Kinematic Lensing using HI or Optical Linewidths

Kinematic Weak Lensing with Spatially Unresolved Line Widths

Project Status — December 26, 2025

Executive Summary

This project articulates a formalism for kinematic lensing using spatially unresolved line widths (HI 21cm or optical) combined with imaging survey ellipticities, building on Huong et al 2025. The Tully-Fisher relation constrains a disk galaxy's inclination, which determines the magnitude $|\varepsilon_{\rm incl}|$ of its intrinsic ellipticity but not its orientation $\phi_n$. This constrains the gravitational shear $\gamma$ to lie on a ring of radius $R$ in the $(\gamma_1, \gamma_2)$ plane, partially breaking the shape-noise degeneracy.

In particular we incorporate that the information content depends on the ratio $R/\sigma_m$: face-on galaxies ($R \ll \sigma_m$) provide full two-dimensional constraints with no information loss, while only edge-on galaxies suffer the factor-of-4 reduction in power found in variance-based treatments. Since face-on and moderately inclined systems contribute ~95% of the Fisher information, we find gain factors $G \approx 3$ over standard weak lensing for realistic DSA-2000 + Euclid/Roman parameters—corresponding to factor of $\approx 3$ reduction in required sample size for a given shear precision.

Context and Goals

The Shape Noise Problem

Weak gravitational lensing measures the coherent distortion of background galaxy shapes by foreground mass concentrations. The cosmological shear signal is small ($|\gamma| \sim 0.01$–$0.03$), while galaxies have intrinsic ellipticities with dispersion $\sigma_{\rm int} \approx 0.25$–$0.3$ per component. This shape noise is the dominant source of statistical uncertainty in current and future surveys. Precision scales as $\sigma_\gamma \propto \sigma_{\rm int}/\sqrt{N}$, requiring very large samples.

The Kinematic Lensing Idea

If we knew each galaxy's intrinsic ellipticity $\varepsilon_{\rm src}$, we could measure shear directly: $\gamma = \varepsilon_{\rm obs} - \varepsilon_{\rm src}$. For disk galaxies, the intrinsic ellipticity is set by the viewing geometry—specifically the inclination angle $i$. An independent measurement of $i$ (e.g., from the Tully-Fisher relation) partially breaks the shape noise degeneracy.

Two regimes exist:

• Resolved kinematics: Spatially resolved velocity fields (IFU spectroscopy) determine both inclination $i$ and position angle $\phi_n$, fully specifying $\varepsilon_{\rm src}$. Limited to bright, nearby galaxies.
• Unresolved line widths: Integrated HI 21cm profiles constrain $\sin i$ via Tully-Fisher, giving $|\varepsilon_{\rm incl}|$ but not $\phi_n$. Applicable to tens of millions of galaxies with forthcoming radio surveys.

The Opportunity: DSA-2000 + Euclid/Roman

The Deep Synoptic Array (DSA-2000) will detect HI 21cm emission from $>3 \times 10^7$ galaxies over $\sim$30,000 deg², with substantial overlap with Euclid ($\sim$15,000 deg²) and Roman imaging. This creates an unprecedented opportunity:

• Euclid/Roman provide precise shape measurements ($\sigma_m \sim 0.15$–$0.25$) and stellar masses $M_\star$ for Tully-Fisher calibration
• DSA-2000 provides HI line widths (inclination constraints) and precise spectroscopic redshifts
• Combined: Kinematic lensing with $>10^7$ galaxies, independent systematics from radio vs. optical

Goals of This Work

1. Develop the likelihood formalism for shear inference from the ring constraint, going beyond variance-based approximations
2. Quantify the information content as a function of inclination, revealing which galaxies contribute most
3. Calculate realistic gain factors $G$ over standard weak lensing, including finite disk thickness and non-axisymmetry
4. Compare with existing work (Huang et al. 2025) and clarify what is new
5. Provide a foundation for optimal shear estimation pipelines combining radio and optical data

Core Formalism

Ellipticity and Shear

We adopt the standard weak lensing convention for complex ellipticity. For axis ratio $q \leq 1$ and position angle $\phi$:

$$\varepsilon = \varepsilon_1 + i\varepsilon_2 = |\varepsilon| \, e^{2i\phi}, \qquad |\varepsilon| = \frac{1-q}{1+q}$$

In the weak lensing regime, the observed ellipticity relates to source ellipticity and shear by:

$$\varepsilon_{\rm obs} = \gamma + \varepsilon_{\rm src} + \eta$$

where $\eta$ represents measurement noise with variance $\sigma_m^2$.

Inclination from Tully-Fisher

For a thin axisymmetric disk at inclination $i$, the intrinsic ellipticity is:

$$\varepsilon_{\rm src} = \varepsilon_{\rm incl} = |\varepsilon_{\rm incl}| \, e^{2i\phi_n}, \qquad |\varepsilon_{\rm incl}| = \frac{1 - \cos i}{1 + \cos i} \equiv R$$

The Tully-Fisher relation constrains $\sin i$ via the observed HI line width:

$$W_{\rm obs} = W_{\rm TF}(M_\star) \sin i \quad \Rightarrow \quad \sin i = \frac{W_{\rm obs}}{W_{\rm TF}(M_\star)}$$

with intrinsic scatter $\sigma_{\rm TF}$ propagating to fractional uncertainty $\sigma_f$ on $\sin i$.

The Ring Constraint

Combining these relations, the shear is:

$$\gamma = \varepsilon_{\rm obs} - R \, e^{2i\phi_n}$$

Since $\phi_n$ is unknown (uniformly distributed on $[0,\pi)$ for unresolved observations), the shear is constrained to a circle:

$$|\gamma - \varepsilon_{\rm obs}| = R$$

with center at $\varepsilon_{\rm obs}$ and radius $R = |\varepsilon_{\rm incl}|$ determined from the line width.

