Although the spin parameter of dark matter halos is well known to follow a log-normal distribution at fixed epoch, its quantitative redshift evolution - encompassing both the mean and the dispersion - remains only partially explored. Prior studies either lack the mass resolution required to establish reliable evolutionary trends or do not provide analytical relations that enable forward modelling. Using a suite of ΛCDM N-body simulations with controlled resolution across the redshift range 0 ≤ z ≤ 5, we characterise the evolution of the mean and dispersion of the Peebles (λ) and Bullock (λ′) definitions of spin. We find a mild but statistically robust linear evolution for ln λ and a non-monotonic trend with a turnover at z ≈ 1 – 2 for ln λ′, which we verify are unaffected by mass resolution of choice of halo definition. We provide closed-form fitting functions for these trends that allow modellers to draw physically motivated spin values at any redshift within our range of validity. This is a practical, redshift-dependent alternative to the common assumption of a constant spin distribution, and provides a useful input to semi-empirical and semi-analytic models of galaxy formation.

Post-red and post-asymptotic giant stars in binary systems with main sequence companions, have periods in the range ∼50-2000 days, eccentricities as high as 0.6 and are surrounded by a circumbinary disc. Their orbital separations are small enough that the system must have gone through Roche lobe overflow when the primary was a full blown giant; Roche lobe overflow between a giant and a more compact companion tend to lead to a common envelope inspiral, leaving a circular orbit with periods much shorter than observed in these systems. In this first work in a series we explore to what extent a high mass ratio, q ≡ M2/M1, can lead to wider orbital separations, by carrying out a series of 3D, hydrodynamical CE binary interaction simulations with the smoothed particle hydrodynamics code PHANTOM. The giant is a 0.88 M⊙, 90 R⊙, red giant branch star and the companions have a range of masses such that q = 0.68 − 1.5. While larger q values result in wider post-CE separations, the upper limit we predict is only ∼ 50 R⊙, smaller than the observed range. The pre-CE mass transfer phase is longer for larger companion masses and around q ≳ 1 the nature of the CE inspiral changes significantly, showing greater stability, as predicted by analytical theory. However, this phase is not converged with respect to simulation resolution and it is expected that a higher resolution would lead to even more stability and a longer pre-inspiral phase. Despite more material flowing through the L2 and L3 Lagrange points for higher q values, with the potential for the formation of a circumbinary disc structure in this way, we conclude that, for our parameters, circumbinary discs are more likely to form from fall back of leftover bound envelope. Fall-back times are short (a few × 100 years) and fall-back discs extend between 0.5 and 5 au (outside the binary orbit), at which point the discs are likely to spread farther on short timescales via viscous torques. These discs have characteristics in line with those observed.

Observations indicate that high-redshift galaxies undergo episodic star formation bursts, driving strong outflows that expel gas and suppress accretion. We investigate the consequences for metal and dust content of galaxies at $z\geq\!5$ using our semi-analytical model, Ashvini. We track gas-phase and stellar metallicities ($Z_{\textrm g}, Z_{\star}$) and dust mass ($M_{\textrm{d}}$) in dark matter haloes spanning $M_{\textrm{h}} = 10^6{-}10^{11}\,\text{M}_{\odot}$, comparing continuous and bursty star formation scenarios – which reflect underlying assumptions of instantaneous and delayed feedback – and we allow for metallicity-dependent feedback efficiency. Delayed feedback induces oscillations in $Z_{\textrm{g}}$ and $Z_{\star}$, with $Z_{\textrm{g}}$ declining sharply at low stellar and halo masses; the mass scale of this decline increases towards lower redshift. Reionisation introduces significant scatter in $Z_{\textrm{g}}$, producing an upturn followed by rapid decline. Metallicity-dependent feedback moderates this decline at $z=7{-}10$, flattening the $Z_{\textrm{g}}$–mass relation to $\simeq$$0.03$–$0.04\,\text{Z}_{\odot}$. Dust production tracks $Z_{\textrm{g}}$ but is sensitive to burst history, causing delayed enrichment. Our results show that burst-driven feedback decouples $Z_{\textrm{g}}$ and $Z_{star}$, imprints intrinsic scatter in mass–metallicity relations, and delays dust growth. These effects are strongest in low-mass halos ($M_{\textrm{h}}\sim 10^7\,\text{M}_{\odot}$), where shallow potentials amplify the impact of feedback. Our results are consistent with recent hydrodynamical and semi-analytical simulations and provide context for interpreting James Webb Space Telescope metallicity and dust measurements, highlighting the importance of episodic star formation in early galaxy chemical evolution.

