Galaxy Shapes and Intrinsic Alignments in The MassiveBlack-II Simulation
Ananth Tenneti, Rachel Mandelbaum, Tiziana Di Matteo, Yu Feng, Nishikanta Khandai
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 18 March 2014 (MN L A TEX style file v2.2)
Galaxy Shapes and Intrinsic Alignments in theMassiveBlack-II Simulation
Ananth Tenneti ∗ , Rachel Mandelbaum † , Tiziana Di Matteo ‡ , Yu Feng ,Nishikanta Khandai McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA
18 March 2014
ABSTRACT
The intrinsic alignment of galaxy shapes with the large-scale density field is a contami-nant to weak lensing measurements, as well as being an interesting signature of galaxyformation and evolution (albeit one that is difficult to predict theoretically). Here weinvestigate the shapes and relative orientations of the stars and dark matter of halosand subhalos (central and satellite) extracted from the MassiveBlack-II simulation,a state-of-the-art high resolution hydrodynamical cosmological simulation which in-cludes stellar and AGN feedback in a volume of (100 h − Mpc) . We consider redshiftevolution from z = 1 to 0 .
06 and mass evolution within the range of subhalo masses,10 − . × . h − M (cid:12) . The shapes of the dark matter distributions are generallymore round than the shapes defined by stellar matter. The projected root-mean-square(RMS) ellipticity per component for stellar matter is measured to be e rms = 0 .
28 at z = 0 . M subhalo > . h − M (cid:12) , which compares favourably with observationalmeasurements. We find that the shapes of stellar and dark matter are more roundfor less massive subhalos and at lower redshifts. By directly measuring the relativeorientation of the stellar matter and dark matter of subgroups, we find that, on aver-age, the misalignment between the two components is larger for less massive subhalos.The mean misalignment angle varies from ∼ ◦ − ◦ for M ∼ − h − M (cid:12) and shows a weak dependence on redshift. We also compare the misalignment an-gles in central and satellite subhalos at fixed subhalo mass, and find that centrals aremore misaligned than satellites. We present fitting formulae for the shapes of darkand stellar matter in subhalos and also the probability distributions of misalignmentangles. Key words: methods: numerical – hydrodynamics – gravitational lensing: weak –galaxies: star formation
Weak gravitational lensing is a useful probe to constrain cos-mological parameters since it is sensitive to both luminousand dark matter (Hu 2002; Benabed & van Waerbeke 2004;Ishak et al. 2004; Takada & White 2004; Bernstein & Jain2004; Huterer 2010). In particular, weak lensing surveys canbe used to probe theories of modified gravity and provideconstraints on the properties of dark matter and dark energy(Albrecht et al. 2006; Weinberg et al. 2013). Many upcom-ing surveys like Large Synoptic Survey Telescope(LSST) ∗ [email protected] † [email protected] ‡ [email protected] and Euclid aim to determine the constant and dynamicalparameters of the dark energy equation of state to a veryhigh precision using weak lensing.However, constraining cosmological parameters withsub-percent errors in future cosmological survey requires thesystematic errors to be well below those in typical weaklensing measurements with current datasets. The intrinsicshapes and orientations of galaxies are not random but cor-related with each other and the underlying density field.This is known as intrinsic galaxy alignments. The intrinsicalignment (IA) of galaxy shapes with the underlying densityfield is an important theoretical uncertainty that contami-nates weak lensing measurements (Heavens et al. 2000; Croft http://sci.esa.int/euclid/c (cid:13) a r X i v : . [ a s t r o - ph . C O ] M a r Tenneti et al. & Metzler 2000; Jing 2002; Hirata & Seljak 2004). Accuratetheoretical predictions of IA through analytical models and N -body simulations (Hirata & Seljak 2004; Heymans et al.2006; Schneider & Bridle 2010; Joachimi et al. 2013) in theΛCDM paradigm is complicated by the absence of bary-onic physics, which we expect to be important given thatthe alignment of interest is that of the observed, baryoniccomponent of galaxies. So, we either need simulations thatinclude the physics of galaxy formation or N -body simula-tions with rules for galaxy shapes and alignments.Proposed analysis methods to remove IA from weaklensing measurements either involve removing considerableamount of cosmological information (which requires veryaccurate redshift information; nulling methods: Joachimi& Schneider 2008, 2009), or involve marginalizing overparametrized models of how the intrinsic alignments affectobservations as a function of scale, redshift and galaxy type(e.g., Bridle & King 2007; Joachimi & Bridle 2010; Blazeket al. 2012). The simultaneous fitting method, with a rel-atively simple intrinsic alignments model, was used for atomographic cosmic shear analysis of CFHTLenS data (Hey-mans et al. 2013). The latter methods, while preserving morecosmological information than nulling methods, can onlywork correctly if there is a well-motivated intrinsic align-ments model as a function of galaxy properties. Existingcandidates for the intrinsic alignment model to be used insuch an approach include the linear alignment model (Hi-rata & Seljak 2004) or simple modifications of it (e.g., usingthe nonlinear power spectrum: Bridle & King 2007), N -bodysimulations populated with galaxies and stochastically mis-aligned with halos in a way that depends on galaxy type(Heymans et al. 2006), and the halo model (Schneider &Bridle 2010), which includes rules for how central and satel-lite galaxies are intrinsically aligned.In this study, we use the large volume, high-resolutionhydrodynamic simulation, MassiveBlack-II (Khandai et al.2014), which includes a range of baryonic processes to di-rectly study the shapes and alignments of galaxies. In par-ticular, we measure directly the shapes of the dark andstellar matter components of halos and subhalos (modeledas ellipsoids in three dimensional space). We examine howshapes evolve with time and as a function of halo/subhalomass. Previous work used N -body simulations and analyt-ical modeling to study triaxial shape distributions of darkmatter halos as a function of mass and their evolution withredshift (Hopkins et al. 2005; Allgood et al. 2006; Lee et al.2005; Schneider et al. 2012). More recently, hydrodynamiccosmological simulations have also been used to study theeffects of baryonic physics on the shapes of dark matter ha-los (Bailin et al. 2005; Kazantzidis et al. 2006; Knebe et al.2010; Bryan et al. 2013). Here, using a high-resolution hy-drodynamic simulation in a large cosmological volume thatincorporates the physics of star formation and associatedfeedback as well as black hole accretion and AGN feedback,we focus on measuring directly the shapes of the stellar com-ponents of galaxies and examine the misalignments betweenstars and dark matter in galaxies (central and satellite). Wealso measure the projected (2D) shapes for comparison withobservations. This study is important because the measuredintrinsic alignments of galaxies are related to the projectedshape correlations of the stellar component of subgroups(galaxies) by the density-ellipticity and ellipticity-ellipticity correlations (Heymans et al. 2006). By measuring the pro-jected ellipticities of the stellar and dark matter componentof simulated galaxies, we can attempt to understand the dif-ferences between these two. In addition, we can do a basiccomparison of the stellar components with observational re-sults, and validate the realism of the