Halo Spin Research Articles

STOCHASTIC MODEL OF THE SPIN DISTRIBUTION OF DARK MATTER HALOS

We employ a stochastic approach to probing the origin of the log-normal distributions of halo spin in N-body simulations. After analyzing spin evolution in halo merging trees, it was found that a spin change can be characterized by a stochastic random walk of angular momentum. Also, spin distributions generated by random walks are fairly consistent with those directly obtained from N-body simulations. We derived a stochastic differential equation from a widely used spin definition and measured the probability distributions of the derived angular momentum change from a massive set of halo merging trees. The roles of major merging and accretion are also statistically analyzed in evolving spin distributions. Several factors (local environment, halo mass, merging mass ratio, and redshift) are found to influence the angular momentum change. The spin distributions generated in the mean-field or void regions tend to shift slightly to a higher spin value compared with simulated spin distributions, which seems to be caused by the correlated random walks. We verified the assumption of randomness in the angular momentum change observed in the N-body simulation and detected several degrees of correlation between walks, which may provide a clue for the discrepancies between the simulated and generated spin distributions in the voids. However, the generated spin distributions in the group and cluster regions successfully match the simulated spin distribution. We also demonstrated that the log-normality of the spin distribution is a natural consequence of the stochastic differential equation of the halo spin, which is well described by the Geometric Brownian Motion model.

FLOW PATTERNS AROUND DARK MATTER HALOS: THE LINK BETWEEN HALO DYNAMICAL PROPERTIES AND LARGE-SCALE TIDAL FIELD

We study how halo intrinsic dynamical properties are linked to their formation processes for halos in two mass ranges, $10^{12}-10^{12.5}h^{-1}{\rm M_\odot}$ and $\ge 10^{13}h^{-1}{\rm M_\odot}$, and how both are correlated with the large scale tidal field within which the halos reside at present. Halo merger trees obtained from cosmological $N$-body simulations are used to identify infall halos that are about to merge with their hosts. We find that the tangential component of the infall velocity increases significantly with the strength of the local tidal field, but no strong correlation is found for the radial component. These results can be used to explain how the internal velocity anisotropy and spin of halos depend on environment. The position vectors and velocities of infall halos are aligned with the principal axes of the local tidal field, and the alignment depends on the strength of the tidal field. Opposite accretion patterns are found in weak and strong tidal fields, in the sense that in a weak field the accretion flow is dominated by radial motion within the local structure, while a large tangential component is present in a strong field. These findings can be used to understand the strong alignments we find between the principal axes of the internal velocity ellipsoids of halos and the local tidal field, and their dependence on the strength of tidal field. They also explain why halo spin increases with the strength of local tidal field, but only in weak tidal fields does the spin-tidal field alignment follow the prediction of the tidal torque theory. We discuss how our results may be used to understand the spins of disk galaxies and velocity structures of elliptical galaxies and their correlations with large-scale structure.

Structural properties of artificial halos in nonstandard dark matter simulations

Artificial fragmentation of the matter density field causes the formation of spurious groups of particles in N-body simulations of non-standard Dark Matter (DM) models which are characterized by a small scale cut-off in the linear matter power spectrum. These spurious halos alter the prediction of the mass function in a range of masses where differences among DM models are most relevant to observational tests. Using a suite of high resolution simulations we show that the contamination of artificial groups of particles significantly affect the statistics of halo spin, shape and virial state parameters. We find that spurious halos have systematically larger spin values, are highly elliptical or prolate and significantly deviate from virial equilibrium. These characteristics allow us to detect the presence of spurious halos even in non-standard DM models for which the low-mass end of the mass function remains well behaved. We show that selecting halos near the virial equilibrium provides a simple and effective method to remove the bulk of spurious halos from numerical halo catalogs and consistently recover the halo mass function at low masses.

How to bend galaxy disc profiles: the role of halo spin

Abstract The radial density profiles of stellar galaxy discs can be well approximated as an exponential. Compared to this canonical form, however, the profiles in the majority of disc galaxies show downward or upward breaks at large radii. Currently, there is no coherent explanation in a galaxy formation context of the radial profile per se, along with the two types of profile breaks. Using a set of controlled hydrodynamic simulations of disc galaxy formation, we find a correlation between the host halo's initial angular momentum and the resulting radial profile of the stellar disc: galaxies that live in haloes with a low spin parameter λ ≲ 0.03 show an up-bending break in their disc density profiles, while galaxies in haloes of higher angular momentum show a down-bending break. We find that the case of pure exponential profiles (λ ≈ 0.035) coincides with the peak of the spin parameter distribution from cosmological simulations. Our simulations not only imply an explanation of the observed behaviours, but also suggest that the physical origin of this effect is related to the amount of radial redistribution of stellar mass, which is anticorrelated with λ.

