The sign of energy exchange between vacuum and matter

Some analyses of cosmological data favour a dark-energy equation of state above $-1$ today and below $-1$ at high redshift (pattern i), and fits with a free electron mass return values at recombination 0.6 to 1 per cent above the laboratory one (pattern ii). With dark matter treated as conserved and an anchor before the epoch in question, pattern (i) needs energy to flow from matter into the vacuum (Lemma 1), and pattern (ii) needs a particle mass that falls with time. A vacuum energy without kinetic term that tracks the global minimum of its effective potential gives the opposite sign: energy flows from the vacuum to dust and the particle mass does not fall (Theorem 1(a)). A scalar that changes only the electron mass and tracks its minimum in a Standard Model plasma makes the electron lighter at recombination than in a laboratory environment less dense than the plasma (Theorem 2 and Corollary 2).

Two further classes give weaker statements of the same sign. A passive linear relaxation towards the minimum, started at equilibrium, transfers a non-negative amount of energy to matter when counted from the start (Theorem 1(b)), and a unital map acting on a passive state of the matter (a map that preserves the identity, such as dephasing, on a state whose occupations do not increase with energy) does not lower the energy of the matter in that step (Theorem 1(c)). A field that does not track, is light enough to mediate a force across the Earth and has an electron coupling that does not grow towards the past can raise the electron mass at recombination by at most about $1.1\%\sqrt{\langle\Omega_{\rm kin}\rangle}$ (a bound proved under the hypotheses of Theorem 3), where $\langle\Omega_{\rm kin}\rangle\le1$ is its average kinetic share of the energy density. All statements concern the homogeneous background; the tracking fluid of Theorem 1(a) has $c_s^2\le0$ (Proposition 1).

Pavón and Wang argued from the Le Chatelier–Braun principle that energy flows from dark energy to dark matter [PW]. Fardon, Nelson and Weiner considered neutrinos whose mass is set by a field at the minimum of its effective potential in the neutrino background [FNW]; the gradient instability that accompanies such a field was found for neutrinos by Afshordi, Zaldarriaga and Kohri [AZK] and for coupled dark matter by Bean, Flanagan and Trodden [BFT 1], [BFT 2]. Das, Corasaniti and Khoury showed that a reconstruction with conserved matter can turn an interacting vacuum into an apparent phantom component [DCK]. Baryakhtar, Simon and Weiner computed the in-medium force on a hyperlight scalar that modulates the electron mass [BSW 1], [BSW 2]. Inequality (3) below is the simplest case of monotone comparative statics [Topkis], [MS], in which the minimiser of a family of objectives moves monotonically when the parameter enters through decreasing differences. Theorem 2 applies the comparison that gives Theorem 1 to the electron mass in a plasma, for an arbitrary potential and an arbitrary number of couplings.

The page on linear constraints on the cosmological constant treats the analogous sign question for a vacuum energy fixed by a global constraint linear in the stress tensor. For a single weight tied to the past light cone in a flat universe of fluids with pressure between zero and a third of the density, with a constant ratio of its energy component to its trace outside the window from minus one quarter to zero, a separable measure and a log-concave kernel, the magnitude of the vacuum value there does not increase along a worldline, which follows from Chebyshev’s association inequality. On the page on negative neutrino mass in BAO data, one route to a lower inferred neutrino mass sum is a conversion of matter into dark energy, which is the direction that Theorem 1(a) excludes for dust and a vacuum of the class considered here.

