Moduli domination and reheating constraints in a symbolic model

Use units $\mathrm{\hslash }=c=k_B=1$. Let $J\subseteq \mathbb{R}$ be a nonempty open time interval. In the inherited FLRW convention, $t$ and $a_n$ have length dimension, spatial coordinates and $\gamma$ are dimensionless, $[{\phi }_n]=[m_n]=\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}$, $[{\rho }_{\varphi ,n}]=[{\rho }_{r,n}]=[p_{r,n}]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^4$, and $[{\mathrm{\Gamma }}_n]=[M_P]=[T]=\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}$; $g_0$ is dimensionless. A modulus mode here is the label for one retained real scalar coordinate ${\phi }_n:J\to \mathbb{R}$, locally canonical near a quadratic minimum; the label is compactification-motivated, but no particular compactification is assumed. For each $n$, take target $\mathbb{R}$, target metric $G_{11}=1$, scalar coupling $K=1$, mass $m_n>0$ constant on $I$, and potential $V_n(\phi )=m_n^2{\phi }^2\mathrm{/}2$ on the field range. The scalar time profile is ${\phi }_n\in C^{\mathrm{\infty }}(J,\mathbb{R})$ and its homogeneous spacetime field is ${\mathrm{\Phi }}_n(t,x)={\phi }_n(t)$.

Take flat, expanding FLRW model data with $c=1$ on a common nonempty open time interval $J$: $\mathrm{\Sigma }$ is a connected smooth three-manifold with flat Riemannian metric $\gamma$, each $a_n\in C^{\mathrm{\infty }}(J,(0,\mathrm{\infty }))$, $M_n=J\times \mathrm{\Sigma }$, $g_n=-d{t}^2+a_n(t{)}^2\gamma$, and the comoving future-directed unit field is $n_n={\mathrm{\partial }}_t$. Write $H_n={\dot{a}}_n\mathrm{/}{a}_n$. The scalar comoving density and pressure are ${\rho }_{\varphi ,n}=({\dot{\phi }}_n^2+m_n^2{\phi }_n^2)\mathrm{/}2$ and $p_{\varphi ,n}=({\dot{\phi }}_n^2-m_n^2{\phi }_n^2)\mathrm{/}2$.

A pre-decay radiation component means smooth homogeneous profiles ${\tilde{\rho }}_{r,n},{\tilde{p}}_{r,n}\in C^{\mathrm{\infty }}(J,\mathbb{R})$, pulled back to $M_n$ by ${\pi }_J:M_n\to J$, ${\pi }_J(t,x)=t$, with ${\rho }_{r,n}>0$ and $p_{r,n}={\rho }_{r,n}\mathrm{/}3$. It is a perfect fluid carried by $n_n$, with stress tensor $T_{r,n}^{ab}=({\rho }_{r,n}+p_{r,n})n_n^a{n}_n^b+p_{r,n}{g}_n^{ab}$. The scalar and this fluid are the only pre-decay sources, and ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t},n}={\rho }_{\varphi ,n}+{\rho }_{r,n}$ is their total comoving density.

Let $I\subset J$ be a nondegenerate compact interval. Use the centered uniform average of width ${\mathrm{\Delta }}_n>0$ from the averaged-dust result: ${\bar{Q}}_n(t)={\mathrm{\Delta }}_n^{-1}{\int }_{t-{\mathrm{\Delta }}_n\mathrm{/}2}^{t+{\mathrm{\Delta }}_n\mathrm{/}2}{Q}_n(s)ds$ when $W_n(t)=[t-{\mathrm{\Delta }}_n\mathrm{/}2,t+{\mathrm{\Delta }}_n\mathrm{/}2]\subseteq I$. Assume the full smoothness, homogeneous-equation, pointwise adiabatic, total-variation, and window-limit hypotheses of $\mathtt{\text{pthm-r5-quadratic-modulus-averaged-dust}}$. In particular, for a fixed admissible reference center with positive averaged density, that result gives $a_n(t{)}^3{\bar{\rho }}_{\varphi ,n}(t)\mathrm{/}(a_n(t_0{)}^3{\bar{\rho }}_{\varphi ,n}(t_0))\to 1$ uniformly over admissible centers. Choose fixed times $t_i<t_d$ in $I$, independent of $n$, with $W_n(t_i),W_n(t_d)\subseteq I$ and ${\bar{\rho }}_{\varphi ,n}(t_i),{\bar{\rho }}_{\varphi ,n}(t_d)>0$ for all sufficiently large $n$.

