Cavendish experiment with fast radio bursts on cosmological scales

1. Problem Statement

The primary goal of modern cosmology is to test General Relativity (GR) and distinguish between Dark Energy (DE) and Modified Gravity (MG) theories. A key parameter for these tests is the effective gravitational constant for light motion ($G_{\text{light}}$), which relates the Weyl potential ($\Psi + \Phi$) sensed by light to the matter overdensity ($\delta_m$).

• The Bottleneck: In GR, $\delta_m$ can be inferred from weak lensing. However, in MG scenarios, this inference breaks down because the relationship between lensing and matter clustering changes. Consequently, there is currently no unbiased proxy for $\delta_m$ that does not rely on assuming GR is correct.
• The Gap: Existing estimators (like $E_G$) rely on modeling redshift-space distortions and the continuity equation, introducing theoretical dependencies that limit their ability to strictly test gravity. There is a critical need for a method to measure $\delta_m$ directly and independently of gravity models.

2. Methodology

The authors propose a new cosmological "Cavendish experiment" using Fast Radio Bursts (FRBs) as a novel probe. The methodology relies on three main pillars:

A. FRB Dispersion Measure (DM) as a Proxy for $\delta_m$

• Physical Basis: The Weak Equivalence Principle implies that on large scales ($\gtrsim 10$ Mpc), dark matter and baryonic matter share the same spatial distribution ($\delta_m = \delta_b$).
• FRB Connection: FRBs travel through the ionized intergalactic medium (IGM). Their Dispersion Measure (DM) is directly proportional to the integrated electron density along the line of sight. Since ionized electrons constitute $\gtrsim 90\%$ of cosmic baryons, the electron density fluctuation ($\delta_e$) is a close approximation of the baryon fluctuation ($\delta_b$), and thus $\delta_m$.
• Localization: The method requires localized FRBs (identifying the host galaxy redshift) to separate the IGM contribution from the host galaxy and Milky Way contributions using cross-correlation statistics.

B. The $F_G$ Estimator

The authors define a new estimator, $F_G$, which combines three tracers:

1. $\Delta_g$: Galaxy surface overdensity (from galaxy surveys like DESI).
2. $\kappa$: Weak lensing convergence (from surveys like Rubin/LSST).
3. $D$: FRB Dispersion Measure.

The estimator is constructed in Fourier space as the ratio of cross-power spectra:
$$\hat{F}_G \equiv \hat{F} \frac{\hat{C}^{g\kappa}_\ell}{\hat{C}^{gD}_\ell}$$

• $\langle \Delta_g \kappa \rangle$ isolates the Weyl potential ($\Phi + \Psi$).
• $\langle \Delta_g D \rangle$ isolates the electron density ($\delta_e \approx \delta_m$).
• The normalization factor $\hat{F}$ is chosen such that the expectation value $F_G = G_{\text{light}}/G$.

This approach allows for a tomographic measurement of $G_{\text{light}}$ across redshift bins without relying on the linearized continuity equation or specific MG parameterizations.

C. Mitigating Systematic Errors (Electron Bias)

A major systematic concern is the clustering bias of electrons ($b_e$), which may differ from unity ($b_e \neq 1$) due to astrophysical processes (e.g., stars forming in overdense regions, gas depletion in halos).

• Solution: The authors derive an analytical relation (Eq. 10) to estimate $f_e b_e$ (ionization fraction $\times$ bias) using the total baryon budget.
• Logic: Since the total baryon bias is unity ($b_b=1$) on large scales, the electron bias can be inferred by subtracting the contributions of stars ($f_* b_*$) and neutral hydrogen ($f_{HI} b_{HI}$) from the total.
• Validation: Using hydrodynamical simulations (TNG300-1 and Illustris-1), they demonstrate that this "astrophysics-agnostic" approach reduces systematic errors to the $\lesssim 1\%$ level, even with strong feedback models.

3. Key Contributions

1. Novel Proxy for Matter: Established that localized FRB DM is an unbiased tracer of matter overdensity ($\delta_m$), solving the "chicken-and-egg" problem of measuring $\delta_m$ without assuming GR.
2. New Estimator ($F_G$): Proposed a direct, model-independent estimator for $G_{\text{light}}$ that relies solely on the Weak Equivalence Principle and cross-correlations, avoiding the modeling pitfalls of previous methods like $E_G$.
3. Systematic Mitigation Strategy: Developed a robust method to correct for electron clustering bias using independent constraints on stellar and neutral gas distributions, validated against state-of-the-art simulations.
4. Feasibility Study: Demonstrated that upcoming radio arrays (e.g., DSA-2000, BURSTT) and optical surveys (DESI, Rubin) can achieve the necessary sample sizes ($\sim 10^5$ localized FRBs) to make this measurement.

4. Results and Forecasts

The authors performed a forecast analysis assuming:

• Data: DESI-like galaxy catalogs, Rubin-like weak lensing, and $\sim 10^5$ localized FRBs up to $z \sim 1$.
• Scale Cuts: Conservative cuts of $k \le 0.1 \, h/\text{Mpc}$ (linear regime).

Key Findings:

• Precision: The measurement can achieve a precision of $\lesssim 10\% \sqrt{10^5/N_{\text{FRB}}}$ per redshift bin.
• Overall Accuracy: Across 10 redshift bins ($0 \lesssim z \lesssim 1$), the overall precision reaches $\sim 2\%$.
• Systematics:
  • Without mitigation, electron bias induces a systematic shift of 1–5%.
  • With the proposed mitigation (Eq. 10), residual systematics are reduced to $\lesssim 1\%$.
  • This residual is subdominant to the statistical noise for $N_{\text{FRB}} \sim 10^5$.
• Comparison: This precision is competitive with or superior to constraints derived from full-shape Large Scale Structure (LSS) modeling (e.g., Euclid), but with significantly fewer theoretical assumptions.

5. Significance

• Testing Modified Gravity: A detection of $G_{\text{light}} \neq G$ would rule out GR and specific MG models (like $f(R)$ gravity). Combined with the speed of gravity constraints from GW170817, this could rule out the entire Horndeski scalar-tensor theory class.
• Breaking Degeneracies: Accurate measurement of $\delta_m$ alongside $\Psi+\Phi$ (lensing) and $\Psi$ (velocity) allows for the construction of consistency relations that can distinguish between Modified Gravity and Clustered Dark Energy models, which are otherwise degenerate.
• Multi-Messenger Cosmology: This work highlights the transformative potential of FRBs not just as distance indicators, but as fundamental probes of the cosmic matter distribution and gravitational physics across 20 orders of magnitude in scale.

In summary, the paper presents a rigorous framework to use FRBs as a "cosmic Cavendish experiment," offering a path to measure the strength of gravity on cosmological scales with unprecedented model independence and precision.