We present the Dark-ages Reionization and Galaxy-formation Observables from Numerical Simulations (DRAGONS) program and Tiamat, the collisionless N-body simulation program upon which DRAGONS is built. The primary trait distinguishing Tiamat from other large simulation programs is its density of outputs at high redshift (100 from z=35 to z=5; roughly one every 10 Myrs) enabling the construction of very accurate merger trees at an epoch when galaxy formation is rapid and mergers extremely frequent. We find that the friends-of-friends halo mass function agrees well with the prediction of [Watson:2013p2555] at high masses, but deviates at low masses, perhaps due to our use of a different halo finder or perhaps indicating a break from universal behaviour. We then analyse the dynamical evolution of galaxies during the Epoch of Reionization finding that only a small fraction (~20%) of galactic halos are relaxed.


We illustrate this using standard relaxation metrics to establish two dynamical recovery time-scales: i) halos need ~1.5 dynamical times following formation, and ii) ~2 dynamical times following a major (3:1) or minor (10:1) merger to be relaxed. This is remarkably consistent across a wide mass range. Lastly, we use a phase-space halo finder to illustrate that major mergers drive long-lived massive phase-space structures which take many dynamical times to dissipate. This can yield significant differences in the inferred mass build-up of galactic halos and we suggest that care must be taken to ensure a physically meaningful match between the galaxy-formation physics of semi-analytic models and the halo finders supplying their input.


Following cosmological recombination the baryonic gas filling the Universe became predominantly neutral. The fact that this gas is known to be mostly ionized today [Gunn:1965p2569] implies that the intergalactic medium (IGM) under went a significant reionization event at some early point in its history. This fact is responsible for some of the major questions in extragalactic astronomy including: when did this process occur and what were the responsible ionizing sources? Recent observations have begun to provide preliminary answers to this question [Fan:2006p2573, Ouchi:2010p2575, PlanckCollaboration:2015p2562]. Soon measurements of highly redshifted 21-cm radio emission [Furlanetto:2006p2567, Morales:2010p2568] will open an important new observational window for study of the first galaxies, providing the first direct probe of the neutral hydrogen content in the early Universe.


The development of theoretical models that self-consistently include the physics of galaxy formation and intergalactic hydrogen will play a key role in understanding the nature of the first galaxies and in interpreting these observations. This paper is the first in a series describing the Dark-ages Reionization and Galaxy-formation Observables from Numerical Simulations (DRAGONS) project which aims to integrate detailed semi-analytic models constructed specifically to study galaxy formation at high redshift, with semi-numerical models of the galaxy–IGM interaction [Zahn:2007p2570, Mesinger:2007p2571, Geil:2008p2572]. The galaxy-formation modelling for DRAGONS is implemented using a set of large N-body simulations which we refer to as the Tiamat simulation suite. Tiamat provides a framework within which to implement a semi-analytic model for reionization and to study the formation histories, structure and properties of the dark matter halos that dictate the formation sites and assembly histories of the first galaxies.


Over the past decade the requirements for simulations aiming to address the structure of reionization and galaxy formation in the Epoch of Reionization (EoR) have been studied extensively. The consensus from previous N-body studies [Iliev:2007p2581, Zahn:2007p2570, McQuinn:2007p2582, Shin:2008p2583, Lee:2008p2584, Croft:2008p2585] and analytic models [Furlanetto:2004p2578, Wyithe:2007p2579, Barkana:2009p2580] is that large-scale over-dense regions near bright sources ionize first with clustered neighbouring sources contributing to increase the size of ionised regions. Simulations on the scale of 100 Mpc are found to be large enough to correctly capture the structure and duration of reionization, although volumes up to 500^3 Mpc are required to capture all large scale power due to clustering of HII regions [Iliev:2014p2577]. The challenge is to model galaxy formation in volumes of this size with sufficient resolution.


In the cold neutral IGM prior to reionization, molecular cooling may proceed within minihalos with masses ~10^6 solar masses. However, the processes principally responsible for regulating galaxy formation are expected to be active in halos with virial temperatures greater than T_min ~ 10^4 K, above which atomic hydrogen cooling becomes efficient. On the other hand, the growth of HII regions during reionization is also expected to be influenced by radiative feedback due to suppression of galaxy formation below the cosmological Jeans mass within a heated IGM [Dijkstra:2004p2576]. Together these constraints indicate that sufficient resolution is required to identify halo masses down to ~5x10^7 solar masses within a volume of ~100 Mpc. In addition to this dynamic range of scales, for the DRAGONS program we aim to accurately resolve the relevant time-scales of high-redshift galaxy formation putting an additional constraint on the cadence with which simulation outputs must be generated.


At z~6 the dynamical time of a galactic disc falls below the lifetime of the least massive Type-II supernova progenitor (~4x10^7 yr). As a result, snapshots with a cadence of ~10^7 years are required to follow galaxy formation correctly during the EoR with a semi-analytic model. This interval is an order of magnitude shorter than needed to describe galaxy formation at redshifts z~0. In this paper we present the Tiamat suite of collisionless N-body simulations which we have run to satisfy these requirements and upon which the DRAGONS program will be constructed. Given its critical importance as the foundation of the program, we take this opportunity to present the methodology of constructing this set of simulations and to characterise the populations of galactic halos obtained. In particular, we shall carefully examine the dynamical evolution of galactic halos during the reionization era.


We seek this understanding because of its potential impact on the structure of galactic halos, which is of fundamental importance to the physics of galaxy formation at any epoch, including the EoR. In particular, halo concentrations and angular momenta are generally believed to dictate the size and surface density of the disc-like structures in which the majority of star formation occurs. Dynamical disturbances can additionally drive starbursts or affect the stability of these disc-like structures, strongly affecting star formation and forcing morphological transformations which contribute to the assembly of galactic spheroids. This can in turn affect observable galaxy sizes or alter UV fluxes and escape fractions with important effects on the reionization history of the Universe. At low redshift, it has been shown that a halo's dynamical state can systematically affect the structure and gravitational potential of galactic halos [Thomas:1998p2559, Neto:2007p2556, Power:2012p2560, Ludlow:2014p2558].


These studies have collectively established a set of criteria (which we refer to henceforth as standard relaxation criteria) capable (at low redshifts at least) of separating halos with disturbed structure from those with relaxed structure. These standard criteria consist of cuts on three metrics for each halo: the separation of its dense centre from its centre of mass (x_off), its pseudo-virial ratio constructed from its velocity dispersion and gravitational binding energy (T/|U|), and its substructure fraction (f_sub). When low-redshift halos are separated into relaxed and unrelaxed samples in this way, substantial effects on halo concentration and (to a lesser extent) spin have been demonstrated. This is of particular importance to studies which aim to understand the processes which establish the universal density profiles of halos extracted from collisionless N-body simulations [Navarro:1997p2543]. While there is broad agreement at low redshift as to the dependance of halo structure on mass, redshift and dynamical state, recent studies which have attempted to push this understanding to the EoR [Prada:2012p2538, Diemer:2015p2657, Dutton:2014p2658, Hellwing:2015p2659] have found less consensus.


At high redshifts where simulations predict that merger rates are very high, halos significantly less concentrated, and merger orbital properties quite different, the influence of dynamical state on halo structure may differ from local trends. It is unclear to what degree dynamical disturbance may play a role in the differences in high-redshift halo structure reported in the literature since these studies have not been consistent in their treatment of this issue. Unfortunately, the details of how the standard relaxation metrics evolve following dynamical disturbances has not been properly explored at any redshift, nor has their efficacy at separating relaxed systems from unrelaxed systems been demonstrated at high redshift. Before presenting a detailed analysis of halo structure at high redshift to understand the discrepancies present in the literature, we aim first to address both of these issues here. Given its fine snapshot temporal resolution, Tiamat represents a unique resource for exploring these issues across the full range of masses most relevant to galaxy formation in the early Universe.


We will find that the standard relaxation criteria are effective at identifying systems that are recovering from their formation or from recent significant mergers. With this methodology properly validated at high redshift, we will subsequently perform a detailed analysis of the structure of both relaxed and unrelaxed high redshift dark matter halos -- including spin parameter and concentration-mass relations -- in a companion paper (Angel et al. 2015; PAPER-II). The Tiamat N-body simulation hosts a semi-analytic model of galaxy formation named Meraxes, which has been integrated within a semi-numerical model for ionization structure. In subsequent papers we will present this model (Mutch et al. 2015; Paper-III) and use it in a range of studies including high redshift galaxy luminosity functions (Liu et al. 2015, PAPER-IV) and the ionization structure of the intergalactic medium (Geil et al 2015, PAPER-V).


Complementary high resolution hydrodynamics simulations called Smaug [already presented in][]{Duffy:2014p2561} will characterise the basic scaling relationships of early galaxy formation. There will then be a detailed comparison of Meraxes to the results of Smaug with suggested constraints of the semi-analytic model based on hydrodynamics (Qin et al, in prep). This section examines the time-scales by which galactic halos at high redshift relax following formation and mergers. We relate these time-scales to their dynamical age and to intervals between merger events and will find that only towards the end of the epoch of reionization do significant numbers of halos exist in relaxed states. Lastly, we will also present some preliminary results about their phase-space structure that may be of consequence for the application of semi-analytic models at high redshift. It has long since been shown that the structure of halos extracted from collisionless N-body simulations has a significant dependance on the dynamical state of the system [Thomas:1998p2559, Neto:2007p2556, Power:2012p2560, Ludlow:2014p2558].


At low redshifts at least, cuts on three metrics for quantifying the dynamical state of halos have found success at separating systems with disturbed structure from those with relaxed structure: the offset parameter (x_off), given by the displacement of the densest centre of a halo from its centre of mass; the virial ratio (T/|U|), given by 2K/|U|, where K is the kinetic energy and U is the halo's gravitational binding energy [see Section 5.1 of][and references therein, for a detailed description of virialisation]Poole:2006p41; and the substructure fraction (f_sub), which we take here to be the ratio of the particle count of all but the most massive of a FoF halo's substructures to its total particle count. Each have simple physical interpretations as measures of dynamical state. Elevated values of f_sub naturally arise during the earliest stages of a merger when a halo is naturally split between multiple similarly sized substructures.