Resonance lines encode rich information about astrophysical sources and their environments, yet fully analytic treatments of multi-line radiative transfer remain almost entirely unexplored. We present exact, closed-form solutions for steady-state resonant-line radiativeP transfer in “V-shaped” atomic networks, where a single ground state couples to multiple transitions. Starting from the full angle-dependent transfer equation, we generalise absorption and emission coefficients to an arbitrary number of lines, derive a modified Fokker–Planck expansion of the frequency-redistribution integral, and use a judicious change of variables to reduce the problem to a Helmholtz equation with point-like sources in frequency space. This transformation admits analytic solutions for arbitrary sets of lines with fixed frequency offsets and strengths in both slab and spherical geometries. We implement V-shaped line networks in the colt Monte Carlo radiative transfer code and find excellent agreement with the analytic predictions across a wide range of line separations, optical depths, and damping parameters, establishing our solutions as stringent validation benchmarks. For concrete applications related to the Lyman-alpha (Ly$\alpha$) transition of neutral hydrogen, we examine how fine-structure splitting and deuterium injection modify the emergent spectra, internal radiation field, and radiative force multiplier. We show that these effects leave previous conclusions about Ly$\alpha$ feedback in the early universe essentially unchanged. Even when direct observational diagnostics are subtle, our framework provides novel analytic and numerical insights into coupled resonance-line transport and facilitates progress in general modelling of multi-line radiative transfer in diverse astrophysical settings.

Asymptotic giant branch (AGB) stars are important to chemical evolution at metallicity $Z \sim 0.0001$ ($\text{[Fe/H]} \approx -2.2$) as they contribute significantly to the production of nitrogen, lead, and dust in the early Universe. The contribution of AGB stars to the chemical evolution of the Universe is often quantified using the chemical yields from single AGB stars. Binary evolution challenges our understanding of chemical evolution as binary phenomena such as mergers and mass transfer episodes can significantly alter the stellar evolution pathways and yields. In this work, we use binary population synthesis code binary_c to model populations of low and intermediate-mass ($\sim 0.7$–$7\,\mathrm{M}_{\odot}$) stars at metallicity $Z = 0.0001$. Our binary star populations predict $\sim 37\%$ fewer thermally pulsing AGB stars than our single star populations, leading to a $\sim 40\%$ decrease in the amount of ejected C and a $\sim 35$–40% reduction in elements synthesised through the slow neutron capture process. The uncertainty introduced by the mass-loss from stellar winds on the AGB makes the impact of binary evolution on the total amount of ejected N uncertain. The total N yield ejected by our binary star populations ranges from a 17% to a 36% decrease compared to our single star populations. However, our binary populations overproduce N by over an order of magnitude during the period $300\text{--}700\, {\rm Myr}$ after formation.

The potential of a galaxy to host habitable planets is one of the most important questions in astrobiology. It is tightly connected to the evolution of galaxy-scale properties and the underlying cosmological processes. Using the improved cosmological simulation IllustrisTNG, we revisit the claim that a population of small, metal-rich, star-forming galaxies (‘Cloudlet’), forms a local peak on the mass-metallicity relation, reflecting an enhanced galactic habitability potential. We refine the earlier analysis by applying updated filtering criteria to identify a more refined sample, further selecting objects based on their history. This process resulted in a confirmed sample of 97 dwarf galaxies, alongside 519 additional structures of uncertain origin, potentially comprising both numerical artefacts and unrecognised physical systems. Under these stricter conditions, the proposed bimodality in galactic habitability is strongly diminished. However, the astrobiological potential of metal-rich dwarfs, most of which are compact remnants of more massive galaxies that underwent tidal stripping, is a thrilling area of exploration. Although dense stellar environments are traditionally seen as inhospitable, recent studies highlight the role of dynamic environments in enhancing the distribution of biological material. Furthermore, the potential habitability of tidal structures formed in the aftermath of galactic interactions is a fascinating possibility. Our findings suggest that non-traditional structures support conditions favourable for life, opening up exciting new avenues for astrobiological research. This research underscores the need for a holistic approach to studying habitability that moves beyond planetary and stellar-focused frameworks to incorporate the broader galactic environment. Understanding the interactions between galaxies, their evolution, and the influence of their surroundings is essential to developing a more comprehensive model of how and where life might emerge and persist across the Universe.

