---
title: Multi-Tracer Technique in Cosmology & Imaging
url: https://www.emergentmind.com/topics/multi-tracer-technique
type: topic
---

# Multi-Tracer Technique in Cosmology & Imaging

The multi-tracer technique denotes a class of methods in which two or more tracers are analysed jointly so that relative responses rather than absolute amplitudes carry the signal of interest. In large-scale structure, it “employs a ratio of densities of two differently biased galaxy samples that trace the same underlying matter density field,” and was proposed to alleviate the cosmic variance problem [1904.10857]. In laboratory X-ray and PET settings, related multi-tracer strategies use energy thresholds, K-edges, prompt-gamma triple coincidences, or tracer-conditioned synthesis to distinguish or reconstruct multiple tracer classes within a single acquisition or model [2207.07724]. This suggests a broader methodological family in which multiple tracers are used either to cancel a shared variance term, disentangle source populations, or recover complementary biological and dynamical information.

## 1. Conceptual basis and information content

In the cosmological formulation, the central idea is that different tracer populations sample the same underlying matter field but with different biases. Cosmic variance is common to all tracers, whereas the relative clustering amplitudes depend on the tracer-dependent response. Abramo and Leonard showed that galaxy surveys mapping multiple species of tracers can improve constraints on some cosmological parameters “far beyond the limits imposed by a simplistic interpretation of cosmic variance,” and that the relative clustering amplitudes between tracers are eigenvectors of the multi-tracer Fisher matrix [1302.5444].

A compact statement of the formalism uses the effective powers
\[
\mathcal{P}_{\alpha}(\vec{k};\vec{x}) = \bar{n}_\alpha(\vec{x}) P_\alpha(\vec{k};\vec{x}), \qquad
\mathcal{P}(\vec{k}; \vec{x}) = \sum_{\alpha=1}^{N_t} \mathcal{P}_\alpha .
\]
For parameters $\log \mathcal{P}_\alpha$, $\log \mathcal{P}_\beta$, the Fisher information density per mode is
\[
F_{\alpha \beta} (\vec{k};\vec{x}) =
\frac{1}{4}
\left[
\delta_{\alpha \beta}  \, \frac{\mathcal{P}_\alpha  \, \mathcal{P}}{1+\mathcal{P}}
+
\frac{\mathcal{P}_\alpha  \mathcal{P}_\beta  \, (1-\mathcal{P})}{(1+\mathcal{P})^2}
\right].
\]
After diagonalization, the total clustering amplitude is still cosmic-variance limited, but the relational variables are not. In one scenario analysed in that work, the enhancement was “as large as a factor of $\sim 3$ for the accuracy in the determination of the redshift distortion parameter, and a factor $\sim 5$ for the local non-Gaussianity parameter” [1302.5444].

The same logic appears in alternative guises outside standard galaxy clustering. When two differently biased fields can be ratioed, or when two tracer species have separable detector signatures, the dominant common fluctuation can be suppressed and the relative information isolated. This is explicit in 21-cm applications, in multi-population radio surveys, and in energy-resolved X-ray or PET schemes discussed below.

## 2. Estimators, likelihoods, and survey pipelines

The multi-tracer formalism has been implemented both in Fourier space and in harmonic space. In Fourier analysis, Abramo, Secco, and Loureiro presented optimal quadratic estimators for surveys with several tracer species. Their estimators can “simultaneously fit the matter power spectrum and the biases of the tracers - as well as redshift-space distortions (RSDs), non-Gaussianities (NGs), or any other effects that are manifested through differences between the clusterings of distinct species of tracers,” reduce to Feldman–Kaiser–Peacock for a single tracer, and project to the Percival–Verde–Peacock estimator when only the underlying power spectrum is estimated [1505.04106]. The weighted fields are built from
\[
w_{\sigma\alpha}(\vec{x},\vec{k}) =
\Bigg[\delta_{\sigma\alpha} - \frac{\mathcal{P}_\sigma}{1+\mathcal{P}} \Bigg] \bar{n}_\alpha B_\alpha(\vec{x},\vec{k}),
\qquad
f_\sigma (\vec{x},\vec{k}) = \sum_\alpha w_{\sigma\alpha}(\vec{x},\vec{k}) \delta_\alpha(\vec{x}),
\]
and the covariance of the resulting estimators is the inverse of the multi-tracer Fisher matrix [1505.04106].

