Hot stars in old stellar populations: a continuing need for intermediate ages

We present numerical investigations into the formation of massive stars from turbulent cores of density structure  ρ∝ r ^−1.5  . The results of five hydrodynamical simulations are described, following the collapse of the core, fragmentation and the formation of small clusters of protostars. We generate two different initial turbulent velocity fields corresponding to power-law spectra   P ∝ k ⁻⁴  and   P ∝ k ^−3.5  , and we apply two different initial core radii. Calculations are included for both completely isothermal collapse, and a non-isothermal equation of state above a critical density  (10⁻¹⁴ g cm⁻³)  . Our calculations reveal the preference of fragmentation over monolithic star formation in turbulent cores. Fragmentation was prevalent in all the isothermal cases. Although disc fragmentation was largely suppressed in the non-isothermal runs due to the small dynamic range between the initial density and the critical density, our results show that some fragmentation still persisted. This is inconsistent with previous suggestions that turbulent cores result in the formation of a single massive star. We conclude that turbulence cannot be measured as an isotropic pressure term.     

Evolution of the phase-space density of dark matter haloes and mixing effects in merger events

Cosmological N -body simulations were performed to study the evolution of the phase-space density   Q =ρ/σ³  of dark matter haloes. No significant differences in the scale relations   Q ∝σ^−2.1  or   Q ∝ M ^−0.82  are seen for the 'cold' or 'warm' dark matter models. The follow-up of individual haloes from   z = 10  up to the present time indicate the existence of two main evolutionary phases: an early and fast one  (10 > z > 6.5)  , in which Q decreases on the average by a factor of 40 as a consequence of the randomization of bulk motions, and a late and long one  (6.5 > z ≥ 0)  , in which Q decreases by a factor of 20 because of mixing induced by merger events. The study of these haloes has also evidenced that rapid and positive variations of the velocity dispersion, induced by merger episodes, are related to a fast decrease of the phase-space density Q .     

Scaling up tides in numerical models of galaxy and halo formation

The purpose of this article is to show that when dynamically cold, dissipationless self-gravitating systems collapse, their evolution is a strong function of the symmetry in the initial distribution. We explore with a set of pressureless homogeneous fluids the time evolution of ellipsoidal distributions and map the depth of potential achieved during relaxation as function of initial ellipsoid axis ratios. We then perform a series of N -body numerical simulations and contrast their evolution with the fluid solutions. We verify an analytic relation between collapse factor and particle number N in spherical symmetry, such that  ∝ N ^1/3  . We sought a similar relation for axisymmetric configurations, and found an empirical scaling relation such that  ∝ N ^1/6  in these cases. We then show that when mass distributions do not respect spherical or axial symmetry, the ensuing gravitational collapse deepens with increasing particle number N but only slowly: 86 per cent of triaxial configurations may collapse by a factor of no more than 40 as   N →∞  . For   N ≈10⁵  and larger, violent relaxation develops fully under the Lin–Mestel–Shu instability such that numerical N -body solutions now resolve the different initial morphologies adequately.     

Dark matter distribution in dwarf spheroidal galaxies

We study the distribution of dark matter in dwarf spheroidal galaxies by modelling the moments of their line-of-sight velocity distributions. We discuss different dark matter density profiles, both cuspy and possessing flat density cores. The predictions are made in the framework of standard dynamical theory of two-component (stars and dark matter) spherical systems with different velocity distributions. We compare the predicted velocity dispersion profiles to observations in the case of Fornax and Draco dwarfs. For isotropic models the dark haloes with cores are found to fit the data better than those with cusps. Anisotropic models are studied by fitting two parameters, dark mass and velocity anisotropy, to the data. In this case all profiles yield good fits, but the steeper the cusp of the profile, the more tangential is the velocity distribution required to fit the data. To resolve this well-known degeneracy of density profile versus velocity anisotropy, we obtain predictions for the kurtosis of the line-of-sight velocity distribution for models found to provide best fits to the velocity dispersion profiles. It turns out that profiles with cores typically yield higher values of kurtosis which decrease more steeply with distance than the cuspy profiles, which will allow us to discriminate between the profiles once the kurtosis measurements become available. We also show that with present quality of the data the alternative explanation of velocity dispersions in terms of Modified Newtonian Dynamics cannot yet be ruled out.     

