Microlensing in dark matter haloes

Using eight dark matter haloes extracted from fully self-consistent cosmological N -body simulations, we perform microlensing experiments. A hypothetical observer is placed at a distance of 8.5 kpc from the centre of the halo measuring optical depths, event durations and event rates towards the direction of the Large Magellanic Cloud. We simulate 1600 microlensing experiments for each halo. Assuming that the whole halo consists of massive astronomical compact halo objects (MACHOs),   f = 1.0  , and a single MACHO mass is   m _M= 1.0 M_⊙  , the simulations yield mean values of  τ= 4.7^+5.0_−2.2× 10⁻⁷  and  Γ= 1.6^+1.3_−0.6× 10⁻⁶  events star⁻¹ yr⁻¹. We find that triaxiality and substructure can have major effects on the measured values so that τ and Γ values of up to three times the mean can be found. If we fit our values of τ and Γ to the MACHO collaboration observations, we find   f = 0.23^+0.15_−0.13  and   m _M= 0.44^+0.24_−0.16  . Five out of the eight haloes under investigation produce f and m _M values mainly concentrated within these bounds.     

The physical parameters of Markarian 501 during flaring activity

We analyse an N -body simulation of the Small Magellanic Cloud (SMC), that of Gardiner & Noguchi, to determine its microlensing statistics. We find that the optical depth owing to self-lensing in the simulation is low, 0.4×10⁻⁷, but still consistent (at the 90 per cent level) with that observed by the EROS and MACHO collaborations. This low optical depth is due to the relatively small line-of-sight thickness of the SMC produced in the simulation. The proper motions and time-scales of the simulation are consistent with those observed assuming a standard mass function for stars in the SMC. The time-scale distribution from the standard mass function generates a significant fraction of short time-scale events: future self-lensing events towards the SMC may have the same time-scales as events observed towards the Large Magellanic Cloud (LMC). Although some debris was stripped from the SMC during its collision with the LMC about 2×10⁸ yr ago, the optical depth of the LMC owing to this debris is low, a few ×10⁻⁹, and thus cannot explain the measured optical depth towards the LMC.     

Deep galaxy counts, extragalactic background light and the stellar baryon budget

We assess the constraints imposed by the observed extragalactic background light (EBL) on the cosmic history of star formation and the stellar-mass density today. The logarithmic slope of the galaxy number–magnitude relation from the Southern Hubble Deep Field imaging survey is flatter than 0.4 in all seven UBVIJHK optical bandpasses, i.e. the light from resolved galaxies has converged from the UV to the near-IR. We find a lower limit to the surface brightness of the optical extragalactic sky of about 15 nW m⁻² sr⁻¹, comparable to the intensity of the far-IR background from COBE data. Assuming a Salpeter initial mass function with a lower cut-off consistent with observations of M subdwarf disc stars, we set a lower limit of Ω_g+s h ²>0.0013  I ₅₀ to the visible (processed gas + stars) mass density required to generate an EBL at a level of 50  I ₅₀ nW m⁻² sr⁻¹; our 'best-guess' value is Ω_g+s h ²≈0.0031  I ₅₀. Motivated by the recent microlensing results of the MACHO collaboration, we consider the possibility that massive dark haloes around spiral galaxies are composed of faint white dwarfs, and show that only a small fraction (≲5 per cent) of the nucleosynthetic baryons can be locked in the remnants of intermediate-mass stars forming at z _F≲5, as the bright early phases of such haloes would otherwise overproduce the observed EBL.     

