Spherical Collapse Model Research Articles

Cosmological Implications of the Fundamental Relations of X-Ray Clusters

Based on the two-parameter family nature of X-ray clusters of galaxies obtained in a separate paper, we discuss the formation history of clusters and cosmological parameters of the universe. Utilizing the spherical collapse model of cluster formation, and assuming that the cluster X-ray core radius is proportional to the virial radius at the time of the cluster collapse, the observed relations among the density, radius, and temperature of clusters imply that cluster formation occurs in a wide range of redshift. The observed relations favor the low-density universe. Moreover, we find that the model of $n\sim -1$ is preferable.

The APM cluster--galaxy cross-correlation function: constraints on and galaxy bias

We estimate the cluster-galaxy cross-correlation function (ξcg), from the APM galaxy and galaxy cluster surveys. We obtain estimates both in real space from the inversion of projected statistics and in redshift space using the galaxy and cluster redshift samples. The amplitude of ξcg is found to be almost independent of cluster richness. At large separations, r ≥ 5 h−1Mpc (h=H0/100 km s−1 Mpc−1, where H0 is the Hubble constant), ξcg has a similar shape to the galaxy-galaxy and cluster-cluster autocorrelation functions. ξcg in redshift space can be related to the real-space ξcg by convolution with an appropriate velocity field model. Here we apply a spherical collapse model, which we have tested against N-body simulations, finding that it provides an accurate description of the averaged infall velocity of matter into galaxy clusters. We use this model to estimate β (β =Ω0.6/b, where b is the linear bias parameter), and find that it tends to overestimate the true result in simulations by only ∼ 10–30 per cent. Application to the APM results yields β =0.46 with β < 0.73 at 95 per cent confidence. This measure is complementary to the estimates made of the density parameter from larger scale bulk flows and from the virialized regions of clusters on smaller scales. We also compare the APM ξcg and galaxy autocorrelations directly with the mass correlation and cluster-mass correlations in COBE-normalized simulations of popular cosmological models, and derive two independent estimates of the galaxy biasing expected as a function of scale. This analysis reveals that both low-density and critical-density COBE-normalized cold dark matter (CDM) models require anti-biasing by a factor ∼ 2 on scales r ≤ 2 h−1 Mpc, and that the mixed dark matter (MDM) model is consistent with a constant biasing factor on all scales. The critical-density CDM model also suffers from the usual deficit of power on large scales (r ≥ 20 h−1 Mpc). We use the velocity fields predicted from the different models to distort the APM real-space cross-correlation function. Comparison with the APM redshift-space ξcg yields an estimate of the value of Ω0.6 needed in each model. We find that only the low-Ω model is fully consistent with observations, with MDM marginally excluded at the ∼ 2 σ level.

Gravitational instabilities and faster evolving density perturbations

The evolution of inhomogeneities in a spherical collapse model is studied by expanding the Einstein equation in powers of inverse radial parameter. In the linear regime, the density contrast is obtained for flat, closed and open universes. In addition to the usual modes, an infinite number of new growing modes are contained in the solutions for pressureless open and closed universes. In the nonlinear regime, we obtain the leading growing modes in closed forms for a flat universe and also, in the limits of small and large times, for an open universe.

Cosmological perturbation theory and the spherical collapse model — III. The velocity divergence field and the Ω dependence

ABSTRA C T Cosmological perturbation theory (PT) is a useful tool to study the cumulants of the density and velocity fields in the large-scale structure of the Universe. In Papers I and II of this series we saw that the spherical collapse (SC) model provides the exact solution to PT at tree-level and gives a good approximation to the loop corrections (next-to-leading orders), indicating negligible tidal effects. Here, we derive predictions for the (smoothed) cumulants of the velocity divergence field v ; =·v for an irrotational fluid in the SC model. By comparing these with the exact analytic results of Scoccimarro & Frieman, it is shown that, at least for the unsmoothed case, the loop corrections to the cumulants of v are dominated by tidal effects. However, most of the tidal contribution seems to cancel out when computing the hierarchical ratios, TJ oh v J i = h v 2 i J π 1 . We also extend the work presented in Papers I and II to give predictions for the cumulants of the density and velocity divergence fields in non-flat spaces. In particular, we show the equivalence between the spherically symmetric solution to the equations of motion in the SC model (given in terms of the density) and that of the Lagrangian PT approach (given in terms of the displacement field). It is shown that the Q dependence is very weak for both cosmic fields even at one loop (a 10 per cent effect at most), except for the overall factor f ﷿Qfi that couples to the velocity divergence.