simulated galaxy pop-ulation.Another aspect of the problem that we consider in thispaper is the relative orientation of the stellar component ofthe halo with its dark matter component. Many dark matter-only simulations have illustrated that dark matter halosexhibit large-scale intrinsic alignments (e.g., Faltenbacheret al. 2002; Hopkins et al. 2005; Altay et al. 2006; Heymanset al. 2006), but the prediction of galaxy intrinsic align-ments from halo intrinsic alignments requires a statisticalunderstanding of the relationship between galaxy and haloshapes. To date, there has been no direct measurement ofgalaxy versus halo misalignment with a large statistical sam-ple of galaxies through hydrodynamic simulations. Recently,Dubois et al. (2014) studied the alignment between the spinof galaxies and their host filament direction using a hydro-dynamical cosmological simulation of box size 100 h − Mpc.Studies of misalignment based on SPH simulations of smallervolumes detected misalignments between the baryonic anddark matter component of halos (van den Bosch et al. 2003;Sharma & Steinmetz 2005; Hahn et al. 2010; Deason et al.2011). These studies considered the correlation of spin andangular momentum of the baryonic component with darkmatter. The spin correlations are arguably more relevantfor the intrinsic alignments of spiral galaxies (Hirata & Sel-jak 2004), whereas the observed intrinsic alignments in realgalaxy samples are dominated by red, pressure-supported,elliptical galaxies (Mandelbaum et al. 2011; Joachimi et al.2011); hence a study of the correlation of projected shapesis more relevant for the issue of weak lensing contamination.However, to make precise predictions based on the halo orsubhalo mass at different redshifts, we need a hydrodynamicsimulation of very large volume and high resolution. TheMassiveBlack-II SPH simulation meets those requirements,making it a good choice for this kind of study.Others arrived at constraints on misalignments using N -body simulations and calibrating the misalignments byadopting a simple parametric form to agree with observa-tionally detected shape correlation functions (Faltenbacheret al. 2009; Okumura et al. 2009). There are also studies ofthe alignment of a central galaxy with its host halo whereit is assumed that the satellites trace the dark matter dis-tribution (e.g., Wang et al. 2008). By using hydrodynamicsimulations, we can directly calculate the misalignment dis-tributions for all galaxies as a function of halo mass andcosmic time. Resolution of the galaxies into centrals andsatellites also helps to understand the effect of local envi-ronment.This paper is organized as follows. In Section 2, we de-scribe the SPH simulations used for this work and the meth-ods used to obtain the shapes and orientations of groups andsubgroups. In Section 3, we give the axis ratio distributionsof dark matter and stellar matter of subgroups. In Section 4,we show our results for misalignments of the stellar compo-nent of subgroups with their host dark matter subgroups. InSection 5 we compare the shape distributions and misalign-ment angle between centrals and satellites. Finally, we sum- c (cid:13)000
Weak gravitational lensing is a useful probe to constrain cos-mological parameters since it is sensitive to both luminousand dark matter (Hu 2002; Benabed & van Waerbeke 2004;Ishak et al. 2004; Takada & White 2004; Bernstein & Jain2004; Huterer 2010). In particular, weak lensing surveys canbe used to probe theories of modified gravity and provideconstraints on the properties of dark matter and dark energy(Albrecht et al. 2006; Weinberg et al. 2013). Many upcom-ing surveys like Large Synoptic Survey Telescope(LSST) ∗ [email protected] † [email protected] ‡ [email protected] and Euclid aim to determine the constant and dynamicalparameters of the dark energy equation of state to a veryhigh precision using weak lensing.However, constraining cosmological parameters withsub-percent errors in future cosmological survey requires thesystematic errors to be well below those in typical weaklensing measurements with current datasets. The intrinsicshapes and orientations of galaxies are not random but cor-related with each other and the underlying density field.This is known as intrinsic galaxy alignments. The intrinsicalignment (IA) of galaxy shapes with the underlying densityfield is an important theoretical uncertainty that contami-nates weak lensing measurements (Heavens et al. 2000; Croft http://sci.esa.int/euclid/c (cid:13) a r X i v : . [ a s t r o - ph . C O ] M a r Tenneti et al. & Metzler 2000; Jing 2002; Hirata & Seljak 2004). Accuratetheoretical predictions of IA through analytical models and N -body simulations (Hirata & Seljak 2004; Heymans et al.2006; Schneider & Bridle 2010; Joachimi et al. 2013) in theΛCDM paradigm is complicated by the absence of bary-onic physics, which we expect to be important given thatthe alignment of interest is that of the observed, baryoniccomponent of galaxies. So, we either need simulations thatinclude the physics of galaxy formation or N -body simula-tions with rules for galaxy shapes and alignments.Proposed analysis methods to remove IA from weaklensing measurements either involve removing considerableamount of cosmological information (which requires veryaccurate redshift information; nulling methods: Joachimi& Schneider 2008, 2009), or involve marginalizing overparametrized models of how the intrinsic alignments affectobservations as a function of scale, redshift and galaxy type(e.g., Bridle & King 2007; Joachimi & Bridle 2010; Blazeket al. 2012). The simultaneous fitting method, with a rel-atively simple intrinsic alignments model, was used for atomographic cosmic shear analysis of CFHTLenS data (Hey-mans et al. 2013). The latter methods, while preserving morecosmological information than nulling methods, can onlywork correctly if there is a well-motivated intrinsic align-ments model as a function of galaxy properties. Existingcandidates for the intrinsic alignment model to be used insuch an approach include the linear alignment model (Hi-rata & Seljak 2004) or simple modifications of it (e.g., usingthe nonlinear power spectrum: Bridle & King 2007), N -bodysimulations populated with galaxies and stochastically mis-aligned with halos in a way that depends on galaxy type(Heymans et al. 2006), and the halo model (Schneider &Bridle 2010), which includes rules for how central and satel-lite galaxies are intrinsically aligned.In this study, we use the large volume, high-resolutionhydrodynamic simulation, MassiveBlack-II (Khandai et al.2014), which includes a range of baryonic processes to di-rectly study the shapes and alignments of galaxies. In par-ticular, we measure directly the shapes of the dark andstellar matter components of halos and subhalos (modeledas ellipsoids in three dimensional space). We examine howshapes evolve with time and as a function of halo/subhalomass. Previous work used N -body simulations and analyt-ical modeling to study triaxial shape distributions of darkmatter halos as a function of mass and their evolution withredshift (Hopkins et al. 2005; Allgood et al. 2006; Lee et al.2005; Schneider et al. 2012). More recently, hydrodynamiccosmological simulations have also been used to study theeffects of baryonic physics on the shapes of dark matter ha-los (Bailin et al. 2005; Kazantzidis et al. 2006; Knebe et al.2010; Bryan et al. 2013). Here, using a high-resolution hy-drodynamic simulation in a large cosmological volume thatincorporates the physics of star formation and associatedfeedback as well as black hole accretion and AGN