SPIN ALIGNMENTS OF SPIRAL GALAXIES WITHIN THE LARGE-SCALE STRUCTURE FROM SDSS DR7

Using a sample of spiral galaxies selected from the Sloan Digital Sky Survey Data Release 7 (SDSS DR7) and Galaxy Zoo 2 (GZ2), we investigate the alignment of spin axes of spiral galaxies with their surrounding large scale structure, which is characterized by the large-scale tidal field reconstructed from the data using galaxy groups above a certain mass threshold. We find that the spin axes of only have weak tendency to be aligned with (or perpendicular to) the intermediate (or minor) axis of the local tidal tensor. The signal is the strongest in a \cluster environment where all the three eigenvalues of the local tidal tensor are positive. Compared to the alignments between halo spins and local tidal field obtained in N-body simulations, the above observational results are in best agreement with those for the spins of inner regions of halos, suggesting that the disk material traces the angular momentum of dark matter halos in the inner regions.

Swirling around filaments: are large-scale structure vortices spinning up dark haloes?

The kinematic analysis of dark matter and hydrodynamical simulations suggests that the vorticity in large-scale structure is mostly confined to, and predominantly aligned with their filaments, with an excess of probability of 20 per cent to have the angle between vorticity and filaments direction lower than 60 degrees relative to random orientations. The cross sections of these filaments are typically partitioned into four quadrants with opposite vorticity sign, arising from multiple flows, originating from neighbouring walls. The spins of halos embedded within these filaments are consistently aligned with this vorticity for any halo mass, with a stronger alignment for the most massive structures up to an excess of probability of 165 per cent. On large scales, adiabatic/cooling hydrodynamical simulations display the same vorticity in the gas as in the dark matter. The global geometry of the flow within the cosmic web is therefore qualitatively consistent with a spin acquisition for smaller halos induced by this large-scale coherence, as argued in Codis et al. (2012). In effect, secondary anisotropic infall (originating from the vortex-rich filament within which these lower-mass halos form) dominates the angular momentum budget of these halos. The transition mass from alignment to orthogonality is related to the size of a given multi-flow region with a given polarity. This transition may be reconciled with the standard tidal torque theory if the latter is augmented so as to account for the larger scale anisotropic environment of walls and filaments.

HIghMass—HIGH H I MASS, H I-RICH GALAXIES ATz∼ 0 HIGH-RESOLUTION VLA IMAGING OF UGC 9037 AND UGC 12506

We present resolved HI observations of two galaxies, UGC 9037 and UGC 12506, members of a rare subset of galaxies detected by the ALFALFA extragalactic HI survey characterized by high HI mass and high gas fraction for their stellar masses. Both of these galaxies have M$_*>10^{10}$ M$_\odot$ and M$_\text{HI}>$ M$_*$, as well as typical star formation rates for their stellar masses. How can such galaxies have avoided consuming their massive gas reservoirs? From gas kinematics, stability, star formation, and dark matter distributions of the two galaxies, we infer two radically different histories. UGC 9037 has high central HI surface density ($>10$ M$_\odot$ pc$^{-2}$). Its gas at most radii appears to be marginally unstable with non-circular flows across the disk. These properties are consistent with UGC 9037 having recently acquired its gas and that it will soon undergo major star formation. UGC 12506 has low surface densities of HI, and its gas is stable over most of the disk. We predict its gas to be HI-dominated at all except the smallest radii. We claim a very high dark matter halo spin parameter for UGC 12506 ($\lambda=0.15$), suggesting that its gas is older, and has never undergone a period of star formation significant enough to consume the bulk of its gas.