Gravitational lensing of type Ia supernovae by galaxy clusters

We propose a method to remove the mass-sheet degeneracy that arises when the mass of galaxy clusters is inferred from gravitational shear. The method utilizes high-redshift standard candles that undergo weak lensing. Natural candidates for such standard candles are type Ia supernovae (SNe Ia).
When corrected with the light-curve shape (LCS), the peak magnitude of SNe Ia provides a standard candle with an uncertainty in apparent magnitude of Δ m ≃0.1–0.2. Gravitational magnification of a background SN Ia by an intervening cluster would cause a mismatch between the observed SN Ia peak magnitude compared with that expected from its LCS and redshift. The average detection rate for SNe Ia with a significant mismatch of ≥2Δ m behind a cluster at z ≃0.05–0.15 is about 1–2 supernovae per cluster per year at J , I , R ≲25–26.
Since SNe are point-like sources for a limited period, they can experience significant microlensing by massive compact halo objects (MACHOs) in the intracluster medium. Microlensing events caused by MACHOs of ∼10⁻⁴ M⊙ are expected to have time-scales similar to that of the SN light curve. Both the magnification curve by a MACHO and the light curve of a SN Ia have characteristic shapes that allow us to separate them. Microlensing events caused by MACHOs of smaller mass can unambiguously be identified in the SN light curve if the latter is continuously monitored. The average number of identifiable microlensing events per nearby cluster ( z ≲0.05) per year is ∼0.02 ( f /0.01), where f is the fraction of the cluster mass in MACHOs of masses 10⁻⁷< M _macho/M⊙<10⁻⁴.     

Dark matter subhaloes in numerical simulations

We use cosmological Λ cold dark matter (CDM) numerical simulations to model the evolution of the substructure population in 16 dark matter haloes with resolutions of up to seven million particles within the virial radius. The combined substructure circular velocity distribution function (VDF) for hosts of 10¹¹ to  10¹⁴ M_⊙  at redshifts from zero to two or higher has a self-similar shape, is independent of host halo mass and redshift, and follows the relation  d n /d v = (1/8)( v _cmax/ v _cmax,host)⁻⁴  . Halo to halo variance in the VDF is a factor of roughly 2 to 4. At high redshifts, we find preliminary evidence for fewer large substructure haloes (subhaloes). Specific angular momenta are significantly lower for subhaloes nearer the host halo centre where tidal stripping is more effective. The radial distribution of subhaloes is marginally consistent with the mass profile for   r ≳ 0.3 r _vir  , where the possibility of artificial numerical disruption of subhaloes can be most reliably excluded by our convergence study, although a subhalo distribution that is shallower than the mass profile is favoured. Subhalo masses but not circular velocities decrease towards the host centre. Subhalo velocity dispersions hint at a positive velocity bias at small radii. There is a weak bias towards more circular orbits at lower redshift, especially at small radii. We additionally model a cluster in several power-law cosmologies of   P ∝ kⁿ   , and demonstrate that a steeper spectral index, n , results in significantly less substructure.     

Microlensing by cosmic strings

We consider the signature and detectability of gravitational microlensing of distant quasars by cosmic strings. Because of the simple image configuration such events will have a characteristic lightcurve, in which a source would appear to brighten by exactly a factor of 2, before reverting to its original apparent brightness. We calculate the optical depth and event rate, and conclude that current predictions and limits on the total length of strings on the sky imply optical depths of ≲ 10⁻⁸ and event rates of fewer than one event per 10⁹ sources per year. Disregarding those predictions but replacing them with limits on the density of cosmic strings from the cosmic microwave background fluctuation spectrum, leaves only a small region of parameter space (in which the sky contains about 3 × 10⁵ strings with deficit angle of the order of 0.3 milli-seconds) for which a microlensing survey of exposure 10⁷ source years, spanning a 20–40-year period, might reveal the presence of cosmic strings.     

N-body simulations for testing the stability of triaxial galaxies in MOND

We perform a stability test of triaxial models in Modified Newtonian Dynamics (MOND) using N -body simulations. The triaxial models considered here have densities that vary with   r ⁻¹  in the centre and   r ⁻⁴  at large radii. The total mass of the model varies from 10⁸ to  10¹⁰ M_⊙  , representing the mass scale of dwarfs to medium-mass elliptical galaxies, respectively, from deep MOND to quasi-Newtonian gravity. We build triaxial galaxy models using the Schwarzschild technique, and evolve the systems for 200 Keplerian dynamical times (at the typical length-scale of 1.0 kpc). We find that the systems are virial overheating, and in quasi-equilibrium with the relaxation taking approximately 5 Keplerian dynamical times (1.0 kpc). For all systems, the change of the inertial (kinetic) energy is less than 10 per cent (20 per cent) after relaxation. However, the central profile of the model is flattened during the relaxation and the (overall) axis ratios change by roughly 10 per cent within 200 Keplerian dynamical times (at 1.0 kpc) in our simulations. We further find that the systems are stable once they reach the equilibrium state.     