feedback,we focus on measuring directly the shapes of the stellar com-ponents of galaxies and examine the misalignments betweenstars and dark matter in galaxies (central and satellite). Wealso measure the projected (2D) shapes for comparison withobservations. This study is important because the measuredintrinsic alignments of galaxies are related to the projectedshape correlations of the stellar component of subgroups(galaxies) by the density-ellipticity and ellipticity-ellipticity correlations (Heymans et al. 2006). By measuring the pro-jected ellipticities of the stellar and dark matter componentof simulated galaxies, we can attempt to understand the dif-ferences between these two. In addition, we can do a basiccomparison of the stellar components with observational re-sults, and validate the realism of the simulated galaxy pop-ulation.Another aspect of the problem that we consider in thispaper is the relative orientation of the stellar component ofthe halo with its dark matter component. Many dark matter-only simulations have illustrated that dark matter halosexhibit large-scale intrinsic alignments (e.g., Faltenbacheret al. 2002; Hopkins et al. 2005; Altay et al. 2006; Heymanset al. 2006), but the prediction of galaxy intrinsic align-ments from halo intrinsic alignments requires a statisticalunderstanding of the relationship between galaxy and haloshapes. To date, there has been no direct measurement ofgalaxy versus halo misalignment with a large statistical sam-ple of galaxies through hydrodynamic simulations. Recently,Dubois et al. (2014) studied the alignment between the spinof galaxies and their host filament direction using a hydro-dynamical cosmological simulation of box size 100 h − Mpc.Studies of misalignment based on SPH simulations of smallervolumes detected misalignments between the baryonic anddark matter component of halos (van den Bosch et al. 2003;Sharma & Steinmetz 2005; Hahn et al. 2010; Deason et al.2011). These studies considered the correlation of spin andangular momentum of the baryonic component with darkmatter. The spin correlations are arguably more relevantfor the intrinsic alignments of spiral galaxies (Hirata & Sel-jak 2004), whereas the observed intrinsic alignments in realgalaxy samples are dominated by red, pressure-supported,elliptical galaxies (Mandelbaum et al. 2011; Joachimi et al.2011); hence a study of the correlation of projected shapesis more relevant for the issue of weak lensing contamination.However, to make precise predictions based on the halo orsubhalo mass at different redshifts, we need a hydrodynamicsimulation of very large volume and high resolution. TheMassiveBlack-II SPH simulation meets those requirements,making it a good choice for this kind of study.Others arrived at constraints on misalignments using N -body simulations and calibrating the misalignments byadopting a simple parametric form to agree with observa-tionally detected shape correlation functions (Faltenbacheret al. 2009; Okumura et al. 2009). There are also studies ofthe alignment of a central galaxy with its host halo whereit is assumed that the satellites trace the dark matter dis-tribution (e.g., Wang et al. 2008). By using hydrodynamicsimulations, we can directly calculate the misalignment dis-tributions for all galaxies as a function of halo mass andcosmic time. Resolution of the galaxies into centrals andsatellites also helps to understand the effect of local envi-ronment.This paper is organized as follows. In Section 2, we de-scribe the SPH simulations used for this work and the meth-ods used to obtain the shapes and orientations of groups andsubgroups. In Section 3, we give the axis ratio distributionsof dark matter and stellar matter of subgroups. In Section 4,we show our results for misalignments of the stellar compo-nent of subgroups with their host dark matter subgroups. InSection 5 we compare the shape distributions and misalign-ment angle between centrals and satellites. Finally, we sum- c (cid:13)000 , 000–000 hapes Alignments marize our conclusions in Section 6. The functional formsfor our results are provided in the Appendix. We use the MassiveBlack-II (MBII) simulation to measureshapes and alignments of dark matter and stellar compo-nents of halos and subhalos. MBII is a state-of-the-art highresolution, large volume, cosmological hydrodynamic simu-lation of structure formation. An extensive description of thesimulation and major predictions for the halo and subhalomass functions, their clustering, the galaxy stellar mass func-tions, galaxy spectral energy distribution and properties ofthe AGN population is presented by Khandai et al. (2014).We refer the reader to this publication for details on MBIIand briefly summarize the major relevant aspects here.The MBII simulation was performed with the cosmolog-ical TreePM-Smooth Particle Hydrodynamics (SPH) code p-gadget . It is a hybrid version of the parallel code, gad-get2 (Springel et al. 2005a) that has been upgraded to runon Petaflop scale supercomputers. In addition to gravity andSPH, the p-gadget code also includes the physics of multi-phase ISM model with star formation (Springel & Hernquist2003a), black hole accretion and feedback (Springel et al.2005a; Di Matteo et al. 2012). Radiative cooling and heat-ing processes are included (as in Katz et al. 1996), as isphotoheating due to an imposed ionizing UV background.The interstellar medium (ISM), star formation and super-novae feedback as well as black hole accretion and associatedfeedback are treated by means of previously developed sub-resolution models. In particular, the multiphase model forstar forming gas we use, developed by Springel & Hernquist(2003b), has two principal ingredients: (1) a star formationprescription and (2) an effective equation of state (EOS).A thermal instability is assumed to operate above a criticaldensity threshold ρ th , producing a two phase medium con-sisting of cold clouds embedded in a tenuous gas at pressureequilibrium. Stars form from the cold clouds, and short-livedstars supply an energy of 10 ergs to the surrounding gasas supernovae. This energy heats the diffuse phase of theISM and evaporates cold clouds, thereby establishing a self-regulation cycle for star formation. ρ th is determined self-consistently in the model by requiring that the EOS is con-tinuous at the onset of star formation. Stellar feedback in theform of stellar winds is also included. The prescription forblack hole accretion and associated feedback from massiveblack holes follows the one developed by Di Matteo et al.(2005); Springel et al. (2005b). We represent black holes bycollisionless particles that grow in mass by accreting gas (atthe local dynamical timescale) from their environments. Ifthe accretion rates reach the critical Eddington limit theyare then capped at that value. A fraction f (fixed to 5% to fitthe local black-hole galaxy relations) of the radiative energyreleased by the accreted material is assumed to couple ther-mally to nearby gas and influence its thermodynamic state.Black holes merge when they approach the spatial resolutionlimit of the simulation (Springel & Hernquist 2003b)MBII contains N part = 2 × dark matter and gasparticles in a cubic periodic box of length 100 h − Mpc on a side, with a gravitational smoothing length (cid:15) = 1 . h − kpcin comoving units. A single dark matter particle has a mass m DM = 1 . × h − M (cid:12) and the initial mass of a gas parti-cle is m gas = 2 . × h − M (cid:12) . The cosmological param-eters used in the simulation are as follows: amplitude ofmatter fluctuations σ = 0 . η s = 0 . m = 0 . Λ = 0 . b = 0 . h = 0 .
