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We study the cosmological power spectra (PS) of the differential and integral galaxy volume number densities $\gamma_i$ and $\gamma_i^{*}$, constructed with the cosmological distances $d_i$ $(i=A,G,L,Z)$, where $d_A$ is the angular diameter distance, $d_G$ is the galaxy area distance, $d_L$ is the luminosity distance and $d_z$ is the redshift distance. Theoretical and observational quantities were obtained in the FLRW spacetime with a non-vanishing $\Lambda$. The radial correlation $\Xi_i$, as defined in the context of these densities, is discussed in the wave number domain. All observational quantities were computed using luminosity function (LF) data obtained from the FORS Deep Field galaxy survey. The theoretical and observational PS of $\gamma_i$, $\gamma_i^{\ast}$, $\Xi_i$ and $\gamma_i / \gamma_i^\ast$ were calculated by performing Fourier transforms on these densities previously derived by Iribarrem et al. (2012) from the observed values $\gamma_{obs}$ and ${\gamma^\ast}_{obs}$ obtained using the galactic absolute magnitudes and galaxy LF Schechter's parameters presented in Gabasch et al. (2004, 2006) in the range $0.5 \le z \le5.0$. The results show similar behavior of the PS obtained from $\gamma$ and $\gamma^{\ast}$ using $d_L$, $d_z$ and $d_G$ as distance measures. The PS of the densities defined with $d_A$ have a different and inconclusive behavior, as this cosmological distance reaches a maximum at $z\approx 1.6$ in the adopted cosmology. For the other distances, our results suggest that the PS of ${\gamma_i}_{obs}$, ${\gamma^\ast_i}_{obs}$ and ${\gamma_i / \gamma^{\ast}_i}_{obs}$ have a general behavior approximately similar to the PS obtained with the galaxy two-point correlation function and, by being sample size independent, they may be considered as alternative analytical tools to study the galaxy distribution., Comment: In memory of William R. Stoeger (1943-2014). LaTeX. 15 pages, 15 figures, 35 graphs. Additional comments at the end of Secs. 2.3, 3.2, 4, 5 and caption of Fig. 2. Accepted for publication in "The European Physical Journal Plus"

This article reviews, and offers supportive reflections on, the main points of Ernan McMullin's provocative 1998 article, “Cosmic Purpose and the Contingency of Human Evolution,’’ reprinted in this issue of Zygon. In it he addresses the important science-theology issue of how the Creator's purpose and intention to assure the emergence of human beings is consonant with the radical contingency of the evolutionary process. After discussing cosmic and biological evolution and critically summarizing recent solutions to this question by Keith Ward, John Polkinghorne, Arthur Peacocke, Alvin Plantinga, and others, who presuppose in different ways that God is subject to time, McMullin compellingly argues for the traditional position, that God is unconditioned by time, and this enables God to work purposefully through contingency, randomness, and chance just as easily as through law-like regularity.

The present-day Universe appears to be homogeneous on very large scales. Yet when the casual structure of the early Universe is considered, it becomes apparent that the early Universe must have been highly inhomogeneous. The current paradigm attempts to answer this problem by postulating the inflation mechanism However, inflation in order to start requires a homogeneous patch of at least the horizon size. This paper examines if dynamical processes of the early Universe could lead to homogenization. In the past similar studies seem to imply that the set of initial conditions that leads to homogenization is of measure zero. This essay proves contrary: a set of initial conditions for spontaneous homogenization of cosmological models can form a set of non-zero measure., 7 pages. Fifth Award in the 2010 Gravity Research Foundation essay competition

The causal limit usually considered in cosmology is the particle horizon, delimiting the possibilities of causal connection in the expanding universe. However it is not a realistic indicator of the effective local limits of important interactions in spacetime. We consider here the matter horizon for the Solar System, that is,the comoving region which has contributed matter to our local physical environment. This lies inside the effective domain of dependence, which (assuming the universe is dominated by dark matter along with baryonic matter and vacuum-energy-like dark energy) consists of those regions that have had a significant active physical influence on this environment through effects such as matter accretion and acoustic waves. It is not determined by the velocity of light c, but by the flow of matter perturbations along their world lines and associated gravitational effects. We emphasize how small a region the perturbations which became our Galaxy occupied, relative to the observable universe -- even relative to the smallest-scale perturbations detectable in the cosmic microwave background radiation. Finally, looking to the future of our cosmic domain, we suggest simple dynamical criteria for determining the present domain of influence and the future matter horizon. The former is the radial distance at which our local region is just now separating from the cosmic expansion. The latter represents the limits of growth of the matter horizon in the far future., 18 pages and 2 figures

