Rivers of Gas I.: Unveiling The Properties of High Redshift Filaments
Marius Ramsøy, Adrianne Slyz, Julien Devriendt, Clotilde Laigle, Yohan Dubois
MMNRAS , 1–17 (2019) Preprint 8 January 2021 Compiled using MNRAS L A TEX style file v3.0
Rivers of Gas I.: Unveiling The Properties of High RedshiftFilaments
Marius Ramsøy ★ , Adrianne Slyz , Julien Devriendt , Clotilde Laigle , andYohan Dubois Sub-department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK Sorbonne Universités, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
At high redshift, the cosmic web is widely expected to have a significant impact on the mor-phologies, dynamics and star formation rates of the galaxies embedded within it, underscoringthe need for a comprehensive study of the properties of such a filamentary network. With thisgoal in mind, we perform an analysis of high- 𝑧 gas and dark matter (DM) filaments arounda Milky Way-like progenitor simulated with the ramses adaptive mesh refinement (AMR)code from cosmic scales ( ∼ ∼
20 kpcat 𝑧 = 𝑟 fromthe simulation are consistent with that of isothermal cylinders in hydrostatic equilibrium. Suchan analytic model also predicts a redshift evolution for the core radius of filaments in fairagreement with the measured value for DM ( 𝑟 ∝ ( + 𝑧 ) − . ± . ). Gas filament cores growas ( 𝑟 ∝ ( + 𝑧 ) − . ± . ). In both gas and DM, temperature and vorticity sharply drop at theedge of filaments, providing an excellent way to constrain the outer filament radius. Whenfeedback is included the gas temperature and vorticity fields are strongly perturbed, hinderingsuch a measurement in the vicinity of the galaxy. However, the core radius of the filaments asmeasured from the gas density field is largely unaffected by feedback, and the median centraldensity is only reduced by about 20%. Key words: galaxies: formation — galaxies: evolution — cosmology: large-scale structure ofUniverse — methods: numerical
Galactic surveys have revealed the presence of anisotropic structureon scales of Mpc, made up of nodes, voids, sheets and filaments (e.g.Davis et al. 1982; de Lapparent et al. 1986; Geller & Huchra 1989).Cosmological simulations are able to reproduce this network, theso-called cosmic web (Bond et al. 1996; Pogosyan et al. 1998), andunveil its existence not just for the distribution of galaxies but alsofor the underlying gas and DM density, as a consequence of the hi-erarchical growth of structures in Λ CDM. Gravity amplifies smallanisotropies, resulting in a near homogenous background collapsingto form sheets which can collapse again along another axis to formfilaments. Halos form at filament intersections where, according tocosmological hydrodynamics simulations, galaxies at high redshiftgrow in mass and angular momentum primarily by material trans-ported along these filaments (Birnboim & Dekel 2003; Kereš et al. ★ E-mail: [email protected] © a r X i v : . [ a s t r o - ph . GA ] J a n Marius Ramsøy et al. of extended gas disks around galaxies (co-rotating with the stellardisk), either directly in emission (e.g. from Lyman- 𝛼 , Prescott et al.2015) or indirectly in absorption (e.g. from galaxy-quasar pairs,as studied in Zabl et al. 2019; Ho & Martin 2019), all suggestingfilamentary accretion from the cosmic web.Rather than directly pursuing the filament properties them-selves, it is possible to infer them through indirect methods. Onlarge-scales, many authors have measured halo or galaxy spin align-ment with cosmic filaments both in simulations (see e.g. Aragón-Calvo et al. 2007; Codis et al. 2012; Dubois et al. 2014; Laigle et al.2015; Ganeshaiah Veena et al. 2018; Kraljic et al. 2019) and low- 𝑧 spectroscopic observations (e.g. Tempel & Libeskind 2013; Chenet al. 2019; Krolewski et al. 2019, among others). These resultshighlight a redshift and mass dependence of the alignment signal,with halos with masses above 𝑀 h > M (cid:12) displaying spins per-pendicularly oriented with respect to the nearest filament, whereasspins of halos with masses below 𝑀 h < M (cid:12) align with thenearest filament. At low masses this is thought to be due to accre-tion of vorticity rich gas that drive spins to align with the filament.At high masses this behaviour is overcome by mergers, or as Laigleet al. (2015) argues, the accretion of material from multiple vortic-ity domains. This dichotomy in galaxy spins shows the profoundimpact of cosmic filaments on the galaxies embedded within them.The inverse problem has also been studied (Pandya et al. 2019) at-tempting to use the alignment of galaxies to detect the cosmic webwith the CANDELS survey (Grogin et al. 2011; Koekemoer et al.2011). The non-detection of the alignment signal is likely due tothe number of prolate galaxies with spectroscopically determinedredshifts with stellar masses 9 < log ( M ∗ / M (cid:12) ) <
10 in the survey,as well as these galaxies’ nearest neighbours. .On smaller scales, the misalignment of gas and DM angularmomenta in simulations has been attributed to different redistri-bution processes during halo virialisation (e.g. Kimm et al. 2011;Stewart et al. 2013). However, it has also been argued that instabili-ties within the filaments could develop, leading to their fragmenta-tion and breakup, thereby preventing cold gas from being smoothlyaccreted by the host galaxy. In such a scenario, the angular mo-mentum segregation between DM and gas could be construed as anartefact of poor numerical resolution in filaments. Several authors(Freundlich et al. 2014; Mandelker et al. 2016; Padnos et al. 2018;Mandelker et al. 2019; Berlok & Pfrommer 2019) carried out ide-alised simulations of filaments entering a halo, and concluded thatthey should be stable, given their width and velocity. Cornuault et al.(2018) used a phenomenological model of a gas stream to explorethe possibility of a turbulent, multi-phase filament. The accretion ef-ficiency of such a filament would be reduced, but it remains unclearas to whether such a multi-phase model constitutes an acceptabledescription of cosmological filaments. Using a cosmological zoomsimulation tailored to achieve maximum resolution in the filaments,Rosdahl & Blaizot (2012) find that they remain stable within haloswith masses of up to a few 10 M (cid:12) at least as to low as 𝑧 = 𝛼 blobs (LABs)and emitters (LAEs) (e.g. Kikuta et al. 2019; Umehata et al. 2019),which will be observable by the NIRspec instrument (Latif et al.2011). In addition, LAEs should be detectable with the proposedBlueMUSE instrument. On larger scales, filamentary gas can bedetected in the radio with the Square Kilometer Array (Kooistraet al. 2019). Lyman- 𝛼 forest tomography also allows the probing ofthe cosmic web in the IGM, with the feasibility of observations forthe Very Large Telescope investigated by (Lee et al. 2014) and theEuropean Extremely Large Telescope by (Japelj et al. 2019). Thiswill enable the detection and exploration of the full 3 dimensionalstructure of the cosmic web.Efforts to understand observed filament properties are corre-spondingly mirrored by simulations (e.g. Gheller et al. 2015, 2016).On large scales, filaments of the cosmic web are reported to havea radial power law profile in density with a power law index com-prised between -1 and -2 (see e.g. Colberg et al. 2005; Dolag et al.2006; Aragón-Calvo et al. 2010). Smaller scale studies have beenperformed by e.g. Ocvirk et al. (2016), who determined the outerradii of filaments in their simulation to be about 50 ℎ − kpc at 𝑧 = . well-resolved filaments and take a step in this direction by measuringfilament profiles from the density, vorticity, and temperature fieldinformation available in a zoom-in cosmological simulation. Ourfocus is on intermediate-scale filaments, that is, those connecting toa M ★ galaxy, at moderate to high redshift ( 𝑧 ≥ ∼ ) with similar resolution (Park et al. 2019, Dubois et al.in prep), where a statistical sample of filaments can be obtained,connecting a more diverse ensemble of galaxies.The structure of this paper is as follows: in section 2.1 we MNRAS , 1–17 (2019) igh Redshift Filaments Figure 1.
Zooming in on the NUT galaxy gas density field at 𝑧 =
4. The leftmost panel shows a gas density projection of the entire simulation volume (12.5comoving Mpc), with the high resolution zoom region enclosed in the square located in the bottom right corner of the first panel. Each subsequent panel, goingfrom left to right, displays a projection of 1/8 th of the volume of the previous panel. The size of each volume in physical units is indicated. The middle panelshows the region within which the analysis in this paper is performed, chosen so as to maximize the length of the studied filament. outline the simulation set up. In section 2.2 we describe how weidentify the filaments and perform the analysis. Section 3 presentsthe results of our work, compares filament properties to an analyticmodel and discusses the robustness of our measurements vis-à-visresolution. We summarize our results in section 4. The analysis is performed on two simulations of the nut suite (Pow-ell et al. 2011), a series of cosmological zoom-in simulations of aMilky Way like galaxy designed to study the effects of resolutionand various physical processes on its formation and evolution us-ing the Adaptive Mesh Refinement (AMR) code ramses (Teyssier2002). Initial conditions are generated at redshift 𝑧 =
499 using theMPGrafic code (Prunet et al. 2008) with cosmological parametersset in accordance with the WMAP5 results (Dunkley et al. 2009).The simulation volume is a cubic box 9 ℎ − Mpc on a side and acoarse root grid of 128 cells. A series of three nested grids are thencentred on a sphere with radius 2.7 ℎ − Mpc which encompasses theLagrangian volume occupied by the galaxy (host dark matter halomass of 𝑀 vir = × M (cid:12) by 𝑧 = . × 𝑀 (cid:12) ,whereas the gas evolution equations are solved on the AMR gridby means of a Godunov method (HLLC Riemann solver) with aMinMod limiter to reconstruct variables at cell interfaces. The gasdensity field in the simulation is shown in Fig. 1 at 𝑧 =
4, graduallyzooming in from the full box onto the central galaxy itself.In this paper, we use a nut simulation with no feedback, andone with mechanical supernova feedback as defined in Kimm et al.(2015). In the following, we refer to these two simulations as the “no-feedback" and “feedback" runs respectively. The feedback recipe ofKimm et al. (2015) ensures that the appropriate energy or momen-tum is deposited into the cells around the supernova, depending onwhether the Sedov-Taylor phase of the blast wave is resolved or not.This prevents the supernova energy from being artificially radiatedaway, as would happen if solely thermal energy was injected (theso called over-cooling problem described in Katz 1992). Both runsunder study use cooling tables calculated by Sutherland & Dopita(1993), down to 10 K, and the Rosen & Bregman (1995) approxi-mation for temperatures below this threshold. A UV background is
Level
Figure 2.
