Searcharxiv⌕ Search

arXiv subjects

Jeremy Miller

Publications and source records attributed to Jeremy Miller.

At least 37 records · Page 2Linked to original sources
arXiv2024

Accuracy Requirements: Assessing the Importance of First Post-Adiabatic Terms for Small-Mass-Ratio Binaries

We investigate the impact of post-adiabatic (1PA) terms on parameter estimation for extreme and intermediate mass-ratio inspirals using state-of-the-art waveform models. Our analysis is the first to employ Bayesian inference to assess systematic errors for 1PA waveforms. We find that neglecting 1PA terms introduces significant biases for the (small) mass ratio $ε\gtrsim 10^{-5}$ for quasi circular orbits in Schwarzschild spacetime, which can be mitigated with resummed 3PN expressions at 1PA order. Moreover, we show that the secondary spin is strongly correlated with the other intrinsic parameters, and it can not be constrained for $ε\lesssim 10^{-5}$. Finally, we highlight the need for addressing eccentric waveform systematics in the small-mass-ratio regime, as they yield stronger biases than the circular limit in both intrinsic and extrinsic parameters.

gr-qc↗
arXiv2024

Partial bases and homological stability of $\operatorname{GL}_{n}(R)$ revisited

Let $R$ be a unital ring satisfying the invariant basis number property, that every stably free $R$-module is free, and that the complex of partial bases of every finite rank free module is Cohen--Macaulay. This class of rings includes every ring of stable rank $1$ (e.g. any local, semi-local or Artinian ring), every Euclidean domain, and every Dedekind domain $\mathcal{O}_S$ of arithmetic type where $|S| > 1$ and $S$ contains at least one non-complex place. Extending recent work of Galatius--Kupers--Randal-Williams and Kupers--Miller--Patzt, we prove that the sequence of general linear groups $\operatorname{GL}_n(R)$ satisfies slope-$1$ homological stability with $\mathbb{Z}[1/2]$-coefficients.

math.AT↗
arXiv2023

A degree theorem for the simplicial closure of Auter space

The degree of a based graph is the number of essential nonbasepoint vertices after generic perturbation. Hatcher--Vogtmann's degree theorem states that the subcomplex of Auter space of graphs of degree at most d is (d-1)-connected. We extend the definition of degree to the simplicial closure of Auter space and prove a version of Hatcher--Vogtmann's result in this context.

math.AT↗
arXiv2023

Gravitational waveforms for compact binaries from second-order self-force theory

We produce gravitational waveforms for nonspinning compact binaries undergoing a quasicircular inspiral. Our approach is based on a two-timescale expansion of the Einstein equations in second-order self-force theory, which allows first-principles waveform production in tens of milliseconds. Although the approach is designed for extreme mass ratios, our waveforms agree remarkably well with those from full numerical relativity, even for comparable-mass systems. Our results will be invaluable in accurately modelling extreme-mass-ratio inspirals for the LISA mission and intermediate-mass-ratio systems currently being observed by the LIGO-Virgo-KAGRA Collaboration.

gr-qc↗
arXiv2023

SpaceTx: A Roadmap for Benchmarking Spatial Transcriptomics Exploration of the Brain

Mapping spatial distributions of transcriptomic cell types is essential to understanding the brain, with its exceptional cellular heterogeneity and the functional significance of its spatial organization. Spatial transcriptomics techniques are hoped to accomplish these measurements, but each method uses different experimental and computational protocols, with different trade-offs and optimizations. In 2017, the SpaceTx Consortium was formed to compare these methods and determine their suitability for large-scale spatial transcriptomic atlases. SpaceTx work included progress in tissue processing, taxonomy development, gene selection, image processing and data standardization, cell segmentation, cell type assignments, and visualization. Although the landscape of experimental methods has changed dramatically since the beginning of SpaceTx, the need for quantitative and detailed benchmarking of spatial transcriptomics methods in the brain is still unmet. Here, we summarize the work of SpaceTx and highlight outstanding challenges as spatial transcriptomics grows into a mature field. We also discuss how our progress provides a roadmap for benchmarking spatial transcriptomics methods in the future. Data and analyses from this consortium, along with code and methods are publicly available at https://spacetx.github.io/.