702 as perWMAP7 (Komatsu et al. 2011).Fig. 1 shows snapshots of the MBII simulation withdark matter and stellar matter distributions at redshift z = 0 .
06. From the top figure, we can see the formationof cosmic web with galaxies extending over the whole lengthof the simulation volume. The bluish-white colored region inthe figure represents the density of the dark matter distri-bution and the red lines show the direction of the major axisof ellipse for the projected shape defined by the stellar com-ponent. The figures in the bottom panel, which are zoomedsnapshots of individual halos of different masses, show thedensity distribution of dark matter and stellar matter. Theover plotted blue and red ellipses depict the projected shapesof dark matter and stellar matter of subhalos respectively.To generate group catalogs of particles in the simula-tion, we used the friends of friends (FoF) group finder algo-rithm (Davis et al. 1985). This algorithm identifies groupson the fly using linking length of 0 . z = 1 . z = 0 .
06. We find good agreement with thetheoretical prediction given in Tinker et al. (2008) based onSpherical Overdensity (SO) approach. This gives an idea ofthe mass range we are exploring by the use of this simulation.To generate subgroup catalogs, the subfind code (Springelet al. 2001) is used on the group catalogs. The subgroupsare defined as locally overdense, self-bound particle groups.Groups of particles are defined as subgroups when they haveat least 20 gravitationally bound particles. A comparison be-tween the properties of halos and subhalos recovered usingdifferent halo and subhalo finders can be found in Knebeet al. (2011), where it is concluded that the properties ofhalos and subhalos, like mass, position, velocity, two-pointcorrelation returned by different finders agree within errorbars to each other. In all the discussions in this paper, halosand subhalos are interchangeable for groups and subgroupsrespectively.
Here we describe the method adopted to determine theshapes and orientations of groups and subgroups for darkmatter and stellar components. For each group and sub-group, the dark matter and stellar shapes are determinedby using the positions of dark matter and star particlesrespectively. By using the positions of all particles of thecorresponding type, the halo and subhalo shapes in 3D aremodelled as ellipsoids. For projected shapes, the positionsof particles of corresponding type projected onto the XY plane are used to model the shapes as ellipses. We use theunweighted inertia tensor given by c (cid:13) , 000–000 Tenneti et al.
Figure 1.
Top:
Snapshot of the MBII simulation in a slice of thickness 2 h − Mpc at redshift z = 0 .
06. The bluish-white colored regionrepresents the density of the dark matter distribution and the red lines show the direction of the major axis of ellipse for the projectedshape defined by the stellar component.
Bottom Left:
Dark matter (shown in gray) and stellar matter (shown in red) distribution in themost massive group at z = 0 .
06 of mass 7 . × h − M (cid:12) . The blue and red ellipses show the projected shapes of dark matter andstellar matter of subhalos respectively. Bottom Middle:
Dark matter and stellar matter distribution in a group of mass 3 . × h − M (cid:12) . Bottom Right:
Dark matter and stellar matter distribution in a group of mass 1 . × h − M (cid:12) . I ij = (cid:80) n m n x ni x nj (cid:80) n m n , (1)where m n represents the mass of the n th particle and x ni , x nj represent the position coordinates of the n th parti-cle with 0 (cid:54) i, j (cid:54) (cid:54) i, j (cid:54) N -body simulation by considering only parti-cles within a given fraction of the virial radius. In this paper,we are only concerned with the standard unweighted inertiatensor definition for determining shapes and defer investiga-tion of other definitions for a future study.Consider the 3D case. Let the eigenvectors of the iner-tia tensor be ˆ e a , ˆ e b , ˆ e c and the corresponding eigenvalues be λ a , λ b , λ c , where λ a > λ b > λ c . The eigenvectors represent c (cid:13) , 000–000 hapes Alignments Figure 2.
Dark matter and stellar mass function for FOF groups(halos) at z = 0 . , .
0, compared with the SO-based predictionfrom Tinker et al. (2008) generated with ∆ = 0 . the principal axes of the ellipsoids with the lengths of theprincipal axes ( a, b, c ) given by the square roots of the eigen-values ( √ λ a , √ λ b , √ λ c ). We now define the 3D axis ratiosas q = ba , s = ca (2)In 2D, the eigenvectors are ˆ e (cid:48) a , ˆ e (cid:48) b with the correspondingeigenvalues λ (cid:48) a , λ (cid:48) b , where λ (cid:48) a > λ (cid:48) b . The lengths of major andminor axes are a (cid:48) = √ λ (cid:48) a , b (cid:48) = (cid:112) λ (cid:48) b with axis ratio, q (cid:48) = b (cid:48) /a (cid:48) as defined before.Our predictions from SPH simulations can be com-pared with those from N -body simulations using the full 3Dshapes, while the projected shapes are useful for comparisonwith results from observational data. In all our results, weused groups and subgroups with a minimum of 1000 darkmatter and star particles each. We describe the convergencetests performed to arrive at this cutoff in Section 2.3. The reliability of statements about the shapes of matter dis-tributions depends on the number of particles used to tracethose distributions. Thus, we made a convergence test to fixthe minumum number of particles needed to measure shapesof halos and subhalos reliably. In Fig. 3, we show the his-tograms of shapes measured using all the dark matter parti-cles in a given subhalo, and compared it with the histogramsobtained by using a random subsample of 50, 300, 500 and1000 particles in the subhalo. This is done in a mass rangewhere we have enough subhalos with > (cid:104) q (cid:105) is 0 .
83 and (cid:104) s (cid:105) is0 .
70 using all particles. (cid:104) q (cid:105) varies as 0 . , . , . , . , .
83 using 50 , , , (cid:104) s (cid:105) are 0 . , . , . , . , .
70. Al-though the mean axis ratios show good convergence with300 or 500 particles, from the plots we can see that the his-tograms have not converged. Hence, we choose a minimum of1000 particles for our analysis. In Figure 4, we show a con-tour plot of the number of dark matter particles and starparticles in subgroups at z = 0 .
06. The two different densitypeaks in the contour plot are due to different dark matter tostellar mass ratios in centrals and satellite subgroups. Theright density peak corresponds to central subhalos while theleft one is for satellite subgroups, which exhibit strippingof the dark matter subhalo and hence fewer dark matterparticles. The lines show a cutoff of 1000 particles for darkmatter and star particles. By choosing this cutoff, we areexcluding subhalos of low stellar to halo mass ratio in sub-halos around the low mass range 10 − . h − M (cid:12) . Soin this mass range, we are excluding a significant fraction ofsubhalos with low stellar mass from our analysis. However,in the high mass range, we are able to analyze a fair sampleof subhalos. In this section, we show the axis ratio distributions of theshapes of dark matter and stellar matter component of ha-los and subhalos modeled as ellipsoids as described in Sec-tion 2.2. We investigate their dependence on the mass rangeof subgroups and their evolution with redshift. We also com-pare the relative axis ratio distributions of dark matter andstellar matter in subhalos.
