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NASA NTRS1994

Large-scale structure after COBE: Peculiar velocities and correlations of cold dark matter halos

Large N-body simulations on parallel supercomputers allow one to simultaneously investigate large-scale structure and the formation of galactic halos with unprecedented resolution. Our study shows that the masses as well as the spatial distribution of halos on scales of tens of megaparsecs in a cold dark matter (CDM) universe with the spectrum normalized to the anisotropies detected by Cosmic Background Explorer (COBE) is compatible with the observations. We also show that the average value of the relative pairwise velocity dispersion sigma(sub v) - used as a principal argument against COBE-normalized CDM models-is significantly lower for halos than for individual particles. When the observational methods of extracting sigma(sub v) are applied to the redshift catalogs obtained from the numerical experiments, estimates differ significantly between different observation-sized samples and overlap observational estimates obtained following the same procedure.

Zurek, Wojciech H.↗
NASA NTRS1994

On determining the spectrum of primordial inhomogeneity from the COBE DMR sky maps: Results of two-year data analysis

A new technique of Fourier analysis on a cut sky has been applied to the two-year Cosmic Background Explorer (COBE) Differential Microwave Radiometer (DMR) 53 and 90 GHz sky maps. The Bayesian power spectrum estimation results are consistent with the Harrison-Zel'dovich n = 1 model. The maximum likelihood estimates of the usual parameters defining the power spectrum of primordial perturbations are n = 1.22 (1.02) and Q(sub rms-PS) = 17 (20) microK including (excluding) the quadrupole. A spectral-index-independent normalization is naturally expressed for the two-year maps in terms of the multipole amplitude a(sub 9) = 8.2 (8.3) microK (to approximately 12 sigma significance). The marginal likelihood function on n obtained by intergration with respect to a(sub 9) renders n = 1.17 +/- 0.31 (0.96 +/- 0.36).

Gorski, K. M.↗
NASA NTRS1994

COBE DIRBE near-infrared polarimetry of the zodiacal light: Initial results

This Letter describes near-infrared polarimetry of the zodiacal light at 2.2 micrometers, measured with the Diffuse Infrared Background Experiment (DIRBE) aboard the Cosmic Background Explorer (COBE) spacecraft. The polarization is due to scattering of sunlight. The polarization vector is perpendicular to the scattering plane, and its observed amplitude on the ecliptic equator at an elongation of 90 deg and ecliptic longitude of 10 deg declines from 12.0 +/- 0.4% at 1.25 micrometers to 8.0 +/- 0.6% at 3.5 micrometers (cf. 16% in the visible); the principal source of uncertainty is photometric noise due to stars. The observed near-infrared colors at this location are redder than Solar, but at 3.5 micrometers this is due at least in part to the thermal emission contribution from the interplanetary dust. Mie theory calculations show that both polarizations and colors are important in constraining models of interplanetary dust.

Berriman, G. B.↗
NASA NTRS1994

Limits on three-point correlations in the COBE DMR first-year anisotropy maps

We compute the three-point temperature correlation function of the Cosmic Background Explorer (COBE) Differential Microwave Radiometer (DMR) first-year sky maps to search for non-Gaussian temperature fluctuations. The level of fluctuations seen in the computed correlation function are too large to be attributable solely to instrument noise. However the fluctuations are consistent with the level expected to result from a superposition of istrument noise and sky signal arising from a Gaussian power-law model of initial fluctuations, with a quadrupole normalized amplitude of 17 micro K and a power-law spectral index n = 1. We place limits on the amplitude of intrinsic three-point correlations with a variety of predicted functional forms.

