Abundance of cosmic voids in EFT of dark energy

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This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The Universe is a Sponge, Not a Solid Block

Imagine the universe not as a solid block of cheese, but as a giant, cosmic sponge.

The Holes: The empty spaces in the sponge are called Cosmic Voids. These are vast regions where there is almost no matter (no stars, no galaxies). They make up most of the volume of the universe.
The Cheese: The solid parts of the sponge are clusters of galaxies and filaments.

Scientists have long used these "holes" to test how gravity works. But there's a mystery: Why is the universe expanding faster and faster? The standard answer is "Dark Energy," but we don't really know what it is. Some scientists think maybe gravity itself works differently on huge scales.

This paper asks: If gravity works differently than Einstein said it does, how would the "holes" in our cosmic sponge change?

The Toolkit: A "Universal Remote" for Gravity

The authors didn't just pick one specific theory of "weird gravity." Instead, they used a tool called the Effective Field Theory (EFT) of Dark Energy.

The Analogy: Imagine you are trying to tune a radio to find a specific station.

Old way: You have to build a brand-new radio for every single theory you want to test (one radio for Theory A, another for Theory B).
This paper's way: They built a "Universal Remote." This remote has knobs (parameters) that can tweak the signal to represent any theory of modified gravity within a certain family. They didn't have to rebuild the radio; they just turned the knobs to see what happens.

The most important knob they turned was called $\alpha_B$ (Alpha-B). Think of this as a "friction" or "mixing" knob that changes how the scalar field (a hidden force driving expansion) interacts with gravity.

The Experiment: The Spherical Shell Game

To see how these "knobs" affect the voids, the authors modeled a void as a set of concentric onion layers (spherical shells) of empty space.

The Setup: Imagine a bubble of nothingness in a sea of matter. The edges of this bubble are made of invisible shells.
The Rule: In our normal universe (General Relativity), these shells expand outward because the universe is expanding, but they slow down because the gravity of the surrounding matter is pulling them back.
The "Void Formation" Moment: A void is officially "born" when the outer shells expand so much that they cross over each other (like two rings of a chainmail armor sliding past one another). This is called Shell-Crossing.

The authors asked: If we turn the "Alpha-B" knob, does the shell-crossing happen sooner or later? And does the void end up bigger or smaller?

The Surprising Findings

Here is what they discovered, using our analogies:

1. The "Tug-of-War" Effect

When they turned up the gravity knob ( $\alpha_B$ ), two things happened at the same time, fighting against each other:

Effect A: Gravity got stronger. This pulled the shells back harder, making them collapse (or cross) sooner.
Effect B: The "growth" of the voids got faster. The universe's expansion pushed the shells apart more aggressively.

The Result: These two effects almost perfectly canceled each other out! The final size of the void didn't change as much as the authors expected. It was like a tug-of-war where both teams were equally strong; the rope barely moved. The change in the void's "birth certificate" (the critical density) was tiny—about 10 times smaller than the knob they turned.

2. The "Small vs. Big" Difference

When they looked at the abundance of voids (how many exist at different sizes), they found a split personality:

Tiny Voids (Small Scales): The number of small voids changed mostly because the background music changed. In physics terms, the "linear matter power spectrum" (which describes how matter is distributed in the universe) shifted. The gravity knob changed the ingredients of the universe, which changed how many small bubbles formed.
Huge Voids (Large Scales): The number of giant voids changed because of the rules of the game. Here, the specific way gravity pulls on the shells (the critical density) mattered more.

3. The "Time Travel" Factor

The authors also looked at when these changes happen. They found that if the gravity knob changes slowly over time (a specific mathematical parameter called $p$ ), the effects are more dramatic.

Analogy: If you turn up the volume on a song suddenly, it's jarring. If you turn it up gradually over a long time, the song feels different. They found that models where the gravity change happens "earlier" in the universe's history (smaller $p$ ) create bigger differences in the number of voids we see today.

Why Does This Matter?

Imagine you are a detective trying to solve a crime. You have a suspect (Dark Energy) and a crime scene (the Cosmic Voids).

If you assume the standard rules of gravity, you expect to find a certain number of "holes" in the sponge.
If gravity is actually "weird" (modified), the number and size of those holes will be different.

