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Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed May 3, 2011 (MN LATEX style file v2.2)

Rigging dark halos: why is hierarchical galaxy formationconsistent with the inside-out build-up of thin discs?

C. Pichon1,2,3, D. Pogosyan4, T. Kimm2, A. Slyz2, J. Devriendt2,3 and Y. Dubois21 Institut d’Astrophysique de Paris, 98 bis boulevard Arago, 75014 Paris, France2 Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK.3 Observatoire de Lyon (UMR 5574), 9 avenue Charles Andre, F-69561 Saint Genis Laval, France.4Department of Physics, University of Alberta, 11322-89 Avenue, Edmonton, Alberta, T6G 2G7, Canada.

May 3, 2011

ABSTRACTState-of-the-art hydrodynamical simulations show that gas inflow through the virialsphere of dark matter halos is focused (i.e. has a preferred inflow direction), con-sistent (i.e. its orientation is steady in time) and amplified (i.e. the amplitude of ad-vected specific angular momentum increases with time). This is a consequence of thedynamics of the cosmic web within the neighbourhood of the halo, which producessteady, angular momentum rich, filamentary inflow of cold gas. On large scales, thedynamics within neighbouring patches drives matter out of the surrounding voids,into walls and filaments before it finally gets accreted onto virialised dark matterhalos. As these walls/filaments constitute the boundaries of asymmetric voids, theynaturally acquire a net transverse motion, which explains the angular momentumrich nature of the later infall which comes from further away (lever effect). We ar-gue that this large-scale driven consistency explains why cold flows are so efficientat building up thin discs from the inside out.

1 INTRODUCTION

One of the persistent puzzles of the standard paradigm ofgalaxy formation is the following: why do we observe thingalactic discs when hierarchical clustering naively suggeststhese galaxies should undergo repetitive random interac-tions with satellites and incoherent gas infall from their en-vironment? Indeed whilst one can argue that the probabil-ity of a head on collision with satellites should be small,incoherent but continuous gas infall poses a much greaterthreat to the ability of a bottom-up scenario of galaxy forma-tion to form ubiquitous thin galactic discs. Historically, as-tronomers (Rees & Ostriker 1977; Silk 1977) have invoked theprimordial monolithic collapse of a spheroidal body of gaswhich is shock-heated to its virial temperature. In this sce-nario, subsequent in-falling gas (Fillmore & Goldreich 1984;Bertschinger 1985, now described as secondary infall) shock-heats as it hits the virial radius, while the inner, denser regionof the hot gaseous sphere secularly rains onto the central discas it radiatively cools. This process has been coined the ”hotmode” of gas accretion and implies a clear correlation be-tween the spin of the hot galactic corona and that of the discwhich is effectively assumed to be shielded from its cosmicenvironment.

Over the last few years it has been (re)realized(Birnboim & Dekel (2003), Keres et al. (2005), Ocvirk et al.(2008), Brooks et al. (2009)) that most of the gas feedinggalaxies arrives cold (≈ 104K) from the large scale structuresalong filaments, and does not create such a halo filling coronaof hot gas, as the radiative shock it must necessarily generate

to do so is unstable to cooling processes. These investigationscorrespond to an update in the current Λ-CDM cosmologicalframework of the basic prediction of Binney (1977) who ar-gued that for all but the most massive galaxies, the accretedgas, provided it was dense enough, would never shock-heatto temperatures where Bremsstrahlung dominates cooling asit would first cool by atomic transitions. In the author’s ownwords, the accretion shock would be ’isothermal’ rather than’adiabatic’ and consequently, only a negligible fraction of thegas would ever reach temperatures T ∼ Tvir. In parallel tothe work of Aubert et al. (2004) on anisotropic dark matter in-fall onto galactic halos, Katz et al. (2003) Keres et al. (2005),numerically confirmed that a large fraction of the gas is in-deed accreted through filamentary streams, where it remainscold before it reaches the galaxy (see also Kay et al. (2000);Fardal et al. (2001) for early indication that this gas was notshock-heated). This “cold-mode“ accretion dominates theglobal growth of all galaxies at high redshifts (z > 3) andthe growth of low mass (Mhalo 6 5 × 1011 M⊙) objects at latetimes (Dekel & Birnboim 2006).

The addition of accretion through cold streams to thestandard galaxy formation framework has received muchattention (Keres et al. 2009; Brooks et al. 2009, e.g.) becauseof its potential implications for the star formation historyof galaxies (although see Benson & Bower (2011) for a de-fense of the opposite point of view). In the traditional ’hotmode’ picture, star formation is delayed as accreted gas isshock-heated and requires time to cool onto the central ob-ject. In contrast, if this material comes in cold, star forma-

2 C. Pichon, D. Pogosyan, T. Kimm, A. Slyz, J. Devriendt and Y. Dubois

tion can be fueled on a halo free-fall time. Cold-mode accre-tion should also have an important impact on the proper-ties (scale length, scale height, rotational velocity) of galac-tic discs, if as conjectured by Keres et al. (2005), cold streamsmerge onto disks ”like streams of cars entering an express-way”, converting a significant fraction of their infall veloc-ity to rotational velocity. Dekel et al. (2009) argued along thesame lines in their analysis of the MareNostrum simulation:the stream carrying the largest coherent flux with an impactparameter of a few kiloparsecs may determine the disc’s spinand orientation. Powell et al. (2010) spectacularly confirmedthese conjectures by showing that indeed, the filaments con-nect rather smoothly to the disc: they appear to join from dif-ferent directions, coiling around one another and forming athin extended disc structure, their high velocities driving itsrotation.

The way angular momentum is advected through thevirial sphere as a function of time is expected to play a keyrole in re-arranging the gas and dark matter within dark mat-ter halos. The pioneer works of Peebles (1969); Doroshkevich(1970); White (1984) addressed the issue of the original spinup of collapsed halos, explaining its linear growth up to thetime the initial overdensity decouples from the expansion ofthe Universe through the re-alignment of the primordial per-turbation’s inertial tensor with the shear tensor. However, lit-tle theoretical work has been devoted to analyzing the out-skirts of the Lagrangian patches associated with virialiseddark matter halos, which account for the later infall of gasand dark matter onto the already formed halos. In this pa-per, we quantify how significant this issue is and present aconsistent picture of the time evolution of angular momen-tum accretion based on our current theoretical understand-ing of the large scale structure dynamics. More specifically,the paper attempts to resolve the conundrum of why coldgas flows in Λ-CDM universes are consistent with thin diskformation. Indeed, as far as galactic disc formation is con-cerned, the heart of the matter lies in understanding how andwhen gas is accreted onto the disc. In other words, what arethe geometry and temporal evolution of the gas accretion?