Understanding the halo-mass and galaxy-mass cross-correlation functions

We use the Millennium Simulation (MS) to measure the cross-correlation between halo centres and mass (or equivalently the average density profiles of dark haloes) in a Lambda cold dark matter (ΛCDM) cosmology. We present results for radii in the range  10  h ⁻¹ kpc < r < 30  h ⁻¹ Mpc  and for halo masses in the range  4 × 10¹⁰ < M ₂₀₀ < 4 × 10¹⁴  h ⁻¹ M_⊙  . Both at   z = 0  and at   z = 0.76  these cross-correlations are surprisingly well fitted if the inner region is approximated by a density profile of NFW or Einasto form, the outer region by a biased version of the linear mass autocorrelation function, and the maximum of the two is adopted where they are comparable. We use a simulation of galaxy formation within the MS to explore how these results are reflected in cross-correlations between galaxies and mass. These are directly observable through galaxy–galaxy lensing. Here also we find that simple models can represent the simulation results remarkably well, typically to ≲10 per cent. Such models can be used to extend our results to other redshifts, to cosmologies with other parameters, and to other assumptions about how galaxies populate dark haloes. Our galaxy formation simulation already reproduces current galaxy–galaxy lensing data quite well. The characteristic features predicted in the galaxy–galaxy lensing signal should provide a strong test of the ΛCDM cosmology as well as a route to understanding how galaxies form within it.     

On the prospect of inferring the halo structure and the masses of dark objects through parallax microlensing

We study the proposed use of parallax microlensing in the direction of the Large Magellanic Cloud (LMC) to separate the effects of the mass function of dark massive halo objects (MHOs or 'machos') on the one hand, and their spatial distribution and kinematics on the other. This disentanglement is supposed to allow a much better determination of the two than could be achieved entirely on the basis of the durations of events. We restrict our treatment to the same class of power-law spherical models for the halo of MHOs studied in a previous paper by Marković 38 Sommer-Larsen, and assume that one can eliminate microlensing events caused by massive objects outside the halo (e.g., the LMC halo). Whereas the duration-based error in the average MHO mass, μ¯ ≡  M ¯/M, exceeds (at N  = 100 events) μ¯ by a factor of 2 or more, parallax microlensing remarkably brings it down to 15–20 per cent of μ¯, regardless of the shape of the mass function. In addition, the slope α of the mass function, d n /dμ ∝ μ^α, can be inferred relatively accurately (σ_α < 0.4) for a broader range, −3 < α < 0. The improvement in the inference of the halo structure is also significant: the index γ of the density profile ( ρ ∼  R ^−γ) can be obtained with the error σ_γ < 0.4. While in a typical situation the errors for the parameters specifying the velocity dispersion profile are of about the same magnitude as the parameters themselves, virtually all the uncertainty is 'concentrated' in linear combinations of the parameters that may have little influence on the profile, thus allowing its reasonably accurate inference.     

Self-similar spherical collapse with non-radial motions

We derive the asymptotic mass profile near the collapse centre of an initial spherical density perturbation, δ ∝ M ^{− ε} , of collisionless particles with non-radial motions. We show that angular momenta introduced at the initial time do not affect the mass profile. Alternatively, we consider a scheme in which a particle moves on a radial orbit until it reaches its turnaround radius, r ∗. At turnaround the particle acquires an angular momentum L =ℒ√ GM _* r _* per unit mass, where M ∗ is the mass interior to r ∗. In this scheme, the mass profile is M ∝ r ^{3/(1+3 ε )} for all ε >0 , in the region r / r _t ≪ℒ , where r _t is the current turnaround radius. If ℒ≪1 then the profile in the region ℒ≪ r / r _t ≪1 is M ∝ r for ε <2/3 , and remains M ∝ r ^{3/(1+3 ε )} for ε ≥2/3 . The derivation relies on a general property of non-radial orbits which is that the ratio of the pericentre to apocentre is constant in a force field k ( t ) r ⁿ with k ( t ) varying adiabatically.     