Resolution map for a slice of thickness 300pc, across a (625kpc) region of the computational domain at 𝑧 =
4, with each colour representinga different resolution level as indicated on the figure. At this redshift, thefilament is uniformely sampled at 1.2 kpc resolution (AMR level 11: green)and partly at 0.61 kpc (AMR level 12: yellow) around the most massivehalos embedded in it. Even though the highest spatial resolution reached inthe simulation is 10 pc, which corresponds to AMR level 20, levels above13 are not shown as they are confined to the galaxies themselves and theirimmediate vicinity. instantaneously turned on at 𝑧 = . × H . cm − with an efficiencyof 1% per free-fall time, calibrated on observations by Kennicutt(1998). A detailed description of the implementation of star for-mation used in this version of ramses may be found in Rasera &Teyssier (2006) and Dubois & Teyssier (2008). For the feedbackrun, a Chabrier initial mass function Chabrier (2003) is adopted,with 31.7% of the mass fraction of each star particle ending up asa single type II supernovae and releasing 10 erg M − (cid:12) of energyafter a 10 Myr time delay and expelling heavy elements with a 5%yield. As we aim to measure the properties of the cosmic web filaments,both in the DM and gas density fields, we now describe how weidentify these structures in the simulations.
MNRAS , 1–17 (2019)
Marius Ramsøy et al. kp c kpc DM Gas (no feedback) Gas (feedback) C o l u m n D e n s it y ( g c m − ) T e m p e r a t u r e ( K ) V o r ti c it y ( 𝑠 − ) Figure 3.
DM (left column) and gas (no-feedback run, middle column; feedback run, right column), with each row showing column density ( top ), temperature( middle ) and vorticity ( bottom ) in a slice 625 kpc across and 1 kpc thick at 𝑧 =
4. The main filament, as extracted from the DM density field, is overplotted ( b luesolid line) on the column density maps. The virial radii of the 50 largest halos are marked as circles. The differences between the feedback and no-feedbackskeletons are caused by small differences in the noise level associated with DM particles: they yield slightly different paths which have a very similar length,so that either path can be chosen by the algorithm described in the text. The colour bar for the density represents the gas. To estimate it for the DM, onesimply needs to divide the numbers shown by the universal baryon fraction. For the DM temperature, velocity dispersion is used as a proxy, with dark bluecorresponding to regions of ∼ .
02 km s − and deep red with ∼
100 km s − . In the vorticity panels, red represents matter swirling counter clockwise aroundthe filament, and blue is for matter rotating in the opposite direction. The DM particle distribution is tessellated using the Delaunay Tes-sellation Field Estimator tool (Schaap 2007) and fed to the codeDisPerSE (Sousbie 2011). DisPerSE computes stationary points(maxima, minima and saddle points) of the density field using theHessian matrix and assigns to each pair of critical points (e.g.maxima-saddle) a persistence, namely a measure of how signifi-cant it is with respect to a Poisson distribution. The persistencethreshold is the single parameter that determines which features areconsidered as noise and which robustly pertain to the topology ofthe underlying density field. From this set of stationary points thatcharacterize the topology of the field, DisPerSE connects saddles to maxima following the direction of least gradient to create a networkof filaments which will be referred to in this paper as the “skeleton”,and we will call “nodes" the maxima of the density field.
For each simulation, filaments are extracted from the Delaunay tes-sellation reconstruction of the DM density field, setting a persistencethreshold of 10 𝜎 . This persistence threshold is chosen such that theobserved skeleton is in good visual agreement with the DM densityfield. Our results are in fact insensitive to the exact value chosen forthis threshold, as we are only studying the main filaments feeding the MNRAS , 1–17 (2019) igh Redshift Filaments galaxy (see Section 2.2.3). The skeleton is additionally processedwith skelconv (see DisPerSE manual ) using the breakdown andsmooth functions. breakdown removes duplicate segments enter-ing a node from two different starting points. These segments can beso close as to be indistinguishable from one another and as such areremoved to prevent their over-representation in the final skeleton.The skeleton is then smoothed by averaging over the positions of the30 nearest neighbours of each segment. This mitigates the effectsof Poisson noise on the skeleton, ensuring that individual segmentslocally follow the global direction of the filament they belong to.In both the feedback and no-feedback runs, 1.22 physical kpc isthe maximum spatial resolution reached in filaments, defined asthe size of an individual cell on the highest AMR grid level thatentirely maps the filament (see Fig. 2). As is clear from Figure2, higher refinement levels are triggered within filaments but theircoverage is patchy, and mostly concentrated around halos/galaxiesembedded within these elongated structures. As we argue in ourconvergence analysis (Section 3), we believe 1.22 kpc is enough toresolve the radial structure of filaments, at least those that connectto halos/galaxies with masses similar (or larger) to the one we studyin this paper (roughly M ★ ). We emphasize that this is a much higherresolution than that currently reached in large-scale cosmologicalhydrodynamics simulations, where (cid:39) The gas and DM distributions differ significantly even for the no-feedback run (compare left and middle panels of Fig. 3), with thegas density field presenting much fewer filamentary structures thanthe DM . In addition, even though dwarf galaxies residing withinfilaments are affected by feedback, the impact of this feedback onthe growth of the central galaxy is minimal (it does not lead tothe disruption of the main filament) as the majority of the gasfeeding it at high redshift is accreted via filaments, and not frommergers (Danovich et al. 2012; Tillson et al. 2015). However, the gasdensity field in the run with feedback will be more perturbed dueto interactions with galaxy winds and shocks (see middle and right This is a consequence of re-ionisation reheating the gas of the IGM andpreventing accretion into the shallow potentials of DM filaments (Katz et al.2019). panels of Fig. 3), making the comparison between feedback and no-feedback runs difficult. Furthermore, DisPerSE is designed to workwith particle data, as it allows in this case a meaningful definitionof persistence (the very concept of which relies on quantifying thesignificance of a feature with respect to Poisson noise). For thesereasons, and given that we are not interested in probing the existenceof filaments within the virial radius of DM halos in this work, theDM density field seems more appropriate to carry out filamentextraction.We therefore elect to extract the skeleton from the DM densityfield, but trim it in order to keep only the main filament, along whichmost material flows onto the galaxy. For an M ★ central galaxy, themain filament traced in the gas clearly coincides with its DM coun-terpart (see top panels of Fig. 3). As we are analyzing a filamentconnecting to a single object, we identify the approximate regionwhere it begins and ends by eye, and select the highest density pointin this region as its starting/end point . We then use Dijkstra’s al-gorithm (Dijkstra 1959) to compute the shortest path (following theskeleton) between the start and end points. This works by assigningto each segment a distance from the start point, travelling along allthe various possible paths of the skeleton. Whenever a shorter pathto a given segment, 𝑠 , is found, then the selected path is updatedup to 𝑠 , and the distances to all segments connected to 𝑠 along thispath which have a longer path length, are updated. This processis iterated until the network is traversed, yielding the shortest pathbetween the given start and end point. The method is valid providedthe main filament flows mostly straight onto the galaxy, which, inturn, holds until the filament gets close to the galaxy disk (Powellet al. 2011).In order to avoid the filament passing through halos, filamentsegments located in regions with densities higher than 130 timesthe mean density of the Universe were excluded . This densitythreshold is chosen empirically, but the resulting skeleton does notdepend very sensitively on the chosen value provided this latter is onthe order of 100 times the mean density of the Universe. The entireinitial filamentary network and the resulting main filament extractedafter post-processing are shown in Fig. 4. Fig. 3 highlights that theskeletons extracted from the DM density fields of the feedbackand no-feedback runs are slightly different. In this Figure, one canclearly see a pair of filaments on the left side of the central galaxy,which are in the final stages of merging. As a result, our algorithmidentifies two possible paths along which the main filament wouldhave essentially the same length. Small changes in the noise levelassociated with the DM particles in the two different runs change theexact way that segments connect, resulting in the algorithm pickingone of these paths in one run and the other path in the other run.Our results are, by and large, independent of such small randomlyinduced differences. For larger volume cosmological simulations where an ensemble of fila-ments is available one can forgo the inspection by eye and simply use theclosest pair of galaxies with similar masses which are linked by the skeletonas the starting and end points of a filament. This value is lower than 200 times the critical density of the Universewhich is commonly used in the literature to define virialised structures. Thisreflects the fact that the density of halos at the virial radius is lower thantheir average density by about a factor 3.MNRAS000
