arXiv subjects

Michael Joyce

Publications and source records attributed to Michael Joyce.

At least 19 recordsLinked to original sources

arXiv2023

Convergence of halo statistics: code comparison between Rockstar and CompaSO using scale-free simulations

In this study, we perform a halo-finder code comparison between Rockstar and CompaSO. Based on our previous analysis aiming at quantifying resolution of $N$-body simulations by exploiting large (up to $N=4096^3$) simulations of scale-free cosmologies run using Abacus, we focus on convergence of the HMF, 2PCF and mean radial pairwise velocities of halo centres selected with the aforementioned two algorithms. We establish convergence, for both Rockstar and CompaSO, of mass functions at the $1\%$ precision level and of the mean pairwise velocities (and also 2PCF) at the $2\%$ level. At small scales and small masses, we find that Rockstar exhibits greater self-similarity, and we also highlight the role played by the merger-tree post-processing of CompaSO halos on their convergence. Finally, we give resolution limits expressed as a minimum particle number per halo in a form that can be directly extrapolated to LCDM.

astro-ph.CO↗

arXiv2023

Testing growth rate dependence in cosmological perturbation theory using scale-free models

We generalize previously derived analytic results for the one-loop power spectrum (PS) in scale-free models (with linear PS $P(k) \propto k^n$) to a broader class of such models in which part of the matterlike component driving the Einstein de Sitter expansion does not cluster. These models can be conveniently parametrized by $\alpha$, the constant logarithmic linear growth rate of fluctuations (with $\alpha=1$ in the usual case). For $-3< n<-1$, where the one-loop PS is both infrared and ultraviolet convergent and thus explicitly self-similar, it is characterized conveniently by a single numerical coefficient $c(n, \alpha)$. We compare the analytical predictions for $c(n=-2, \alpha)$ with results from a suite of $N$-body simulations with $\alpha \in [0.25, 1]$ performed with an appropriately modified version of the GADGET code. Although the simulations are of small ($256^3$) boxes, the constraint of self-similarity allows the identification of the converged PS at a level of accuracy sufficient to test the analytical predictions for the $\alpha$ dependence of the evolved PS. Good agreement for the predicted dependence on $\alpha$ of the PS is found. To treat the UV sensitivity of results which grows as one approaches $n =-1$, we derive exact results incorporating a regularization $k_c$ and obtain expressions for $c(n, \alpha, k_c/k)$. Assuming that this regularization is compatible with self-similarity allows us to infer a predicted functional form of the PS equivalent to that derived in effective field theory (EFT). The coefficient of the leading EFT correction at one loop has a strong dependence on $\alpha$, with a change in sign at $\alpha \approx 0.16$, providing a potentially stringent test of EFT.

astro-ph.CO↗

arXiv2022

Constraining accuracy of the pairwise velocities in $N$-body simulations using scale-free models

We present a continuation of an analysis that aims to quantify resolution of $N$-body simulations by exploiting large (up to $N=4096^3$) simulations of scale-free cosmologies run using Abacus. Here we focus on radial pairwise velocities of the matter field, both by direct estimation and through the cumulative-2PCF (using the pair conservation equation). We find that convergence at the $1\%$ level of the mean relative pairwise velocity can be demonstrated over a range of scales, evolving from a few times the grid spacing at early times to slightly below this scale at late times. We show the analysis of two different box sizes as well as from averaging results from the smaller boxes, and compare the power of the two aforementioned estimators in constraining accuracy at each scale. Down to scales of order of the smoothing parameter, convergence is obtained at $\sim5\%$ precision, and shows a behaviour indicating asymptotic stable clustering. We also infer for LCDM simulations conservative estimates on the evolution of the lower cut-off to resolution (at $1\%$ and $5\%$ precision) as a function of redshift.

astro-ph.CO↗

arXiv2022

Cosmological perturbation theory using generalized Einstein de Sitter cosmologies