A directly applied version is the Multi-Tracer Optimal Estimator (MTOE). In VIPERS mocks, MTOE provided more accurate measurements than FKP “independently of the tracer-selection strategy adopted, on both small and large scales.” The reported gains were an average error reduction of “$\sim$ 40$\%$ at $k \, [h$ Mpc$^{-1}]\gtrsim 0.3$,” a gain of “$\sim$ 10$\%$ for the ratios of $P^{(0)}_\alpha(k)$” on large scales, and—when extending to $0.3 < k \, [h$ Mpc$^{-1}]< 0.5$—average improvements of “$\sim$ 30 $\%$ for the amplitudes of the monopoles, $\sim$ 70 $\%$ for the monopole ratios, and $\sim$ 20 $\%$ for the galaxy biases” [1909.00010].

In harmonic space, Tanidis and Camera developed a likelihood-based pipeline for angular power spectra that includes single and multiple tracers, density fluctuations, redshift-space distortions, and weak lensing magnification. The observable is the full set of auto- and cross-spectra,
\[
\langle g^{A,i}_{\ell m} g^{B,j*}_{\ell' m'} \rangle = \delta^{\rm K}_{\ell\ell'} \delta^{\rm K}_{mm'} C^{\rm g}_\ell(z^A_i, z^B_j),
\]
with Gaussian covariance generalized to the multi-tracer, multi-bin case. Their publicly released Modified CosmoSIS implementation produced an “enhancement of $44\%$ on the $2\sigma$ upper bound on the sum of neutrino masses” [2009.05584].

## 3. Cosmological parameter inference

A major application of the multi-tracer technique is neutrino mass inference from scale-dependent clustering. In the presence of massive neutrinos,
\[
\delta_h = b\, \delta_X, \qquad
\delta_m = (1 - f_\nu) \delta_{bc} + f_\nu \delta_\nu ,
\]
and the bias with respect to total matter displays a step-like scale dependence around the neutrino free-streaming scale. Fonseca and collaborators combined SKAO-MID 21cm intensity mapping, LSST-like photometric clustering, and CMB-S4 lensing in a joint Fisher analysis of the full set of angular auto- and cross-spectra. The forecasted $1\sigma$ uncertainties were $\sigma(M_\nu) \simeq 234$ meV for SKAO-MID alone, $\sigma(M_\nu) \simeq 782$ meV for LSST-like photometric clustering alone, $\sigma(M_\nu) \simeq 45$ meV for SKAO-MID $\times$ CMB-S4, and $\sigma(M_\nu) \simeq 12$ meV for the full SKAO-MID + LSST + CMB-S4 multi-tracer combination, using only linear clustering information and without a prior on optical depth [2109.03763].

For local primordial non-Gaussianity, the modern optimization criterion is not the naive bias difference. Sailer, Ferraro, and White showed that for two galaxy samples $A$ and $B$ the constraining power is
\[
\propto |b_1^B b_\phi^A - b_1^Ab_\phi^B|,
\]
rather than the traditional expectation $\propto |b_1^A - b_1^B|$. Using IllustrisTNG galaxy simulation data, they found that different equal galaxy number splits lead to different values of this combination, and that “splitting by $g-r$ color is the most promising, more than doubling the significance of detecting $f_{\rm NL}b_\phi \neq 0$” [2302.09066].

Radio surveys provide a complementary route because they naturally contain multiple populations with different halo biases. Forecasts for future radio continuum surveys that include star-forming galaxies and radio AGN populations found that in the most realistic case the $1-\sigma$ error on $f_{\rm NL}$ “falls within the range 4.07 and 6.58” [1912.08362]. An earlier Fisher analysis based on simulated redshift distributions for star-forming galaxies, starburst galaxies, radio-quiet quasars, and FRI/FRII populations reported $\sigma_{f_{\rm NL}}=3.6$ for a galaxy detection flux limit of $10\,\mu$Jy and $\sigma_{f_{\rm NL}}=2.2$ for $1\,\mu$Jy, compared with $\sigma_{f_{\rm NL}}=48$ and $\sigma_{f_{\rm NL}}=12$ for the corresponding undifferentiated populations [1402.2290].

## 4. Beyond linear theory: EFT, non-linear bias, and higher-order statistics

The multi-tracer programme has been extended beyond linear theory in two main directions: perturbative bias expansion and higher-order correlators. In the Effective Field Theory of Large-Scale Structure, the tracer density contrast is written as
\[
\delta_g(\mathbf{x}, \tau) = \sum_\mathcal{O} b_\mathcal{O}(\tau) \mathcal{O}(\mathbf{x}, \tau) + \epsilon(\mathbf{x}, \tau) + \sum_{\mathcal{O}} \epsilon_{\mathcal{O}}(\mathbf{x}, \tau)\mathcal{O}(\mathbf{x}, \tau),
\]
and cross-spectra are modelled with independent bias parameters for each tracer. Kokron, De Rose, and collaborators found that the constraints in $\omega_{\rm cdm}$ and $h$ using multi-tracer are “less biased and approximately $60\%$ better than those obtained for a single tracer,” while bias-expansion parameters typically have errors “half of the single-tracer case” [2108.11363].