For each simulation, filaments are extracted from the Delaunay tes-sellation reconstruction of the DM density field, setting a persistencethreshold of 10 𝜎 . This persistence threshold is chosen such that theobserved skeleton is in good visual agreement with the DM densityfield. Our results are in fact insensitive to the exact value chosen forthis threshold, as we are only studying the main filaments feeding the MNRAS , 1–17 (2019) igh Redshift Filaments galaxy (see Section 2.2.3). The skeleton is additionally processedwith skelconv (see DisPerSE manual ) using the breakdown andsmooth functions. breakdown removes duplicate segments enter-ing a node from two different starting points. These segments can beso close as to be indistinguishable from one another and as such areremoved to prevent their over-representation in the final skeleton.The skeleton is then smoothed by averaging over the positions of the30 nearest neighbours of each segment. This mitigates the effectsof Poisson noise on the skeleton, ensuring that individual segmentslocally follow the global direction of the filament they belong to.In both the feedback and no-feedback runs, 1.22 physical kpc isthe maximum spatial resolution reached in filaments, defined asthe size of an individual cell on the highest AMR grid level thatentirely maps the filament (see Fig. 2). As is clear from Figure2, higher refinement levels are triggered within filaments but theircoverage is patchy, and mostly concentrated around halos/galaxiesembedded within these elongated structures. As we argue in ourconvergence analysis (Section 3), we believe 1.22 kpc is enough toresolve the radial structure of filaments, at least those that connectto halos/galaxies with masses similar (or larger) to the one we studyin this paper (roughly M ★ ). We emphasize that this is a much higherresolution than that currently reached in large-scale cosmologicalhydrodynamics simulations, where (cid:39) The gas and DM distributions differ significantly even for the no-feedback run (compare left and middle panels of Fig. 3), with thegas density field presenting much fewer filamentary structures thanthe DM . In addition, even though dwarf galaxies residing withinfilaments are affected by feedback, the impact of this feedback onthe growth of the central galaxy is minimal (it does not lead tothe disruption of the main filament) as the majority of the gasfeeding it at high redshift is accreted via filaments, and not frommergers (Danovich et al. 2012; Tillson et al. 2015). However, the gasdensity field in the run with feedback will be more perturbed dueto interactions with galaxy winds and shocks (see middle and right This is a consequence of re-ionisation reheating the gas of the IGM andpreventing accretion into the shallow potentials of DM filaments (Katz et al.2019). panels of Fig. 3), making the comparison between feedback and no-feedback runs difficult. Furthermore, DisPerSE is designed to workwith particle data, as it allows in this case a meaningful definitionof persistence (the very concept of which relies on quantifying thesignificance of a feature with respect to Poisson noise). For thesereasons, and given that we are not interested in probing the existenceof filaments within the virial radius of DM halos in this work, theDM density field seems more appropriate to carry out filamentextraction.We therefore elect to extract the skeleton from the DM densityfield, but trim it in order to keep only the main filament, along whichmost material flows onto the galaxy. For an M ★ central galaxy, themain filament traced in the gas clearly coincides with its DM coun-terpart (see top panels of Fig. 3). As we are analyzing a filamentconnecting to a single object, we identify the approximate regionwhere it begins and ends by eye, and select the highest density pointin this region as its starting/end point . We then use Dijkstra’s al-gorithm (Dijkstra 1959) to compute the shortest path (following theskeleton) between the start and end points. This works by assigningto each segment a distance from the start point, travelling along allthe various possible paths of the skeleton. Whenever a shorter pathto a given segment, 𝑠 , is found, then the selected path is updatedup to 𝑠 , and the distances to all segments connected to 𝑠 along thispath which have a longer path length, are updated. This processis iterated until the network is traversed, yielding the shortest pathbetween the given start and end point. The method is valid providedthe main filament flows mostly straight onto the galaxy, which, inturn, holds until the filament gets close to the galaxy disk (Powellet al. 2011).In order to avoid the filament passing through halos, filamentsegments located in regions with densities higher than 130 timesthe mean density of the Universe were excluded . This densitythreshold is chosen empirically, but the resulting skeleton does notdepend very sensitively on the chosen value provided this latter is onthe order of 100 times the mean density of the Universe. The entireinitial filamentary network and the resulting main filament extractedafter post-processing are shown in Fig. 4. Fig. 3 highlights that theskeletons extracted from the DM density fields of the feedbackand no-feedback runs are slightly different. In this Figure, one canclearly see a pair of filaments on the left side of the central galaxy,which are in the final stages of merging. As a result, our algorithmidentifies two possible paths along which the main filament wouldhave essentially the same length. Small changes in the noise levelassociated with the DM particles in the two different runs change theexact way that segments connect, resulting in the algorithm pickingone of these paths in one run and the other path in the other run.Our results are, by and large, independent of such small randomlyinduced differences. For larger volume cosmological simulations where an ensemble of fila-ments is available one can forgo the inspection by eye and simply use theclosest pair of galaxies with similar masses which are linked by the skeletonas the starting and end points of a filament. This value is lower than 200 times the critical density of the Universewhich is commonly used in the literature to define virialised structures. Thisreflects the fact that the density of halos at the virial radius is lower thantheir average density by about a factor 3.MNRAS000 , 1–17 (2019)
Marius Ramsøy et al. kp c kpc F il a m e n t g a s d e n s it y ( g / c m ) Figure 4.
The left plot shows the raw skeleton extracted by DisPerSE, which traces all the filaments of the DM density field, coloured according to the relativedensity (with low density in red and higher density in blue) . Using Dijkstra’s algorithm we then obtain the skeleton on the right, where we have removedfilament segments from regions with densities greater than 130 times the mean density, resulting in gaps around virialized halos and sub-halos (indicated bycircles enclosing their virial radii on Fig. 3). In both panels, the skeleton is overplotted on a 𝑧 = Due to the discrete Lagrangian nature of the numerical techniqueused to evolve the DM density field, a simple cloud-in-cell interpo-lation onto a reasonably sized regular grid generates a non-smoothdensity field in poorly sampled, low density regions. To get aroundthis difficulty, a Delaunay tessellation (Schaap & van de Weygaert2000) is computed from the DM density and velocity fields (see e.g.Schaap 2007), which ensures their spatial continuity. The Delaunaygrid is then projected onto a regular uniform grid, coinciding withAMR grid level 11, which corresponds to the maximum resolutionmapping of the entire filament (cubic cells 1.22 kpc on a side, seeFig. 2). This uniform grid is used for measurement of all quantitiesin this paper unless otherwise stated. The DM velocity dispersionfield – used as a proxy for temperature – is then obtained by com-puting the square of the difference between each particle velocityand the value of its nearest neighbour grid cell and re-applying theDelaunay tessellation with this dispersion as the weight. Every timethe Delaunay tessellation is projected onto the grid, we average allthe tetrahedra (or volume fractions of) that co-exist in each gridcell. The vorticity, on the other hand, is simply calculated by takingthe curl of the velocity field on the uniform grid. As this latter is ex-tremely noisy, a Gaussian smoothing is applied prior to computingvorticity, with a width of 2 cells.
For each segment of the skeleton, a field (density, temperature orvorticity) is linearly interpolated in a plane, the thickness of whichis equal to the skeleton segment length (typically 0.3 kpc, thoughthis depends on the local density). This plane is perpendicular tothe segment and centred on it. An example of individual cross-sections in the density, temperature and vorticity fields is displayedin Fig. 5. Note that the position of the DM or gas density peakdoes not necessarily lie exactly at the centre of the plane due tothe smoothing of the skeleton. Smoothing is required to ensure thatindividual segments point along the filament direction, and thus that the extracted planes are truly perpendicular to the filament.The gas density maximum is not tied to the DM density maximumand thus is also unlikely to be at the centre of the plane. In orderto correct for such small offsets which nevertheless do affect pro-file measurements, each plane is shifted using a method similar tothe ’shrinking sphere’ method outlined in Power et al. (2003). Thecentre of mass of a circle centred (with a radius greater than thetruncation radius) on the initial guess from DisPerSE is calculated.The circle is moved to the centre of mass before the procedure isrepeated with a smaller circle. This method is more robust to thepresence of additional substructure within the filament, particularlyas cells with 𝜌 > (cid:104) 𝜌 (cid:105) have had their density reduced to 40 (cid:104) 𝜌 (cid:105) forthe calculation of the centre of mass. This prevents halos existingwithin or near the filament from being chosen as the filament centreand distorting the filament profile. DM and gas planes are there-fore translated independently. This procedure allows us to align allsegments when stacking cross-sections.Vorticity and temperature fields interpolated onto the planeperpendicular to the segments are then translated with the sameshift as the density field. When looking in the plane perpendicularto them, filaments appear as strong peaks in the projected densityfield (see top row of Fig. 5). Alongside this, the major walls asso-ciated with these filaments is often visible extending out from thepeaks, forming thick elongated structures which are not necessarilystraight. In the temperature field (middle row and middle columnof Fig. 5), strong radial shocks are observed around the filamentsthemselves, with weaker shocks also present at the wall boundariesand where the walls intersect to form the filaments. In the vorticityfield (bottom row, middle column of Fig. 5) both filaments and wallsare identified with the regions of highest vorticity amplitude. TheDM filaments (left column of Figs. 3 and 5) appear wider than theirgaseous counterparts. Supernovae feedback (right column of Figs. 3and 5) renders filaments and walls imperceptible in the gas vorticityfield (bottom right panels) although radial shocks are still presentat the filament edges (middle right panel) and the gas density peakremains clearly visible (top right panel).Radial profiles are measured from the 2D cross-sections bycomputing the azimuthal average in concentric shells centred on MNRAS , 1–17 (2019) igh Redshift Filaments kp c kpc DM Gas (no feedback) Gas (feedback) D e n s it y ( g c m − ) T e m p e r a t u r e ( K ) V o r ti c it y ( k m s − kp c − ) Figure 5.
