Cosmological peculiar velocities in general relativity

This paper refutes recent claims that the 1+3 covariant approach predicts anomalously strong growth of galaxy peculiar velocities by demonstrating that a consistent application of the covariant equations yields results identical to standard metric-based perturbation theory, thereby confirming that the standard treatment is fully relativistic and does not produce unphysical bulk flows or mimic accelerated expansion.

The Gist

The Big Picture: A Cosmic Speed Limit Debate

Imagine the universe as a giant, expanding ocean. Inside this ocean, there are islands (galaxies). Because the ocean is expanding, the islands drift apart. But sometimes, islands also have their own "currents" pushing them in specific directions relative to the water. These are called peculiar velocities.

Recently, a group of scientists claimed they found a "secret super-current" in the universe. They argued that if you use a specific, fancy mathematical tool (called the 1+3 covariant approach) to calculate how fast these islands drift, the speed grows massively over time—so fast that galaxies would eventually be moving faster than light! They suggested this super-fast drift could explain why the universe seems to be accelerating, potentially removing the need for "Dark Energy."

Chris Clarkson and Roy Maartens, the authors of this paper, are here to say: "Hold on. That math is broken."

They show that when you use that fancy tool correctly, you get the exact same result as the standard, well-tested method. There is no super-current. The universe isn't accelerating because of weird galaxy drifts; it's slowing down, just like standard physics predicts.

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The Core Conflict: Two Ways to Measure the Drift

To understand the mistake, let's use an analogy of a moving walkway at an airport.

1. The Standard View (The Longitudinal Gauge): Imagine you are standing still on the floor, watching people walk on a moving walkway. You measure how fast they are walking relative to the floor. This is the standard way cosmologists have measured galaxy speeds for decades. It gives a slow, steady growth in speed.
2. The "New" View (The Covariant Approach): Imagine you decide to stand on a different moving walkway that is moving at a slightly different speed. You measure the people's speed relative to your walkway.

The problem arises because the "New" group claimed that by standing on their specific walkway, they could prove the people were accelerating wildly. They argued that their method was "fully relativistic" (using Einstein's full rules) while the standard method was just "Newtonian" (simplified).

The Authors' Rebuttal:
Clarkson and Maartens say: "You can't just pick a walkway and ignore the rules of the airport."

In the "New" group's math, they treated the difference between the two walkways as an external force (like a wind blowing the people). They set this "wind" to zero to see what happens, which made the math predict wild speeds.

The Reality:
The "wind" isn't an external force you can just turn off. It is tied to the people walking. If the people speed up, the wind changes. If you turn off the wind in the equation, you are breaking the laws of physics. When you fix the math so the "wind" and the "walkers" stay connected, the wild speed disappears, and you get back to the slow, steady growth we already knew.

The "Ghost" Acceleration

The "New" group also claimed that because these galaxies were moving so fast, they could trick us into thinking the universe is accelerating (expanding faster and faster), even if it's actually slowing down.

The Analogy:
Imagine you are in a car that is braking (slowing down). You look out the window at a friend running alongside you.

• If you run with the car, you see the car slowing down normally.
• If you run against the car, or if you are on a different vehicle entirely, your view of the car's speed changes.

The "New" group confused what the car is doing (the physical universe slowing down) with how the driver sees it (the observer's perspective).

The authors explain that the "acceleration" they found was just an illusion created by looking at the universe from a weird, non-standard angle. If you translate that view back to the "real" perspective (the galaxies themselves), the universe is clearly slowing down due to gravity. The "illusion" doesn't change the physical reality.

Why This Matters

1. No "Super-Speed" Galaxies: If the "New" math were right, galaxies today would be moving at millions of kilometers per second (faster than light!). We don't see that. We see them moving at normal speeds. The authors prove the math that predicted the super-speed was flawed.
2. Dark Energy is Still Needed: Since the "super-drift" isn't real, we can't use it to explain why the universe is accelerating. We still need Dark Energy to explain that phenomenon.
3. Math is Consistent: The paper reassures us that the "fancy" relativistic math and the "standard" math actually agree with each other when used correctly. There is no secret new physics hiding in the equations.

The Takeaway

Think of the universe as a giant, slowing-down train.

• The Standard View: We measure how fast the passengers are walking inside the train. They walk slowly.
• The "New" Claim: Someone looked at the train from a helicopter and claimed the passengers were sprinting so fast they were breaking the sound barrier, and that this sprinting was making the train speed up.
• The Authors' Conclusion: The helicopter observer made a math error. They treated the wind from the helicopter as a separate force. When they fixed the math, they realized the passengers are just walking slowly, and the train is still slowing down.

In short: The universe is behaving exactly as we thought it was. The "faster growth" was a mathematical ghost, not a physical reality.

Technical Summary

1. Problem Statement

Recent literature (specifically references [1–7]) has claimed that the 1+3 covariant approach to cosmological perturbation theory predicts a significantly stronger growth rate for galaxy peculiar velocities compared to standard metric-based perturbation theory.

• The Claim: In a perturbed Einstein-de Sitter (EdS) universe, the covariant approach yields a peculiar velocity growth law of $v_a \propto t^{4/3}$ (or $v \propto a^2$), whereas standard longitudinal gauge theory predicts $v_a \propto t^{1/3}$ (or $v \propto a^{1/2}$).
• The Implication: If the $t^{4/3}$ growth were correct, peculiar velocities at the present day would be orders of magnitude larger than observed (potentially exceeding the speed of light) and could mimic the effects of accelerated expansion (dark energy) in a matter-dominated universe.
• The Authors' Stance: Clarkson and Maartens argue that these claims are incorrect. They assert that the standard result is fully relativistic and that the "stronger growth" arises from an inconsistent mathematical treatment of the coupled covariant system, specifically by treating constrained variables as independent source terms.

