Studying spherical collapse and its implications in the Eddington-inspired Born-Infeld gravity theory

This paper investigates spherical collapse in Eddington-inspired Born-Infeld gravity, demonstrating that the theory's matter-gradient corrections necessitate regularized density profiles and result in a lower linear collapse threshold alongside higher turnaround and virial overdensities compared to the standard $\Lambda$CDM model.

The Gist

Imagine the universe as a giant, expanding balloon. For a long time, scientists have used a standard rulebook called General Relativity (Einstein's theory) to predict how clumps of matter on that balloon—like galaxies and galaxy clusters—should behave as they try to collapse under their own gravity.

This paper investigates a different, slightly more complex rulebook called Eddington-inspired Born-Infeld (EiBI) gravity. The author, Velásquez-Toribio, asks: If we use these new rules instead of Einstein's, how does the "crunching" of matter change?

Here is the breakdown of the paper's findings using simple analogies:

1. The Problem with the "Perfect Sphere"

In the standard rulebook (General Relativity), scientists often use a mental shortcut called the "Top-Hat" model. Imagine a perfect, solid sphere of dough. The inside is perfectly smooth, and the edge is a sharp, sudden cut. When this sphere collapses, the math is easy because the edge is a clean line.

However, the EiBI rulebook has a twist: It cares about gradients.
Think of the EiBI theory like a very sensitive chef who doesn't just care about how much dough you have, but also how steeply the dough slopes at the edges.

• The Issue: If you use the "Top-Hat" (a perfect sphere with a sharp edge), the slope at the edge is infinite (it goes from full dough to no dough instantly). In the EiBI rulebook, this creates a mathematical explosion (a singularity). The theory breaks down because it can't handle a sharp edge.
• The Solution: The author had to replace the sharp "Top-Hat" with a smooth, fuzzy sphere. Imagine the dough gently fading out at the edges rather than stopping abruptly. This "smoothing" is essential for the math to work in this new theory.

2. The Two Shapes Tested

To see how this smoothing affects the collapse, the author tested two different "fuzzy" shapes:

1. The Tanh Profile: A mathematically smooth, S-shaped curve that fades out gently.
2. The Peak Profile: A shape based on the statistical "peaks" of a random field (like the highest points on a bumpy landscape).

Even though both shapes were calibrated to have the same total mass and size, they had different internal "textures." The paper found that the internal texture matters. In EiBI gravity, two clouds of the same mass but different internal shapes will collapse slightly differently. This is a big deal because, in the old rules, only the total mass mattered.

3. The Results: How the Collapse Changes

The author simulated how these fuzzy spheres collapse and compared the results to the standard "Top-Hat" model (which represents our current best guess, the $\Lambda$CDM model). Here is what happened:

• The "Start" Line (Linear Threshold): The amount of initial "push" needed to start a collapse is lower in EiBI gravity. It's easier to get the ball rolling.
• The "Turnaround" (The Peak of Expansion): Imagine a ball thrown upward. The "turnaround" is the exact moment it stops going up and starts falling down.
  • In EiBI gravity, the ball falls down sooner (at a smaller radius) than in the standard model.
  • However, at that moment, the density of the matter is higher. It's like the ball is more compressed when it finally turns around.
• The "Final State" (Virial Overdensity): After the collapse settles into a stable clump (like a galaxy cluster), the final density is higher in EiBI gravity. The clumps end up denser than we would expect under Einstein's rules.
• The Size: The physical size of the turnaround point is slightly smaller, but this effect is less dramatic than the changes in density.

4. The "Mass Dependence" Surprise

In the standard model, a small clump of matter and a huge clump of matter behave in almost the same way (they are "universal").
In this EiBI study, size matters.

• The author found that the way a clump collapses depends on its mass. A small clump behaves differently than a giant one.
• Analogy: Think of it like falling through water. A small pebble and a large boulder fall differently because the water's resistance interacts with their size. In EiBI gravity, the "resistance" of the universe's geometry interacts with the size of the matter clump, making them collapse in unique ways.

