihaloes, and were thereby transformed into D-type I-fronts, preceded by shock waves. For a minihalo with virial temperature below 10 4 K, the I-front gradually burned its way through the minihalo which trapped it, removing all of its baryonic gas by causing a supersonic, evaporative wind to blow backwards into the IGM, away from the exposed layers of minihalo gas just behind the advancing I-front. We describe this process in detail, along with some of its observable consequences, for the illustrative case of a minihalo of total mass 10 7 M ○. , exposed to a distant source of ionizing radiation with either a stellar or quasar-like spectrum, after it was overtaken at redshift z = 9 by the weak, R-type I-front which ionized the IGM surrounding the source. For a source at z = 9 which emits 10 56 ionizing photons per second at 1 Mpc (or, equivalently, 10 52 ionizing photons per second at 10 kpc), the photoevaporation of this minihalo takes about 100-150 Myr, depending on the source spectrum, ending at about z = 7.5. Such hitherto neglected feedback effects were widespread during the reionization epoch. N-body simulations and analytical estimates of halo formation in the ACDM model suggest that sub-kpc minihaloes such as these, with virial temperatures below 10 4 K, were so common as to cover the sky around larger-mass source haloes and possibly dominate the absorption of ionizing photons during reionization. This means that previous estimates of the number of ionizing photons per hydrogen atom required to complete reionization which neglected this effect may be too low. Regardless of their effect on the progress of reionization, however, the minihaloes were so abundant that random lines of sight through the high-z Universe should encounter many of them, which suggests that it may be possible to observe the processes described here in the absorption spectra of distant sources.

The H II Region of the First Star

Simulations predict that the first stars in a ΛCDM universe formed at redshifts z > 20 in minihalos with masses of about 106 M☉. We have studied their radiative feedback by simulating the propagation of ionization fronts (I-fronts) created by these first Population III stars (M* = 15-500 M☉) at z = 20, within the density field of a cosmological simulation of primordial star formation, outward through the host minihalo and into the surrounding gas. A three-dimensional ray-tracing calculation tracks the I-front once the H II region evolves a champagne flow inside the minihalo, after the early D-type I-front detaches from the shock and runs ahead, becoming R type. We take account of the hydrodynamic back-reaction by an approximate model of the central wind. We find that the escape fraction of ionizing radiation from the host halo increases with stellar mass, with 0.7 fesc 0.9 for 80 M*/M☉ 500. To quantify the ionizing efficiency of these stars as they begin cosmic reionization, we find that for M* 80 M☉, the ratio of gas mass ionized to stellar mass is ~60,000, roughly half the number of ionizing photons released per stellar baryon. Nearby minihalos are shown to trap the I-front, so their centers remain neutral. This is contrary to the recent suggestion that these stars would trigger formation of a second generation by fully ionizing neighboring minihalos, stimulating H2 formation in their cores. Finally, we discuss how the evacuation of gas from the host halo reduces the growth and luminosity of miniquasars that may form from black hole remnants of the first stars.

Cosmological radiative transfer codes comparison project - I. The static density field tests

Radiative transfer (RT) simulations are now at the forefront of numerical astrophysics. They are becoming crucial for an increasing number of astrophysical and cosmological problems; at the same time their computational cost has come within reach of currently available computational power. Further progress is retarded by the considerable number of different algorithms (including various flavours of ray tracing and moment schemes) developed, which makes the selection of the most suitable technique for a given problem a non-trivial task. Assessing the validity ranges, accuracy and performances of these schemes is the main aim of this paper, for which we have compared 11 independent RT codes on five test problems: (0) basic physics; (1) isothermal H II region expansion; (2) H II region expansion with evolving temperature; (3) I-front trapping and shadowing by a dense clump and (4) multiple sources in a cosmological density field. The outputs of these tests have been compared and differences analysed. The agreement between the various codes is satisfactory although not perfect. The main source of discrepancy appears to reside in the multifrequency treatment approach, resulting in different thicknesses of the ionized-neutral transition regions and the temperature structure. The present results and tests represent the most complete benchmark available for the development of new codes and improvement of existing ones. To further this aim all test inputs and outputs are made publicly available in digital form.