A typical filament cross-section, extracted 200 kpc away from the central galaxy in DM (left column) and gas (no-feedback run, middle column;feedback run, right column) at 𝑧 =
4. The thickness of the slice is of order 1 kpc. Note how the central filament (density peak in the 2D slice) is embedded in aweaker wall structure (which appears as a thick elongated tube encompassing the peak). From top to bottom row: density, temperature (or velocity dispersionfor DM, running from 0 to 25 kms − , dark blue to red) and vorticity along the filament, with red representing matter rotating counter-clockwise and blue in theopposite direction. the highest density point. When discussing the effects of resolutionon the filament profile, we take the median value for the distributionof all filament segments at a given resolution and for each radius, as asingle profile is required. However, for the rest of the measurementsin this paper, we consider individual profiles fitted to each cross-section over the entire radius range. In Fig. 6 the median profileobtained in that way is indicated by the filled red disk symbolsjoined by the red solid line, with the 1 𝜎 scatter around the medianprofile indicated by the shaded area. The advantage of this secondmethod (fitting the whole profile) is that we can easily bin resultsaccording to other filament properties, such as distance to the centralgalaxy. This should more accurately reflect the underlying filamentproperty distribution. Vorticity being a vector, the structure of the vorticity field is far morecomplex than density or temperature (as illustrated by the bottompanels of Figs. 3 and 5) and, as a result, it not easy to stack indi-vidual vorticity profiles obtained for each skeleton segment. Whenstacking is required, we therefore use the modulus of the vorticityparallel to the direction of the filament and ignore azimuthal varia-tions in vorticity. The vorticity field in the direction of the filamentis extracted in the same way as described in the previous sectionfor the density and temperature. As shown in the bottom panels ofFig. 5, the vorticity has a multipolar structure, with several rotat-ing and counter-rotating vortices surrounding the filament. Outsidethe filament the amplitude of vorticity rapidly declines. Within fil-aments, the geometry of the vorticity field is mainly quadrupolar(see Laigle et al. 2015, though we found that dipoles and higher
MNRAS000
MNRAS000 , 1–17 (2019)
Marius Ramsøy et al. order structures are not uncommon reflecting that the flow haveshell-crossed several times). This larger diversity in the structure ofthe vorticity field probably reflects the fact that the analysis in thispaper looks at smaller scale vorticity than Laigle et al. (2015), andextends the measurement to gas. We recall that primordial vortic-ity is destroyed in an expanding Universe, and therefore voids areextremely vorticity poor. Vorticity can later be produced by shocksor shell-crossing (respectively for gas and DM), and as result ischiefly confined to walls, filaments and nodes (see e.g. Pichon &Bernardeau 1999).
In the following, we first derive analytically the radial profiles offilaments under the assumption that they are in hydrostatic equilib-rium, and then compare them to the profiles directly measured inthe simulation.
To obtain our analytic solution, we make the simple assumption thatfilaments may be modelled as infinite self-gravitating isothermalcylinders. Fig. 7 presents the sound speed and velocity dispersionprofiles in filaments. We have been careful to subtract the bulk ve-locity of the material when extracting this data. Within the filament,the sound speed and velocity dispersion are flat and dominate overthe accretion velocity onto the filament, which suggests – for thecentre of the filament at least – that the filament may indeed betreated as an isothermal cylinder in hydrostatic equilibrium , i.e. ∇ 𝜙 = − ∇ 𝑃𝜌 , (1)where 𝑃 = K 𝜌 , and K = k B 𝑇 /( 𝜇 m p ) , with 𝜌 the density, 𝑇 thetemperature, k B the Boltzmann constant, m p the proton mass and 𝜇 the mean molecular weight of the gas. Stodólkiewicz (1963) solvedthis equation in the case of cylindrical symmetry (see also Ostriker1964), and we will discuss the solution shortly. However, before wedo, we briefly outline why it also applies to the collisionless DMfluid. Let us consider the time independent Jeans equations (Jeans1915) for such a collisionless system: 𝜕𝜕𝑥 𝑖 ( 𝑛𝑣 𝑖 ) = 𝑛𝑣 𝑖 𝜕𝑣 𝑗 𝜕𝑥 𝑖 = − 𝑛 𝜕𝜙𝜕𝑥 𝑗 − 𝜕 ( 𝑛𝜎 𝑖 𝑗 ) 𝜕𝑥 𝑖 . (2)where 𝑣 𝑖 are the velocities, 𝜎 𝑖 𝑗 are velocity dispersions and 𝑛 is theDM number density. Under cylindrical symmetry, we may neglectall but the radial component of these equations. Further assumingsteady state (i.e. that 𝑣 𝑟 =
0, such that accretion onto the filament isnegligible compared to internal pressure support) and that velocitydispersion is isotropic, the equations simplify to: 𝑑𝑑𝑟 ( 𝑛𝑣 𝑟 ) = 𝑑𝜙𝑑𝑟 = − 𝑛 𝑑 ( 𝑛𝜎 ) 𝑑𝑟 . (3)The second equation in (3) is entirely analogous to equation (1),with K = 𝜎 , though it is clear that accretion flow onto the filament At least in the plane perpendicular to the filament, as we know thateventually, DM and gas flow along the filament into dark matter halos. In thesteady state regime however, such a flow should not perturb the equilibrium. cannot be ignored at large radii (see Fig. 7). The solution to bothequations is therefore that given by Stodólkiewicz (1963): 𝜌 ( 𝑟 ) = 𝜌 (cid:0) + ( 𝑟 / 𝑟 ) (cid:1) with 𝑟 = √︄ K 𝜋𝐺 𝜌 . (4)where 𝐺 is Newton’s gravitational constant, 𝜌 is the central densityand 𝑟 the core radius of the filament. Note however that it is thegravitational potential common to both components which shouldappear on the left hand side in both the second equation in (3) andin eq. (1), so that technically speaking only the DM is truly close toa self-gravitating isothermal cylinder, the gas being in hydrostaticequilibrium in the potential well of the DM filament.As a preliminary test of the model, we can use a typical DMfilament central density of 𝜌 ∼ . × − g cm − , i.e. ∼
100 timesthe mean density of the Universe (central filament density is subjectto significant variations but this is typical of the median DM density,see left panel of Fig. 6) at 𝑧 = 𝜎 ∼
10 km/s (typical of themedian DM velocity dispersion we measure, see Fig. 7). Pluggingthese values in the second equation (4), we find a scale radius 𝑟 ∼ 𝜌 in equation(4) rather than that of the gas to obtain an estimate of 𝑟 for thislatter (as shown on Fig 6 the gas density is about a factor 5 lowerthan that of the DM throughout the filament, in agreement with theuniversal value Ω DM / Ω 𝐵 ). As the DM velocity dispersion for theparticular filament we study is comparable to the gas sound speed(i.e. ∼
10 km / s or 10 K which corresponds the bottom temperatureof the cooling curve for atomic hydrogen, see Fig. 7) we expect acore radius for the gas similar to that of the DM, i.e. 𝑟 ∼ 𝑦 perpendicular to the plane (Spitzer1978): 𝜌 ( 𝑦 ) = 𝜌 𝑦 sech ( 𝑦 / ℎ ) , (5)with the scale height ℎ = √︁ K/( 𝜋𝐺 𝜌 𝑦 ) taking a very similarfunctional form as 𝑟 in eq. (4), and 𝜌 𝑦 standing for the density inthe mid-plane ( 𝑦 =
0) of the wall. However, we need to integratethis wall profile over concentric cylindrical shells to evaluate howit modifies the filament profile. Unfortunately, this integral doesnot possess a simple analytic closed form, so we approximate theazimuthally averaged density of the wall by: 𝜌 ( 𝑟 ) = 𝜌 𝑦 𝛼 tanh ( 𝛼𝑟 / ℎ ) 𝑟 / ℎ , with 𝛼 = 𝜋 /
2. Such an approximation captures the asymptotic
MNRAS , 1–17 (2019) igh Redshift Filaments l og D e n s it y ( g / c m ) Radius (kpc)DM gas
Figure 6.
Fits of the median DM (left panel) and gas (right panel) density profiles (red solid line and red disk symbols) at 𝑧 =
4, using a filament plus wall M fil + wall (green dashed line)or a filament only M fil (yellow dashed line) model. The blue lines shows the M fil + wall decomposed in the filament (dot-dashed)and wall (dotted). The shaded area represents a 1 sigma deviation estimated by bootstrap re-sampling the values at each radial distance. behaviour of the correct solution for 𝑟 (cid:29) ℎ , and is accurate to betterthan 14% for all values of 𝑟 . As can be seen in Fig. 6, the inclusion ofa wall modifies the shape of the outer filament. This might (at leastpartially) explain the discrepancy between filament density profilespreviously reported in the literature, with power-law slopes rangingbetween -1 and -2 (see e.g. Colberg et al. 2005; Dolag et al. 2006;Aragón-Calvo et al. 2010). However we caution that these studieswere performed on much larger scales and so may not be directlycomparable to our work as they might potentially be affected bydifferent biases.One can easily show that in the case where the gas isothermalsound speed 𝑐 𝑠 = √︁ k B 𝑇 /( 𝜇 m p ) equals the DM dispersion velocity 𝜎 , the density profiles of the gas and DM have the same exactshape, differing only by their normalisation, i.e. the value of thecentral density. In the more general case where these two velocitiesdiffer, one can write the gas density profile: 𝜌 𝑔 ( 𝑟 ) = 𝜌 𝑔 (cid:16) + ( 𝑟 / 𝑟 ) (cid:17) − 𝜎 / 𝑐 𝑠 , so that if 𝑐 𝑠 > 𝜎 , it is shallower than that of the DM, and vice-versa.Katz et al. (2019) measured this effect by comparing two versionsof the same cosmological simulation with and without reionisation.They find that narrow streams are widened by the photo-heating ofthe gas, and that the gas counterparts of the lightest DM filamentscan even be entirely erased.Note that this reasoning also applies to the isothermal gasdensity profile of a DM dominated isolated wall: when 𝑐 𝑠 and 𝜎 differ, it becomes 𝜌 𝑔 ( 𝑦 ) = 𝜌 𝑔𝑦 sech 𝜎 / 𝑐 𝑠 ( 𝑦 / ℎ ) , The first assumption we have made concerns the isothermality ofthe filament-wall system and that accretion onto the filament pro-vides it with negligible support. In Fig. 7 we can see that for theno feedback run both the gas sound speed (black solid line andsolid disk symbols) and the DM velocity dispersion (red solid lineand solid disk symbols) stay constant over most of the width of thefilament, indicating that the isothermal approximation does indeedhold rather well. In addition, the gas and DM accretion velocities areconsiderably lower than the sound speed and velocity dispersion re-spectively, and as such the dynamics of the system should be mainlydriven by the pressure support. Moreover, the accretion shock itselfis also non-adiabatic. Indeed, the upstream mach number (beyond20 kpc) is M 𝑢 ≈ / =
5, thus the Rankine-Hugoniot jump con-ditions lead to a downstream Mach number M 𝑑 = .
