String axiverse candidates and cosmological production channels

Consider a declared compactification EFT with at least two retained periodic axion coordinates. For the Type IIB realization described in the cited source, let $Y$ be a specified compact six-dimensional internal space, let $[{\mathrm{\Sigma }}_A]\in H_4(Y;\mathbb{Z})$ be chosen integral four-cycle classes, and let $C_4$ be the Ramond–Ramond four-form gauge potential. The angular coordinate ${\theta }_A=(q_A{\int }_{{\mathrm{\Sigma }}_A}{C}_4)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2\pi$ is the normalized period on a chosen cycle representative, with the normalization $q_A$ and large-gauge identifications supplied as model data. Other compactifications must supply their own internal space, gauge-potential degree, cycle classes and period normalizations. In every case specify the joint positive-definite kinetic matrix, charge lattice and separate source-generated potential for each mode.

For the cold-matter branch, require a nonzero homogeneous mode that is a decoupled canonical eigen-coordinate with a positive time-independent mass for each member of a controlled family satisfying every hypothesis of A homogeneous harmonic axion has a controlled averaged cold-matter limit, including its member-dependent-mass, small-$H_n\mathrm{/}{m}_{a,n}$, total-variation, and admissible-window limits; after a specified reheating history, the mode then has the theorem's leading averaged cold-matter limit. For any other populated mode, use a specified spatially flat FLRW chart with metric $d{s}^2=-d{t}^2+a(t{)}^2{\delta }_{ij}d{x}^{i}d{x}^j$, $a>0$, and unit comoving observer $n={\mathrm{\partial }}_t$. A spatial Fourier mode $e^{i\mathbf{k}\cdot \mathbf{x}}$ has constant comoving wave vector $\mathbf{k}$ and physical momentum magnitude $k\mathrm{/}a(t)$ measured by $n$, where $k=\mathrm{\mid }\mathbf{k}\mathrm{\mid }$. A mode with physical momentum much greater than its mass is relativistic; a mode with momentum much less than its mass is nonrelativistic. For a homogeneous hidden perfect-fluid component with density ${\rho }_D$ measured by $n$, impose $p_D={\rho }_D\mathrm{/}3$ and separate conservation ${\dot{\rho }}_D+3H({\rho }_D+p_D)=0$, $H=\dot{a}\mathrm{/}a$. Then $d(a^4{\rho }_D)\mathrm{/}dt=a^4({\dot{\rho }}_D+4H{\rho }_D)=0$, so its energy density redshifts as $a^{-4}$; call such a component dark radiation. Here a modulus means a retained real scalar coordinate of the compactification, and modulus decay means decay of its excitation through separately specified couplings; this may instead produce a relativistic or nonrelativistic axion population, depending on the decay kinematics and history.

When inflationary perturbations are included, use a declared two-field trajectory ${\varphi }^I(t)$ with positive field-space metric $G_{IJ}$ and Hubble rate $H_{\mathrm{H}\mathrm{u}\mathrm{b}}:=\dot{a}\mathrm{/}a$. Put $\dot{\sigma }=(G_{IJ}{\dot{\varphi }}^I{\dot{\varphi }}^J{)}^{1\mathrm{/}2}>0$, ${\widehat{\sigma }}^I={\dot{\varphi }}^I\mathrm{/}\dot{\sigma }$, choose a $G$-unit vector ${\widehat{s}}^I$ orthogonal to ${\widehat{\sigma }}^I$, and let $Q^I=\delta {\varphi }^I+({\dot{\varphi }}^I\mathrm{/}{H}_{\mathrm{H}\mathrm{u}\mathrm{b}})\psi$ be the source's gauge-invariant Mukhanov–Sasaki field perturbation, where $\psi$ is the scalar spatial-metric perturbation on equal-time slices. In the source convention define $\mathcal{R}=H_{\mathrm{H}\mathrm{u}\mathrm{b}}{\widehat{\sigma }}_I{Q}^I\mathrm{/}\dot{\sigma }$ and $S=H_{\mathrm{H}\mathrm{u}\mathrm{b}}{\widehat{s}}_I{Q}^I\mathrm{/}\dot{\sigma }$; $S$ is the single independent linear isocurvature mode in this two-field limit. Which branch occurs depends on cycle data, masses, initial angles and perturbations, decay channels and thermal history. Here an axiverse means a model with multiple axion-like modes and distinct field parameters. This is a source-backed family of possibilities, not a universal abundance or a claim that every compactification produces an observed axiverse.

A homogeneous harmonic axion has a controlled averaged cold-matter limit, Reheating temperature in a sudden-decay modulus model