Amplification of primordial magnetic fields by anisotropic gravitational collapse

If a magnetic field is frozen into a plasma that undergoes spherical compression, then the magnetic field B varies with the plasma density ρ according to   B ∝ρ^2/3  . In the gravitational collapse of cosmological density perturbations, however, quasi-spherical evolution is very unlikely. In anisotropic collapses the magnetic field can be a much steeper function of gas density than in the isotropic case. We investigate the distribution of amplifications in realistic gravitational collapses from Gaussian initial fluctuations using the Zel'dovich approximation. Representing our results using a relation of the form   B ∝ρ^α  , we show that the median value of α can be much larger than the value  α= 2/3  resulting from spherical collapse, even if there is no initial correlation between magnetic field and principal collapse directions. These analytic arguments go some way towards understanding the results of numerical simulations.     

Parameters of core collapse

This paper considers the phenomenon of deep core collapse in collisional stellar systems, with stars of equal mass. The collapse takes place on some multiple,  ξ⁻¹  , of the central relaxation time, and produces a density profile in which  ρ∝ r ^−α  , where α is a constant. The parameters α and ξ have usually been determined from simplified models, such as gas and Fokker–Planck models, often with the simplification of isotropy. Here we determine the parameters directly from N -body simulations carried out using the newly completed GRAPE-6.     

Dark matter in elliptical galaxies – I. Is the total mass density profile of the NFW form or even steeper?

Elliptical galaxies are modelled as Sérsic luminosity distributions with density profiles (DPs) for the total mass adopted from the DPs of haloes within dissipationless ΛCDM (cold dark matter) N -body simulations. Ellipticals turn out to be inconsistent with cuspy low-concentration NFW models representing the total mass distribution, neither are they consistent with a steeper −1.5 inner slope, nor with the shallower models proposed by Navarro et al., nor with NFW models 10 times more concentrated than predicted, as deduced from several X-ray observations – the mass models, extrapolated inwards, lead to local mass-to-light ratios that are smaller than the stellar value inside an effective radius ( R _e), and to central aperture velocity dispersions that are much smaller than observed. This conclusion remains true as long as there is no sharp steepening (slope < −2) of the dark matter DPs just inside 0.01 virial radii.
The very low total mass and velocity dispersion produced within R _e by an NFW-like total mass profile suggests that the stellar component should dominate the dark matter component out to at least R _e. It should then be difficult to kinematically constrain the inner slope of the DP of ellipticals. The high-concentration parameters deduced from X-ray observations appear to be a consequence of fitting an NFW model to the total mass DP made up of a stellar component that dominates inside and a dark matter component that dominates outwards.
An appendix gives the virial mass dependence of the concentration parameter, central density and total mass of the Navarro et al. model. In a second appendix are given single integral expressions for the velocity dispersions averaged along the line of sight, in circular apertures and in thin slits, for general luminosity density and mass distributions, with isotropic orbits.     

The survival and disruption of cold dark matter microhaloes: implications for direct and indirect detection experiments

If the dark matter particle is a neutralino, then the first structures to form are cuspy cold dark matter (CDM) haloes collapsing after redshifts   z ≈ 100  in the mass range  10⁻⁶–10⁻³ M_⊙  . We carry out a detailed study of the survival of these microhaloes in the Galaxy as they experience tidal encounters with stars, molecular clouds, and other dark matter substructures. We test the validity of analytic impulsive heating calculations using high-resolution N -body simulations. A major limitation of analytic estimates is that mean energy inputs are compared to mean binding energies, instead of the actual mass lost from the system. This energy criterion leads to an overestimate of the stripped mass and an underestimate of the disruption time-scale, since CDM haloes are strongly bound in their inner parts. We show that a significant fraction of material from CDM microhaloes can be unbound by encounters with Galactic substructure and stars; however, the cuspy central regions remain relatively intact. Furthermore, the microhaloes near the solar radius are those which collapse significantly earlier than average and will suffer very little mass-loss. Thus, we expect a fraction of surviving bound microhaloes, a smooth component with narrow features in phase space, which may be uncovered by direct detection experiments, as well as numerous surviving cuspy cores with proper motions of arcminutes per year, which can be detected indirectly via their annihilation into gamma-rays.     

