Testing the equivalence principle across the Universe: a model-independent approach with galaxy multi-tracing

This paper proposes a model-independent method using galaxy multi-tracing and relativistic corrections to test the equivalence principle on cosmological scales, forecasting that while current surveys can detect the necessary corrections, only the Square Kilometre Array (SKA) will achieve the precision required to constrain potential violations.

The Gist

The Big Question: Does Gravity Play Favorites?

Imagine you are in a giant, invisible elevator falling through space. According to Einstein's famous Equivalence Principle, if you drop a bowling ball and a feather inside this elevator, they should fall at the exact same speed and hit the floor at the same time. It doesn't matter what they are made of; gravity treats them equally.

We have tested this with incredible precision in our solar system (using atoms, metals, and even astronauts). But there is a huge mystery in the universe: Dark Matter.

Dark Matter is the invisible "glue" that holds galaxies together. We can't see it, but we know it's there because of its gravity. The big question this paper asks is: Does gravity treat Dark Matter the same way it treats normal matter (like stars and gas), or does it play favorites?

If gravity treats Dark Matter differently, it means our current understanding of physics (General Relativity) is incomplete, and there might be a new, invisible "fifth force" acting on the universe.

The Problem: How Do We Test Something We Can't See?

Usually, to test gravity, scientists look at how galaxies move. But there's a catch: to understand how they move, you have to know how they are biased (how they cluster) and how the universe is expanding. It's like trying to judge a runner's speed without knowing the length of the track or the wind speed. You end up needing to guess a lot of things, which makes the test shaky.

The authors of this paper wanted to build a test that doesn't need to guess anything. They wanted a "Null Test."

• Analogy: Imagine a scale that is perfectly balanced. If you put a feather on one side and a bowling ball on the other, and the scale stays perfectly level, you know something is wrong with gravity. If it tips, you know gravity is working normally. You don't need to know the weight of the feather or the ball beforehand; you just look at the balance.

The Solution: Listening to the "Echoes" of the Universe

The authors came up with a clever way to listen to the universe using Galaxy Surveys (like the DESI and SKA telescopes).

1. The Two Groups: They propose looking at two different groups of galaxies at the same time: "Bright" galaxies and "Faint" galaxies. Think of them like two different species of birds in the same forest.
2. The Cross-Check: Instead of just counting how many birds are in a tree, they look at how the two groups relate to each other.
3. The Secret Sauce (Relativistic Corrections): This is the magic part. When light travels from a distant galaxy to us, it gets slightly distorted by the gravity it passes through. It's like looking at a straw in a glass of water; the straw looks bent.
   • In the universe, this "bending" includes a tiny effect called Gravitational Redshift. It's a whisper from the universe saying, "Hey, time is moving slower here because of gravity!"
   • Normally, this whisper is too quiet to hear. But by comparing the "Bright" and "Faint" galaxies, the authors found a way to amplify this whisper.

The Magic Number: $E_P$

The team created a specific number, which they call $E_P$.

• If the Equivalence Principle is true (gravity treats everyone the same), this number will be exactly 1.
• If the Equivalence Principle is broken (Dark Matter feels a different force), this number will not be 1.

The beauty of their method is that they don't need to know the shape of the universe, how fast it's expanding, or exactly how biased the galaxies are. They just measure the relationship between the two galaxy groups, and the math automatically cancels out all the messy unknowns, leaving only the $E_P$ number.

The Results: Who Can Hear the Whisper?

The authors ran simulations to see if current and future telescopes could actually hear this whisper.

• DESI (The Current/Next-Gen Telescope): They found that DESI is very good at hearing the "relativistic corrections" (the gravitational echoes). It can confirm that these weird effects exist. However, it's not quite sensitive enough to tell us exactly if the Equivalence Principle is broken. It's like having a microphone that can hear a whisper, but not loud enough to distinguish the words.
• SKA (The Square Kilometre Array - The Future Giant): This is the big winner. The SKA is a massive radio telescope array being built in Africa and Australia. The authors predict that SKA will be able to measure the $E_P$ number with 7% to 15% precision.
  • Analogy: If the Equivalence Principle is a song playing at a specific volume, SKA will be able to hear it clearly enough to say, "Yes, that's the song," or "No, the pitch is slightly off."

Why This Matters

If SKA finds that $E_P$ is not 1, it would be a massive discovery. It would mean:

1. Dark Matter is weird: It might be interacting with a new force we've never seen.
2. Einstein needs an update: General Relativity might need a tweak to explain the whole universe, not just our solar system.

In summary: This paper proposes a clever, "model-independent" way to test if gravity treats Dark Matter fairly. By comparing two types of galaxies and listening to the subtle "echoes" of gravity (relativistic corrections), we can finally check if the universe plays favorites. While current telescopes can hear the echo, the future Square Kilometre Array will be the first to tell us if the song is out of tune.

Technical Summary

1. Problem Statement

The Weak Equivalence Principle (EP) is a cornerstone of General Relativity (GR), stating that all bodies fall identically in a gravitational potential regardless of their composition. While the EP has been tested with extreme precision for Standard Model particles (baryons) within the Solar System, its validity for Dark Matter (DM) on cosmological scales remains an open question.

• The Gap: Current cosmological tests of gravity often assume the EP holds or rely on specific theoretical models (e.g., Horndeski scalar-tensor theories) that fix the power spectrum shape and background evolution.
• The Goal: The authors aim to develop a model-independent test to determine if Dark Matter experiences the same unscreened long-range gravitational force as baryons, or if it is subject to additional "fifth forces" or non-gravitational interactions.

2. Methodology

The authors propose a novel approach using galaxy multi-tracing and large-scale relativistic corrections to construct a measurable null test.