Set $a_{i,n}=a_n(t_i)$, $a_{d,n}=a_n(t_d)$, and $r_{i,n}={\bar{\rho }}_{\varphi ,n}(t_i)\mathrm{/}{\rho }_{r,n}(t_i)$, with $0<r_{i,n}<1$. Between $t_i$ and $t_d$, assume exact separate radiation conservation, ${\rho }_{r,n}(t)a_n(t{)}^4={\rho }_{r,n}(t_i)a_{i,n}^4$, with no energy exchange between the components. Define $q_n=r_{i,n}{a}_{d,n}\mathrm{/}{a}_{i,n}$ and suppose $q_n\to q\in [0,\mathrm{\infty })$. Define the averaged-density diagnostic $r_n(t_d^{-})={\bar{\rho }}_{\varphi ,n}(t_d)\mathrm{/}{\rho }_{r,n}(t_d)$, using the prescribed smooth pre-decay family on the full centered window $W_n(t_d)$; this is only a pre-decay branch diagnostic, not post-decay scalar data. The averaged-dust result and radiation conservation give $r_n(t_d^{-})=q_n(1+o(1))$, hence $r_n(t_d^{-})\to q$. Define averaged modulus domination at $t_d^{-}$ by $r_n(t_d^{-})>1$ and averaged subdominance by $r_n(t_d^{-})<1$. Thus $q>1$ implies eventual averaged modulus domination and $q<1$ eventual averaged subdominance; if $q=1$, the limit gives no exact sign conclusion.

At $t_d$, impose the total-source sudden-decay prescription: $H_n(t_d^{-})={\mathrm{\Gamma }}_n>0$ and all pre-decay energy is instantaneously matched to an ideal thermal radiation bath with density ${\rho }_{\mathrm{r}\mathrm{a}\mathrm{d}}(T)={\pi }^2{g}_{\ast }(T)T^4\mathrm{/}30$. Let $M_P>0$ be the reduced Planck mass, $M_P^2=(8\pi G_N{)}^{-1}$, and take $g_{\ast }(T)=g_0>0$ for every $T>0$. A reheating temperature $T_{\mathrm{R}\mathrm{H},n}>0$ satisfies ${\pi }^2{g}_{\ast }(T_{\mathrm{R}\mathrm{H},n})T_{\mathrm{R}\mathrm{H},n}^4\mathrm{/}30={\rho }_{\mathrm{t}\mathrm{o}\mathrm{t},n}(t_d^{-})$. For this globally constant-$g_0$ model the matching equation has a unique positive solution. For a declared model requirement $T_{\mathrm{R}\mathrm{H},n}\ge T_{\min }>0$, the equivalent condition is ${\mathrm{\Gamma }}_n\ge ({\pi }^2{g}_0\mathrm{/}90{)}^{1\mathrm{/}2}{T}_{\min }^{}\mathrm{/}{M}_P$. Here $T_{\min }$ is a model requirement, not a reported BBN threshold. The example gives no universal modulus mass or lifetime, decay-width law, branching ratio, dark-radiation abundance, or exclusion of string compactifications; its density comparison is averaged and leading-order only.

A quadratic modulus has leading averaged dust behavior

Reheating temperature in a sudden-decay modulus model

The calculation uses no decay-width formula such as $\mathrm{\Gamma }\propto m^3\mathrm{/}{M}_P^2$; such a coefficient is channel- and model-dependent. A numerical BBN, baryogenesis, dark-radiation or gravitino constraint would require its own sourced temperature threshold, decay products and thermal-history assumptions. This example only shows how specified requirements propagate through the two conditional A-page results.