Summary

The paper concludes that EiBI gravity is not just a simple tweak to Einstein's theory (like just changing the strength of gravity). Instead, it introduces a new sensitivity to the shape and smoothness of matter.

• Key Takeaway: You cannot just look at "how much" matter is there; you must also look at "how it is arranged."
• The Verdict: If EiBI gravity is the correct description of the universe, then galaxy clusters will be denser and form slightly differently than we currently predict, and these differences will depend on the specific shape of the matter inside them.

The author notes that this work is just the foundation. Now that they understand how a single sphere collapses under these rules, the next step (for future papers) would be to use this knowledge to predict how many galaxies and clusters we should see in the real universe.

Technical Summary

1. Problem Statement

The paper addresses the challenge of modeling spherical gravitational collapse within Eddington-inspired Born-Infeld (EiBI) gravity, a modified gravity theory formulated in the Palatini approach.

• The Core Issue: In General Relativity (GR), the standard "top-hat" model (a discontinuous, uniform overdensity) is a robust tool for deriving collapse benchmarks (e.g., linear collapse threshold $\delta_c$, turnaround overdensity $\delta_t$, virial overdensity $\Delta_{vir}$). However, EiBI gravity introduces a modification to the Poisson equation in the weak-field limit that depends explicitly on spatial derivatives of the matter density ($\nabla \rho$ and $\nabla^2 \rho$).
• The Consequence: A discontinuous top-hat profile generates singular (distributional) contributions at the boundary in EiBI gravity, rendering the standard collapse dynamics ill-defined. Therefore, the collapse problem in EiBI is intrinsically tied to the smoothing scale and the internal radial structure of the overdensity profile, requiring a regularization prescription that is absent in standard GR treatments.

2. Methodology

The author develops a self-consistent spherical-collapse framework under the sub-horizon, pressureless, and quasi-static approximations.

A. Theoretical Framework

• Field Equations: Starting from the EiBI action in the Palatini formalism, the author derives the modified Poisson equation for the Newtonian potential $\Phi$:
  $$\nabla^2_r \Phi = 4\pi G_N \rho_m + \frac{\kappa_{BI}}{4} \nabla^2_r \rho_m$$
  This is recast into an effective acceleration equation where the EiBI correction acts as a force proportional to the density gradient: $a_{EiBI} \propto -\nabla \rho$.
• Evolution Equation: By combining the continuity and Euler equations, a master evolution equation for the density contrast $\delta$ is derived. Crucially, the EiBI source term involves the Laplacian of the density contrast ($\nabla^2 \delta$).

B. Regularization and Initial Conditions

To handle the gradient singularity, the author rejects the ideal top-hat model in favor of regularized profiles. Two distinct initial configurations are compared, calibrated to share the same characteristic radius and cumulative mass proxy:

1. Regularized Tanh Profile: A phenomenological smoothing using a hyperbolic tangent function to ensure finite derivatives.
2. Peak-Based Profile: A statistically motivated profile derived from the conditional average of a Gaussian random field around a peak (using the BBKS formalism). This provides a more physical interpretation of the initial overdensity structure.

C. Numerical Implementation

• Closure: An "effective physical-gradient closure" is adopted, approximating $\nabla^2_r \delta \simeq -k_{phys}^2 P \delta$, where $P$ is a dimensionless profile factor encoding the shape dependence.
• Diagnostics: The code solves the nonlinear evolution equation to determine:
  • Linear collapse threshold ($\delta_c$).
  • Turnaround epoch ($a_t$), radius ($R_t$), and overdensity ($\delta_t$).
  • Virial overdensity ($\Delta_{vir}$) and the virial-to-turnaround radius ratio ($\eta = R_{vir}/R_t$).
• Virial Prescription: An effective virial theorem is derived for EiBI, which includes bulk terms ($\int \rho^2 dV$) and surface terms arising from the density gradient at the boundary. This replaces the standard $2T + W = 0$ relation.

