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: $\alpha$0 where $\alpha$1 is the underlying matter power spectrum, $\alpha$2 is mean line intensity (e.g., in $\alpha$3K), $\alpha$4 and $\alpha$5 are the respective linear biases, and $\alpha$6 denotes Poisson contributions.

2. Noise Modeling and Signal-to-Noise Estimation

Accurate forecasts require comprehensive noise modeling for both LIM and Ly-$\alpha$7 observables. For LIM, the instrumental noise per 3D voxel (volume $\alpha$8) is: $\alpha$9 with a beam/channel window function $\delta I_{\rm LIM}(\mathbf{k})$0.

For COMAP-Y5 at $\delta I_{\rm LIM}(\mathbf{k})$1, $\delta I_{\rm LIM}(\mathbf{k})$2K, $\delta I_{\rm LIM}(\mathbf{k})$3, and $\delta I_{\rm LIM}(\mathbf{k})$4 determined by beam and spectral resolution.

Ly-$\delta I_{\rm LIM}(\mathbf{k})$5 tomography noise is dominated by finite effective sightline density $\delta I_{\rm LIM}(\mathbf{k})$6,

$\delta I_{\rm LIM}(\mathbf{k})$7

assuming pixel noise S/N per Å ≈ 2.

The cross-power spectrum variance per $\delta I_{\rm LIM}(\mathbf{k})$8-mode is

$\delta I_{\rm LIM}(\mathbf{k})$9

Summing in inverse variance across $\delta F_{{\rm Ly}\alpha}(\mathbf{k})$0-bins gives total S/N: $\delta F_{{\rm Ly}\alpha}(\mathbf{k})$1

3. Simulation Methodology

Cross-frame intensity analyses employ large-volume cosmological hydrodynamic simulations to model both LIM and Ly-$\delta F_{{\rm Ly}\alpha}(\mathbf{k})$2 signals. The ASTRID suite is a prominent example:

• Modified GADGET-3 code with SPH and tree+PM gravity.
• Volume: $\delta F_{{\rm Ly}\alpha}(\mathbf{k})$3 cMpc, $\delta F_{{\rm Ly}\alpha}(\mathbf{k})$4 particles.
• Star formation: Springel & Hernquist multiphase ISM, molecular-H$\delta F_{{\rm Ly}\alpha}(\mathbf{k})$5 correction.
• Cooling: Katz, Weinberg & Hernquist rates; UV background as per Faucher-Giguère, rescaled to match $\delta F_{{\rm Ly}\alpha}(\mathbf{k})$6.
• Reionization: patchy H I ($\delta F_{{\rm Ly}\alpha}(\mathbf{k})$7) via Battaglia map, patchy He II ($\delta F_{{\rm Ly}\alpha}(\mathbf{k})$8) via 30 $\delta F_{{\rm Ly}\alpha}(\mathbf{k})$9 cMpc stochastic bubbles.
• Black holes: seeded in $\alpha$0, Bondi accretion with $\alpha$1 thermal feedback.

Mock Ly-$\alpha$2 forest sightlines are generated using FAKESPECTRA on a 250 $\alpha$3 ckpc grid, enforcing alignment with SDSS DR14 1D flux power to ≤10%. Molecular emission (e.g., CO) utilizes a subhalo–SFR–$\alpha$4 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$\alpha$5PFS Ly-$\alpha$6 tomography (mean sightline separation $\alpha$7–3.7 $\alpha$8 cMpc) achieves a $\alpha$9200–300% increase in COMAP detection S/N relative to auto-only LIM.
• COMAP$$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$0eBOSS or COMAP$$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$1DESI ($$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$2–13 $$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$3 cMpc) yields a 50–75% S/N improvement over LIM alone.
• For [C II] intensity mapping: EXCLAIM$$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$4(DESI/eBOSS Ly-$$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$5 forest) achieves $$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$6, reflecting a substantial gain due to higher sightline density and the negative Ly-$$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$7 bias ($$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$8).

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

Probe Pair	Sightline Sep ($$P_{\rm LIM}(\mathbf{k}) \equiv \langle \delta I_{\rm LIM}(\mathbf{k})\, \delta I_{\rm LIM}^*(\mathbf{k})\rangle$$9 cMpc)	Expected S/N Improvement
COMAP × PFS Ly-$\alpha$0	2.5–3.7	200–300%
COMAP × (eBOSS or DESI)	10–13	50–75%
EXCLAIM × (DESI/eBOSS Ly-$\alpha$1) vs. EXCLAIM × quasar	—	$\alpha$210×

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 ($\alpha$3) and are uncorrelated with Ly-$\alpha$4 transmission fluctuations; these contaminants vanish in the cross-spectrum $\alpha$5. Conversely, Ly-$\alpha$6 forest systematics (e.g., continuum fitting, damped Ly-$\alpha$7 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$8 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$\alpha$9Ly-$$P_{{\rm Ly}\alpha}(\mathbf{k}) \equiv \langle \delta F_{{\rm Ly}\alpha}(\mathbf{k})\, \delta F_{{\rm Ly}\alpha}^*(\mathbf{k}) \rangle$$0) in early phases of LIM surveys could expedite cosmological signal confirmation, influence survey strategy, and prioritize resources for overlap with dense Ly-$$P_{{\rm Ly}\alpha}(\mathbf{k}) \equiv \langle \delta F_{{\rm Ly}\alpha}(\mathbf{k})\, \delta F_{{\rm Ly}\alpha}^*(\mathbf{k}) \rangle$$1 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-$$P_{{\rm Ly}\alpha}(\mathbf{k}) \equiv \langle \delta F_{{\rm Ly}\alpha}(\mathbf{k})\, \delta F_{{\rm Ly}\alpha}^*(\mathbf{k}) \rangle$$2 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).