The reason peculiar velocities grow faster in general relativity than in Newtonian gravity

This paper demonstrates that peculiar velocities grow significantly faster in general relativity than in Newtonian gravity because relativistic theory accounts for the gravitational contribution of kinetic energy (peculiar flux), a factor absent in Newtonian models, thereby offering a potential explanation for observed large-scale bulk flows that challenge the standard $\Lambda$CDM paradigm.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Mystery: The Universe is Moving Faster Than Expected

Imagine the universe as a giant, expanding ocean. Most of the water (galaxies) is flowing away from us because the ocean itself is stretching out (this is the Hubble expansion). But, just like in a real ocean, there are also currents and eddies where water moves on top of the main flow. In astronomy, we call these extra movements "peculiar velocities" or "bulk flows."

For decades, astronomers have been measuring these currents. Recently, they found a problem: The currents are moving much faster and covering much larger distances than our best theories predicted.

It's like a weather forecast saying a wind will blow at 10 mph, but the anemometer is recording 100 mph. This has caused a crisis in cosmology because the current standard model (called ΛCDM) says these fast flows shouldn't exist.

The Old Theory: The "Flat Earth" Map (Newtonian Gravity)

For a long time, scientists tried to solve this using Newtonian gravity (the physics of apples falling and planets orbiting, as described by Isaac Newton).

• The Analogy: Think of Newtonian gravity like a flat map. It's great for driving around town, but it doesn't account for the curvature of the Earth when you try to fly a plane across the globe.
• The Prediction: Using this "flat map" physics, the theory predicts that these cosmic currents should grow very slowly over time. Specifically, the speed should increase by the cube root of time ($t^{1/3}$). It's a slow, steady creep.
• The Flaw: This prediction is too weak. It cannot explain the "super-highway" speeds astronomers are actually seeing.

The New Theory: The "Curved Earth" Map (General Relativity)

The authors of this paper, Erick Pasten and Christos Tsagas, argue that we've been using the wrong map. We need General Relativity (Einstein's theory), which accounts for the fact that space and time are curved and interconnected.

• The Analogy: Imagine you are running on a treadmill.
  • Newton's View: Only your weight matters. If you are heavy, you push down hard. If you are just moving your legs but not pushing down, you don't affect the treadmill.
  • Einstein's View: Everything that has energy affects the treadmill. Even if you are just running in place (moving energy), that motion itself creates a "push" or a gravitational effect.

The Secret Ingredient: "Moving Energy"

Here is the core discovery of the paper:

In Newton's world, only mass (stuff) creates gravity. If matter is moving, that motion doesn't create extra gravity.
In Einstein's world, energy creates gravity. And when matter moves, it carries energy flux (a flow of energy).

The Analogy:
Imagine a crowd of people walking through a hallway.

• Newtonian View: The hallway only feels the weight of the people standing still. If they start walking, the hallway doesn't feel any extra "pull" from their movement.
• Relativistic View: The hallway feels the weight of the people plus the "oomph" of their movement. The fact that they are moving creates a new kind of gravitational pull that the hallway feels.

The authors realized that in the universe, the "moving matter" (the bulk flows) creates its own gravitational boost. This boost acts like a self-reinforcing engine.

The Result: A Speed Boost

Because of this extra "moving energy" gravity:

1. Newtonian Physics: The driving force (the push) gets weaker over time. The speed grows slowly ($v \propto t^{1/3}$).
2. Relativistic Physics: The driving force stays strong or even gets stronger because the moving matter keeps feeding energy back into the gravitational field.
   • Minimum Result: The speed grows linearly ($v \propto t$). This is much faster than Newton predicted.
   • Maximum Result: When you add in the clumping of galaxies (structure formation), the speed can grow even faster ($v \propto t^{4/3}$).

The Metaphor:

• Newton is like a car coasting downhill. As it goes, friction slows it down, and it doesn't gain much speed.
• Einstein is like a car with a turbocharger that kicks in because the car is moving. The faster it goes, the more fuel it injects, and the faster it accelerates.

Why This Matters

The authors show that we don't need to invent new physics, dark matter, or change the rules of the universe to explain these fast flows. We just need to stop using the "flat map" (Newton) for a problem that requires the "curved map" (Einstein).

• The Conclusion: The "Bulk Flow Puzzle" isn't a bug in the universe; it's a bug in our calculation method. When we correctly account for the gravity of moving energy, the universe naturally allows for these fast, deep currents. The standard model (ΛCDM) is fine; we just need to calculate it using the full power of Einstein's General Relativity.

Summary in One Sentence

The universe's currents are faster than we thought not because the laws of physics are broken, but because we forgot that moving matter creates its own extra gravity, which acts like a turbocharger to speed things up.

Technical Summary

Here is a detailed technical summary of the paper "The reason peculiar velocities grow faster in general relativity than in Newtonian gravity" by Erick Pasten and Christos G. Tsagas.

1. Problem Statement

The paper addresses the "bulk flow" puzzle in cosmology. Observational surveys (e.g., CosmicFlows) have increasingly reported large-scale peculiar motions (bulk flows) with sizes and velocities that exceed the theoretical limits predicted by the standard $\Lambda$CDM concordance model.