A. Theoretical Framework

The test is based on the Euler equation in Fourier space, which relates the velocity field ($V$) to the gravitational potential ($\Psi$). In the presence of EP violation, this equation is modified by free functions $\Theta(k, z)$ (friction) and $\Gamma(k, z)$ (fifth force):
$$V' + [1 + \Theta(k, z)]V - \frac{k}{H}[1 + \Gamma(k, z)]\Psi = 0$$
They also allow for modifications to the Poisson equation via a function $\mu_G(k, z)$.

B. The Null Test Quantity ($E_P$)

The authors define a specific quantity, $E_P$, which acts as a null test. If the EP holds, $E_P = 1$. Any deviation from unity signals a violation, regardless of the physical mechanism:
$$E_P \equiv 1 + \Theta - \frac{3\Omega_m \mu_G \Gamma}{2f}$$
where $f$ is the growth rate of cosmic structure.

• Key Insight: $E_P \neq 1$ if DM is subject to non-gravitational forces ($\Theta, \Gamma \neq 0$) or if gravity couples differently to DM and baryons.

C. Observational Strategy: Galaxy Multi-Tracing

To measure $E_P$, the authors utilize the cross-correlation of two different galaxy populations (Bright $B$ and Faint $F$) with different linear biases ($b_B \neq b_F$).

1. Relativistic Corrections: Standard galaxy clustering analyses focus on density and Redshift-Space Distortions (RSD). However, the authors include relativistic corrections (Doppler terms and Gravitational Redshift) which are suppressed by $H/k$ but are crucial for testing the EP.
2. Breaking Symmetry: Relativistic corrections introduce anti-symmetric contributions to the cross-power spectrum ($P_{\Delta F \Delta B}$) that are absent in auto-spectra.
3. The Observable: The cross-spectrum contains a term proportional to $\tau_1$, which depends linearly on the difference in biases and the parameter $E_P$:
   $$\tau_1 = \beta_B \alpha_B - \beta_F \alpha_F - E_P (\beta_B - \beta_F)$$
   Here, $\beta = f/b$. Since $\alpha$ (related to magnification and evolution bias) is measurable, and $\beta$ can be constrained, $E_P$ becomes the only unknown if two populations with different biases are used.

D. Analysis Technique

• Fisher Matrix Analysis: The authors perform a Fisher forecast to estimate the precision of $E_P$ constraints.
• Model Independence: The method does not require assumptions about:
  • The shape of the matter power spectrum ($P(k)$).
  • The background cosmological evolution.
  • The growth rate of structure ($f$).
  • The specific galaxy bias function.
  • The specific model of EP violation.
• Surveys: Forecasts are generated for DESI (Dark Energy Spectroscopic Instrument) and SKA2 (Square Kilometre Array Phase 2).

3. Key Contributions

1. First Fully Model-Independent EP Test: Unlike previous studies that relied on fixed power spectrum shapes (often derived from CMB constraints at high redshift), this approach allows for arbitrary power spectrum shapes and background evolutions. It can test theories where GR is not recovered at high redshift or where DM has complex interactions.
2. Utilization of Relativistic Effects: The paper demonstrates that gravitational redshift and Doppler terms, often treated as noise or sub-dominant, are the primary signal carriers for testing the EP on cosmological scales.
3. Multi-Tracer Necessity: The authors rigorously show that measuring $E_P$ requires the cross-correlation of two populations with distinct biases ($\beta_B \neq \beta_F$). Without this bias difference, the EP-violating term vanishes from the observable.
4. Robustness Checks: The analysis accounts for wide-angle corrections, Alcock-Paczyński effects, and non-linear damping (Finger-of-God), showing that the method remains robust even when these complexities are included.

4. Results

The authors provide forecasts for the precision of $E_P$ constraints:

• DESI (Dark Energy Spectroscopic Instrument):

  • Relativistic Corrections: Can be detected with high significance (Signal-to-Noise ratio $\sim 7$ at $z=0.15$).
  • EP Constraint: DESI is unable to constrain $E_P$ to an acceptable level (errors $> 100\%$) with standard bias splits. Even with extreme bias differences, the constraints remain weak ($\sim 176\%$ error).
  • Reason: Limited volume and galaxy density compared to SKA.

• SKA2 (Square Kilometre Array Phase 2):

  • Relativistic Corrections: Detectable with precision between 0.8% and 13%.
  • EP Constraint: SKA2 can constrain $E_P$ to 7–15% precision in the redshift range $0.25 < z < 0.6$.
  • Reason: Larger survey volume, higher galaxy number densities, and the ability to achieve a larger difference in bias parameters ($\Delta \beta$) between the two populations.

• Bias Dependence: The precision of $E_P$ is inversely proportional to the difference in bias between the two populations. Larger $\Delta b$ yields tighter constraints.

5. Significance

This work represents a paradigm shift in testing fundamental physics on cosmological scales:

• New Physics Window: It provides a direct, model-independent probe of whether Dark Matter interacts via forces other than gravity. A detection of $E_P \neq 1$ would imply new physics beyond the Standard Model and General Relativity (e.g., dark photons, self-interacting DM, or modified gravity).
• Future Surveys: It establishes that next-generation radio surveys like SKA2 are essential for fundamental physics tests, moving beyond just measuring dark energy parameters to testing the laws of gravity themselves.
• Methodological Advance: By successfully decoupling the test from assumptions about the power spectrum and background evolution, the method avoids "circular reasoning" often found in modified gravity tests, making the results more robust against theoretical biases.

In conclusion, the paper demonstrates that while current and near-future optical surveys (DESI) can detect relativistic effects, only the massive scale of SKA2 will allow for a precise, model-independent test of the Equivalence Principle for Dark Matter.