Cosmological perturbation theory and the spherical collapse model — II. Non-Gaussian initial conditions

In Paper I of this series, we introduced the spherical collapse (SC) approximation in Lagrangian space as a way of estimating the cumulants ξJ of density fluctuations in cosmological perturbation theory (PT). Within this approximation, the dynamics is decoupled from the statistics of the initial conditions, so we are able to present here the cumulants for generic non-Gaussian initial conditions, which can be estimated to arbitrary order including the smoothing effects. The SC model turns out to recover the exact leading-order non-linear contributions up to terms involving non-local integrals of the J-point functions. We argue that for the hierarchical ratios SJ, these non-local terms are subdominant and tend to compensate each other. The resulting predictions show a non-trivial time evolution that can be used to discriminate between models of structure formation. We compare these analytic results with non-Gaussian N-body simulations, which turn out to be in very good agreement up to scales where σ ≲ 1.

Cosmological Perturbation Theory and the Spherical Collapse model — I. Gaussian initial conditions

We present a simple and intuitive approximation for solving the perturbation theory (PT) of small cosmic fluctuations. We consider only the spherically symmetric or monopole contribution to the PT integrals, which yields the exact result for tree-graphs (i.e. at leading order). We find that the non-linear evolution in Lagrangian space is then given by a simple local transformation over the initial conditions, although it is not local in Euler space. This transformation is found to be described by the spherical collapse (SC) dynamics, as it is the exact solution in the shearless (and therefore local) approximation in Lagrangian space. Taking advantage of this property, it is straightforward to derive the one-point cumulants, ξJ, for both the unsmoothed and smoothed density fields to arbitrary order in the perturbative regime. To leading-order this reproduces, and provides us with a simple explanation for, the exact results obtained by Bernardeau. We then show that the SC model leads to accurate estimates for the next corrective terms when compared with the results derived in the exact perturbation theory making use of the loop calculations. The agreement is within a few per cent for the hierarchical ratios . We compare our analytic results with N-body simulations, which turn out to be in very good agreement up to scales where σ ≈ 1. A similar treatment is presented to estimate higher order corrections in the Zel'dovich approximation. These results represent a powerful and readily usable tool to produce analytical predictions that describe the gravitational clustering of large-scale structure in the weakly non-linear regime.

Quantum stress tensor in self-similar spherical dust collapse

We calculate the quantum stress tensor for a massless scalar field in the 2-d self-similar spherical dust collapse model which admits a naked singularity. We find that the outgoing radiation flux diverges on the Cauchy horizon. This may have two consequences. The resultant back reaction may prevent the naked singularity from forming, thus preserving cosmic censorship through quantum effects. The divergent flux may lead to an observable signature differentiating naked singularities from black holes in astrophysical observations.

The generalized Poisson distribution and a model of clustering from Poisson initial conditions

This paper presents a new derivation of the generalized Poisson distribution. The derivation is based on the barrier crossing statistics of random walks associated with the Poisson distribution. A simple interpretation of this model in terms of a single server queue is also included. In the astrophysical context, the generalized Poisson distribution is interesting because it provides a good fit to the evolved, Eulerian counts-in-cells distribution measured in numerical simulations of hierarchical clustering from Poisson initial conditions. The new derivation presented here can be used to construct a useful analytic model of the evolution of clustering measured in these simulations. The model is consistent with the assumption that, as the Universe expands and the comoving sizes of regions change as a result of gravitational instability, the number of such expanding and contracting regions is conserved. The model neglects the influence of external tides on the evolution of such regions. Indeed, in the context of this model, the generalized Poisson distribution can be thought of as arising from a simple variant of the well-studied spherical collapse model, in which tidal effects are also neglected. This has the following implication: insofar as the generalized Poisson distribution derived from this model is a reasonable fit to the numerical simulation results, the counts-in-cells statistic must be relatively insensitive to such effects. This may be a consequence of the Poisson initial condition. The model can be understood as a simple generalization of the excursion set model which has recently been used to estimate the number density of collapsed, virialized haloes. The generalization developed here allows one to estimate the evolution of the spatial distribution of these haloes, as well as their number density. For example, it provides a framework within which the halo-halo correlation functions, at any epoch, can be computed analytically. In the model, when haloes first virialize, they are uncorrelated with each other. This is in good agreement with the simulations. Since it allows one to describe the spatial distribution of the haloes and the mass simultaneously, the model allows one to estimate the extent to which these haloes are biased tracers of the underlying matter distribution.