Hinshaw, G.↗
NASA NTRS1994

The evolution of voids in the adhesion approximation

We apply the adhesion approximation to study the formation and evolution of voids in the universe. Our simulations-carried out using 128(exp 3) particles in a cubical box with side 128 Mpc-indicate that the void spectrum evolves with time and that the mean void size in the standard Cosmic Background Explorer Satellite (COBE)-normalized cold dark matter (CDM) model with H(sub 50) = 1 scals approximately as bar D(z) = bar D(sub zero)/(1+2)(exp 1/2), where bar D(sub zero) approximately = 10.5 Mpc. Interestingly, we find a strong correlation between the sizes of voids and the value of the primordial gravitational potential at void centers. This observation could in principle, pave the way toward reconstructing the form of the primordialpotential from a knowledge of the observed void spectrum. Studying the void spectrum at different cosmological epochs, for spectra with a built in k-space cutoff we find that the number of voids in a representative volume evolves with time. The mean number of voids first increases until a maximum value is reached (indicating that the formation of cellular structure is complete), and then begins to decrease as clumps and filaments erge leading to hierarchical clustering and the subsequent elimination of small voids. The cosmological epoch characterizing the completion of cellular structure occurs when the length scale going nonlinear approaches the mean distance between peaks of the gravitaional potential. A central result of this paper is that voids can be populated by substructure such as mini-sheets and filaments, which run through voids. The number of such mini-pancakes that pass through a given void can be measured by the genus characteristic of an individual void which is an indicator of the topology of a given void in intial (Lagrangian) space. Large voids have on an average a larger measure than smaller voids indicating more substructure within larger voids relative to smaller ones. We find that the topology of individual voids is strongly epoch dependent, with void topologies generally simplifying with time. This means that as voids grow older they become progressively more empty and have less structure within them. We evaluate the genus measure both for individual voids as well as for the entire ensemble of voids predicted by CDM model. As a result we find that the topology of voids when taken together with the void spectrum is a very useful statistical indicator of the evolution of the structure of the universe on large scales.

Sahni, Varun↗
NASA NTRS1994

Power spectrum constraints from spectral distortions in the cosmic microwave background

Using recent experimental limits on chemical potential distortions from Cosmic Background Explorer (COBE) Far Infrared Astronomy Satellite (FIRAS), and the large lever-arm spanning the damping of sub-Jeans scale fluctuations to the COBE DMR fluctuations, we set a constraint on the slope of the primordial power spectrum n. It is possible to analytically calculate the contribution over the full range of scales and redshifts, correctly taking into account fluctuation growth and damping as well as thermalization processes. Assuming conservatively that mu is less than 1.76 x 10(exp -4), we find that the 95% upper limit on n is only weakly dependent on other cosmological parameters, e.g., n is less than 1.60 (h=0.5) and n is less than 1.63 (h=1.0) for Omega(sub 0) = 1, with marginally weaker constraints for Omega(sub 0) is less than 1 in a flat model with a cosmological constant.

Hu, Wayne↗
NASA NTRS1994

The N (II) 205 micron line in M82: The warm ionized medium

Detection of the 205 micrometer fine structure line of N II in the nearby starburst galaxy M82 is reported. The intensity wihin a 54 sec Full width at Half Maximum (FWHM) beam is (7.1 +/- 1.2) x 10(exp -19) W cm(exp -2). The ratio of the intensity of the recently detected 122 micrometer line to that of the 2.5 micrometer lines is = (4.2) (sup =1.6) (sub -1.2), significantly larger than the corresponding Galactic value of 1.6 +/- 0.3, reflecting higher electron densities within the central 850 pc of M82 in comparison to the Cosmic Background Explorer (COBE) Galactic average. The 2.5 micrometer line profile is consistent with other far-infrared fine-structure line profiles observed in M82. The observations are interpreted in the context of a two-component model of the ionized medium in M82. We find that a component of density as low as approximately 50 cm(exp -3) can comprise up to 70% of the total mass of warm ionized gas within the beam. The balance of the ionized mass is comprised of a component of density approximately greater than 100 cm(exp -3). A model is explored in which the dneser ionized medium constitute the boundaries of neutral surfaces which border the expanding hot plasma from the nuclear region.

Petuchowski, S. J.↗
NASA NTRS1994

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.