HIghMass-HIGH H I MASS, H I-RICH GALAXIES ATz∼ 0 SAMPLE DEFINITION, OPTICAL AND Hα IMAGING, AND STAR FORMATION PROPERTIES

We present first results of the study of a set of exceptional HI sources identified in the 40% ALFALFA extragalactic HI survey catalog alpha.40 as being both HI massive (M_HI > 10^10 Msun) and having high gas fractions for their stellar masses: the HIghMass galaxy sample. We analyze UV- and optical-broadband and Halpha images to understand the nature of their relatively underluminous disks in optical and to test whether their high gas fractions can be tracked to higher dark matter halo spin parameters or late gas accretion. Estimates of their star formation rates (SFRs) based on SED-fitting agree within uncertainties with the Halpha luminosity inferred SFRs. The HII region luminosity functions have standard slopes at the luminous end. The global SFRs demonstrate that the HIghMass galaxies exhibit active ongoing star formation (SF) with moderate SF efficiency, but relative to normal spirals, a lower integrated SFR in the past. Because the SF activity in these systems is spread throughout their extended disks, they have overall lower SFR surface densities and lower surface brightness in the optical bands. Relative to normal disk galaxies, the majority of HIghMass galaxies have higher Halpha equivalent widths and are bluer in their outer disks, implying an inside-out disk growth scenario. Downbending double exponential disks are more frequent than upbending disks among the gas-rich galaxies, suggesting that SF thresholds exist in the downbending disks, probably as a result of concentrated gas distribution.

The Vainshtein mechanism in the cosmic web

We investigate the dependence of the Vainshtein screening mechanism on the cosmic web morphology of both dark matter particles and halos as determined by ORIGAMI. Unlike chameleon and symmetron screening, which come into effect in regions of high density, Vainshtein screening instead depends on the dimensionality of the system, and screened bodies can still feel external fields. ORIGAMI is well-suited to this problem because it defines morphologies according to the dimensionality of the collapsing structure and does not depend on a smoothing scale or density threshold parameter. We find that halo particles are screened while filament, wall, and void particles are unscreened, and this is independent of the particle density. However, after separating halos according to their large scale cosmic web environment, we find no difference in the screening properties of halos in filaments versus halos in clusters. We find that the fifth force enhancement of dark matter particles in halos is greatest well outside the virial radius. We confirm the theoretical expectation that even if the internal field is suppressed by the Vainshtein mechanism, the object still feels the fifth force generated by the external fields, by measuring peculiar velocities and velocity dispersions of halos. Finally, we investigate the morphology and gravity model dependence of halo spins, concentrations, and shapes.

HALO SPIN PARAMETER IN COSMOLOGICAL SIMULATIONS

Using a cosmological ΛCDM simulation, we analyze the differences between the widely-used spin parameters suggested by Peebles and Bullock. The dimensionless spin parameter λ proposed by Peebles is theoretically well-justified but includes an annoying term, the potential energy, which cannot be directly obtained from observations and is computationally expensive to calculate in numerical simulations. The Bullock’s spin parameter λ′ avoids this problem assuming the isothermal density profile of a virialized halo in the Newtonian potential model. However, we find that there exists a substantial discrepancy between λ and λ′ depending on the adopted potential model (Newtonian or Plummer) to calculate the halo total energy and that their redshift evolutions differ to each other significantly. Therefore, we introduce a new spin parameter, λ′′, which is simply designed to roughly recover the value of λ but to use the same halo quantities as used in λ′. If the Plummer potential is adopted, the λ′′ is related to the Bullock’s definition as λ′′ = 0.80 × (1 + z)? 1/12 λ′. Hence, the new spin parameter λ′′ distribution becomes consistent with a log-normal distribution frequently seen for the λ′ while its mean value is much closer to that of λ. On the other hand, in case of the Newtonian potential model, we obtain the relation of λ′′ = (1 + z)? 1/8 λ′; there is no significant difference at z = 0 as found by others but λ′ becomes more overestimated than λ or λ′′ at higher redshifts. We also investigate the dependence of halo spin parameters on halo mass and redshift. We clearly show that although the λ′ for small-mass halos with Mh 12 M ⊙ seems redshift independent after z = 1, all the spin parameters explored, on the whole, show a stronger correlation with the increasing halo mass at higher redshifts.