In the ’standard’ paradigm of disc formation, this ques-tion was split in two. The dark matter and gas present in thevirialised halo both acquired angular momentum throughtidal torques in the pre-virialisation stage, i.e. until turn-around (e.g White 1984). The gas was later shock-heated asit collapsed, and secularly cooled and condensed into a disk(Fall & Efstathiou 1980) having lost most of the connectionwith its anisotropic cosmic past. In the modern cold modeaccretion picture which now seems to dominate all but themost massive halos, these questions need to be re-addressed.

The outline of the paper is as follows: in section 2, us-ing hydrodynamical simulations, we report evidence that fil-amentary flows advect an ever increasing amount of angularmomentum through the halo virial sphere. We also demon-strate that the orientation of these flows is consistent, i.e.maintained over long periods of time. Section 3 presents re-sults obtained through simplified pure dark matter simula-tions of the collapse of a Lagrangian patch associated witha virialised halo as these have the merit of better illustratingthe dynamics of matter flows in the outskirts of the halo. Sec-tion 4 is devoted to validation, conclusions and prospects.

Figure 1. Top panel: a typical galaxy residing in a high mass halo(M ∼ 2 × 1012 M⊙) from the MareNostrum simulation and its sur-rounding filaments as seen in the gas (top left panel, colour coded bydensity) and dark matter (top right panel). The virial sphere is alsoshown here. Gas filaments are significantly thinner than their darkmatter counterpart. Note the extent and the coherence of the largescale gaseous filaments surrounding that galaxy. The bottom panelsdisplay a time sequence representing the evolution of the halo be-tween z = 4 and z = 2.5 colour coded with density (red) metallicity(green), and temperature (blue). The consistency of the direction ofthe infalling gas (in red) is obvious.

0.0

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0.6

w(θ

)

1010 - 1011 Msun

0 20 40 60 80θ

z=6.1→5.0

z=5.0→3.8

z=3.8→2.5

Figure 2. The covariances (thick line) between different redshifts(as labeled) of the thresholded density maps on the virial sphere,Rvir, together with the corresponding dispersion (inter-quartile, dot-ted lines). The lower bound of the thresholded density is chosensuch that filamentary structures stand out, while the upper boundis adopted to minimise the signal from the satellites (see the text,Section 2). The orientation of filaments is temporally coherent, as isqualitatively illustrated in Figure 1 (bottom panel).

2 HYDRODYNAMICAL EVIDENCE

Let us start by briefly reporting the relevant hydrodynam-ical results we have obtained. We statistically analysed theadvected specific angular momentum of both gas and darkmatter at the virial radius of dark haloes in the MareNostrumcosmological simulation at redshift 6.1, 5.0, 3.8, 2.5 and 1.5(see Figure 1, Details can be found in (Kimm 2011, in prep.)).

The MareNostrum simulation (Ocvirk et al. 2008;Devriendt et al. 2010) was carried out using the Eulerianhydrodynamic code, RAMSES (Teyssier 2002), which uses anAdaptive Mesh Refinement (AMR) technique. It followed

Rigging dark halos 3

10 11 12 13log Mvir

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Figure 3. Left: advected specific momentum as a function of cosmic time and mass in the MareNostrum simulation. Note that more and moreangular momentum is being advected as a function of time and mass. Right: advected spin parameter as a function of cosmic time and mass; aredshift trend persists, while the amplitude of the advected spin parameter is larger than the inner value of 0.04 for the dark matter component.

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0.6

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Figure 4. The ratio of the component of the advected gas’s spin alongthe axis defined by the spin of dark matter, to the amplitude ofthe advected gas’s spin, as a function of mass and redshift, for the

dense (solid)/diffuse (dashed) component in the MareNostrum sim-ulation. This measurement corresponds to the median of 3119, 11357,and 15999 halos. The cold flow is more aligned than the diffuse flow.This supports the view that the momentum rich gas flows along thefilaments and preserves most of its orientation. See Figure 6 for amap illustrating the concentration of momentum in the filaments.

the evolution of a cubic cosmological volume of 50h−1 Mpcon a side (comoving), containing 10243 dark matter particlesand an Eulerian root grid of 10243 gas cells. A ΛCDMconcordance universe (Ωm = 0.3, ΩΛ = 0.7, Ωb = 0.045, h =H0/[100 kms−1Mpc-1] = 0.7, σ8 = 0.9, n = 1) correspondingto the WMAP 1 best fit cosmology was adopted, resultingin a dark matter particle mass mp = 1.41 × 107 M⊙. Aquasi Lagrangian refinement policy was enforced to keepthe spatial resolution fixed at about 1 h−1kpc in physicalcoordinates. A uniform UV background instantaneously

-1.0 -0.5 0.0 0.5 1.0cos θGas-Gas

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osθ)

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Figure 5. PDF of the cosine of relative angle between the spin ofthe advected gas, 〈j〉vr , for different pairs of redshifts as labeled. Themean of the PDF is also represented as a vertical arrow for the vari-ous pairs of redshifts considered. The orientation of the spin of fila-ments is temporally coherent.

turned on at z = 8.5 was adopted (Haardt & Madau 1996).Gas was allowed to radiatively cool (Sutherland & Dopita1993) and sink in the potential well of DM halos. Wheneverthe gas exceeded a threshold density of nH = 0.1cm−3,five percent of it was turned into stars per local free-falltime. Massive young stars exploded as supernova aftera time delay of 10 Myr corresponding to their averagelifetime, and we modeled these explosions using a Sedovblast wave solution (Dubois & Teyssier 2008). Potentiallyimportant physics such as AGN feedback, magnetic fields,and radiative transfer were ignored in the simulation.