The Ring Likelihood

Single Galaxy

The likelihood for shear $\gamma$ given observed ellipticity and ring radius is obtained by marginalizing over $\phi_n$:

$$\mathcal{L}(\gamma \,|\, \varepsilon_{\rm obs}, R) = \int_0^\pi \frac{d\phi_n}{\pi} \, \frac{1}{2\pi\sigma_m^2} \exp\left[-\frac{|\varepsilon_{\rm obs} - \gamma - R e^{2i\phi_n}|^2}{2\sigma_m^2}\right]$$

Defining $d \equiv |\varepsilon_{\rm obs} - \gamma|$ and evaluating the integral yields the Bessel likelihood:

$$\mathcal{L}(\gamma) \propto \exp\left[-\frac{d^2 + R^2}{2\sigma_m^2}\right] \, I_0\left(\frac{dR}{\sigma_m^2}\right)$$

where $I_0$ is the modified Bessel function of the first kind. This likelihood is not Gaussian—it is ring-shaped, peaked at $d = R$.

Limiting Regimes

Face-on limit ($R \ll \sigma_m$): Using $I_0(x) \approx 1$ for small $x$:

$$\mathcal{L}(\gamma) \propto \exp\left[-\frac{d^2}{2\sigma_m^2}\right]$$

This is a standard 2D Gaussian—the ring degenerates to a point.

Edge-on limit ($R \gg \sigma_m$): Using $I_0(x) \approx e^x/\sqrt{2\pi x}$ for large $x$:

$$\log \mathcal{L}(\gamma) \approx -\frac{(d - R)^2}{2\sigma_m^2} + \text{const}$$

The likelihood is sharply peaked on the ring $d = R$, constraining only the radial direction.

Including Tully-Fisher Uncertainty

The TF scatter $\sigma_f$ propagates to uncertainty on $R$:

$$\sigma_{\sin i} = \sin i \cdot \sigma_f, \qquad \sigma_{\cos i} = \frac{\sin^2 i}{\cos i} \sigma_f$$

Through the chain $\sin i \to \cos i \to R$:

$$\sigma_R = \frac{2R(1+R)}{1-R} \cdot \sigma_f \cdot \frac{\sin i}{\cos i}$$

This diverges for edge-on galaxies ($\cos i \to 0$), explaining their negligible information contribution.

The total radial uncertainty combines measurement noise and TF scatter:

$$\sigma_{\rm rad}^2 = \sigma_m^2 + \sigma_R^2$$

Combined Likelihood for Multiple Galaxies

For $N$ galaxies in a local region probing a common shear $\gamma$:

$$\log \mathcal{L}(\gamma) = \sum_{i=1}^N \left[ -\frac{d_i^2 + R_i^2}{2\sigma_m^2} + \log I_0\left(\frac{d_i R_i}{\sigma_m^2}\right) \right]$$

This is a sum of "ring potentials"—the maximum likelihood $\hat{\gamma}$ occurs where the rings best intersect.

Fisher Information

The Rank-1 Structure

In the sharp-ring limit, the Fisher matrix for a single galaxy is rank-1:

$$\mathbf{F} = \frac{1}{\sigma_{\rm rad}^2} \, \hat{n} \otimes \hat{n}, \qquad \hat{n} = (\cos 2\phi_n, \sin 2\phi_n)$$

This constrains only the radial direction; the tangential direction (along the ring) is unconstrained.

The $n_{\rm eff}$ Formula

The Fisher information per galaxy depends on $R/\sigma_m$, interpolating between face-on and edge-on limits:

$$F = \frac{n_{\rm eff}}{2(\sigma_m^2 + \sigma_R^2)}, \qquad n_{\rm eff} = 1 + \frac{1}{1 + R^2/\sigma_m^2}$$

Face-on ($R \to 0$):
$n_{\rm eff} = 2$
$F = 1/(\sigma_m^2 + \sigma_R^2)$
Full 2D information

Edge-on ($R \gg \sigma_m$):
$n_{\rm eff} \to 1$
$F = 1/[2(\sigma_m^2 + \sigma_R^2)]$
Rank-1 (factor of 2 in Fisher)

Combining $N$ Galaxies

For random orientations $\{\phi_{n,i}\}$, the averaged Fisher matrix becomes diagonal:

$$\langle \mathbf{F}_{\rm tot} \rangle = \sum_{i=1}^N \frac{n_{{\rm eff},i}}{2(\sigma_m^2 + \sigma_{R,i}^2)} \, \mathbf{I}$$

Individual rank-1 constraints sum to full-rank when galaxies have diverse $\phi_n$.

Standard Weak Lensing Comparison

Without kinematic information, the intrinsic ellipticity is unknown with population variance $\sigma_{\rm int}^2$:

$$F_{\rm std} = \frac{N}{\sigma_m^2 + \sigma_{\rm int}^2}, \qquad \sigma_{\rm int}^2 = \langle R^2 \rangle = 3 - 4\ln 2 \approx 0.227$$

The Gain Factor

$$G \equiv \frac{F_{\rm kin}}{F_{\rm std}} = (\sigma_m^2 + \sigma_{\rm int}^2) \left\langle \frac{n_{\rm eff}}{2(\sigma_m^2 + \sigma_R^2)} \right\rangle$$

For an isotropic inclination distribution ($\cos i$ uniform on $[0,1]$), the population average becomes:

$$G = (\sigma_m^2 + \sigma_{\rm int}^2) \int_0^1 \frac{n_{\rm eff}(x)}{2[\sigma_m^2 + \sigma_R^2(x)]} \, dx$$

where $x = \cos i$, $R(x) = (1-x)/(1+x)$, and $\sigma_R(x) = 2R(1+R)/(1-R) \cdot \sigma_f \sqrt{1-x^2}/x$.

Realistic Disk Properties

Finite Thickness

Real disks have intrinsic thickness $q_0 \sim 0.2$. The observed axis ratio becomes:

$$q^2 = q_0^2 + (1 - q_0^2) \cos^2 i$$

modifying the ellipticity–inclination relation:

$$R(i; q_0) = \frac{1 - q(i; q_0)}{1 + q(i; q_0)}$$

Key effects: $R_{\rm max} \approx 0.67$ for $q_0 = 0.2$ (not 1.0); face-on disks have $R_{\rm min} \approx 0.09$ (not 0).

Non-Axisymmetry

Bars, spirals, and warps add ellipticity not captured by the inclination model:

$$\sigma_{\rm rad}^2 \to \sigma_m^2 + \sigma_R^2 + \sigma_{\rm disk}^2$$

with $\sigma_{\rm disk} \sim 0.05$ from empirical constraints.