SECULAR DAMPING OF STELLAR BARS IN SPINNING DARK MATTER HALOS

We demonstrate that growth of stellar bars in spinning dark matter halos is heavily suppressed in the secular phase of evolution, using numerical simulations of isolated galaxies. In a representative set of models, we show that for values of the cosmological spin parameter lambda > 0.03, bar growth (in strength and size) becomes increasingly quenched. Furthermore, slowdown of bar pattern speed weakens substantially with increasing `lambda,' until it ceases completely. The terminal structure of bars is affected as well, including extent and shape of their boxy/peanut bulges. The essence of this effect lies in the modified angular momentum exchange between the disk and the halo facilitated by the bar. For the first time we have demonstrated that a dark matter halo can emit and not purely absorb angular momentum. Although the halo as a whole is not found to emit, the net transfer of angular momentum from the disk to the halo is significantly reduced or completely eliminated. The paradigm shift implies that the accepted view that disks serve as sources of angular momentum and halos serve as sinks, must be revised. Halos with lambda > 0.03 are expected to form a substantial fraction, based on lognormal distribution of lambda. Dependence of secular bar evolution on halo spin, therefore, implies profound corollaries for the cosmological evolution of galactic disks.

Hydrodynamical simulations of coupled and uncoupled quintessence models – II. Galaxy clusters

We study the $z=0$ properties of clusters (and large groups) of galaxies within the context of interacting and non-interacting quintessence cosmological models, using a series of adiabatic SPH simulations. Initially, we examine the average properties of groups and clusters, quantifying their differences in LCDM, uncoupled Dark Energy (\ude) and coupled Dark Energy (\cde) cosmologies. In particular, we focus upon radial profiles of the gas density, temperature and pressure, and we also investigate how the standard hydrodynamic equilibrium hypothesis holds in quintessence cosmologies. While we are able to confirm previous results about the distribution of baryons, we also find that the main discrepancy (with differences up to $20\%$) can be seen in cluster pressure profiles. We then switch attention to individual structures, mapping each halo in quintessence cosmology to its \LCDM\ counterpart. We are able to identify a series of small correlations between the coupling in the dark sector and halo spin, triaxiality and virialization ratio. When looking at spin and virialization of dark matter haloes, we find a weak ($5\%$) but systematic deviation in fifth force scenarios from \LCDM.

Hydrodynamical simulations of coupled and uncoupled quintessence models – I. Halo properties and the cosmic web

We present the results of a series of adiabatic hydrodynamical simulations of several quintessence models (both with a free and an interacting scalar field) in comparison to a standard \LCDM\ cosmology. For each we use $2\times1024^3$ particles in a $250$\hMpc\ periodic box assuming WMAP7 cosmology. In this work we focus on the properties of haloes in the cosmic web at $z=0$. The web is classified into \emph{voids}, \emph{sheets}, \emph{filaments} and \emph{knots} depending on the eigenvalues of the velocity shear tensor, which are an excellent proxy for the underlying overdensity distribution. We find that the properties of objects classified according to their surrounding environment shows a substantial dependence on the underlying cosmology; for example, while $V_{\rm max}$ shows average deviations of $\approx5$ per cent across the different models when considering the full halo sample, comparing objects classified according to their environment, the size of the deviation can be as large as $20$ per cent. We also find that halo spin parameters are positively correlated to the coupling, whereas halo concentrations show the opposite behaviour. Furthermore, when studying the concentration-mass relation in different environments, we find that in all cosmologies underdense regions have a larger normalization and a shallower slope. While this behaviour is found to characterize all the models, differences in the best-fit relations are enhanced in (coupled) dark energy, thus providing a clearer prediction for this class of models.

DM haloes in the fifth-force cosmology

We investigate how long-range scalar interactions affect the properties of dark matter haloes.For doing so we employ the ReBEL model which implements an additional interaction between dark matter particles.On the phenomenological level this is equivalent to a modification of gravity. We analyse the differences betweenfive ReBEL models and ΛCDM using a series of high resolution cosmological simulations. Emphasis is placed oninvestigating how halo properties change in the presence of a fifth force. We report that the density profileof ReBEL haloes is well described by the NFW profile but with mean concentrations from 5%to a few times higher than the standard ΛCDM value. We also find a slight increase of the halo spin for haloes more massivethan 5 × 1011M☉, reflecting a higher rotational support of those haloes due to scalar forces.In addition, the dark matter haloes in our models are more spherical than their counterparts in ΛCDM.The ReBEL haloes are also more virialised, with a large difference from ΛCDM for strong fifth forces and a much smaller changefor weak scalar interactions.