The distributions of axis ratios, q ( b/a ) and s ( c/a ) for darkmatter and stellar matter of subgroups at redshift z = 0 . . − . h − M (cid:12) ,10 . − . h − M (cid:12) , and > . h − M (cid:12) . For convenience,we refer to these mass bins as M , M M M
3, the largest subhalo mass is1 . × h − M (cid:12) at z = 1 .
0, with a host halo mass of1 . × h − M (cid:12) ; it grows to 6 . × h − M (cid:12) at z = 0 . . × h − M (cid:12) . In Figure 6, we compare the distribution of axis ratios forgroups and subgroups at redshift z = 0 . c (cid:13) , 000–000 Tenneti et al.
Figure 3.
Normalized histograms of axis ratios at z = 0 .
06 showing a comparison between shapes determined by using all particles inthe subhalo with those obtained using a random subsample of 50, 300, 500 and 1000 particles in the subhalo.
Left: q ( b/a ); Right: s ( c/a ). Figure 4.
Distribution of the number of dark matter and starparticles in subgroups at z = 0 .
06, where the colorbar indicatesthe number density of subhalos. the shapes of subgroups are more round when compared togroups in any given mass bin, in agreement with the findingsof Kuhlen et al. (2007) using dark matter-only simulation.Even in hydrodynamic simulations, Kazantzidis et al. (2006)found that dark matter subhalos are more round than ha-los. We can also see that as we go to higher mass bins, theaxis ratios of dark matter halos and subhalos decrease inagreement with the findings of Knebe et al. (2008).To investigate the mass dependence of axis-ratio distri- butions for the stellar matter component of subgroups, weplot the axis ratios ( q, s ) at redshifts z = 0 . M , M M z = 1 .
0, and 0 .
06 for the middle mass bin,10 . − . h − M (cid:12) . The lines show that at lower redshifts,the shapes tend to become rounder. Hopkins et al. (2005),Allgood et al. (2006) and Schneider et al. (2012) used N -body simulations and considered the axis ratio distributionsas a function of mass and redshift. Their results show thatat a given mass, halos are more round at lower redshift, andmore massive halos are more flattened which is consistentwith our findings. In Fig. 8, we show the average axis ra-tios, (cid:104) q (cid:105) and (cid:104) s (cid:105) as a function of mass at different redshifts z = 1 . , .
3, and 0 .
06 for the dark matter and stellar compo-nent. We also provide fitting functions for the average axisratios of the dark matter and stellar component of subha-los as a function of mass and redshift in Appendix A. Theplots for average axis ratios of the dark matter componentcan be compared against Allgood et al. (2006). Our resultsagree with theirs qualitatvely in that the average axis ratios, (cid:104) q (cid:105) and (cid:104) s (cid:105) , increase as we go to lower redshifts and lowermasses for the dark matter component. Their curves showa lower average (cid:104) s (cid:105) which may be because of the differentcriteria used in the determination of halo shapes, changes indark matter shapes from the effect of baryons, and differentcosmological parameters. Also, they measured average axisratios for halos, while our results are for subhalos. For thestellar matter, we can see that in general, the average axisratios decrease with subhalo mass. However, there is an in-crease in the intermediate mass range around ∼ h − M (cid:12) c (cid:13) , 000–000 hapes Alignments Figure 5.
3d axis ratio distributions of dark matter and stellar matter in subhalos at z = 0 .
06, for masses of subhaloes in the range10 . − . h − M (cid:12) . followed by a decreasing trend once again. We will investi-gate the dependence of this trend on the type and color ofgalaxies in a future study to understand the significance ofthis mass scale.To compare the axis ratio distributions of projectedshapes defined by stellar matter of subhalos with resultsfrom observational measurements, we use the statistic, rmsellipticity. The rms ellipticity per single component, e rms , is given by e = (cid:80) i ( − q (cid:48) i q (cid:48) i ) N , (3)where q (cid:48) i = b (cid:48) i a (cid:48) i for the i th subgroup and N is the total num-ber of subgroups considered. In Fig 9, we show the projectedrms ellipticity e rms as a function of cumulative mass of sub-halos (by considering all subhalos of mass greater a givenmass) for redshifts z = 1 . , .
3, and 0 .
06. Our results canbe compared against those from observations in the Sloan c (cid:13) , 000–000 Tenneti et al.
Figure 6.
Comparison of axis ratios, q ( b/a ) (left panel) and s ( c/a ) (right panel) between dark matter subgroups and groups at z = 0 . Figure 7.
Axis ratios q ( b/a ) (left panel) and s ( c/a ) (right panel) for stellar matter of subhalos at z = 0 . M , M M
3) and at z = 1 . , .
06 for the central mass bin, M Digital Sky Survey (SDSS) given in Reyes et al. (2012).For stellar matter, we obtained e rms = 0 .
28 at z = 0 . M subhalo > h − M (cid:12) , which is smaller than the observedvalue of 0 .
36, but reasonably close (and larger than thatexpected for dark matter component). The catalogue usedby Reyes et al. (2012) has been corrected for measurementnoise, but it has some selection effects that bias it slightlyin the direction of eliminating small round galaxies, thusboosting the RMS ellipticity in the sample of galaxies se-lected in the data compared to a fair sample. In addition toSDSS, we also made a comparison with observational results obtained using data from COSMOS survey. An e rms = 0 . ∼ .
67. These galaxiescorrespond to a mass of ∼ h − M (cid:12) at the median red-shift. We made further comparison with measurements onrms ellipticity presented in Joachimi et al. (2013). For a closecomparison, we used the results presented for late-type diskdominated galaxies at z = 1 with the assumption that thesample of galaxies in the simulation is dominated by disks atthis redshift. The observational measurements give an rmsellipticity per component of ∼ .
39 at z = 1 which is higher c (cid:13) , 000–000 hapes Alignments Figure 8.
Average axis ratios, (cid:104) q (cid:105) (left panel) and (cid:104) s (cid:105) (right panel) for dark matter and stellar component of subhalos as a function ofmass, at redshifts z = 1 . , .
3, and 0 . Figure 9.
RMS ellipticity per component for projected shapes, e rms , for dark matter and stellar matter at z = 1 . , .
3, and 0 . than our values which are in the range of 0 . − .
35. Thelower rms ellipticity may be due to a lower fraction of disk-dominated galaxies in the simulations, or due to the disksnot being perfectly realistic. Another comparison is madewith a sample of elliptical red galaxies ( S LRG ) given inJoachimi et al. (2013) where the rms ellipticity per singlecomponent is measured to be ∼ .
31 at z = 1 .