Shapiro, Paul R.↗
NASA NTRS1994

The Local Supercluster as a test of cosmological models

The Local Supercluster kinematic properties (the Local Group infall toward the Virgo Cluster and galaxy density distribution about the Virgo Cluster) in various cosmological models are examined utilizing large-scale N-body (PM) simulations 500(exp 3) cells, 250(exp 3) particles, and box size of 400 h(exp -1) Mpc) and are compared to observations. Five models are investigated: (1) the standard, Cosmic Background Explorer Satellite (COBE)-normalized cold dark matter (CDM) model with omega = 1, h = 0.5, and sigma(sub 8) = 1.05; (2) the standard Hot Dark Matter (HDM) model with omega = 1, h = 0.75, and sigma(sub 8) = 1; (3) the tilted CDM model with omega = 1, h = 0.5, n = 0.7, and sigma(sub 8) = 0.5; (4) a CDM + lambda model with omega = 0.3, lambda = 0.7, h = 2/3, and sigma(sub 8) = 2/3; (5) the PBI model with omega = 0.2, h = 0.8, x = 0.1, m = -0.5, and sigma(sub 8) = 0.9. Comparison of the five models with the presently available observational measurements v(sub LG) = 85 - 305 km/s (with mean of 250 km/s), delta(n(sub g))/(n(sub g)-bar) = 1.40 + or - 0.35) suggests that an open universe with omega approximately 0.5 (with or without lambda) and sigma(sub 8) approximately 0.8 is preferred, with omega = 0.3-1.0 (with or without lambda) and sigma(sub 8) = 0.7-1.0 being the acceptable range. At variance with some previous claims based on either direct N-body or spherical nonlinear approaches, we find that a flat model with sigma(sub 8) approximately 0.7-1.0 seems to be reasonably consistent with observations. However, if one favors the low limit of v(sub LG) = 85 km/s, then an omega approximately 0.2-0.3 universe seems to provide a better fit, and flat (omega = 1) models are ruled out at approximately 95% confidence level. On the other hand, if the high limit of v(sub LG) = 350 km/s is closer to the truth, then it appears that omega approximately 0.7-0.8 is more consistent. This test is insensitive to the shape of the power spectrum, but rather sensitive to the normalization of the perturbation amplitude on the relevant scale (e.g., sigma(sub 8)) and omega. We find that neither linear nor nonlinear relations (with spherical symmetry) are good approximations for the relation between radial peculiar velocity and density perturbation, i.e., nonspherical effects and gravitational tidal field are important. The derived omega using either of the two relations is underestimated. In some cases, this error is as large as a factor of 2-4.

Cen, Renyue↗
NASA NTRS1994

COBE diffuse infrared background experiment observations of the galactic bulge

Low angular resolution maps of the Galactic bulge at 1.25, 2.2, 3.5, and 4.9 micrometers obtained by the Diffuse Infrared Background Experiment (DIRBE) onboard NASA's Cosmic Background Explorer (COBE) are presented. After correction for extinction and subtraction of an empirical model for the Galactic disk, the surface brightness distribution of the bulge resembles a flattened ellipse with a minor-to-major axis ratio of approximately 0.6. The bulge minor axis scale height is found to be 2.1 deg +/- 0.2 deg for all four near-infrared wavelengths. Asymmetries in the longitudinal distribution of bulge brightness contours are qualitatively consistent with those expected for a triaxial bar with its near end in the first Galactic quadrant (0 deg less than l less than 90 deg). There is no evidence for an out-of-plane tilt of such a bar.

Weiland, J. L.↗
NASA NTRS1994

COBE diffuse infrared background experiment observations of Galactic reddening and stellar populations

This Letter describes the results of an initial study of Galactic extinction and the colors of Galactic stellar populations in the near-IR using the Diffuse Infrared Background Experiment (DIRBE) aboard the Cosmic Background Explorer (COBE) spacecraft. The near-IR reddening observed by DIRBE is consistent with the extinction law tabulated by Rieke & Lebofsky (1985). The distribution of dust and stars in most of the first and fourth quadrants of the Galactic plane (0 deg less than l less than 90 deg, and 270 deg less than l less than 360 deg, respectively) can be modeled as a stellar background source seen through up to approximately 4 mag of extinction at 1.25 micrometers. The unreddened near-IR colors of the Galactic disk are similar to those of late-K and M giants. The Galactic bulge exhibits slightly bluer colors in the 2.2-3.5 micrometers range, as noted by Terndrup et al. (1991). Star-forming regions exhibit colors that indicate the presence of a approximately 900 K continuum produced by hot dust or polycyclic aromatic hydrocarbons (PAHs) contributing at wavelengths as short as 3.5 micrometers.

Arendt, R. G.↗
NASA NTRS1994

The motions of clusters and group of galaxies

The distributions of peculiar velocities of rich clusters and of groups of galaxies are investigated for different cosmological models and are compared with observations. Four cosmological models are studied: standard cold dark matter (CDM) (omega = 1); low-density CDM (omega = 0.3); hot dark matter (HDM) (omega = 1); and primeval baryonic isocurvature (PBI) (omega = 0.3). All models are normalized to the microwave background fluctuations observed by Cosmic Background Explorer (COBE). We find that rich clusters of galaxies exhibit a Maxwellian distribution of peculiar velocities in all models, as expected from a Gaussian initial density field. The clusters appear to be fundamental and efficient tracers of the large-scale velocity field. The cluster three-dimensional velocity distribution typically peaks at v approximately 600 km/s and extends to high cluster velocities of v approximately 2000 km/s. The low-density CDM model exhibits somewhat lower velocities: it peaks at approximately 400 km/s and extends to approximately 1200 km/s. Approximately 10% (approximately 1% for low-density CDM) of all model rich clusters move with high peculiar velocities of V greater than or = 10(exp 3) km/s. The highest velocity clusters frequently originate in dense superclusters. The model velocity distributions of rich clusters are compared with the model velocity distributions of small groups of galaxies, and of the total matter. The group velocity distribution is, in general, similar to the velocity distribution of the rich clusters. The matter velocity distribution is similar to that of the rich clusters for the omega = 0.3 models; these models exhibit Maxwellian velocity distributions for clusters, for groups, and for matter that are all similar to one another. The mass distribution in the omega = 1 models, however, exhibits a longer tail of high velocities than do the clusters. This high-velocity tail originates mostly from the high velocities that exist within rich clusters. The model velocity distributions of groups and clusters of galaxies are compared with observations. The data exhibit a larger high-velocity tail, to v(sub r) greater than or = 2000 km/s, than is seen in the model simulations (except for HDM). Because of the large observational uncertainties, however, the data are consistent at a approximately 1-3 sigma level with the model predictions and with a Gaussian initial density field. Accurate observations of cluster peculiar velocities, especially at the high-velocity tail, shoud provide powerful constraints on the cosmological models.