1. Vacuum, matter and what a reconstruction sees

We work in a spatially flat Friedmann–Robertson–Walker universe with scale factor $a(t)$ , Hubble rate $H=\dot a/a>0$ , and signature $(-,+,+,+)$ . A component with energy density $\rho$ and pressure $p$ that exchanges no energy with others obeys $\dot\rho+3H(\rho+p)=0$ . The vacuum component has $T^V_{ab}=-\rho_vg_{ab}$ , so $p_v=-\rho_v$ , and it is non-propagating: it has no kinetic term, and its energy density is a function of auxiliary variables that carry no energy of their own. The matter component is dust, with conserved particle number density $n$ ( $na^3$ constant) and mass per particle $\mu$ , so that $\rho_M=n\mu$ . Total energy is conserved, which defines the exchange rate $Q$ :

\dot\rho_v=-Q,\qquad \dot\rho_M+3H\rho_M=Q. \tag{1}

A positive $Q$ means that energy flows from the vacuum to matter. If $\rho_v>0$ , the vacuum density has the equation of state $w_{\rm vac}=-1-\dot\rho_v/(3H\rho_v)$ , the one it would need if it were conserved separately; the symbol $w_{\rm eff}$ below refers to the reconstruction defined next. For dust, $Q=n\dot\mu$ , so $Q\ge0$ is the statement that the particle mass does not decrease. Where the transferred energy is deposited as heat (Theorem 1(b) and (c), Proposition 2), $Q$ is still defined by $Q=-\dot\rho_v$ , and the second relation in (1) holds if the heat is counted as dust-like energy that dilutes as $a^{-3}$ ; the statements below say which reading they use.

A cosmological analysis typically assumes that cold dark matter is conserved, fixes its density at some epoch $a_*$ , and attributes the rest of the energy budget to dark energy. We call the result the reconstruction anchored at $a_*$ : $\rho_c^{\rm rec}(a)=\rho_M(a_*)(a_*/a)^3$ and $\rho_{\rm DE,eff}=\rho_v+\rho_M-\rho_c^{\rm rec}$ . Early anchors correspond to normalising the dark-matter density to the cosmic microwave background (CMB), late anchors to a low-redshift measurement.

Lemma 1 (reconstruction). Under (1), with $\Delta(t)=a^{-3}\int_{t_*}^{t}Q\,a^3\,dt'$ ,

\dot\rho_{\rm DE,eff}=-3H\Delta,\qquad 1+w_{\rm eff}=\frac{\Delta}{\rho_{\rm DE,eff}}, \tag{2}

where $w_{\rm eff}$ is defined by $\dot\rho_{\rm DE,eff}=-3H(1+w_{\rm eff})\rho_{\rm DE,eff}$ and $\rho_{\rm DE,eff}>0$ . If $Q\ge0$ at all times, then for an anchor in the past of $t$ ( $t_*<t$ ) $\Delta\ge0$ and $w_{\rm eff}\ge-1$ , and for an anchor in the future of $t$ ( $t_*>t$ ) $\Delta\le0$ and $w_{\rm eff}\le-1$ . The comoving dark-matter densities satisfy $(a^3\rho_M)(t_1)\le(a^3\rho_M)(t_2)$ for $t_1<t_2$ .

Proof. From (1), $a^3\rho_M$ has derivative $a^3Q$ , so $\rho_M-\rho_c^{\rm rec}=\Delta$ . Then $\rho_{\rm DE,eff}=\rho_v+\Delta$ and $\dot\rho_{\rm DE,eff}=-Q+(-3H\Delta+Q)=-3H\Delta$ . The sign statements follow from the sign of the integral over $[t_*,t]$ , which is negative if the limits are reversed. The last statement integrates $a^3Q\ge0$ . $\square$

The sign statements use only the sign of $\int_{t_*}^tQ\,a^3dt'$ , not the sign of $Q$ at each time. A value $w_{\rm eff}<-1$ at time $t$ in a fit anchored at $t_*<t$ is, by (2), the same as $\Delta(t)<0$ : the comoving matter density at $t$ is lower than at the anchor, so $Q<0$ on part of $[t_*,t]$ . With a late anchor, a vacuum that feeds matter looks phantom in the past, the effect described in [DCK], so a comparison with observations needs the anchor to be stated. The dark-energy preference of the DESI second data release combined with CMB and supernova data is a preference for $w_0>-1$ and $w_a<0$ in $w(a)=w_0+w_a(1-a)$ , at $2.8$ – $4.2\sigma$ over a cosmological constant depending on the supernova sample [DESI].