Reionization in a cold dark matter universe: The feedback of galaxy formation on the intergalactic medium

We study the coupled evolution of the intergalactic medium (IGM) and the emerging structure in the universe in the context of the cold dark matter (CDM) model, with a special focus on the consequences of imposing reionization and the Gunn-Peterson constraint as a boundary condition on the model. We have calculated the time-varying density of the IGM by coupling our detailed, numerical calculations of the thermal and ionization balance and radiative transfer in a uniform, spatially averaged IGM of H and He, including the mean opacity of an evolving distribution of gas clumps which correspond to quasar absorption line clouds, to the linearized equations for the growth of density fluctuations in both the gaseous and dark matter components in a CDM universe. We use the linear growth equations to identify the fraction of the gas which must have collapsed out at each epoch, an approach similar in spirit to the so-called Press-Schechter formalism. We identify the IGM density with the uncollapsed baryon fraction. The collapsed fraction is postulated to be a source of energy injection into the IGM, by radiation or bulk hydrodynamical heating (e.g., via shocks) or both, at a rate which is marginally enough to satisfy the Gunn-Peterson constraint at z less than 5. Our results include the following: (1) We find that the IGM in a CDM model must have contained a substantial fraction of the total baryon density of the universe both during and after its reionization epoch. (2) As a result, our previous conclusion that the observed Quasi-Stellar Objects (QSOs) at high redshift are not sufficient to ionize the IGM enough to satisfy the Gunn-Peterson constraint is confirmed. (3) We predict a detectable He II Gunn-Peterson effect at 304(1 + z) A in the spectra of quasars at a range of redshift z greater than or approx. 3, depending on the nature of the sources of IGM reionization. (4) We find, moreover, that a CDM model with high bias parameter b (i.e., b greater than or approx. 2) cannot account for the baryon content of the universe at z approximately 3 observed in quasar absorption line gas unless Omega (sub B) significantly exceeds the maximum value allowed by big bang nucleocynthesis. (5) For a CDM model with bias parameter within the allowed range of (lower) values, the lower limit to Omega(sub B) imposed by big bang nucleosynthesis (Omega(sub B) h(sup 2) greater than or equal to 0.01) combines with our results to yield the minimum IGM density for the CDM fodel. For CDM with b = 1 (Cosmic Background Explorer (COBE) normalization), we find Omega(sub IGM)(sup min) (z approximately 4) approx. equal 0.02-0.03, and Omega(sub IGM)(sup min)(z approximately 0) approx. equal 0.005-0.03, depending upon the nature of the sources of IGM reionization. (6) In general, we find that self-consistent reionization of the IGM by the collapsed baryon fraction has a strong effect on the rate of collapse. (7) As a further example, we show that the feedback effect on the IGM of energy release by the collapsed baryon fraction may explain the slow evolution of the observed comoving QSO number density between z = 5 and z = 2, followed by the sharp decline after z = 2.

C2-ray: A new method for photon-conserving transport of ionizing radiation

Abstract We present a new numerical method for calculating the transfer of ionizing radiation, called C2-ray (conservative, causal ray-tracing method). The transfer of ionizing radiation in diffuse gas presents a special challenge to most numerical methods which involve time- and spatial-differencing. Standard approaches to radiative transport require that grid cells must be small enough to be optically-thin while time steps are small enough that ionization fronts do not cross a cell in a single time step. This quickly becomes prohibitively expensive. We have developed an algorithm which overcomes these limitations and is, therefore, orders of magnitude more efficient. The method is explicitly photon-conserving, so the depletion of ionizing photons by bound-free opacity is guaranteed to equal the photoionizations these photons caused. As a result, grid cells can be large and very optically-thick without loss of accuracy. The method also uses an analytical relaxation solution for the ionization rate equations for each time step which can accommodate time steps which greatly exceed the characteristic ionization and ionization front crossing times. Together, these features make it possible to integrate the equation of transfer along a ray with many fewer cells and time steps than previous methods. For multi-dimensional calculations, the code utilizes short-characteristics ray tracing. The method scales as the product of the number of grid cells and the number of sources. C2-ray is well-suited for coupling radiative transfer to gas and N-body dynamics methods, on both fixed and adaptive grids, without imposing additional limitations on the time step and grid spacing. We present several tests of the code involving propagation of ionization fronts in one and three dimensions, in both homogeneous and inhomogeneous density fields. We compare to analytical solutions for the ionization front position and velocity, some of which we derive here for the first time. As an illustration, we apply C2-ray to simulate cosmic reionization in three-dimensional inhomogeneous cosmological density field. We also apply it to the problem of I-front trapping in a dense clump, using both a fixed and an adaptive grid.