0, in agree-ment with our result, but no significant redshift evolutionof e rms is detected for this sample. However, the fraction of galaxies in our simulated sample that are red is likely tobe a function of redshift. Also, in some of the observations(Reyes et al. 2012; Joachimi et al. 2013), the shape esti-mator is weighted towards the inner part of the luminositydistribution in a galaxy, while our shape measurements areobtained by considering all the particles of a given type inthe subhalo, emphasizing the shape of stellar matter at largeradii (similar to the shape estimates in Mandelbaum et al.2013 from fitting light profiles to galaxy models). Given theknown differences between how the measurements in dataand simulations were carried out, it is difficult to make aquantitative comparison, however, there are no red flags fora major discrepancy. In this section, we compare the major axis orientations of thestellar components and dark matter components of subhalos,in 3D and 2D, in order to quantify the degree of misalign-ment between them. We investigate the dependence of theprobability distribution of the misalignments on the massrange of subhalos and redshift. We also discuss the changein misalignments in going from 3D, as defined by the physics,to 2D, which is what we observe for real galaxies. Finally,the misalignments are compared for centrals and satellitesubgroups.
For each subgroup, we determined the relative orientationof the major axis of its dark matter subhalo with its stel-lar component. If ˆ e ga and ˆ e da are the major axes of thestellar and dark matter components, respectively, then the c (cid:13) , 000–000 Tenneti et al.
Table 1.
Mean 3D misalignments in subgroups at redshifts z =1 . , .
3, and 0 .
06 in the mass bins M , M M h − M (cid:12) ) Mean 3D misalignment angle z = 1 . z = 0 . z = 0 . M . − . . ◦ . ◦ . ◦ M . − . . ◦ . ◦ . ◦ M > . . ◦ . ◦ . ◦ Table 2.
Mean 2D misalignments in subgroups at redshifts z =1 . , .
3, and 0 .
06 in mass bins M , M M h − M (cid:12) ) Mean 2D misalignment angle z = 1 . z = 0 . z = 0 . M . − . . ◦ . ◦ . ◦ M . − . . ◦ . ◦ . ◦ M > . . ◦ . ◦ . ◦ misalignment angle is given by θ m = arccos( | ˆ e da · ˆ e ga | ) (4)The same definition can be used to determine the mis-alignment angle in 2D. It is to be noted here that the majoraxis is not well defined for ellipsoids which are nearly spher-ical. However, we verified that our results for misalignmentangles do not change significantly when we exclude subhaloswith q and s > .
95 for shapes defined by the dark matteror stellar matter.
In Fig. 10, we show the misalignment probability distribu-tions for subgroups at redshifts z = 1 . , .
3, and 0 .
06 inmass bins M , M M
3. From the plots, we see thatin the massive bins, the stellar component is more stronglyaligned with its dark matter subhalos. The mean 3D mis-alignments for each mass bin are listed in Table 1. As wego from lower to higher mass bins, the mean misalignmentsdecrease from 34 . ◦ to 13 . ◦ . For a given mass bin, themisalignment strength increases towards lower redshifts, asshown in the plot and table; however, the trend with mass isfar stronger than the trend with redshift. When comparing3D and 2D misalignments, we find that the misalignmentsare more prominent in the 3D situation. This is mainly dueto a decrease in misalignment angle by projecting along aparticular direction. Also, if we consider random distributionof misalignment angles, it can be inferred geometrically thatthe probability increases with angle of misalignment in 3D,while the distribution is uniform in 2D. In Appendix B, wegive fitting functions for the probability distributions of 3Dand 2D misalignment angles in different mass bins at red-shifts z = 1 . , . , .
06. These probability distributions ofmisalignment angles are useful in predicting intrinsic align-ment signals and estimating the C parameter (overall align-ment strength) in the linear alignment model (Blazek et al.2011). Table 2 shows the mean misalignments in 2D. Thefitting functions for mean misalignment angles as a functionof mass are given in Appendix C. The misalignment distri-bution for masses M subhalo > h − M (cid:12) shows that the stellar shapes are well aligned with their host dark mattersubhalos with a mean misalignment angle of 10 . ◦ at z = 1and 13 . ◦ at z = 0 .
06. In a similar mass range, using N -body simulations, Okumura et al. (2009) assumed a gaussiandistribution of misalignment angle with zero mean and con-strained the width, σ θ , to be around 35 ◦ so as to match theobserved ellipticity correlation functions for central LRGs.This corresponds to an absolute mean misalignment angleof ∼ ◦ . The galaxies used by Okumura et al. (2009) havemasses corresponding to our highest mass bin, for which wepredict a stronger alignment between dark matter halo andgalaxy; however, because of the different methodology usedto indirectly derive their misalignment angle compared toour direct prediction from simulations, a detailed compari-son is difficult. Here we consider the axis ratio distributions and misalign-ment probability distributions for central and satellite sub-groups in different mass bins, divided in two ways: based onthe parent halo mass and based on the individual subhalomass.In the top panel of Fig. 11, we show normalized his-tograms of q and s for centrals and satellites binned accord-ing to their parent halo mass, for the bins, M , M M s , the parent halomass. These trends go in the opposite direction: satellitestend to have a lower value of s when their parent halo massis low, or when their subhalo mass is high. If we comparethe top and bottom right figures, the minor-to-major axis ra-tio distributions for centrals exhibit little mass dependencewhen binning by subhalo mass, but more mass dependencewhen binning by parent halo mass, suggesting an interestingenvironment dependence.In Fig. 12, we show the distributions of the misalign-ment angles for central and satellite subgroups in differentmass bins at redshifts z = 1 . , .
3, and 0 .
06. In the rightpanel, the binning is based on halo mass, while in the leftpanel, the binning is according to subhalo mass. We can seethat both centrals and satellites exhibit the same qualita-tive features in the distributions of misalignment angles asthe whole sample of subgoups in Fig. 10. Tables 3 and 4show the mean misalignment angles of centrals and satel-lites binned binned according to their subhalo and parenthalo masses, respectively. Considering mass bins based onindividual masses of subhalos, we see that in general, thedegree of alignment is larger for satellites than for centralsfor all mass bins. However, if we bin based on the mass of theparent halo, then at higher halo masses, central subgroupstend to have larger alignments than the satellite subgroups.This effect may be due to the centrals having higher massesthan the satellites, which tends to correlate with having ahigher degree of alignment. c (cid:13) , 000–000 hapes Alignments Figure 10.
Histogram of 3D (left panel) and 2D (right panel) misalignments at redshifts z = 1 . , .
3, and 0 .
06 in the mass bins M , M M Table 3.
Mean 3D misalignments in central and satellite subgroups at redshifts z = 1 . , .
3, and 0 .