The dependence in equations of motion

We show that the equations of motion governing the evolution of a collisionless gravitating system of particles in an expanding universe can be cast in a form which is almost independent of the cosmological density parameter, $\Omega$, and the cosmological constant, $\Lambda$. The new equations are expressed in terms of a time variable $\tau\equiv \ln D$, where $D$ is the linear rate of growth of density fluctuations. The weak dependence on the density parameter is proportional to $\epsilon=\Omega^{-0.2}-1$ times the difference between the peculiar velocity (with respect to $\tau$) of particles and the gravity field. In the general case, the effect of this weak $\Omega$ dependence is to enhance the rate of evolution of density perturbations in dense regions. In a flat universe with $\Lambda\ne 0$, this enhancement is less pronounced than in an open universe with $\Lambda=0$ and the same $\Omega$. Using the spherical collapse model, we find that the increase of the $rms$ density fluctuations in a low $\Omega$ universe relative to that in a flat universe with the same linear normalization is $\sim 0.01 \epsilon(\Omega) < \delta^3 >$, where $\delta$ is the density field in the flat universe. The equations predict that the smooth average velocity field scales like $\Omega^{0.6}$ while the local velocity dispersion (rms value) scales, approximately, like $\Omega^{0.5}$. High resolution N-body simulations confirm these results and show that density fields, when smoothed on scales slightly larger than clusters, are insensitive to the cosmological model. Halos in an open model simulation are more concentrated than halos of the same $M/\Omega$ in a flat model simulation.

The Omega dependence in equations of motion

We show that the equations of motion governing the evolution of a collisionless gravitating system of particles in an expanding universe can be cast in a form which is almost independent of the cosmological density parameter, $\Omega$, and the cosmological constant, $\Lambda$. The new equations are expressed in terms of a time variable $\tau\equiv \ln D$, where $D$ is the linear rate of growth of density fluctuations. The weak dependence on the density parameter is proportional to $\epsilon=\Omega^{-0.2}-1$ times the difference between the peculiar velocity (with respect to $\tau$) of particles and the gravity field. In the general case, the effect of this weak $\Omega$ dependence is to enhance the rate of evolution of density perturbations in dense regions. In a flat universe with $\Lambda\ne 0$, this enhancement is less pronounced than in an open universe with $\Lambda=0$ and the same $\Omega$. Using the spherical collapse model, we find that the increase of the $rms$ density fluctuations in a low $\Omega$ universe relative to that in a flat universe with the same linear normalization is $\sim 0.01 \epsilon(\Omega) $, where $\delta$ is the density field in the flat universe. The equations predict that the smooth average velocity field scales like $\Omega^{0.6}$ while the local velocity dispersion (rms value) scales, approximately, like $\Omega^{0.5}$. High resolution N-body simulations confirm these results and show that density fields, when smoothed on scales slightly larger than clusters, are insensitive to the cosmological model. Halos in an open model simulation are more concentrated than halos of the same $M/\Omega$ in a flat model simulation.

Sheet Models of Protostellar Collapse

view Abstract Citations (77) References (60) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Sheet Models of Protostellar Collapse Hartmann, Lee ; Calvet, Nuria ; Boss, Alan Abstract Recognizing that protostellar clouds are unlikely to be completely spherical, we explore some effects of initial cloud geometry by considering collapse from a sheet initially in hydrostatic equilibrium. A qualitatively different feature of sheet collapse compared with spherical contraction is the development of relatively evacuated cavities in the infalling dusty cloud, which arise because material falls in first along the shortest dimension to the central gravitating mass. Using analytic models of collapse, which reproduce the main features of our previous numerical time-dependent simulations, we perform detailed radiative transfer calculations, which suggest that these collapse cavities can naturally explain the morphological appearance of many reflection nebulae around young stars on small distance scales without requiring initially diverging outflows. Sheet collapse models can simultaneously explain small-scale reflection nebula morphologies and dust envelope emission properties of many young stellar objects more easily than the standard spherical collapse models. The sheet collapse picture suggests that protostars, i.e., young stellar objects still accreting a large fraction of their mass from infalling envelopes, may be optically visible over a substantial range of system inclinations to the line of sight. These results may be especially relevant to cases where fragmentation and collapse has been triggered by an external impulse, such as a shock wave. We show how many properties of the flat-spectrum T Tauri star HL Tau can be interpreted in terms of flattened protostellar cloud collapse and draw some distinctions between the flattened toroids resulting in our calculations and the "pseudodisk" of Galli & Shu. Publication: The Astrophysical Journal Pub Date: June 1996 DOI: 10.1086/177330 Bibcode: 1996ApJ...464..387H Keywords: STARS: CIRCUMSTELLAR MATTER; STARS: FORMATION; STARS: PRE-MAIN-SEQUENCE full text sources ADS | data products SIMBAD (4)