The separable analytical solution in standard perturbation theory for an Einstein de Sitter (EdS) universe can be generalized to the wider class of such cosmologies (``generalized EdS'', or gEdS) in which a fraction of the pressure-less fluid does not cluster. We derive the corresponding kernels in both Eulerian perturbation theory (EPT) and Lagrangian perturbation theory, generalizing the canonical EdS expressions to a one-parameter family where the parameter can be taken to be the exponent $\alpha$ of the growing mode linear amplification $D(a) \propto a^{\alpha}$. For the power spectrum (PS) at one loop in EPT, the contribution additional to standard EdS is given, for each of the `13' and `22' terms, as a function of two infra-red safe integrals. In the second part of the paper we show that the calculation of cosmology-dependent corrections in perturbation theory in standard (e.g. LCDM-like) models can be simplified, and their magnitude and parameter dependence better understood, by relating them to our analytic results for gEdS models. At second order the time dependent kernels are equivalent to the analytic kernels of the gEdS model with $\alpha$ replaced by a single redshift dependent effective growth rate $\alpha_2(z)$. At third order the time evolution can be conveniently parametrized in terms of two additional such effective growth rates. For the PS calculated at one loop order, the correction to the PS relative to the EdS limit can be expressed in terms of just $\alpha_2(z)$, one additional effective growth rate function and the four infra-red safe integrals of the gEdS limit. This is much simplified compared to expressions in the literature that use six or eight red-shift dependent functions and are not explicitly infra-red safe. Using the analytic gEdS expression for the PS with $\alpha=\alpha_2(z)$ gives a good approximation (to $\sim 25 \%$) for the exact result.

astro-ph.CO↗

arXiv2021

Accuracy of power spectra in dissipationless cosmological simulations

We exploit a suite of large \emph{N}-body simulations (up to N=$4096^3$) performed with \Abacus, of scale-free models with a range of spectral indices $n$, to better understand and quantify convergence of the matter power spectrum. Using self-similarity to identify converged regions, we show that the maximal wavenumber resolved at a given level of accuracy increases monotonically as a function of time. At the 1\% level it starts at early times from a fraction of $k_\Lambda$, the Nyquist wavenumber of the initial grid, and reaches at most, if the force softening is sufficiently small, $\sim 2-3 k_\Lambda$ at the very latest times we evolve to. At the $5\%$ level, accuracy extends up to wavenumbers of order $5k_\Lambda$ at late times. Expressed as a suitable function of the scale-factor, accuracy shows a very simple $n$-dependence, allowing a extrapolation to place conservative bounds on the accuracy of \emph{N}-body simulations of non-scale free models like LCDM. We note that deviations due to discretization in the converged range are not well modelled by shot noise, and subtracting it in fact degrades accuracy. Quantitatively our findings are broadly in line with the conservative assumptions about resolution adopted by recent studies using large cosmological simulations (e.g. Euclid Flagship) aiming to constrain the mildly non-linear regime. On the other hand, we remark that conclusions about small scale clustering (e.g. concerning the validity of stable clustering) obtained using PS data at wavenumbers larger than a few $k_\Lambda$ may need revision in light of our convergence analysis.

astro-ph.CO↗

arXiv2021

Gravitational collapse from cold uniform asymmetric initial conditions

Using controlled numerical N-body experiments, we show how, in the collapse dynamics of an initially cold and uniform distribution of particles with a generic asymmetric shape, finite $N$ fluctuations and perturbations induced by the anisotropic gravitational field compete to determine the physical properties of the asymptotic quasi-stationary state. When finite $N$ fluctuations dominate the dynamics, the particle energy distribution changes greatly and the final density profile {decays outside its core} as $r^{-4}$ with an $N$-dependent amplitude. On the other hand, in the limit where the anisotropic perturbations dominate, the collapse is softer and the density profile shows a decay as $r^{-3}$, as is typical of halos in cosmological simulations. However, even in this limit, convergence with $N$ of the macroscopic properties of the virialized system, such as the particle energy distributions, the bound mass, and the density profile, is very slow and not clearly established, including for our largest simulations (with $N \sim 10^6$). Our results illustrate the challenges of accurately simulating the first collapsing structures in standard-type cosmological models

astro-ph.CO↗

arXiv2021

Good and Proper: Self-similarity of N-body Simulations with Proper Force Softening