In some well-known scenarios of open-universe eternal inflation, developed by Vilenkin and co-workers, a large number of universes nucleate and thermalize within the eternally inflating mega-universe. According to the proposal, each universe nucleates at a point, and therefore the boundary of the nucleated universe is a space-like surface nearly coincident with the future light cone emanating from the point of nucleation, all points of which have the same proper-time. This leads the authors to conclude that at the proper-time t = t_{nuc} at which any such nucleation occurs, an infinite open universe comes into existence. We point out that this is due entirely to the supposition of the nucleation occurring at a single point, which in light of quantum cosmology seems difficult to support. Even an infinitesimal space-like length at the moment of nucleation gives a rather different result -- the boundary of the nucleating universe evolves in proper-time and becomes infinite only in an infinite time. The alleged infinity is never attained at any finite time., 13 pages and 6 figures

We study the galaxy cosmological mass function (GCMF) in a semi-empirical relativistic approach using observational data provided by galaxy redshift surveys. Starting from the theory of Ribeiro & Stoeger (2003, arXiv:astro-ph/0304094) between the mass-to-light ratio, the selection function obtained from the luminosity function (LF) data and the luminosity density, the average luminosity $L$ and the average galactic mass $\mathcal{M}_g$ are computed in terms of the redshift. $\mathcal{M}_g$ is also alternatively estimated by a method that uses the galaxy stellar mass function (GSMF). Comparison of these two forms of deriving the average galactic mass allows us to infer a possible bias introduced by the selection criteria of the survey. We used the FORS Deep Field galaxy survey sample of 5558 galaxies in the redshift range $0.5 < z < 5.0$ and its LF Schechter parameters in the B-band, as well as this sample's stellar mass-to-light ratio and its GSMF data. Assuming ${\mathcal{M}_{g_0}} \approx 10^{11} \mathcal{M}_\odot$ as the local value of the average galactic mass, the LF approach results in $L_{B} \propto (1+z)^{(2.40 \pm 0.03)}$ and $\mathcal{M}_g \propto (1+z)^{(1.1\pm0.2)}$. However, using the GSMF results produces $\mathcal{M}_g \propto (1+z)^{(-0.58 \pm 0.22)}$. We chose the latter result as it is less biased. We then obtained the theoretical quantities of interest, such as the differential number counts, to calculate the GCMF, which can be fitted by a Schechter function. The derived GCMF follows theoretical predictions in which the less massive objects form first, being followed later by more massive ones. In the range $0.5 < z < 2.0$ the GCMF has a strong variation that can be interpreted as a higher rate of galaxy mergers or as a strong evolution in the star formation history of these galaxies., In memory of William R. Stoeger (1943-2014). LaTeX, 8 pages, 7 figures. Minor changes to match version sent to publisher. To appear in "Astronomy and Astrophysics"

Using the framework for solving the spherically symmetric field equations in observational coordinates given in Araujo and Stoeger (1999), their formulation and solution in the perturbed FLRW sperically symmetric case with observational data representing galaxy redshifts, number counts and observer area distances, both as functions of redshift on our past light cone, are presented. The importance of the central conditions, those which must hold on our world line C, is emphasized. In detailing the solution for these perturbations, we discuss the gauge problem and its resolution in this context, as well as how errors and gaps in the data are propagated together with the genuine perturbations. This will provide guidance for solving, and interpreting the solutions of the more complicated general perturbation problem with observational data on our past light cone., Comment: Latex 23 pages, no figures, submitted to Astrophysical Journal

In this paper we use the integration scheme for the exact spherically symmetric Einstein field equations in observational coordinates recently developed by Araujo and Stoeger, and input data functions for area distance and galaxy number counts that would be obtained if the universe were exactly represented by nonflat Friedmann-Lemaitre-Robertson-Walker models.