Elevated values of the virial ratio are found prior to the dissipation of orbital energy following a merger. Lastly, elevated values of x_off are a natural result of the movement of a halo's dense core as it orbits the centre of mass of its system following even minor disturbances. The standard values for relaxed systems which we adopt are those proposed originally by [Neto:2007p2556] and recently confirmed to be successful in the study of halo density profiles [Ludlow:2014p2558]; specifically, x_off < 0.07, T/|U| < 1.35 and f_sub < 0.1. To date, a careful examination of how these metrics evolve following dynamical disturbances has not been performed however, leaving the physical nature of these cuts unclear. Additionally, it is unclear how appropriate they are for high-redshift studies. In what follows we shall study the evolution of these relaxation metrics following three sorts of mass accretion event capable of driving dynamical disturbances: halo formation (defined as the point at which a halo last reached 50% of its present mass), mergers between a primary halo and a secondary halo at least one third its mass (so-called `major', or 3:1 mergers) or mergers between a primary halo and a secondary halo at least one tenth its mass (so-called `minor', or 10:1 mergers).


Throughout our analysis we will measure intervals of time for a halo at redshift z in units of its dynamical time, which we take to be 10% of the Hubble time at that redshift. Times in this dimensionless system of units will be denoted by tau. At all redshifts, the Hubble time is tau=10 in this system. The three times since a halo last experienced each of these events will be referred to as dynamical ages. The dynamical ages required for our relaxation criteria (x_off, T/|U| and f_sub) to return to and maintain standard values for relaxed halos following these events are used to motivate two recovery times: a formation recovery time and a merger recovery time. We find that x_off starts with high values of ~0.2 at the time of halo formation, declining to our relaxed level of 0.07 at a formation age of ~1.5 and then to baseline levels of x_off~0.04 afterwards.


Following 3:1 and 10:1 mergers, peak levels occur roughly one dynamical time after a merger begins with relaxed levels obtained at a merger age of ~2. Peak values of 0.2 and roughly 0.07 are reached following 3:1 and 10:1 mergers, suggesting that mergers are progressively less likely to excite the system above our x_off~0.07 relaxation criterion as mass ratios drop below 10%. Interestingly, the virial ratio shows significantly less evolution following both formation and merger events. In all cases, the distribution peak sits at levels similar to our virial ratio ~1.35 relaxation criteria at times when x_off lies above its relaxation criteria. Once x_off is found to drop below relaxation levels (or shortly before) the virial ratio can be seen to decline somewhat from values of ~1.35 to ~1. Generally however, the virial ratio exhibits much less sensitivity to dynamical disturbances and relaxes to baseline levels quicker than x_off, suggesting that it is a much less robust discriminator of dynamical state.


Lastly, the substructure fraction shows a very simple and well defined behaviour following dynamical events. At formation, a wide range of values are seen about a distribution peak of ~0.2. A slow decline to baseline values follows. After merger events, the substructure fraction increases by expected amounts: 30% for 3:1 mergers and 10% for 10:1 mergers. The subsequent decline in the substructure fraction is more rapid than what is seen following formation, with levels dropping at a rate of approximately 20% per dynamical time. We also find that substructure fractions return to standard relaxed values 30 to 50% faster than core offsets following dynamical disturbances. Despite this, because the substructure fraction is most sensitive to dynamical disturbance in the earliest stages of mergers, it is an effective compliment to the x_off statistic which exhibits a slight delay in reacting during merger events.


We conclude then that, of the three metrics we study here, the x_off statistic is the most effective single measure of dynamical state. It is sensitive to disturbances from mergers greater than approximately 10:1 and retains this sensitivity for approximately 2 dynamical times afterwards. The virial ratio is significantly less discriminating than these statistics but evolves in ways consistent with the relaxation of x_off and f_sub. With the exception of f_sub, all metrics are essentially independent of mass throughout the period of relaxation following halo formation or mergers larger than 10:1. The differing trends of f_sub with mass for each simulation is a numerical effect arising from their differing resolutions as a function of mass and is an expected result. Despite this one numerical effect, this clearly illustrates that high-redshift halos recover from formation and merger events within a time which is highly insensitive to their mass.


These results suggest that following formation or mergers greater than 10:1, a small and fixed number of pericentric passages of the material disturbed at large radius in the merger remnant are required for relaxation. If this is the case, the mass independence of relaxation could be seen as a product of the fact that halo crossing times depend only on their mean density, which is defined in terms of a fixed overdensity, and independent of mass. Secondary factors which could influence halo relaxation include halo concentrations, shapes and merger orbital properties. From these results, we define two mass-independent recovery times separating relaxed and unrelaxed systems at high redshift: a formation age of 1.5 and a merger age of 2. Our expectation is that high-redshift halos which have doubled their mass within one and a half dynamical times or which have experienced mergers larger than 10:1 within two dynamical times are likely to be disturbed.


How then do the fractions of halos meeting these recovery criteria evolve with redshift? The distribution of all three dynamical ages evolves very little from z=15 to z=5. Over this redshift range, the distribution of formation ages is very narrow and peaked very close to our formation recovery timescale of 1.5. The near constant value of the formation age during this epoch is consistent with early mass accretion histories which are exponential, as found previously by several other authors [Wechsler:2002p1783, McBride:2009p1230, Correa:2015p2654]. The distribution of times since 3:1 mergers is much broader and is also peaked near our merger recovery timescale of 2. This tells us that typical halos at high redshifts across all galactic masses are doubling their mass on timeframes that only barely permit relaxation while simultaneously, major mergers are occurring at rates which only barely permit recovery between events.


The situation is importantly different for minor mergers. In this case we find that halos experience minor mergers at rates which are much too rapid (on average) to permit dynamical relaxation between events. The narrow distribution of formation ages, broad distribution of merger ages and short times between 10:1 mergers persists almost unchanged until approximately z~2. At this time, we find that 10^12 solar mass halos begin to become progressively older, as typical formation ages and times since major mergers increase and times since minor mergers creep above merger recovery times of 2 by z=0. The increase of formation age at lower redshifts corresponds to a transition in these halos' mass accretion histories from an exponential form to a linear form, as discussed already in the literature [McBride:2009p1230, Correa:2015p2654].


The fractions of halos which meet these recovery criteria as a function of redshift is presented explicitly. Here we see the disappearance of halos with formation times less than 1.5 and the sustained low levels of halos having had sufficient time to recover from their most recent mergers. We have added to these plots the fraction of halos that simultaneously satisfy our standard x_off, T/|U|,and f_sub relaxation criteria. Remarkably, the fraction of relaxed halos and the fraction having had sufficient time to recover from their last 10:1 (or larger) merger are very similar across a wide range of masses and redshifts. We conclude from this that the standardised relaxation criteria of [Neto:2007p2556] are effectively identifying systems that have been disturbed by 10:1 (or larger) mergers. It should be noted however that our recovery criteria of 1.5 and 2 have been calibrated at high redshift and may need adjustment at low redshift, where halos are substantially more concentrated and the orbital properties of merging systems are significantly different.


This is likely the reason why our estimates of the recovered fraction exceeds the relaxed population at z<2. Taken together, we see that at high redshifts (z>5), the fraction of relaxed halos drops to levels of ~20% at all galactic masses. Combined with the rapid decline in the number density of halos with redshift at this time, we conclude that the abundance of relaxed galactic halos prior to the epoch of reionization drops to very low levels. This should make it very challenging to assemble large populations of relaxed halos at z>10, which is of particular concern for studies seeking to understand the processes acting to establish universal density profiles for collisionless systems at high redshift. As an exercise during the development phase of the Tiamat simulations, we analysed one of our simulations (TinyTiamatW) with the ROCKSTAR halo finder, allowing us to study the effect of halo finding on our semi-analytic modelling campaign.


Doing so has yielded an interesting new insight into the dynamical lives of high-redshift galactic halos. There is a stark difference between the results from ROCKSTAR and SUBFIND. We can see clearly that while the halo appears relatively undisturbed with unremarkable substructure, it in fact consists primarily of two very massive subhalos which are distinct in phase space. Phase-space halo finders such as ROCKSTAR are of course designed to separate halo substructures in this way, but it is not entirely clear that this is a desired result for applications in galaxy formation modelling. While approaches differ in detail, the central premise of all semi-analytic galaxy formation models is that the total matter assembly provided by their merger tree inputs can be reliably mapped to a faithful description of the baryonic assembly of galactic halos. Problems may arise if the collisional fluids (particularly the hot halos) associated with multiple collisionless systems oscillating through each other for >3 dynamical times can not follow the collisionless material of their initial hosts.


Substantial amounts of this gas will be stripped or rapidly coalesce into one hot halo, loosing its association with its original collisionless component while that material continues to orbit. This is the case with the Bullet Cluster for instance, albeit at a different mass scale and redshift. It is also the situation modelled by [McCarthy:2008p2661] who find that the stripping of a galaxy's hot halo (due to tides, ram pressure stripping, and hydrodynamic instabilities) is extremely efficient up to and during its first pericentric passage. The amount of material removed varies with halo mass, concentration and orbit, but is substantial and typically in the range of 60 to 80% for the broad range of cases they examine. If such structures were short lived, the impact on our galaxy formation model would likely be insignificant. However, they are in fact long lived in dynamical terms.


A comparison of the evolving substructure fractions of FoF halos extracted from TinyTiamatW using Subfind to those obtained from ROCKSTAR as functions of the dynamical ages shows that while we see the familiar decline of f_sub following formation and mergers in the Subfind trees, the ROCKSTAR trees exhibit a much slower decline, reaching constant levels only after approximately 5 dynamical times, sustaining levels well above our standard relaxation criteria even after that. On the other hand, if these substructures were rare, their impact on galaxy formation modelling would again be minimal. They are in fact very common. The FoF halo mass functions for the two catalogs are virtually identical (except at the highest masses where the larger linking length used by ROCKSTAR unsurprisingly yields more systems, presumably due to overlinking), the substructure fractions at the highest (and most resolved) masses of the two catalogs are very different.


Substructure fractions are 50 to 60% at the highest masses in ROCKSTAR indicating that only around half of the mass in these systems is assigned to the most massive component of the system. This is a consequence of a very different splitting of the top level of the FoF group's substructure hierarchy. Suggestions of this effect can be seen in the recent work of [Behroozi:2015p2660]. While these authors find that substructure properties like position and velocity generally agree between configuration and phase-space halo finders, they find that substantial differences in masses can occur. They also find strong disagreements in the frequency and duration of major mergers, particularly at redshifts z>1. We emphasise that we make no attempt here to advocate for one halo finding approach over another. Rather, we seek to make the point that care should be taken to ensure that each semi-analytic model is matched, in a physically meaningful way, to the nature of the substructure hierarchy supplied by the halo finder contributing to its input.