Analysis of self-similarity in scale-free $N$-body simulations reveals the spatial and temporal scales for which statistics measured in cosmological simulations are converged to the physical continuum limit. We examine how the range of scales in which the two-point correlation function is converged depends on the force softening length and whether it is held constant in comoving or proper coordinates. We find that a proper softening that reaches roughly 1/30th of the inter-particle spacing by the end of the simulation resolves the same spatial and temporal scales as a comoving softening of the same length while using a third fewer time steps, for a range of scale factors typical to $\Lambda$CDM simulations. We additionally infer an inherent resolution limit, set by the particle mass and scaling as $a^{-1/2}$, beyond which reducing the softening does not improve the resolution. We postulate a mapping of these results with spectral index $n=-2$ to $\Lambda$CDM simulations.

astro-ph.CO↗

arXiv2020

Quantifying resolution in cosmological N-body simulations using self-similarity

We demonstrate that testing for self-similarity in scale-free simulations provides an excellent tool to quantify the resolution at small scales of cosmological N-body simulations. Analysing two-point correlation functions measured in simulations using ABACUS, we show how observed deviations from self-similarity reveal the range of time and distance scales in which convergence is obtained. While the well-converged scales show accuracy below 1 percent, our results show that, with a small force softening length, the spatial resolution is essentially determined by the mass resolution. At later times the lower cut-off scale on convergence evolves in comoving units as $a^{-1/2}$ ($a$ being the scale factor), consistent with a hypothesis that it is set by two-body collisionality. A corollary of our results is that N-body simulations, particularly at high red-shift, contain a significant spatial range in which clustering appears converged with respect to the time-stepping and force softening but has not actually converged to the physical continuum result. The method developed can be applied to determine the resolution of any clustering statistic and extended to infer resolution limits for non-scale-free simulations.

astro-ph.CO↗

arXiv2020

Schubert polynomial analogues for degenerate involutions

We survey the recent study of involution Schubert polynomials and a modest generalization that we call degenerate involution Schubert polynomials. We cite several conditions when (degenerate) involution Schubert polynomials have simple factorization formulae. Such polynomials can be computed by traversing through chains in certain weak order posets, and we provide explicit descriptions of such chains in weak order for involutions and degenerate involutions. As an application, we give several examples of how certain multiplicity-free sums of Schubert polynomials factor completely into very simple linear factors.

math.CO↗

arXiv2019

An accurate physical model for halo concentrations

The relation between halo mass, M, and concentration, c, is a critical component in our understanding of the structure of dark matter halos. While numerous models for this relation have been proposed, almost none of them attempt to derive the evolution of the relation analytically. We build on previous efforts to model the c-M relation as a function of physical parameters such as the peak height, $ν$, and the effective power spectrum slope, $n_{\rm eff}$, which capture the dependence of $c$ on halo mass, redshift, and cosmology. We present three major improvements over previous models. First, we derive an analytical expression for the c-M relation that is valid under the assumption of pseudo-evolution, i.e., assuming that the density profiles of halos are static in physical coordinates while the definition of their boundary evolves. We find that this ansatz is highly successful in describing the evolution of the low-mass end of the c-M relation. Second, we employ a new physical variable, the effective exponent of linear growth, $α_{\rm eff}$, to parameterize deviations from an Einstein-de Sitter expansion history. Third, we combine an updated definition of $n_{\rm eff}$ with the additional dependence on $α_{\rm eff}$ and propose a phenomenological extension of our analytical framework to include all halo masses. This semianalytical model matches simulated concentrations in both scale-free models and LambdaCDM to 5% accuracy with very few exceptions and differs significantly from all previously proposed models. We present a publicly available code to compute the predictions of our model in the python toolkit Colossus, including updated parameters for the model of Diemer and Kravtsov.

