An EFT Map of Axion Dark Radiation from Reheating

The Big Picture: The Universe's "Reheating" Party

Imagine the early Universe right after the Big Bang. It went through a period of rapid expansion called inflation, driven by a heavy, invisible field called the inflaton. When inflation stopped, the inflaton field started to vibrate like a plucked guitar string.

This vibration had to stop eventually. The energy stored in those vibrations had to be transferred to other particles to create the hot soup of matter and light we see today. This process is called reheating. It's like the inflaton is a giant battery that needs to be drained to power the rest of the universe.

Usually, scientists think this battery drains into visible particles (like protons and electrons). But what if some of that energy leaked out into a "dark sector"—particles we can't see, like axions (a type of light, ghostly particle)? If axions are produced, they act as Dark Radiation, adding a tiny bit of extra heat to the cosmic background. We measure this extra heat as $\Delta N_{eff}$ .

The Problem: Looking at Only One Leak

Previously, scientists looked at how axions are made in two separate ways, like looking at a bucket with two holes but only checking one at a time:

The "Decay" Hole: The inflaton particle breaks apart directly into two axions (like a battery cell popping open and spilling its contents).
The "Annihilation" Hole: Two inflaton particles crash into each other and turn into axions (like two batteries colliding and sparking a fire).

The problem is that these two holes behave differently depending on how fast the "draining" happens (the Reheating Temperature, or $T_{rh}$ ).

Decay is strongest when the draining is slow.
Annihilation is strongest when the draining is fast (because the particles are packed tighter together).

If you only look at the "Decay" hole, you might think the universe must have drained slowly. If you only look at "Annihilation," you might think it drained fast. You miss the whole picture.

The Solution: The "Kinetic Map"

This paper introduces a new way to look at the problem using a tool called an Effective Field Theory (EFT). Think of this as a master blueprint that connects the two holes into one single system.

The authors imagine the axion's ability to move (its "kinetic term") is controlled by the inflaton field. They write a mathematical formula where the inflaton acts like a dial that changes how easy it is for axions to move.

The Linear Dial ( $c_1$ ): Controls the direct decay (one inflaton $\to$ two axions).
The Quadratic Dial ( $c_2$ ): Controls the collisions (two inflatons $\to$ two axions).

Crucially, the paper shows that you can't just pick one dial. The "collision" process is actually a mix of the direct collision plus a subtle interference from the single-particle decay. It's like a choir where the sound of the soloist changes how the duet sounds. You have to measure the whole choir to get the right note.

The "U-Turn" Discovery

The most exciting finding is how the total amount of Dark Radiation ( $\Delta N_{eff}$ ) changes as the Reheating Temperature ( $T_{rh}$ ) changes.

At Low Temperatures: The "Decay" hole dominates. As the temperature goes up, the amount of Dark Radiation goes down (because the visible particles soak up more energy, leaving less for the axions).
At High Temperatures: The "Annihilation" hole dominates. As the temperature goes up, the amount of Dark Radiation goes up (because the inflatons are so crowded that they crash into each other more often).

The Result: If you plot this on a graph, the line doesn't just go up or down; it makes a U-shape (or a checkmark). It goes down, hits a minimum, and then goes back up.

This is a game-changer. In the past, scientists thought Dark Radiation could only tell them an "upper limit" (e.g., "The temperature couldn't have been higher than X"). But because of this U-shape, the new map says:

If the temperature is too low, the decay is too strong (ruled out).
If the temperature is too high, the collisions are too strong (ruled out).
Therefore, the temperature must be in a specific "Goldilocks zone" in the middle.

The "EFT Map"

The authors created a 2D map (like a treasure map) with two axes:

$c_1$ (How strong the decay dial is).
$c_2$ (How strong the collision dial is).

On this map, there is a "forbidden zone" (shaded in orange) where the amount of Dark Radiation would be too high for our current universe.

If you are in the bottom-left, you are safe.
If you move too far right (too much collision), you get caught.
If you move too far up (too much decay), you get caught.

Because the two processes fight each other, the "safe zone" is a curved strip, not just a simple line. This allows scientists to use measurements of Dark Radiation to pin down exactly what the microscopic rules of the early universe were, and even estimate the temperature of the reheating era with a specific lower and upper bound.

Summary in a Metaphor

Imagine you are trying to guess how fast a car was driving by looking at the skid marks on the road.

Old Method: You only looked at the front tires. If the marks were long, you thought, "It must have been going slow." If they were short, "It must have been fast." But you were wrong because you ignored the back tires.
This Paper's Method: You realize the front and back tires interact. The front tires leave marks that get shorter as speed increases, but the back tires leave marks that get longer as speed increases.
The Conclusion: When you combine them, you see a specific pattern. If the marks are too short or too long, the car couldn't have been driving at that speed. The only speed that fits the pattern is a specific range in the middle.

This paper builds that combined map for the early universe, showing us that the "speed" of the Big Bang's reheating phase was likely trapped in a specific, narrow window, determined by the delicate balance between axion decay and axion collisions.

Technical Summary: An EFT Map of Axion Dark Radiation from Reheating

Problem Statement
The microscopic details of the reheating epoch—the transition from the inflaton-dominated era to the radiation-dominated hot Big Bang—remain largely unknown. Standard cosmological probes often fail to access this era because visible particles produced during reheating rapidly thermalize, erasing information about the specific operators that drained the inflaton's energy. Weakly coupled relics, which do not thermalize, offer a potential window into this history. While axions and axion-like particles (ALPs) produced during reheating have been studied as candidates for cold dark matter or dark radiation, previous analyses often treated production mechanisms (such as direct inflaton decay or inflaton annihilation) in isolation. This separation can obscure the full physical picture, particularly when the axion coupling arises from a shift-symmetric kinetic term where decay and annihilation channels are intrinsically linked.

