Unclustered tracers remain unclustered: the lack of primordial non-Gaussianity response of bias-zero tracers

1. Problem Statement

The primary goal of next-generation large-scale structure (LSS) surveys (e.g., DESI, Euclid, SPHEREx) is to constrain Primordial Non-Gaussianities (PNG), specifically the local type parameter $f_{NL}$.

• The Hypothesis: Castorina et al. (2018) proposed that unclustered tracers (galaxies or halos with linear bias $b_1 = 0$) selected by binning them according to their local density ($\delta_{t,R}$) would be optimal for constraining PNG.
• Theoretical Basis: In a Gaussian universe, $b_1=0$ implies no clustering. However, in a PNG universe, the bias becomes scale-dependent ($b(k) \propto f_{NL} b_\phi / k^2$). Castorina et al. suggested that for $b_1=0$, the PNG response parameter $b_\phi$ would be non-zero (following the universality relation $b_\phi = 2\delta_{crit}(b_1 - 1)$), creating a strong $1/k^2$ signal in the power spectrum that could tightly constrain $f_{NL}$.
• The Conflict: This paper aims to verify if this theoretical prediction holds in state-of-the-art simulations, or if the complex selection method (local density binning) introduces non-linearities that suppress the PNG signal.

2. Methodology

The authors utilized high-resolution N-body simulations from the PNG-UNITsim suite to test the hypothesis.

• Simulations:
  • PNG-UNIT: A twin of the UNIT simulation ($L=1 h^{-1}\text{Gpc}$, $N=4096^3$ particles) with local $f_{NL} = 100$.
  • PNG-UNITsim-XL: A larger volume ($L=3 h^{-1}\text{Gpc}$) with $f_{NL} = -20$.
  • Both were populated with Halo Occupation Distribution (HOD) models to simulate realistic galaxy samples: LRGs (Luminous Red Galaxies) and QSOs (Quasi-Stellar Objects), matching DESI number densities.
• Tracer Selection:
  • Parent catalogues included: All halos ($M > 2\times 10^{10} h^{-1}M_\odot$), halos split into three mass bins, and HOD-based galaxies.
  • Density Binning: Tracers were classified into bins based on their local overdensity $\delta_{t,R}$ calculated in spheres of radius $R = 8 h^{-1}\text{Mpc}$.
  • Alternative Selection: They also reproduced Castorina's method of splitting the parent sample into two subsamples based on a threshold $\delta_{t,R}^{max}$ to find the bin where $b_1 \approx 0$.
• Analysis:
  • Computed the tracer auto-power spectrum ($P_{tt}$) and tracer-matter cross-power spectrum ($P_{tm}$).
  • Fitted the parameters $b_1$ (linear bias), $b_\phi$ (PNG response), and $A_{sn}$ (shot noise deviation from Poisson) using the cross-power spectrum up to $k_{max} = 0.08 h \text{Mpc}^{-1}$.
  • Used MCMC (emcee) with a Gaussian diagonal covariance matrix.

3. Key Contributions

• First rigorous test of bias-zero tracers in PNG simulations: Unlike previous works relying on separate-universe simulations or Gaussian ICs, this study uses direct PNG simulations ($f_{NL} \neq 0$) to measure the response.
• Validation of HOD models: The study extends the analysis from dark matter halos to realistic galaxy populations (LRGs and QSOs) using HOD models.
• Quantification of Shot Noise: The paper highlights that density-selected tracers exhibit super-Poissonian shot noise, a critical factor often overlooked in theoretical optimality calculations.

4. Results

The results fundamentally contradict the hypothesis that bias-zero tracers are optimal for PNG constraints.

• Vanishing PNG Response ($b_\phi \approx 0$):
  • For all halos and low-mass halos ($10.5 < \log M < 11$), the measured $b_\phi$ is consistent with zero across all density bins, regardless of $b_1$.
  • For high-mass halos ($\log M > 12$) and HOD galaxies (LRG/QSO), a trend closer to the universality relation ($b_\phi = 2\delta_{crit}(b_1 - 1)$) is recovered only when $|b_1| > 1$.
  • Crucially, as $b_1$ approaches 0, the curve flattens, and $b_\phi$ drops to 0. Thus, the "unclustered" tracers ($b_1 \approx 0$) remain unclustered even in the presence of PNG.
• Deviation from Universality: The complex selection based on local density ($\delta_{t,R}$) causes the tracers to deviate strongly from the simple universality relation assumed in theoretical models.
• Super-Poissonian Shot Noise:
  • The selected tracers exhibit significantly higher shot noise than the Poisson prediction ($1/n$).
  • For the bin closest to $b_1=0$ in the all-halo sample, the shot noise factor $A_{sn}$ was found to be $\sim 27$ (a factor of ~30 increase).
  • Even for realistic galaxy samples, $A_{sn} \approx 1.2-1.3$, which is non-negligible.
• Fisher Forecast Impact:
  • Due to $b_\phi \approx 0$ and high shot noise, the Fisher information for $f_{NL}$ vanishes.
  • For LRGs/QSOs, the forecasted error $\sigma(f_{NL})$ degrades from $\sim 4$ (using the full sample) to $\sim 100-130$ when using the specific density bin where $b_1 \approx 0$.

5. Significance and Conclusion

• Refutation of Optimality: The paper concludes that bias-zero tracers selected by local density are not optimal for constraining PNG. They do not generate the expected scale-dependent bias signal ($1/k^2$) because their PNG response ($b_\phi$) vanishes when $b_1 \to 0$.
• Physical Explanation: The discrepancy with Castorina et al. (2018) is likely due to the selection method. Selecting by local halo density introduces higher-order terms in the density and bias expansion that depend on cosmology (including the amplitude of fluctuations $A_s$), breaking the assumptions of the separate-universe approach used in previous studies.
• Implications for Future Surveys:
  • Strategies relying on finding $b_1=0$ tracers to maximize PNG constraints should be abandoned.
  • The high shot noise associated with these tracers further hinders their utility.
  • The authors emphasize the need for realistic simulation suites (like PNG-UNITsim) to validate non-standard statistics before applying them to observational data.

In summary, the study demonstrates that "unclustered tracers remain unclustered" in the context of PNG, rendering them ineffective for measuring primordial non-Gaussianity.