The dark matter distribution in disc galaxies

We use high-quality optical rotation curves of nine low-luminosity disc galaxies to obtain the velocity profiles of the surrounding dark matter haloes. We find that they increase linearly with radius at least out to the edge of the stellar disc, implying that, over the entire stellar region, the density of the dark halo is about constant.
The properties of the mass structure of these haloes are similar to those found for a number of dwarf and low surface brightness galaxies, but provide a more substantial evidence of the discrepancy between the halo mass distribution predicted in the cold dark matter scenario and those actually detected around galaxies. We find that the density law proposed by Burkert reproduces the halo rotation curves, with halo central densities ( ρ ₀∼1–4×10⁻²⁴ g cm⁻³) and core radii ( r ₀∼5–15 kpc) scaling as ρ ₀∝ r ₀^−2/3.     

Optical Gravitational Lensing Experiment OGLE-1999-BUL-32: the longest ever microlensing event – evidence for a stellar mass black hole?

We describe the discovery of the longest microlensing event ever observed, OGLE-1999-BUL-32, also independently identified by the MACHO collaboration as MACHO-99-BLG-22. This unique event has an Einstein radius crossing time of 640 d. The high-quality data obtained with difference image analysis shows a small but significant parallax signature. This parallax effect allows one to determine the Einstein radius projected on to the observer plane as     . The transverse velocity projected on to the observer plane is about 79 km s⁻¹. We argue that the lens is likely to have a mass of at least a few solar masses, i.e. it could be a stellar black hole. The black hole hypothesis can be tested using the astrometric microlensing signature with the soon-to-be installed Advanced Camera for Surveys on board the Hubble Space Telescope . Deep X-ray and radio images may also be useful for revealing the nature of the object.     

The edge-on spiral gravitational lens B1600+434

We present new observations of the gravitational lens (GL) system B1600+434, strongly suggesting that the lens is an edge-on spiral galaxy. These observations are used to constrain the mass model of the system, in particular the oblateness and velocity dispersion of the dark matter halo around the lensing galaxy. From an analytical model we find a lower limit on the halo oblateness q _halo=( c/a )_ρ≳0.4; more detailed numerical models give a lower limit of q _halo≳0.5. We determine an average halo velocity dispersion of σ_halo=190±15 km s⁻¹ over all non-singular isothermal elliptical (NIE) halo models. Constraining the models to larger and more massive discs decreases this average by only 10 km s⁻¹. A lower limit of σ_halo≳150 km s⁻¹ is found, even for disc masses larger than the mass inside the Einstein radius. This lower limit indicates the need for a massive dark matter halo, contributing at least half of the mass inside the Einstein radius. Time-delay calculations give (54±3)/ h ₅₀ d for the NIE halo model and (70±4)/ h ₅₀ d for the modified Hubble profile (MHP) halo model. Although the time delay for both NIE and MHP halo models is well constrained on our parameter grid, it strongly depends on the halo surface density profile. We furthermore find that the presence of a flat luminous mass distribution can severely alter the statistical properties of the lens.     

