⚛️ phenomenology

A Multi-Axion Ladder Across Cosmic History: From Inflation, BBN, and Early Dark Energy to Late-Time Accelerated Expansion

This paper proposes a homogeneous multi-axion cosmological model where four scalar fields with specific potentials sequentially drive transient dark energy episodes during Big Bang nucleosynthesis and the early universe, as well as the current late-time accelerated expansion, offering a unified framework for recurrent dark energy across cosmic history.

Original authors: Amartya Sengupta, Dejan Stojkovic, Adam G. Riess

Published 2026-09-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Amartya Sengupta, Dejan Stojkovic, Adam G. Riess

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

The universe is not a static stage; it is a dynamic story written in the expansion of space itself. For decades, astronomers have mapped this story using a standard model that treats the cosmos as a simple mixture of ordinary matter, invisible dark matter, and a mysterious force called dark energy that pushes galaxies apart. This model works well for most of history, but it faces a stubborn puzzle: measurements of how fast the universe is expanding today do not match the predictions made by looking back at the infant universe. This discrepancy, known as the Hubble tension, suggests that something might be missing from our cosmic timeline. Perhaps the force driving the universe apart is not a constant, unchanging background, but a dynamic player that has appeared and disappeared at different moments in history, leaving subtle fingerprints on the evolution of stars, gas, and light.

In a new study, researchers propose a specific mechanism to fill these gaps, suggesting that the universe has been visited by a "ladder" of invisible fields that activate at different times. These fields are a type of particle known as axions, which are light and ghostly, capable of existing in a state of high potential energy before slowly rolling down to a lower state. The researchers constructed a model where four such fields exist simultaneously. Instead of acting all at once, they are tuned to wake up one after another as the universe expands and slows down. The first field stirs during the first few minutes of the universe, the next two wake up just before the first stars form, and the final one is the force we feel today, driving the current acceleration of the cosmos.

The team simulated the entire history of the universe, from a fraction of a second after the Big Bang to the present day, tracking how these four fields interact with the expansion of space. They found that the first field creates a brief, tiny burst of extra energy during Big Bang nucleosynthesis, the era when the first atomic nuclei were forged. This burst speeds up the expansion just enough to alter the balance of light elements like deuterium and helium, but it fades away quickly. The next two fields are more substantial. They rise to prominence around the time when the universe transitions from being dominated by radiation to being dominated by matter. Together, these two fields can account for nearly ten percent of the total energy in the universe at that specific moment, effectively speeding up the expansion rate before the first light of the cosmic microwave background is released. This temporary speed-up helps resolve the tension between early and late measurements of the universe's expansion.

Crucially, the model explains why these early bursts do not ruin the universe today. The fields are designed with a specific shape in their energy landscape that causes them to dilute much faster than normal radiation once they begin to oscillate. After their brief moments of dominance, they lose their energy so rapidly that they become almost invisible, leaving behind only a tiny, harmless residue. The fourth field, however, behaves differently. It is a "thawing" field that has been frozen in place for billions of years by the friction of the expanding universe. Only recently, as the expansion has slowed, has it begun to roll, providing the dark energy we observe today. This single framework connects the earliest moments of the cosmos with its current accelerated expansion, suggesting that dark energy is not a single constant but a recurring feature of cosmic evolution.

The researchers tested their model against known constraints, such as the abundance of light elements and the detailed patterns of the cosmic microwave background. They found that their four-field setup is consistent with current observations, provided the initial conditions are set correctly. The study highlights that the timing of these events is sensitive to how far the fields were initially displaced from their resting points. A small change in this starting position can significantly alter the height and timing of the energy peaks. While the model remains a theoretical construction, it offers a concrete way to test whether the universe has indeed hosted these transient episodes. If future observations of the cosmic microwave background or the distribution of galaxies can detect the specific signatures of these overlapping energy bursts, it would confirm that the expansion history of the universe is far more complex and layered than previously thought.

The work also opens the door to searching for other hidden episodes in cosmic history. If the universe can host a ladder of fields, there may be other rungs we have not yet seen, active at times between the formation of the first stars and the current era. The researchers suggest that by looking for localized increases in the expansion rate at different epochs, astronomers could map out the full spectrum of these axion fields. This approach treats the expansion history of the universe as a diagnostic tool, where every deviation from the standard timeline could reveal a new particle or a new chapter in the story of the cosmos. The study does not claim to have solved the Hubble tension definitively, but it provides a robust, testable framework that links the physics of the very small to the fate of the very large, suggesting that the dark energy shaping our future may have been whispering its presence since the very beginning.

Technical Summary: A Multi-Axion Ladder Across Cosmic History

Problem and Motivation
Current cosmological observations present a tension between the expansion rate inferred from Cosmic Microwave Background (CMB) data under the standard $\Lambda$ CDM model and local distance-ladder measurements (the Hubble tension). Additionally, Big Bang Nucleosynthesis (BBN) and late-time structure growth provide constraints on the expansion history at widely separated epochs. While Early Dark Energy (EDE) models have been proposed to address the Hubble tension by introducing a transient scalar field before recombination, the question remains whether a single particle-physics mechanism can naturally explain scalar-field contributions across multiple cosmic eras, from BBN to the present day. This paper investigates whether a spectrum of axion-like fields, characterized by a hierarchy of potential curvature scales, can generate transient energy density contributions at specific epochs (BBN, pre-recombination, and late-time) within a single homogeneous cosmology.