The non-linear-bias perspective sharpens the optimization problem. Cabass, Ivanov, Philcox, and Simonović showed that, within non-linear bias expansion, “directly splitting the non-linear bias generally leads to smaller error bars in $A_s$, $h$, and $\omega_{\rm cdm}$ compared to a simple split in $b_1$.” For a characteristic bias split, the reported reductions in marginalized $1\sigma$ errors were $13\%$, $18\%$, and $24\%$ for $\omega_{\rm cdm}$, $h$, and $A_s$ with a $b_1$ split in EFT, and $34\%$, $38\%$, and $56\%$ when splitting all non-linear biases together. The same study found that substantial gains persist down to $\bar{n} \sim 10^{-4}~h^3\mathrm{Mpc}^{-3}$, that improvements hold for unbalanced splits with one subsample containing as little as $10$–$20\%$ of the total, and that extending to more than two tracers “does not provide further improvements” [2504.18245].

At the level of higher-order statistics, Gualdi, Gil-Marín, and Verde developed a bispectrum multi-tracer formalism. For two tracers, the power-spectrum vector is
\[
\bm{D_P} = [P^{tt},\,P^{tt'},\,P^{t't'}],
\]
while the bispectrum vector is
\[
\bm{D_B} = [B^{ttt},\,B^{(ttt')},\,B^{(tt't')},\,B^{t't't'}].
\]
Their Fisher analysis suggested that optimal constraints arise from maximizing “the ratio of number densities, the difference between the linear biases, the difference between the quadratic biases, and the difference between the products $b_1\,b_\Phi$ for each tracer” [2305.04028].

## 5. Extensions to 21-cm cosmology, stochastic backgrounds, and diffuse skies

A distinctive variant applies the multi-tracer ratio directly to the 21-cm field. During cosmic dawn and reionization, the redshift-space ratio of 21-cm brightness fluctuations to galaxy density fluctuations can be written as
\[
R^{\rm lin}_{\bf k}(z) \equiv \frac{\delta_{21}({\bf k}, z)}{\delta_{\rm gal}({\bf k}, z)} =
\frac{T_{21}(z)\big[W(k, z) + \mu^2_{\bf k}\big]}{b + \mu^2_{\bf k}},
\]
where $T_{21}(z)$ is the sky-averaged 21-cm signal, $W(k,z)$ is a source window function, and $b$ is the galaxy bias. Fialkov and Barkana argued that the anisotropy of this ratio can be used to measure the global 21-cm signal, probe the spectral energy distribution of radiative sources, and extract large-scale properties of the second tracer. In idealized galaxy surveys the method works well, but in more realistic surveys limited to highly biased luminous galaxies, “the inevitable Poisson noise makes the reconstruction far more challenging” [1904.10857].

Simulated multi-tracer analyses with HI intensity mapping and photometric galaxies show both the power and the fragility of cosmic-variance cancellation. Full-sky simulations of an SKA-like 21cm survey combined with an LSST-like photometric survey found that, in the absence of foregrounds and with realistic noise levels, multi-tracer estimators improve sensitivity by a factor of “$2$-$4$.” When foregrounds are included, estimators using the 21cm auto-correlation become biased, but cross-correlation estimators are “immune to this” and remain better than the cosmic-variance-contaminated scenario, even though foreground cleaning and poor radial resolution reduce the gain [1808.03093].

The same conceptual structure now appears in other sky fields. Multi-Tracer Correlated Stacking for the nano-Hz stochastic gravitational wave background stacks the SGWB signal over sky pixels coincident with overdensities of AGNs, quasars, or bright galaxies. In simulations, the technique “uniquely distinguishes between isotropic and anisotropic distributions of SGWB source, surpassing the capabilities of angular power spectrum-based methods” [2501.01499]. For the unresolved $\gamma$-ray background, cross-correlations with DES galaxies gave a signal-to-noise ratio of $7.96$, and a multi-tracer combination with DES weak lensing increased the total significance to $8.6$, “firmly establishing the extragalactic origin of the UGRB” [2601.13312].

## 6. Imaging, laboratory X-ray velocimetry, and PET

In detector-based imaging, the multi-tracer idea is realized by exploiting tracer-dependent instrumental signatures. In laboratory X-ray particle tracking velocimetry, energy-threshold photon counting detectors can “directly measure the deposited photon energy,” and systems with “two or more thresholds” enable K-edge material detection. The study introducing tungsten-coated hollow carbon spheres of $O(50~\mu\mathrm{m})$ diameter showed that laboratory XPTV is practical with a single-threshold PCD and that “energy-thresholding identification of different classes of tracers is feasible.” The tungsten-coated particles had average SNR $\sim 47$ versus $\sim 25$ for AGSF-33 silver-coated particles, while the K-edge values listed were $69.5$ keV for tungsten and $25.5$ keV for silver [2207.07724].