astro-ph.CO↗

arXiv2019

Long-lived transient structure in collisionless self-gravitating systems

The evolution of self-gravitating systems, and long-range interacting systems more generally, from initial configurations far from dynamical equilibrium is often described as a simple two phase process: a first phase of violent relaxation bringing it to a quasi-stationary state in a few dynamical times, followed by a slow adiabatic evolution driven by collisional processes. In this context the complex spatial structure evident, e.g., in spiral galaxies is understood either in terms of instabilities of quasi-stationary states, or a result of dissipative non-gravitational interactions. We illustrate here, using numerical simulations, that purely self-gravitating systems evolving from quite simple initial configurations can in fact give rise easily to structures of this kind of which the lifetime can be large compared to the dynamical characteristic time, but short compared to the collisional relaxation time scale. More specifically, for a broad range of non-spherical and non-uniform rotating initial conditions, gravitational relaxation gives rise quite generically to long-lived non-stationary structures of a rich variety, characterized by spiral-like arms, bars and even ring-like structures in special cases. These structures are a feature of the intrinsically out-of-equilibrium nature of the system's collapse, associated with a part of the system's mass while the bulk is well virialized. They are characterized by predominantly radial motions in their outermost parts, but also incorporate an extended flattened region which rotates coherently about a well virialized core of triaxial shape with an approximately isotropic velocity dispersion. We discuss the possible relevance of these simple toy models to the observed structure of real galaxies emphasizing the difference between dissipative and dissipationless disc formation.

astro-ph.GA↗

arXiv2017

Wonderful Symmetric Varieties and Schubert Polynomials

Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of $GL_n$ on the flag variety $GL_n/B$. Putting this together with a slight extension of the results of Can-Joyce-Wyser, we arrive at a family of polynomial identities which show that certain explicit sums of Schubert polynomials factor as products of linear forms.

math.AG↗

arXiv2017

Transient spiral arms from far out of equilibrium gravitational evolution

We describe how a simple class of out of equilibrium, rotating and asymmetrical mass distributions evolve under their self-gravity to produce a quasi-planar spiral structure surrounding a virialized core, qualitatively resembling a spiral galaxy. The spiral structure is transient, but can survive tens of dynamical times, and further reproduces qualitatively noted features of spiral galaxies as the predominance of trailing two-armed spirals and large pitch angles. As our models are highly idealized, a detailed comparison with observations is not appropriate, but generic features of the velocity distributions can be identified to be potential observational signatures of such a mechanism. Indeed, the mechanism leads generically to a characteristic transition from predominantly rotational motion, in a region outside the core, to radial ballistic motion in the outermost parts. Such radial motions are excluded in our Galaxy up to 15 kpc, but could be detected at larger scales in the future by GAIA. We explore the apparent motions seen by external observers of the velocity distributions of our toy galaxies, and find that it is difficult to distinguish them from those of a rotating disc with sub-dominant radial motions at levels typically inferred from observations. These simple models illustrate the possibility that the observed apparent motions of spiral galaxies might be explained by non-trivial non-stationary mass and velocity distributions without invoking a dark matter halo or modification of Newtonian gravity. In this scenario the observed phenomenological relation between the centripetal and gravitational acceleration of the visible baryonic mass could have a simple explanation.

astro-ph.GA↗

arXiv2017

Combinatorial Models for the Variety of Complete Quadrics

We develop several combinatorial models that are useful in the study of the $SL_n$-variety $\mathcal{X}$ of complete quadrics. Barred permutations parameterize the fixed points of the action of a maximal torus $T$ of $SL_n$, while $μ$-involutions parameterize the orbits of a Borel subgroup of $SL_n$. Using these combinatorial objects, we characterize the $T$-stable curves and surfaces on $\mathcal{X}$, compute the $T$-equivariant $K$-theory of $\mathcal{X}$, and describe a Białynicki-Birula cell decomposition for $\mathcal{X}$. Furthermore, we give a computational characterization of the Bruhat order on Borel orbits in $\mathcal{X}$.

math.AG↗

arXiv2017

Stable clustering and the resolution of dissipationless cosmological N-body simulations