47, and it thusfollows that the downstream sound speed (within 10 kpc of filamentcentre) should be 40 km s − , i.e. twice the value of that measured.This indicates that the filament accretion shock is radiative ratherthan adiabatic. Finally, we also plot on Fig. 7, the circular velocity 𝑉 𝑐 ≡ ( 𝐺 𝑀 / 𝑟 ) / (blue solid line with solid disk symbols) measuredfor the filament, where 𝑀 is the mass enclosed by a cylindrical shellof radius 𝑟 . We find that it is comparable to or lower than the soundspeed/velocity dispersion, which further indicates that the filamentis chiefly supported by pressure rather than by rotation, contrarilyto what is argued in Mandelker et al. (2018).In Fig. 6 we present 2 models, pure filament ( M fil ), filamentwith wall ( M fil + wall ). In practice, this means that along each indi-vidual skeleton segment, we fit the radial density using the formula: 𝜌 ( 𝑟 ) = 𝜌 (cid:0) + ( 𝑟 / 𝑟 ) (cid:1) + 𝜌 tanh ( 𝛼𝑟 / 𝑟 ) 𝑟 𝑟 , (6) MNRAS000
47, and it thusfollows that the downstream sound speed (within 10 kpc of filamentcentre) should be 40 km s − , i.e. twice the value of that measured.This indicates that the filament accretion shock is radiative ratherthan adiabatic. Finally, we also plot on Fig. 7, the circular velocity 𝑉 𝑐 ≡ ( 𝐺 𝑀 / 𝑟 ) / (blue solid line with solid disk symbols) measuredfor the filament, where 𝑀 is the mass enclosed by a cylindrical shellof radius 𝑟 . We find that it is comparable to or lower than the soundspeed/velocity dispersion, which further indicates that the filamentis chiefly supported by pressure rather than by rotation, contrarilyto what is argued in Mandelker et al. (2018).In Fig. 6 we present 2 models, pure filament ( M fil ), filamentwith wall ( M fil + wall ). In practice, this means that along each indi-vidual skeleton segment, we fit the radial density using the formula: 𝜌 ( 𝑟 ) = 𝜌 (cid:0) + ( 𝑟 / 𝑟 ) (cid:1) + 𝜌 tanh ( 𝛼𝑟 / 𝑟 ) 𝑟 𝑟 , (6) MNRAS000 , 1–17 (2019) Marius Ramsøy et al. V e l o c it y ( k m s − ) Radius (kpc) c 𝑠 v 𝑟,𝑔 𝜎 v 𝑟,𝐷𝑀 v 𝑐 Figure 7.
Median values for gas sound speed (black) from the no feedbackrun, DM velocity dispersion (red), gas (green) and DM (yellow) accretionvelocity and circular velocity (blue) profiles at 𝑧 =
4, with shaded regionsrepresenting the 1 sigma scatter about the mode for each data point. Notethat in the inner filament region, one measures a near constant sound speedand velocity dispersion, which indicates the filament is, to a large extent,isothermal. This breaks down at larger radii, due to both higher rates ofradial inflow and falling sound speed and velocity dispersion. where 𝑟 = ℎ , and 𝜌 = 𝜌 𝑦 𝛼 − . In principle, the values for 𝜎 and 𝑐 𝑠 could be different for the wall and embedded filament. However,for sake of simplicity and since we expect these two quantities toroughly behave in a similar manner, at least in the vicinity of thefilament, we ignore the possible change in the ratio of 𝜎 / 𝑐 𝑠 in our M fil + wall model (see Fig 7 for the validity of this assumption). Thefit is performed using the Levenberg-Marquardt algorithm wherethe wall is first fit to the outer half of the profile with the filamentcontribution set to zero. The wall parameters are then frozen inplace while fitting the filament parameters. In the case of a purefilament model (i.e. M fil ), the filament is fit to the entire profile,setting 𝜌 =
0. The procedure was tested by applying it to a sampleof artificial profiles, which typically returned radii within 1 cell ofthe input radius, but does break down when the filament is too wide(i.e. extends into the region where the wall is fitted). While this is asuitable range for the purposes of this paper, filaments continue togrow as time progresses, and this method may become unsuitableat later times. In Fig. 6, we show how each of the two models faresagainst the measured median density profile of the DM (left panel)or gas (right panel) at 𝑧 =
4. Errors on the radius are estimatedby considering the full distribution of density profiles measuredfrom individual skeleton segments, and fitting this distribution withthe best matched normal distribution to evaluate the value of thestandard deviation. For errors on the density, we use the best matchedlog-normal distribution instead, which is better suited to densitydistributions in filaments (see e.g. Cautun et al. 2014).Looking at Fig 6, it is not possible to distinguish the twomodels, M fil + wall (green curve) and M fil (yellow curve), in theinner region ( 𝑟 ≤
20 kpc). When the profiles are stacked as in thisfigure the fits work equally well with or without the wall. Howeverthe size of the core radius that these models return are very different when considering individual profiles: 𝑟 = . ± .
15 kpc for M fil compared to 𝑟 = . ± .
82 kpc for M fil + wall for the DM filament.This factor of 2 discrepancy is also present for the gas filament: 𝑟 = . ± .
86 kpc for M fil compared to 𝑟 = . ± .
96 kpc for M fil + wall . The core radii given by the M fil model can be rejected bysimple visual inspection of Fig. 3: they are comparable to the outeredge radius of the filaments.Although we only show the 𝑧 = 𝜒 𝜈 which peaks at 3 for the DM density profile and0.5 for the gas in the preferred model M fil + wall (dashed and solidgreen lines in Fig 8 respectively). For the M fil model these samedistributions are much less strongly peaked around 𝜒 𝜈 = 𝜒 𝜈 = 𝜒 suggest a fit to the simulation data whichlies somewhat on the poor side, it is unclear that the validity ofthe model should be measured by 𝜒 statistics in the first place.Indeed individual profiles deviations from the model are very likelycorrelated with one another when substructures residing within thefilament perturb its density field.For sake of completeness, let us mention that at 𝑧 =
4, the fitsof the full set of skeleton segments using the M fil + wall model to theDM component for the no feedback run returns 𝑟 = . ± .
82 kpcas the mode and width of the fitted gaussian for the core radius of thefilament (as previously mentioned), and 𝑟 = . ± .
80 kpc for thescale height of the wall. Similarly we fit a log normal distributionto the DM central densities to obtain log ( 𝜌 / gcm − ) = − . ± .
49 for the filament and log ( 𝜌 / gcm − ) = − . ± .
45 forthe wall. As for the gas, we obtain 𝑟 = . ± .
96 kpc and 𝑟 = . ± .
00 kpc, with densities of log ( 𝜌 / gcm − ) = − . ± . ( 𝜌 / gcm − ) = − . ± .
37. A list of values for thefilament radii and densities at other redshifts is provided in tablesA1 and A3.As gas filament temperature remains around 10 K at all timesafter re-ionization, their density profile flattens rapidly as the mass oftheir DM counterpart decreases and the sound speed approaches thecritical value of 𝑐 𝑠 = √ 𝜎 . This means that low mass filaments willonly exist in the DM component (compare the top left and middlepanels in Fig 3), as a 10 K gas has too much pressure to be trappedin the DM potential well in that case, and thus, talking about a gas 𝑟 becomes quite meaningless. On the other side of the mass range,we expect more massive filaments, where DM has a larger velocitydispersion, to have better defined cores in the gas than DM, as thisformer should still radiatively cool down to ∼ K and thus havea much steeper density profile than its DM counterpart. As a resultof this cooling, it is possible that the central gas density of massivefilaments will become comparable to that of the DM, in which caseour assumption that the DM sets the gravitational potential wouldcease to be valid and the core radii of the two components might thendiffer substantially. However, for the filament system considered in
MNRAS , 1–17 (2019) igh Redshift Filaments p 𝜒 𝜈 Figure 8.