• The Discrepancy: While many surveys agree on the direction of bulk flows, recent data suggests speeds and depths significantly higher than the $\Lambda$CDM model allows.
• The Root Cause: The standard $\Lambda$CDM predictions for peculiar velocities are based on Newtonian gravity (or "quasi-Newtonian" approximations). These theories predict a relatively slow linear growth rate for peculiar velocities ($v \propto t^{1/3}$) during the matter-dominated (Einstein-de Sitter) epoch.
• The Hypothesis: The authors propose that the discrepancy arises because Newtonian theory fails to account for the gravitational contribution of the peculiar energy flux (kinetic energy of moving matter), a factor that is inherently present in General Relativity (GR).

2. Methodology

The authors perform a unified, covariant comparison of three theoretical frameworks applied to a flat Friedmann-Robertson-Walker (FRW) background:

1. Newtonian Treatment: Standard linear perturbation theory using the Euler equation and Poisson's equation.
2. Quasi-Newtonian Treatment: A relativistic approach that imposes strict constraints (zero vorticity, zero shear) to simplify the math, effectively reducing the system to Newtonian equations.
3. Fully Relativistic Treatment: A linear perturbation analysis within General Relativity that imposes no artificial constraints and fully accounts for the stress-energy tensor contributions of moving matter.

Key Technical Steps:

• Frame Definition: They define two frames: the Cosmic Microwave Background (CMB) rest frame and the bulk-flow matter frame.
• Flux Analysis: They analyze the energy-momentum tensor transformation between these frames. In GR, relative motion generates an effective energy flux vector ($q_a$) even if the fluid is perfect in its own rest frame.
• Evolution Equations: They derive the linear evolution equations for the peculiar velocity field ($v$) and the 4-acceleration ($A_a$) in all three frameworks.
• Source Term Comparison: They specifically isolate the "peculiar flux" term in the conservation laws and Einstein's equations to see how it drives the velocity growth.
• Recovery Attempt: They attempt to recover the relativistic growth rate from the Newtonian setup by selectively including "source-free" integration terms in the Poisson equation, demonstrating the mathematical link between the two.

3. Key Contributions

• Identification of the Driving Agent: The paper identifies that the fundamental difference lies in the 4-acceleration ($A_a$), which acts as the driving force for peculiar velocities.
  • In Newtonian/Quasi-Newtonian theories, the driving force decays with time ($A_a \propto t^{-2/3}$).
  • In General Relativity, the peculiar flux contributes to the gravitational field, resulting in a driving force that is either constant ($A_a \propto \text{const}$) or increasing ($A_a \propto t^{1/3}$) during the Einstein-de Sitter phase.
• Unified Covariant Framework: The authors provide a direct, covariant comparison showing exactly where the Newtonian and relativistic treatments diverge (specifically at the level of the conservation laws and the stress-energy tensor).
• Recovery of Relativistic Results: They demonstrate that the relativistic growth rate can be mathematically recovered from the Newtonian Poisson equation by retaining specific solenoidal (divergence-free) integration terms ($V_\alpha$) that are typically discarded or set to zero by boundary conditions.

4. Results

The study yields distinct growth rates for the peculiar velocity field ($v$) during the matter-dominated (Einstein-de Sitter) epoch:

Framework	Driving Force Behavior	Velocity Growth Rate ($v$)
Newtonian	Decays as $t^{-2/3}$	$v \propto t^{1/3}$
Quasi-Newtonian	Decays as $t^{-2/3}$	$v \propto t^{1/3}$
General Relativity (Minimal)	Constant ($A_a = \text{const}$)	$v \propto t$
General Relativity (Full)	Grows as $t^{1/3}$ (with structure formation)	$v \propto t^{4/3}$

Specific Findings:

• Relativistic Enhancement: The inclusion of the peculiar flux's gravitational input in GR leads to a growth rate of at least $v \propto t$, which is significantly faster than the Newtonian $t^{1/3}$.
• Structure Formation: When density gradients and structure formation effects are included in the relativistic treatment, the growth rate accelerates further to $v \propto t^{4/3}$.
• Vorticity and Shear: The paper notes that while the velocity field grows rapidly, the peculiar vorticity and shear grow much slower ($v \propto t^{1/3}$), meaning the bulk flow is dominated by the velocity magnitude rather than rotational or shear distortions.
• Late-Time Suppression: The authors acknowledge that in the late-time $\Lambda$-dominated era, cosmic acceleration suppresses velocity growth. However, the enhanced growth accumulated during the earlier Einstein-de Sitter phase should survive to the present day, explaining the observed fast bulk flows.

5. Significance

• Resolution of the Bulk Flow Puzzle: The paper suggests that the "anomalously fast" bulk flows reported in recent surveys do not necessarily require new physics, exotic dark energy, or a failure of the $\Lambda$CDM model. Instead, the discrepancy arises from using an inappropriate gravitational theory (Newtonian) for a problem that requires General Relativity.
• Theoretical Consistency: It demonstrates that GR naturally accommodates faster and deeper bulk flows within the standard $\Lambda$CDM paradigm without introducing new free parameters.
• Implications for Simulations: The authors highlight a gap in current numerical simulations, which often rely on Newtonian codes or specific gauge choices that ignore the relativistic peculiar flux. They call for the development of fully relativistic numerical codes to properly test these analytical predictions.
• Observational Outlook: The results provide a theoretical basis for interpreting high-redshift peculiar velocity surveys (like CosmicFlows-4). The model predicts that peculiar velocities should have been higher at earlier times (higher redshifts) and suppressed at lower redshifts, a trend consistent with some recent observational data.

In conclusion, the paper argues that the "peculiar flux" is the missing gravitational link in Newtonian cosmology. By accounting for the fact that moving matter gravitates in General Relativity, the predicted growth of bulk flows aligns with the faster, deeper flows observed in the universe.