The determination of the resolution of cosmological N-body simulations, i.e., the range of scales in which quantities measured in them represent accurately the continuum limit, is an important open question. We address it here using scale-free models, for which self-similarity provides a powerful tool to control resolution. Such models also provide a robust testing ground for the so-called stable clustering approximation, which gives simple predictions for them. Studying large N-body simulations of such models with different force smoothing, we find that these two issues are in fact very closely related: our conclusion is that the accuracy of two point statistics in the non-linear regime starts to degrade strongly around the scale at which their behaviour deviates from that predicted by the stable clustering hypothesis. Physically the association of the two scales is in fact simple to understand: stable clustering fails to be a good approximation when there are strong interactions of structures (in particular merging) and it is precisely such non-linear processes which are sensitive to fluctuations at the smaller scales affected by discretisation. Resolution may be further degraded if the short distance gravitational smoothing scale is larger than the scale to which stable clustering can propagate. We examine in detail the very different conclusions of studies by Smith et al. (2003) and Widrow et al. (2009} and find that the strong deviations from stable clustering reported by these works are the results of over-optimistic assumptions about scales resolved accurately by the measured power spectra, and the reliance on Fourier space analysis. We emphasise the much poorer resolution obtained with the power spectrum compared to the two point correlation function..

astro-ph.CO↗

arXiv2017

Particle number dependence in the non-linear evolution of N-body self-gravitating systems

Simulations of purely self-gravitating N-body systems are often used in astrophysics and cosmology to study the collisionless limit of such systems. Their results for macroscopic quantities should then converge well for sufficiently large N. Using a study of the evolution from a simple space of spherical initial conditions - including a region characterised by so-called "radial orbit instability" - we illustrate that the values of N at which such convergence is obtained can vary enormously. In the family of initial conditions we study, good convergence can be obtained up to a few dynamical times with N $ \sim 10^3$ - just large enough to suppress two body relaxation - for certain initial conditions, while in other cases such convergence is not attained at this time even in our largest simulations with N $\sim 10^5$. The qualitative difference is due to the stability properties of fluctuations introduced by the N-body discretisation, of which the initial amplitude depends on N. We discuss briefly why the crucial role which such fluctuations can potentially play in the evolution of the N-body system could, in particular, constitute a serious problem in cosmological simulations of dark matter.

astro-ph.CO↗

arXiv2017

Stationary ordered non-equilibrium states of long-range interacting systems

Long-range interacting Hamiltonian systems are believed to relax generically towards non-equilibrium states called "quasi-stationary" because they evolve towards thermodynamic equilibrium very slowly, on a time-scale diverging with particle number. We show here that, by applying a suitable perturbation operator for a finite time interval, we obtain, in a family of long-range systems, non-equilibrium states which appear to be strictly stationary. They exist even in the case of a harmonic potential, and are characterised by an ordered microscopic phase space structure. We give some simple heuristic arguments which predict reasonably well some properties of these states.

cond-mat.stat-mech↗

arXiv2017

Formation and relaxation of quasi-stationary states in particle systems with power law interactions

We explore the formation and relaxation of so-called quasi-stationary states (QSS) for particle distributions in three dimensions interacting via an attractive radial pair potential $V(r \rightarrow \infty) \sim 1/r^γ$ with $γ> 0$, and either a soft-core or hard-core regularization at small $r$. In the first part of the paper we generalize, for any spatial dimension $d \geq 2$, Chandrasekhar's approach for the case of gravity to obtain analytic estimates of the rate of collisional relaxation due to two body collisions. The resultant relaxation rates indicate an essential qualitative difference depending on the integrability of the pair force at large distances: for $γ>d-1$ the rate diverges in the large particle number $N$ (mean field) limit, unless a sufficiently large soft core is present; for $γ< d-1$, on the other hand, the rate vanishes in the same limit even in the absence of any regularization. In the second part of the paper we compare our analytical predictions with the results of extensive parallel numerical simulations in $d=3$, for a range of different exponents $γ$ and soft cores leading to the formation of QSS. We find, just as for the previously well studied case of gravity (which we also revisit), excellent agreement between the parametric dependence of the observed relaxation times and our analytic predictions. Further, as in the case of gravity, we find that the results indicate that, when large impact factors dominate, the appropriate cut-off is the size of the system (rather than, for example, the mean inter-particle distance). Our results provide strong evidence that the existence of QSS is robust only for long-range interactions with a large distance behavior $γ< d-1$; for $γ\geq d-1$ the existence of such states will be conditioned strongly on the short range properties of the interaction.

cond-mat.stat-mech↗