Numerical Results

Gain Factors — Idealized Thin Disks

$\sigma_f$	$\sigma_m = 0.10$	$\sigma_m = 0.15$	$\sigma_m = 0.20$	$\sigma_m = 0.25$
0.00	14.6	7.3	4.6	3.4
0.05	8.5	4.8	3.3	2.6
0.10	7.4	4.3	3.0	2.4
0.15	6.7	4.0	2.8	2.3
0.20	6.2	3.7	2.7	2.2

Highlighted: DSA-2000 realistic range ($\sigma_f \approx 0.10$–$0.15$)

Gain Factors — Realistic Disks ($q_0=0.2$, $\sigma_{\rm disk}=0.05$)

$\sigma_f$	$\sigma_m = 0.10$	$\sigma_m = 0.15$	$\sigma_m = 0.20$	$\sigma_m = 0.25$
0.00	8.0	4.6	3.2	2.4
0.05	5.7	3.7	2.7	2.1
0.10	4.8	3.2	2.4	2.0
0.15	4.2	2.9	2.2	1.9
0.20	3.9	2.7	2.1	1.8

Information Budget by Inclination

For $\sigma_m = 0.2$, $\sigma_f = 0.1$ (thin disk):

Inclination Range	Fraction of Total Information
Face-on ($i < 26°$)	25%
Low inclination ($26° < i < 46°$)	44%
Moderate ($46° < i < 60°$)	26%
High inclination ($60° < i < 73°$)	5%
Edge-on ($i > 73°$)	<1%

Key result: Face-on and moderate-inclination galaxies ($i < 60°$) provide ~95% of the total information, despite edge-on systems comprising 30% of the sample by solid angle. This is because $\sigma_R \to \infty$ as $\cos i \to 0$.

Comparison with Huang, Krause et al. (2025)

Reference: arXiv:2510.18011

Aspect	Huang et al.	This Work
Basic constraint	$\gamma = \varepsilon_{\rm obs} - |\varepsilon_{\rm incl}|e^{2i\phi}$ Same	Identical equation
Methodology	Correlation functions $\xi_\pm$	Per-galaxy Bessel likelihood New
Information loss	Uniform factor-of-4 in $C_\ell$	$R$-dependent $n_{\rm eff}$ New
Face-on galaxies	Same penalty as edge-on	Full 2D, no penalty New
Galaxy weighting	Uniform	Optimal: $\propto n_{\rm eff}/\sigma_{\rm rad}^2$ New

Both approaches recover consistent results when properly compared. The likelihood formalism reveals that the factor-of-4 applies only to edge-on systems; face-on galaxies escape this penalty entirely. Since they dominate the information budget, the population-averaged gain exceeds naive factor-of-4 estimates.

File Inventory

LaTeX Paper

• kinematic_lensing_paper_v2.tex — Main paper (13 pages)
• kinematic_lensing_paper_v2.pdf — Compiled PDF

Figures

• figure1_ring_constraint.png/pdf — Ring geometry schematic
• figure2_gain_factor_v2.png/pdf — $G$ vs $\sigma_f$

Code & Documentation

• kinematic_lensing_figures_v2.ipynb — Figure generation notebook
• CLAUDE.md — Context file for Claude Code

Fiducial Parameters

Parameter	Symbol	Value	Source
Shape measurement noise	$\sigma_m$	0.15–0.25	Euclid/Roman
TF fractional scatter	$\sigma_f$	0.10–0.15	HI surveys
Disk thickness	$q_0$	0.15–0.25	Observations
Non-axisymmetry	$\sigma_{\rm disk}$	0.03–0.07	Bars/spirals
Galaxy count	$N$	$>10^7$	DSA-2000
Sky overlap	—	~15,000 deg²	Euclid $\cap$ DSA

Fiducial result: $\sigma_m = 0.20$, $\sigma_f = 0.12$, $q_0 = 0.2$, $\sigma_{\rm disk} = 0.05$ $\;\Rightarrow\;$ $G \approx 2.3$

Future Directions

Theory

• Full likelihood MCMC shear inference
• Modified $\xi_\pm$ with $n_{\rm eff}$ weighting
• Intrinsic alignment mitigation
• TF evolution $\sigma_f(z)$

Applications

• DSA-2000 detailed forecasts
• SKA comparison
• Optical emission lines (H$\alpha$, [OII])
• Cosmological parameter forecasts

Key References

• Huang et al. (2025), arXiv:2510.18011 — One-component KL, SKA2 forecasts
• Huff et al. (2013), MNRAS 431, 1629 — "Kinematic lensing" terminology
• Blain (2002), ApJ 570, L51 — First KL proposal
• Gurri et al. (2020), MNRAS 499, 4591 — First KL measurement
• Hallinan et al. (2019), BAAS 51, 255 — DSA-2000

---

Developed in conversation between H.-W. Rix and Claude (Anthropic), December 2025.

RR Lyrae in the inner Galaxy

At the last group meeting we discussed several issues revolving around RRL in the galaxy. a) Maddie Lucey's paper: she did the selection function using the Rybizki and Drimmel trick. Do we have to do it a la Cantat-Gauding, with the VIVACE and DES as the 'ground-truth' sample b) the poor old heart and RRL. The Vivace sample shows a tight know of RRL at the very center. The distance modulus spread (after extinction correction) is only sigma = 0.25mag. Or 12% of 1.0 kpc at 8.1 kpc. Plan: do light-curved based metallicities and get the spatial structure...

Constraining chemical yield delay-time distributions with abundance-age data

Constraining Delay Time Distributions from Observed Age-Abundance Relations: A Single-Zone Chemical Evolution Approach

Abstract

We present a method for constraining the delay time distribution (DTD) of element X from observed stellar abundance patterns. Using the single-zone chemical evolution framework of Weinberg et al. (2017), we derive the DTD that produces a linear increase in \([\text{X/Mg}]\) with stellar age for stars at solar magnesium abundance (\([\text{Mg/H}] = 0\)). We find that achieving such a linear trend requires a DTD that increases with delay time, compensating for the declining star formation history. This mathematical framework provides a direct link between observable age-abundance relations and the underlying nucleosynthetic delay times.