THE CLUSTERING OF ALFALFA GALAXIES: DEPENDENCE ON H I MASS, RELATIONSHIP WITH OPTICAL SAMPLES, AND CLUES OF HOST HALO PROPERTIES

We use a sample of ~6000 galaxies detected by the Arecibo Legacy Fast ALFA (ALFALFA) 21cm survey, to measure the clustering properties of HI-selected galaxies. We find no convincing evidence for a dependence of clustering on the galactic atomic hydrogen (HI) mass, over the range M_HI ~ 10^{8.5} - 10^{10.5} M_sun. We show that previously reported results of weaker clustering for low-HI mass galaxies are probably due to finite-volume effects. In addition, we compare the clustering of ALFALFA galaxies with optically selected samples drawn from the Sloan Digital Sky Survey (SDSS). We find that HI-selected galaxies cluster more weakly than even relatively optically faint galaxies, when no color selection is applied. Conversely, when SDSS galaxies are split based on their color, we find that the correlation function of blue optical galaxies is practically indistinguishable from that of HI-selected galaxies. At the same time, SDSS galaxies with red colors are found to cluster significantly more than HI-selected galaxies, a fact that is evident in both the projected as well as the full two-dimensional correlation function. A cross-correlation analysis further reveals that gas-rich galaxies "avoid" being located within ~3 Mpc of optical galaxies with red colors. Next, we consider the clustering properties of halo samples selected from the Bolshoi LambdaCDM simulation. A comparison with the clustering of ALFALFA galaxies suggests that galactic HI mass is not tightly related to host halo mass, and that a sizable fraction of subhalos do not host HI galaxies. Lastly, we find that we can recover fairly well the correlation function of HI galaxies by just excluding halos with low spin parameter. This finding lends support to the hypothesis that halo spin plays a key role in determining the gas content of galaxies.

GALAXY SPIN ALIGNMENT IN FILAMENTS AND SHEETS: OBSERVATIONAL EVIDENCE

Galaxies properties are known to be affected by their environment. One important question is how their angular momentum reflects the surrounding cosmic web. We use the SDSS to investigate the spin axes of spiral and elliptical galaxies relative to their surrounding filament/sheet orientations. To detect filaments a marked point process with interactions (the "Bisous model") is used. Sheets are found by detecting "flattened" filaments. The minor axes of ellipticals are found to be preferentially perpendicular to hosting filaments. A weak correlation is found with sheets. These findings are consistent with the notion that elliptical galaxies formed via mergers which predominantly occurred along the filaments. The spin axis of spiral galaxies is found to align with the host filament, with no correlation between spiral spin and sheet normal. When examined as a function of distance from the filament axis, a much stronger correlation is found in outer parts, suggesting that the alignment is driven by the laminar infall of gas from sheets to filaments. When compared with numerical simulations our results suggest that the connection between DM halo and galaxy spin is not straightforward. Our results provide an important input to the understanding of how galaxies acquire their angular momentum.

Spinning dark matter haloes promote bar formation

Stellar bars are the most common non-axisymmetric structures in galaxies and their impact on the evolution of disc galaxies at all cosmological times can be significant. Classical theory predicts that stellar discs are stabilized against bar formation if embedded in massive spheroidal dark matter halos. However, dark matter halos have been shown to facilitate the growth of bars through resonant gravitational interaction. Still, it remains unclear why some galaxies are barred and some are not. In this study, we demonstrate that co-rotating (i.e., in the same sense as the disc rotating) dark matter halos with spin parameters in the range of $0 \le \lambda_{\mathrm{dm}} \le 0.07$ - which are a definite prediction of modern cosmological models - promote the formation of bars and boxy bulges and therefore can play an important role in the formation of pseudobulges in a kinematically hot dark matter dominated disc galaxies. We find continuous trends for models with higher halo spins: bars form more rapidly, the forming slow bars are stronger, and the final bars are longer. After 2 Gyrs of evolution, the amplitude of the bar mode in a model with $\lambda_{\mathrm{dm}} = 0.05$ is a factor of ~6 times higher, A_2/A_0 = 0.23, than in the non-rotating halo model. After 5 Gyrs, the bar is ~ 2.5 times longer. The origin of this trend is that more rapidly spinning (co-rotating) halos provide a larger fraction of trailing dark matter particles that lag behind the disc bar and help growing the bar by taking away its angular momentum by resonant interactions. A counter-rotating halo suppresses the formation of a bar in our models. We discuss potential consequences for forming galaxies at high-redshift and present day low mass galaxies which have converted only a small fraction of their baryons into stars.