Let us first address the issue of the consistency of the ori-entation of large scale filaments that pierce the virial sphereof dark halos. Figure 2 displays the cross correlation of nHI

4 C. Pichon, D. Pogosyan, T. Kimm, A. Slyz, J. Devriendt and Y. Dubois

Figure 6. The distribution of the z component (i.e. along the spin axisof their dark matter halo) of the spin parameter (λz, right panels) atz=3.8 in the MareNostrum simulation for two different mass haloes:5 × 1012 (top panels) and 3 × 1012 M⊙(bottom panels). Projections ofthe hydrogen density are shown in the left panels. The field of viewis 4 virial radii on a side. It is apparent that some filaments displaylarge values of λz (colour coded in yellow).

maps on the virial sphere, Rvir for different pairs of red-shifts as labeled on the figure. This figure corresponds to thestatistical average over HEALPIX (Gorski et al. 2005) mapsof thresholded gas density measured at Rvir. Since we areprimarily interested in filaments, wall-like structures andvoid regions are excluded by imposing a redshift-dependentthreshold density. This is chosen as 6.6 times the critical den-sity of the universe, which identifies the filamentary struc-tures reasonably well at the resolution of the Mare Nostrumsimulation. The contribution from satellite galaxies is min-imised by replacing the region occupied by satellite galaxieswith gas at the density threshold for star formation: nH =0.1cm−3. The cross-correlation coefficient plotted is definedas

w(θ) = ∑l,m

〈almb∗lm〉Pl(cos(θ))/√

(∑l,m

〈|alm|2〉∑l,m

〈|blm|2〉) ,

where alm and blm are the harmonic transforms of the mapsat the two given redshifts, Pl is the Legendre polynomial and〈 〉 stands for the statistical median over halos within theconsidered mass range.

Figure 2 shows that for halos with M ∼ 1010 − 1011 M⊙there is a significant correlation of the directions of the gasinfall between markedly different redshifts, w(0) ≈ 0.3− 0.4.This is especially true at high redshifts. Since halos of thismass have accumulated the bulk of their matter over ∆z ≈1.2 by redshift z ≈ 3.5, this value yields a lower bound on theamount of temporal coherence achieved over finer time slicestaken through the epoch of assembly of the outer regions ofa galactic patch.

Let us now turn to the impact such a temporal coherencehas on the advection of gas angular momentum through the

virial sphere. As the accretion-weighted specific angular mo-mentum at virial radius reads:

〈j〉vr=

∣

∣

∣

∣

∑i mivrΘ(−vr)~ri ×~vi

∑i mivrΘ(−vr)

∣

∣

∣

∣

, (1)

where Θ is the Heaviside step function, we measure it byadding the contribution of the infalling (vr < 0) gas cells orparticles within a shell with 0.95 6 r/Rvir 6 1.05. Note that~vi and~ri are measured relative to the center of the halo andmi is the mass of cell i. The values we quote are averagedover 3119, 6655, 11357, 15419, and 15999 halos at z = 6.1, 5.0.3.8, 2.5, and 1.5, respectively.

Figure 3, left panel, shows that the advected angularmomentum modulus of the gas is increasing with cosmictime and halo mass according to a trend qualitatively con-sistent with that expected in the spherical collapse picture(Quinn & Binney 1992). The right panel represents the spin

parameter, (λ = j/√

2RvirVc, Bullock et al. 2001), for thesame halos. Note that, in this definition, the virial radius,Rvir, and the circular velocity, Vc, are set by the dark mat-ter component alone. A residual trend in the evolution ofgas spin as a function of redshift is clearly visible, along withlarger values of λ gas compared to λ DM. Such an increaseof gas angular momentum as a function of cosmic time wasalso noticed by Brooks et al. (2009).

Figure 4 displays the ratio of the z-component of the ad-vected spin of the gas (where the z-axis is chosen alignedwith the spin of the dark halo) to the modulus of that spin.The dense component of the advected (filamentary) flow,solid line, is distinguished from the diffuse material dashedline using the following density thresholds: ×10−3, 5× 10−4,and 8 × 10−5 H/cc for z = 6.1, 3.5 and 1.8 respectively. Fromthe plot, it is clear that the dense infalling gas has a spin fairlywell aligned with that of the dark matter whatever the massor the redshift considered and that in any case, this densergas arrives much more aligned than the diffuse gas at thevirial radius. Note that the z-component of the diffuse gasincreases with redshift. Figure 6 shows examples of typicalmaps of λz and gives a good idea of the alignment of ma-terial with large values of λz with the main direction of thefilaments. As a matter of fact, such maps are quite represen-tative of the statistical result presented in Figure 4.

Finally, Figure 5 represents the Probability DistributionFunction (PDF) of the cosine of the relative angle between theadvected angular momentum of the gas at different redshifts,as labeled on the plot. It quantifies the correlation of the ori-entation of the momentum of the infalling gas as a function ofcosmic time. This figure complements the correlation foundin Figures 1 and 2 for the orientation of the filaments them-selves and corresponds, in redshift space, to the spatial corre-lation of the relative orientation of angular momentum alongthe filaments described in Appendix A.

Together, Figures 2, 3, 4 and 5 allow us to draw the fol-lowing statistically robust conclusions: for the range of red-shifts and mass considered and the corresponding sub-gridphysics implemented in the MareNostrum simulation, coldgas is advected through filaments at the virial radius with anincreasing spin parameter along a consistent direction. Thecorresponding advected angular momentum orientation issignificantly correlated in time and its orientation is consis-tent with that of the dark matter halo’s spin. These trends arefurther established in a companion work (Kimm 2011) which

Rigging dark halos 5

exploits much (minimum a hundred times) better resolvedindividual galaxies in the NUT suite of zoom simulations. Inthat paper, it is also convincingly demonstrated that the darkmatter and gas are initially advected with the same specificangular momentum (when averaged over the whole virialsphere), whereas the gas within Rvir has more specific mo-mentum than its DM counterpart. Let us now explain thesestatistical measurements at the virial radius through the dy-namics of the surrounding cosmic web.

3 GRAVITATIONAL COLLAPSE OF GALACTICPATCH

3.1 The origin of the angular momentum

It is now accepted that the origin of the angular momentumof galaxies is in the initial distribution of the matter velocitiesin the patches from which galaxies are formed. The shapeof a typical protogalactic patch is neither spherically sym-metric, nor is bounded by equipotential surface, which re-sults in a total (i.e. integrated over the patch) non-zero an-gular momentum that grows during linear stage of struc-ture formation due to torques on the patch from the gravi-tational tidal field (Peebles 1969; Doroshkevich 1970; White1984). The angular momentum growth ∝ a2D(t), where D(t)is the growing mode of gravitational instability, extends tothe nonlinear Zeldovich regime. At this stage, the specific an-gular momenta of baryons and dark matter are equal. Whilethe global properties of the following nonlinear collapse ofthe patch are well captured by the spherical model or, withmore precision, by the elliptical approximation of the Peak-Patch-Theory (Bond & Myers 1996), the detailed distributionof matter and motions in the patch are much more complex.At intermediate mildly nonlinear stages, the dark matter inthe patch assembles into the hierarchical cosmic web. The gasfollows the potential formed by the dark matter but is subjectto cooling and heating processes whose balance determinesthe level at which gas tracks the dark matter. Numerical sim-ulations (see Figure 1) show that the cold (T ∼ 104K) gasforms rather narrow and smooth filaments that follow theless well defined, wider and clumpier dark matter filamen-tary overdensities.

