Clues from $\mathcal{Q}$--A null test designed for line intensity mapping cross-correlation studies

TL;DR

The paper tackles the challenge of obtaining auto-power spectra in line intensity mapping (LIM) when cross-spectrum estimators are preferred for mitigating noise and systematics. It introduces the Q-estimator, a data-driven null test built from cross-spectra of four tracers, to assess the validity of the B19 cross-spectrum auto-spectrum reconstruction under linear bias and strong tracer correlation assumptions. Through toy models and halo-based LIM simulations (including star-formation lines and 21 cm), it shows that when $\\mathcal{Q} \\approx 1$ the B19 estimator is reliable; deviations signal decorrelation, non-linear bias, or interloper contamination. The results provide a practical, survey-level diagnostic to define trustworthy scales and redshift ranges for multi-line LIM analyses and guide future extensions to interloper handling and angular statistics.

Abstract

Estimating the auto power spectrum of cosmological tracers from line-intensity mapping (LIM) data is often limited by instrumental noise, residual foregrounds, and systematics. Cross-power spectra between multiple lines offer a robust alternative, mitigating noise bias and systematics. However, inferring the auto spectrum from cross-correlations relies on two key assumptions: that all tracers are linearly biased with respect to the matter density field, and that they are strongly mutually correlated. In this work, we introduce a new diagnostic statistic, \(\mathcal{Q}\), which serves as a data-driven null test of these assumptions. Constructed from combinations of cross-spectra between four distinct spectral lines, \(\mathcal{Q}\) identifies regimes where cross-spectrum-based auto-spectrum reconstruction is unbiased. We validate its behavior using both analytic toy models and simulations of LIM observables, including star formation lines ([CII], [NII], [CI],[OIII]) and the 21-cm signal. We explore a range of redshifts and instrumental configurations, incorporating noise from representative surveys. Our results demonstrate that the criterion \( \mathcal{Q} \approx 1 \) reliably selects the modes where cross-spectrum estimators are valid, while significant deviations are an indicator that the key assumptions have been violated. The \( \mathcal{Q} \) diagnostic thus provides a simple yet powerful data-driven consistency check for multi-tracer LIM analyses.

Figures (16)

• Figure 1: The observed frequency $\nu_{\rm obs}$ of selected star formation lines as a function of redshift $z$. The shaded regions denote the frequency coverage of various LIM experiments, as shown in the top legend. The five vertical dashed lines correspond to $z=1,2,3, 6$ and $8$. At a given redshift, solid lines passing through a given shaded region will be observed by that corresponding experiment. The horizontal dotted straight lines represent the $10-1000$ GHz frequency coverage of the hypothetical Super-LIM experiment considered in this paper.
• Figure 2: The $\mathcal{Q}$ estimator as a function of spatial wavenumber $k$ for two toy models. The blue horizontal line represents $\mathcal{Q}$ for four perfectly correlated realizations of density fields generated from a realization of a matter density field at $z=6$, while the red fluctuating line shows the result for four independent random Gaussian fields that are completely uncorrelated by construction. The $\mathcal{Q}$ statistic is computed using a single combination of the cross-power spectra $(P_{\rm AB}P_{\rm CD})/(P_{\rm AC}P_{\rm BD})$; it is equal to unity at all $k$ for the perfectly correlated fields, which is expected. However, this is not true for the uncorrelated fields, and we see large fluctuations in $\mathcal{Q}$ due to the presence of zeroes in the denominator.
• Figure 3: The $\mathbf{\mathcal{Q}}$ estimator (top row) and the B19 estimator (bottom row) both as a function of $k$ for one of the five illustrative configurations (Cases I–V, as described in Section \ref{['sec:param_explore']}) obtained by varying the SFR–luminosity slope parameters $\beta_{\rm SFR}$ of four mock lines while keeping $\alpha_{\rm SFR}$ and $M_{\rm min}$ fixed. One sees that different patterns of halo weighting across the four tracers can lead to qualitatively different behaviours of the B19 estimator, and that no single configuration of lines is universally optimal for computing $\mathcal{Q}$: some arrangements are intrinsically more prone to false positive or false negative outcomes than others.
• Figure 4: Performance of the $\mathcal{Q}$–estimator and the B19 power–spectrum estimator for four star–formation–tracing lines ([C${\rm \textsc{ii}}$], [N${\rm \textsc{ii}}$], [C${\rm \textsc{i}}$], and [O${\rm \textsc{iii}}$]) in the absence of instrumental noise at $z=2$. The left panel shows the three $\mathcal{Q}$ combinations, $\mathcal{Q}_1$, $\mathcal{Q}_2$, and $\mathcal{Q}_3$ (defined in Eq. \ref{['eq:Qi_defs']}), as functions of $k$. On large scales all three combinations are consistent with $\mathcal{Q}\simeq1$, while $\mathcal{Q}_2$ and $\mathcal{Q}_3$ begin to deviate at the few percent level for $k\gtrsim0.2\,{\rm Mpc}^{-1}$, reflecting small differences in halo weighting between the tracers. The middle and right panels show the ratios of the B19–reconstructed power spectra to the true spectra for [C${\rm \textsc{ii}}$] and [N${\rm \textsc{ii}}$], respectively, for all tri–line combinations: the B19 estimator is accurate to better than $\sim5\%$ on scales where the corresponding $\mathcal{Q}_i$ remain close to unity, and gradually departs from unity once the $\mathcal{Q}$ combinations begin to drift. The rightmost panel displays the cross–correlation coefficients $r_{ab}(k)$ between all pairs of lines, demonstrating that the tracers are very highly correlated ($r_{ab}\simeq1$) at $k\lesssim0.2\,{\rm Mpc}^{-1}$, with correlation degrading on smaller scales where the departures in both $\mathcal{Q}$ and B19 become visible.
• Figure 5: Same as Fig. \ref{['fig:LIM_no_noise']}, but including Gaussian instrumental noise appropriate for the fiducial Super–LIM configuration with $N_{\rm det}\,t_{\rm survey}=2\times10^5\,{\rm hr}$. The left panel shows $\mathcal{Q}_1$, $\mathcal{Q}_2$, and $\mathcal{Q}_3$ as functions of $k$, with shaded bands indicating the $1\sigma$ scatter over many noise realizations. On large scales ($k\lesssim0.3\,{\rm Mpc}^{-1}$) all three combinations remain consistent with $\mathcal{Q}=1$ within the uncertainties, while at smaller scales $\mathcal{Q}_2$ and $\mathcal{Q}_3$ exhibit statistically significant departures, signalling the breakdown of the assumptions underlying the B19 estimator. The middle and right panels show the B19–reconstructed power spectra for [C${\rm \textsc{ii}}$] and [N${\rm \textsc{ii}}$] (normalized by the true spectra) for different tri–line combinations; for [C${\rm \textsc{ii}}$] the reconstruction remains accurate up to $k\simeq0.3\,{\rm Mpc}^{-1}$, whereas for the fainter [N${\rm \textsc{ii}}$] line the deviations become more pronounced and combination–dependent. The scales where $\mathcal{Q}_2$ and $\mathcal{Q}_3$ depart from unity coincide with the onset of bias in the B19 reconstructions, illustrating the utility of $\mathcal{Q}$ as a noise–robust null test.