1. Introduction

The chemical abundances of stars encode the enrichment history of their birth environments. For elements produced on different timescales, abundance ratios can serve as "chemical clocks" that trace the temporal evolution of galaxies. A particularly powerful diagnostic is the ratio \([\text{X/Mg}]\), where Mg is produced essentially instantaneously by core-collapse supernovae (CCSNe), while element X may have both instantaneous and delayed production channels.

The key question we address is: Can chemical evolution models constrain the delay time distribution (DTD) of element X from observed age-abundance relations? Specifically, we consider stars with solar magnesium abundance (\([\text{Mg/H}] = 0\)) and examine how their \([\text{X/Mg}]\) ratios vary with stellar age. This approach builds on the single-zone chemical evolution framework developed by Weinberg et al. (2017), which provides analytic solutions for abundance evolution with realistic delay time distributions.

In this work, we:

1. Adapt the Weinberg et al. (2017) framework to the specific case of \([\text{X/Mg}]\) evolution
2. Derive the mathematical relationship between observed age-\([\text{X/Mg}]\) trends and the underlying DTD
3. Solve for the DTD that produces a linear increase in \([\text{X/Mg}]\) with stellar age

2. Single-Zone Chemical Evolution Framework

2.1 Basic Equations

Following Weinberg et al. (2017), we consider a one-zone model where metals are instantaneously mixed throughout the star-forming gas. The evolution of element masses is governed by:

\[\dot{M}_{\text{Mg}} = m^{\text{cc}}_{\text{Mg}} \dot{M}_* - (1 + \eta - r) \dot{M}_* Z_{\text{Mg}}\]

\[\dot{M}_{\text{X}} = m^{\text{cc}}_{\text{X}} \dot{M}_* + m^{\text{delayed}}_{\text{X}} \langle\dot{M}_*\rangle_f - (1 + \eta - r) \dot{M}_* Z_{\text{X}}\]

where:

• \(\dot{M}_*\) is the star formation rate
• \(m^{\text{cc}}_{\text{Mg}}\) and \(m^{\text{cc}}_{\text{X}}\) are the instantaneous yields from CCSNe
• \(m^{\text{delayed}}_{\text{X}}\) is the delayed yield of element X
• \(\eta\) is the outflow mass loading factor
• \(r\) is the recycling fraction
• \(Z_i = M_i/M_g\) is the mass fraction of element \(i\)

The delayed production term involves the time-averaged star formation rate:

\[\langle\dot{M}_*(t)\rangle_f = \frac{\int_0^t \dot{M}_*(t') f(t-t') dt'}{\int_0^{\infty} f(\tau) d\tau}\]

where \(f(\tau)\) is the delay time distribution we seek to constrain.

2.2 Solutions for Exponential Star Formation History

For an exponentially declining star formation history, \(\dot{M}_*(t) = \dot{M}_{*,0} e^{-t/\tau_{\text{sfh}}}\), Weinberg et al. (2017) show that the equilibrium abundances are:

\[Z_{\text{Mg,eq}} = \frac{m^{\text{cc}}_{\text{Mg}}}{1 + \eta - r - \tau_*/\tau_{\text{sfh}}}\]

\[Z_{\text{X,eq}} = \frac{m^{\text{cc}}_{\text{X}} + m^{\text{delayed}}_{\text{X}} \cdot \text{DF}_{\infty}}{1 + \eta - r - \tau_*/\tau_{\text{sfh}}}\]

where \(\tau_* = M_g/\dot{M}_*\) is the star formation efficiency timescale and \(\text{DF}_{\infty}\) is the asymptotic value of the delayed factor.

2.3 The Delayed Factor

The delayed factor, which encodes the contribution from delayed production, is:

\[\text{DF}(t) = \frac{\langle\dot{M}_*(t)\rangle_f}{\dot{M}_*(t)} = e^{t/\tau_{\text{sfh}}} \int_0^t f(\tau) e^{-\tau/\tau_{\text{sfh}}} d\tau\]

This factor evolves from 0 at \(t=0\) (no delayed contribution) to some asymptotic value as \(t \to \infty\).

3. Connecting Observables to Theory

3.1 The Age-Metallicity Relation

Stars inherit the gas-phase abundances at their formation time. Therefore:

• Old stars (large age \(\tau_{\text{age}}\)) formed at small \(t\) when \([\text{X/Mg}]\) was low
• Young stars (small \(\tau_{\text{age}}\)) formed at large \(t\) when \([\text{X/Mg}]\) was high

The relationship between stellar age and formation time is:

\[t_{\text{form}} = t_{\text{now}} - \tau_{\text{age}}\]

3.2 The [X/Mg] Ratio

The abundance ratio at any time is:

\[\text{[X/Mg]}(t) = \log_{10}\left(\frac{Z_{\text{X}}(t)}{Z_{\text{Mg}}(t)}\right) = \log_{10}\left(\frac{m^{\text{cc}}_{\text{X}} + m^{\text{delayed}}_{\text{X}} \cdot \text{DF}(t)}{m^{\text{cc}}_{\text{Mg}}}\right)\]

This can be rewritten as:

\[\text{[X/Mg]}(t) = \text{[X/Mg]}_{\text{initial}} + \log_{10}\left(1 + \frac{m^{\text{delayed}}_{\text{X}}}{m^{\text{cc}}_{\text{X}}} \cdot \text{DF}(t)\right)\]

4. Constraining the DTD from Observations

4.1 The Inverse Problem

Given an observed relation \([\text{X/Mg}] = F(\tau_{\text{age}})\) for stars with \([\text{Mg/H}] = 0\), we want to find the DTD \(f(\tau)\) that reproduces this relation.

For stars with \([\text{Mg/H}] = 0\), we know that:

\[Z_{\text{Mg}}(t_{\text{form}}) = Z_{\text{Mg},\odot}\]

This constraint determines when each star formed, allowing us to convert the age-abundance relation into a time-abundance relation.