Methodology
The authors construct a homogeneous multi-axion cosmology involving four canonical pseudoscalar fields ( $\phi_i$ ) evolving in a Friedmann-Lemaître-Robertson-Walker background. The fields are labeled by their dominant epoch of activity: BBN, two Early Dark Energy fields (EDE1, EDE2), and a late-time Dark Energy (DE) field.

Potential Structure:
- The three transient fields (BBN, EDE1, EDE2) utilize a third-power cosine potential: $V_i(\phi_i) = \Lambda_i^4 [1 - \cos(\phi_i/f_i)]^3$ . Near their minima, these potentials are sextic ( $\propto \phi^6$ ), leading to an equation of state $\langle w \rangle = 1/2$ and a dilution rate $\rho \propto a^{-9/2}$ during rapid, small-amplitude oscillations. This allows the fields to dilute faster than radiation, preventing them from dominating the universe at late times.
- The late-time field utilizes an ordinary cosine potential ( $n=1$ ), behaving as a thawing quintessence field with a vacuum mass $m_{DE} \sim H_0$ .
Dynamics and Coupling:
- The fields are coupled only gravitationally through the Friedmann equation. As the Hubble rate $H$ decreases, fields with progressively smaller curvature scales ( $m_{eff}$ ) overcome Hubble friction and begin to roll.
- The rolling of an earlier field modifies the Hubble friction experienced by later fields, creating a coupled evolution where the expansion history is determined by the sum of all scalar energy densities.
- The authors numerically integrate the coupled Klein-Gordon and Friedmann equations from a high redshift ( $z \sim 10^{11}$ ) to the present day, starting with the fields displaced from their potential maxima (hilltop) with zero initial velocity.
Inflationary Extension:
- The paper briefly explores an "aligned" two-axion sector that could serve as an inflaton. By aligning two periodic potentials, the model generates a super-Planckian effective decay constant ( $f_{eff} \approx 7.43 M_{Pl}$ ) despite sub-Planckian microscopic decay constants, providing a candidate for the inflationary epoch that sets the initial conditions for the post-inflationary axion ladder.

Key Results
The authors present a benchmark model with specific parameters (decay constants, potential scales, and initial misalignment angles) that yields distinct transient contributions:

BBN Era: The first field peaks at $z \approx 1.04 \times 10^9$ with an energy fraction $\Omega_{peak} \approx 0.99\%$ . This transient increases the expansion rate by approximately 0.5% during nucleosynthesis. Due to the sextic dilution, its residual density today is negligible ( $\sim 10^{-11}$ ).
Early Dark Energy (EDE): Two fields peak around matter-radiation equality. EDE1 peaks at $z \approx 2.43 \times 10^3$ ( $\Omega \approx 7.8\%$ ) and EDE2 at $z \approx 7.93 \times 10^3$ ( $\Omega \approx 8.6\%$ ). Their combined contribution reaches a maximum of $\approx 9.7\%$ near $z \approx 2.54 \times 10^3$ . This overlap broadens the duration of the expansion enhancement compared to single-field EDE models.
Late-Time Dark Energy: The fourth field is normalized to provide the current dark energy density ( $\Omega_{DE,0} \approx 0.685$ ). It acts as a thawing field with a present-day equation of state $w_{DE} \approx -0.925$ .
Residuals: The combined residual density of the three transient fields at the present epoch is extremely small ( $\approx 6.0 \times 10^{-7}$ ), ensuring consistency with late-time observations.
Sensitivity: A scan of the initial hilltop offsets reveals that the peak energy fractions are highly sensitive to the initial displacement (varying by factors of 8–10), while the peak redshifts are less sensitive (varying by factors of 1.3–1.5).

Significance and Claims
The paper claims to demonstrate a "mass-ladder" picture where a hierarchy of axion scales naturally generates scalar-field contributions at distinct cosmic epochs without fine-tuning the timing of each field's onset, provided the initial displacements are set.

Unified Framework: It offers a unified mechanism where the same particle-physics principles (shift symmetry breaking and non-perturbative potentials) explain transient energy densities at BBN, pre-recombination, and late times.
EDE Improvement: The authors suggest that distributing the EDE contribution between two staggered fields (rather than a single field) may improve the fit to CMB data by altering the expansion history and gravitational potentials over a broader redshift interval, though they note that a full likelihood analysis including perturbations is required to confirm this.
Observational Constraints: The paper outlines how future observations of light-element abundances (BBN), CMB anisotropies, and large-scale structure can test this specific multi-field scenario. It also proposes a method to search for "additional rungs" in the axion ladder by constraining transient expansion features at other epochs using a Gaussian profile in the expansion rate.
Dilution Mechanism: The use of sextic minima is highlighted as a crucial feature that allows early transient fields to dilute faster than radiation, solving the problem of relic overproduction that plagues standard axion models with quadratic minima.

The authors conclude that while the benchmark model is consistent with current constraints, its viability depends on future quantitative comparisons with BBN reaction networks and joint CMB/BAO/SN analyses, particularly regarding whether the two-field EDE configuration offers a statistically significant improvement over single-field models.

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