How does the surface density and size of disc galaxies measured in hydrodynamic simulations correlate with the halo spin parameter?

Late-type low surface brightness galaxies (LSBs) are faint disc galaxies with central maximum stellar surface densities below 100 M⊙ pc−2. The currently favoured scenario for their origin is that LSBs have formed in fast-rotating haloes with large angular momenta. We present the first numerical evidence for this scenario using a suite of self-consistent hydrodynamic simulations of a 2.3 × 1011 M⊙ galactic halo, in which we investigate the correlations between the disc stellar/gas surface densities and the spin parameter of its host halo. A clear anticorrelation between the surface densities and the halo spin parameter λ is found. That is, as the halo spin parameter increases, the disc cutoff radius at which the stellar surface density drops below 0.1 M⊙ pc−2 monotonically increases, while the average stellar surface density of the disc within that radius decreases. The ratio of the average stellar surface density for the case of λ = 0.03 to that for the case of λ = 0.14 reaches more than 15. We demonstrate that the result is robust against variations in the baryon fraction, confirming that the angular momentum of the host halo is an important driver for the formation of LSBs.

COSMIC VORTICITY AND THE ORIGIN HALO SPINS

In the standard model of cosmology, structure emerges out of non-rotational flow and the angular momentum of collapsing halos is induced by tidal torques. The growth of halo angular momentum in the linear and quasi-linear phases is associated with a shear, curl-free, flow and it is well described within the linear framework of tidal torque theory (TTT). However, TTT is rendered irrelevant as haloes approach turn around and virialization. At that stage the flow field around halos has non-zero vorticity. Using a cosmological simulation, we have examined the importance of the curl of the velocity field (vorticity) in determining halo spin, finding a strong alignment between the two. We have also examined the alignment of vorticity with the principle axes of the shear tensor, finding that it tends to be perpendicular to the axis along which material is collapsing fastest (e1). This behavior is independent of halo masses and cosmic web environment. Our results agree with previous findings on the tendency of halo spin to be perpendicular to e1, and of the spin of (simulated) halos and (observed) galaxies to be aligned with the large-scale structure. Our results imply that angular momentum growth proceeds in two distinct phases. In the first phase angular momentum emerges out of a shear, curl-free, potential flow, as described by TTT. In the second phase, in which haloes approach virialization, the angular momentum emerges out of a vortical flow and halo spin becomes strongly aligned with the vorticity of the ambient flow field.

The impact of baryons on the spins and shapes of dark matter haloes

We use numerical simulations to investigate how the statistical properties of dark matter (DM) haloes are affected by the baryonic processes associated with galaxy formation. We focus on how these processes influence the spin and shape of a large number of DM haloes covering a wide range of mass scales, from galaxies to clusters at redshifts zero and one, extending to dwarf galaxies at redshift two. The haloes are extracted from the OverWhelmingly Large Simulations, a suite of state-of-the-art high-resolution cosmological simulations run with a range of feedback prescriptions. We find that the median spin parameter in DM-only simulations is independent of mass, redshift and cosmology. At z=0 baryons increase the spin of the DM in the central region (<=0.25 r_200) by up to 30 per cent when feedback is weak or absent. This increase can be attributed to the transfer of angular momentum from baryons to the DM, but is no longer present at z=2. We also present fits to the mass dependence of the DM halo shape at both low and high redshift. At z=0 the sphericity (triaxiality) is negatively (positively) correlated with halo mass and both results are independent of cosmology. Interestingly, these mass-dependent trends are markedly weaker at z=2. While the cooling of baryons acts to make the overall DM halo more spherical, stronger feedback prescriptions tend to reduce the impact of baryons by reducing the central halo mass concentration. More generally, we demonstrate a strongly positive (negative) correlation between halo sphericity (triaxiality) and galaxy formation efficiency, with the latter measured using the central halo baryon fraction. In conclusion, our results suggest that the effects of baryons on the DM halo spin and shape are minor when the effects of cooling are mitigated, as required by realistic models of galaxy formation, although they remain significant for the inner halo.