4.2 Linear Age-[X/Mg] Relation

Consider the specific case where \([\text{X/Mg}]\) increases linearly with age:

\[\text{[X/Mg]}(\tau_{\text{age}}) = A + B \cdot \tau_{\text{age}}\]

where \(A\) and \(B\) are constants determined by observations. Converting to formation time:

\[\text{[X/Mg]}(t_{\text{form}}) = A + B \cdot (t_{\text{now}} - t_{\text{form}}) = (A + B \cdot t_{\text{now}}) - B \cdot t_{\text{form}}\]

This requires \([\text{X/Mg}]\) to decrease linearly with formation time.

4.3 Required Delayed Factor Evolution

From the expression for \([\text{X/Mg}]\), we need:

\[\log_{10}\left(1 + \frac{m^{\text{delayed}}_{\text{X}}}{m^{\text{cc}}_{\text{X}}} \cdot \text{DF}(t)\right) = C - B \cdot t\]

where \(C\) is a constant. Taking the antilog:

\[1 + \frac{m^{\text{delayed}}_{\text{X}}}{m^{\text{cc}}_{\text{X}}} \cdot \text{DF}(t) = 10^{C - B \cdot t}\]

Solving for the delayed factor:

\[\text{DF}(t) = \frac{m^{\text{cc}}_{\text{X}}}{m^{\text{delayed}}_{\text{X}}} \left(10^{C - B \cdot t} - 1\right)\]

5. Deriving the Delay Time Distribution

5.1 The Integral Equation

From the definition of the delayed factor:

\[\text{DF}(t) = e^{t/\tau_{\text{sfh}}} \int_0^t f(\tau) e^{-\tau/\tau_{\text{sfh}}} d\tau\]

Taking the derivative:

\[\frac{d\text{DF}}{dt} = \frac{1}{\tau_{\text{sfh}}} \text{DF}(t) + f(t)\]

Solving for \(f(t)\):

\[f(t) = \frac{d\text{DF}}{dt} - \frac{1}{\tau_{\text{sfh}}} \text{DF}(t)\]

5.2 Solution for Linear [X/Mg] Growth

For the delayed factor derived above:

\[\frac{d\text{DF}}{dt} = -B \ln(10) \cdot \frac{m^{\text{cc}}_{\text{X}}}{m^{\text{delayed}}_{\text{X}}} \cdot 10^{C - B \cdot t}\]

Substituting into the expression for \(f(t)\):

\[f(t) = \frac{m^{\text{cc}}_{\text{X}}}{m^{\text{delayed}}_{\text{X}}} \left[-B \ln(10) \cdot 10^{C - B \cdot t} - \frac{1}{\tau_{\text{sfh}}} \left(10^{C - B \cdot t} - 1\right)\right]\]

5.3 Key Mathematical Insight

For typical parameter values where \(B > 0\) (older stars have lower \([\text{X/Mg}]\)):

• The first term dominates and is negative
• This implies \(f(t) < 0\) for some range of \(t\)
• A negative DTD is unphysical!

This paradox arises because we're trying to achieve a decreasing \([\text{X/Mg}]\) with formation time despite an exponentially declining SFH. The resolution requires a modified approach.

6. Physical DTD Solutions

6.1 Compensating for Declining SFH

To achieve increasing \([\text{X/Mg}]\) with decreasing age (decreasing with formation time), the DTD must compensate for the exponentially declining star formation. The required form is:

\[f(\tau) = g(\tau) \cdot \frac{e^{\tau/\tau_{\text{sfh}}}}{\tau_{\text{sfh}}}\]

where \(g(\tau)\) is a normalized distribution. The exponential factor exactly cancels the weighting from the declining SFH, giving:

\[\text{DF}(t) = \int_0^t g(\tau) d\tau = G(t)\]

where \(G(t)\) is the cumulative distribution function of \(g(\tau)\).

6.2 Power-Law Family of Solutions

For a smooth transition from \(\text{DF}(0) = 0\) to \(\text{DF}(t_{\text{max}}) = 1\), consider:

\[g(\tau) = \frac{(n+1)\tau^n}{t_{\text{max}}^{n+1}} \quad \text{for } 0 < \tau < t_{\text{max}}\]

This gives:

\[G(t) = \left(\frac{t}{t_{\text{max}}}\right)^{n+1}\]

The complete DTD is:

\[\boxed{f(\tau) = \frac{(n+1)\tau^n}{t_{\text{max}}^{n+1}} \cdot \frac{e^{\tau/\tau_{\text{sfh}}}}{\tau_{\text{sfh}}}}\]

6.3 Linear [X/Mg] Growth

For exactly linear growth in \([\text{X/Mg}]\) with age, we need \(\text{DF}(t) \propto t\), which requires \(n = 0\):

\[f(\tau) = \frac{1}{t_{\text{max}}} \cdot \frac{e^{\tau/\tau_{\text{sfh}}}}{\tau_{\text{sfh}}}\]

This DTD:

• Increases exponentially with delay time
• Peaks at late times
• Provides uniform enrichment rate despite declining SFH

7. Discussion

7.1 Physical Interpretation

The derived DTD that increases with delay time is unusual compared to typical astrophysical delay distributions (e.g., Type Ia SNe, which decline as \(t^{-1.1}\)). Possible physical scenarios include:

1. Metallicity-dependent yields: If element X production efficiency increases with metallicity, later generations contribute more
2. Multiple sources: A combination of sources with different characteristic timescales
3. Mass-dependent delays: If lower-mass stars (with longer lifetimes) are more efficient X producers

7.2 Observational Tests

To apply this framework:

1. Measure \([\text{X/Mg}]\) vs age for a sample of stars with \([\text{Mg/H}] \approx 0\)
2. Fit the functional form of the age-abundance relation
3. Use the derived expressions to constrain the DTD
4. Compare with theoretical predictions for various nucleosynthetic sources

7.3 Limitations

This analysis assumes:

• Perfect mixing (one-zone approximation)
• Constant star formation efficiency
• No radial flows or stellar migration
• Metallicity-independent yields

Relaxing these assumptions would require more complex models but could provide additional constraints on the DTD.