High-order correlations of peaks and haloes: a step towards understanding galaxy biasing

We develop an analytic model for the hierarchical correlation amplitudes is the jth order connected moment of counts in spheres of radius R) of density peaks and dark matter haloes in the quasi-linear regime. The statistical distributions of density peaks and dark matter haloes within the initial density field (assumed Gaussian) are determined by the peak formalism of Bardeen et al. and by an extension of the Press-Schechter formalism, respectively. Modifications of these distributions caused by gravitationally induced motions are treated using a spherical collapse model. We test our model against results for S3,g(R) and S4,g(R) from a variety of N-body simulations. The model works well for peaks even on scales where the second moment of mass is significantly greater than unity. The model also works successfully for haloes that are identified earlier than the time when the moments are calculated. Because haloes are spatially exclusive at the time of their identification, our model is only qualitatively correct for haloes identified at the same time as the moments are calculated. For currently popular initial density spectra, the values of Sj,g at R ∼ 10 h−1 Mpc are significantly smaller for both haloes and peaks than those for the mass, unless the linear bias parameterb [defined by for large R] is comparable to or less than unity. The Sj,g depend only weakly on b for large b but increase rapidly with decreasing b at b ∼1. Thus if galaxies are associated with peaks in the initial density field, or with dark haloes formed at high redshifts, a measurement of Sj,g in the quasi-linear regime should determine whether galaxies are significantly biased relative to the mass. We use our model to interpret the observed high-order correlation functions of galaxies and clusters. We find that if the values of Sj,gfor galaxies are as high as those given by the APM survey, then APM galaxies should not be significantly biased.

The structure of dark matter haloes in hierarchical clustering models

We use a set of large cosmological N-body simulations to study the internal structure of dark matter haloes which form in scale-free models. We find that the radius r_178 corresponding to a mean interior overdensity of 178 accurately delineates the quasi-static halo interior from the surrounding infalling material, in agreement with the simple spherical collapse model. The interior velocity dispersion correlates with mass, again in good agreement with the spherical collapse model. Interior to the virial radius r_178, the spherical averaged density, circular velocity and velocity dispersion profiles are well fit by a simple 2-parameter analytic model proposed by Navarro etal (1995). This model has density going as 1/r at small radii, steepening to 1/r^3 at large radii, and fits our haloes to the resolution limit of the simulations. The two model parameters, scalelength and mass, are tightly correlated. Lower mass haloes are more centrally concentrated, and so have scalelengths which are a smaller fraction of their virial radius than those of their higher mass counterparts. This reflects the earlier formation times of low mass haloes. The haloes are moderately aspherical, with typical axial ratios 1:0.8:0.65 at their virial radii. These shapes are maintained by an anisotropic velocity dispersion tensor. The median value of the spin parameter is lambda=0.04, with a weak trend for lower lambda at higher halo mass. We also investigate how the halo properties depend on the algorithm used to identify them, using both friends-of-friends and spherical overdensity methods. We find that for groups selected at mean overdensities 100-400 by either method the properties are insensitive to how the haloes are selected, if the halo centre is taken as the position of the most bound particle

Merger rates in hierarchical models of galaxy formation – II. Comparison with N-body simulations