Bahcall, Neta A.↗
NASA NTRS1994

X-ray clusters in a cold dark matter + lambda universe: A direct, large-scale, high-resolution, hydrodynamic simulation

A new, three-dimensional, shock-capturing, hydrodynamic code is utilized to determine the distribution of hot gas in a cold dark matter (CDM) + lambda model universe. Periodic boundary conditions are assumed: a box with size 85/h Mpc, having cell size 0.31/h Mpc, is followed in a simulation with 270(exp 3) = 10(exp 7.3) cells. We adopt omega = 0.45, lambda = 0.55, h identically equal to H/100 km/s/Mpc = 0.6, and then, from the cosmic background explorer (COBE) and light element nucleosynthesis, sigma(sub 8) = 0.77, omega(sub b) = 0.043. We identify the X-ray emitting clusters in the simulation box, compute the luminosity function at several wavelength bands, the temperature function and estimated sizes, as well as the evolution of these quantities with redshift. This open model succeeds in matching local observations of clusters in contrast to the standard omega = 1, CDM model, which fails. It predicts an order of magnitude decline in the number density of bright (h nu = 2-10 keV) clusters from z = 0 to z = 2 in contrast to a slight increase in the number density for standard omega = 1, CDM model. This COBE-normalized CDM + lambda model produces approximately the same number of X-ray clusters having L(sub x) greater than 10(exp 43) erg/s as observed. The background radiation field at 1 keV due to clusters is approximately the observed background which, after correction for numerical effects, again indicates that the model is consistent with observations.

Cen, Renyue↗
NASA NTRS1994

Angular power spectrum of the microwave background anisotropy seen by the COBE differential microwave radiometer

The angular power spectrum estimator developed by Peebles (1973) and Hauser & Peebles (1973) has been modified and applied to the 2 yr maps produced by the Cosmic Background Explorer Satellite Differential Microwave Radiometer (COBE DMR)). The power spectrum of the real sky has been compared to the power spectra of a large number of simulated random skies produced with noise equal to the observed noise and primordial density fluctuation power spectra of power-law form, with P(k) proportional to k(exp n). Within the limited range of spatial scales covered by the COBE DMR, corresponding to spherical harmonic indices 3 less than or = l is less than or approximately = 30, the best-fitting value of the spectral index is n = 1.25(sup +0.39 sub -0.44) with the Harrisson-Zel'dovich value n = 1 approximately 0.5 sigma below the best fit. For 3 less than or = l less than or approximately = 19, the best fit is n = 1.46(sup +0.39 sub -0.44). Comparing the COBE DMR delta-T/T at small l to the delta-T/T at l approximately = 50 from degree scale anisotropy experiments gives a smaller range of acceptable spectral indices which includes n = 1.

Wright, E. L.↗
NASA NTRS1994

Correlated noise in the COBE DMR sky maps

The Cosmic Background Explorer Satellite Differential Radiometer (COBE DMR) sky maps contain low-level correlated noise. We obtain estimates of the amplitude and pattern of the correlated noise from three techniques: angular averages of the covariance matrix, Monte Carlo simulations of two-point correlation functions and direct analysis of the DMR maps. The results from the three methods are mutually consistent. The noise covariance matrix of a DMR sky maps is diagonal to an accuracy of better than 1%. For a given sky pixel, the dominant noise covariance occure with the ring of pixels at an angular separation of 60 deg due to the 60 deg separation of the DMR horns. The mean covariance at 60 deg is 0.45%((sup +0.18)(sub -0.14)) of the mean variance. Additionally, the variance in a given pixel is 0.7% greater than would be expected from a single beam experiment with the same noise properties. Autocorrelation functions suffer from a approximately 1.5 sigma positive bias at 60 deg while cross-correlations have no bias. Published COBE DMR results are not significantly affected by correlated noise.