On the origin of cold dark matter halo density profiles

N -body simulations predict that cold dark matter (CDM) halo-assembly occurs in two phases: (i) a fast-accretion phase with a rapidly deepening potential well; and (ii) a slow-accretion phase characterized by a gentle addition of mass to the outer halo with little change in the inner potential well. We demonstrate, using one-dimensional simulations, that this two-phase accretion leads to CDM haloes of the Navarro, Frenk & White (NFW) form and provides physical insight into the properties of the mass-accretion history that influence the final profile. Assuming that the velocities of CDM particles are effectively isotropized by fluctuations in the gravitational potential during the fast-accretion phase, we show that gravitational collapse in this phase leads to an inner profile  ρ( r ) ∝ r ⁻¹  . Slow accretion on to an established potential well leads to an outer profile with  ρ( r ) ∝ r ⁻³  . The concentration of a halo is determined by the fraction of mass that is accreted during the fast-accretion phase. Using an ensemble of realistic mass-accretion histories, we show that the model predictions of the dependence of halo concentration on halo formation time and, hence, the dependence of halo concentration on halo mass, and the distribution of halo concentrations all match those found in cosmological N -body simulations. Using a simple analytic model that captures much of the important physics, we show that the inner   r ⁻¹  profile of CDM haloes is a natural result of hierarchical mass assembly with an initial phase of rapid accretion.     

Structures in a class of magnetized scale-free discs

We construct analytically stationary global configurations for both aligned and logarithmic spiral coplanar magnetohydrodynamics (MHD) perturbations in an axisymmetric background MHD disc with a power-law surface mass density  Σ₀∝ r ^−α  , a coplanar azimuthal magnetic field   B ₀∝ r ^−γ  , a consistent self-gravity and a power-law rotation curve   v ₀∝ r ^−β  , where v ₀ is the linear azimuthal gas rotation speed. The barotropic equation of state  Π∝Σ ⁿ   is adopted for both MHD background equilibrium and coplanar MHD perturbations where Π is the vertically integrated pressure and n is the barotropic index. For a scale-free background MHD equilibrium, a relation exists among  α, β, γ  and n such that only one parameter (e.g. β) is independent. For a linear axisymmetric stability analysis, we provide global criteria in various parameter regimes. For non-axisymmetric aligned and logarithmic spiral cases, two branches of perturbation modes (i.e. fast and slow MHD density waves) can be derived once β is specified. To complement the magnetized singular isothermal disc analysis of Lou, we extend the analysis to a wider range of  −1/4 < β < 1/2  . As an illustrative example, we discuss specifically the  β= 1/4  case when the background magnetic field is force-free. Angular momentum conservation for coplanar MHD perturbations and other relevant aspects of our approach are discussed.     

Prolate Jaffe models for galaxies

We introduce a class of prolate Jaffe models for elliptical galaxies, which are a further extension of Jaffe's spherical models of axisymmetric elliptical systems, and study the properties of their densities, circular velocities, velocity dispersions and two-integral even distribution functions. The form of the potential allows the density to be expressed simply as a function of the potential and radial coordinate R . The models have finite total mass and their densities at large distances decay radially as r ⁻⁴, except on the major axis, where the densities decay as r ⁻³. It is known from Hunter's formulae that the velocity dispersions for prolate models can be expressed in terms of elementary functions of R and z , unlike those for the oblate Jaffe models recently given by Jiang, and that the prolate models have anisotropic velocity distributions. Thus the prolate models are easier to study than the oblate models. It is also found that the two-integral even distribution functions on the physical boundary of the galaxies increase monotonically with the relative energy, for the prolate models. Furthermore, numerical calculation shows that the two-integral even distribution functions generated from their densities are non-negative, even for very 'squeezed' prolate Jaffe models. However, the edge-on projected surface densities for these prolate models cannot be expressed as simply as for the oblate models.     

Gravitational wave signals from the collapse of the first stars

We study the gravitational wave emission from the first stars, which are assumed to be very massive objects (VMOs). We take into account various feedback (both radiative and stellar) effects regulating the collapse of objects in the early Universe and thus derive the VMO initial mass function and formation rate. If the final fate of VMOs is to collapse, leaving very massive black hole remnants, then the gravitational waves emitted during each collapse would be seen as a stochastic background. The predicted spectral strain amplitude in a critical density cold dark matter (CDM) universe peaks in the frequency range ν ≈5×10⁻⁴–5×10⁻³ Hz, where it has a value in the range ≈10⁻²⁰–10⁻¹⁹ Hz^−1/2, and might be detected by the Laser Interferometer Space Antenna ( LISA ). The expected emission rate is roughly 4000 event yr⁻¹, resulting in a stationary discrete sequence of bursts, i.e. a shot-noise signal.     