2. Methodology

The authors employ the 1+3 covariant formalism (following Ellis, van Elst, and others) to analyze scalar perturbations in a dust-dominated universe. Their methodology involves:

• Frame Definition: They define two 4-velocity fields:
  1. $\tilde{u}_a$: The matter 4-velocity (comoving with galaxies, geodesic, shear-free, irrotational).
  2. $u_a$: A chosen "cosmic rest frame" (observer congruence) which is shear-free and irrotational but moves relative to the matter.
• Linearization: They restrict the analysis to first-order scalar perturbations, neglecting non-linear terms and assuming the matter is pressure-free dust ($p=0, \pi_{ab}=0$).
• System Closure: They derive a closed system of linear differential equations for the peculiar velocity $v_a$ and the 4-acceleration $A_a$ of the observer frame.
• Comparison: They compare the results of this consistent covariant derivation against:
  • Standard metric-based perturbation theory in the longitudinal (Newtonian) gauge.
  • The specific derivations in the controversial papers [1–7] to identify the mathematical error.
• Appendix A: They explicitly map the covariant variables to the Newtonian gauge variables to prove that the shear-free, irrotational observer congruence in the covariant approach corresponds exactly to the longitudinal gauge observer.

3. Key Contributions and Technical Derivations

A. Identification of the "Spurious" Growth

The authors identify the root cause of the conflicting claims in how the covariant equations are rearranged.

• The covariant system can be rearranged into second-order differential equations for $v_a$ that look like forced oscillators:
  $$\ddot{v}_a + \dots = S_a$$
  where $S_a$ contains terms involving density gradients ($\Delta_a$) and expansion gradients ($Z_a$).
• The Error: The controversial papers treat $S_a$ as an external source that can be set to zero to find a "homogeneous solution," leading to $v_a \propto t^{4/3}$.
• The Correction: The authors demonstrate that $S_a$ is not independent of $v_a$. The terms in $S_a$ are constrained by the coupled field equations (Raychaudhuri, shear propagation, and Weyl evolution). Setting $S_a = 0$ arbitrarily breaks the consistency of the coupled system.

B. Derivation of the Correct Growth Law

By consistently solving the coupled subsystem for $(v_a, A_a)$:

1. Momentum Conservation: Relates the acceleration of the observer frame to the velocity:
   $$A_a = -\dot{v}_a - H v_a$$
2. Shear and Weyl Evolution: Using the shear-free condition ($\sigma_{ab}=0$) and irrotational condition ($\omega_{ab}=0$), the authors derive a constraint equation linking the time derivative of acceleration to the velocity:
   $$\dot{A}_a + 2H A_a + 4\pi G \rho v_a = 0$$
3. Closed System: Eliminating $A_a$ from these two equations yields a single second-order differential equation for $v_a$:
   $$\ddot{v}_a + 3H \dot{v}_a + (\dot{H} + 2H^2 - 4\pi G \rho)v_a = 0$$
4. Solution: In an Einstein-de Sitter universe ($\dot{H} = -3/2 H^2$ and $4\pi G \rho = 3/2 H^2$), this simplifies to:
   $$\ddot{v}_a + 3H \dot{v}_a - H^2 v_a = 0$$
   The growing mode solution is:
   $$v_a \propto t^{1/3} \propto a^{1/2}$$
   This matches the standard result from metric-based perturbation theory exactly.

C. Clarification of Kinematics vs. Dynamics

The paper addresses the claim that bulk flows can mimic accelerated expansion.

• Distinction of Congruences: The authors emphasize the difference between the matter congruence (galaxies, $\tilde{u}_a$) and the observer congruence ($u_a$).
• Physical Deceleration: The physical expansion and deceleration of the universe are properties of the matter worldlines. Using the Raychaudhuri equation for the matter frame, they show the deceleration parameter $\tilde{q}$ is strictly positive (decelerating) for dust.
• Observer Artifacts: The kinematics of the arbitrary observer frame ($u_a$) may differ, but this does not alter the physical expansion of the matter. Claims that bulk flows create "apparent acceleration" confuse the kinematics of a mathematical reference frame with the physical dynamics of the matter distribution.

4. Results

• Reproduction of Standard Results: The 1+3 covariant approach, when applied consistently, reproduces the standard relativistic growth law $v \propto a^{1/2}$.
• Refutation of Faster Growth: The growth laws $v \propto t^{4/3}$ and $v \propto t$ claimed in recent literature are mathematically spurious artifacts of inconsistent truncation of the coupled system.
• No Anomalous Bulk Flows: There is no new relativistic mode that allows for bulk flows large enough to mimic dark energy or exceed observed velocity limits.
• Gauge Equivalence: The covariant shear-free/irrotational frame is shown to be the exact counterpart of the longitudinal gauge in metric perturbation theory, confirming that the disagreement is physical, not merely a gauge artifact.

5. Significance

• Validation of Standard Cosmology: The paper reinforces that standard linear perturbation theory is fully relativistic and does not require "corrections" from the covariant formalism to explain peculiar velocities.
• Methodological Warning: It serves as a critical warning against treating constrained variables in coupled differential systems as independent sources, a common pitfall in complex relativistic fluid dynamics.
• Resolution of Controversy: It definitively closes the debate regarding "stronger growth" in the covariant approach, clarifying that the 1+3 formalism is consistent with General Relativity and does not predict anomalous large-scale flows or apparent acceleration in a dust universe.
• Implications for Observations: The results imply that observed bulk flows must be explained within the standard cosmological framework (e.g., via non-linear effects or statistical variance) rather than invoking new relativistic growth mechanisms or alternative explanations for dark energy.