The goal of this work is to compute the number density of far-IR selected galaxies in the comoving frame and along the past lightcone of observationally constrained LTB models and to compare those results with their standard model counterparts. We derived integral number densities and differential number densities using different cosmological distance definitions in the LTB models. Then, we computed selection functions and consistency functions for the luminosity functions in the combined fields of the Herschel/PACS evolutionary probe (PEP) survey in both standard and void cosmologies, from which we derived the observed values of the above-mentioned densities. We used the Kolmogorov-Smirnov statistics to study both the evolution of the consistency functions and its connection to the evolution of the comoving density of sources. Finally, we fitted the power-law behaviour of the densities along the observer's past lightcone. The analysis of the comoving number density shows that the increased flexibility of the LTB models is not enough to fit the observed redshift evolution of the number counts, if it is specialised to a recent best-fit giant void parametrisation. The results for the power-law fits of the densities along the observer's past lightcone show general agreement across both cosmological models studied here around a slope of -2.5 $\pm$ 0.1 for the integral number density on the luminosity-distance volumes. The differential number densities show much bigger slope discrepancies. We conclude that the differential number densities on the observer's past lightcone were still rendered dependent on the cosmological model by the flux limits of the PEP survey. In addition, we show that an intrinsic evolution of the sources must be assumed to fit the comoving number-density redshift evolution in the giant void parametrisation for the LTB models used in this work., 11 pages, 6 figures. Accepted for publication in A&A

This paper studies intermediate homogenization of inhomogeneous cosmological models. It shows that spherically symmetric models, regardless of the equation of state, can undergo intermediate homogenization, i.e. a model can approach a homogeneous and isotropic state (which acts as a saddle point) from a relatively wide range of initial inhomogeneous conditions. The homogenization is not permanent - just temporary. Eventually the model evolves toward a future inhomogeneous state. We also looked at the problem of the gravitational entropy. All definitions of entropy that we checked give decreasing gravitational entropy during the homogenization process. Thus, we should either accept that gravitational entropy can decrease or try to define it in other ways than just via density gradients, as these decrease during homogenization., 8 pages, 3 figures, accepted for publication in Phys. Rev. D

Recently we presented limits on the deviation of the Universe from a Friedmann-Robertson-Walker form, which are implied directly by anisotropies in the cosmic background radiation, without assuming inflationary or other models of evolution. Here we weaken some of our previous assumptions, ensuring that all assumptions are in principle observational, and obtain improved limits.

This paper discusses a connection between the relativistic number counts of cosmological sources and the observed galaxy luminosity function (LF). Observational differential number densities are defined and obtained from published LF data using such connection. We observe a distortion in the observational quantities that increases with higher redshift values as compared to the theoretical predictions. The use of different cosmological distance measures plays a role in such a distortion, Comment: 3 pages, 3 figures. Abridged version of arXiv:1201.5571

This paper studies the connection between the relativistic number density of galaxies down the past light cone in a Friedmann-Lemaitre-Robertson-Walker spacetime with non-vanishing cosmological constant and the galaxy luminosity function (LF) data. It extends the redshift range of previous results presented in Albani et al. (2007:astro-ph/0611032) where the galaxy distribution was studied out to z=1. Observational inhomogeneities were detected at this range. This research also searches for LF evolution in the context of the framework advanced by Ribeiro and Stoeger (2003:astro-ph/0304094), further developing the theory linking relativistic cosmology theory and LF data. Selection functions are obtained using the Schechter parameters and redshift parametrization of the galaxy luminosity functions obtained from an I-band selected dataset of the FORS Deep Field galaxy survey in the redshift range 0.5, Comment: 14 pages, 12 figures, 6 tables, LaTeX. Minor changes to match corrected proofs. Accepted for publication in "Astronomy and Astrophysics"

For many years now theoretical cosmologists have been building on the rich legacy of Georges Lemaitre’s ideas and accomplishments in mathematically modelling the structure and dynamics of the universe. His recognition and careful demonstration of the viability and attractiveness of expanding models of the universe, along with his seminal idea of the “primeval atom,” which would later be referred to as the Big Bang, are very well known. This has led directly, through other important contributions by Friedmann, Robertson and Walker, to the standard perfectly isotropic and spatially homogeneous (smooth) Friedmann-Lemaitre-Robertson-Walker (FLRW) models of our universe. These are certainly a key part of Lemaitre’s legacy. After reviewing those models, we discuss another very important, but less well-known, contribution to fundamental cosmology, his general spherically symmetric cosmological solutions with pressure and a cosmological constant (vacuum energy, which is a leading candidate for dark energy). These models are generalizations of the FLRW models and, in general, are inhomogeneous. In recent years they have stimulated a great deal of research in the quest for further confirming the standard model, understanding the limitations of its perturbed versions, or possibly replacing it with something more adequate. Their importance for advancing our understanding and modelling of the universe, and the ways in which they are presently being studied and deployed, is described and discussed.