Before a binary system enters into a common envelope (CE) phase, accretion from the primary star onto the companion star through Roche Lobe overflow (RLOF) will lead to the formation of an accretion disk, which may generate jets. Accretion before and during the CE may alter the outcome of the interaction. Previous studies have considered different aspects of this physical mechanism. Here we study the properties of an accretion disk formed via 3D hydrodynamic simulations of the RLOF mass transfer between a 7 M$_{\odot}$, red supergiant star and a 1.4 M$_{\odot}$, neutron star companion. We simulate only the volume around the companion for improved resolution. We use a 1D implicit mesa simulation of the evolution of the system during 30 000 yr between the on-set of the RLOF and the CE to guide the binary parameters and the mass-transfer rate, while we simulate only 21 yr of the last part of the RLOF in 3D using an ideal gas quasi-isothermal equation of state. We expect that a pre-CE disk under these parameters will have a mass of $\sim 5\times 10^{-3}$ M$_{\odot}$ and a radius of $\sim40\ R_\odot$ with a scale height of $\sim 5\ R$$_{\odot}$. The temperature profile of the disk is shallower than that predicted by the formalism of Shakura and Sunyaev, but more reasonable cooling physics would need to be included. We stress test these results with respect to a number of physical and numerical parameters, as well as simulation choices, and we expect them to be reasonable within a factor of a few for the mass and 15% for the radius. We also contextualise our results within those presented in the literature, in particular with respect to the dimensionality of simulations and the adiabatic index. We discuss the measured accretion rate in the context of the Shakura and Sunyaev formalism and debate the viscous mechanisms at play, finishing with a list of prospects for future work.

Post-asymptotic giant branch (post-AGB) binaries are surrounded by dusty circumbinary disks and exhibit unexpected orbital properties resulting from poorly understood binary interaction processes. Re-accreted gas from the circumbinary disk alters the photospheric chemistry of the post-AGB star, producing a characteristic underabundance of refractory elements that correlates with condensation temperature – a phenomenon known as chemical depletion. This work investigates how re-accretion from a disk drives chemical depletion, and the impact accreted matter has on post-AGB evolution. We used the MESA code to evolve 0.55 and 0.60 M$_{\odot}$ post-AGB stars with the accretion of refractory element-depleted gas from a circumbinary disk. Our study adopts observationally-constrained initial accretion rates and disk masses to reproduce the chemical depletion patterns of six well-studied post-AGB binary stars: EP Lyr, HP Lyr, IRAS 17038-4815, IRAS 09144-4933, HD 131356, and SX Cen. We find high accretion rates ($\gt 10^{-7}$ M$_{\odot}\,\mathrm{yr}^{-1}$) and large disk masses ($\gtrsim10^{-2}$ M$_{\odot}$) necessary to reproduce observed depletion, particularly in higher-mass, hotter post-AGB stars ($T_{\textrm{eff}}\gtrsim$ 6 000 K). A slower evolution (lower core mass) is required to reproduce cooler ($T_{\textrm{eff}}\lesssim$ 5 000 K) depleted post-AGB stars. Rapid accretion significantly impacts post-AGB evolution, stalling stars at cooler effective temperatures and extending post-AGB lifetimes by factors of around 3 to 10. Despite this, extended post-AGB timescales remain within or below the planetary nebula visibility timescale, suggesting accretion cannot account for the observed lack of ionised PNe in post-AGB binaries. Our findings constrain accretion-flow parameters and advance our understanding of disk-binary interactions in post-AGB systems.

Asymptotic giant branch (AGB) stars play a significant role in our understanding of the origin of the elements. They contribute to the abundances of C, N, and approximately 50% of the abundances of the elements heavier than iron. An aspect often neglected in studies of AGB stars is the impact of a stellar companion on AGB stellar evolution and nucleosynthesis. In this study, we update the stellar abundances of AGB stars in the binary population synthesis code binary_c and calibrate our treatment of the third dredge-up using observations of Galactic carbon stars. We model stellar populations of low- to intermediate-mass stars at solar-metallicity and examine the stellar wind contributions to C, N, O, Sr, Ba, and Pb yields at binary fractions between 0 and 1. For a stellar population with a binary fraction of 0.7, we find $\sim$20–25% less C and s-process elements ejected than from a population composed of only single stars, and we find little change in the N and O yields. We also compare our models with observed abundances from Ba stars and find our models can reproduce most Ba star abundances, but our population estimates a higher frequency of Ba stars with a surface [Ce/Y] > $+0.2\,$dex. Our models also predict the rare existence of Ba stars with masses $ \gt 10\,\textrm{M}_{\odot}$.