q-bio.NC↗
arXiv2022

Comparing second-order gravitational self-force, numerical relativity and effective one body waveforms from inspiralling, quasi-circular and nonspinning black hole binaries

We present the first systematic comparison between gravitational waveforms emitted by inspiralling, quasi-circular and nonspinning black hole binaries computed with three different approaches: second-order gravitational self-force (2GSF) theory, as implemented in the 1PAT1 model; numerical relativity (NR), as implemented by the SXS collaboration; and the effective one body (EOB) formalism, as implemented in the TEOBResumS waveform model. To compare the models we use both a standard, time-domain waveform alignment and a gauge-invariant analysis based on the dimensionless function $Q_ω(ω)\equiv ω^2/\dotω$, where $ω$ is the gravitational wave frequency. We analyse the domain of validity of the 1PAT1 model, deriving error estimates and showing that the effects of the final transition to plunge, which the model neglects, extend over a significantly larger frequency interval than one might expect. Restricting to the inspiral regime, we find that, while for mass ratios $q = m_1/m_2\le 10$ TEOBResumS is largely indistinguishable from NR, 1PAT1 has a significant dephasing $\gtrsim 1$rad; conversely, for $q\gtrsim 100$, 1PAT1 is estimated to have phase errors $<0.1$rad on a large frequency interval, while TEOBResumS develops phase differences $\gtrsim1$rad with it. Most crucially, on that same large frequency interval we find good agreement between TEOBResumS and 1PAT1 in the intermediate regime $15\lesssim q\lesssim 64$, with $<0.5$rad dephasing between them. A simple modification to the TEOBResumS flux further improves this agreement for $q\gtrsim 30$, reducing the dephasing to $\approx0.27$rad even at $q=128$. Our results pave the way for the construction of GSF-informed EOB models for both intermediate and extreme mass ratio inspirals for the next generation of gravitational wave detectors.

gr-qc↗
arXiv2022

Comparing second-order gravitational self-force and effective one body waveforms from inspiralling, quasi-circular and nonspinning black hole binaries II: the large-mass-ratio case

We compare recently computed waveforms from second-order gravitational self-force (GSF) theory to those generated by a new, GSF-informed, effective one body (EOB) waveform model for (spin-aligned, eccentric) inspiralling black hole binaries with large mass ratios. We focus on quasi-circular, nonspinning, configurations and perform detailed GSF/EOB waveform phasing comparisons, either in the time domain or via the gauge-invariant dimensionless function $Q_ω\equiv ω^2/\dotω$, where $ω$ is the gravitational wave frequency. The inclusion of high-PN test-mass terms within the EOB radiation reaction (notably, up to 22PN) is crucial to achieve an EOB/GSF phasing agreement below 1~rad up to the end of the inspiral for mass ratios up to 500. For larger mass ratios, up to $5\times 10^4$, the contribution of horizon absorption becomes more and more important and needs to be accurately modeled. Our results indicate that our GSF-informed EOB waveform model is a promising tool to describe waveforms generated by either intermediate or extreme mass ratio inspirals for future gravitational wave detectors

gr-qc↗
arXiv2022

Extremal stability for configuration spaces

We study stability patterns in the high dimensional rational homology of unordered configuration spaces of manifolds. Our results follow from a general approach to stability phenomena in the homology of Lie algebras, which may be of independent interest.

math.AT↗
arXiv2022

Mapping class group actions on configuration spaces and the Johnson filtration

Let $F_n(Σ_{g,1})$ denote the configuration space of $n$ ordered points on the surface $Σ_{g,1}$ and let $Γ_{g,1}$ denote the mapping class group of $Σ_{g,1}$. We prove that the action of $Γ_{g,1}$ on $H_i(F_n(Σ_{g,1});\mathbb{Z})$ is trivial when restricted to the $i^{th}$ stage of the Johnson filtration $\mathcal{J}(i)\subset Γ_{g,1}$. We give examples showing that $\mathcal{J}(2)$ acts nontrivially on $H_3(F_3(Σ_{g,1}))$ for $g\ge 2$, and provide two new conceptual reinterpretations of a certain group introduced by Moriyama.

math.GT↗
arXiv2021

Gravitational-wave energy flux for compact binaries through second order in the mass ratio