In this paper we show that the mass-energy density of the Universe can be fully determined in terms of the cosmological redshifts, their time drifts and angular-diameter distance (observer area distance). Besides providing an indirect measurement of the mass-energy density of the Universe, we show how one can use the time-drift of the cosmological redshifts as a replacement for the mass-energy density element in the minimally required data set to construct an spherically symmetric Lema\^{\i}tre-Tolman-Bondi (LTB) model for the Universe in observational coordinates., Comment: 17 pages

Through astronomy, physics, cosmology, biology and the other sciences we have come to understand a great deal about nature and our amazing universe. We can sketch their main features in very broad strokes – evolution, relationality, nested levels of organization, complexity and potentiality for life and consciousness, contingency, transience and fragility, and order with some directionality. The universe itself has developed from a very simple, homogeneous system into one of incredible complexity – through the efficacy of the interactions which dominate it, and the emerging, highly differentiated relationships which operate within the macroscopic and microscopic systems and subsystems which form within it. But the universe, life and we ourselves are sources of wonder and inspiration – there is an enormous surplus of understanding, meaning, and significance which is nourished by scientific understanding but transcends it. They present themselves as inexhaustibly rich and deep – ultimately as Mystery. Though scientific rationality encounters limits to its capabilities to probe this Mystery more deeply, its incredible success within its competencies points to and validates our rational and personal quest for understanding and meaning beyond the cosmological limit – through the arts, philosophy, theology and the personal engagement from which they arise.

Over the past ninety years we have come to understand and appreciate the world, the universe which embraces it, and their emergence and development in completely new ways. Thanks to astronomy and physics - particularly to the speciality known as cosmology - we now know that the universe we inhabit began expanding and cooling from an extremely hot and dense, homogeneous, simple state about 13.7 billion years ago. That initial state, often now referred to as the Planck era, was so extreme that our current physics is completely unable to describe it. Space and time as we know them had not yet emerged, and the fundamental forces of gravity, electromagnetism, and the strong and weak nuclear interactions were undoubtedly unified, and thus indistinguishable from one another. Only a thorough and complete quantum description of reality, including space-time and gravity - a quantum cosmology - would be adequate. That is something we do not yet possess, although many people are expending tremendous efforts to develop the components of such a description by exploring superstrings, loop-quantum gravity and non-commutative geometry, as well as exploiting semi-classical approaches to quantizing space-time, gravity and the universe itself.

We systematically review the solutions of the vacuum Einstein equations for the axially symmetric stationary case which are harmonic maps. In particular, we show that the interesting part of the Kerr solution is a composition of a harmonic map intoH12 with a totally geodesic map fromH12 into SS(1,1). We also point out, relying on Sanchez' results, that there is an analogous structure for the Lorentz-domain cases involving cylindrical gravitational waves and colliding plane waves.

The authors describe the fluid-ray (FR) tetrad, which is based on the four-velocity u of the cosmological fluid and the null vector k which lies along the generators of the null geodesics in the past light cones C-(p0) centred on the worldline C. Then, they formulate the field equations in terms of this tetrad, giving also the Jacobi identities, the metric variable equations for observational coordinates, and the contracted Bianchi identities. The equivalent set of equations, involving the Ricci identities, the full Bianchi identities, along with the contracted Bianchi identities, is also given. This formulation facilitates the integration of the field equations in certain cosmological contexts.

The authors give the exact spherically symmetric solution of the Einstein field equations for dust (p=0) in observational coordinates for data on the past light cone C-(p0) consisting of redshifts, observer area distances, and galaxy number counts. A Bondi potential is found, which facilitates the integration. Also they show in detail how FLRW models can be constructed via these solutions from FLRW data. Finally, they discuss their approach with respect to other observationally oriented treatments, particularly those of Ehlers and Rindler (1987), and Rindler and Suson (1989).

The author intends this chapter as a "trial balloon". The chapter sketches God's action can be described in the world, accepting with critical seriousness both the present and projected knowledge of reality as from the sciences, philosophy and other non-theological disciplines, and the present knowledge of God, his/her relationship with people and world, and his/her activity within it. From revelation, and partially from reason, it is know that God exists, created the universe and all that is in it, reveals him/herself to people, loves and cares for people, continually acts within material creation, particularly now through Jesus and the presence of his spirit-a promise which will be fulfilled only after our deaths. There are a series of unsolved problems related to divine action, which flow from this discussion of the impossibility of adequately articulating or modeling God's direct action towards a creature. Keywords: God's direct action; revelation; scientific knowledge of reality