This paper tells us: "Don't just look at the size of the holes; look at the distribution of their sizes."

If we see a lot of small voids but fewer big ones (or vice versa) compared to our predictions, it might be a sign that gravity isn't working exactly as Einstein said.
The paper provides a "cheat sheet" for astronomers. It tells them exactly what to look for in future telescope data to see if our theory of gravity needs a software update.

The Bottom Line

The universe is full of giant empty bubbles. By studying how these bubbles form and grow, we can test if gravity works differently on the largest scales. This paper used a "universal remote" to test many theories at once and found that while the changes are subtle, they leave a specific fingerprint on the size distribution of cosmic voids. If we can measure the voids precisely enough, we might finally crack the code of Dark Energy.

1. Problem Statement

The standard $\Lambda$ CDM model, while successful, faces theoretical challenges (fine-tuning of the cosmological constant) and observational tensions. Modified gravity theories, particularly scalar-tensor theories like the Horndeski family, offer alternatives. However, testing these theories is complicated by screening mechanisms (e.g., Vainshtein screening) that hide deviations from General Relativity (GR) in high-density regions.

Cosmic voids (vast underdense regions) are promising laboratories for testing gravity because the density contrast $\delta$ is bounded ( $\delta \ge -1$ ), meaning the Vainshtein mechanism may only operate partially or not at all, potentially leaving observable signatures of modified gravity. While previous studies have analyzed voids in specific modified gravity models (e.g., $f(R)$ , Galileon), there is a need for a model-independent analysis. This paper addresses this gap by investigating void evolution and abundance using the Effective Field Theory (EFT) of Dark Energy, a unified framework for scalar-tensor theories.

2. Methodology

A. Theoretical Framework: EFT of Dark Energy

The authors employ the EFT of dark energy, which expands the Horndeski action around a cosmological background.

Assumptions:
- The background evolution follows the standard $\Lambda$ CDM model.
- Gravitational wave speed equals the speed of light ( $\alpha_T = 0$ ).
- The effective Planck mass is time-independent ( $\alpha_M = 0$ ).
- The analysis focuses on the kinetic braiding effect, characterized by the time-dependent parameter $\alpha_B(t)$ .
- The time dependence of EFT parameters is modeled as $\alpha_i(t) = \alpha_{i0} (H_0/H)^{2p}$ , where $p$ is a constant.
Simplification: The study restricts itself to manifestly second-order scalar-tensor theories, neglecting DHOST (Degenerate Higher-Order Scalar-Tensor) extensions for this specific analysis.

B. Modeling Void Formation

The authors model void formation using the spherical shell dynamics approach:

Shell Evolution: They solve the equation of motion for a spherical shell of non-relativistic matter in a modified gravitational potential derived from the EFT action.
Nonlinear Dynamics: The full nonlinear equations are solved to track the shell radius $R(t)$ until shell-crossing occurs (when neighboring shells intersect). This intersection time ( $t_{sc}$ ) defines the moment of void formation.
Critical Density Contrast: They calculate the linearly extrapolated critical density contrast ( $\delta_{sc}$ ) at the time of shell-crossing. This is the threshold used in excursion set theory to predict void abundance.
Overdensity (Halo) Formation: To account for the "void-in-cloud" problem (voids embedded in collapsing overdensities), they also compute the critical density contrast for halo collapse ( $\delta_c$ ) using a spherical collapse model.

C. Void Size Function (VSF)

Using the Sheth–van de Weygaert (SvdW) model, the authors compute the Void Size Function, $dn/d\ln r$ , which describes the number density of voids as a function of their volume.

The calculation incorporates four key modified gravity components:
1. The linear matter power spectrum ( $P(k)$ ), affected by $\alpha_K$ and $\alpha_B$ .
2. The critical density for void formation ( $\delta_{sc}$ ).
3. The critical density for halo collapse ( $\delta_c$ ).
4. The expansion factor ( $R$ ) of the shell at the time of shell-crossing.
They use the hi-class Einstein-Boltzmann solver to compute the linear matter power spectrum.