In this paper we are concerned with understanding thelater time infall of momentum rich material on an earlyformed proto-halo. The deviation of the matter motion fromsemi-radial is especially significant in the outer layers of thepatch. Let us consider a smoothed picture of the patch, witha smoothing scale, Rsmooth selecting the inner halo regionthat collapsed at high redshift z = zin. An example of suchsmoothed patch is shown in Figure 7 ( see section 3.2 belowfor details about this experiment). The exterior region con-tains the mass that will be accreted during the further evolu-tion to redshift zcol < zin by which the whole patch collapses.

The dominant feature of the geometry of the outer massdistribution are filamentary bridges that emanate from thecentral halo and extend to the neighbouring peaks. Sad-dle points along the filaments mark the boundary of ourpatch. Calculations of the peak-saddle cross-correlations inthe Gaussian initial field (Pichon & et al 2011, in prep.) showthat the typical separation between a peak and the bound-ary saddle is ≈ 2Rsmooth, with little sensitivity to the power

spectrum slope. Hence, our choice of the inner halo scale tobe half the radius of the patch 1 allows us to just resolve thefilamentary environment of the proto-halo.

While the dynamics of the smoothed inner halo canbe approximated by the ellipsoidal collapse with matter in-falling along mostly radial trajectories, the filaments repre-sent the structures in which the outer matter with signifi-cant transverse velocities accumulates. For this matter thevelocity component parallel to the filaments is statisticallyreduced so the drain of matter from filaments dominates theaccretion onto the inner halo at the later times. Note that thetime it takes for the inner halo to collapse is similar to thetime for the outer filaments of the same scale to turn around(Pogosyan et al. 1998), reaching overdensities δ ∼ 10 andforming potential troughs to channel cold gas.

For illustration, let us first observe the typical initialproperties of a patch around the peak in 2D, as shownin Figure 8. The theory of the skeleton of the cosmic web(Novikov et al. 2006; Pogosyan et al. 2009) provides an ap-proximate boundary of the collapsing patch by defining it assurfaces that pass through the saddle points on the filamentslinked to the central peak. This partitioning is not exact inview of the future non-linear evolution of the structure, butis closely related to the partitioning of the initial velocity flowpointing to the peak. Importantly, it illustrates the generallynon-spherical nature of the initial proto-halo.

The map of the angular momentum in Figure 8 showsthat at the initial stage both the central region and primarydense filaments (the filaments that form bridges to neigh-bouring haloes, in our example tracked by branches 2 and6 of the skeleton) typically contain matter with initially littleangular momentum. Particles with high angular momentumare spread outside the dense structures in regions with alter-nating sense of rotation. As the evolution of the patch pro-gresses, these filaments will collect the high angular momen-tum from the outside material and, being responsible for thelate accretion towards the central peak, supply angular mo-mentum after the central halo forms. The angular momen-tum in a filament is dynamically contained in the displace-ment of its apex w.r.t. the center of the mass of the halo, andits residual transverse motion. Both are generic features of fil-amentary structures formed in gravitational collapse, whoserelative impact may vary with the mass of the filament andthe structure of the patch. Note finally that the amount of mo-mentum cancellation expected during shell crossing is boundto be less important in 3D than for this oversimplified setting.

3.2 The 3D dynamics of the collapsing patch

The fate of the total momentum within the patch is encodedwithin the initial condition and the later evolution of the sur-rounding tidal field (Doroshkevich 1970). As a larger andlarger fraction of the patch turns around, it becomes insensi-tive to the latter, hence its enclosed momentum is conserved.

1 Thus, in contrast to the classical tidal torque theory (TTT) frame-work, we investigate the properties of our patch in greater detail,effectively using a filtering scale roughly twice as small as what isused in TTT. This constitutes departure from the quadratic represen-tation of the density and tidal fields that, we argue, is important indriving the anisotropy of the delayed infall.

6 C. Pichon, D. Pogosyan, T. Kimm, A. Slyz, J. Devriendt and Y. Dubois

Figure 7. Colour coded density (left column) and modulus momentum (right column) of the patch at early, z > zin, time (top row) and atthe redshift of the inner halo collapse z = zin (bottom row). This figure illustrates the structure of the angular momentum distribution and itslater accretion in a collapsing (protogalactic) patch. The progenitors of the filamentary structures in the initial conditions , i.e. the central halo,∼ Rsmooth in size, and the filaments emanating from it, are initially momentum poor, since they are the regions where velocity is predominantlyradial. The high momentum particles (in blue) are in the voids outside (not so much at the centers of voids but closer to walls/filaments) wherethe velocities are more transverse w.r.t. the filaments. At the linear stage the structures remain unmodified, with amplitudes of velocities and,hence, angular momenta of the particles growing, but displacement of the matter negligible. When the patch enters its non-linear dynamicalstage, the high momentum matter starts accumulating in the filaments, making the filaments the locus of momentum rich material. Filamentschange from being momentum poor to momentum rich during the time needed for the matter to travel from voids to filaments, which is similarto the time the inner halo takes to collapse. We observe that at redshift zin, exactly when the inner halo virializes, filaments have changed totheir momentum rich status. This picture also shows that it takes time t > tin for the momentum rich particles from the outskirts to reach thehalo, since the velocity of particles parallel to the filament is statistically lower. Filaments provides momentum rich material at the accretion

stage following initial halo formation.

