Impact of lensing magnification on the power spectrum turnover

Technical Summary: Impact of Lensing Magnification on the Power Spectrum Turnover

Problem Statement
The turnover scale ($k_0 \approx 0.0163 \, h/\text{Mpc}$) in the matter power spectrum marks the transition between modes that entered the Hubble horizon during radiation domination and those during matter domination. Because this scale is determined by the physics of matter–radiation equality, it serves as a fundamental standard ruler, complementary to Baryon Acoustic Oscillations (BAO), capable of constraining the physical matter density ($\Omega_{m0}h^2$) and the Hubble constant ($H_0$).

While high-redshift spectroscopic surveys (e.g., Euclid H$\alpha$ and the proposed MegaMapper Lyman-break galaxy survey) offer increased cosmic volume to enhance the precision of turnover measurements, they introduce a critical accuracy challenge. In these deep surveys, the observed galaxy power spectrum monopole receives a scale-dependent correction from gravitational lensing magnification. Standard analyses often rely on the Kaiser approximation, which neglects lensing. The authors investigate whether ignoring this lensing correction biases the measurement of the turnover scale, potentially undermining its utility as a standard ruler at high redshifts.

Methodology
The authors employ a forecasting framework using mock surveys modeled after the Euclid H$\alpha$ survey ($0.9 \le z \le 1.8$) and the proposed MegaMapper survey ($2.1 \le z \le 5$). The analysis proceeds as follows:

1. Theoretical Modeling:

   • The observed galaxy density contrast is modeled as $\delta_{\text{obs}} = \delta_S + \delta_L$, where $\delta_S$ is the standard Kaiser term and $\delta_L$ is the lensing magnification contribution dependent on the magnification bias $s$ and lensing convergence $\kappa$.
   • The power spectrum monopole is computed including the cross-power between the standard term and lensing, as well as the lensing auto-power.
   • Calculations utilize the Planck 2018 flat $\Lambda$CDM parameters, with power spectra computed via CLASS and numerical integration of lensing kernels performed using CosmoWAP.

2. Parameter Recovery and Bias Estimation:

   • Standard Case (No Lensing): The authors fit an asymmetric parabola model to the standard monopole near the turnover to recover $k_0$ and its error $\sigma(k_0)$. In this scenario, $k_0$ is treated as redshift-independent, allowing for the stacking of all redshift bins to maximize constraining power.
   • Lensing Case (Correct Theory): When lensing is included, the turnover position becomes redshift-dependent ($k_0 \to \tilde{k}_0(z)$). The authors fit the lensing-corrected monopole in individual redshift bins, treating the turnover scale and curvature parameters as distinct for each bin.
   • Bias Calculation: The bias is quantified as the difference between the "true" lensing-corrected turnover ($\tilde{k}_0$) and the turnover recovered by the incorrect (no-lensing) model, expressed in units of the precision ($\sigma$) inferred from the no-lensing analysis.

Key Results
The study reveals that lensing magnification significantly alters the shape of the power spectrum monopole, particularly on large scales ($k \lesssim k_{\text{eq}}$), leading to a redshift-dependent shift in the observed turnover scale.

• Euclid-like Survey ($0.9 \le z \le 1.8$):

  • The lensing effect induces a shift in the turnover scale, but the turnover remains detectable.
  • The maximum bias is approximately 0.4$\sigma$ at the highest redshift ($z \approx 1.69$).
  • The authors conclude that for Euclid-like surveys, the bias is small enough that neglecting lensing may be acceptable, though modeling it is preferable for precision.

• MegaMapper-like Survey ($2.1 \le z \le 5$):

  • The impact of lensing is severe. The shift in the turnover scale grows rapidly with redshift.
  • A bias exceeding 1$\sigma$ occurs for $z \gtrsim 2.9$.
  • The maximum bias reaches 3.6$\sigma$ at $z \approx 3.55$.
  • Crucially, for $z \gtrsim 3.7$ (depending on the magnification bias model $s(z)$), the turnover feature vanishes entirely in the observed power spectrum monopole. The spectrum becomes too flat to identify a peak.
  • Consequently, data in the range $2.9 \lesssim z \le 5$ does not provide a reliable recovery of the intrinsic turnover unless lensing is accurately modeled and removed.

• DESI Quasar Sample:

  • A preliminary theoretical check on the DESI DR1 quasar catalogue ($0.8 \le z \le 3.5$) suggests a negligible lensing shift based on current estimates of magnification bias, though the authors note this requires further investigation as the catalogue develops.

Significance and Claims
The paper asserts that the lensing correction to the monopole must be included and carefully modeled when using high-redshift spectroscopic surveys to measure the cosmological turnover.

• Validity of the Standard Ruler: The authors demonstrate that the turnover scale is not a fixed standard ruler in the observed galaxy power spectrum at high redshifts; it is distorted by lensing.
• Limits on Stacking: While stacking redshift bins improves precision in the standard model, this strategy is invalid in the presence of lensing unless the redshift-dependent relation between bins is perfectly modeled. Without such modeling, stacking high-redshift bins (specifically for MegaMapper-like surveys) leads to biased or non-existent turnover detections.
• Survey Design Implications: For surveys targeting the turnover at high redshifts (like MegaMapper), there is a strict redshift limit ($z \approx 2.9$) beyond which the intrinsic turnover cannot be reliably recovered without precise modeling of magnification bias. The authors emphasize that accuracy in theory is as critical as precision in measurement for these ultra-large scale probes.