Self-similar shocked accretion of collisional gas with radiative cooling

We describe similarity solutions that characterize the collapse of collisional gas on to scale-free perturbations in an Einstein–de Sitter universe. We consider the effects of radiative cooling and derive self-similar solutions under the assumption that the cooling function is a power law of density and temperature, Λ( T , ρ )∝ ρ ^3/2 T . We use these results to test the ability of smooth particle hydrodynamics (SPH) techniques to follow the collapse and accretion of shocked, rapidly cooling gas in a cosmological context. Our SPH code reproduces the analytical results very well in cases that include or exclude radiative cooling. No substantial deviations from the predicted central mass accretion rates or from the temperature, density and velocity profiles are observed in well-resolved regions inside the shock radius. This test problem lends support to the reliability of SPH techniques to model the complex process of galaxy formation.     

Density cusps: restrictions on non-axisymmetric models

Galactic nuclei are now generally thought to have density cusps in their centres, and the evidence is mounting that as a consequence they are unlikely to be triaxial. Self-consistent stellar dynamical models of non-axisymmetric cusps would be an interesting counter-argument to this conclusion. We consider 2D analogues of triaxial cusps: a sequence of non-axisymmetric, cuspy discs first described by Sridhar & Touma. Scale-free models with potential Φ ∝  r ^α are examined in detail. It is shown analytically for 0 < α ≲ 0.43 that self-consistent models with positive phase-space density do not exist. Numerical solutions of the combined Vlasov and Poisson equations suggest that the whole sequence of models with 0 < α < 1 are also unphysical. Together with existing work on cusps, we conclude on purely theoretical grounds that galactic nuclei are not expected to be triaxial.     

Gravitational lensing of anisotropic sources

In strong gravitational lensing, the multiple images we see correspond to light rays that leave the source in slightly different directions. If the source emission is anisotropic, the images may differ from conventional lensing predictions (which assume isotropy). To identify scales on which source anisotropy may be important, we study the angle δ between the light rays emerging from the source, for different lensing configurations. If the lens has a power-law profile   M ∝ R ^γ  , the angle δ initially increases with lens redshift and then either diverges (for a steep profile  γ < 1  ), remains constant (for an isothermal profile  γ= 1  ), or vanishes (for a shallow profile  γ > 1  ) as   z _l→ z _s  . The scaling with lens mass is roughly  δ∝ M ^1/(2−γ)  . The results for an Navarro–Frenk–White (NFW) profile are qualitatively similar to those for a shallow power law, with δ peaking at about half the redshift of the source (not half the distance). In practice, beaming could modify the statistics of beamed sources lensed by massive clusters: for an opening angle  θ_jet  , there is a probability as high as   P ∼ 0.02–0.07(θ_jet/0.5°)⁻¹  that one of the lensed images may be missed (for  2 ≲ z _s≲ 6  ). Differential absorption within active galactic nuclei (AGNs) could modify the flux ratios of AGNs lensed by clusters; a sample of AGNs lensed by clusters could provide further constraints on the sizes of absorbing regions. Source anisotropy is not likely to be a significant effect in galaxy-scale strong lensing.     

The distribution function of dark matter in massive haloes

We study the distribution function (DF) of dark matter particles in haloes of mass range  10¹⁴–10¹⁵ M_⊙  . In the numerical part of this work we measure the DF for a sample of relaxed haloes formed in the simulation of a standard Λ cold dark matter (ΛCDM) model. The DF is expressed as a function of energy E and the absolute value of the angular momentum L , a form suitable for comparison with theoretical models. By proper scaling we obtain the results that do not depend on the virial mass of the haloes. We demonstrate that the DF can be separated into energy and angular momentum components and propose a phenomenological model of the DF in the form     . This formulation involves three parameters describing the anisotropy profile in terms of its asymptotic values (β₀ and  β_∞  ) and the scale of transition between them ( L ₀). The energy part   f _E ( E )  is obtained via inversion of the integral for spatial density. We provide a straightforward numerical scheme for this procedure as well as a simple analytical approximation for a typical halo formed in the simulation. The DF model is extensively compared with the simulations: using the model parameters obtained from fitting the anisotropy profile, we recover the DF from the simulation as well as the profiles of the dispersion and kurtosis of radial and tangential velocities. Finally, we show that our DF model reproduces the power-law behaviour of phase-space density   Q =ρ( r )/σ³( r )  .     