The attenuation model is the Beer–Lambert law,
\[
I = I_0 \exp(-\mu(E) x),
\]
so multi-tracer discrimination follows from material-dependent jumps in $\mu(E)$ at the K-edge. The same work noted present PCD count-rate limits of nominally $\sim 10^6$–$10^7$ photons/pixel/s, larger pixel sizes of $O(100~\mu\mathrm{m})$, and the importance of tracer monodispersity, but also emphasized that future multiple-threshold PCDs could track “multiple tracer species and scalar fields simultaneously” [2207.07724].

A PET analogue uses isotopes with distinct coincidence structure. In multiplexed PET with $^{55}$Co and $^{18}$F, $^{55}$Co emits a prompt gamma in cascade with a positron in $75\%$ of decays, producing triple coincidences, whereas $^{18}$F produces only double coincidences. The unmixing procedure was
\[
\text{F-18 image} = \text{Double image} - g \times \text{Triple image},
\qquad
\text{Co-55 image} = \text{Triple image} - f \times \text{Double image},
\]
with empirically measured values $g=12.6$ and $f=0.005813$. Phantom studies found linear relationships between coincidence counts and activity, and hot-to-background ratios in unmixed images were typically within $<15\%$ error; the study also demonstrated simultaneous in vivo imaging of two tracers in a single PET session [2411.08237].

A different branch of the literature uses “multi-tracer PET” to denote joint synthesis or translation of multiple PET tracers from other modalities. The 3D unified anatomy-aware cyclic adversarial network (UCAN) translates among $^{18}$F-FDG, $^{11}$C-UCB-J, and $^{11}$C-PiB using a single conditional model with MRI guidance, reporting “NMSE less than 15% for all PET tracers” [2107.05491]. DIReCT$++$ synthesizes $^{18}$F-AV-45 and $^{18}$F-FDG from T1-weighted MRI plus clinical information using a 3D rectified flow architecture and BiomedCLIP conditioning; the reported downstream performance for MRI + synthesized multi-tracer PET was $93.5\%$ ACC and $95.7\%$ AUC for AD vs. CN, and $81.96\%$ accuracy for EMCI vs. LMCI compared with $77.65\%$ for MRI alone [2604.11176]. RelA-Diffusion extends this paradigm to TAU, PBR, and PIB synthesis from T1-weighted and T2-FLAIR MRI with tracer-label conditioning, reporting PSNR values of $28.314$, $29.324$, and $26.270$ for TAU-PET, PBR-PET, and PIB-PET on the NFL-LONG dataset [2602.21345].

## 7. Limitations, optimization criteria, and outlook

Across domains, the technique is powerful only when the tracer responses differ and the shared modes are actually observed in common. In cosmological survey combinations, the overlap sky area is crucial because cosmic variance cancellation requires observing the same modes, and the main practical limitations are cross-survey systematics, foregrounds and noise, and redshift accuracy [2109.03763]. In HI intensity mapping, foreground cleaning removes long-wavelength radial modes, and when combined with the low redshift resolution of photometric surveys it reduces the sensitivity of the multi-tracer estimator, even though cross-correlation estimators remain robust to foreground bias [1808.03093].

Tracer selection is therefore an optimization problem rather than a purely classificatory one. For power spectra and bispectra, the formal analyses suggest maximizing the ratio of number densities, the difference between linear biases, the difference between quadratic biases, and the difference between the products $b_1 b_\Phi$ [2305.04028]. Beyond linear theory, the most efficient split need not be the one with the largest $b_1$ contrast, because directly splitting non-linear bias coefficients can yield smaller error bars in $A_s$, $h$, and $\omega_{\rm cdm}$, and more than two tracers need not help once shot noise rises as the sample is subdivided [2504.18245].

In detector physics, the limiting factors are different but structurally analogous: count-rate limits, spatial resolution, and tracer manufacturability or monodispersity in X-ray PCD systems; triple-coincidence statistics in multiplexed PET; and the availability of paired data and faithful conditioning variables in synthetic multi-tracer PET models [2207.07724]. A plausible implication is that the multi-tracer technique is best viewed not as a single algorithm but as a design principle: construct tracer classes whose shared latent field is common, whose responses are measurably distinct, and whose joint analysis exposes relative information that a single tracer cannot isolate.

Source: https://www.emergentmind.com/topics/multi-tracer-technique