3. Key Contributions

1. Formalization of Gradient-Dependent Collapse: The paper establishes that in EiBI gravity, collapse dynamics cannot be determined solely by the overdensity amplitude; the radial profile shape is a fundamental physical parameter.
2. Rejection of the Top-Hat Idealization: It demonstrates that the discontinuous top-hat is mathematically inadmissible in EiBI without regularization, necessitating a coarse-graining prescription.
3. Effective Virial Theorem: The derivation of a modified virial balance (Eq. 77) that explicitly accounts for the EiBI matter-gradient coupling, showing that virial quantities become profile-dependent observables rather than universal functions of redshift.
4. Comparative Analysis: The first systematic comparison of phenomenological (Tanh) vs. statistical (Peak-based) initial conditions in EiBI spherical collapse, isolating the "profile-shape effect" from normalization artifacts.

4. Key Results

The numerical results, compared against the $\Lambda$CDM reference model, reveal the following trends as the dimensionless coupling $\hat{\kappa}_{BI}$ increases:

• Linear Collapse Threshold ($\delta_c$):

  • Result: $\delta_c$ is lowered relative to $\Lambda$CDM.
  • Implication: Collapse is reached from a slightly smaller linear amplitude because the EiBI gradient force aids the collapse.
  • Profile Dependence: A slight but distinct difference exists between Tanh and Peak-based profiles, confirming sensitivity to internal structure.

• Turnaround Overdensity ($\delta_t$):

  • Result: $\delta_t$ is enhanced relative to $\Lambda$CDM.
  • Implication: The perturbation reaches maximum expansion at a higher density contrast. This is a stronger nonlinear diagnostic of EiBI than $\delta_c$.

• Turnaround Radius ($R_t$):

  • Result: $R_t$ shows a modest reduction compared to $\Lambda$CDM.
  • Implication: The effect is weaker on geometric quantities than on density-based quantities, as the EiBI correction is sourced by density gradients.

• Virial Overdensity ($\Delta_{vir}$):

  • Result: $\Delta_{vir}$ is enhanced, mirroring the behavior of $\delta_t$.
  • Implication: The final virialized state is denser than in GR.

• Virial-to-Turnaround Ratio ($\eta = R_{vir}/R_t$):

  • Result: $\eta$ deviates slightly from the standard GR value of $0.5$ (increasing to $\sim 0.5 + 10^{-6}$).
  • Implication: The virialization process is modified but remains close to the standard benchmark for the parameter ranges tested.

• Mass Dependence:

  • Unlike in GR where turnaround quantities are nearly universal, EiBI results show a visible mass dependence. Changing the mass alters the characteristic size of the perturbation, thereby changing the effective strength of the gradient term.

5. Significance

• Theoretical Foundation: This work provides the necessary theoretical foundation for applying EiBI gravity to large-scale structure formation. It clarifies that any prediction for halo mass functions or bias in EiBI must account for the profile dependence of the collapse threshold.
• Observational Probes: The results identify turnaround overdensity ($\delta_t$) and virial overdensity ($\Delta_{vir}$) as the most sensitive nonlinear diagnostics for distinguishing EiBI from $\Lambda$CDM.
• Nature of Modification: The study confirms that EiBI gravity does not act as a simple rescaling of Newton's constant (which would preserve universality). Instead, it introduces a local matter-curvature response controlled by the spatial structure of the density field.
• Future Directions: The paper sets the stage for subsequent work connecting these collapse benchmarks to halo abundance statistics, cluster number counts, and observational constraints from galaxy surveys.

In summary, the paper argues that the "matter-gradient" nature of EiBI gravity fundamentally alters the spherical collapse problem, making the internal structure of dark matter halos a critical variable in cosmological modeling and offering new avenues for testing modified gravity theories.