Normalised distributions of reduced 𝜒 𝜈 obtained when fitting thefilament only (yellow), or filament plus wall (green) models to filament den-sity projections in individual slices perpendicular to each skeleton segmentat 𝑧 = this paper, the approximation of similar DM and gas density profilesseems to hold quite well (see Fig 6).In light of the previous discussion, we interpret the differencebetween the measured and predicted median values of 𝑟 as a depar-ture from the isothermal/hydrostatic approximations for the filament(see Fig. 9, middle panels), rather than to asymmetry or a systematicvariation of core size as a function of distance to the galaxy (seesection 11 for more detail concerning this latter variation). For theDM, the filament median velocity dispersion varies by 10% within ∼ − 𝑟 and increases 𝜌 asthe filament approaches the DM halo to which it is connected. Aspreviously mentioned, we will come back to this latter point in sec-tion 3.4 of our results devoted to the temporal and spatial evolutionof filament properties, but already note that such a focusing effectis not as pronounced in our simulations. Having extracted the main filament from the simulation and mea-sured the characteristic radius of its core through the use of a sim-plified model of hydrostatic equilibrium for its density profile, we now turn to the question of determining its outer size, or truncation radius, as the analytic profile cannot extend to infinity in the radialdirection.Beyond a certain radius it is no longer true that sound anddispersion velocities dominate over the accretion velocity, as canbe seen in Fig. 7. A failure of the hydrostatic model will thus oc-cur, leading to a potential definition of the truncation radius, whichalso coincides with the position of the accretion shock onto thefilament for the gas. We have opted not to use the peak tempera-ture position as a definition of the truncation radius, as individualskeleton segment profiles, both in temperature and vorticity areoften asymmetric and/or distorted by their environment, and maycontain multiple peaks when averaged over concentric radial shellsas a result (see Fig 5 for an example). Moreover, such a definitionwould not apply to DM velocity dispersion profiles. We have there-fore chosen to use a universal method for all physical quantitiesand types of filament (gas or DM), which also has the benefit ofproviding internal consistency between measurements. We thus de-fine the truncation radius as the point where the steepest descentin the temperature/vorticity/velocity dispersion profile is attained.In the DM, this is analogous to the splashback radius for halos asdefined in Diemer et al. (2017), as vorticity and velocity dispersionis only generated in the DM where shell crossing has occurred At 𝑧 = . ± . . ± . . ± . . ± . not a study wherewe change the resolution and re-run the simulation, but simply apost-processing of the same simulation at different resolutions, sowe expect to achieve better agreement than if we had done a properresolution study. As resolution increases progressively from 10 kpcto 1.22 kpc (level 8 to 11), the profiles are seen to converge acrossevery panel of Fig 9. Note that for comparison, our lowest levelof resolution, i.e. 10 kpc roughly corresponds to the highest levelof resolution available to capture filaments in current cosmologicalsimulations with volumes on the order of 100 Mpc on the side, likeMare Nostrum (Ocvirk et al. 2008), Horizon-AGN (Dubois et al.2014), MassiveBlackII (Khandai et al. 2015), Eagle (Schaye et al.2015), IllustrisTNG (Nelson et al. 2018) or SIMBA (Davé et al.2019).Looking first at the density profiles both of the DM and gasfilaments (top panels of Fig. 9), one can see that in going from thehighest resolution level to the lowest one, the central density (insidethe core) is underestimated by about an order of magnitude, andone becomes unable to measure the core radius of the profile with areasonable accuracy. On the other hand, the DM velocity dispersionprofiles (middle left panel of Fig 9) seem to converge faster thanthe density ones, with the lower resolution estimates compatiblewith the higher resolution ones at all radii. This seemingly rapidconvergence is induced by the shape of the isothermal profiles whichare, by definition, flat, especially in the case of the DM. For thegas (middle right panel of Fig 9), the temperature does not showas marked a convergence as the DM velocity dispersion becauseof the presence of the accretion shock: the low resolution data(black curve), which barely resolves the truncation radius of thefilament underestimates the shock temperature and overestimates MNRAS , 1–17 (2019) Marius Ramsøy et al. l og 𝜌 ( g c m − ) l og 𝜎 ( k m s − ) l og 𝜔 ( k m s − kp c − ) kpc l og 𝜌 ( g c m − ) l og T ( K ) l og 𝜔 ( k m s − kp c − ) DM Gas
Level 8Level 9Level 10Level 11
Figure 9.
The median radial profiles of the filaments in DM (left column) and gas in the no-feedback run (right column), for density (top row), temperature(middle row) and vorticity (bottom row) at 𝑧 =
4. Displayed profiles (from black to yellow) represent data extracted at different spatial resolutions of the AMRsimulation grid (vertical dashed lines or levels 8 to 11 respectively, see text for detail). Error bars are generated by bootstrapping the distribution of individualfilaments profiles, and taking the root mean square at each radius. MNRAS , 1–17 (2019) igh Redshift Filaments the core temperature by a similar amount. Having said that, the shockposition is fairly robust to resolution changes despite being radiallyasymmetric, which leads to its ‘smearing’. A minimum resolutionof 2.4 kpc is required to correctly capture both the temperature ofthe accretion shock and that of the gas filament core. Having focused, so far, the discussion of the filament profile at 𝑧 =
4, we now address the issue of its temporal evolution. Since 𝑟 = √︁ K/ 𝜋𝐺 𝜌 , we naively expect that 𝑟 ∝ ( + 𝑧 ) − / , providedthe filament central density scales with that of the background Uni-verse — which we measure to be the case (see Table A3) — andits central temperature/velocity dispersion remains roughly con-stant with redshift. Conversely, we can deduce the scaling of fil-ament temperature/velocity dispersion with redshift by measuringthe departure of 𝑟 from this specific power law scaling. In oursimulation, we find that for the gaseous filament, the central ra-dius grows as 𝑟 ∝ ( + 𝑧 ) − . ± . , which means that the soundspeed should scale like 𝑐 𝑠 ∝ ( + 𝑧 ) − . ± . whereas we measure 𝑐 𝑠 ∝ ( + 𝑧 ) − . ± . , i.e. an evolution quite consistent with thenaive expectation.For the DM filament counterpart, the growth of 𝑟 is faster,with a measurement of 𝑟 ∝ ( + 𝑧 ) − . ± . (see Fig 10), a fasterrate than the approximate size of the central galaxy ( 𝑟 gal = . 𝑟 vir ,blue solid line on the Figure). This implies that 𝜎 ∝ ( + 𝑧 ) − . ± . as redshift decreases, whereas we measure in the simulation that 𝜎 scales as ( + 𝑧 ) − . ± . . The evolution of both gas and DM fila-ment core radii are therefore consistent with the naive expectationat a ∼ 𝜎 confidence level. Strictly speaking, any evolution of thecentral sound speed and velocity dispersion is in contradiction withthe underlying assumption of isothermality used to derive the fila-ment profiles, as this latter requires no change in either quantity withredshift. However, as the evolution is slow compared to the soundcrossing time of the central region, an instantaneous isothermalprofile fits the data fairly well.The explanation for the somewhat faster growth of the coreradius of the filaments than the radius of the central halo to which itis connected is that the ’old’ core material is preferentially drainedby halos residing within the filament, while a ’new’ core forms outof more freshly accreted matter onto the filament (see e.g. Pichonet al. 2011). As a result, the filament core radius is more sensi-tive to the recent accretion history onto the filament than the halo.Such a behaviour is reminiscent, at least qualitatively, to that ofthe Navarro–Frenk–White density profile scale radius, 𝑟 𝑠 , found bye.g. Muñoz-Cuartas et al. (2011) whose time evolution also differssignificantly from that of the virial radius of the DM halo (exceptin that case it is the opposite: 𝑟 𝑠 , which is less sensitive to the halorecent accretion history, starts decreasing with redshift earlier than 𝑟 vir , see their Figure 5). As the gas can be considered, to first order,in hydrostatic equilibrium in the DM filament potential well, weexpect the evolution of its core radius to be somewhat influencedby that of the DM, i.e. that its growth also be sped up. We intendto explore this effect in more detail and with a larger sample offilaments to better assess the universality of this behaviour.As for the truncation radius for the gas/DM filaments, deter-mined from either the vorticity or the temperature/velocity disper-sion, it represents the locus where fresh material is accreting, andas such is the rough equivalent of the halo virial radius. Fig. 10shows the evolution of this radius as a function of redshift, along with the size of the main halo embedded in the filament ( 𝑟 vir , or-ange solid line). For the DM filament, the truncation radius evolvesas 𝑟 𝜔 ∝ ( + 𝑧 ) − . ± . or 𝑟 𝑇 ∝ ( + 𝑧 ) − . ± . , depend-ing on whether ones uses vorticity or velocity dispersion to de-fine it. This is a growth rate very similar to that of the halo size 𝑟 vir ∝ ( + 𝑧 ) − . ± . in this range of redshifts. However, thegas truncation radius, either derived from the vorticity or temper-ature of the gas filament which scale as 𝑟 𝜔 ∝ ( + 𝑧 ) − . ± . and 𝑟 𝑇 ∝ ( + 𝑧 ) − . ± . respectively, grows significantly fasterthan its DM counterpart. This is reminiscent of the stability drivenargument for the propagation of a radiative shock within DM halosadvanced by Birnboim & Dekel (2003), but this time applied tothe filament: as time progresses and density drops the shock is ableto propagate outwards and ends up filling the entire DM filamentvolume. Practically, this means that even though the gas filamentstarts off being smaller than the central halo embedded within it(see Fig. 10) at high redshift, the truncation radius rapidly catchesup with the virial radius. In our specific case, they are essentiallythe same size by 𝑧 = . 𝑧 = ∼ Stellar feedback has a profound impact on the region surroundingthe galaxy and filament. Given enough time, the superbubbles itgenerates extend most of the way up the filament, as can be seen inthe central and bottom right panels of Fig. 3. These galactic windsinject vorticity on large scales, and as such, this physical quantity is
MNRAS , 1–17 (2019) Marius Ramsøy et al. R a d i u s ( kp c ) Redshift zDM gas r r 𝑇 r 𝜔 r 𝑣𝑖𝑟 Δ 𝑥 r 𝑔𝑎𝑙 Figure 10.