We have made a detailed comparison of the results of large N-body simulations with the analytical description of the merging histories of dark matter halos presented in Lacey & Cole 1993, which is based on an extension of the Press- Schechter method (Bond etal 1991,Bower 1991). We find the analytical predictions for the halo mass function, merger rates and formation times to be remarkably accurate. The analytical model is however expected to apply for arbitrary Omega and general power spectra. Dark matter halos were identified in the simulations using two different methods. For halos selected at mean overdensities 100-200, the analytical mass function was found to provide a good fit to the simulations with a collapse threshold close to that predicted by the spherical collapse model, with a typical error of <30% over a range of 1000 in mass, which is the full dynamic range of our N-body simulations. This was insensitive to the type of filtering used. Over a range of 100 - 1000 in mass, there was also good agreement with the analytical predictions for merger rates, including their dependence on the masses of the two halos involved and the time interval being considered, and for formation times, including the dependence on halo mass and formation epoch. The analytical Press-Schechter mass function and its extension to halo lifetimes and merger rates thus provide a very useful description of the growth of dark matter halos through hierarchical clustering, and should providea valuable tool in studies of the formation and evolution of galaxies and galaxy clusters.

Internal velocity and mass distributions in simulated clusters of galaxies for a variety of cosmogonic models

The mass and velocity distributions in the outskirts (0.5-3.0 h^-1^ Mpc) of simulated clusters of galaxies are examined for a suite of cosmogonic models (two {OMEGA}_0_ = 1 and two {OMEGA}_0_ = 0.2 models) utilizing large-scale particle-mesh (PM) simulations (500^3^ cells, 250^3^ particles and box size of 100 h^-1^ Mpc, giving a nominal resolution of 0.2 h^-1^ Mpc with the true resolution ~0.5 h^-1^ Mpc). Through a series of model computations, designed to isolate the different effects, we find that both {OMEGA}_0_ and P_k_ (λ &lt;= 16 h^-1^ Mpc) are important to the mass distributions in clusters of galaxies. There is a correlation between power, P_k_, and density profiles of massive clusters; more power tends to point to the direction of a stronger correlation between α and M(r &lt; 1.5 h^-1^ Mpc) (see eq. [1] for definitions); i.e., massive clusters being relatively extended and small mass clusters being relatively concentrated. A lower {OMEGA}_0_ universe tends to produce relatively concentrated massive clusters and relatively extended small mass clusters compared to their counterparts in a higher {OMEGA}_0_ model with the same power. Models with little (initial) small-scale power, such as the HDM model, produce more extended mass distributions than the isothermal distribution for most of the mass clusters. But the CDM models show mass distributions of most of the clusters more concentrated than the isothermal distribution. X-ray and gravitational lensing observations are beginning providing useful information on the mass distribution in and around clusters; some interesting constraints on {OMEGA}_0_ and/or the (initial) power of the density fluctuations on scales λ &lt;= 16 h^-1^ Mpc (where linear extrapolation is invalid) can be obtained when larger observational data sets, such as the Sloan Digital Sky Survey, become available. With regard to the velocity distribution, we find two interesting points. First, in 0.5 &lt; r &lt; 3.0 h^-1^ Mpc region, all four velocity dispersions (one-dimensional [1D], radial, tangential, line-of-sight) show decreasing distributions as a function of clustercentric distance in the three CDM models; but the HDM model shows just the opposite: weakly increasing velocity dispersions outward. The CDM models can reasonably fit the observed galaxy velocity dispersions in the Coma cluster of galaxies but the HDM model provides a poor fit. Second, we find that for the scales 0.5 &lt; r &lt; 3.0 h^-1^ Mpc, the tangential velocity dispersion is always larger than the radial component by a factor of 1.2-1.6 in the CDM models and 1.3-2.0 in the HDM model. In all models the ratio of radial to tangential velocity dispersions is a decreasing function from 0.5 h^-1^ Mpc to 3.0 h^-1^ Mpc for massive clusters (smaller mass clusters tend to show a minimum for that ratio around 1.5-2.0 h^-1^ Mpc in the CDM models). While the velocity dispersions among the three Cartesian directions are isotropic on average, a large scatter (40%) exists in all models. We also examine the infall issue in detail. Lower {OMEGA}_0_ models are found to have larger turnaround radius for a fixed-mass clump than high {OMEGA}_0_ models; this conclusion is insensitive to P_k_. But we find that the following relation (between the turnaround radius, R_ta_, and the mass within R_ta_, M_ta_), log_10_ R_ta_ = a + b log_10_ M_ta_ (a = -5.2 +/- 0.2, b = 0.40 +/- 0.02 R_ta_ and M_ta_ are in h^-1^ Mpc and h^-1^ M_sun_, respectively) holds for all the models (the uncertainties in a and b indicate the variations among models). In addition, the relation between the overdensity inside the turnaround radius, δ_ta_, and M_ta_ is fitted by log_10_ δ_ta_ = c + d log_10_ M_ta_ (cf. Table 1 for values of c and d). We show that the spherical top-hat collapse model in an Einstein-de Sitter universe, having δ_ta_ = 9π^2^/16 = 5.55, gives a fair fit to results (~4-10) of the nonlinear, nonspherical simulations performed here. Lower {OMEGA}_0_ models have considerably higher δ_ta_~10-30, as expected. Finally, we find that the isothermal approximation (cf. eq. [10]) appears to underestimate the true masses within the Abell radius by 10%- 30% with a scatter of ~50% around the estimated mean (in the three hierarchical models).