Molecular hydrogen emission in Cygnus A

We present J , H and K -band spectroscopy of Cygnus A, spanning 1.0–2.4 μm in the rest-frame and hence several rovibrational H₂, H recombination and [Fe  ii ] emission lines. The lines are spatially extended by up to 6 kpc from the nucleus, but their distinct kinematics indicate that the three groups (H, H₂ and [Fe  ii ]) are not wholly produced in the same gas. The broadest line, [Fe  ii ] λ 1.644, exhibits a non-Gaussian profile with a broad base (FWHM≃1040 km s⁻¹), perhaps because of the interaction with the radio source. Extinctions to the line-emitting regions substantially exceed earlier measurements based on optical H recombination lines.
Hard X-rays from the quasar nucleus are likely to dominate the excitation of the H₂ emission. The results of Maloney, Hollenbach & Tielens are thus used to infer the total mass of gas in H₂ v=1–0 S(1)-emitting clouds as a function of radius, for gas densities of 10³ and 10⁵ cm⁻³, and stopping column densities N _H=10²²–10²⁴ cm⁻². Assuming azimuthal symmetry, at least 2.3×10⁸ M_⊙ of such material is present within 5 kpc of the nucleus, if the line-emitting clouds see an unobscured quasar spectrum. Alternatively, if the bulk of the X-ray absorption to the nucleus inferred by Ueno et al. actually arises in a circumnuclear torus, the implied gas mass rises to ∼10¹⁰ M_⊙. The latter plausibly accounts for 10⁹ yr of mass deposition from the cluster cooling flow, for which within this radius.     

The clustering of Lyman-break galaxies

We calculate the statistical clustering of Lyman-break galaxies predicted in a selection of currently fashionable structure formation scenarios. These models are all based on the cold dark matter model, but vary in the amount of dark matter, the initial perturbation spectrum, the background cosmology and the presence or absence of a cosmological constant term. If Lyman-break galaxies form as a result of hierarchical merging, the amplitude of clustering depends quite sensitively on the minimum halo mass that can host such a galaxy. Interpretation of the recent observations by Giavalisco et al. would therefore be considerably clarified by a direct determination of the relevant halo properties. For a typical halo mass around 10¹¹  h ⁻¹ M⊙ the observations do not discriminate strongly between cosmological models, but if the appropriate mass is larger, say 10¹²  h ⁻¹ M⊙ (which seems likely on theoretical grounds), then the data strongly favour models with a low matter density.     

Non-Gaussian cosmic microwave background temperature fluctuations from peculiar velocities of clusters

We use numerical simulations of a (480 Mpc  h ⁻¹)³ volume to show that the distribution of peak heights in maps of the temperature fluctuations from the kinematic and thermal Sunyaev–Zeldovich (SZ) effects will be highly non-Gaussian, and very different from the peak-height distribution of a Gaussian random field. We then show that it is a good approximation to assume that each peak in either SZ effect is associated with one and only one dark matter halo. This allows us to use our knowledge of the properties of haloes to estimate the peak-height distributions. At fixed optical depth, the distribution of peak heights resulting from the kinematic effect is Gaussian, with a width that is approximately proportional to the optical depth; the non-Gaussianity comes from summing over a range of optical depths. The optical depth is an increasing function of halo mass and the distribution of halo speeds is Gaussian, with a dispersion that is approximately independent of halo mass. This means that observations of the kinematic effect can be used to put constraints on how the abundance of massive clusters evolves, and on the evolution of cluster velocities. The non-Gaussianity of the thermal effect, on the other hand, comes primarily from the fact that, on average, the effect is larger in more massive haloes, and the distribution of halo masses is highly non-Gaussian. We also show that because haloes of the same mass may have a range of density and velocity dispersion profiles, the relation between halo mass and the amplitude of the thermal effect is not deterministic, but has some scatter.     

Evolution of massive binary black holes

Since many or most galaxies have central massive black holes (BHs), mergers of galaxies can form massive binary black holes (BBHs). In this paper we study the evolution of massive BBHs in realistic galaxy models, using a generalization of techniques used to study tidal disruption rates around massive BHs. The evolution of BBHs depends on BH mass ratio and host galaxy type. BBHs with very low mass ratios (say, ≲0.001) are hardly ever formed by mergers of galaxies, because the dynamical friction time-scale is too long for the smaller BH to sink into the galactic centre within a Hubble time. BBHs with moderate mass ratios are most likely to form and survive in spherical or nearly spherical galaxies and in high-luminosity or high-dispersion galaxies; they are most likely to have merged in low-dispersion galaxies (line-of-sight velocity dispersion ≲90 km s⁻¹) or in highly flattened or triaxial galaxies.
The semimajor axes and orbital periods of surviving BBHs are generally in the range  10^-3–10 pc  and  10–10⁵ yr;  they are also larger in high-dispersion galaxies than in low-dispersion galaxies, larger in nearly spherical galaxies than in highly flattened or triaxial galaxies, and larger for BBHs with equal masses than for BBHs with unequal masses. The orbital velocities of surviving BBHs are generally in the range  10²–10⁴ km s^-1  . The methods of detecting surviving BBHs are also discussed.
If no evidence of BBHs is found in AGNs, this may be either because gas plays a major role in BBH orbital decay or because nuclear activity switches on soon after a galaxy merger, and ends before the smaller BH has had time to spiral to the centre of the galaxy.     

