---
title: Default Spin Magnitude Population Model
url: https://www.emergentmind.com/topics/default-spin-magnitude-population-model
type: topic
---

# Default Spin Magnitude Population Model

The Default spin magnitude population model is the standard LIGO–Virgo–KAGRA population model for binary black hole component spin magnitudes used through GWTC-3. In its spin-magnitude sector, it assumes that the two component spins are independently and identically distributed and that each spin magnitude is drawn from a Beta distribution on the physical interval \([0,1]\). In the broader LVK default spin model, spin tilts are modeled separately by a mixture of an isotropic distribution and a truncated Gaussian in \(\cos\theta\) peaked at alignment, but the defining feature of the Default spin magnitude population model is the IID Beta prescription for the component magnitudes \(\chi_1\) and \(\chi_2\). Recent work has emphasized both its utility and its formal limitation: it is simple, convenient, and standard, yet it cannot assign finite probability mass to exactly zero spin, which becomes consequential when interpreting evidence for nonspinning subpopulations [2507.23663].

## 1. Formal definition within LVK population inference

In hierarchical inference, the model treats the mass-ordered component spin magnitudes \(\chi_1,\chi_2\) as latent variables and fits them with a population prior \(\pi_{\mathrm{pop}}(\theta\mid\Lambda)\), where \(\theta\) includes \(\{\chi_1,\chi_2\}\) and the spin-magnitude hyperparameters are
\[
\Lambda=\{\alpha_\chi,\beta_\chi\}.
\]
The defining assumption is that both component spins are drawn IID from a common Beta distribution on \([0,1]\). Equivalently, in the notation used as a baseline in later population studies,
\[
\pi(\chi_1,\chi_2\mid \alpha_\chi,\beta_\chi)
=
\mathrm{Beta}(\chi_1\mid\alpha_\chi,\beta_\chi)\,
\mathrm{Beta}(\chi_2\mid\alpha_\chi,\beta_\chi).
\]

This structure makes the model compact and exchangeable at the level of component spin magnitudes. It is also the baseline or “Vanilla model” in subsequent analyses of mass–spin correlations, where it represents the default mass-independent hypothesis against which more elaborate mass-dependent or mixture models are compared. In that later usage, the Default spin model remains a single-population model with uncorrelated mass, spin, and redshift distributions [2406.01679].

## 2. Boundary behavior, singularity, and the spike-at-zero problem

The central mathematical feature of the Beta family is its flexibility on bounded support, together with sharp limiting behavior at the endpoints. The model description emphasizes that
\[
\text{as } \alpha_\chi\to 0,\ \beta_\chi\to\infty,\quad p(\chi)\to \delta(\chi=0),
\]
and similarly
\[
\text{as } \alpha_\chi\to\infty,\ \beta_\chi\to 0,\quad p(\chi)\to \delta(\chi=1).
\]
This limiting property allows the inferred distribution to become extremely concentrated near zero or one. However, except in a limiting sense, the Beta family does not place finite probability mass exactly at \(\chi=0\) or \(\chi=1\).

The same analysis distinguishes between “singular” and “nonsingular” uses of the Beta family. When \(\alpha_\chi<1\), the density diverges at \(\chi=0\); when \(\beta_\chi<1\), it diverges at \(\chi=1\). This does not create a delta function. It instead creates a boundary divergence without finite point mass, which is precisely why the model can mimic but cannot formally represent a true nonspinning subpopulation.

| Variant | Hyperpriors | Consequence |
|---|---|---|
| Singular Beta | \(\alpha_\chi\sim \mathrm{U}(0.1,10)\), \(\beta_\chi\sim \mathrm{U}(0.1,10)\) | Allows boundary-divergent densities |
| Nonsingular Beta | \(\alpha_\chi\sim \mathrm{U}(1,10)\), \(\beta_\chi\sim \mathrm{U}(1,10)\) | Excludes boundary divergence |
| Truncated Gaussian alternative | \(\mu_\chi\sim \mathrm{U}(0,1)\), \(\sigma_\chi\sim \mathrm{U}(0.05,1)\) | Finite density at boundaries |

The paper’s terminology of “model mismatch” refers exactly to cases in which the true population includes a literal spike at zero spin. A fully nonspinning population has \(\chi_1=\chi_2=0\) for every event; a mixed population may contain a true subpopulation with \(\chi=0\); and a one-spinning-black-hole-per-binary population may have one component exactly zero in every binary. None of these are formally contained in a smooth IID Beta family. The model therefore “splits the difference” by concentrating density near zero, which can induce posterior railing against hyperprior boundaries and can bias posterior predictive distributions away from the true population shape [2507.23663].