06 in subhalo mass bins M , M M z = 1 . z = 0 . z = 0 . h − M (cid:12) ) Centrals Satellites Centrals Satellites Centrals Satellites M . − . . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ M . − . . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ M > . . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ In this study, we used the MBII cosmological hydrodynamicsimulation to study halo and galaxy shapes and alignments,which are relevant for determining the intrinsic alignmentsof galaxies, an important contaminant for weak lensing mea-surements with upcoming large sky surveys. While N -bodysimulations have been used in the past to study intrinsicalignments, it is also important to study the effects due toinclusion of the physics of galaxy formation; this includes ef-fects both on the overall shapes (ellipticities) of the galaxiesand halos, but also on any misalignment between them. Inorder to study this particular issue, we measured the shapesof dark matter and stellar component of groups and sub-groups.Previous studies have used N -body simulations to studythe mass dependence and redshift evolution of the shapesof dark matter halos and subhalos (Lee et al. 2005; Allgoodet al. 2006; Kuhlen et al. 2007; Wang et al. 2008; Knebe et al.2008; Schneider et al. 2012). Our results are qualitativelyconsistent with several trends identified in previous work.The first such trend that we confirm using SPH simulationsis that subhalos are more round than halos (Kazantzidiset al. 2006; Kuhlen et al. 2007). The second trend that weconfirm is that the shapes of less massive subhalos are moreround than more massive subhalos (Knebe et al. 2008; Wang et al. 2008) and as we go to lower redshifts, the subhalos alsotend to become rounder (Hopkins et al. 2005; Allgood et al.2006; Schneider et al. 2012).The effect of including baryonic physics on the shapesof dark matter halos was studied previously using hydro-dynamic simulations in a box of smaller size and resolutioncompared to ours (Kazantzidis et al. 2006; Knebe et al. 2010;Bryan et al. 2013). Kazantzidis et al. (2006) found that theaxis ratios of dark matter halos increase due to the inclusionof gas cooling, star formation, metal enrichment, thermalsupernovae feedback and UV heating. Bryan et al. (2013)found that there is no major effect on shapes under strongfeedback, but they observed a significant change in the in-ner halo shape distributions. Knebe et al. (2010) found thatthere is no effect on the shapes of dark matter subhaloes,where they included gas dynamics, cooling, star formationand supernovae feedback. Here, we took advantage of the ex-tremely high resolution of MBII to directly study the massdependence and redshift evolution of the shapes of the stellarcomponent of subhalos in addition to dark matter. However,we did not study the effect of baryonic physics on dark mat-ter shapes by comparison with a reference dark matter onlysimulation in this work.We found that the shapes of the dark matter compo-nent of subhalos are more round than the stellar component.Similar to dark matter subhalo shapes, the shapes of the stel- c (cid:13) , 000–000 Tenneti et al.
Figure 11.
Axis ratio distributions of stellar matter in subgroups for centrals and satellites in mass bins M , M M Top panel:
Results when dividing based on the parent halo mass; bottom panel: when dividing based on the subhalo mass. In both rows, the leftand right panels show results for q and s , respectively. Table 4.
Mean 3D misalignments in central and satellite subgroups at redshifts z = 1 . , .
3, and 0 .
06 in parent halo mass bins M , M M z = 1 . z = 0 . z = 0 . h − M (cid:12) ) Centrals Satellites Centrals Satellites Centrals Satellites M . − . . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ M . − . . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ M > . . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ c (cid:13)000
06 in parent halo mass bins M , M M z = 1 . z = 0 . z = 0 . h − M (cid:12) ) Centrals Satellites Centrals Satellites Centrals Satellites M . − . . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ M . − . . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ M > . . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ c (cid:13)000 , 000–000 hapes Alignments Figure 12.
Histograms of misalignment angles for central and satellite subgroups in mass bins M , M M Left:
Results whendividing based on the subhalo mass; right: when dividing based on the halo mass. lar component also become more round as we go to lowermasses of subhalos and lower redshifts. We are also ableto calculate the projected rms ellipticity per single compo-nent for stellar matter of subhalos, which can be directlycompared with observational results in Reyes et al. (2012).While the observed result is 0.36 at the given mass range,from our simulation, we measured a value of 0.28 at z = 0 . M > h − M (cid:12) , which is close, particularly given theuncertainties that result from observational selection effectsthat are not present in the simulations and that drive theRMS ellipticity to larger values, and given the different ra-dial weighting in the two measurements.By modelling subhalos as ellipsoids in 3D, we are ableto calculate the misalignment angle between the orienta-tion of dark matter and stellar component. Previous studiesof misalignments in simulations used either low-resolutionhydrodynamic simulations, or N -body simulations with ascheme to populate halos with galaxies and assign a stochas-tic misalignment angle and other assumptions (Sharma &Steinmetz 2005; Heymans et al. 2006; Faltenbacher et al.2009; Okumura et al. 2009; Hahn et al. 2010; Deason et al.2011). By direct calculation from our high-resolution simu-lation data, we found that in massive subhalos, the stellarcomponent is more aligned with that of dark matter, quali-tatively similar to results that have been inferred previouslythrough other means. For instance, at z = 0 .
06, the meanmisalignment angles in mass bins from 10 . − . h − M (cid:12) ,10 . − . h − M (cid:12) , and 10 . − . h − M (cid:12) are 34 . ◦ ,27 . ◦ , 13 . ◦ , respectively. The amplitude of misalignmentincreases as we go to lower redshifts. The total mean mis-alignment angle of 30 . ◦ , 30 . ◦ , 32 . ◦ at z = 1 .
0, 0 .
3, 0 . ◦ for their sample of red andblue centrals. Okumura et al. (2009) used N -body simula-tions and an HOD model for assigning galaxies to halos. Thealignment of central LRG’s with host halos is assumed to fol-low a Gaussian distribution with zero mean. Okumura et al.arrived at a standard deviation of 35 ◦ to match the observedellipticity correlation. Our predictions of misalignments forcentral and satellite subgroups are direct measurements thatcould be done through hydrodynamic simulations which in-clude the physics of star formation.In conclusion, we found that the axis ratios of the shapesof stellar component of subhalos are smaller when comparedto that of dark matter. The shapes of both dark matter andstellar component tend to become more round at low massesand low redshifts. We measured the misalignment betweenthe shapes of dark matter and stellar component and foundthat the misalignment angles are larger at lower masses andincrease slightly towards lower redshifts. We found that the c (cid:13) , 000–000 Tenneti et al. dependence is more on the mass of subhalo than redshift.Finally, we split our subhalos sample into centrals and satel-lites and found that in similar mass range, the satellites havesmaller misalignment angles.We initiated this study with the goal of predicting in-trinsic alignments and constraining their impact on weakgravitational lensing measurements. In this paper, we pre-sented our results on the axis ratios and orientations of boththe dark matter and stellar matter of subhalos. Future workwill include the dependence of these results on the radialweighting function used to measure the inertia tensor (as inSchneider et al. 2012), galaxy type and the difference be-tween the shape of the stellar mass versus of the luminositydistribution. We will also present our results on the intrinsicalignment two-point correlation functions in a future paper.Finally, future work should include investigation of the im-pact of changes in the prescription for including baryonicphysics in the simulations.