Such differences may lead to significant systematics with mass in the evolution of merger trees which could masquerade as physical processes as diverse as mass dependancies in dust properties, photon escape fractions, feedback and cooling. A detailed account of how the cooling and feedback modelling of DRAGONS compares to the Smaug hydrodynamic simulations of [Duffy:2014p2561] will be presented in Qin et al (2015, PAPER-VII), where a direct halo-by-halo comparison of the two methodologies will be presented. We take this opportunity to point out one other possible important astrophysical consequence of large bulk phase-space structures such as this. Recent studies have begun to investigate the possibility that heating from dark matter annihilation may be observable in the redshifted 21-cm background from z>30 [Furlanetto:2006p2566, Evoli:2014p2655, Mack:2014p2655, Schon:2014p2564]. If phase-space structures such as these prove to be common at this epoch, important changes to inferred annihilation cross sections may result.


We have introduced the Dark-ages Reionization and Galaxy-formation Observables from Numerical Simulations (DRAGONS) program and presented the Tiamat collisionless N-body simulation suite upon which it is constructed. The abundance of friends-of-friends (FoF) structures populating Tiamat is a good match to the universal model proposed by [Watson:2013p2555] at high masses, but we find a supression of low-mass systems, possibly due to differences in our halo finding procedure or perhaps indicating a deviation from universal behaviour, at least at large redshifts. Using Tiamat we have also illustrated the dynamically violent conditions experienced by galactic halos at large redshift. We find that across a wide range of galactic mass (10^8 to 10^11 solar masses) above z=5, halos relax from their formation and from mergers in essentially the same way and in the same amount of time: within one and a half dynamical times in the case of their formation and within two dynamical times following mergers involving a primary and a secondary larger than 10% of its mass.


The distribution of formation times and times since major mergers maintain approximately these time-scales across all redshifts above z=5 while the time between minor mergers is typically significantly less. Relaxed fractions maintain levels of less than 20% at z>5 as a result. Using the GiggleZHR simulation we find that this remains true for 10^12 solar mass halos until z~2. It appears that the rate of minor mergers principally regulate a halo population's relaxed fraction, as measured by standard metrics. Combined with the rapid decline of the halo mass function at redshifts z>10, the abundance of relaxed halos prior to the epoch of reionization must be extremely low. Using the phase-space halo finder ROCKSTAR, we also demonstrate that high-redshift halos host large and long-lived substructures that go undetected to halo finders such as Subfind which utilise configuration-space information only.


This results in substructure fractions that are much higher for ROCKSTAR than for Subfind, with probable implications for semi-analytic models of galaxy formation at high redshift. Taken together, these results illustrate the dynamically violent circumstances under which galaxy formation proceeds in the early Universe. The consequences are many and significant, including implications for photon escape fractions, efficiencies of feedback from winds (both stellar and AGN) and the efficiency of spheroid assembly. These in turn can have important consequences for the reionization history of the Universe during the EoR and observed galaxy sizes.


We use high resolution N-Body simulations to study the concentration and spin parameters of dark matter haloes in the mass range between 10 to the power of 8 and 10 to the power of 11 solar masses per h and redshifts between 5 and 10, corresponding to the haloes of galaxies thought to be responsible for reionization. We build a sub-sample of equilibrium haloes and contrast their properties to the full population that also includes unrelaxed systems. Concentrations are calculated by fitting both NFW and Einasto profiles to the spherically-averaged density profiles of individual haloes. After removing haloes that are out-of-equilibrium, we find a concentration-mass (c(M)) relation at redshifts greater than 5 that is almost flat and well described by a simple power-law for both NFW and Einasto fits. The intrinsic scatter around the mean relation is approximately a delta c_vir of 1 (or 20 per cent) at redshift z=5. We also find that the analytic model proposed by [2014MNRAS.441..378L] reproduces the mass and redshift-dependence of halo concentrations.


Our best-fit Einasto shape parameter, alpha, depends on peak height, nu, in a manner that is accurately described by the equation alpha equals 0.0070 times nu squared plus 0.1839. The distribution of the spin parameter, lambda, has a weak dependence on equilibrium state; lambda peaks at roughly 0.033 for our relaxed sample, and at approximately 0.04 for the full population. The spin-virial mass relation has a mild negative correlation at high redshift. In the current cosmological paradigm cold dark matter (CDM) collapses to form gravitationally bound structures within an expanding background universe. Known as dark matter (DM) haloes, these objects are initially small but undergo repeated merging to form ever larger systems. Galaxies form within these haloes as in-falling gas cools and converts to stars [1978MNRAS.183..341W]. Their evolution and structural properties therefore underpin those of the embedded galaxies. These ideas have evolved into the field of semi-analytic modelling in which galaxies are grown within an evolving population of dark-matter haloes extracted from purely N-Body simulations [2006RPPh...69.3101B, 2006MNRAS.365...11C, 2008MNRAS.391..481S, 2008MNRAS.388..587L].


The characteristics of DM haloes have been the subject of extensive research. Mass determines the overall size of the halo, but several other important parameters have also been identified. For example, using N-Body simulations [1997ApJ...490..493N] found that the density profiles of virialised haloes can be well described by rescaling a simple formula (hereafter known as the NFW profile), which states that the density rho at radius r over the critical density rho_c equals the characteristic density delta_c divided by (r/r_s) times (1+r/r_s) squared. Here r_s is the characteristic scale radius at which the logarithmic density slope is equal to -2; delta_c is the characteristic density contrast, and rho_c is the critical background density at redshift z. These parameters can be expressed in a variety of forms. One common approach is to use a virial mass and concentration, c_vir, defined as the ratio of the halo's virial radius to its scale radius, R_vir/r_s. The virial mass of a halo is defined as that enclosed by the radius R_vir within which the density is a specific factor times the background density [1998ApJ...495...80B].


While the NFW profile is a common description, several recent studies [2004MNRAS.349.1039N, 2008MNRAS.388....2H, 2010MNRAS.402...21N] have shown that the density profiles of simulated haloes exhibit small but systematic deviations from the NFW equation. The Einasto profile [1965TrAlm...5...87E], provides a better approximation to the radial density profile [2004MNRAS.349.1039N, 2013MNRAS.432.1103L]. Its equation is defined by the natural logarithm of rho(r) over rho_-2 equals -2/alpha times [(r/r_-2) to the power of alpha minus 1]. Like the NFW profile, the Einasto equation has two scaling parameters, r_-2 and rho_-2, and an additional shape parameter, alpha. Note that r_-2 and r_s are equivalent. At low redshift the concentration parameter decreases with increasing halo mass. NFW interpreted this finding as a result of hierarchical clustering: smaller haloes form earlier than more massive objects, when the universe was denser [1997ApJ...490..493N]. They suggested that concentration reflects the background density of the Universe at the halo's formation time. The same negative trend was also seen in subsequent N-Body simulations [2001ApJ...554..114E, 2002ApJ...568...52W, 2008MNRAS.390L..64D].


One approach relates the characteristic density to the past accretion history of the halo's main progenitor. [2002ApJ...568...52W], for example, calculated the mass assembly histories (MAHs) of simulated haloes and used a proportionality constant to relate the concentration to background density at the halo's time of formation. The redshift dependence of the c(M) relation was later studied by [2003ApJ...597L...9Z], who found a weakening of the relation for the highest mass haloes at any redshift. By redshift of approximately 3-4 the negative trend is no longer present in the simulations of [2008MNRAS.387..536G], who focus on masses greater than 10 to the power of 11 solar masses per h. The flattening of the c(M) relation was also reported by [2009ApJ...707..354Z] who connected halo concentrations to the period at which halo growth transitions from a rapid to a slow phase. These models provide a clear interpretation of why concentration depends only weakly on mass for the most massive systems: because these haloes are forming today, they share the same formation time, and therefore concentration.


[2007MNRAS.381.1450N] studied the z=0 c(M) relation in the Millennium simulation [2005Natur.435..629S], while [2011ApJ...740..102K] and [2012MNRAS.423.3018P] extended the analysis to z=6 using both the Millennium and Bolshoi simulations. In agreement with previous work, these authors each find a decline of concentration with mass. However, both [2012MNRAS.423.3018P] and [2011ApJ...740..102K] have also reported an upturn of the c(M) relation at the high mass end. On the other hand, Ludlow et al. (2012) demonstrated that there is no upturn amongst relaxed haloes, and showed how the transient dynamical states of merging systems can result in a non-monotonic c(M) relation. While the details continue to be debated, it is clear overall that the diversity of halo formation histories play a critical role in establishing the shape and evolution of the c(M) relation [2014MNRAS.441..378L, 2015arXiv150200391C]. While the low redshift mass-concentration relation is well studied, at high z the relation is poorly constrained.


For example, [2012MNRAS.423.3018P] and [2014arXiv1407.4730D] find a high-mass upturn (above a few times 10 to the power of 10 solar masses per h) at z=5 amongst the full halo population. At similar masses, [2014arXiv1402.7073D], find a relation with a slightly positive slope, whereas [2015arXiv150506436H] report a weak negative slope that flattens by z=9. These authors, however, imposed different equilibrium cuts on their halo samples, which hampers a direct comparison with their results. Given this unsettled state of affairs it is clear that there is some debate on the precise nature of the high redshift c(M) relation and the role played by unrelaxed haloes. For example, a halo suffering a merger is unlikely to have a simple, smooth density profile, and will take time to settle back into equilibrium. This situation becomes increasingly important at high redshift due to the elevated merger rates of potentially star-forming haloes. These sorts of concerns led [2007MNRAS.381.1450N] to introduce three physically-motivated parameters to identify systems far from equilibrium: 1) the mass-fraction in substructure f_sub; 2) the offset between the halo's center of mass and its most-bound particle, x_off, and 3) the pseudo-virial-ratio of kinetic and potential energies, phi.