Within the framework of self-force theory, we compute the gravitational-wave energy flux through second order in the mass ratio for compact binaries in quasicircular orbits. Our results are consistent with post-Newtonian calculations in the weak field and they agree remarkably well with numerical-relativity simulations of comparable-mass binaries in the strong field. We also find good agreement for binaries with a spinning secondary or a slowly spinning primary. Our results are key for accurately modelling extreme-mass-ratio inspirals and will be useful in modelling intermediate-mass-ratio systems.

gr-qc↗
arXiv2021

Representation stability, secondary stability, and polynomial functors

We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two generalizations of classical homological stability theorems with twisted coefficients. We apply our results to prove homological stability for hyperelliptic mapping class groups with twisted coefficients, prove new representation stability results for congruence subgroups, establish secondary homological stability for groups of diffeomorphisms of surfaces viewed as discrete groups, and improve the known stable range for homological stability for general linear groups of the sphere spectrum.

math.AT↗
arXiv2021

Two-timescale evolution of extreme-mass-ratio inspirals: waveform generation scheme for quasicircular orbits in Schwarzschild spacetime

Extreme-mass-ratio inspirals, in which a stellar-mass compact object spirals into a supermassive black hole in a galactic core, are expected to be key sources for LISA. Modelling these systems with sufficient accuracy for LISA science requires going to second (or {\em post-adiabatic}) order in gravitational self-force theory. Here we present a practical two-timescale framework for achieving this and generating post-adiabatic waveforms. The framework comprises a set of frequency-domain field equations that apply on the fast, orbital timescale, together with a set of ordinary differential equations that determine the evolution on the slow, inspiral timescale. Our analysis is restricted to the special case of quasicircular orbits around a Schwarzschild black hole, but its general structure carries over to the realistic case of generic (inclined and eccentric) orbits in Kerr spacetime. In our restricted context, we also develop a tool that will be useful in all cases: a formulation of the frequency-domain field equations using hyperboloidal slicing, which significantly improves the behavior of the sources near the boundaries. We give special attention to the slow evolution of the central black hole, examining its impact on both the two-timescale evolution and the earlier self-consistent evolution scheme.

gr-qc↗
arXiv2021

On the generalized Bykovskii presentation of Steinberg modules

We study presentations of the virtual dualizing modules of special linear groups of number rings, the Steinberg modules. Bykovskii gave a presentation for the Steinberg modules of the integers, and our main result is a generalization of this presentation to the Gaussian integers and the Eisenstein integers. We also show that this generalization does not give a presentation for the Steinberg modules of several Euclidean number rings.

math.AT↗
arXiv2020

On the top dimensional cohomology groups of congruence subgroups of $\text{SL}_n(\mathbb{Z})$

Let $Γ_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(\mathbb{Z})$. Borel-Serre proved that the cohomology of $Γ_n(p)$ vanishes above degree $\binom{n}{2}$. We study the cohomology in this top degree $\binom{n}{2}$. Let $\mathcal{T}_n(\mathbb{Q})$ denote the Tits building of $\text{SL}_n(\mathbb{Q})$. Lee-Szczarba conjectured that $H^{\binom{n}{2}}(Γ_n(p))$ is isomorphic to $\widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/Γ_n(p))$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{\binom{n}{2}}(Γ_n(p)) \rightarrow \widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/Γ_n(p))$ is always surjective, but is only injective for $p \leq 5$. In particular, we completely calculate $H^{\binom{n}{2}}(Γ_n(5))$ and improve known lower bounds for the ranks of $H^{\binom{n}{2}}(Γ_n(p))$ for $p \geq 5$.

math.NT↗
arXiv2020

Stability in the high-dimensional cohomology of congruence subgroups

We prove a representation stability result for the codimension-one cohomology of the level three congruence subgroup of $\mathbf{SL}_n(\mathbb{Z})$. This is a special case of a question of Church-Farb-Putman which we make more precise. Our methods involve proving several finiteness properties of the Steinberg module for the group $\mathbf{SL}_n(K)$ for $K$ a field. This also lets us give a new proof of Ash-Putman-Sam's homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church-Putman's homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_n(\mathbb{Z})$.

math.AT↗