Nevertheless, since the Vlasov-Euler-Poisson set of equationsis non-linear, the detailed timing and the geometry of theflow matters in determining how and where this momentumis redistributed locally. In particular, as we saw above, thefate of advected momentum is a matter of timing and subtlepartial cancellation. Indeed momentum is an additive vectorquantity, so that advected momenta from different geomet-rical regions, when kinematically brought together via shellcrossing, will produce a net quantity. In the first stage of thegravitational collapse, momentum is mostly in the transversemotion within the expanding voids, then its partially cancelsout in walls as the void shell cross; it cancels out again infilaments, and again towards the center along filaments. Ineffect, the measured flux of momenta at various radii all de-pends i) on the subtle in-balance in transverse motion therewas to start with (i.e. on the relative geometry of surroundingvoids), ii) on the relative mass in these different structures,

iii) on the external and internal torquing between these com-ponents, and finally iv) on the fraction lost to spinning upsubstructures forming along. It is therefore not straightfor-ward to visualize the different stages of the transport in threedimensions.

In order to partially circumvent this difficulty, let uscarry the following idealised numerical experiment. We gen-erate the initial conditions of a 100h−1Mpc Λ-CDM simula-tion using MPgrafic (Prunet et al. 2008) with 2563 dark mat-ter particles. The initial density and velocity cubes are thensmoothed with a Gaussian filter of σ = 1.5 Mpc/h. Thepurpose of this smoothing is to focus visualization on thelarge scale dynamics. The smoothed IC simulation is thenrun down to z = 0 where a friend of friend (FOF) catalogis constructed. As an illustration, we chose within that cata-log a somewhat massive halo in order to represent the innerflow observed for gaseous filaments. We define the patch as

Rigging dark halos 7

Figure 8. left panel: the density field and initial velocity field measured w.r.t. to the central peak. Right panel: the corresponding angularmomentum w.r.t. the central peak position. The green colour corresponds to small angular momentum, as, for example, near the central peak.Both deep blue and red colours are regions of high magnitude angular momenta, of opposite signs, clockwise and anti-clockwise, respectively.In both panels the skeleton of the progenitors of the filamentary structure is superimposed as gold lines. Green lines map the boundary of thepatch according to the density gradient flow. The most massive filaments (#2 and #6) to be formed are those that form overdense bridges to theneighbouring halos. These filaments are indeed the loci of the divergence of the velocity field and are the progenitors of long-lived structuresalong which the matter drains onto the central halo with a time delay relative to the isotropic infall. They define the directions along which much

of the outer material will reach the halo. (a note of caution: one should not consider the initial velocity field as delineating flow lines for thefuture infall of material). In the example given, these directions are not generally coincident with the shape of the central elliptical proto-halo,but are defined by the position of the neighbouring halos and voids and the resulting velocity field structure. However, it is when there is analignment of the peak and the shear of the velocity field that the most prominent filaments arise (Bond et al. 1996). These dense filaments providedeep enough potentials to contain the T ∼ 104K gas. The other (e.g., #1,#3,#4 and #5) filaments of the skeleton, although mapping critical linesof the density gradient by constuction, may not be associated with particularly notable overdensities. They reflect short-lived anisotropies ofthe collapse of the halo, as, for example, might arise from a mismatch of the orientation of the original peak with the velocity flow.

the Lagrangian extension of all particles which end up withinthe FOF halo at z = 0. Figure 7 already made use of this haloto illustrate the partial cancelation of momentum within theearly stage of formation of the filaments.

Figure 9 displays a time sequence projection of the log-density within that patch. The filaments connecting the cen-tral halo to the edge of the patch are clearly visible, and theirrelative motion does reflect the tidal field within that patch.In the first stage of the gravitational collapse, both the centralpeak and the surrounding voids compete to, respectively, at-tract and repel the dark matter in the outskirt of the patch.From a distance, this figure suggests that indeed the orienta-tion of the filaments does not change much over the courseof the collapse of that patch (see also Figure 11 below).

Figure 10 displays the trace of particles colour-coded byredshift for some shifting range of cosmic time. Inspectingthese trails in 3D is instructive and led us to the scenario pre-sented in this paper. It appears quite clearly that the trajecto-ries of the dark matter particles initially within the voids ofthe patch present a sequence of inflections. These inflectionscorrespond to shell crossings, when the flow either reachesa wall or a filament or finally the central peak. For instance,the right panel of Figure 10, which corresponds to a zoom ofthe north-east filament, presents such whirling trajectories,where a fraction of the transverse flow within the surround-ing walls coils up and is converted into a spinning, sink-

ing feature within the filament. Through this process, a frac-tion of the orbital momentum within the large scale structureis transformed into spin, while the residual transverse mo-tion is converted into the drift of the filament. 2 Indeed, Fig-ure 11 demonstrates this on the smoothed IC patch by trac-ing the filaments of different snapshots using the skeleton(Sousbie et al. 2010), a code which basically traces the ridgesconnecting peaks and saddle points of the density field. Herethe “persistent skeleton“ was computed from the dark matterparticles within the patch, and co-added for a range of red-shift while keeping fixed the position of the most bound par-ticle. The residual distortion from one snapshot to anotherreflects the drift of the momenta rich filaments.

Let us now look at the actual momentum advectedand canceled in the flow. Figure 12 displays vector fields ofthe momenta for three snapshots, roughly corresponding totimes when (left panel) most of the momenta still lays in thevoids, (middle panel) a significant fraction had started mi-grating in the filaments and walls and (right panel) most ofthe flow had converged into the central object and filaments.On this figure, the colour coding reflects the amplitude of themomentum and spans the same range of values across the

2 This cosmic flow within filaments is also consistent with the mea-sured spin of dark halos in filaments (codis et al., in preparation.)

8 C. Pichon, D. Pogosyan, T. Kimm, A. Slyz, J. Devriendt and Y. Dubois

Figure 10. Left: example of trajectories of dark matter particles, colour coded by redshift. These trails first converge towards the filamentsand then along the filaments towards the central halo: their motion account for both the orbital momentum of the filamentary flow and thespin of satellites formed within those filaments. Right: a zoom of the left panel corresponding to the shell crossing between two walls leadingto the formation of the north east filament. Some of the momentum lost in the transverse motion is given to the spin of structures formingwithin that filament; the rest is converted into the angular momentum of the filament. The corresponding animation is available for downloadat http://www.iap.fr/users/pichon/rig/.

three panels. It follows that the amplitude of the advectedmomentum is lower along the filaments than in the voids(owing to the cancellation while it is being formed), but dis-plays a significant gradient along the north and the west fila-ment on the right panel. Of course these idealized numericalexperiments raise the so called “the cloud in cell problem”:here we focused on a “hot dark matter” picture of thingswere competing processes on smaller scales do not disruptthe larger scale structure of the flow. This assumption is rea-sonable, as we expect the larger scale distribution of matterto impact most the later angular momentum advection. Thehydrodynamical results of Section 2 suggest they are not sta-tistically dominant, at least beyond redshift 3 1.5.