Phase-space mixing and the merging of cusps

Collisionless stellar systems are driven towards equilibrium by mixing of phase-space elements. I show that the excess-mass function     [where     is the coarse-grained distribution function] always decreases on mixing . D ( f ) gives the excess mass from values of     . This novel form of the mixing theorem extends the maximum phase-space density argument to all values of f . The excess-mass function can be computed from N -body simulations and is additive: the excess mass of a combination of non-overlapping systems is the sum of their individual D ( f ). I propose a novel interpretation for the coarse-grained distribution function, which avoids conceptual problems with the mixing theorem.
As an example application, I show that for self-gravitating cusps (  ρ∝ r ^−γ  as   r → 0  ) the excess mass   D ∝ f ^{−2(3−γ)/(6−γ)}  as   f →∞  , i.e. steeper cusps are less mixed than shallower ones, independent of the shape of surfaces of constant density or details of the distribution function (e.g. anisotropy). This property, together with the additivity of D ( f ) and the mixing theorem, implies that a merger remnant cannot have a cusp steeper than the steepest of its progenitors. Furthermore, I argue that the cusp of the remnant should not be shallower either, implying that the steepest cusp always survives.     

Degeneracy of semi-analytic solution for axially symmetric lens equations

We derive a simple semi-analytical approximation for lens equations with an arbitrary radially symmetric mass density ρ( r ), when   r /ξ₀≪ 1  and ξ₀ is the scalelength of the density profile. At the strong lensing regime, which is mostly constrained by the inner part of the mass density profile, we assume ρ∝ r ^α.
A dark matter (DM) haloes (GNFW model) are parametrized through a shape parameter α, a concentration parameter c ₁ and the total mass M . We apply our semi-analytical model to show how the solutions of the axially symmetric lens equations are degenerated in respect to the parameters α and c ₁.
In the case of an asymmetric dual image lens system, similar effective degeneracy is produced when the geometry of the lens is relaxed. Because it is impossible to determine the exact location of the source image, a family of solutions is acquired when the mass of the lens object and location of the observed images are fixed.
Our results indicate that the amount of degeneration is only weakly affected by the asymmetry in the lensing geometry set-up, e.g. the observational effective degeneracy is very close to the true physical degeneracy of the Einstein ring solutions. Basically with high-enough values for the concentration parameter, the degeneracy spawns the whole range for the shape parameter  α=[−2.0, −1.0]  .     

Models of the Cosmic Horseshoe gravitational lens J1004+4112

We model the extremely massive and luminous lens galaxy in the Cosmic Horseshoe Einstein ring system J1004+4112, recently discovered in the Sloan Digital Sky Survey. We use the semilinear method of Warren & Dye, which pixelizes the source surface brightness distribution, to invert the Einstein ring for sets of parametrized lens models. Here, the method is refined by exploiting Bayesian inference to optimise adaptive pixelization of the source plane and to choose between three differently parametrized models: a singular isothermal ellipsoid, a power-law model and a Navarro, Frenk & White (NFW) profile. The most probable lens model is the power law with a volume mass density  ρ∝ r ^{−1.96±0.02}  and an axis ratio of ∼0.8. The mass within the Einstein ring (i.e. within a cylinder with projected distance of ∼30 kpc from the centre of the lens galaxy) is  (5.02 ± 0.09) × 10¹² M _⊙  , and the mass-to-light ratio is ∼30. Even though the lens lies in a group of galaxies, the preferred value of the external shear is almost zero. This makes the Cosmic Horseshoe unique amongst large separation lenses, as almost all the deflection comes from a single, very massive galaxy with little boost from the environment.     

A multiwavelength view at the heart of the superwind in NGC 253

We present a new method of studying quadruple lenses in elliptical power-law potentials parametrized by ψ ( x , y )∝( x ²+ y ² q ²) ^{β /2} β (0 β <2). For this potential, the moments of the four image positions weighted by signed magnifications (magnification times parity) have very simple properties. In particular, we find that the zeroth moment – the sum of four signed magnifications satisfies ≃2/(2− β ); the relation is exact for β =0 (point-lens) and β =1 (isothermal potential), independent of the axial ratio. Similar relations can be derived when a shear is present along the major or minor axes. These relations, however, do not hold well for the closely related elliptical density distributions. For a singular isothermal elliptical density distribution without shear, the sum of signed magnifications for quadruple lenses is ≈2.8, again nearly independent of the ellipticity. For the same distribution with shear, the total signed magnification is around 2–3 for most cases, but can be significantly different for some combinations of the axial ratio and shear where six or eight images can appear.