Hidden Sector Relic Abundances from an Effective Planck Mass

Dark matter is the invisible scaffolding of the universe, a substance that holds galaxies together through gravity but refuses to interact with light. While we know it exists, we do not know what it is made of. One compelling idea is that dark matter lives in a "hidden sector," a parallel world of particles and forces that are completely separate from the atoms and light we see every day. These hidden worlds are not just science fiction; they appear naturally in many advanced theories of physics, including those trying to unify gravity with quantum mechanics. A critical question for cosmologists is how these hidden worlds were born. In the earliest moments of the universe, a period of rapid expansion known as inflation ended with a burst of energy that reheated the cosmos. If this energy spilled over into a hidden sector, it would create a population of dark matter particles. The amount of energy that spills over is measured by a branching ratio, a number that tells us what fraction of the total energy went into the hidden world versus our visible one. Intuitively, one might assume that if only a tiny fraction of energy went into a hidden sector, the dark matter produced there would be negligible and harmless.

A team of researchers at the University of Michigan, along with collaborators in India, has challenged this intuition. They developed a new way to calculate how much dark matter survives from these hidden sectors, focusing on how the energy density of the universe evolves as it expands. Their work reveals that the relationship between the initial energy spill and the final amount of dark matter is far more complex than previously thought. They found that for certain types of dark matter, specifically those that were once in thermal equilibrium with their environment, the final amount of dark matter depends very weakly on how much energy was initially injected. Even if the hidden sector received a minuscule fraction of the universe's energy, it could still produce enough dark matter to overwhelm the cosmos, a scenario known as overclosure. This finding forces a reevaluation of which hidden sector models are viable, suggesting that the universe is far more sensitive to these hidden populations than simple energy accounting would suggest.

To understand their discovery, one must first look at how the universe cools. After the initial burst of energy, the universe is filled with a hot soup of particles. As it expands, it cools down, and particles that were once zipping around freely begin to slow down and interact less. In the visible sector, we have a standard model of how particles like protons and electrons behave. The researchers realized that for a hidden sector, the math describing this cooling process looks exactly the same as the visible sector, but with a crucial twist: the "speed limit" of the universe's expansion, governed by gravity, appears different to the hidden particles. They introduced a concept called an effective Planck mass, which acts as a modified gravitational constant for the hidden sector. This adjustment accounts for the fact that the hidden sector is much colder and less energetic than our own. By using this effective mass in their equations, they could track how the number of dark matter particles changes over time without needing to solve the entire problem from scratch every time.

The team applied this framework to two main scenarios: "freeze-out" and "freeze-in." In a freeze-out scenario, dark matter particles are initially abundant and interact frequently with each other. As the universe cools, they stop interacting and their numbers "freeze" at a specific level. The researchers found that for these models, the final amount of dark matter scales with the fourth root of the initial energy fraction. This means that even if the hidden sector received only a tiny fraction of the energy, the resulting dark matter density remains surprisingly high. They tested this with computer simulations involving particles with masses ranging from one gigaelectronvolt to ten teraelectronvolts, confirming that the energy density of the surviving particles grows as the initial energy fraction increases, but only very slowly. Consequently, a hidden sector with a branching ratio as small as one part in ten thousand could still produce enough dark matter to fill the universe to the point of collapse. This result holds true even for complex models with multiple types of interacting particles, where one type might decay into another.

In contrast, the "freeze-in" scenario tells a different story. Here, dark matter particles are never abundant; they are slowly created from the thermal bath of other particles and then stop being produced as the universe cools. For these models, the researchers found that the final amount of dark matter scales with the branching ratio to the power of five-fourths. This is a much steeper relationship. It means that if the initial energy injection is small, the resulting dark matter is vanishingly small. In these cases, a tiny branching ratio effectively protects the universe from being overfilled with dark matter. The team confirmed this with numerical integrations, showing that for freeze-in particles, the energy density drops off rapidly as the branching ratio decreases. This distinction is vital: it means that not all hidden sectors are dangerous. Some are naturally suppressed, while others, specifically those that undergo freeze-out, pose a significant risk of overclosure even with minimal energy input.

The implications of this work extend to the most ambitious theories of physics, such as string theory, which often predict the existence of many hidden sectors. If a theory contains dozens or hundreds of these hidden worlds, the cumulative effect could be enormous. The researchers calculated that if there are many sectors producing freeze-out dark matter, the total amount of dark matter in the universe would grow significantly, scaling with the number of sectors to the power of three-fourths. This suggests that a universe with many hidden sectors could easily be overfilled with dark matter, placing strict limits on how many such sectors can exist. Furthermore, the study highlights that the initial conditions of the universe, specifically how the inflaton field decays, are critical. Model builders can no longer assume that a small branching ratio is safe; they must calculate the specific dynamics of each hidden sector to ensure the universe does not end up with too much dark matter.

The study also touches on the production of magnetic monopoles, heavy particles that can form during phase transitions in the early universe. Previous work had shown that these monopoles also scale with the fourth root of the branching ratio, similar to the freeze-out particles. The researchers confirmed that their new framework unifies these findings, showing that the effective Planck mass approach applies broadly to various mechanisms of dark matter production. This includes scenarios where unstable particles decay into stable dark matter, or where different types of dark matter exchange energy with one another. In every case where the particles were once in thermal equilibrium, the weak dependence on the branching ratio remained a dominant feature.

Ultimately, this research provides a new set of tools for understanding the dark side of the universe. It shifts the focus from simply counting how much energy went into a hidden sector to understanding how that energy evolves over time. The discovery that a small energy input can lead to a large dark matter abundance is a sobering reminder of the sensitivity of cosmic history. For physicists constructing theories of the universe, the message is clear: the safety of a model cannot be judged by the size of the energy injection alone. The dynamics of how particles interact and freeze out determine the fate of the cosmos, and even the smallest hidden sectors can leave the largest footprints. This work does not solve the mystery of dark matter, but it provides a rigorous method to test which theories of hidden worlds are compatible with the universe we observe today.