⚛️ general relativity

Cosmic strings, domain walls and environment-dependent clustering

This paper introduces "norns," a new relativistic simulation code to study environment-dependent clustering in phantom-crossing dark energy models, revealing that non-minimally coupled scalar fields driving late-time phase transitions can generate cosmic strings that suppress structure growth in voids while enhancing it in overdense regions, leaving distinctive signatures in the matter distribution detectable in low-redshift data.

Original authors: Øyvind Christiansen, Julian Adamek, Martin Kunz

Published 2026-01-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Øyvind Christiansen, Julian Adamek, Martin Kunz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

Imagine the universe as a giant, expanding balloon. For decades, scientists have thought this balloon is being inflated by a mysterious, steady force called "Dark Energy," which acts like a constant pressure pushing everything apart. This is the standard story, known as ΛCDM.

But new data suggests the inflation might be behaving strangely—maybe it's speeding up in a way that breaks the rules of the standard story. This paper explores a wilder idea: what if Dark Energy isn't a constant force, but a shape-shifting field that reacts to its surroundings?

Here is the breakdown of what the authors did and found, using simple analogies.

1. The Shape-Shifting Field (The "Symmetron")

Think of the universe as a room filled with air. In some parts of the room (dense clusters of galaxies), the air is thick and heavy. In other parts (vast empty spaces called "voids"), the air is thin.

The authors studied a hypothetical field (a kind of invisible energy) that acts like a mood ring or a thermostat:

In crowded places (Overdensities): The field "hides" or "screens" itself. It acts like a shy person in a crowded room, staying quiet and not interfering with gravity. This is why we don't see weird forces messing up our solar system.
In empty places (Underdensities/Voids): The field "wakes up." It becomes active and exerts a new, invisible force (a "fifth force") that pushes and pulls on matter.

2. The Great "Snap" (Phase Transition)

The paper focuses on a specific moment in the universe's history (around 6 billion years ago) when this field suddenly "snapped" into a new state. This is called a Structure-Induced Phase Transition.

Imagine a glass of water that is super-cooled but hasn't frozen yet. If you drop a single ice crystal in, the whole thing freezes instantly. In this model, the "ice crystals" were the empty voids of the universe. As the universe expanded and these voids grew large enough, the field suddenly changed its behavior only in those empty spots, while staying hidden in the dense galaxy clusters.

3. Cosmic Scars: Strings vs. Walls

When this field "snapped," it didn't happen perfectly everywhere at once. Just like cracks forming in drying mud or ice forming on a pond, the universe developed defects where the transition didn't line up.

The authors compared two types of these cosmic scars:

Domain Walls (The "Walls"): Imagine a sheet of paper tearing. The tear is a 2D surface. This happens if the field is simple (like a real number).
Cosmic Strings (The "Strings"): Imagine a rope getting tangled. The defect is a 1D line. This happens if the field is more complex (like a complex number).

The authors built a super-computer simulation (named norns) to watch these "strings" and "walls" form and move. They found that these strings act like invisible fishing lines that get pinned to the dense galaxy clusters, while the empty voids between them get pushed apart.

4. What Happened to the Universe?

The simulation revealed some surprising results:

The Voids Got Emptier: In the empty spaces, the new "fifth force" acted like a vacuum cleaner, sucking matter out of the voids and pushing it toward the dense galaxy clusters. This made the voids even more empty and the clusters even denser than in the standard model.
The "Halo" Effect: Galaxies live in "halos" of dark matter. The simulation showed that in these new models, the halos formed differently. In some cases, the extra force helped clumps of matter stick together faster. In others (specifically the "string" models), it actually prevented small clumps from forming because the force was too chaotic.
Speeding Up: The particles in these simulations were moving faster. The new force gave them an extra kick, creating a "tail" of very fast-moving particles that you wouldn't see in the standard model.