## 3. Spin sorting and order-statistics reinterpretation

A major reinterpretive step is to move from the usual mass ordering to spin sorting. Standard gravitational-wave analyses define \(\chi_1\) and \(\chi_2\) by mass, with \(\chi_1\) attached to the more massive black hole and \(\chi_2\) to the less massive one. The alternative is to define
\[
\chi_A\ge \chi_B,
\]
where \(A\) denotes the higher-spin component and \(B\) the lower-spin component. This is especially natural when astrophysical formation scenarios predict asymmetric spin magnitudes, for example tidal spin-up in isolated binaries or binaries containing one higher-generation merger remnant.

Within the Default framework, spin sorting is not a fundamental parametrization but a derived construction. If \(p(\chi_{1/2}\mid\Lambda)\) is the common component-spin distribution and \(\mathrm{CDF}(\chi_{1/2}\mid\Lambda)\) its cumulative distribution function, then the spin-sorted distributions are the order statistics
\[
p(\chi_A)=2\,p(\chi_{1/2}\mid\Lambda)\,\mathrm{CDF}(\chi_{1/2}\mid\Lambda),
\]
\[
p(\chi_B)=2\,p(\chi_{1/2}\mid\Lambda)\left[1-\mathrm{CDF}(\chi_{1/2}\mid\Lambda)\right].
\]
These formulas matter because the larger of two low-spin draws need not itself peak at zero. Even when the underlying IID Beta density is concentrated near zero, the distribution of \(\chi_A\) can peak at appreciably nonzero values, whereas \(\chi_B\) remains much more sensitive to a low-spin excess.

This reinterpretation is deliberately imperfect in singly-spinning populations. If exactly one black hole per binary has nonzero spin, then the true component spins are not IID. The paper makes this explicit by noting that an implied marginal surrogate could be written as
\[
p(\chi)=\frac{1}{2}\left(\delta(\chi=0)+\mathrm{Beta}(\chi;\alpha_{\rm true},\beta_{\rm true})\right),
\]
but the order statistics of this IID surrogate do not recover the true sorted-spin laws. That deliberate mismatch is not treated as a flaw in itself; rather, it is the mechanism by which spin sorting exposes phenomenology that would be obscured in the original mass-ordered parameterization [2507.23663].

## 4. Simulated populations and discriminatory power under mismodeling

The model’s behavior was tested on several simulated binary black hole populations observable in an O4-like LIGO Hanford–Livingston network. Across all populations, the mass and distance distributions were held fixed: \(q\sim \mathrm{U}(0.25,1)\), detector-frame chirp mass distributed as \(\pi(\mathcal{M})\propto \mathcal{M}^{-3.5}\) on \([35,200]\,M_\odot\), together with standard distance and extrinsic priors. Spins were the only feature varied.

Three anchor populations were studied. The fully nonspinning population set \(\chi_1=\chi_2=0\) for every event and produced 156 detected events. The singly-spinning population assigned one component exactly zero in each binary and drew the other from \(\mathrm{Beta}(\alpha_{\rm true},\beta_{\rm true})\) with
\[
\alpha_{\rm true}=1.014,\qquad \beta_{\rm true}=3.402,
\]
producing 154 detected events; in spin-sorted variables its true distributions are \(\pi(\chi_A)=\mathrm{Beta}(\chi_A;\alpha_{\rm true},\beta_{\rm true})\) and \(\pi(\chi_B)=\delta(0)\). The fully spinning population drew both components IID from that same Beta distribution and produced 149 detected events. In addition, 11 mixed populations were constructed by combining the fully nonspinning and fully spinning populations at mixture fractions \(f_{\rm nospin}\in[0,1]\), with 149 events each.

For the fully nonspinning population, the Default model recovered distributions strongly concentrated near zero but never an actual spike. When singular Beta distributions were allowed, the recovered \(\chi_A\) distribution peaked at
\[
\chi_A=0.03^{+0.04},
\]
while the nonsingular version peaked at
\[
\chi_A=0.09^{+0.03}.
\]
These are substantially smaller than the GWTC-3 Default-model result
\[
\chi_A=0.29^{+0.13}.
\]
The paper further reports that for the nonspinning population recovered with the nonsingular Beta model,
\[
\chi_{A,99\%}=0.44^{+0.05},
\]
whereas GWTC-3 gives
\[
\chi_{A,99\%}=0.73^{+0.15},
\]
with the difference being inconsistent at \(>90\%\) credibility.