The properties of cooling flows in X-ray luminous clusters of galaxies

We discuss the X-ray properties of the cooling flows in a sample of 30 highly X-ray luminous clusters of galaxies, observed using the ASCA and ROSAT satellites. We demonstrate the need for multiphase models to consistently explain the spectral and imaging X-ray data for the clusters. The mass deposition rates of the cooling flows, independently determined from the ASCA spectra and ROSAT images, exhibit good agreement and exceed 1000 M_⊙ yr⁻¹ in the largest systems. We confirm the presence of intrinsic X-ray absorption in the clusters using a variety of spectral models. The measured equivalent hydrogen column densities of absorbing material are sensitive to the spectral models used in the analysis, varying from average values of a few 10²⁰ atom cm⁻² for a simple isothermal emission model, to a few 10²¹ atom cm⁻² using our preferred cooling flow models, assuming in each case that the absorber lies in a uniform foreground screen. The true intrinsic column densities are likely to be even higher if the absorbing medium is distributed throughout the clusters. We summarize the constraints on the nature of the X-ray absorber from observations in other wavebands. Much of the X-ray absorption may be caused by dust.     

Dark matter halo profiles in scale-free cosmologies

We explore the dependence of the central logarithmic slope of dark matter halo density profiles α on the spectral index n of the linear matter power spectrum P ( k ) using cosmological N -body simulations of scale-free models [i.e. P ( k ) ∝ k ⁿ ]. These simulations are based on a set of clear, reproducible and physically motivated criteria that fix the appropriate starting and stopping times for runs, and allow one to compare haloes across models with different spectral indices and mass resolutions. For each of our simulations we identify samples of well-resolved haloes in dynamical equilibrium and we analyse their mass profiles. By parametrizing the mass profile using a 'generalized' Navarro, Frenk & White profile in which the central logarithmic slope α is allowed to vary while preserving the r ⁻³ asymptotic form at large radii, we obtain preferred central slopes for haloes in each of our models. There is a strong correlation between α and n , such that α becomes shallower as n becomes steeper. However, if we normalize our mass profiles by r ₋₂, the radius at which the logarithmic slope of the density profile is −2, we find that these differences are no longer present. This is apparent if we plot the maximum slope     as a function of r / r ₋₂– we find that the profiles are similar for haloes forming in different n models. This reflects the importance of concentration, and reveals that the concentrations of haloes forming in steep- n cosmologies tend to be smaller than those of haloes forming in shallow- n cosmologies. We conclude that there is no evidence for convergence to a unique central asymptotic slope, at least on the scales that we can resolve.     

The coarse geometry of merger trees in Λ cold dark matter

We introduce the contour process to describe the geometrical properties of merger trees. The contour process translates a tree structure into a one-dimensional object: the contour walk. We characterize the contour walk measuring its length and action. The length is proportional to the number of progenitors in the tree, and the action is a proxy for the mean length of a branch in the tree.
We construct the contour walk for merger trees extracted from the public data base of the Millennium Run and also for merger trees constructed with a public Monte Carlo code which implements a Markovian algorithm. The trees correspond to haloes of final masses between 10¹¹ and  10¹⁴  h ⁻¹ M_⊙  . We study how the length and action scale with the mass of the final halo.
In all cases, except for the action measured from Markovian trees, we find a transitional scale around  1–6 × 10¹²  h ⁻¹ M_⊙  , which can be interpreted as the point where the halo growth starts to be dominated by mergers and not accretion. The transitional scale depends weakly on the mass resolution and time discretization of the merging trees.