We introduce adaptive particle refinement for compressible smoothed particle hydrodynamics (SPH). SPH calculations have the natural advantage that resolution follows mass, but this is not always optimal. Our implementation allows the user to specify local regions of the simulation that can be more highly resolved. We test our implementation on practical applications including a circumbinary disc, a planet embedded in a disc, and a flyby. By comparing with equivalent globally high-resolution calculations, we show that our method is accurate and fast, with errors in the mass accreted onto sinks of less than 9% and speed ups of 1.07–6.62$\times$ for the examples shown. Our method is adaptable and easily extendable, for example, with multiple refinement regions or derefinement.

Observations of the intracluster medium (ICM) in the outskirts of galaxy clusters reveal shocks associated with gas accretion from the cosmic web. Previous work based on non-radiative cosmological hydrodynamical simulations have defined the shock radius, $r_{\text{shock}}$, using the ICM entropy, $K \propto T/{n_\mathrm{e}}^{2/3}$, where T and $n_{\text{e}}$ are the ICM temperature and electron density, respectively; the $r_{\text{shock}}$ is identified with either the radius at which K is a maximum or at which its logarithmic slope is a minimum. We investigate the relationship between $r_{\text{shock}}$, which is driven by gravitational hydrodynamics and shocks, and the splashback radius, $r_{\text{splash}}$, which is driven by the gravitational dynamics of cluster stars and dark matter and is measured from their mass profile. Using 324 clusters from The Three Hundred project of cosmological galaxy formation simulations, we quantify statistically how $r_{\text{shock}}$ relates to $r_{\text{splash}}$. Depending on our definition, we find that the median $r_{\text{shock}} \simeq 1.38 r_{\text{splash}} (2.58 R_{200})$ when K reaches its maximum and $r_{\text{shock}} \simeq 1.91 r_{\text{splash}} (3.54 R_{200})$ when its logarithmic slope is a minimum; the best-fit linear relation increases as $r_{\text{shock}} \propto 0.65 r_{\text{splash}}$. We find that $r_{\text{shock}}/R_{200}$ and $r_{\text{splash}}/R_{200}$ anti-correlate with virial mass, $M_{200}$, and recent mass accretion history, and $r_{\text{shock}}/r_{\text{splash}}$ tends to be larger for clusters with higher recent accretion rates. We discuss prospects for measuring $r_{\text{shock}}$ observationally and how the relationship between $r_{\text{shock}}$ and $r_{\text{splash}}$ can be used to improve constraints from radio, X-ray, and thermal Sunyaev-Zeldovich surveys that target the interface between the cosmic web and clusters.

Fast radio bursts (FRBs) are millisecond-duration radio waves from the Universe. Even though more than 50 physical models have been proposed, the origin and physical mechanism of FRB emissions are still unknown. The classification of FRBs is one of the primary approaches to understanding their mechanisms, but previous studies classified conventionally using only a few observational parameters, such as fluence and duration, which might be incomplete. To overcome this problem, we use an unsupervised machine-learning model, the Uniform Manifold Approximation and Projection to handle seven parameters simultaneously, including amplitude, linear temporal drift, time duration, central frequency, bandwidth, scaled energy, and fluence. We test the method for homogeneous 977 sub-bursts of FRB 20121102A detected by the Arecibo telescope. Our machine-learning analysis identified five distinct clusters, suggesting the possible existence of multiple different physical mechanisms responsible for the observed FRBs from the FRB 20121102A source. The geometry of the emission region and the propagation effect of FRB signals could also make such distinct clusters. This research will be a benchmark for future FRB classifications when dedicated radio telescopes such as the square kilometer array or Bustling Universe Radio Survey Telescope in Taiwan discover more FRBs than before.

