Density reconstruction from biased tracers: Testing the equivalence principle through consistency relations

TL;DR

The paper proposes a cosmological test of the weak equivalence principle (EP) based on consistency relations in large-scale structure, showing that EP violation would generate a distinctive anti-symmetric, 1/K dipole in the equal-time squeezed bispectrum. It introduces quadratic estimators that reconstruct the long-wavelength density mode from cross-spectra of distinct tracers, focusing on the anti-symmetric shift response as the smoking-gun signal. Applying this framework to DESI-like surveys, the authors forecast competitive constraints on the EP-violation amplitude, especially when mildly nonlinear scales are included, and demonstrate that QE-based analyses can approach the capabilities of direct bispectrum methods with practical advantages. The study also discusses the limitations posed by poorly constrained EP-violating bias parameters and outlines strategies to combine QE with additional observables to bolster sensitivity. Overall, the work offers a practical, near-term route to testing EP on cosmological scales using upcoming galaxy surveys.

Abstract

Consistency relations of large-scale structure offer a unique and powerful test of the weak equivalence principle (EP) on cosmological scales. If the EP is violated, different tracers will undergo different accelerations in response to a uniform gravitational field, and this loss of universality manifests as a dipole with a characteristic $1/K$ scale dependence in the squeezed limit of the bispectrum. In this work we show that such a violation can be identified with a particular anti-symmetric {modulation} in the local cross-power spectrum of distinct tracers. Based on this observation, we propose to test the EP using quadratic estimators as a more practical alternative to the conventional approach of directly estimating the bispectrum. We apply our quadratic estimator to a DESI-like survey and forecast constraints on the overall amplitude of EP violation. Including mildly nonlinear scales in our reconstruction ($k_\mathrm{max}\simeq0.15\, h\,\mathrm{Mpc}^{-1}$), we find that our estimator is competitive with the more exhaustive direct bispectrum approach. This means surveys like DESI can already benefit from the quadratic estimator approach.

Figures (16)

• Figure 1: The transport of two distinct galaxies under a (near) uniform gravitational field, e.g. sourced by a long-wavelength potential $\phi_L\propto\delta_L/K^2$. If the EP holds, then over a given time interval the gravitational field induces identical accelerations, resulting in identical displacements for both galaxies, independent of their mass, type, composition, etc. If the EP does not hold, each galaxy suffers a different displacement and a relative shift $\Delta{\mathbf{r}}_{AB}\neq0$ develops between them, directed along the gradient $-\nabla\phi_L$. This relative shift---which we call the anti-symmetric shift due to the dipolar nature of the effect---can be targeted as a signature of EP violation.
• Figure 2: Application of displacement estimator $\widehat{h}^\mathcal{D}_{XY}$ [Eq. \ref{['eq:estimatorfourier']}] to DESI LRG- and ELG-like mocks from the AbacusSummit simulations ($X=\mathrm{LRG}$, $Y=\mathrm{ELG}$), for redshift $z = 0.5$. Predictions are given by Eqs. \ref{['eq:forecast_autorspec']} and \ref{['eq:forecast_crossspec']}, using rough approximate galaxy bias fits (see Appendix \ref{['sec:sims']} for details). Left panel: The cross correlation between the reconstruction and input matter field. Center panel: The cross-correlation between the reconstruction and the LRG simulation. Right panel: The auto correlation of the reconstruction. This is mainly explained by Gaussian variance contributions (dashed pink). As the cross- and auto- correlations of the reconstruction effectively measure three- and four-point functions, respectively, we also include the effect of shot noise from higher-order components (mixed bispectrum and trispectrum). The modes used in the reconstruction range from $k_{\mathrm{min,rec}}=0.05h\,\mathrm{Mpc}^{-1}$ to $k_{\mathrm{max,rec}}=0.2h\,\mathrm{Mpc}^{-1}$, as indicated by the grey bands. Error bars are given by the standard deviation of the mean.
• Figure 3: Left panel: Example of unnormalized symmetric and anti-symmetric QE responses $R^{\mathcal{D}}_{\pm\alpha}$ [Eq. \ref{['eq:QE-response']}] for the growth, shift, and tidal terms. We clearly see a strong scale dependence in the displacement estimator due to the anti-symmetric shift response (dashed green). Right panel: Biased response $B^\mathcal{D}_{\alpha,XY}$ [Eq. \ref{['eq:RD']}] of the displacement estimator. By construction, the anti-symmetric shift component is flat on large scales (dashed green). This is also reflected in the effective bias $b_\mathcal{D}=\sum_\alpha B^\mathcal{D}_{\alpha,XY}$ (solid black). For reference, we also show in solid grey $b_\mathcal{D}$ when the $\epsilon$ amplitude is a factor of ten smaller. In this example we set $\epsilon=10^{-2}$, $b_{1X}=1.6,b_{1Y}=1.2$, $b_{2X}=b_{2Y}=-0.3$, and $b_{\epsilon,\alpha X}=b_{\epsilon,\alpha Y}= 1$, for each $\alpha \in\{\mathrm{G,S,T}\}$.
• Figure 4: Forecasted constraints on EP violation expected from DESI. Unmarginalized (green) and marginalized (over $b_{1X}, b_{2X}, b_{s^2X}$, blue) constraints on the combination $\epsilon \times C^\mathrm{S}_{[AB]}$ as a function of the maximum wavenumber $k_\mathrm{max,rec}$ used in the QE reconstruction. These are the constraints expected from a combined analysis of all cross-correlations $P_{\mathrm{cross}}^{XY}$ for $X,Y \in \{A,B,\mathcal{D}\}$, $X\neq Y$, i.e. between external tracers $A$, $B$ and the matter modes reconstructed from the displacement estimator (or $\mathcal{D}\otimes \mathrm{Galaxies}$ for short). For comparison, the grey band shows the $1\sigma$ bounds from Planck CMB+DESI BAO reported in Ref. Bottaro:2024pcb. To make this comparison, we assume that both $C^\mathrm{S}_{[AB]}$ and the fraction of fifth-force interacting dark matter are of order unity.
• Figure 5: QE variance and normalization. Comparing variance (solid) and normalization (dashed) for different estimators. The displacement estimator (yellow) has a variance comparable to the full anti-symmetric shift estimator (light-blue). It increases more rapidly as we go towards smaller scales, though most of the signature we care about is on large-scales. We over-plot the linear matter power spectrum (black) for reference (though care is required as the true recovered spectrum will be biased in a scale-dependent way). Here we use $b_{1X}=1.6,b_{1Y}=1.2$ and $\bar{n}_A = 3\cdot 10^{-4} h^3 \mathrm{Mpc}^{-3},\ \bar{n}_B = 4\cdot 10^{-4} h^3 \mathrm{Mpc}^{-3}$.