Evolution of the core (black curves) and truncation radii (red and green curves for estimates based on the vorticity and temperature respectively)of the filaments with redshift. The left panel represents the DM filament, and its gas counterpart is on the right. Dashed lines represent the feedback run. Thevirial radius is shown in orange and the approximate extent of the central galaxy (20% of virial radius) in yellow. Finally, the spatial resolution of the simulationin the filament is indicated by the solid blue line at the bottom of each panel. no longer confined to the filamentary gas. Note that, in spite of this,larger scale cosmic web filaments (i.e. larger than the superbubble)could still have well defined vorticity quadrants. More importantly,vorticity in the dark matter filament counterpart (bottom left panelof Fig. 3) remains by-and-large unaffected, to the point that we donot deem it necessary to plot it on Fig. 11 for the run with feedback.The CGM/IGM gas is also strongly heated by this stellar feed-back, and so the temperature signature of the accretion shock ontothe filament is lost as well, as the middle right panel of Fig 3 demon-strates. Once again, this signature survives in the velocity dispersionof the DM component (middle left panel of Fig 3). Despite suchsignificant perturbations, the presence of a DM filament potentialwell coupled to the relatively high density of the gas ensures thatthe cooling time within the filament remains short. As a result, thefilament is still visible as a cold stream cutting through the hotsuperbubble in the middle right panel of Fig. 3.Due to these consequent perturbations induced by the stellarfeedback, we cannot use either the temperature or vorticity to definethe gas filament truncation radii in the feedback run. It should also benoted that our assumption of isothermality of the filaments becomesless valid than in the no stellar feedback case as stronger temperaturegradients develop between core and outer envelope. To be morespecific, in the no feedback case, the temperature varies betweencore and truncation radius by about a factor of two, but in thefeedback case in can reach an order of magnitude. However, mostof this gradient is localised in the outer parts of the filament, so thatthe central region retains a significantly large isothermal core. Thiscan be understood by performing the following simple calculation.Neglecting the presence of the wall, we may integrate both the gasand DM density profiles (from Eq. 4) to obtain the filament mass per unit length, 𝜇 , and its half-mass radius: 𝜇 ( 𝑟 ) = 𝜌 𝜋𝑟 + ( 𝑟 / 𝑟 ) and 𝑟 / = 𝑟 . (7)The fact that the (small) core radius contains half of the massmakes the filamentary material relatively impervious to the stellarfeedback/filament interaction: provided the core is shielded fromit, there can only be a minor change in the amount of gas massthe filament carries. It has been suggested in the literature thatKelvin-Helmholtz instabilities could be triggered at the interfacebetween cold filament gas and the feedback powered, hot, galac-tic wind (e.g. Mandelker et al. 2016, and subsequent work). Thesewill depend non trivially on redshift and the distance of a fila-ment segment to the central galaxy, so it is quite difficult to de-fine a unique characteristic timescale, 𝑡 KH . Nevertheless, writing 𝑡 KH ( 𝑟 ) = ( 𝑟 / 𝑣 𝑤 ) √︁ 𝜌 ( 𝑟 )/ 𝜌 𝑤 where 𝑣 𝑤 is the relative velocity be-tween the wind and the gas filament and 𝜌 𝑤 the density of the wind,we can see that given the steepness of the filament density profile wemeasure, 𝑡 KH becomes larger as the perturbation progresses deeperin the filament. This means that the timescale is ultimately set by 𝑡 KH ( 𝑟 ) . Plugging in typical numbers for our feedback run at 𝑧 = 𝑟 ∼ 𝜌 ( 𝑟 ) ∼ × − g cm − , 𝑣 𝑤 ∼
100 km/s, and 𝜌 𝑤 ∼ × − g cm − , we thus get 𝑡 KH ∼
20 Myr which is aboutan order of magnitude shorter than the infall time from the virialradius of the embedded halo. The conclusion is thus that our gasfilaments should not survive the interaction.Notwithstanding that this does not happen in our simulations,which might admittedly be of too low a resolution to capture theinstability properly, the calculation ignores both the importanceof radiative cooling within the filament which might confine the
MNRAS , 1–17 (2019) igh Redshift Filaments R a d i u s ( kp c ) l og 𝜌 ( g / c m ) Distance to central galaxy (kpc)r r 𝑇 r 𝜔 r 𝑣𝑖𝑟 𝜌 𝑔 𝜌 𝐷𝑀 Figure 11.
Top:
Gas filament radii as a function of distance to the galaxy,in the no-feedback ( solid line) and feedback runs ( dashed line) at 𝑧 = Bottom:
Central density of thefilament as a function of distance to the galaxy. Gas is black, DM is green,with solid and dashed lines representing no feedback and feedback runsrespectively. The vertical blue dashed line indicates the virial radius of thehalo. perturbations at the surface Vietri et al. (1997), and the importantfact that, as we have previously discussed, gas filaments are not self-gravitating but are located within a dominant DM filamentpotential well. Because of this, it is unclear as to whether Kelvin-Helmholtz instabilities can impart to the gas a radial velocity (as inperpendicular to the filament axis) larger than the escape velocitynecessary to climb out of this potential well. Should they not, theywould simply render the gas flow within the filament turbulentwithout affecting the filamentary nature of gas accretion onto halos.In the feedback case, we measure that the core radius of the gasfilament evolves with redshift as 𝑟 ∝ ( + 𝑧 ) − . ± . , i.e. with ascaling very similar to the no feedback run (see Fig 10). Neverthe-less, given the importance of the stellar feedback perturbations, oneexpects gas accretion onto the filament to be reduced in their pres-ence. To quantify this effect, we plot the ratio of median feedback to 𝜌 f b / 𝜌 no f b Radius (kpc) z Figure 12.
Ratio of the median gas density profiles in the feedback and nofeedback runs, 𝜌 fb / 𝜌 nofb , as a function of distance to the filament center.Curves of different colours represent different redshifts, as indicated on thefigure. Very early in the simulation ( 𝑧 =
8) feedback enhances the densityof gas in the filament. However, at almost all other redshifts, the reversehappens: the filament is depleted of gas in the feedback run as comparedto the no-feedback run. The amplitude of the effect is not monotonic withredshift. no feedback gas density profiles along the filament as a function ofredshift in Fig. 12. From the figure, one can see that while the sizeof core radius is not significantly affected by feedback, the centraldensity is, to a larger extent. At 𝑧 = 𝑧 = .
6, at which point the feedbackceases to have an effect on the filament core. We emphasize thatcontrary to the growth of the core/truncation radii, the impact offeedback does not scale monotonically with redshift, as it dependsboth on the global properties of the IGM/filament and the star for-mation history of the galaxy which drives the feedback. Indeed, asshown on Fig. 12, at early times ( 𝑧 ∼
8) the filament core density iseven enhanced by the action of feedback. It is possible that some ofthis extra gas will be entrained in the filament, but another possibil-ity is that it will act as a shield from fresh feedback at later times. Ina future paper, we plan to use tracer particles developed in Cadiouet al. (2019) to distinguish between these two situations. Outside thefilament, the density is seen to be enhanced in the simulation withfeedback, which is somehow expected from mass conservation ofthe filamentary gas and the presence of the extra material broughtby the galactic winds.It should be noted that the stellar feedback implemented inour simulation is the supernova prescription of Kimm et al. (2015),which ensures that the correct energy/momentum is given to the gasirrespective of whether the Taylor-Sedov phase of the supernovais spatially resolved. As such if the filaments are not destroyedby this supernova feedback then they are unlikely to be destroyedby any ’realistic’ supernova feedback. Yet, other types of stellarfeedback are also present which could alter filament properties,whether by direct action of the feedback on the filaments or throughsuppression of star formation and thereby the supernova feedback(e.g. resonant scattering of Lyman alpha photons in high redshiftdwarf galaxies Kimm et al. 2018). There is, of course, photo-heatingdue to ionising radiation which can induce an important gas densitydepletion especially in filaments connecting low mass halos (seeKatz et al. 2019, for detail). We believe that this effect is, by-
MNRAS000
MNRAS000 , 1–17 (2019) Marius Ramsøy et al. and-large, captured by the UV background model implementationpresent in both the stellar feedback and no-feedback runs. However,another mode of stellar feedback which we do not account for, mightbe more effective at filament disruption as it is less confined to thegalaxy: cosmic rays (see e.g. Pfrommer et al. 2017), . Finally, forfilaments connecting halos of higher mass, Dubois et al. (2013)showed that AGN are also very effective at disrupting filaments,and can even destroy their cores.