5. How Do We Spot This?

The authors realized that if you look at the universe with a standard telescope (looking at the big picture of galaxy distribution), these changes might look very small—only a few percent different from the standard model. You might miss them entirely.

However, if you look at the details, the signal is huge:

The "Empty" Tail: If you count how many empty spaces exist, the new models predict way more extremely empty spaces than the standard model. It's like finding a room that is 99% empty instead of 90% empty.
Marked Statistics: The authors suggest a new way to look at data. Instead of just counting galaxies, we should "mark" them based on how empty their neighborhood is. If you do this, the difference between the standard model and their new model becomes very loud and clear.

The Bottom Line

The paper argues that the universe might be undergoing a "late-time phase transition" where a hidden force wakes up specifically in the empty voids. This creates a network of cosmic strings that reshapes the universe, making voids emptier and galaxy clusters denser.

While the overall map of the universe looks mostly the same, the details of the empty spaces and the speed of the galaxies hold the smoking gun. The authors conclude that future surveys (like DESI or Euclid) should look specifically at these "environment-dependent" clues—like the distribution of empty voids—to see if this exotic physics is real.

In short: The universe might have a secret switch that turns on a new force only in the empty spaces, and we need to look at the emptiness to find it.

Technical Summary: Cosmic Strings, Domain Walls, and Environment-Dependent Clustering

Problem and Motivation
Recent cosmological data suggesting a phantom-crossing dark energy equation of state motivate models featuring non-minimal couplings that induce fifth forces on structure formation. Reconciling such models with local tests typically requires strong screening mechanisms, leading to environment-dependent clustering. While the symmetron model offers a relativistic realization where a scalar field $\phi$ couples to matter stress-energy, standard implementations often assume homogeneous phase transitions. However, recent work indicates that structure-induced phase transitions (SIPT), where the transition occurs at redshifts $z_{SSB} \gg z_*$ (where $z_*$ is the transition redshift in a homogeneous universe), can address the dark energy coincidence problem.

A critical distinction in topological defect formation arises from the symmetry of the scalar field. A real-valued symmetron ( $\phi \in \mathbb{R}$ ) breaks a $Z_2$ symmetry, forming domain walls. A complex-valued symmetron ( $\phi \in \mathbb{C}$ ) breaks a global $U(1)$ symmetry, forming cosmic strings. The impact of these distinct defect types (strings vs. walls) on late-time structure formation, particularly in the SIPT regime where defects pin to overdensities, had not been previously simulated in a fully relativistic cosmological context.

Methodology
The authors introduce norns (NOn-linear and Relativistic Numerical Strings), a fully relativistic cosmological particle-mesh code based on the scheme of gevolution. This code self-consistently evolves a complex scalar field coupled to matter, generalizing the symmetron model to produce global $U(1)$ strings rather than domain walls. For comparison, the evolution of the real-valued symmetron is performed using the existing AsGRD code.

Model: The scalar field is governed by an effective potential $V_{eff}(\phi)$ derived from a conformal transformation between the Jordan and Einstein frames. The model includes a tachyonic mass $\mu$ , self-interaction $\lambda$ , and conformal coupling $M$ . The study focuses on the SIPT regime where the phase transition is triggered by local underdensities (voids) rather than the background density.
Defect Identification: Strings are identified by calculating the winding number of the complex phase around closed loops on the grid. Domain walls are identified by locating zero-crossings of the real field.
Simulations: A suite of simulations was run with box size $B = 500 \, h^{-1}\text{Mpc}$ and resolution $N = 1280^3$ . Parameters were chosen to target the SIPT regime ( $0.5 \lesssim z_{SSB} \lesssim 0.8$ ) with strong screening ( $z_* \leq 0.1$ ). Both complex ( $\phi \in \mathbb{C}$ ) and real ( $\phi \in \mathbb{R}$ ) field variants were compared across different coupling strengths ( $\beta$ ) and Compton wavelengths ( $L_C$ ).