ACKNOWLEDGMENTS
RM’s work on this project is supported in part by the Al-fred P. Sloan Foundation. We thank Alina Kiessling, MichaelSchneider, and Jonathan Blazek for useful discussions of thiswork. The simulations were run on the Cray XT5 supercom-puter Kraken at the National Institute for ComputationalSciences. This research has been funded by the National Sci-ence Foundation (NSF) PetaApps programme, OCI-0749212and by NSF AST-1009781.
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APPENDIX A: FUNCTIONAL FORMS FORDARK MATTER AND STELLAR MATTERSHAPES
Here, we give the functional forms for the average axis ratios( q , s ) of shapes defined by dark matter and stellar matter insubhalos as a function of mass and redshift. The parametersare given in Table A1. The plots showing fits for mean axis ratios of the shapes of dark matter and stellar matter aregiven in Figs. A1 and A2 respectively.The fitting functions for average axis ratios are givenby, (cid:104) q, s (cid:105) = (1 + z ) γ (cid:88) i a i (cid:20) log( MM piv ) (cid:21) i (A1)where, M piv is 10 h − M (cid:12) .The fitting functions are linear in log( MM piv ) for shapesof dark matter with i = 0 , th degreein log( MM piv ) with i = 0 , , , , , , APPENDIX B: FUNCTIONAL FORMS FORPROBABILITY DISTRIBUTIONS OF 3D AND2D MISALIGNMENT ANGLES
The probability distributions for 3D misalignment angles inthe two lower mass bins 10 . − . h − M (cid:12) and 10 . − . h − M (cid:12) are given by dpdθ = A z (1 − e − γ z θ ) e − B z θ + (1 − e − α z θ ) C z (B1)In the highest mass bin, 10 . − . h − M (cid:12) the fittingfunction is, dpdθ = A z e − B z θ (B2)The probability distributions for 2D misalignment an-gles in different mass bins are given by dpdθ = A z e − B z θ + C z (B3)The fits for the probability distributions in 3D and 2Dare shown in Fig. B1 and Fig. B2 respectively and the pa-rameters are given in Tables B1 and B2. APPENDIX C: FUNCTIONAL FORMS FORMEAN MISALIGNMENT ANGLES IN 3D AND2D
The mean misalignment angles in 3D and 2D are given by, θ ( M ) = ( a z − a z e − ( log( M ) − d zb z ) )( c z log( M ) + c z ) (C1)The plots showing fits for mean misalignments in 3Dand 2D are shown in Fig. C1 and Fig. C2 respectively. Thecorresponding parameters are given in Tables C1 and C2. c (cid:13) , 000–000 Tenneti et al.
Figure A1.
Fits for the axis ratios of shape defined by dark matter in subhalos as a function of mass and redshift
Figure A2.
Fits for the axis ratios of shape defined by stellar matter in subhalos as a function of mass and redshift
Table A1.
Parameters, γ and a i for mean axis ratios, (cid:104) q (cid:105) and (cid:104) s (cid:105) in mass range, 10 . − . h − M (cid:12) Axis ratio γ a a a a a a a q (Dark Matter) -0.12 0.797 -0.049 - - - - - s (Dark Matter) -0.19 0.663 -0.059 - - - - - q (Stellar Matter) -0.14 0.771 -0.004 -0.068 -0.017 -0.061 -0.003 -0.015 s (Stellar Matter) -0.19 -0.585 0.031 -0.089 -0.034 0.075 -0.001 -0.016c (cid:13)000
Parameters, γ and a i for mean axis ratios, (cid:104) q (cid:105) and (cid:104) s (cid:105) in mass range, 10 . − . h − M (cid:12) Axis ratio γ a a a a a a a q (Dark Matter) -0.12 0.797 -0.049 - - - - - s (Dark Matter) -0.19 0.663 -0.059 - - - - - q (Stellar Matter) -0.14 0.771 -0.004 -0.068 -0.017 -0.061 -0.003 -0.015 s (Stellar Matter) -0.19 -0.585 0.031 -0.089 -0.034 0.075 -0.001 -0.016c (cid:13)000 , 000–000 hapes Alignments Table B1.
Parameters for probability distributions of 3D misalignment angles at redshifts z = 1 . , .
3, and 0 .
06 for subhalos in the massbins M . − . h − M (cid:12) , M . − . h − M (cid:12) and M > . h − M (cid:12) . z = 1 . z = 0 . z = 0 . A z B z C z γ z α z A z B z C z γ z α z A z B z C z γ z α z M .
211 0 .
079 0 .
004 0 .
023 100 0 .
146 0 .
071 0 .
005 0 .
028 100 0 .
055 0 .
052 0 .
004 0 .
071 100 M .
122 0 .
088 0 .
002 0 .
134 100 0 .
091 0 .
074 0 .
003 0 .
121 100 0 .
058 0 .
057 0 .
003 0 .
166 100 M .
115 0 .
119 0 . − − .
073 0 . − − − .
064 0 . − − − Table B2.
Parameters for probability distributions of 2D misalignment angles at redshifts z = 1 . , .
3, and 0 .
06 for subhalos in the massbins M . − . h − M (cid:12) , M . − . h − M (cid:12) and M > . h − M (cid:12) . z = 1 . z = 0 . z = 0 . A z B z C z A z B z C z A z B z C z M .
044 0 .
060 0 .
003 0 .
042 0 .
060 0 .
003 0 .
041 0 .
056 0 . M .
077 0 .
089 0 .
001 0 .
064 0 .
075 0 .
002 0 .
056 0 .
069 0 . M . .
211 0 . .
146 0 .
162 0 . .
133 0 .
137 0 . Figure B1.
Fits for probability distributions of 3D misalignmentangles at z = 0 . Table C1.
Parameters for mean misalignment angles in 3D atredshifts z = 1 . , . .
06 for subhalos in the mass range,10 . − . h − M (cid:12) . z a z a z b z c z c z d z . . − .
35 0 .
79 1 . − .
70 10 . . . − .
72 0 .
97 1 . − .
46 10 . .
06 1 . − .
58 0 .
96 1 . − .
74 10 . Figure B2.
Fits for probability distributions of 2D misalignmentangles at z = 0 . Table C2.
Parameters for mean misalignment angles in 2D atredshifts z = 1 . , . .
06 for subhalos in the mass range,10 . − . h − M (cid:12) . z a z a z b z c z c z d z . . − .
84 0 .
82 0 . − .
77 9 . . . − .
50 0 .
89 0 . − .
75 9 . .
06 2 . − .
40 0 .
91 0 . − .
99 10 . (cid:13) , 000–000 Tenneti et al.
Figure C1.
Fits for mean misalignment angles in 3D as a func-tion of mass
Figure C2.
Fits for mean misalignment angles in 2D as a func-tion of mass c (cid:13)000