Extending the approach developed by Ara\'ujo and Stoeger [1] and improved in Ara\'ujo {\it et al} [2], we have shown how to construct dust-filled $\Lambda \neq 0$ Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) cosmological models from FLRW cosmological data on our past light cone. Apart from being of interest in its own right -- demonstrating how such data fully determines the models -- it is also illustrated in the flat case how the more general spherically symmetric (SS) Einstein field equations can be integrated in observational coordinates with data fit to FLRW forms arrayed on our past light cone, thus showing how such data determines a FLRW universe -- which is not {\it a priori} obvious. It is also shown how to integrate these exact SS equations, in cases where the data are not FLRW, and the space-time is not known to be flat. It is essential for both flat and non-flat cases to have data giving the maximum of the observer area (angular-diameter) distance, and the redshift $z_{max}$ at which that occurs. This enables the determination of the vacuum-energy density $\mu_{\Lambda}$, which would otherwise remain undetermined., Comment: 20 pages

We introduce a simple and straight-forward averaging procedure, which is a generalization of one which is commonly used in electrodynamics, and show that it possesses all the characteristics we require for linearized averaging in general relativity and cosmology -- for weak-field and perturbed FLRW situations. In particular we demonstrate that it yields quantities which are approximately tensorial in these situations, and that its application to an exact FLRW metric yields another FLRW metric, to first-order in integrals over the local coordinates. Finally, we indicate some important limits of any linearized averaging procedure with respect to cosmological perturbations which are the result of averages over large amplitude small and intermediate scale inhomogeneities, and show our averaging procedure can be approximately implemented by that of Zotov and Stoeger in these cases., Comment: 17 pages, prl revtex style

This paper uses data obtained from the galaxy luminosity function (LF) to calculate two types of radial number densities statistics of the galaxy distribution as discussed in Ribeiro (2005), namely the differential density $\gamma$ and the integral differential density $\gamma^\ast$. By applying the theory advanced by Ribeiro and Stoeger (2003), which connects the relativistic cosmology number counts with the astronomically derived LF, the differential number counts $dN/dz$ are extracted from the LF and used to calculate both $\gamma$ and $\gamma^\ast$ with various cosmological distance definitions, namely the area distance, luminosity distance, galaxy area distance and redshift distance. LF data are taken from the CNOC2 galaxy redshift survey and $\gamma$ and $\gamma^\ast$ are calculated for two cosmological models: Einstein-de Sitter and an $\Omega_{m_0}=0.3$, $\Omega_{\Lambda_0}=0.7$ standard cosmology. The results confirm the strong dependency of both statistics on the distance definition, as predicted in Ribeiro (2005), as well as showing that plots of $\gamma$ and $\gamma^\ast$ against the luminosity and redshift distances indicate that the CNOC2 galaxy distribution follows a power law pattern for redshifts higher than 0.1. These findings bring support to Ribeiro's (2005) theoretical proposition that using different cosmological distance measures in statistical analyses of galaxy surveys can lead to significant ambiguity in drawing conclusions about the behavior of the observed large scale distribution of galaxies., Comment: LaTeX, 37 pages, 6 tables, 10 figures. Accepted for publication in "The Astrophysical Journal"

Not only Einstein’s general ideas about gravity, but also his specific field equation (E) have been excellently confirmed in astronomy. Hence one also applies it to other macroscopic problems (hydrodynamic stellar models) and to quantum gravity. There, however, (E) can only be tested in a tiny region just off the mass shell K2 = 0 of free gravitons. One nevertheless believes (E) to hold even at ‘high energies’ (the popular expression for large, Lorentz‐invariant √|K2|), perhaps for all √|K2| < MPl with Planck’s MPl = (8π GNewton)−1/2. We rather accept (E) only as the infrared limit of a more basic equation (W), which involves Weyl’s conformal curvature tensor. We cannot justify (W) or specify its behavior at low and high K by observations or experiments; only connections with particle theories can help. We thus have examined the effect of gravity on the infrared structure of the electron. As a most striking and firm conclusion, we find the huge M Pl totally irrelevant at high ‘energies’. It merely arises ...