8. Conclusions

We have developed a mathematical framework for constraining delay time distributions from observed age-abundance relations in stellar populations. The key findings are:

1. Observable \([\text{X/Mg}]\) trends with stellar age directly encode information about the DTD of element X
2. For stars at fixed \([\text{Mg/H}]\), the age-abundance relation can be inverted to derive the required DTD
3. Achieving a linear increase in \([\text{X/Mg}]\) with age requires a DTD that increases exponentially with delay time
4. This unusual form compensates for the declining star formation history to maintain steady enrichment

This framework provides a direct link between observable stellar abundances and the underlying physics of nucleosynthesis, offering a new tool for understanding the origin of elements.

Acknowledgments

We thank...

References

Weinberg, D. H., Andrews, B. H., & Freudenburg, J. 2017, ApJ, 837, 183

Binary Stars and Gravitational Waves

Gravitational waves (GWs) offer a unique probe into compact binary systems. Non-merging white dwarf (WD) binaries—systems that emit continuous, nearly monochromatic GWs without merging within a Hubble time—are significant sources in the milli-Hertz frequency range, observable by LISA. Here we summarize results for GW strain amplitude from such binaries and details how LISA determines their distance and sky position, with exhaustive derivations and examples.

1. Derivation of the GW Signal

Consider two white dwarfs with masses \( M_1 \) and \( M_2 \), orbiting circularly with period \( P \), at a luminosity distance \( D \) from Earth. We use the quadrupole approximation in the non-relativistic limit to compute the GW strain.

Step 1: Define Orbital Parameters

The orbital dynamics are foundational to the GW signal. Define the following:

• Orbital Angular Frequency (\( \omega \)):
  \[ \omega = \frac{2\pi}{P}, \]
  where \( P \) is the orbital period in seconds, and \( \omega \) is in radians per second. This relates the frequency of rotation to the time for one complete orbit.
• Total Mass (\( M \)):
  \[ M = M_1 + M_2, \]
  the sum of the individual masses in kilograms, governing the system's gravitational interaction.
• Reduced Mass (\( \mu \)):
  \[ \mu = \frac{M_1 M_2}{M_1 + M_2}, \]
  a measure of the effective mass for the two-body problem, also in kilograms, which influences the GW amplitude.
• Orbital Separation (\( a \)): Apply Kepler's third law for a circular orbit:
  \[ P = 2\pi \sqrt{\frac{a^3}{G M}}, \]
  where \( G \) is the gravitational constant (\( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)). Square both sides:
  \[ P^2 = \frac{4\pi^2 a^3}{G M}. \]
  Rearrange for \( a^3 \) and take the cube root:
  \[ a = \left( \frac{G M P^2}{4\pi^2} \right)^{1/3}, \]
  yielding the semi-major axis \( a \) as function of the period and total mass.
• Orbital Inclination (\( \iota \)): The angle between the orbital angular momentum vector and the line of sight to the observer. For \( \iota = 0 \), the binary is face-on, while for \( \iota = \pi/2 \), it is edge-on.

Step 2: Compute the Quadrupole Moment

The GW signal arises from the changing mass quadrupole moment. The orbit lies in a plane that can be tilted with respect to the observer's line of sight. For a general inclination, we first define the positions in the orbital plane and then transform to the observer's frame.

In the orbital plane (with center of mass at the origin), the positions are:

• WD 1 Position (\( \mathbf{r}_1 \)):
  \[ \mathbf{r}_1 = \left( \frac{M_2}{M} a \cos \omega t, \frac{M_2}{M} a \sin \omega t, 0 \right), \]
  where \( \frac{M_2}{M} a \) is the distance from the center of mass to WD 1.
• WD 2 Position (\( \mathbf{r}_2 \)):
  \[ \mathbf{r}_2 = \left( -\frac{M_1}{M} a \cos \omega t, -\frac{M_1}{M} a \sin \omega t, 0 \right), \]
  with \( \frac{M_1}{M} a \) as the distance to WD 2, and the negative sign ensuring the center of mass condition \( M_1 \mathbf{r}_1 + M_2 \mathbf{r}_2 = 0 \).

The mass quadrupole tensor is:

\[ I_{ij} = \sum_k m_k \left( r_k^i r_k^j - \frac{1}{3} \delta_{ij} r_k^2 \right), \]

where \( \delta_{ij} \) is the Kronecker delta, and \( r_k^2 = (r_k^x)^2 + (r_k^y)^2 + (r_k^z)^2 \). Compute key components:

• \( I_{xx} \):
  \[ I_{xx} = M_1 \left( \frac{M_2}{M} a \cos \omega t \right)^2 + M_2 \left( -\frac{M_1}{M} a \cos \omega t \right)^2, \]
  \[ = M_1 \frac{M_2^2 a^2 \cos^2 \omega t}{M^2} + M_2 \frac{M_1^2 a^2 \cos^2 \omega t}{M^2}, \]
  \[ = \frac{M_1 M_2^2 + M_2 M_1^2}{M^2} a^2 \cos^2 \omega t = \frac{M_1 M_2 (M_1 + M_2)}{M^2} a^2 \cos^2 \omega t, \]
  \[ = \mu a^2 \cos^2 \omega t. \]
  Adjust for the trace:
  \[ r_1^2 = \left( \frac{M_2}{M} a \right)^2 (\cos^2 \omega t + \sin^2 \omega t) = \left( \frac{M_2}{M} a \right)^2, \]
  \[ I_{xx} = \mu a^2 \cos^2 \omega t - \frac{1}{3} \mu a^2 \delta_{xx} = \mu a^2 \left( \cos^2 \omega t - \frac{1}{3} \right). \]
• \( I_{yy} \):
  \[ I_{yy} = \mu a^2 \sin^2 \omega t - \frac{1}{3} \mu a^2 = \mu a^2 \left( \sin^2 \omega t - \frac{1}{3} \right). \]
• \( I_{xy} = I_{yx} \):
  \[ I_{xy} = M_1 \left( \frac{M_2}{M} a \cos \omega t \right) \left( \frac{M_2}{M} a \sin \omega t \right) + M_2 \left( -\frac{M_1}{M} a \cos \omega t \right) \left( -\frac{M_1}{M} a \sin \omega t \right), \]
  \[ = \mu a^2 \cos \omega t \sin \omega t. \]

Step 3: Second Time Derivative of the Quadrupole Moment

GWs depend on the second time derivative \( \ddot{I}_{ij} \):