4 DISCUSSION

This paper has presented a series of results concerning thenature of the momentum rich dynamical gas flows at thevirial radius of collapsed dark matter halos. Measurementswere carried out using the AMR Mare Nostrum simulation(Ocvirk et al. 2008) and allowed us to draw the followingconclusions: at redshift 1.5 and above, gas inflow through thevirial sphere is focused (preferred inflow direction) consis-tent (orientation of advected momentum steady in time) andamplified (increasing amplitude of advected momentum astime goes by). The qualitative analysis of very simple 2D ini-tial conditions and idealised dark matter simulations have

3 Substructures on smaller scales may become relevant at lowerredshift when dark energy kicks in and dries out the gas supply,for a subset of massive halos embedding elliptical galaxies (seeDomınguez-Tenreiro et al. (2010) and the discussion below).

allowed us to explain this coherent flow in terms of the dy-namics of the corresponding gravitational patch.

4.1 Advection of momentum along cold flows

In view of these findings we sketched the following scenariofor the gas flow along cold streams. In the outskirts of a form-ing halo, large scale flows arise as the surrounding voids ex-pel gas and dark matter. As the flows coming from oppositedirections meet, dark matter undergoes shell-crossing whilstthe gas shocks, cools and collapses into the newly createdwalls and filaments at the boundary between voids. Each ofthese boundaries acquires a net transverse velocity which re-flects the asymmetry between the voids it divides. Within theboundary, the longitudinal component of the flow then dragsgas towards the growing halo, advecting momentum in theprocess. As the transverse velocity should be similar alongthe boundary, the later the infall onto the halo, the further itoriginates from, the larger the momentum it brings (lever ef-fect). Thus the orientation of the advected momentum at thevirial radius of the halo is steady in time, as it is piloted bythis large-scale transverse motion which in effect is encodedin the initial conditions of the patch. It also means that gashits the virial sphere of the halo along a preferred direction:that of the incoming filaments/walls.

We are now in a position to answer the two questionsraised in the introduction, namely ”how and why is gas ac-creted onto the galactic disc?”. The answer to “how” is: soas to build up the circumgalactic medium through the directaccretion of cold gas with ever increasing and (fairly well)aligned angular momentum. The answer to why is: becausethe internal dynamics of the cosmic web within the peakpatch produces such a coherent flow. Momentum rich inflowis delayed compared to radial inflow as a fraction of the patch

Rigging dark halos 9

Figure 9. Projected view of the dark matter density in a collapsingpatch with the smoothed out ICs as a function of cosmic time fromtop to bottom and right to left. As these maps reflect the entire merg-

ing history of a given halo, they clearly show that the incoming di-rection of the infall remains fairly constant with time, with a residualdrift, which reflects the advected momentum along the filament (seealso Figures 1 and 11).

first flows away from voids and into the walls and filaments.Only then is this material brought back in the direction of thehalo.

Hence the anisotropic distribution of angular momen-tum outside of the collapsed halo is critical to explain theformation of thin galactic discs. It means that the gas arrivesat Rvir in two distinct flavours: dense versus diffuse, as out-comes of different dynamical histories within the peak patch.The denser phase is produced as gas collapses into walls andfilaments where it cools and which further feed the centralhalo. The diffuse phase is either gas that is accreted directlyonto the halo, or the hot gas which was unable to cool due tolow density, and was not confined to filamentary structures.As the result, the hot gas does not display a preferred align-ment w.r.t. the dark halo’s spin (Figure 4), as this coherenceis only achieved via filamentary infall in which it did notparticipate. Even though the mass involved in filamentaryflows can be small compared to the total gas mass involvedin the formation of the central galaxy at any given time, itshould play a critical role in supplying the circumgalacticregion constructively with increasingly momentum-rich gas,while at the same time minimizing the destructive impact ofincoming substructures on the existing disc. In short, discsare in fact produced by, rather than shielded from the cosmicenvironment.

4.2 The dynamics of the gas within Rvir

Though the dynamics of the infalling gas within Rvir is com-plex and is the topic of companion papers (Kimm 2011;Tillson 2011; Powell 2011, in prep.), we qualitatively describe

Figure 11. The sweeping skeleton within the inner region of thepatch shown in Figure 9 color coded by redshift from light to dark.The central peak is in the middle of the figure. To zeroth order, thisfigure suggests that the direction of the filament does not changemuch. Looking more closely, from one redshift to another, the vari-ous branches of the skeleton do slide with cosmic time. Indeed, thenet transverse motion following shell crossing of the filament willinduce a residual drift of the filament, which the skeleton captures.At the top of this plot, note the top left-most fork before a satellitewhich will merge with the central halo at a later time.

here what happens to the cold gas brought by filamentarystreams into Rvir as it reaches the outskirt of the central disc(the circumgalactic medium). Typically, this flow will main-tain coherency. Simulations show it will reach ∼ Rvir/10on a infalling orbit, while roughly preserving its radial ve-locity and width but radiating away its kinetic energy ac-quired during the free fall. It will cross ∼ Rvir/10 with an im-pact parameter corresponding to its momentum at Rvir andovershoot the center until it encounters counter-falling dif-fuse gas against which it will radiatively shock (Powell et al.2010). As it remains dense and can therefore cool efficiently,the net effect of this shock will be kinematic: the componentof linear momentum normal to the shock will be lost, whilethe parallel component will correspond to its new post shockvelocity. It will then resume its plunging orbit on a much lessradial orbit (as the rate of change of energy with momentum,dE/dJ ∼ v/R, (with v the velocity), is large close to pericen-ter, R), and spiral in towards the inner disc. Hence the actualconfiguration of the shock is expected to induce a change inangular momentum.