Simulating cosmic reionization at large scales – II. The 21-cm emission features and statistical signals

We present detailed predictions for the redshifted 21-cm signal from the epoch of reionization. These predictions are obtained from radiative transfer calculations on the results of large-scale (100 h -1 Mpc), high dynamic range, cosmological simulations. We consider several scenarios for the reionization history, of both early and extended reionizations. From the simulations, we construct and analyse a range of observational characteristics, from the global signal, via detailed images and spectra, to statistical representations of rms fluctuations, angular power spectra, and probability distribution functions to characterize the non-Gaussianity of the 21-cm signal. We find that the different reionization scenarios produce quite similar observational signatures, mostly differing in the redshifts of 50 per cent reionization, and of final overlap. All scenarios show a gradual transition in the global signatures of mean signal and rms fluctuations, which would make these more difficult to observe. Individual features, such as deep gaps and bright peaks, are substantially different from the mean, and mapping these with several arcminutes and 100 s of kHz resolution would provide a direct measurement of the underlying density field and the geometry of the cosmological H II regions, although significantly modified by peculiar velocity distortions. The presence of late emission peaks suggests these to be a useful target for observations. The power spectra during reionization are strongly boosted compared to the underlying density fluctuations. The strongest statistical signal is found around the time of 50 per cent reionization and displays a clear maximum at an angular scale of l ∼ 3000-5000. We find the distribution function of emission features to be strongly non-Gaussian, with an order of magnitude higher probability of bright emission features. These results suggest that, observationally, it may be easier to find individual bright features than deriving the power spectra, which, in turn, is easier than observing individual images.

Minihalo photoevaporation during cosmic reionization: Evaporation times and photon consumption rates

ABSTRACT The weak, R-type ionization fronts (I-fronts) which swept across the intergalacticmedium (IGM) during the reionization of the universe often found their paths blockedby cosmological minihaloes (haloes with virial temperatures T vir 6 10 4 K). When thishappened, the neutral gas which filled each minihalo was photoevaporated; as theI-front burned its way through the halo, decelerating from R-type to D-type, all thehalo gas was eventually blown back into the IGM as an ionized, supersonic wind. Ina previous paper (Shapiro, Iliev & Raga 2004, hereafter Paper I), we described thisprocess and presented our results of the first simulations of it by numerical gas dy-namics with radiation transport in detail. For illustration we focused on the particularcase of a 10 7 M ⊙ minihalo which is overrun at z = 9 by an intergalactic I-front causedby a distant source of ionizing radiation, for different types of source spectra (eitherstellar from massive Pop. II or III stars, or QSO-like) and a flux level typical of that ex-pected during reionization. In a Cold Dark Matter (CDM) universe, minihaloes formedin abundance before and during reionization and, thus, their photoevaporation is animportant, possibly dominant, feature of reionization, which slowed it down and cost itmany ionizing photons. In view of the importance of minihalo photoevaporation, bothas a feedback mechanism on the minihaloes and as an effect on cosmic reionization,we have now performed a larger set of high-resolution simulations to determine andquantify the dependence of minihalo photoevaporation times and photon consumptionrates on halo mass, redshift, ionizing flux level and spectrum. We use these results toderive simple expressions for the dependence of the evaporation time and photon con-sumption rate on these halo and external flux parameters which can be convenientlyapplied to estimate the effects of minihaloes on the global reionization process in bothsemi-analytical calculations and larger-scale, lower-resolution numerical simulationswhich cannot adequately resolve the minihaloes and their photoevaporation. We findthat the average number of ionizing photons each minihalo atom absorbs during itsphotoevaporation is typically in the range 2-10. For the collapsed fraction in mini-haloes expected during reionization, this can add ≈1 photon per total atom to therequirements for completing reionization, potentially doubling the minimum numberof photons required to reionize the universe.Key words: hydrodynamics—radiative transfer—galaxies: halos—galaxies: high-redshift—intergalactic medium—cosmology: theory