• \( \ddot{I}_{xx} \):
  \[ \dot{I}_{xx} = \mu a^2 \frac{d}{dt} (\cos^2 \omega t) = \mu a^2 \cdot 2 \cos \omega t (-\omega \sin \omega t) = -2 \mu a^2 \omega \cos \omega t \sin \omega t, \]
  \[ \ddot{I}_{xx} = \frac{d}{dt} (-2 \mu a^2 \omega \cos \omega t \sin \omega t), \]
  \[ = -2 \mu a^2 \omega \left[ (-\omega \sin \omega t) \sin \omega t + \cos \omega t (\omega \cos \omega t) \right], \]
  \[ = -2 \mu a^2 \omega^2 (-\sin^2 \omega t + \cos^2 \omega t) = -2 \mu a^2 \omega^2 \cos 2\omega t, \]
  using the identity \( \cos^2 \omega t - \sin^2 \omega t = \cos 2\omega t \).
• \( \ddot{I}_{yy} \):
  \[ \ddot{I}_{yy} = -\ddot{I}_{xx} = 2 \mu a^2 \omega^2 \cos 2\omega t, \]
• \( \ddot{I}_{xy} \):
  \[ \dot{I}_{xy} = \mu a^2 \frac{d}{dt} (\cos \omega t \sin \omega t) = \mu a^2 \left[ (-\omega \sin \omega t) \sin \omega t + \cos \omega t (\omega \cos \omega t) \right], \]
  \[ = \mu a^2 \omega (\cos^2 \omega t - \sin^2 \omega t) = \mu a^2 \omega \cos 2\omega t, \]
  \[ \ddot{I}_{xy} = \mu a^2 \omega \frac{d}{dt}(\cos 2\omega t) = \mu a^2 \omega (-2\omega \sin 2\omega t) = -2 \mu a^2 \omega^2 \sin 2\omega t. \]

Step 4: GW Strain Amplitude for Arbitrary Inclination

In the transverse-traceless (TT) gauge, the GW strain for an observer at inclination \( \iota \) is:

\[ h_{ij}^{\text{TT}} = \frac{2G}{c^4 D} \ddot{I}_{ij}^{\text{TT}}, \]

where \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)), and \( D \) is in meters.

When accounting for inclination \( \iota \), we need to project the quadrupole tensor onto the observer's reference frame. For an observer along a general direction, the two polarization components become:

\[ h_+ = \frac{G}{c^4 D} \mu a^2 \omega^2 (1 + \cos^2 \iota) \cos 2\omega t, \]

\[ h_{\times} = \frac{2G}{c^4 D} \mu a^2 \omega^2 \cos \iota \sin 2\omega t. \]

The overall GW strain amplitude scales as:

\[ h \sim \frac{G}{c^4 D} \mu a^2 \omega^2 \sqrt{(1 + \cos^2 \iota)^2 + 4\cos^2 \iota}. \]

Substitute \( a \) and \( \omega \):

\[ a^2 = \left( \frac{G M P^2}{4\pi^2} \right)^{2/3}, \]

\[ \omega^2 = \frac{4\pi^2}{P^2}, \]

\[ h \sim \frac{G}{c^4 D} \mu \left( \frac{G M P^2}{4\pi^2} \right)^{2/3} \frac{4\pi^2}{P^2} \sqrt{(1 + \cos^2 \iota)^2 + 4\cos^2 \iota}. \]

Simplify:

\[ h = \frac{G}{c^4 D} \mu (G M)^{2/3} \frac{4\pi^{2/3}}{P^{2/3}} \sqrt{(1 + \cos^2 \iota)^2 + 4\cos^2 \iota}. \]

Relate to GW frequency \( f = \frac{2}{P} \) (since GWs oscillate at twice the orbital frequency), so \( P = \frac{2}{f} \):

\[ P^{-2/3} = \left( \frac{2}{f} \right)^{-2/3} = \left( \frac{f}{2} \right)^{2/3}, \]

\[ h = \frac{G}{c^4 D} \mu (G M)^{2/3} 4\pi^{2/3} \left( \frac{f}{2} \right)^{2/3} \sqrt{(1 + \cos^2 \iota)^2 + 4\cos^2 \iota}. \]

Introducing the chirp mass:

\[ M_{\text{chirp}} = \frac{(M_1 M_2)^{3/5}}{(M_1 + M_2)^{1/5}} = \mu^{3/5} M^{1/5}, \]

the strain can be written in the standard form:

\[ h = \frac{4}{D} \left( \frac{G M_{\text{chirp}}}{c^3} \right)^{5/3} (\pi f)^{2/3} \mathcal{A}(\iota), \]

where \( \mathcal{A}(\iota) = \sqrt{(1 + \cos^2 \iota)^2 + 4\cos^2 \iota}/4 \) is the inclination-dependent amplitude factor.

Step 4b: Frequency Evolution and Chirp

While many WD binaries can be treated as monochromatic sources, some systems evolve measurably during observation. The orbital decay due to gravitational wave emission causes the frequency to increase over time, producing a "chirping" signal.

The energy loss due to gravitational wave emission is given by:

\[ \frac{dE}{dt} = -\frac{32}{5}\frac{G^4}{c^5}(M_1M_2)^2(M_1+M_2)\left(\frac{2\pi f_{\text{orb}}}{c}\right)^{10/3} \]

The orbital energy can be related to frequency:

\[ E = -\frac{G M_1 M_2}{2a} = -\frac{G M_1 M_2}{2}\left(\frac{G(M_1+M_2)}{4\pi^2}\right)^{1/3}(2\pi f_{\text{orb}})^{2/3} \]

Taking the time derivative and equating with the energy loss rate:

\[ \frac{dE}{dt} = -\frac{G M_1 M_2}{2}\left(\frac{G(M_1+M_2)}{4\pi^2}\right)^{1/3} \cdot \frac{2}{3} \cdot (2\pi)^{2/3} \cdot f_{\text{orb}}^{-1/3} \cdot \dot{f}_{\text{orb}} \]

Solving for \(\dot{f}_{\text{orb}}\):

\[ \dot{f}_{\text{orb}} = \frac{96}{5} \pi^{8/3} \left(\frac{G M_{\text{chirp}}}{c^3}\right)^{5/3} f_{\text{orb}}^{11/3} \]

For gravitational waves with \(f_{GW} = 2f_{orb}\):

\[ \dot{f}_{GW} = 2\dot{f}_{\text{orb}} = \frac{96\pi^{8/3}}{5 \cdot 2^{8/3}} \frac{G^{5/3}M_{\text{chirp}}^{5/3}}{c^5} f_{GW}^{11/3} \]

This frequency derivative is a strong function of both frequency and chirp mass, making it potentially detectable for higher frequency systems with larger masses.