This paper aims to connect the theory of relativistic cosmology number counts with the astronomical data, practice, and theory behind the galaxy luminosity function (LF). We treat galaxies as the building blocks of the Universe, but ignore most aspects of their internal structures by considering them as point sources. However, we do consider general morphological types in order to use data from galaxy redshift surveys, where some kind of morphological classification is adopted. We start with a general relativistic treatment for a general spacetime, and then link the derived expressions with the LF definition adopted in observational cosmology. Then equations for differential number counts, the related relativistic density per source, and observed and total relativistic energy densities of the universe, and other related quantities are written in terms of the luminosity and selection functions. As an example of how these theoretical/observational relationships can be used, we apply them to test the LF parameters determined from the CNOC2 galaxy redshift survey, for consistency with the Einstein-de Sitter (EdS) cosmology, which they assume, for intermediate redshifts. We conclude that there is a general consistency for the tests we carried out, namely both the observed relativistic mass-energy density, and the observed relativistic mass-energy density per source, which is equivalent to differential number counts, in an EdS Universe. In addition, we find clear evidence of a large amount of hidden mass, as has been obvious from many earlier investigations. At the same time, we find that the CNOC2 LF give differential galaxy counts somewhat above the EdS predictions, indicating that this survey observes more galaxies at 0.1 < z < 0.4 than the model's predictions., 46 pages, 2 figures, 7 tables, AASTeX. Accepted for publication in "The Astrophysical Journal"

The idea of a multiverse -- an ensemble of universes -- has received increasing attention in cosmology, both as the outcome of the originating process that generated our own universe, and as an explanation for why our universe appears to be fine-tuned for life and consciousness. Here we carefully consider how multiverses should be defined, stressing the distinction between the collection of all possible universes, and ensembles of really existing universes that are essential for an anthropic argument. We show that such realised multiverses are by no means unique. A proper measure on the space of all really existing universes or universe domains is needed, so that probabilities can be calculated, and major problems arise in terms of realised infinities. As an illustration we examine these issues in the case of the set of Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) universes. Then we briefly summarise scenarios like chaotic inflation, which suggest how ensembles of universe domains may be generated, and point out that the regularities which must underlie any systematic description of truly disjoint multiverses must imply some kind of common generating mechanism. Finally, we discuss the issue of testability, which underlies the question of whether multiverse proposals are really scientific propositions., Comment: 32 pages, 2 figures Completely revised text addressing physics issues more fully and omitting philosophical discussions previously included

Assuming that the cosmological principle holds, Maartens, Ellis and Stoeger (MES) recently constructed a detailed scheme linking anisotropies in the cosmic background radiation (CMB) with anisotropies and inhomogeneities in the large scale structure of the universe and showed how to place limits on those anisotropies and inhomogeneities simply by using CMB quadrupole and octupole limits. First we indicate and discuss the connection between the covariant multipole moments of the temperature anisotropy used in the MES scheme and the quadrupole and octupole results from COBE. Then we introduce those results into the MES limit equations to obtain definite quantitative limits on the complete set of cosmological measures of anisotropy and inhomogeneity. We find that all the anisotropy measures are less than 10^{-4} in the case of those not affected by the expansion rate H, and less than 10^{-6} Mpc^{-1} in the case of those which are. These results quantitatively demonstrate that the observable universe is indeed close to Friedmann-Lemaitre-Robertson-Walker (FLRW) on the largest scales, and can be adequately modelled by an almost-FLRW model -- that is, the anisotropies and inhomogeneities characterizing the observable universe on the largest scales are not too large to be considered perturbations to FLRW., Original paper with corrections. ApJ 476 435 (1997) erratum to appear ApJ Sept 1999

We consider directly the equations by which matter imposes anisotropies on freely propagating background radiation, leading to a new way of using anisotropy measurements to limit the deviations of the Universe from a Friedmann-Robertson-Walker (FRW) geometry. This approach is complementary to the usual Sachs-Wolfe approach: the limits obtained are not as detailed, but they are more model-independent. We also give new results about combined matter-radiation perturbations in an almost-FRW universe, and a new exact solution of the linearised equations., Comment: 18 pages Latex.

The extent to which individual holes in a cluster of black holes with a mass spectrum can liberate and accrete the resulting material by tidally disrupting stars they encounter, or by capturing stars as binary companions is studied. It is found that the smaller black holes in 'the halo' of such clusters can adequately supply themselves to the level M-dot sub h or greater than 0.0001(M-dot sub h) sub crit, and up to 0.05(M-dot sub h)sub crit for the smallest holes, by tidal disruption, as long as the cluster is embedded in a distribution of stars of relatively high density (not less than 0.1M sub cl/cu pc), and as long as the entire cluster of stars is not too compact (not less than 0.5 pc). Consideration is given to modifications this 'internal' mode of supply introduces in the spectrum emitted by such black hole clusters, and to the current status of their viability as models for AGN and QSOs in light of dynamical studies by Quinlan and Shapiro (1987, 1989).