Comparing the motion of dark matter and standard model particles on cosmological scales

Technical Summary: Comparing the Motion of Dark Matter and Standard Model Particles on Cosmological Scales

Problem Statement
The nature of dark matter remains one of the primary challenges in cosmology and particle physics. While the standard "cold dark matter" (CDM) paradigm assumes dark matter consists of collisionless particles interacting solely via gravity—thereby obeying Euler's equation and following the same geodesics as standard matter—this assumption has not been directly verified. Since dark matter has never been directly detected, it is necessary to test whether it falls into gravitational potential wells in the same manner as baryonic matter. A violation of Euler's equation could indicate a breakdown of the weak equivalence principle or the presence of non-gravitational interactions (often termed a "fifth force") between dark matter and the standard model, dark radiation, dark energy, or itself. This paper focuses on testing the validity of Euler's equation for dark matter at cosmological scales, assuming General Relativity (GR) holds, to search for deviations caused by non-gravitational dark matter interactions.

Methodology
The authors propose a direct test of Euler's equation by combining measurements of galaxy peculiar velocities with measurements of the gravitational potential. The methodology relies on the following theoretical and observational framework:

1. Theoretical Framework: The motion of dark matter is described by a modified Euler's equation in Fourier space:
   $$V' + (1 + \theta)V - \frac{k}{H}(1 + \Gamma)\Psi = 0$$
   where $V$ is the galaxy velocity potential, $\Psi$ is the time distortion (gravitational potential), $H$ is the Hubble parameter, and primes denote derivatives with respect to the logarithm of the scale factor. In the standard CDM scenario, $\Gamma = 0$ and $\theta = 0$. The parameter $\Gamma$ encodes the strength of a potential fifth force. The friction term $\theta$ is assumed negligible in the quasi-static approximation.

2. Observables:

   • Growth Rate ($\hat{f}$): Redshift-space distortions (RSD) in galaxy surveys provide direct measurements of the growth rate of structure, $\hat{f}(z) = f(z)\sigma_8(z)$, which encodes the time evolution of the velocity field $V$.
   • Weyl Potential Evolution ($\hat{J}$): The Weyl potential, $\Psi_W \equiv (\Phi + \Psi)/2$, governs light deflection. Under GR at late times, $\Phi = \Psi$, implying $\Psi_W = \Psi$. Recent analyses of galaxy-galaxy lensing and galaxy clustering allow for the direct measurement of the time evolution of the Weyl potential, encoded in the parameter $\hat{J}(z)$.

3. Constraint Equation: By combining the measurements of $\hat{f}$ and $\hat{J}$, the strength of the fifth force $\Gamma$ can be isolated without assuming a specific dark matter interaction model:
   $$1 + \Gamma(z) = \frac{2\hat{f}(z)}{3\hat{J}(z)} \left( 1 - \frac{d \ln H(z)}{d \ln(1+z)} - \frac{d \ln \hat{f}(z)}{d \ln(1+z)} \right)$$
   This equation allows for the direct inference of $\Gamma$ at specific redshifts using observed data.

4. Data Analysis:

   • Current Data: The authors utilize 22 measurements of $\hat{f}$ from various spectroscopic surveys (redshifts $z \in [0.001, 1.944]$) and four measurements of $\hat{J}$ from the Dark Energy Survey (DES) Year 3 data at effective redshifts $z \in \{0.295, 0.467, 0.626, 0.771\}$.
   • Reconstruction: A cubic spline interpolation is used to reconstruct $\hat{f}(z)$ and its derivative $d \ln \hat{f}/d \ln(1+z)$ at the specific redshifts where $\hat{J}$ is measured.
   • Assumptions: The analysis assumes General Relativity is valid (ensuring $\Psi_W = \Psi$) and that the background evolution follows a $\Lambda$CDM model. It also assumes the fifth force is negligible at high redshifts (consistent with CMB constraints on the matter power spectrum).

Key Results

• Current Constraints: The analysis finds that current data are consistent with Euler's equation ($\Gamma = 0$) across all measured redshifts.
  • At individual redshift bins, $\Gamma$ is compatible with zero with uncertainties ranging from $0.17$ to $0.29$.
  • Assuming a constant fifth force strength over the observed redshift range, the amplitude is constrained to lie within $-21\%$ and $+7\%$ of the gravitational interaction strength.
  • If the fifth force is restricted to be strictly positive (as in some coupled quintessence models), the upper limit is tightened to $11\%$ (68% confidence) or $24\%$ (95% confidence).
• Future Projections: The authors forecast constraints using data from the Dark Energy Spectroscopic Instrument (DESI) and the Vera C. Rubin Observatory (LSST).
  • Combining DESI growth rate forecasts with LSST Weyl potential forecasts will significantly reduce uncertainties.
  • Future surveys are expected to detect a departure from pure gravitational interaction at the level of $3-6\%$ per redshift bin over the range $z \in [0.51, 1.35]$.
  • Assuming a constant strength, future data could constrain the fifth force amplitude to $2\%$ of the gravitational interaction strength.

Significance and Claims
The paper claims that this approach offers a model-independent test of dark matter interactions. Unlike constraints derived from the background evolution of the universe (which can be degenerate with dynamical dark energy models) or specific particle physics models, this method directly probes the validity of Euler's equation.

• Model Independence: The method does not require specifying the nature of the fifth force (scalar vs. vector) or its time evolution. It relies solely on the kinematic relationship between velocity and potential.
• Distinctiveness: The constraints are derived from the motion of dark matter rather than its density evolution. This distinguishes the results from constraints on parameters like $\mu$ (modifications to Poisson's equation), which affect density growth but not necessarily Euler's equation.
• Robustness: The authors note that while the analysis assumes GR, any violation of Euler's equation detected in this framework would uniquely point to non-gravitational dark matter interactions, as modifications to gravity that preserve the equivalence principle would not produce this specific signature in the velocity-potential relation.

The work establishes a new observational pathway to test the fundamental assumption that dark matter interacts only gravitationally, providing current limits and a clear roadmap for next-generation surveys to improve these constraints by an order of magnitude.