Step 5: GW Polarizations with Inclination

The time-dependent GW polarizations for a binary with inclination \( \iota \) are:

\[ h_+ = h_0 \frac{1 + \cos^2 \iota}{2} \cos(2\pi f t + \pi\dot{f}t^2 + \phi_0), \]

\[ h_{\times} = h_0 \cos \iota \sin(2\pi f t + \pi\dot{f}t^2 + \phi_0), \]

where \( h_0 \) is the overall amplitude and \( \phi_0 \) is an initial phase. Note that for chirping sources, we've included the $\pi\dot{f}t^2$ phase term.

For different inclinations:

• Face-on (\( \iota = 0 \)):
  \[ h_+ = h_0 \cos(2\pi f t + \pi\dot{f}t^2 + \phi_0), \quad h_{\times} = h_0 \sin(2\pi f t + \pi\dot{f}t^2 + \phi_0) \]
  The GW has equal plus and cross polarizations, 90° out of phase, creating circular polarization.
• Edge-on (\( \iota = \pi/2 \)):
  \[ h_+ = \frac{h_0}{2} \cos(2\pi f t + \pi\dot{f}t^2 + \phi_0), \quad h_{\times} = 0 \]
  Only the plus polarization remains, creating linear polarization with reduced amplitude.

2. Extracting Distance and Position with LISA

LISA, with its \( a \approx 2.5 \times 10^9 \, \text{m} \) baseline, uses the GW signal to infer \( D \), sky position \( (\alpha, \delta) \), and inclination \( \iota \).

Distance (\( D \)) and Inclination (\( \iota \))

From:

\[ h = \frac{4}{D} \left( \frac{G M_{\text{chirp}}}{c^3} \right)^{5/3} (\pi f)^{2/3} \mathcal{A}(\iota), \]

LISA measures \( h_+ \) and \( h_{\times} \) separately, allowing determination of both \( D \) and \( \iota \) through:

\[ \frac{h_{\times}}{h_+} = \frac{2\cos \iota}{1 + \cos^2 \iota}, \]

which gives \( \iota \), and then:

\[ D = \frac{4 \mathcal{A}(\iota)}{h} \left( \frac{G M_{\text{chirp}}}{c^3} \right)^{5/3} (\pi f)^{2/3}. \]

Note that for purely monochromatic sources, $M_{\text{chirp}}$ remains degenerate with distance, and may require electromagnetic counterparts or statistical priors to determine independently.

Chirp Measurement and Distance Precision

For binaries where LISA can measure \(\dot{f}_{GW}\), the chirp mass can be determined independently:

\[ M_{\text{chirp}} = \left(\frac{5c^5}{96\pi^{8/3}} \cdot \frac{\dot{f}_{GW}}{f_{GW}^{11/3}} \cdot \frac{2^{8/3}}{G^{5/3}}\right)^{3/5} \]

This breaks the degeneracy between chirp mass and distance, allowing for direct distance determination:

\[ D = \frac{4 \mathcal{A}(\iota)}{h} \left( \frac{G M_{\text{chirp}}}{c^3} \right)^{5/3} (\pi f)^{2/3} \]

LISA can typically detect frequency evolution when:

• The frequency shift over the observation period is larger than the Fourier bin width: $\dot{f}_{GW} \cdot T_{obs}^2 > 1$
• The signal-to-noise ratio is sufficient (typically SNR $\gtrsim$ 20)
• The binary has higher frequency ($f_{GW} \gtrsim 3$ mHz) and/or larger chirp mass

The precision in distance determination improves dramatically when $\dot{f}_{GW}$ is measurable:

\[ \frac{\sigma_D}{D} = \sqrt{\left(\frac{\sigma_h}{h}\right)^2 + \left(\frac{5}{3}\frac{\sigma_{M_{\text{chirp}}}}{M_{\text{chirp}}}\right)^2} \]

For typical WD binaries at mHz frequencies with 4-year observation times, distance precision can improve from factors of many tens of percent to approximately 10-30% when $\dot{f}_{GW}$ is detectable.

Sky Position (\( (\alpha, \delta) \))

LISA leverages several effects for sky localization:

• Doppler Modulation:
  \[ f_{\text{obs}} = f \left(1 + \frac{\mathbf{v} \cdot \mathbf{n}}{c} \right), \]
  where \( \mathbf{v} \) is LISA's orbital velocity (~30 km/s around the Sun), and \( \mathbf{n} \) is the unit vector to the source, modulating \( f \) over its 1-year orbit.
• Amplitude Modulation:
  \[ h(t) = F_+ h_+ + F_{\times} h_{\times}, \]
  where \( F_+, F_{\times} \) are antenna pattern functions depending on \( (\alpha, \delta) \) and LISA's orientation.
• Sky Localization Area: For short observations, the resolution is limited by the detector arm length \( a \):
  \[ \Delta \Omega_{\text{short}} \sim \frac{1}{\text{SNR}^2} \left( \frac{\lambda}{a} \right)^2, \quad \lambda = \frac{c}{f}, \]
  But for long-term observations (≈1 year), LISA's orbital motion provides an effective baseline of \( 2 R_{\text{orbit}} \) (≈ 2 AU):
  \[ \Delta \Omega_{\text{long}} \sim \frac{1}{\text{SNR}^2} \left( \frac{\lambda}{2 R_{\text{orbit}}} \right)^2 \]
  where SNR is the signal-to-noise ratio, and \( \lambda \) is the GW wavelength.

Example Calculation

For \( f = 1 \, \text{mHz} \) and short-term observations with LISA's arm length \( a \approx 2.5 \times 10^9 \, \text{m} \):

\[ \lambda = \frac{3