Cross-Frame Intensity Mapping

• Cross-frame intensity mapping is a technique that cross-correlates LIM observations with Ly-α forest data to enhance signal detection and mitigate systematics.
• It employs a Fourier power-spectrum framework along with noise modeling and simulations to forecast significant S/N improvements in high-redshift studies.
• The method effectively suppresses uncorrelated foregrounds, enabling robust constraints on cosmological parameters through enhanced cross-correlation analyses.

Cross-frame intensity refers to the technique of cross-correlating molecular or atomic line intensity mapping (LIM) observations with independent large-scale structure tracers, primarily the Ly-$\alpha$ forest, to enhance measurement fidelity and detect faint cosmological emission backgrounds. This methodology exploits the statistical independence of systematic contaminants between LIM maps and Ly-$\alpha$ absorption, providing a robust avenue for improving signal-to-noise ratios (S/N) and constraining physical parameters in the high-redshift universe (Qezlou et al., 2023).

1. Power-Spectrum Framework

The cross-frame intensity mapping analysis is conducted in Fourier space, using the fluctuation fields:

• $\delta I_{\rm LIM}(\mathbf{k})$ — LIM-measured line-intensity fluctuation,
• $\delta F_{{\rm Ly}\alpha}(\mathbf{k})$ — fluctuation in Ly-$\alpha$ transmitted flux.

The key power spectra are:

• LIM auto-power: $$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$
• Ly-$\alpha$ auto-power: $$P_{{\rm Ly}\alpha}(\mathbf{k}) \equiv \langle \delta F_{{\rm Ly}\alpha}(\mathbf{k})\, \delta F_{{\rm Ly}\alpha}^*(\mathbf{k}) \rangle$$
• Cross-power: $$P_{\rm LIM \times Ly\alpha}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta F_{{\rm Ly}\alpha}^*(\mathbf{k}) \rangle$$

Under a linear-bias plus shot noise model at redshift $z$, these can be written: $$\begin{align*} P_{\rm LIM}(k) &\simeq [T_{\rm CO} b_{\rm CO}]^2\,P_m(k) + P_{\rm shot,CO}\ P_{{\rm Ly}\alpha}(k) &\simeq [b_{{\rm Ly}\alpha}]^2\,P_m(k) + P_{\rm shot,\,Ly\alpha}\ P_{\rm LIM\times Ly\alpha}(k) &\simeq T_{\rm CO}\,b_{\rm CO}\,b_{{\rm Ly}\alpha}\,P_m(k) \end{align*}$$ where $P_m(k)$ is the underlying matter power spectrum, $T_{\rm CO}$ is mean line intensity (e.g., in $\mu$K), $b_{{\rm CO}}$ and $b_{{\rm Ly}\alpha}$ are the respective linear biases, and $P_{\rm shot}$ denotes Poisson contributions.

2. Noise Modeling and Signal-to-Noise Estimation

Accurate forecasts require comprehensive noise modeling for both LIM and Ly-$\alpha$ observables. For LIM, the instrumental noise per 3D voxel (volume $V_{\rm vox}$) is: $$P_{\rm LIM,\,noise}(k,\mu) = \frac{\sigma_N^2}{V_{\rm vox}} W(k,\mu)^{-1}$$ with a beam/channel window function $$W(k,\mu)=\exp[-(k_\perp\sigma_\perp)^2-(k_\parallel\sigma_\parallel)^2]$$.

For COMAP-Y5 at $z \sim 2.5$, $\sigma_N \approx 17.8\,\mu$K, $$V_{\rm vox}\approx 2.64\times4.66\,h^{-1}\,{\rm cMpc}^3$$, and $\sigma_{\perp,\parallel}$ determined by beam and spectral resolution.

Ly-$\alpha$ tomography noise is dominated by finite effective sightline density $n_{2D,\,{\rm eff}}$,

$$P_{{\rm Ly}\alpha,\,{\rm noise}}(k) \approx \frac{1}{ n_{2D,\,{\rm eff}} }$$

assuming pixel noise S/N per Å ≈ 2.

The cross-power spectrum variance per $k$-mode is

$$\sigma^2[P_{\rm LIM\times Ly\alpha}] = [P_{\rm LIM}+P_{\rm LIM,\,noise}][P_{{\rm Ly}\alpha}+P_{{\rm Ly}\alpha,\,{\rm noise}}] + [P_{\rm LIM\times Ly\alpha}]^2$$

Summing in inverse variance across $k$-bins gives total S/N: $$\left(\frac{S}{N}\right)^2 = \sum_k \frac{[P_{\rm LIM\times Ly\alpha}(k)]^2}{[P_{\rm LIM}(k)+P_{\rm LIM,\,noise}(k)][P_{{\rm Ly}\alpha}(k)+P_{{\rm Ly}\alpha,\,{\rm noise}}(k)] + [P_{\rm LIM\times Ly\alpha}(k)]^2}$$

3. Simulation Methodology

Cross-frame intensity analyses employ large-volume cosmological hydrodynamic simulations to model both LIM and Ly-$\alpha$ signals. The ASTRID suite is a prominent example:

• Modified GADGET-3 code with SPH and tree+PM gravity.
• Volume: $250\,h^{-1}$ cMpc, $2\times5500^3$ particles.
• Star formation: Springel & Hernquist multiphase ISM, molecular-H$_2$ correction.
• Cooling: Katz, Weinberg & Hernquist rates; UV background as per Faucher-Giguère, rescaled to match $\langle F\rangle_{\rm obs}$.
• Reionization: patchy H I ($z > 6$) via Battaglia map, patchy He II ($z > 2.8$) via 30 $h^{-1}$ cMpc stochastic bubbles.
• Black holes: seeded in $M_{\rm halo}>5\times 10^{9}\,h^{-1}M_\odot$, Bondi accretion with $5\%$ thermal feedback.

Mock Ly-$\alpha$ forest sightlines are generated using FAKESPECTRA on a 250 $h^{-1}$ ckpc grid, enforcing alignment with SDSS DR14 1D flux power to ≤10%. Molecular emission (e.g., CO) utilizes a subhalo–SFR–$L_{\rm CO}$ double power law calibrated to COMAP Early Science, distributed by cloud-in-cell mapping and converted to brightness via the standard luminosity–temperature formula.

4. Quantitative Forecasts and Empirical Results

Forecasted signal-to-noise enhancements from cross-frame methods derive from simulated observations:

• COMAP$\times$PFS Ly-$\alpha$ tomography (mean sightline separation $d_\perp\sim2.5$–3.7 $h^{-1}$ cMpc) achieves a $\sim$200–300% increase in COMAP detection S/N relative to auto-only LIM.
• COMAP$\times$eBOSS or COMAP$\times$DESI ($d_\perp\sim10$–13 $h^{-1}$ cMpc) yields a 50–75% S/N improvement over LIM alone.
• For [C II] intensity mapping: EXCLAIM$\times$(DESI/eBOSS Ly-$\alpha$ forest) achieves $$(S/N)_{\rm [CII]\times Ly\alpha} \sim 10\times (S/N)_{\rm [CII]\times quasar}$$, reflecting a substantial gain due to higher sightline density and the negative Ly-$\alpha$ bias ($b_{{\rm Ly}\alpha}\approx -0.20$).

A summary of these quantitative improvements is provided in the following table:

Probe Pair	Sightline Sep ($h^{-1}$ cMpc)	Expected S/N Improvement
COMAP × PFS Ly-$\alpha$	2.5–3.7	200–300%
COMAP × (eBOSS or DESI)	10–13	50–75%
EXCLAIM × (DESI/eBOSS Ly-$\alpha$) vs. EXCLAIM × quasar	—	$\sim$10×

Achievements observed in simulation are directly relevant to ongoing and planned surveys due to the overlap of eBOSS Stripe 82 with multiple LIM projects.

5. Suppression of Foregrounds and Systematics

A central attribute of cross-frame intensity mapping is systematic error suppression through cross-correlation. LIM foregrounds (e.g., Galactic continuum, interloper lines) contribute only to the LIM auto spectrum ($P_{\rm LIM} + P_{\rm LIM,\,noise}$) and are uncorrelated with Ly-$\alpha$ transmission fluctuations; these contaminants vanish in the cross-spectrum $P_{\rm LIM\times Ly\alpha}$. Conversely, Ly-$\alpha$ forest systematics (e.g., continuum fitting, damped Ly-$\alpha$ masking) are uncorrelated with LIM measurements. Thus, the cross-power spectrum delivers an unbiased probe of large-scale structure, substantially reducing the influence of instrument- or sky-specific contaminants on detection significance.

This suggests that cross-frame analysis provides a foundational method for achieving early, robust detections of LIM signals that would otherwise be dominated by systematic uncertainties.

6. Scientific Implications and Applications

Cross-frame intensity mapping enables precise characterization of large-scale structure, the clustering of molecular emission, and the underlying matter distribution at high redshift. It achieves superior S/N relative to standard auto-spectrum techniques and is competitive with even spectroscopic galaxy survey cross-correlations in raw S/N. The straightforward modeling of the Ly-$\alpha$ absorption power spectrum further tightens physical constraints, especially on parameters such as line bias and shot noise.

A plausible implication is that the deployment of cross-frame cross-correlations (e.g., LIM$\times$Ly-$\alpha$) in early phases of LIM surveys could expedite cosmological signal confirmation, influence survey strategy, and prioritize resources for overlap with dense Ly-$\alpha$ forest fields.

7. Future Prospects

The effectiveness of cross-frame intensity mapping for foreground and systematic mitigation portends significant advances in LIM cosmology as survey capabilities expand. Future improvements in sightline density (e.g., through next-generation Ly-$\alpha$ tomography) and enhancements in LIM instrumental sensitivity will further amplify the achievable S/N gains. Overlapping survey fields such as those in eBOSS Stripe 82 provide immediate opportunities for empirical validation of forecasts and for refining cosmological parameter constraints with multi-tracer tomographic analyses (Qezlou et al., 2023).