Freezing-in the Axiverse

The presence of multiple light axions in the infrared is a generic feature of many ultraviolet (UV) scenarios. In many cases the number of axions ${\cal N}$ is ${\cal O}(10-100)$ or more. Even in the scenario where these axions interact very weakly with the Standard Model (SM), the presence of ${\cal N}$ light axions poses a challenge to the stringent constraint on the number of relativistic degrees of freedom $N_{\rm eff}$. In order to remain agnostic about the UV, we adopt an effective field theory (EFT) approach, and parametrize the interactions of ${\cal N}$ axions with the SM to quantify the contribution to $N_{\rm eff}$. We consider operators up to dimension six, uncovering one previously-unconsidered charge radius operator, and pay particular attention to the flavor structure of the axion-SM fermion couplings and consider EFTs based on anarchy, textures, and minimal flavor violation. For various choices of such EFTs, we identify the discovery space for current and future cosmic microwave background surveys, including the Simons Observatory and CMB-HD. We show this discovery space depends sensitively on the flavor structure and exhibits a rich interplay with terrestrial and astrophysical probes.

💡 Research Summary

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The paper “Freezing‑in the Axiverse” addresses a pressing cosmological tension that arises when a large number (𝒩 ≈ 10–100 or more) of ultra‑light axion‑like particles (ALPs) are present in the low‑energy spectrum, as is generic in many ultraviolet completions such as string compactifications. Even if each axion couples only very weakly to the Standard Model (SM), the combined contribution of many relativistic species to the effective number of neutrino species, ΔN_eff, can easily exceed the stringent bounds from the Cosmic Microwave Background (CMB) and Big‑Bang Nucleosynthesis (BBN).

To treat this problem in a model‑independent way, the authors adopt an effective field theory (EFT) framework that parametrises all possible interactions between 𝒩 axions ϕ_i and the SM up to dimension six. They first review the well‑known dimension‑5 operators (∂_μ ϕ · O_SM^μ) that couple a single axion linearly to gauge bosons, the Higgs doublet, and SM fermion currents. In doing so they identify a previously overlooked “charge‑radius” operator at dimension six, of the schematic form (∂ϕ_i ∂ϕ_j) H† H, which mediates axion production through Higgs‑portal interactions.

A key conceptual point is that dimension‑5 operators are linear in the axion fields, so each independent operator defines a distinct linear combination a_O that couples to a given SM operator O. Consequently, the number of axion degrees of freedom that actually interact with the SM is not 𝒩 but N_ind, the number of independent dimension‑5 operators after accounting for field redefinitions and the redundancies associated with U(1)_Y, U(1)_B, and lepton‑flavour rotations. In contrast, every dimension‑6 operator is quadratic in the axion fields, meaning that all 𝒩 axions generically participate in the interaction. This leads to two distinct scaling behaviours for the contribution to ΔN_eff:

• For dimension‑5 couplings, ΔN_eff ∝ N_ind (the number of independent operators).
• For dimension‑6 couplings, ΔN_eff ∝ 𝒩 (the total number of axions).

The authors then explore three representative flavour structures for the axion‑fermion couplings:

1. Anarchy – all entries of the coupling matrices are O(1).
2. Froggatt‑Nielsen (FN) textures – hierarchical suppressions arise from a broken U(1) flavour symmetry, strongly damping couplings to the first generation.
3. Minimal Flavour Violation (MFV) – the coupling matrices are aligned with the SM Yukawas, so the heaviest fermions dominate.

These flavour hypotheses dramatically affect the freeze‑in production rates because the thermal bath composition changes with temperature. For example, in the FN case the coupling to electrons and up/down quarks is suppressed, reducing production at temperatures below the electroweak scale, while MFV enhances production through top‑quark and Higgs interactions at higher temperatures.

The cosmological analysis assumes a standard radiation‑dominated universe with an arbitrary reheating temperature T_RH after inflation and a common axion decay constant f_a that controls the overall strength of the dimension‑5 interactions. The authors compute the Boltzmann collision terms for each operator class, obtaining analytic expressions for the axion yield Y_a in the limit T ≫ v (the electroweak scale). Dimension‑5 rates scale as Γ_5 ∼ c^2 T^3/f_a^2, whereas dimension‑6 rates scale as Γ_6 ∼ c^2 T^5/Λ^4, where Λ is the suppression scale of the dimension‑6 operator (often taken to be ≈ 4π f_a). Integrating the Boltzmann equation from T_RH down to the decoupling temperature yields the relic axion energy density ρ_a, which is then translated into ΔN_eff = (8/7)(11/4)^{4/3} ρ_a/ρ_γ.

Using the latest Planck, ACT, and SPT data (ΔN_eff ≲ 0.30 at 95 % CL), the authors derive constraints on the (𝒩, f_a, T_RH) parameter space for each flavour scenario. The main conclusions are:

• If only dimension‑5 operators are present, the bound essentially limits N_ind ≲ 10 for typical f_a ≈ 10^9–10^12 GeV, unless T_RH is significantly below f_a (by two or more orders of magnitude).
• When dimension‑6 operators dominate, the bound becomes far more stringent: even 𝒩 ≈ 10 can be excluded unless the suppression scale Λ is very high or the reheating temperature is low.
• Flavour structures that suppress couplings to light fermions (FN textures) relax the constraints, allowing larger 𝒩 or higher T_RH, while MFV tends to tighten them because of strong top/Higgs couplings.

The paper then projects the sensitivity of upcoming CMB experiments: Simons Observatory, CMB‑S4, and CMB‑HD. These experiments aim for σ(ΔN_eff) ≈ 0.02–0.03, which would probe ΔN_eff down to the level predicted by many of the benchmark models, especially those with MFV or anarchic couplings and moderate 𝒩. The authors illustrate discovery regions in the (𝒩, f_a) plane for several fixed T_RH values, showing that future surveys could either discover a relic axion background or push the viable axiverse parameter space into a narrow corner where either the number of light axions is small or their couplings are highly suppressed.

Finally, the authors discuss complementarity with astrophysical bounds (stellar cooling, supernova 1987A) and terrestrial searches (haloscopes, nuclear magnetic resonance experiments). While those probes are typically sensitive to individual axion couplings, the cosmological ΔN_eff measurement is uniquely sensitive to the collective effect of many weakly coupled states. The interplay between these different avenues can, for instance, rule out a region where CMB data allow a large 𝒩 but stellar cooling excludes the required coupling strength, or vice‑versa.

In summary, “Freezing‑in the Axiverse” provides the first systematic EFT treatment of a multi‑axion sector, clarifies how operator dimension and flavour structure control the cosmological relic abundance, and maps out the discovery potential of next‑generation CMB experiments. The work highlights that a careful top‑down computation of axion‑SM couplings is essential for making robust predictions about ΔN_eff, and it establishes a concrete framework for confronting string‑motivated axiverse scenarios with forthcoming cosmological data.