The effectiveness of these parameters in isolating relaxed DM haloes is further discussed in [2008MNRAS.387..536G] and [2012MNRAS.427.1322L]. A detailed study of these parameters with regard to dynamical relaxation at high redshift is provided by the first paper of the DRAGONS series, Poole et al. (2015b) (hereafter referred to as Paper I). Their results suggest that, across the mass range of our simulations, and for z>5, standard relaxation values for f_sub, x_off and phi obtained from low redshift studies are very effective at identifying systems relaxing from halo formation or recent mergers at high redshift. Concentration is not the only relevant halo property for galaxy formation. Halo spin also plays an important role in semi-analytic models, since angular momentum conservation determines the size of galactic disks [1998MNRAS.295..319M, 2011MNRAS.413..101G], which in turn determine their star formation rates [1959ApJ...129..243S, 1998ApJ...498..541K]. A halo's angular momentum is often expressed as a dimensionless spin parameter, lambda, defined as J_vir divided by sqrt(2) * M_vir * V_vir * R_vir, where J_vir is the total angular momentum within R_vir.


Most studies of the spin parameter have focused on the distribution of spins and its dependence on halo mass [2007MNRAS.376..215B, 2007MNRAS.381.1450N, 2008ApJ...678..621K, 2010crf..work...16M]. At any redshift, halo spins are distributed approximately log-normally, and peak at a lambda of approximately 0.03-0.04. At low redshift, spins are approximately independent of mass but gain a slight negative correlation at higher redshifts [2008ApJ...678..621K, 2010crf..work...16M]. Recently, [2014arXiv1402.7073D] measured the redshift evolution of the lambda - M_vir relation, reporting a weak negative correlation at z=5. In this work we use the Tiamat simulation suite to extend the study of concentration and spin to redshifts z>5. Our simulations were designed to resolve halo masses relevant for galaxy formation during this high-redshift epoch. The purpose of our study is to measure the structural and dynamical properties of haloes that are necessary for forthcoming semi-analytic models of reionization. We organise the paper as follows. In Section 2 we describe the numerical simulations, including halo finding, analysis techniques, and our parametrization of concentration and spin.


In Section 3 we present our concentration--mass relation and its redshift dependence, and in Section 4 the spin distribution, and its mass and redshift dependence. Finally, in Section 5 we summarise our main results. Our analysis focuses on DM haloes identified in three cosmological N-body simulations. These include a 2160-cubed-particle, 67.8 Mpc/h cubed box (the Tiamat simulation) and two smaller but higher resolution volumes of 10 and 22.6 (Mpc/h)-cubed. Each run was carried out with GADGET-2 [2001NewA....6...79S, 2005MNRAS.364.1105S] with RAM (random-access memory) consumption changes in accordance with those detailed in [2015MNRAS.449.1454P]. For each run, the Plummer-equivalent softening length was 1/50th of the mean Lagrangian inter-particle spacing, and the integration accuracy parameter, eta, is set to 0.025, as motivated by the convergence study presented in [2015MNRAS.449.1454P]. Initial conditions were generated using 2nd order perturbation theory using the code 2LPTIC at z=99 and each simulation was run down to z=5; 100 snapshots of particle data were taken equally spaced in time from z=35 to z=5 (one every 11 Myr).


Cosmological parameters for each box were chosen to be consistent with the Planck 2015 data release [2015arXiv150201589P] (h, Omega_m, Omega_b, Omega_Lambda, sigma_8, n_s) = (0.678, 0.308, 0.0484, 0.692, 0.815, 0.968). The relevant numerical parameters are summarised in Table 1. A more detailed discussion of these simulations can be found in Paper I. Summary of simulation parameters. N_p is the total number of particles, L is the length of the box, m_p is the mass of each particle and epsilon is the gravitational force softening length. The ratio between the minimum bin size of the halo profile in dark blue and the size of the virial radius, plotted as a function of particle number for the Tiamat simulation. The solid line is the median while the shaded area is the scatter (68 per cent confidence interval). The largest bin size of 0.09 R_vir indicates our bins begin well inside the halo scale radius for the halo masses considered here.


Haloes were identified in each simulation snapshot using Subfind [2001MNRAS.328..726S]. This produces two outputs: the first contains structures found by a friends-of-friends (FoF) algorithm (we adopt a linking length of 0.2 times the mean inter particle spacing); the second is obtained by dissecting each FoF group into its self-bound 'substructure'. This results is a central 'main halo', typically containing >90 per cent of its virial mass, and a group of lower-mass subhaloes which trace the undigested cores of previous merger events. Each main halo and its substructures were further analyzed to catalog their basic properties and to build their spherically averaged profiles. A (FoF or substructure) halo was required to have a at least 32 particles to be included, resulting in a minimum halo mass of 8.4x10^7 solar masses/h in Tiamat, 2.5x10^7 solar masses/h in MediTiamat and 2.2x10^6 solar masses/h in TinyTiamat. However, when estimating concentrations, a stricter limit of n_p > 5000 within the bound structure was imposed to ensure that the halo's inner regions are well-resolved. For the purpose of estimating spin parameters, this limit is relaxed to 600 particles.


For each main halo, spherically-averaged density profiles were constructed in bins containing an equal number of particles. Only particles considered by Subfind to be bound to the central haloes were used. The number of bins was increased along with the number of halo particles. We imposed a minimum of 5 bins for the smallest haloes, and a maximum of 1000 bins for the largest (reached as particle number tends to 10^6). For example, haloes with 5000 particles have 25 bins, rising to approximately 125 bins for haloes containing 10^5 particles. Best-fit NFW and Einasto profiles were obtained by minimizing a function psi, which is 1/N times the sum of (delta r_i / r_i) * (log rho_i - log rho_model)^2. The factor delta r_i / r_i appropriately weights bins of differing size. Note that only bins in the radial range 0.05 R_vir < r_i < 0.8 R_vir were used. We have verified that the minimum bin radius is always larger than the convergence radius defined in [2003MNRAS.338...14P].


In defining our sample of equilibrium DM haloes we impose both dynamical and resolution criteria following the procedure established by [2007MNRAS.381.1450N], [2008MNRAS.387..536G] and [2012MNRAS.427.1322L]. The behaviour of these equilibrium diagnostics over the mass and redshift range probed by the Tiamat simulation suite is discussed further in Paper I. The criteria defining relaxed haloes include upper limits on the following three quantities: I) The fraction of mass found in satellite subhaloes, f_sub, must be less than 0.1. A high fraction of mass in substructure may be indicative of a recent merger [2007MNRAS.381.1450N]. II) The offset between the position of the most-bound-particle and center-of-mass, x_off, must be less than 0.07. This is complementary to f_sub as it includes mass from unresolved subhaloes. III) The pseudo-virial ratio of kinetic and potential energies, phi = 2K/|U|, must be less than 1.35. This criterion tends to be sensitive to haloes at the pericenter of a merger.


[2007MNRAS.381.1450N] find that these restrictions provide a simple and physically motivated method to exclude haloes that are not well described by an NFW profile. In Paper I these parameters were also shown to be discriminate between haloes that have either recently doubled in mass, or suffered a major or minor merger, within the last 1-2 dynamical times. We therefore adopt them as our standard equilibrium criteria. A further criterion is imposed to select only well-resolved haloes. For the halo concentration analysis a lower limit on particle number of greater than 5000 is imposed. This is derived from work studying convergence of NFW and Einasto profile fits in [2007MNRAS.381.1450N] and [2008MNRAS.387..536G]. This ensures that the inner portion of the halo is well enough resolved to measure the scale radius r_s. We impose a restriction of greater than 600 particles for the spin parameter measurement following [2008ApJ...678..621K]. The dynamics of hierarchical growth means that at the high redshifts studied here, many of our haloes will be far from equilibrium and thus have ill-defined values for concentration and spin.


These sample cuts are designed to remove such systems and to keep transients out of our analysis as much as possible. For example, a halo which has just undergone a major merger may be comprised of two large, high density clumps, and consequently have a high x_off and poorly defined center. The density profile of such a system cannot be captured by simple spherical averages. We stress, however, that there is a continuum of values for x_off, f_sub and phi; the particular values chosen to separate relaxed haloes from unrelaxed are the result of extensive past investigations [2007MNRAS.381.1450N, 2012MNRAS.427.1322L]. To quantify the effects of our sample selection, we note that in Tiamat there are 14391 haloes with more than 5000 particles at redshift z~5, but only 4433 (or ~30 per cent) of these satisfy our relaxation criteria; this reduces to ~15 per cent by z~10. These numbers underline the importance of the dynamical state of haloes at high redshift. Nevertheless parametrised fits to the entire population are useful for many semi-analytic calculations [2014MNRAS.439.2728M, 2015MNRAS.451.2840S] and for this reason we report fits to both our full halo sample as well as the relaxed sub-sample.


Caption: Concentration--mass relation of relaxed central haloes at z=5. Left and Right panels are the NFW and Einasto concentrations respectively. Inner shaded region denotes the bootstrapped 90 per cent confidence interval on the median. The outer shaded region shows the 68 per cent scatter. The line of best fit is fit to the median using the Monte Carlo Markov Chain method implemented in the Python package emcee [2013PASP..125..306F]. This produces c_vir equals 3.8 (+/-0.4) times (M/10^10 solar masses per h) to the power of -0.035 (+/-0.005) at z=5 for NFW fits and c_vir equals 3.8 (+/-0.4) times (M/10^10 solar masses per h) to the power of -0.039 (+/-0.005) for Einasto fits. The fits from [2014arXiv1402.7073D] are also shown with thick solid lines. The Tiamat simulation is designed to study reionization and explores a mass range below previous studies. The dashed black line at c_vir=4 is added for a point of reference. Solid black points represent the [2014MNRAS.441..378L] model where halo concentrations are calculated from the median accretion history in the mass bin.


Caption: The same concentration mass relations for relaxed central haloes as shown in the previous figure, but now at z=7. Inner shaded region denotes the bootstrapped 90 per cent confidence interval on the median. The outer shaded region shows the 68 per cent scatter. The line is fit to the median and gives c_vir = 3.4 (+/-0.6) times (M/10^10 solar masses per h) to the power of -0.019 (+/-0.008) at z=7 for NFW and 3.3 (+/-0.6) times (M/10^10 solar masses per h) to the power of -0.018 (+/-0.008) for Einasto profiles. The dashed black line at c_vir=4 is added for a point of reference. Solid black points represent the [2014MNRAS.441..378L] model where halo concentrations are calculated from the median accretion history in the mass bin. Error bars are derived from the 68 per cent scatter of the accretion histories. The c(M) relations for our equilibrium haloes at z=5 and z=7 are shown in the figures. Both NFW and Einasto concentrations are plotted.