Key Contributions

First Cosmological Simulations of Cosmic Strings: This work presents the first cosmological simulations of non-minimally coupled cosmic strings in the late-time Universe, providing a side-by-side comparison with the domain-wall scenario of the real-valued symmetron.
Development of norns: The release of a fully relativistic code capable of evolving complex scalar fields and tracking topological defects self-consistently within the N-body framework.
Environment-Dependent Probes: The paper demonstrates that traditional summary statistics (like the matter power spectrum) may only show modest deviations from $\Lambda$ CDM, whereas environment-dependent probes (PDFs, marked statistics) reveal significant signatures, particularly in underdense regions.

Results

Defect Dynamics: In the SIPT regime, defects form in underdense voids and subsequently migrate toward and pin to overdense filaments. Complex fields ( $\phi \in \mathbb{C}$ ) form string networks, while real fields ( $\phi \in \mathbb{R}$ ) form domain walls. The defect density is determined by the initial underdensities at the time of transition, not the causal horizon (unlike standard Kibble mechanism transitions). Real-valued fields exhibit better screening properties due to the larger volume coverage of walls and steeper field profiles near overdensities.
Matter Clustering: The matter power spectrum shows modest enhancements relative to $\Lambda$ CDM (typically $\sim 4-15\%$ at $k \sim 0.3-0.5 \, h \text{Mpc}^{-1}$ ), with sub-percent changes at $z \gtrsim 0.2$ . The complex field models generally show slightly larger enhancements than their real-valued counterparts, likely due to less effective screening by strings compared to walls.
Density Distribution: The most dramatic effects occur in underdense regions. The phase transition drives outflows from voids, significantly suppressing structure growth in voids while enhancing it in surrounding overdense regions. This leads to a "hollowing out" of voids compared to $\Lambda$ CDM.
Probability Density Function (PDF): The matter density PDF shows a large enhancement in the low-density tail ( $\delta_R \lesssim -0.65$ ), reaching factors of $\sim 10$ for certain models at $z=0$ . This signal is distinct from $\Lambda$ CDM and is a primary target for constraining these models.
Halo Statistics:
- Mass Function: The halo mass function (HMF) shows a peak enhancement in abundance for intermediate masses, shifting to larger masses over time. However, for strong screening cases (e.g., Model 5C) and low masses in complex models, halo formation is suppressed, likely due to strong acceleration of particles out of underdense regions.
- Velocities: The fifth force induces a large velocity tail in cold dark matter particles. Halo velocity dispersions are enhanced by $\sim 12-30\%$ compared to $\Lambda$ CDM at $z=0$ , potentially causing observable redshift-space distortions.
Marked Statistics: Applying marked power spectra (weighting halos by local density) boosts the signal at specific scales ( $k \sim \pi/2R$ ). While marked statistics improve sensitivity for some models (e.g., Model 3R at $z=0.3$ ), the gains are often offset by increased shot noise and jackknife errors in current catalog sizes.

Significance and Claims
The paper claims that SIPT-type dark sector phase transitions can couple modifications to $\Lambda$ CDM cosmology specifically to nonlinear structure formation. The primary significance lies in the generation of dense, persistent defect networks that evade standard Kibble scaling and produce strong, environment-dependent signatures.

The authors assert that while these models may leave only modest imprints on traditional large-scale observables (like the global matter power spectrum), they produce dramatic modifications in underdense regions, halo properties, and velocity statistics at late times. Consequently, the inclusion of environment-dependent probes—specifically the low-density tail of the matter density PDF and marked halo statistics—is essential for constraining these models. The paper suggests that current and upcoming galaxy surveys (e.g., DESI, SDSS) covering low redshifts ( $z < 0.7$ ) are likely capable of detecting these deviations, particularly the distinct signatures in the void density distribution. The work positions SIPT models as a test case for developing probes sensitive to environment-dependent physics in the dark sector.