For the singly-spinning population, the inferred sorted-spin structure changed qualitatively. The recovered \(\chi_A\) distribution peaked at
\[
\chi_A=0.18^{+0.07}
\]
for the nonsingular Beta case and
\[
\chi_A=0.15^{+0.09}
\]
when singular Betas were allowed; in this case, \(\chi_A\) distributions peaking at zero were ruled out at \(>90\%\) credibility. Meanwhile \(\chi_B\) remained consistent with peaking at zero but less narrowly than in the fully nonspinning population. The paper quotes
\[
\chi_{B,99\%}=0.35^{+0.07}
\]
for the singly-spinning nonsingular-Beta case, compared with
\[
\chi_{B,99\%}=0.23^{+0.04}
\]
for the fully nonspinning case. The resulting conclusion is methodological as much as astrophysical: even under explicit model mismatch, the Default model’s spin-sorted posterior predictive distributions can still distinguish fully nonspinning from singly-spinning populations [2507.23663].

## 5. Mixed populations, comparison with alternatives, and observational conclusions

The mixed-population study addresses the question that the Default model cannot ask directly: how large a truly nonspinning subpopulation could still be compatible with existing observations? Because the model contains no explicit nonspinning-mixture parameter, it does not infer \(f_{\rm nospin}\) directly. Instead, simulated populations with known \(f_{\rm nospin}\) were mapped to recovered signatures in \(\chi_A\) and \(\chi_B\), especially through the summary statistic \(\chi_{A,99\%}\).

The principal results are comparative. Predominantly nonspinning populations with
\[
f_{\rm nospin}\ge 0.8
\]
are distinguishable from predominantly spinning populations with
\[
f_{\rm nospin}\le 0.1
\]
at \(90\%\) credibility. The GWTC-3 value of \(\chi_{A,99\%}\) is inconsistent with mixed populations having
\[
f_{\rm nospin}\gtrsim 0.8,
\]
but the fully spinning case is not ruled out. This yields the paper’s characteristic conclusion: current observations are inconsistent with a fully nonspinning binary black hole population, yet remain compatible with a population in which as much as \(80\%\) of sources are nonspinning.

The paper also compares the Default Beta model with a truncated Gaussian spin-magnitude population model parameterized by \(\Lambda=\{\mu_\chi,\sigma_\chi\}\) on \([0,1]\). The motivation is clear: a truncated Gaussian can peak at \(\chi=0\) or \(\chi=1\) with finite density, unlike a Beta distribution, which must either vanish or diverge at the boundaries. For the nonspinning population, the truncated Gaussian yielded an even narrower inferred distribution, with
\[
\chi_{A,99\%}=0.20^{+0.08}.
\]
For the singly-spinning population, however, the inference was broadly consistent with the Beta recovery. The stated conclusion is therefore not that the Beta family is uniquely problematic, but that smooth parametric models without an explicit delta-function component are generically limited when the true population contains exact zeros [2507.23663].

## 6. Scope, caveats, and related extensions beyond the default assumption

The paper is explicit about the scope of its claims. The hierarchical analysis fits spin magnitudes only; masses, redshift, and tilt hyperdistributions are not jointly inferred. The authors argue that cross-hyperparameter correlations are usually weak, but note that full joint modeling could increase Monte Carlo uncertainty. Selection effects in spin magnitude are neglected by effectively setting
\[
p_{\rm det}(\chi_{1/2})=1,
\]
which is justified in the appendix by showing that detectability is nearly independent of the small-spin regime relevant to the analysis. A further caveat is structural: the order-statistics map from mass-ordered to spin-sorted variables assumes IID component spins, which is violated in singly-spinning populations. The recovered \(\chi_A\) and \(\chi_B\) distributions are therefore effective summaries under the Default model, not exact reconstructions of the true underlying sorted-spin laws.

These caveats motivate extensions rather than abandonment. The analysis argues that the Default model remains useful as a phenomenological diagnostic and can support broad conclusions similar to those from more explicit spike-at-zero or mixture models, provided that spin sorting is used and the model is not overinterpreted as a literal generative law for populations with exact zeros. The specific forward-looking recommendation is to fit \(\chi_A\) and \(\chi_B\) independently, thereby breaking the assumption that the mass-sorted spins are IID.

Related work situates the Default model as a baseline rather than a final description. In analyses of mass–spin correlation, the same Beta–Beta law is used as the canonical mass-independent reference model, but mass-dependent transition models and mixture models are favored by the data. Those studies report strong evidence for a transition between low-mass, low-spin and high-mass, higher-spin binary black hole populations around \(40\text{--}50\,M_\odot\), with the high-mass population comprising \(\sim 2\%\) of the overall population and having spin magnitudes peaking around \(0.7\), though the exact high-mass spin shape is not robustly constrained. This suggests that the Default spin magnitude population model is best understood as a standard baseline: analytically convenient, widely adopted, and informative under spin sorting, but potentially underfit when the astrophysical population contains exact zero-spin components or mass-dependent structure [2406.01679].

Source: https://www.emergentmind.com/topics/default-spin-magnitude-population-model