The inner shaded region shows the bootstrapped 90 per cent confidence interval on the median for mass bins containing at least 20 haloes. The outer shaded area fills the 68 per cent scatter in individual concentration estimates. We find a weak trend of decreasing concentration with mass at z=5 for both NFW and Einasto fits. This trend becomes shallower as redshift increases. By redshift 9 there is no trend apparent for either set of fits. The c(M) relations obtained from both NFW and Einasto fits are similar over this mass range. We note that the systematic difference between NFW and Einasto concentrations is less than 0.1, which is smaller than the change in concentration from the lowest to highest masses studied here. Best-fitting power laws at z=5 are 3.8 (+/-0.4) times (M/10^10 solar masses per h) to the power of -0.039 (+/-0.005) for NFW fits and 3.8 (+/-0.4) times (M/10^10 solar masses per h) to the power of -0.039 (+/-0.005) for Einasto fits. Best-fit parameters are obtained using the Monte Carlo Markov Chain (MCMC) method implemented with the emcee package [2013PASP..125..306F].


The quoted errors are the 68 per cent confidence interval derived from the posterior distribution. We find intrinsic scatter in the c(M) relation of delta c_vir ~ 1.0 (or 20 per cent) for fits to both NFW and Einasto profiles. Best-fit power-law relations are provided in Table 2 for a range of redshifts. In addition to the two scaling variables, the Einasto profile has a shape parameter, alpha. Previous studies have found that alpha depends in a complex way on both halo mass and redshift, but follows a simple relation when expressed in terms of the dimensionless 'peak height' mass parameter, nu = delta_sc / sigma(M,z). Here delta_sc=1.686 is the density threshold for the collapse of a spherical top-hat density perturbation, and sigma(M,z) is the rms density fluctuation in spheres enclosing mass M. We find a similar alpha-nu relation to the previous authors - the fit from [2008MNRAS.387..536G] is shown as a dashed line. Our best fit for the alpha-nu relation is alpha = 0.007 * nu squared + 0.1839.


In order to check for any subtle bias in our fits we have also constructed c(M) and alpha(nu) relations using the median density profiles obtained by stacking haloes in narrow mass bins. This smooths out any unique features of individual systems and allows for a robust estimate of the median structural properties of haloes of a given mass. For our relaxed population we recover the c(M) to within delta c ~ 0.1, for both NFW and Einasto fits. We choose to use the individual fits when computing the best fit c(M) relation. The weak trend in concentration with mass found in our simulation is in qualitative agreement with previous work that found a negative trend at low redshift that becomes progressively shallower with increasing z. For example, both the shallow negative slope and the magnitude of our Einasto concentrations are in good agreement with [2015arXiv150506436H]. In Figure 2 we also plot the c(M) relations from [2014arXiv1402.7073D] (hereafter DM14), also obtained from both NFW and Einasto fits. As DM14 employ different relaxation and resolution criteria, the differences we observe, although slight, are unsurprising.


However, the shape of our trend at redshift 5 is in qualitative disagreement with DM14, who find that a positive trend emerges at z=5 for both Einasto and NFW concentrations. We also find a higher normalization (about 25 per cent) than DM14 at ~10^10 solar masses per h. We do not speculate on the exact combination of these differences that effects the c(M) relation but note that: firstly, halo profiles in DM14 are fit out to 1.2 R_vir while Tiamat profiles are only fit out to 0.8 R_vir, and second, different resolution and relaxation criteria were used. DM14 adopt a minimum halo mass corresponding to 500 particles, and define relaxed haloes as those satisfying x_off < 0.07 and rho_rms < 0.5, where rho_rms is the rms deviation between the haloes density profile and the best-NFW fit. However, we do find good agreement between our c(M) relation results and the model proposed by [2014MNRAS.441..378L], as indicated by the black dots in Figures 2 and 3. We emphasise there is no fit to the halo density profiles here, only the accretion histories are required.


Caption: Best fit values from the relaxed population for NFW-derived and Einasto-derived c(M) relations. N_sample denotes the number of haloes in the sample for Tiamat, MediTiamat, TinyTiamat. Fits and errors are the median and 68 per cent confidence interval using the MCMC package quoted previously. Caption: The Einasto profile alpha-nu relation. At each redshift the median alpha is plotted for bins containing >20 haloes, with the errors derived from the bootstrapped 90 per cent confidence interval on the median. Each symbol denotes a different redshift. We find a similar alpha-nu relation to the low redshift results of previous authors despite the redshift range of our simulations being z>5. Caption: The dependence of concentration on x_off for central subgroups in several mass ranges. Blue lines represent the median. The dashed red line denotes the relaxation criteria cut above which a halo is considered to be out of equilibrium [2007MNRAS.381.1450N, 2008MNRAS.387..536G, 2012MNRAS.427.1322L]. It can be seen that lower x_off parameters (i.e. more relaxed haloes) correlate with higher c_vir for all mass ranges.


Caption: Concentration-mass relation for relaxed haloes at z=5, but now with haloes containing >500 particles. The inner shaded region represents the bootstrapped median value and the outer region the 68 per cent scatter. The inclusion of haloes with particle number <5000 introduces more haloes with lower concentrations at the low mass end of each simulation. Einasto shape parameters, alpha, for haloes with <500 particles are also higher. Caption: The same as Figure 2 but now the case in which no non-equilibrium cuts are enforced and the full sample of haloes with >5000 particles is analysed. The median magnitude of the concentrations has decreased by delta c_vir ~ 1 over all masses in our simulations. To investigate the effect of our equilibrium selection criteria we plot in Figure 5 the variation of concentration with x_off. There is a clear trend that haloes with higher x_off have lower concentrations, representing a delta c_vir ~ 2 decrease for haloes with the largest offsets. The trend is similar at all three of the mass ranges considered. This figure illustrates how important it is to understand the dynamical state of the haloes included in the sample.


In Figures 6 and 7 we again present the c(M) relation but now relax the strict resolution and equilibrium criteria. Firstly, in Figure 6 we show the relation that results from lowering the minimum particle limit for a halo to 500 particles, while maintaining the relaxation criteria. Whereas our n_p > 5000 relation agreed where simulations overlapped, we find a significant discrepancy between the simulations for masses corresponding to haloes with 5000 > n_p > 500. In particular, lower particle numbers result in lower values of c_vir. For Einasto profiles the alpha-nu relation also changes slightly. Many of the 500 < n_p < 5000 haloes have alpha ~ 0.25 and do not follow the previous quadratic alpha-nu relation. We thus caution against over-interpreting Einasto fit parameters at low particle number. In Figure 7 the c(M) relation is plotted for the entire halo sample. At all masses, the median concentrations decrease relative to those of our relaxed haloes. We also note that inclusion of unrelaxed haloes alters the alpha-nu relation slightly. Our best-fit to the full population is alpha = 0.0091 * nu squared + 0.1447.


Table 3 shows the best fit parameters for the full population of resolved haloes with n_p > 500 particles. In the left hand panel of Figure 7 we also plot the models of [2014arXiv1407.4730D] and [2012MNRAS.423.3018P]. [2014arXiv1407.4730D] use the phase space halo finder ROCKSTAR [2013ApJ...762..109B], and do not enforce any relaxation criteria. The mass range covered in [2014arXiv1407.4730D] is equivalent to the mass range covered by Tiamat with halo particle numbers of n_p > 1200. In this mass range we see slight evidence for an upturn in the Tiamat c(M) relation, which is consistent with their findings. Note that this upturn is a feature exclusive to our full halo sample, suggesting its connection to departures from equilibrium. Caption: The Einasto alpha-nu relation but including haloes above n_p > 500 particles. Each symbols denotes a different redshift. These low particle number haloes introduce high alpha parameters which are not in accord with the best fit quadratics found for more resolved haloes in this and previous works.


Our concentrations at these masses and redshifts are lower than those found by [2012MNRAS.423.3018P]. The delta c_vir ~ 1 difference likely originates from a combination of: a) the use of M_200 and c_200 in their definitions, b) the use of maximum circular velocity as a measure of concentration, which is shown in [2013arXiv1303.6158M] to reconcile differences in the c(M) relation, c) a different halo finder (Bound-Density-Maxima), and/or d) a lower particle limit of 500. We also note a similar comparison with [2014arXiv1411.4001K]. When we relax our halo equilibrium criteria we obtain concentrations which are in better agreement with DM14 and [2014arXiv1407.4730D]. Our simulations do not show an upturn after unrelaxed haloes are removed, as pointed out in [2015arXiv150200391C]. In the case of both the relaxed population and the full population we find the weak trend in the M_vir - c_vir to be steady across this mass range. The redshift dependence of the relaxed population of NFW concentrations can be described by an equation. The trend for both NFW and Einasto fits are consistent with being mass-independent at z>8.


In Figure 8 we show the distribution of spin parameters for relaxed haloes in the Tiamat simulation for z=5. The mass range covered is now increased so that particle number n_p > 600 [2008ApJ...678..621K]. Rather than a log-normal, the solid black lines show the best-fitting function of [2007MNRAS.376..215B], which has been shown to better describe the low spin tail. Also shown are log-normal fits by [2008ApJ...678..621K] and [2010crf..work...16M]. Although evaluated at different redshifts from Tiamat, these authors report minimal redshift-evolution in the halo spins. Our results support this, with the distribution being well fit by the Bett et al. equation with a small dependence on redshift. At z=5 we have (lambda_0, alpha) = (0.033 +/- 0.0002, 2.25 +/- 0.04) changing to (lambda_0, alpha) = (0.029 +/- 0.0004, 2.36 +/- 0.1) at z=10. Best fit parameters and errors are again derived using the MCMC method. Parameters for all redshifts are shown in Table 4 along with the numbers of haloes in each sample.