3. Key Contributions

Model-Independent Analysis: This is one of the first studies to analyze void abundance in the general EFT of dark energy framework rather than specific sub-models, allowing for broader constraints on modified gravity.
Quantification of $\delta_{sc}$ Modification: The authors derive how the EFT parameter $\alpha_B$ $α_{B}$ alters the critical density contrast for void formation. They find that the modification is one order of magnitude smaller than the dimensionless EFT parameter $\alpha_B$ $α_{B}$ itself. This is due to a near-perfect cancellation between two competing effects:
- Effect A: Stronger gravity (due to $\alpha_B$ ) causes shells to cross earlier, tending to reduce $|\delta_{sc}|$ .
- Effect B: Enhanced linear growth of perturbations (due to $\alpha_B$ ) tends to increase $|\delta_{sc}|$ .
Scale-Dependent VSF Modifications: The paper demonstrates that the modification to the Void Size Function is scale-dependent.
- Small Scales: Modifications are dominated by changes in the linear matter power spectrum.
- Large Scales: Modifications are driven by a combination of the linear power spectrum and the critical density contrast ( $\delta_{sc}$ ).
Parameter Constraints: The study identifies theoretical bounds on the parameters. For instance, to avoid imaginary scalar fields (which would make the physics unphysical) in underdense regions, the parameter $p$ must satisfy $p \gtrsim 0.87$ for $\alpha_{B0} = 0.1$ .

4. Key Results

Critical Density Contrast ( $\delta_{sc}$ ):
- As $\alpha_{B0}$ increases, $|\delta_{sc}|$ decreases slightly (is suppressed).
- The suppression is small (approx. 10% of the magnitude of $\alpha_{B0}$ ) due to the cancellation of competing effects mentioned above.
- The expansion factor $R$ (comoving radius growth) is suppressed for larger $\alpha_{B0}$ because stronger gravity slows down shell expansion.
Void Size Function (VSF):
- Scale Dependence: The VSF exhibits significant scale-dependent deviations from $\Lambda$ $Λ$ CDM.
  - At small scales ( $r \sim 0.1$ Mpc/h), the VSF is suppressed by $\sim 5\%$ for $\alpha_{B0} = 0.1$ .
  - At large scales ( $r \sim 20$ Mpc/h), the VSF is enhanced by $\sim 40\%$ .
- Dominant Factors:
  - On small scales, the change is almost entirely determined by the modified linear matter power spectrum.
  - On large scales, both the power spectrum and the critical density contrast ( $\delta_{sc}$ ) contribute, though the power spectrum remains the dominant factor.
  - The expansion factor ( $R$ ) and halo critical density ( $\delta_c$ ) have negligible impacts on the VSF.
- Parameter $p$ : The modification to the VSF is enhanced for smaller values of $p$ (e.g., $p=0.87$ vs $p=1$ ) because the modified gravity effects become active earlier in the universe's history.
- Parameter $\alpha_K$ : The VSF is found to be insensitive to the parameter $\alpha_K$ (which affects the sound speed of scalar perturbations) because the variance of the density field remains nearly unchanged by $\alpha_K$ .

5. Significance and Conclusion

This paper provides a crucial bridge between the theoretical framework of EFT of dark energy and observational probes using cosmic voids.

Laboratory for Gravity: It confirms that cosmic voids are sensitive probes for modified gravity, specifically showing that even small deviations in the EFT parameters can lead to observable, scale-dependent changes in void abundance.
Interpretation of Observations: The finding that small-scale VSF modifications are driven by the linear power spectrum, while large-scale modifications involve the critical density contrast, offers a clear guide for interpreting future large-scale structure surveys (e.g., Euclid, DESI, LSST).
Future Directions: The authors note that while the spherical model provides qualitative insights, it ignores complex processes like void merging and ellipticity. They emphasize that N-body simulations incorporating the full EFT of dark energy are the necessary next step to refine these predictions and fully exploit voids as cosmological probes.

In summary, the work establishes that the EFT parameter $\alpha_B$ leaves a distinct, scale-dependent imprint on the cosmic void population, primarily mediated through the linear matter power spectrum and the critical density threshold, offering a robust method to constrain modified gravity models independent of specific Lagrangian choices.