A limited loss of the coherence inherited from the largescale structure is expected to occur in this thermodynami-cal process: as the filamentary inflow has a geometry whichis steady in time, so should the shock. Similarly the grav-itational interaction between infalling satellites and the ex-isting triaxial halo and predating proto-disc should onlyhave a time limited impact on the reshuffling of momentumwithin that region. Therefore we expect the distribution ofpost-shock momenta to reflect this partially retained coher-

10 C. Pichon, D. Pogosyan, T. Kimm, A. Slyz, J. Devriendt and Y. Dubois

Figure 12. Redshift sequence ( z > zin, z = zin and , z < zin) displaying the 3D momentum vector field, colour-coded by amplitude (with thesame dynamical range throughout; dark corresponds to large momentum, light to intermediate momentum; low momentum is not shown).The left panel clearly shows that the voids are momentum rich, the middle panel shows that near zin the filaments have partially advected (andpartially canceled out) a fair fraction of the void’s momenta, while the right panel shows that the remaining momentum indeed is carried inthrough the filaments. Note that the region of high and intermediate momenta do not overlap much between z > zin and z < zin, as expectedgiven the flow. A rough estimate gives that filaments, walls and voids contain 60 %, 30 %, 10 % of the specific momentum at z = zin. Thecorresponding animation is available online at http://www.iap.fr/users/pichon/rig/.

ence and in particular to also display a secular cosmic evo-lution towards larger momenta (though admittedly a par-tially stochastic small change in momentum, dJ, away from aquasi radial, J ∼ 0, orbit may represent an important relativechange of momentum). In a nutshell, we expect the consis-tency of the cosmic flow to overcome, in the long run, theoccasionally messy circumgalactic environment of formingdiscs. In fact, when studying the statistical median merid-ional profile of galaxies in the MareNostrum simulation wedo indeed find (Kimm 2011, in prep.) an excess of advectedmass and momentum in the equatorial plane of the disc.

Using the NUT (Powell et al. 2010, see below) set of highlyresolved hydrodynamical simulations, Kimm (2011) findsthat the specific angular momentum of the gas is at leasttwice as large as that of the dark matter in the circumgalac-tic region Rvir/10 < r < Rvir. More importantly this spe-cific angular momentum is, most of the time, well alignedwith that of the central galactic disc. This region acts like areservoir of rapidly spinning cold gas, which eventually co-herently builds the disc inside out. We therefore argue that,statistically, disc galaxies owe the survival of their thin discto the consistency of the cosmic infall.

4.3 Validation: tracing back the gas flow of a disc galaxy

To substantiate the view that the gas angular momentum ofgalaxies is preferentially advected along filaments and natu-rally leads to the inside-out build up of thin discs, we makeuse of a high resolution hydrodynamical simulation in theNUT suite ((Powell et al. 2010)). The ultimate goal of theNUT suite is to quantify the effect of various physical mecha-nisms (cooling, supernovae/AGN feedback, magnetic fields,radiative transfer) on the properties of well resolved galaxies(several grid cells spanning the scale height of the disc) em-bedded in their cosmological environment. To achieve thisobjective, a cubic volume of the Universe 12.5 comoving Mpcon a side with periodic boundary conditions is tiled with

a uniform 1283 coarse grid. Within this volume, an initialhigh resolution sphere (10243 equivalent) of about 1 Mpcin radius is defined which encompasses the Lagrangian re-gion from which the dark matter particles of a Milky-Waysize halo at z=0 originate. As the simulation progresses, upto 10 additional levels of grid refinement are triggered in aquasi Lagrangian manner so as to keep the size of the small-est grid cell equal to 12 physical pc at all times. The initialconditions of the simulation were generated using MPgrafic

(Prunet et al. 2008) and adopting a WMAP5 cosmology withΩm = 0.258, ΩΛ = 0.742, h ≡ H0/(100kms−1Mpc−1) = 0.72(Dunkley et al. 2009). This yields a dark matter particle massof ≃ 5.5 × 104 M⊙ in the high resolution region. The simula-tion includes radiative cooling, star formation and a uniformredshift dependent UV background instantaneously turnedon at z = 8.5 (Haardt & Madau 1996). The gas density thresh-old for star formation is chosen to be equal to the Jeansthreshold, i.e. nH,th = 400 cm−3 The minimal mass of star

particles is therefore ≃ 2 × 104 M⊙.

To track the evolution of the gas flow in our Eulerian ap-proach, we use tracer particles. These are mass-less particleswhose initial positions and number exactly match those ofthe DM particles in the simulation. However, as the simula-tion proceeds, they are simply advected with the local veloc-ity of the gas calculated by means of a Cloud-In-Cell interpo-lation. Figure 10 shows such gas tracing particles in the NUTsimulation at z = 6.5 and paints them in a colour accord-ing to which concentric spherical shell they belong to (topleft panel). These spherical shells are centered on the diskgalaxy. The background image is a projection of the under-lying gas temperature. Note that most of the tracer particlesare associated with the cold component. The position of thesecoloured particles is then displayed at redshifts z = 8.5 andz = 11.5 in the top middle and right panels, illustrating thattheir radial ordering is preserved over significant periods oftime. The bottom right, middle and left panels present a his-togram of the radii of these particles and their angular mo-

Rigging dark halos 11

z=8.5 z=11.5 z=11.5

Figure 13. The position of the tracer particles superimposed on the gas temperature at redshift z = 6.5 (top left panel), and just the positions ofthe tracer particles at z = 8.5 (top middle panel) and z = 11.5 (top right panel). Note how well the tracer particles follow the shocks and preferthe cold phase. The colour (black, dark blue, light blue, green, orange, red) for the tracer particles encodes their radius at z = 6.5. The (bottomleft panel) shows a histogram of the particles’ angular momenta at z = 8.5 (bottom left panel), z = 11.5 (bottom middle panel) and radii (bottomright panel) (the vertical dashed line represents the mean of each PDF). The progenitor of the different annuli around the disc is clearly foundin the earlier filaments, while the stratification is consistent with an inside out build up of the circumgalactic medium. As expected, the angularmomentum of tracer particles increases with cosmic time.

menta at z = 8.5 and z = 11.5 respectively. Notwithstandingthat the NUT simulation only follows the first couple of Gyrof evolution of a galaxy, it is clear from Figure 10 that, as weargued in this paper, i) the origin of the cold gas in the discand its neighbourhood is filamentary, ii) outer shells of gascome from outer regions which carry more angular momen-tum and iii) the angular momentum of accreted gas increaseswith cosmic time.

4.4 Extensions and perspectives

In this paper, the emphasis was put on the inflow from thecosmic environment onto galactic discs. More specifically, weaddressed the implication of angular momentum transportby cold flows from large scales down to the virial radii of thedark halos hosting these central disc galaxies.