Both [2010crf..work...16M] and [2008ApJ...678..621K] fit a log-normal to their spin distribution, with best-fit parameters given by sigma_0= 0.57 (variance) and lambda_0 = 0.031 (mean), and sigma_0 = 0.53 and lambda_0 =0.035, respectively. Both have slightly higher spins overall. As found by [2007MNRAS.376..215B], the Bett et al. equation provides a better fit to our low spin distribution than does the log-normal. We find that unrelaxed haloes have a noticeable impact on our spin distribution, which is shown in Figure 9. Without removing these haloes we find the best-fit parameters to be (lambda_0, alpha) =( 0.042 +/- 0.0002, 2.70 +/- 0.04). Figure 10 shows the relation between spin and virial mass at z=5, with a power-law best-fit. A slight negative slope of B = -0.01 +/- 0.006 at redshift z=5 decreases to -0.023 +/- 0.016 by z=10. Fits from previous work that study the Bullock spin parameter at high redshift are also shown. We find the scatter in the spin to be roughly constant in each mass bin.


The existence of a small negative trend of spin parameter with mass at these redshifts is in qualitative agreement with [2008ApJ...678..621K], who find no trend at z=0-1 but an emerging trend at z=10. As noted, the virialised halo cut has an affect on our results. However, we find a small negative trend in both the full and relaxed population. For example, the lambda - M_vir relation for the full sample at redshift 5 is shown in Figure 11. Without making cuts we find a relation with slope A=-0.009 at z=5, changing to B=-0.029 at redshift 10. Our spin mass relation with sample cuts is in agreement with the result of B = -0.06 +/- 0.17 from [2008ApJ...678..621K]. Caption: The distribution of Bullock spin parameters for relaxed haloes at redshift 5 in Tiamat. Caption: The distribution of Bullock spin parameters for the full sample of haloes in Tiamat at redshift 5, without cutting unrelaxed haloes. Caption: The spin-virial mass relation for relaxed haloes at z=5. For comparison results for relaxed haloes from [2011MNRAS.411..584M] are also shown.


Caption: The spin-virial mass relation for the full population of haloes at z=5. For comparison fiducial results for relaxed haloes from [2011MNRAS.411..584M] are also shown. We used N-Body simulations to study concentrations and spins of DM haloes at z=5-10 and across the mass range 10^8 to 10^11 solar masses per h; the regime relevant for studies of structure formation during the epoch of reionization. The dependence of these parameters on equilibrium state was investigated by splitting our halo sample into two populations which include i) only relaxed haloes and ii) the full population. We find qualitatively similar results to previous studies. However, we find quantitative differences between our derived c(M) relations and spin-mass relations and those of previous studies, which we attribute to each author's use of different halo finders, to our higher simulation resolution, and to the different relaxation criteria used for our sample. We find the model proposed by [2014MNRAS.441..378L] reproduces both the slope and redshift evolution of our c(M) relation. Our key results are as follows:


Our best-fit concentration-mass relations at z=5 have a slightly negative slope that becomes more shallow towards z=9. Limiting our analysis to equilibrium haloes has a strong impact on the derived c(M) relation due to unrelaxed haloes having lower concentrations at all masses and redshifts. Haloes with larger center-of-mass offset (x_off) typically have lower concentrations. The slope of the c(M) relation becomes shallower at higher redshift, although concentrations decrease at all masses. However, at high redshifts the number of haloes passing the equilibrium criteria is low: only ~30 per cent of haloes in the 67.8 Mpc/h box pass our resolution and relaxation cuts at z=5. Such a high proportion of unrelaxed haloes at the mass scales studied here is a distinct property of the high redshift universe, as discussed in Paper I. We find concentrations of relaxed haloes at z>5 to be well described by specific relations for NFW and Einasto fits. The intrinsic scatter around the c(M) relations is delta c_vir ~ 1 (or 20 per cent). We find the shape parameter of the Einasto profiles to depend on the peak height mass parameter.


Without imposing equilibrium cuts on our sample, the concentrations found in Tiamat have similar values to those reported by [2014arXiv1402.7073D], [2014arXiv1407.4730D] and [2015arXiv150506436H]. Concentrations of haloes in Tiamat are a factor of delta c_vir ~ 0.5-1 lower than reported by [2012MNRAS.423.3018P] and [2014arXiv1411.4001K]. The shallow negative trend in the c(M) relation that flattens from z=5 to z=10, and the overall decrease in the magnitude of our Einasto concentrations agree well with [2015arXiv150506436H]. The distribution of Bullock spin parameters for relaxed haloes at z >= 5 is found to be well fit by the Bett et al. equation with little evolution with redshift. Including unrelaxed haloes results in a spin distribution with a higher mean of lambda_0=0.042. As in previous studies, we find a spin-virial mass relation with a slight negative correlation at high redshift. The trend found here has a slope of approximately -0.02 at z=10. The exclusion of unrelaxed haloes also has the effect of increasing the peak of the spin distribution while the slope of the lambda-M_vir relation remains slightly negative.


Our best-fit power-law relation for relaxed haloes at z=5 is given, as is the relation for the full halo population. The growth of dark matter haloes drives high-z galaxy formation [2013ApJ...768L..37T], while the concentration and spin of haloes are key ingredients for semi-analytic models of galaxy formation [2006MNRAS.365...11C]. This study of these properties for haloes corresponding to the galaxies responsible for reionization will provide a valuable resource for understanding the framework of early galaxy formation. This research was supported by the Victorian Life Sciences Computation Initiative (VLSCI), grant ref. UOM0005, on its Peak Computing Facility hosted at the University of Melbourne, an initiative of the Victorian Government, Australia. Part of this work was performed on the gSTAR national facility at Swinburne University of Technology. gSTAR is funded by Swinburne and the Australian Government’s Education Investment Fund. This research program is funded by the Australian Research Council through the ARC Laureate Fellowship FL110100072 awarded to JSBW. ADL is financed by a COFUND Junior Research Fellowship. We thank Volker Springel for making the GADGET2 and SUBFIND codes available. We also thank N.Gnedin for useful comments on our manuscript.


Correlations between black holes and their host galaxies provide insight into what drives black hole--host co-evolution. We use the Meraxes semi-analytic model to investigate the growth of black holes and their host galaxies from high redshift to the present day. Our modelling finds no significant evolution in the black hole--bulge and black hole--total stellar mass relations out to a redshift of 8. The black hole--total stellar mass relation has similar but slightly larger scatter than the black hole--bulge relation, with the scatter in both decreasing with increasing redshift. In our modelling the growth of galaxies, bulges and black holes are all tightly related, even at the highest redshifts. We find that black hole growth is dominated by instability-driven or secular quasar-mode growth and not by merger-driven growth at all redshifts. Our model also predicts that disc-dominated galaxies lie on the black hole--total stellar mass relation, but lie offset from the black hole--bulge mass relation, in agreement with recent observations and hydrodynamical simulations.


Extensive low-redshift studies reveal a complex interplay between galaxies and the supermassive black holes that reside at their centres, with clear correlations observed between black hole mass and host bulge mass, total stellar mass, velocity dispersion and luminosity [Magorrian1998, Gebhardt2000, Merritt2001, Tremaine2002, Marconi2003, Haring2004, Bentz2009, Kormendy2013, Reines2015]; see the review by [Heckman2014]. These tight correlations suggest a co-evolution between galaxies and supermassive black holes, which may be causal, due to feedback from the active galactic nucleus [AGN; e.g.][Silk1998, Matteo2005, Bower2006, Ciotti2010] or the efficiency with which the galaxy can fuel the black hole [e.g.][Hopkins2010, Cen2015, AnglesAlcazar2017], or coincidental, simply due to mergers causing both black hole and galaxy growth [e.g.][Haehnelt2000, Croton2006b, Peng2007, Gaskell2011, Jahnke2011]. To understand what drives black hole--host co-evolution, it is necessary to study how these correlations change with redshift.


Observing high-redshift black hole--host correlations is fraught with difficulties. Host galaxies are hard to detect since they are often completely outshined by the AGN light, particularly in the rest-frame optical where common stellar mass estimators can be used [Zibetti2009, Taylor2011]. Subtracting the quasar light has resulted in host detections out to redshift z is approximately equal to 2 [Jahnke2009, Mechtley2016], but is yet to be successful for detecting the highest redshift quasars at redshift z is approximately equal to 6 [Mechtley2012]. For these quasars, host masses are often estimated using the widths of observed submillimeter and millimeter emission lines, such as the [CII] 158 micron and CO (6--5) lines [Wang2013]. However, dynamical masses determined from emission line widths are highly dependent on the assumptions made, such as the gas-disc geometries and inclination angles [Valiante2014]. In fact, inclination angle assumptions can change the determined black hole mass to bulge mass ratio measurements by roughly 3 orders of magnitude [Wang2013].


In addition, the emission regions may not trace the spatial distribution of the stellar component of the galaxy, meaning that these dynamical masses may not be representative of the total stellar mass [Narayanan2009]. Determining the black hole masses of high-z quasars is also difficult, with emission-line based estimators relying on calibrations at low redshift. Where these observations are unavailable, Eddington accretion rates are instead often assumed to estimate the black hole mass [as in e.g.][Wang2013, Willott2017], which also leads to large uncertainties. High-redshift studies of the black hole--host mass relations are thus very uncertain. With this in mind, high redshift observations find black holes that are more massive than expected by the local relation, where the canonical black hole--bulge mass ratio is 10 to the power of (-2.31 +/- 0.05) for a bulge mass of 10^11 solar masses [Kormendy2013].


For example, ALMA observations of five redshift z is approximately equal to 6 quasar hosts show black hole to dynamical mass ratios ranging from 10 to the power of -1.9 to 10 to the power of -1.5 [Wang2013]. Similar studies at redshift z is approximately equal to 4--7 [Maiolino2007, Riechers2008, Venemans2012] also give estimates for individual quasars of a black hole mass to dynamical mass ratio greater than or approximately equal to 10 to the power of -2, which is significantly larger than the local value if dynamical masses and bulge masses are assumed to be roughly equivalent. This suggests a faster evolution of the first supermassive black holes relative to their host galaxies [Valiante2014], which could potentially be a result of super-Eddington accretion [Volonteri2015]. The high observed black hole mass to dynamical mass ratio relation at high redshift could, however, be a result of selection effects [Lauer2007, Schulze2011, Schulze2014, DeGraf2015, Willott2017].