Large-amplitude isothermal fluctuations and high-density dark-matter clumps.

Large-amplitude isothermal fluctuations in the dark-matter energy density, parametrized by [Phi][equivalent to][delta][rho][sub DM]/[rho][sub DM], are studied within the framework of a spherical collapse model. For [Phi][approx gt]1, a fluctuation collapses in the radiation-dominated epoch and produces a dense dark-matter object. The final density of the virialized object is found to be [rho][sub [ital F]][approx]140[Phi][sup 3]([Phi]+1)[rho][sub eq], where [rho][sub eq] is the matter density at equal matter and radiation energy density. This expression is valid for the entire range of possible values of [Phi], both for [Phi][much gt]1 and [Phi][much lt]1. Some astrophysical consequences of high-density dark-matter clumps are discussed.

Thermodynamics of a model of nonadiabatic spherical gravitational collapse

In this work, a thermodynamic treatment of a Friedmann-like model of nonadiabatic spherical gravitational collapse is presented. The calculations have been performed according to Eckart's theory of dissipative relativistic fluids, while the diffusion approximation has been adopted for the radiation transport. The conclusions deduced are in agreement with the predictions of the theory of late stellar evolution.

Textures and clusters

We discuss the properties of galaxy clusters expected in a texture-seeded, CDM-dominated, Omega = 1 universe. Assuming that the textures are spherical, we use the spherical collapse model to compute the cluster velocity dispersion (or temperature) distribution function. For objects of mass 10 exp 11 to 10 exp 15 solar masses, we find v varies as M super gamma with gamma of about 0.25. An unbiased (b = 1) texture model predicts too many high-velocity dispersion clusters. A biased texture model appears to be compatible with cluster properties inferred from optical and X-ray observations. In the texture model, the cluster velocity distribution functio does not evolve rapidly; thus, the model predicts the existence of rich clusters at moderate redshift (about 1-2).

Gravitational radiation from even-parity perturbations of stellar collapse: Mathematical formalism and numerical methods.

In this paper we develop the mathematical formalism needed to study even-parity perturbations of spherical stellar collapse models, with the goal of calculating the gravitational radiation emitted during a stellar collapse to a black hole or a type-II supernova explosion. We concentrate on developing a set of gauge-invariant perturbation quantities, both inside the star and outside in the vacuum surrounding the star, their evolution equations, and equations which match the two sets of perturbation quantities at the stellar surface. The spherical background metric is assumed to be diagonal, and the method has been applied to a May-White stellar collapse code. We also discuss the initial conditions which we choose for both the stellar background and the perturbation fields. Finally, we discuss the numerical solution of this system in a computer code, and various code tests against known results.

Spherical vapour bubble collapse

An experimental study of spherical vapour bubble collapse in uniformly subcooled water has been performed, and data have been obtained on bubble radius and pressure histories for subcoolings up to 60°C. A theoretical incompressible equilibrium model of spherical collapse in an infinite sea geometry utilising a finite difference technique to solve the heat conduction equations has been developed, and the predictions are shown to agree with the experimental radius and pressure histories to within 30 per cent, subject to the bubble pressure not exceeding the critical pressure of water. The discrepancies are accounted for in terms of the restricted geometry of the experimental vessel. The noise spectra generated by bubble rebound are shown to extend to beyond 100 kHz in water subcoolings above 40°C.