The baryon mass content (i.e. stellar and gas mass) of dark matter halos in the early Universe depends on both global factors – for example, ionising ultraviolet (UV) radiation background – and local factors – for example, star formation efficiency and assembly history. We use a lightweight semi-analytical model to investigate how both local and global factors impact the halo baryon mass content at redshifts of $z\geq 5$. Our model incorporates a time delay between when stars form and when they produce feedback of $0\leq t^d/\mathrm{Myr} \leq 30$, which can drive bursts of star formation, and a mass and redshift-dependent UV background, which captures the influence of cosmological reionisation on gas accretion onto halos. We use statistically representative halo assembly histories and assume that the cosmological gas accretion rate is proportional to the halo mass accretion rate. Delayed ($t^d$>0) feedback leads to oscillations in gas mass with cosmic time, behaviour that cannot be captured with instantaneous feedback ($t^d$=0). Highly efficient star formation drives stronger oscillations, while strong feedback impacts when oscillations occur; in contrast, inefficient star formation and weak feedback produce similar long-term behaviour to that observed in instantaneous feedback models. If the delayed feedback timescale is too long, a halo retains its gas reservoir but the feedback suppresses star formation. Our model predicts that lower mass systems (halo masses $m_\mathrm{h} \leq 10^7 \mathrm{M}_\odot$) at $z \leq 10$ should be strongly gas deficient ($m_\mathrm{g}\rightarrow 0$), whereas higher mass systems retain their gas reservoirs because they are sufficiently massive to continue accreting gas through cosmological reionisation. Interestingly, in higher mass halos, the median $m_\star/(m_\star+m_\mathrm{g}) \simeq 0.01-0.05$, but is a factor of 3–5 smaller when feedback is delayed. Our model does not include seed supermassive black hole feedback, which is necessary to explain massive quenched galaxies in the early Universe.

This study builds upon our prior work to further explore and unravel the effects of saturated thermal conduction within a viscous resistive MHD framework on the intricate transport mechanisms of angular momentum and energy in disc-jet systems. We conducted a series of 2.5-dimensional non-relativistic time-dependent numerical simulations using the PLUTO code. Employing a saturation parameter spanning [0.002-0.01], our results are consistent with previous investigations that omitted consideration of thermal conduction, affirming the established understanding that kinetic torque plays a predominant role in governing the total accretion angular momentum, surpassing the magnetic contribution within the disc. At the initial time steps of our calculations, we find that thermal conduction enhances this kinetic contribution, while concurrently diminishing the effect of magnetic contribution. In contrast to the prevailing influence of kinetic torque within the disc, we also assert the magnetic torque as the primary contributor to the total ejection angular momentum. We further unveil that doubling the saturation parameter leads to bolstering of approximately $23.7\%$ in the integral dominance of magnetic torque compared to kinetic torque within the jet. Our findings reveal that doubling the effect of thermal conduction improves the integral total accretion power by approximately 2%, thereby slightly amplifying the energy content within the system and increasing overall energy output. We underscore that as the local energy dissipation within the disc intensifies, the significance of the enthalpy accretion flux increases at the expense of the jet power. We reveal that increasing the saturation parameter mitigates enthalpy accumulation within the disc, and further restricts the jet’s energy extraction from the disc. This limitation is determined in our analysis through the decrease in the integral ratio between the bipolar jet and liberated power of approximately $13.8\%$, for twice the strength of the saturation parameter. We identify the Poynting flux as the primary contributor to total jet power, with thermal conduction exerting minimal influence on magnetic contributions. Additionally, we emphasise the integration of jet enthalpy as another significant factor in determining overall jet power, highlighting a distinct correlation between the rise in saturation parameter and heightened enthalpy contribution. Moreover, we observe the promotion of Poynting flux over kinetic flux at advanced time steps of our simulations, a trend supported by the presence of thermal conduction, which demonstrates an integral increase of approximately $11.2\%$ when considering a doubling of the saturation parameter.

To understand the formation and evolution of the Universe, it is crucial to understand how and when the first stars formed. The latest observational data reveal unprecedented information about the chemical enrichment of the early Universe, which seems to behave differently from the local Universe. The first stars, being very massive, enrich their metal-poor environment in an uncertain way. In order to predict the abundances of the first galaxies, we include nucleosynthesis yields from Population III stars up to 300M⊙, including faint supernovae, Wolf-Rayet (WR) stars and Pair-Instability Supernovae (PISN) into our state-of-the-art hydrodynamical cosmological simulations. Our code (based on Gadget-3) also includes the latest nucleosynthesis yields from population II stars (from Kobayashi et al. 2020) for all stellar mass ranges. We predict the chemical abundance evolution of galaxies for different elements from the early Universe to the local Universe. We first test the modelling of stellar feedback by comparing the observed evolution of mass–metallicity relations (MZR) and metallicity gradients of the interstellar medium. We then compare our model including Population III stars with observational data from the James Web Space Telescope (JWST). For elemental abundances, we find that the N/O abundance gives a systematically higher value, comparable to the observational data of very high-redshift galaxies such as GN-z11.