[Willott2017] suggest that since only the most massive z>6 black holes are observed, if the relation has a wide dispersion then one would expect to see a higher value due to the Lauer bias [Lauer2007]: since the luminosity function falls off rapidly at high masses, the most massive black holes occur more often as outliers in galaxies of smaller masses than as typical black holes in the most massive galaxies. Indeed, [Willott2017] found that black holes with mass less than 10^9 solar masses at redshift z>6 fall below the black hole mass--dynamical mass relation for low redshift galaxies, in contrast to the opposite being true for higher mass black holes. Similarly, [Schulze2014] claim that selection effects are the reason for the observed evolution of the black hole mass--bulge mass relation; on applying a fitting method to correct for selection effects, they find no statistical evidence for a cosmological evolution in the black hole mass--bulge mass relation.


A lack of evolution in the black hole--host relations is consistent with the findings of cosmological hydrodynamical simulations such as Horizon-AGN [Volonteri2016], which observes very little evolution in the black hole mass--stellar mass relation from redshift z=0 to 5, and BlueTides [Huang2018], which finds a black hole mass--stellar mass relation at redshift z=8 that is consistent with the local [Kormendy2013] relation. [DeGraf2015], on the other hand, found that the relation evolves slightly for redshift z greater than or equal to 1 for the highest mass black holes, with a steeper slope at the high-mass end at higher redshifts, making selection effects important. The more statistical study of [Schindler2016] found that the ratio of the black hole to stellar mass density is constant within the uncertainties from redshift z=0 to 5, with a slight decrease in the ratio at redshifts between 3 and 5; this is also consistent with no cosmological evolution in the black hole mass--stellar mass relation.


In this work we explore the evolution of the black hole--host relations with the Meraxes semi-analytic model [Mutch2016]. Meraxes is designed specifically to study galaxy formation and evolution at high redshifts, making it ideal for studying the evolution of black holes and their host galaxies. In this work we use Meraxes, a semi-analytic model designed to study galaxy evolution at high redshifts [Mutch2016]. Using the properties of dark matter halos from an N-body simulation, Meraxes analytically models the physics involved in galaxy formation and evolution. We run Meraxes on the collisionless N-body simulations Tiamat and Tiamat-125-HR [Poole2016, Poole2017]. Tiamat is ideal for studying high redshifts, with a high mass and temporal resolution. Tiamat runs from redshift z=35 to z=1.8, with a box size of (67.8 per h Mpc)^3, 2160^3 particles of mass 2.64x10^6 per h solar masses, and a high cadence of 11.1 Myr per output snapshot at redshift z>5.


Tiamat-125-HR is a low-redshift counterpart to Tiamat, running from redshift z=35 to z=0 with the same temporal resolution, but with a lower mass resolution (1080^3 particles of mass 1.33x10^8 per h solar masses) and larger box size of (125 per h Mpc)^3, more suited for low-redshift studies. Throughout this work, we use the higher resolution Tiamat at high-redshifts, and Tiamat-125-HR for redshift z<2, unless otherwise specified. Meraxes assumes that galaxies reside in the centre of dark matter haloes produced by the N-body simulation. Using the properties of these haloes, Meraxes analytically models the baryonic physics involved in galaxy formation and evolution, such as gas cooling, star formation, black hole growth, and supernova and black hole feedback. These analytical prescriptions involve a range of free parameters, which must be calibrated using observations such as the stellar mass function.


In Meraxes, stars in galaxies reside in three components: an exponential disc, a spheroidal merger-driven bulge and a disc-like instability-driven bulge. Bulges grow through both galaxy-galaxy mergers and disc-instabilities. In Meraxes, we assume that galaxy mergers with merger ratio greater than 0.01 trigger a burst of star formation, by causing shocks and turbulence in the cold gas of the parent galaxy. The galaxy will also accumulate the mass of the secondary galaxy. We assume that the dominant mass component of the primary galaxy will regulate where these stars produced by the burst and the secondary's mass will be deposited. If the primary is dominated by a discy component, the mass is added to the instability-driven bulge. Otherwise, we assume that the new stars will accumulate in shells around the spheroidal merger-driven bulge. In major mergers, where the merger ratio is greater than 0.1 or 0.3, we assume that the stellar disc and instability-driven bulges are destroyed, with all stars placed into the merger-driven bulge.


In our model we assume that the galaxy discs are thin, with an exponential surface density and flat rotation curve. Such discs become unstable if the disc mass is greater than the disc velocity squared times the scale radius divided by the gravitational constant, which equals the critical mass [Efstathiou1982, Mo1998]. Here, we take the disc mass as the combined mass of both gas and stars in the disc, and the disc velocity and scale radius as the mass-weighted velocity and scale radius of the stellar and gas discs. If such a disc instability occurs, Meraxes returns the disc to stability by transferring the unstable mass of stars from the disc to the instability-driven bulge. The Meraxes black hole model was introduced in Q17, and updated to include instability-driven growth in M19. In Meraxes, black holes are seeded in every newly-formed galaxy, with a seed mass of 10^4 solar masses. Black holes then grow by accretion of both hot and cold gas, through the radio- and quasar modes, respectively.


We also assume that black holes grow in galaxy mergers, with the black holes in each galaxy merging together. Black holes accrete hot gas from the static hot gas reservoir around the galaxy, at a fraction of the Bondi-Hoyle accretion rate. We consider this fraction a free parameter, which adjusts the efficiency of radio-mode black hole growth [Croton2016]. This accretion is limited by the amount of hot gas in the reservoir and the Eddington limit. A fraction of this accretion mass is radiated away and so during one snapshot, black holes grow through the radio-mode by the remaining mass. We include the effects of radio-mode AGN feedback by assuming that a fraction of the radiated energy is coupled to the surrounding gas, adiabatically heating a mass which is subtracted from the cooling flow, regulating the accretion of new gas onto the black hole [Croton2006a, Croton2016]. This AGN feedback has no significant effect on the results of Tiamat at redshift z greater than or equal to 2.


Black holes accrete cold gas from the galaxy, when triggered by either a galaxy-galaxy merger or a disc instability. During such an event, the black hole mass grows by a certain amount, where the virial velocity and a free parameter adjust the growth efficiency. For merger-triggered growth, we take the efficiency parameter to be proportional to the merger ratio. For instability driven growth, we consider two separate free parameters. During the quasar mode, black holes are assumed to accrete at the Eddington rate, and thus the mass accreted by the black hole during one simulation snapshot is limited. This can result in the mass being accreted over multiple simulation snapshots. We incorporate quasar-mode AGN feedback by considering the energy injected into the gas during a simulation time-step. We assume that this energy generates a wind that heats the cold disc gas and transfers it to the hot gas reservoir, depleting the supply of cold gas available for the black hole to accrete. If sufficient energy is injected by the quasar, this wind can also eject the hot gas.


We calculate the bolometric luminosities of each black hole in the model following the Q17 method, which assumes Eddington luminosity for all accreting black holes, and self-consistently calculates the duty cycle. We consider the luminosities from both the quasar- and radio-modes of accretion. At high-redshifts the contribution from the radio-mode is negligible. At the lowest redshifts (redshift z less than or equal to 2), the radio-mode becomes a more significant growth mechanism for the most massive black holes, and so their luminosities are enhanced slightly by the addition of the radio-mode luminosity. We convert from bolometric to B-band luminosities using the [Hopkins2007] bolometric correction, and then assume a continuum slope of 0.44 to convert to UV luminosities. We also account for obscuration due to quasar orientation, by scaling the UV luminosity function by a factor related to the opening angle of quasar radiation. In our model we assume a constant opening angle, for simplicity, which is a free parameter in our model.


In M19 we calibrated the free parameters in Meraxes to match the observed stellar mass functions at redshift z=0--8, and the black hole--bulge mass relation at redshift z=0. Using this model, we find that the black hole mass function and quasar luminosity functions are much larger than predicted by the observations. In addition, we note that [Shankar2016] find significant selection biases in the black hole--bulge mass relation---a topic of recent debate [see e.g.][Kormendy2019]. Due to the M19 predictions and this potential bias, we assume that the [Shankar2009] redshift z=0 black hole mass function is a less biased indicator of the local black hole population, and retune the model here to better reproduce the black hole observations. Note that we use the same parameter values for Tiamat and Tiamat-125-HR, and use both simulations to tune the model: Tiamat for matching redshift z greater than or equal to 2 observations and Tiamat-125-HR for redshift z<2.


We find that our results from the two simulations are generally consistent at redshift z is approximately equal to 2, with broad qualitative agreement at higher redshifts. We calibrate the free parameters in the model to match the observed stellar mass functions at redshift z=0--8, the [Shankar2009] and [Davis2014] black hole mass function at redshift z=0, and the quasar X-ray luminosity functions from redshift z=5 to 2. Since [Shankar2016] find that the observed black hole--bulge mass relation is biased to high black hole masses, we also require our model to not over-predict this relation, however we do not otherwise tune to it. We note that our best models produce black hole--host mass relations lower than the observations, consistent with the expectations of [Shankar2009], and have steeper slopes. We find that these criteria are met by a range of free parameter values for the merger-driven black hole growth efficiency, and the definition of a major merger.


We note that all of these parameter sets produce very similar results. As a further check of the black hole population, we plot the black hole accretion rate density as a function of redshift for models with these different merger-driven black hole growth efficiencies. We find that the models with lower efficiencies give black hole accretion histories in approximate agreement with the observations. The larger efficiencies overproduce measurements of the black hole accretion rate density [e.g.][Delvecchio2014]. The opening angle of AGN radiation, theta, adjusts the normalization of the UV luminosity function. We tune this to match the observations, finding a preferred theta of 70 degrees, corresponding to an observable fraction of UV quasars of 18 per cent. We show the quasar X-ray luminosity functions at redshift z=5--0, with X-ray luminosities calculated using the [Hopkins2007] bolometric to X-ray correction.


At redshift z=2 the model and the observations agree remarkably well. At redshift z>2 the model over-predicts the observed quasar X-ray luminosity function at intermediate luminosities, by up to ~0.7 dex at redshift z=4, while at redshift z<2 the model under-predicts the luminosity function at these luminosities. Our model shows better agreement with the observations than previous versions of Meraxes. While the observations show a slight increase in the X-ray quasar luminosity functions from redshift z=4 to 2, the model predicts a slight decrease. In fact, we cannot find a combination of black hole parameters that results in a redshift evolution that matches that of the observed X-ray quasar luminosity function at redshift z>2. However, the key quantity of black hole accretion rate density is predicted by the model to peak at redshift z=2 as observed.