In doing so, we purposely overlooked several importantissues. First, our work has shamelessly ignored the well doc-umented dichotomy between galaxies located in dark matterhalos above and below the critical halo mass for shock sta-bility (Keres et al. 2005; Dekel & Birnboim 2006; Ocvirk et al.2008; Brooks et al. 2009). The mass accretion process in thesetwo galaxy populations is known to be thermodynamically

(cold versus hot mode) and geometrically (connected to mul-tiple filaments or embedded in them) different and also todepend sensitively on redshift, to the point that even therelevance of cold flows at low redshift (below z = 1.5) hasbeen questioned. Second, feedback processes internal to thecentral galaxies (massive stars, supernovae, AGN) may re-duce the relevance of anisotropic infall as they may partiallyisotropise outflows, which, in turn, could lead to a disrup-tion of the inflow. Third, along with the cold gas, the cos-mic inflow will advect galaxy/dark halo satellites. The rela-tive fraction of induced minor/major, dry/wet merger willcertainly affect the Hubble type of the central object eventhough the ’cosmic’ consistency of the accretion reported inthis paper should have a statistical bearing on the preferreddirection of satellite infall at z = 0 (Deason et al. 2011). Asdiscussed in Section 4.2, the dynamics within the virial ra-dius of the dark matter host halo is also bound to be signif-icantly more complex: the shorter dynamical time, dynami-cal friction and tidal stripping of satellites, torquing from thecentral triaxial halo and radiative shocking should introducea non-negligible amount of momentum redistribution for thegas within the circumgalactic region.

In a recent paper, Domınguez-Tenreiro et al. (2010),

12 C. Pichon, D. Pogosyan, T. Kimm, A. Slyz, J. Devriendt and Y. Dubois

guided by SPH cosmological simulations, address the massassembly of massive ellipticals in the framework of the ad-hesion model (Gurbatov et al. 1989; Kofman et al. 1990). Inaccordance with this scenario, they also trace back the ori-gin of star forming gas to neighbouring cold filaments (cf.Section 4.3). On the other hand, they also find that hot intra-halo gas originates from diffuse regions where strong shellcrossing is yet to occur. As their focus is on the progeni-tors of massive galaxies, which would correspond to rounder(Pichon & Bernardeau 1999) peak patches associated withrarer events, they measure very little momentum advectionin the first stage of the collapse. As we are concerned withless massive, more common disc-like galaxies, the associatedinitial density peak is more ellipsoidal and early angular mo-mentum advection is therefore more important.

The extension of TTT to quantify the amount of angu-lar momentum expected to be acquired after zin constitutes anatural follow-up of the current paper. Such a theory couldbe constructed from a higher resolution description of thedensity field in the patch (i.e. which includes features suchas ridges and saddle points that appear when the patch issmoothed on scales smaller than its ellipsoidal representa-tion via the inertial tensor), using theoretical tools like theskeleton (Pogosyan et al. 2009). Such an extension involvespredicting the relative dynamical importance of the differentfilaments embedded within the patch, both in terms of ad-vected mass and momentum. One expects that the six fila-ments typically connected to a given 3D peak (Pichon & et al2011, et al, in prep.) mostly cancel each other out in terms ofnet angular momentum flux as all the voids surrounding agiven peak cannot all induce rotation along the same axis.This extension will need to understand filamentary bifurca-tions (Pogosyan et al. 2009, e.g.) inside the peak patch to seewhy only two prevail eventually. One would then be able topossibly predict ab initio some features of the various mor-phological types and the cosmic history of galactic spin-up.Such a theory for the rig of dark halos should be within reach,and will be the topic of further investigation.

ACKNOWLEDGMENTS

We thank J. Binney, S. Codis, M. Haehnelt, J. Magorrian, S.Peirani, S. Prunet and J. F. Sygnet for useful comments dur-ing the course of this work. CP acknowledges support froma Leverhulme visiting professorship at the Astrophysics de-partment of the University of Oxford, and thanks MertonCollege, Oxford for a visiting fellowship. TK acknowledgessupport from a Clarendon DPhil studentship. JD and AS’sresearch is supported by Adrian Beecroft, the Oxford Mar-tin School and STFC. YD is supported by an STFC Post-doctoral Fellowship. We also acknowledge support from theFranco-Korean PHC STAR program and the France CanadaResearch Fund. The MareNostrum simulation was run onthe MareNostrum machine at the Barcelona Supercomput-ing Centre and we would like to warmly thank the staff fortheir support and hospitality. The NUT simulation presentedhere was run on the DiRAC facility jointly funded by STFC,the Large Facilities Capital Fund of BIS and the Universityof Oxford. This research is part of the Horizon-UK project.Let us thank D. Munro for freely distributing his YORICK

programming language and opengl interface (available at

http://yorick.sourceforge.net/) and T. Sousbie for thepersistent skeleton code DISPERSE.

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APPENDIX A: ANGULAR MOMENTUMORIENTATION CORRELATION ALONG FILAMENTS

In Section 2 we found that the orientation of advected angu-lar momentum was fairly stationary w.r.t. cosmic time. Let usunderpin this measurement by investigating the coherencyof this orientation along the large scale structure filaments.Consider a set of filaments (defined here as a set of seg-ments of the skeleton between two peaks, see Sousbie et al.(2009)). Let si and sj be the curvilinear coordinates along

that filament of two segments and ∆θij be the relative an-gle of the two angular momenta, Ji and Jj (i.e. cos(∆θij) =Ji · Ji/|Ji||Jj|). Let us define the expectation, 〈cos(θ)〉(r), ofthe relative angle between the angular momentum orienta-tion as a function of distance along the filament as

〈cos(θ)〉(r) = ∑i,j| |si−sj|=r±∆r

cos(∆θij)/

∑i,j| |si−sj|=r±∆r

1 ,

where the summation is over all pairs, i, j belonging to thesame filament for which the separation in curvilinear coordi-nate falls within ∆r of r. The average of this expectation overall filaments in the simulations is shown on Figure A1 forthe skeleton of a dark matter simulation of 2563 particles in acube of volume (50 Mpc/h)3 smoothed over a 1Mpc/h scale.The orientation of the spin of dark matter particles is clearlycorrelated over scales of the order of 10 Mpc/h. This result isqualitatively consistent with the temporal correlation of Fig-ure 5.