Ram-pressure stripping (RPS) is a process known to remove gas from satellite galaxies. Recent observational studies have found an increased ratio of active galactic nuclei (AGN) among the population of RPS galaxies compared to regular galaxies in the field. To test whether ram pressure (RP) can trigger an AGN, we perform a suite of hydrodynamical wind-tunnel simulations of a massive (Mstar = 1011 Mȯ) galaxy, with inclusion of star formation, stellar feedback and high resolution up to 39 pc. We find that RP increases the inflow of gas to the galaxy centre, which in turn can result in the enhanced BH accretion, as measured by the Bondi-Hoyle model. We also estimate pressure of outflows from our accretion rates and show that AGN feedback would play an important role on the early stages of stripping, while RP itself is not so strong.

We use the rotation curve from Gaia data release (DR) 3 to estimate the mass of the Milky Way. We consider an Einasto density profile to model the dark matter component. We extrapolate and obtain a dynamical mass at 112 kpc. This lower-mass Milky Way is consistent with the significant declining rotation curve, and can provide new insights into our Galaxy and halo inhabitants.

Recent studies of Galactic evolution revealed that the dynamics of the stellar component might be one of the key factors when considering galactic habitability. We run an N-body simulation model of the Milky Way, which we evolve for 10 Gyr, to study the secular evolution of stellar orbits and the resulting galactic habitability related properties, i.e., the density of the stellar component and close stellar encounters. The results indicate that radial migrations are not negligible, even in a simple axisymmetric model with mild levels of dynamical heating, and that the net outward diffusion of the stellar component can populate galactic outskirts with habitable systems. Habitable environment is also likely even at sub-Solar galactocentric radii, because the rate of close encounters should not significantly degrade habitability. Stars that evolve from non-circular to stable nearly circular orbits typically migrate outwards, settling down in a broad Solar neighbourhood. The region between $R \approx 3$ kpc and $R \approx 12$ kpc represents the zone of radial mixing, which can blur the boundaries of the Galactic Habitable Zone (GHZ), as it has been conventionally understood. The present-day stable population of the stars in the Solar neighbourhood originates from this radial mixing zone, with most of the stars coming from the inner regions. The Solar system can be considered as a typical Milky Way habitable system because it migrated outwards from the metal-rich inner regions of the Disk and has a circular orbit in the present epoch. We conclude that the boundaries of the GHZ cannot be sharply confined for a given epoch because of the mixing caused by the stellar migrations and secular evolution of stellar orbits.

In this work, we present a methodology and a corresponding code-base for constructing mock integral field spectrograph (IFS) observations of simulated galaxies in a consistent and reproducible way. Such methods are necessary to improve the collaboration and comparison of observation and theory results, and accelerate our understanding of how the kinematics of galaxies evolve over time. This code, SimSpin, is an open-source package written in R, but also with an API interface such that the code can be interacted with in any coding language. Documentation and individual examples can be found at the open-source website connected to the online repository. SimSpin is already being utilised by international IFS collaborations, including SAMI and MAGPI, for generating comparable data sets from a diverse suite of cosmological hydrodynamical simulations.

Asymmetric emission of gravitational waves during mergers of black holes (BHs) produces a recoil kick, which can set a newly formed BH on a bound orbit around the centre of its host galaxy, or even completely eject it. To study this population of recoiling BHs we extract properties of galaxies with merging BHs from Illustris TNG300 simulation and then employ both analytical and numerical techniques to model unresolved process of BH recoil. This comparative analysis between analytical and numerical models shows that, on cosmological scales, numerically modelled recoiling BHs have a higher escape probability and predict a greater number of offset active galactic nuclei (AGN). BH escaped probability $>$40% is expected in 25$\%$ of merger remnants in numerical models, compared to 8$\%$ in analytical models. At the same time, the predicted number of offset AGN at separations ${>}5$ kpc changes from 58$\%$ for numerical models to 3$\%$ for analytical models. Since BH ejections in major merger remnants occur in non-virialised systems, static analytical models cannot provide an accurate description. Thus we argue that numerical models should be used to estimate the expected number density of escaped BHs and offset AGN.