In addition to published uncertainties in the observations, it may also be the case that at higher redshifts X-ray AGN are more likely to be obscured, which is consistent with evidence from a range of X-ray observations [Treister2006, Vito2014, Buchner2015]. Thus we argue that the inability of our model to match the redshift evolution of the X-ray quasar luminosity function may not represent a significant concern. We show the quasar UV luminosity functions at redshift z=5--0. We find that, as with the X-ray luminosity function, the UV luminosity function decreases from redshift z=5 to 0, though it agrees well with observations at redshift z>2. At redshift z<2, however, we note that the faint-end of the UV luminosity function becomes flat, and by redshift z<1 there is a significant disagreement with the observations, with the model producing too many luminous quasars.


The black hole accretion rate density becomes significantly higher than the observations at redshift z<1, consistent with the quasar luminosities being overestimated at these redshifts. This excess black hole accretion is most likely a result of the model missing important physics required for modelling low-redshift galaxy evolution, particularly in the quenching of massive galaxies, or due to the simplifications assumed in the model such as a constant black hole accretion efficiency. However, as the overall accretion rate density at these redshifts is low, this will not have a significant impact on the black hole mass, an integrated quantity. Thus, while the redshift z<1 black hole accretion rates are overestimated, the black hole mass function and black hole--host mass relations are reliable at low redshifts. Indeed, we find that assuming a lower Eddington ratio significantly improves the match between the model and observed UV luminosity functions at redshift z<1.


However, this causes the model to no longer match the observations at higher redshifts. Thus, some evolving Eddington ratio is necessary for Meraxes to accurately reproduce the redshift z<1 quasar UV luminosity function. We now use the model described to explore black hole growth. We investigate the redshift evolution of the black hole--host scaling relations. To investigate the redshift evolution of the black hole--bulge and black hole--total stellar mass relations we first perform linear least squares fits to the relations at a range of redshifts. We only include galaxies with mass > 10^9.5 solar masses in our fits, so that they are not biased by the large number of low-mass galaxies. Both relations have a slope and normalization that increase with redshift from redshift z=0 to 2, with much weaker evolution for redshift z>2.


Relative to the scatter in the relations, we see minimal evolution in both the black hole--bulge and black hole--total stellar mass relations from redshift z=0 to 6. This lack of evolution in the black hole--host mass relations is consistent with the findings of cosmological hydrodynamical simulations such as Horizon-AGN [Volonteri2016] and BlueTides [Huang2018]. We find that our black hole--total stellar mass relation has similar but slightly larger scatter than the black hole--bulge relation, with the scatter in both decreasing with increasing redshift. While the black hole mass has a slightly stronger relationship with the bulge stellar mass, the black hole and total stellar mass are still tightly correlated. The scatter in the relations is slightly larger than the 0.28 dex observed by [Kormendy2013] locally. However, they are very consistent with those from the BlueTides simulation at high redshift.


The scatter decreases with increasing stellar mass. The median black hole mass to total stellar mass ratio as a function of redshift for galaxies with black hole mass > 10^6 solar masses shows no statistically-significant evolution out to redshift z is approximately equal to 8. This is consistent with current high redshift observations; when selection effects are accounted for, the observations at high redshift are consistent with no cosmological evolution in these relations [Schulze2014]. Our model predicts no significant evolution in the black hole--host mass relations, with the scatter in the relations decreasing at the highest redshifts. This indicates that there is a connection between the growth of black holes and their host galaxies. Indeed, our model includes joint triggering of star formation and black hole growth during galaxy mergers, and black hole feedback which regulates star formation, meaning that the co-evolution of black holes and galaxies is implicit in our model.


This is not consistent with the scenario proposed by [Peng2007] and [Jahnke2011], for example, where the black hole and galaxy growth is uncorrelated and the relationships are generated naturally within a merger driven galaxy evolution framework, due to a central-limit-like tendency. The median black hole mass to total stellar mass ratio as a function of redshift with galaxies split into black hole mass bins shows that lower mass black holes have lower mass ratios than higher mass black holes. This will lead to a notable selection bias, since when observing the most massive black holes, the measured ratio will be higher than that of the entire population. This is generally expected for any sample selected by black hole mass or luminosity where the scatter in the relation is large [e.g.][Lauer2007]. Finally, we note an interesting effect of changing the parameter controlling the black hole efficiency for converting mass to energy.


For a higher efficiency, the median black hole--stellar mass ratio decreases at redshifts z greater than or approximately equal to 6, instead of remaining constant with redshift. We investigate the cause of this high-redshift decrease in the black hole--host relation by considering the Eddington limit. Increasing the efficiency from 0.06 to 0.2 decreases the Eddington limit. This results in many black holes having Eddington-limited growth at the highest redshifts (redshift z greater than or approximately equal to 6), which is not the case for the lower efficiency model. This causes black holes to grow slower than their host galaxies at high redshifts, resulting in a decreased black hole--stellar mass ratio. Observing the high-redshift black hole--stellar mass relation may therefore probe the Eddington limit and the efficiency of black holes in converting mass to energy.


We consider the cumulative fraction of black hole mass formed through each of the mechanisms in our model: black hole seeding, merger-driven quasar-mode accretion, instability-driven quasar-mode accretion, radio-mode accretion and black hole--black hole coalescence in galaxy mergers. The merger-driven growth mode becomes more important at low redshifts, at both low- and high-black hole masses. On average, instabilities grow the majority of mass in black holes at all redshifts, except for galaxies with black hole mass > 10^9 solar masses at redshift z is approximately equal to 0, whose black hole growth becomes dominated by galaxy mergers. Radio-mode growth slowly increases in significance with redshift, yet still has only contributed to a small proportion of the total black hole mass by redshift z=0, except at the highest masses. Note that we consider growth from disc instabilities that are triggered by earlier galaxy mergers as growth via the instability-driven mode.


We also consider the instantaneous growth fractions of black hole mass formed through each mechanism as a function of redshift. As discussed, the model produces unreliable black hole accretion rates at redshift z<1, and so we only consider these black hole growth rates at redshift z>1. The instability-driven growth mode is the dominant growth mechanism, on average, at all redshifts, regardless of black hole mass. The merger-driven quasar mode and black hole--black hole coalescence mode are sub-dominant at all redshifts. The radio-mode grows more mass at low redshift and in the most massive galaxies, with the percentage of total instantaneous black hole growth from this mode increasing from only 0.1 per cent at redshift z=5 to almost 5 per cent at redshift z is approximately equal to 1. Our finding that mergers are not the dominant mechanism for growing black holes is in agreement with a range of observations.


For example, [Koss2010] find that only 25 per cent of local, moderate luminosity X-ray AGN show signs of mergers, though the fraction is much higher for luminous AGN [Hong2015]. From redshift z from approximately 0.3 to 1.0, [Cisternas2010] find that the vast majority (>85 per cent) of X-ray selected AGN do not show signs of mergers, suggesting that the bulk of their black hole accretion has been triggered by some other mechanism. This is also consistent with the findings of [Georgakakis2009], [Villforth2018], [Schawinski2012], [Mechtley2016], [DelMoro2015] and [Marian2019] for AGN at various redshifts. Our result that disc instabilities cause the majority of black hole growth is also consistent with predictions from other simulations. In the GALFORM semi-analytic model, [Fanidakis2011] found that the growth of black holes is dominated by accretion due to disc instabilities.


Using an updated GALFORM model, [Griffin2019] found that accretion of hot gas dominates the growth of black holes at redshift z<2, with disc-instabilities dominant at higher redshifts. [Hirschmann2012] found that instability-driven black hole growth was required to reproduce AGN downsizing, and that while major mergers are the dominant trigger for luminous AGN, especially at high redshift, disc instabilities cause the majority of black hole growth in moderately luminous Seyfert galaxies at low redshift. [Menci2014] find that in their semi-analytic model disc instabilities can provide enough black hole accretion to reproduce the observed AGN luminosity functions up to redshift z is approximately equal to 4.5, but are not likely to be dominant for the highest luminosity AGN or at the highest redshifts. In contrast, [Shirakata2018] find that the primary trigger of AGN at redshift z less than or equal to 4 in their semi-analytic model is mergers.


The hydrodynamical simulation Horizon-AGN found that only ~35 per cent of black hole mass in local massive galaxies is directly attributable to merging, with the majority of black hole growth instead growing via secular processes [Martin2018]. The Magneticum Pathfinder Simulation also found that merger events are not the dominant fuelling mechanism for black holes in redshift z=0--2, with merger fractions less than 20 per cent, except for very luminous quasars at redshift z is approximately equal to 2 [Steinborn2018]. Finally, we comment on the effect of the efficiency parameters for merger-driven and instability-driven black hole growth in the model. We find the instability-driven efficiency from tuning the model, whereas the merger-driven efficiency is less constrained, with several values producing reasonable model results.


Having a merger growth efficiency that is twice, six times or even 18 times larger than the instability-driven growth efficiency may have an effect on the conclusions outlined above. We find, as expected, that models with larger merger efficiencies result in more merger-driven growth. For a merger efficiency twice the instability efficiency, the instability-driven mode still dominates at redshift z=2, while for a six times larger efficiency, the merger-driven mode begins to dominate at the highest black hole masses. For the model with an 18 times larger merger efficiency, the merger-driven mode contributes even more black hole growth, but is still not the dominant growth mode for black holes with mass between 10^6 and 10^9 solar masses. Thus, while the efficiency parameter for merger-driven growth has some effect on the relative distributions of the instability-driven and merger-driven growth modes, the instability-driven mode is still dominant for the majority of black holes, even if the merger growth efficiency is as much as 18 times larger than the secular growth efficiency.


A popular explanation for the black hole--host correlations is that major mergers drive the growth of both black holes and bulges [e.g.][Haehnelt2000, Croton2006b]. If this were the case, one would expect that black holes would only correlate with galaxy properties directly related to the merger process, such as bulge mass, and not, for example, total stellar mass. [Simmons2017] consider a sample of 101 disc-dominated AGN hosts from the SDSS, which they assume must have a major merger-free history since redshift z is approximately equal to 2. They found that these galaxies lie on the typical black hole mass--total stellar mass relation, but lie offset to the left of the black hole mass--bulge mass relation. This indicates that the substantial and ongoing black hole growth in these merger-free disc galaxies must be due to a process other than major mergers, and that major mergers cannot be the primary mechanism behind the black hole--host correlations.