Theory suggests (e.g. Kereš et al. 2005; Dekel & Birnboim 2006;Pichon et al. 2011) that filaments play an extremely important rolein the evolution of galaxies at high redshift. However, their basiccharacteristics are, as yet, not completely understood, and they areextremely hard to detect observationally. We used a suite of highresolution cosmological zoom-in simulations, progressively includ-ing more of the relevant physics, to place constraints on the physicalproperties of such a filament, from large (Mpc) scales to the pointwhere it connects to the virial sphere of the central galaxy. Our mainfindings are as follows: • The filament in both DM and gas simulations can be describedfairly accurately by a universal density profile 𝜌 = 𝜌 𝑜 ( +( 𝑟 / 𝑟 ) ) corresponding to a cylinder in isothermal equilibrium • the filament core radius evolves for the gas grows as 𝑟 ∝ ( + 𝑧 ) − . ± . , with the DM filament core evolving as 𝑟 ∝ ( + 𝑧 ) − . ± . . This evolution of 𝑟 for the gas closely tracks that ofthe size of the galaxy (0 . 𝑟 vir ). • The filament has a second characteristic radius, the truncationradius, which is detectable (at least in simulations) in the temper-ature/velocity dispersion or vorticity fields. This radius scales as 𝑟 tr ∝ ( + 𝑧 ) − . ± . for DM and 𝑟 tr ∝ ( + 𝑧 ) − . ± . for thegas. The DM truncation radius closely matches the virial radius ofthe galaxy. The gas truncation radius is generally thinner at earlytimes. • The filament properties are mildly affected by stellar feedbackfrom the central galaxy. The core radius of the gas filament hardlychanges, but its central density is generally reduced by ∼ ∼ 𝑧 =
6. While the mass brought by theinflowing filament gas is affected to a level of ∼ 𝑧 = .
25, which also allows tocomprehensively extend the redshift range of the analysis. Sucha simulation will thus permit the extraction of a large sampleof filaments from which to derive statistically meaningful quantities.
ACKNOWLEDGEMENTS http://yorick.sourceforge.net/ ). DATA AVAILABILITY STATEMENT
The data underlying this article will be shared on reasonable requestto the corresponding author.
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APPENDIX A: DATA
In Table A1 ( no feedback run) and Table A2 ( feedback run) we presentthe redshift evolution of the filament radius, as fitted from the density,temperature and vorticity field.MNRAS000
In Table A1 ( no feedback run) and Table A2 ( feedback run) we presentthe redshift evolution of the filament radius, as fitted from the density,temperature and vorticity field.MNRAS000 , 1–17 (2019) Marius Ramsøy et al. z DM gas 𝑟 (kpc) 𝑟 𝑇 (kpc) 𝑟 𝜔 (kpc) 𝑟 (kpc) 𝑟 𝑇 (kpc) 𝑟 𝜔 (kpc)3.35 15 . ± .
66 32 . ± .
47 39 . ± .
38 9 . ± .
13 27 . ± .
07 23 . ± . . ± .
48 31 . ± .
94 33 . ± .
23 9 . ± .
92 26 . ± .
21 25 . ± . . ± .
22 33 . ± .
91 35 . ± .
45 7 . ± .
73 25 . ± .
04 17 . ± . . ± .
62 33 . ± .
44 32 . ± .
04 7 . ± .
70 27 . ± .
28 18 . ± . . ± .
59 28 . ± .
84 31 . ± .
66 6 . ± .
19 22 . ± .
36 16 . ± . . ± .
89 27 . ± .
14 29 . ± .
11 4 . ± .
38 20 . ± .
52 14 . ± . . ± .
82 28 . ± .
47 25 . ± .
48 5 . ± .
96 18 . ± .
96 21 . ± . . ± .
64 26 . ± .
79 24 . ± .
05 3 . ± .
53 17 . ± .
12 14 . ± . . ± .
50 20 . ± .
04 22 . ± .
57 4 . ± .
72 13 . ± .
64 13 . ± . . ± .
27 17 . ± .
05 19 . ± .
55 3 . ± .
85 10 . ± .
34 10 . ± . . ± .
53 14 . ± .
99 16 . ± .
37 2 . ± .
23 7 . ± .
13 6 . ± . . ± .
74 11 . ± .
52 15 . ± .
32 2 . ± .
39 7 . ± . . ± . . ± .
75 9 . ± .
58 10 . ± .
76 1 . ± .
79 3 . ± .
49 4 . ± . . ± .
00 6 . ± .
48 8 . ± .
82 1 . ± .
74 3 . ± .
56 3 . ± . Table A1.
Redshift evolution of the filament core radius derived from density ( 𝑟 ) and truncation radii derived from temperature ( 𝑟 𝑇 ) and vorticity ( 𝑟 𝜔 ) asfitted from the density, temperature and vorticity fields extracted from the no-feedback run. See Section 3.2 for details.z DM gas 𝑟 (kpc) 𝑟 𝑇 (kpc) 𝑟 𝜔 (kpc) 𝑟 (kpc) 𝑟 𝑇 (kpc) 𝑟 𝜔 (kpc)3.26 10 . ± .
41 36 . ± .
36 35 . ± .
25 7 . ± .
77 – –3.35 10 . ± .
81 33 . ± .
43 33 . ± .
34 12 . ± .
02 – –3.44 10 . ± .
77 34 . ± .
42 34 . ± .
36 8 . ± .
95 – –3.53 16 . ± .
50 34 . ± .
75 38 . ± .
90 7 . ± .
23 – –3.60 11 . ± .
17 36 . ± .
47 35 . ± .
22 8 . ± .
38 – –3.65 11 . ± .
88 30 . ± .
67 31 . ± .
53 8 . ± .
75 – –3.82 10 . ± .
37 28 . ± .
93 30 . ± .
28 5 . ± .
79 – –3.96 10 . ± .
59 27 . ± .
01 24 . ± .
15 5 . ± .
29 – –4.00 7 . ± .
43 27 . ± .
84 22 . ± .
05 5 . ± .
28 – –4.02 9 . ± .
30 26 . ± .
34 24 . ± .
01 4 . ± .
34 – –4.56 8 . ± .
60 23 . ± .
57 23 . ± .
06 4 . ± .
10 – –5.45 3 . ± .
96 14 . ± .
83 12 . ± .
13 2 . ± .
55 – –5.67 2 . ± .
33 11 . ± .
57 11 . ± .
60 2 . ± .
50 – –6.99 1 . ± .
13 9 . ± .
69 9 . ± .
58 2 . ± .
51 – –8.09 1 . ± .
00 7 . ± .
19 6 . ± .
91 2 . ± .
49 – –
Table A2.
Redshift evolution of the filament core radius derived from density ( 𝑟 ) and truncation radii derived from temperature ( 𝑟 𝑇 ) and vorticity ( 𝑟 𝜔 ) asfitted from the density, temperature and vorticity fields extracted from the feedback run. See Section 3.2 for details. Note the absence of data for the radiiderived from vorticity and temperature, due to the destructive impact of feedback on these fields.z DM gaslog ( 𝜌 / gcm − ) log ( 𝜌 / gcm − ) 𝑟 ( kpc ) log ( 𝜌 / gcm − ) log ( 𝜌 / gcm − ) 𝑟 ( kpc ) − . ± . − . ± .
46 7 . ± . − . ± . − . ± .
36 12 . ± . − . ± . − . ± .
40 9 . ± . − . ± . − . ± .
01 16 . ± . − . ± . − . ± .
44 7 . ± . − . ± . − . ± .
38 8 . ± . − . ± . − . ± .
43 8 . ± . − . ± . − . ± .
38 7 . ± . − . ± . − . ± .
48 8 . ± . − . ± . − . ± .
37 8 . ± . − . ± . − . ± .
41 7 . ± . − . ± . − . ± .
19 7 . ± . − . ± . − . ± .
45 6 . ± . − . ± . − . ± .
39 7 . ± . − . ± . − . ± .
10 6 . ± . − . ± . − . ± .
37 6 . ± . − . ± . − . ± .
41 6 . ± . − . ± . − . ± .
36 7 . ± . − . ± . − . ± .
20 6 . ± . − . ± . − . ± .
36 6 . ± . − . ± . − . ± .
61 6 . ± . − . ± . − . ± .
30 5 . ± . − . ± . − . ± .
25 9 . ± . − . ± . − . ± .
44 5 . ± . − . ± . − . ± .
20 6 . ± . − . ± . − . ± .
07 6 . ± . − . ± . − . ± .
22 6 . ± . − . ± . − . ± .
36 4 . ± . Table A3.
Redshift evolution of the model fitted density as extracted from the no-feedback run. See Section 3.2 for details. MNRAS , 1–17 (2019) igh Redshift Filaments z DM gaslog ( 𝜌 / gcm − ) log ( 𝜌 / gcm − ) 𝑟 ( kpc ) log ( 𝜌 / gcm − ) log ( 𝜌 / gcm − ) 𝑟 ( kpc ) − . ± . − . ± .
41 8 . ± . − . ± . − . ± .
47 13 . ± . − . ± . − . ± .
46 9 . ± . − . ± . − . ± .
45 8 . ± . − . ± . − . ± .
44 8 . ± . − . ± . − . ± .
20 9 . ± . − . ± . − . ± .
51 8 . ± . − . ± . − . ± .
48 7 . ± . − . ± . − . ± .
53 5 . ± . − . ± . − . ± .
56 9 . ± . − . ± . − . ± .
55 8 . ± . − . ± . − . ± .
41 7 . ± . − . ± . − . ± .
62 7 . ± . − . ± . − . ± .
42 8 . ± . − . ± . − . ± .
42 9 . ± . − . ± . − . ± .
07 8 . ± . − . ± . − . ± .
62 7 . ± . − . ± . − . ± .
44 10 . ± . − . ± . − . ± .
03 7 . ± . − . ± . − . ± .
53 9 . ± . − . ± . − . ± .
17 6 . ± . − . ± . − . ± .
42 7 . ± . − . ± . − . ± .
17 6 . ± . − . ± . − . ± .
39 6 . ± . − . ± . − . ± .
30 5 . ± . − . ± . − . ± .
37 6 . ± . − . ± . − . ± .
23 3 . ± . − . ± . − . ± .
36 6 . ± . − . ± . − . ± .
42 3 . ± . − . ± . − . ± .
26 4 . ± . Table A4.
Redshift evolution of the model fitted density as extracted from the feedback run. See Section 3.2 for details.MNRAS000