Methodology
The paper develops a shift-symmetric Effective Field Theory (EFT) framework to systematically organize the production of light axions during reheating. The core assumption is that the inflaton $\phi$ communicates with a light axion $a$ through an inflaton-dependent kinetic metric $Z(\phi)$ , such that the Lagrangian contains the term $\frac{1}{2}Z(\phi)(\partial a)^2$ .

EFT Expansion: The kinetic function $Z(\phi)$ is expanded near the minimum of the inflaton potential (assumed to be quadratic, $V(\phi) = \frac{1}{2}m_\phi^2\phi^2$ ). The expansion is parameterized as $Z(\phi) = 1 + c_1 \frac{\phi}{\Lambda} + \frac{c_2}{2} \frac{\phi^2}{\Lambda^2} + \dots$ , where $\Lambda$ is the suppression scale and $c_1, c_2$ are dimensionless Wilson coefficients.
Production Channels:
- Direct Decay: The linear term ( $c_1$ ) mediates the decay $\phi \to aa$ . This is treated as an invisible branching fraction of the inflaton.
- Inflaton Annihilation: The quadratic term ( $c_2$ ) and two insertions of the linear term contribute to the process $\phi\phi \to aa$ . Crucially, the paper demonstrates that the annihilation amplitude is not determined by $c_2$ alone but by the coherent combination $c_{\text{ann}} \equiv c_2 - c_1^2$ . This arises because the contact diagram from $\phi^2(\partial a)^2$ interferes with the $t$ - and $u$ -channel exchange diagrams generated by the linear operator.
Cosmological Evolution: The authors solve the Boltzmann equations for the axion energy density during the reheating epoch, accounting for the oscillating inflaton background and the expansion of the universe. They derive the contribution to the effective number of relativistic degrees of freedom, $\Delta N_{\text{eff}}$ , by converting the axion-to-radiation ratio at the end of reheating into the standard $\Delta N_{\text{eff}}$ definition.

Key Contributions and Results
The primary result is the derivation of a two-dimensional "EFT map" relating the Wilson coefficients $(c_1, c_2)$ to the observable $\Delta N_{\text{eff}}$ . The paper highlights several critical findings:

Opposite Temperature Dependence: The two production channels exhibit opposite scaling behaviors with respect to the reheating temperature ( $T_{\text{rh}}$ $T_{rh}$ ).
- The direct decay contribution scales as $\Delta N_{\text{eff}}^{\text{dec}} \propto T_{\text{rh}}^{-2}$ . This is because a higher $T_{\text{rh}}$ implies a larger visible decay width, which dilutes the invisible branching fraction into axions.
- The inflaton annihilation contribution scales approximately as $\Delta N_{\text{eff}}^{\text{ann}} \propto T_{\text{rh}}^{4/3}$ (in the regime where the controlled EFT onset occurs well before the end of reheating). A higher $T_{\text{rh}}$ implies reheating completes earlier, when the inflaton energy density is still higher, enhancing the annihilation rate which depends on $\rho_\phi^2$ .
The Crossing Regime: Because of these opposing scalings, the total $\Delta N_{\text{eff}}$ as a function of $T_{\text{rh}}$ exhibits a minimum. This creates a "window" where the total dark radiation is minimized. Treatments that consider only decay or only annihilation miss this crossing, leading to incorrect one-sided bounds (either only low- $T_{\text{rh}}$ or only high- $T_{\text{rh}}$ constraints).
Two-Dimensional Constraints: Current and projected sensitivities to $\Delta N_{\text{eff}}$ (e.g., from Planck 2018, BBN+CMB, SO, and CMB-HD) translate into constraints on the $(c_1, c_2)$ plane. The exclusion contours are curved, reflecting the distinct dependencies: $c_1$ controls the decay direction, while the combination $c_2 - c_1^2$ controls the annihilation direction. The line $c_2 = c_1^2$ represents a "blind line" where the annihilation amplitude vanishes due to the cancellation between contact and exchange diagrams.
Reheating Temperature Bounds: For fixed microscopic parameters, the competition between the two channels can imply both lower and upper bounds on the reheating temperature $T_{\text{rh}}$ . For example, with specific benchmark parameters, current limits could constrain $T_{\text{rh}}$ to a range such as $6.0 \times 10^{11} \text{ GeV} \lesssim T_{\text{rh}} \lesssim 3.0 \times 10^{12} \text{ GeV}$ .

Significance
The paper argues that organizing axion production via a shift-symmetric kinetic EFT reveals that $\Delta N_{\text{eff}}$ is not merely a constraint on an invisible inflaton branching fraction. Instead, it serves as a probe of the microscopic reheating operator and the cosmological history of the inflaton background simultaneously. By consistently treating both direct decay and inflaton annihilation, the framework transforms $\Delta N_{\text{eff}}$ measurements from simple upper limits into a tool capable of constraining the reheating temperature window and mapping the specific structure of the axion-inflaton kinetic coupling. This approach avoids the pitfalls of model-dependent or partial treatments, providing a more robust interpretation of dark radiation data in the context of early universe cosmology.