Lineweaver, C. H.↗
NASA NTRS1994

Large-scale structure effects on the gravitational lens image positions and time delay

We compute the fluctuations in gravitational lens image positions and time delay caused by large-scale structure correlations. We show that these fluctuations can be expressed as a simple integral over the density power spectrum. Using the Cosmic Background Explorer (COBE) normalization we find that positions of objects at cosmological distances are expected to deviate from their true positions by few arcminutes. These deflections are not directly observable. The positions of the images relative to one another fluctuate by a few percent of the relative separation, implying that one does not expect multiple images to be produced by large-scale structure. Nevertheless, the fluctuations are larger than the observational errors on the positions and affect reconstructions of the lens potential. The time delay fluctuations have a geometrical and a gravitational contribution. Both are much larger than the expected time delay from the primary lens, but partially cancel each other. We find that large-scale structure weakly affects the time delay and time delay measurements can be used as a probe of the distance scale in the universe.

Seljak, Uros↗
NASA NTRS1994

On the determination of the cosmic infrared background radiation from the high-energy spectrum of extragalactic gamma-ray sources

In a recent paper Stecker, De Jager, & Salamon have suggested using the observed approximately MeV to TeV spectra of extragalactic gamma-ray sources as probes of the local density of the cosmic infrared background radiation (CIBR) and have subsequently claimed a first possible measurement of the CIBR from the analysis of the gamma-ray spectrum of Mrk 421 (De Jager, Stecker, & Salamon). The CIBR from normal galaxies consists of two components: a stellar emission component (CIBRs), and a thermal dust emission component (CIBRd). Photons with energies in the approximately 0.1-2 TeV range interact primarily with the CIBRs, whereas interactions with CIBRd dominate the absorption of photons in the approximately 2-100 TeV energy range. SDS 92 and DSS94 considered only the interaction of the gamma-rays with the dust emission component of the CIBR. We present here an improved analysis of the absorption of extragalactic TeV gamma rays by the CIBR, taking the dual nature of its origin into account. Applying the analysis to the observed gamma-ray spectrum of Mrk 421, a BL Lac object at z = 0.031, we find agreement with DSS94 tentative evidence for absorption by the CINRs. Our analysis therefore limits the detection of the CIBR to the approximately 15-40 micron wavelength regime which, considering the uncertainties in the highest energy (greater than 4 TeV) data and ion the possibility of absorption inside the source, many turn out to be an upper limit on its energy density. At shorter wavelengths (lambda approximately = 1-15 microns), where the gamma-ray interactions are dominated by the CIBRs, our analysis definitely yields only an upper limit on the energy density of the CIBR. In contrast, DSS94 have claimed a possible first measurement of the CIBR over the entire 1-120 micron wavelength region. The upper limit on the CIBRs and tentative detection of the CIBRd are consistent with normal galaxies contributing most of the energy to the CIBR, and constrain the contribution of some exotic sources. With careful modeling of infrared foreground emissions, these constraints on the CIBR are above the values measurable by the DIRBE experiment on board the Cosmic Background Explorer (COBE) satellite.

Dwek, Eli↗
NASA NTRS1994

Compton scattering of the microwave background by quasar-blown bubbles

At least 10% of quasars drive rapid outflows from the central regions of their host galaxies. The mass and energy flow rates in these winds are difficult to measure, but their kinetic luminosities probably exceed 10(exp 45) ergs/s. This kind of outflow easily sunders the interstellar medium of the host and blows a bubble in the intergalactic medium. After the quasar shuts off, the hot bubble continues to shock intergalactic gas until its leading edge merges with the Hubble flow. The interior hot gas Compton scatters microwave background photons, potentially providing a way to detect these bubbles. Assuming that quasar kinetic luminosities scale with their blue luminosities, we integrate over the quasar luminosity function to find the total distortion (y) of the microwave background produced by the entire population of quasar wind bubbles. This calculation of y distortion is remarkably insensitive to the properties of the intergalactic medium (IGM), quasar lifetimes, and cosmological parameters. Current Cosmic Background Explorer (COBE) limits on y constrain the kinetic luminosities of quasars to be less than several times their bolometric radiative luminosities. Within this constraint, quasars can still expel enough kinetic luminosity to shock the entire IGM by z = 0, but cannot heat and ionize the IGM by z = 4 unless omega(sub IGM) much less than 10(exp -2).

Voit, G. Mark↗