2. The thermodynamic sign theorem

We start with the simplest case. Take a single auxiliary variable $\xi$ , a vacuum energy $g(\xi)=\tfrac12K\xi^2$ with $K>0$ , and dust of particle number density $n$ whose particle mass is $\mu(\xi)=\mu_0+b\,\xi$ . The energy density available to $\xi$ is $\Phi=\tfrac12K\xi^2+n(\mu_0+b\xi)$ . Its minimum is at $\xi=-nb/K$ and the mass there is $\mu=\mu_0-nb^2/K$ . Matter lowers its energy by making its particles lighter, so the denser the dust, the lighter the particles; as the universe expands $n$ falls and $\mu$ rises towards $\mu_0$ . The vacuum energy $g=\tfrac12 n^2b^2/K$ falls with $n$ as well, and the difference is paid to matter: with $\rho_M=n\mu$ and $na^3$ constant, $Q=n\dot\mu=3Hn^2b^2/K>0$ . The sign does not depend on the sign of $b$ .

The general argument replaces the quadratic and the linear coupling by arbitrary functions. Its first step compares two states. Let $g:X\to\mathbb R$ and $e_1,e_2:X\to\mathbb R$ be arbitrary, and let $\xi_j$ be a global minimiser of $\Phi_j=g+e_j$ . Adding $\Phi_1(\xi_1)\le\Phi_1(\xi_2)$ and $\Phi_2(\xi_2)\le\Phi_2(\xi_1)$ cancels $g$ and leaves

D(\xi_2)\le D(\xi_1),\qquad D=e_2-e_1. \tag{3}

The later minimiser lowers the difference of the matter energies, whatever $g$ is. No convexity of $g$ or of $\Phi_j$ enters, and the two minimisers may lie far apart in $X$ . Theorem 1 turns (3) into a bound on the mass in the situations in which a minimiser is reached.

Theorem 1 (thermodynamic sign). Let $\xi$ take values in a set $X$ , let the vacuum energy density be $\rho_v=g(\xi)$ with $g:X\to\mathbb R$ arbitrary, and let the matter depend on $\xi$ only through a mass parameter $\mu(\xi)>0$ , an arbitrary function of $\xi$ . The energy density of the matter in state $j$ , at the entropy and particle numbers that state fixes, is $e_j(\mu)$ , differentiable, with $\partial_\mu e_j>0$ ; for dust of conserved particle number density $n_j$ it is $e_j(\mu)=n_j\mu$ , and section 4 gives it for a plasma. We call $\Phi_j(\xi)=g(\xi)+e_j(\mu(\xi))$ the effective potential in state $j$ .

(a) Global tracking. Let state 1 be earlier and state 2 later, let $\xi_1,\xi_2$ be global minimisers of $\Phi_1,\Phi_2$ , and let $J$ be an interval containing $\mu(\xi_1)$ and $\mu(\xi_2)$ on which $\partial_\mu e_2<\partial_\mu e_1$ (for dust, $n_2<n_1$ ). Then $\mu(\xi_2)\ge\mu(\xi_1)$ . For dust, a field that follows the global minimiser of $\Phi_t$ through an expansion has $a^3n\mu$ non-decreasing, also across first-order jumps, so $Q\ge0$ (as a measure).

(b) Passive linear relaxation. Let $X=\mathbb R^{d_\xi}$ , $g(\xi)=\tfrac12\xi^{\mathsf T}K\xi$ with $K$ symmetric positive definite and constant, $\mu(\xi)=\mu_0+b^{\mathsf T}\xi$ , dust with $n(t)$ non-increasing, and $\dot\xi=-\Gamma(t)\nabla\Phi_t(\xi)$ with $\Gamma(t)$ any matrix function whose symmetric part is positive semi-definite (it may be non-reciprocal and time dependent). If $\xi$ starts at the minimiser of $\Phi_{t_0}$ , then $\mu(t)\ge\mu(t_0)$ for all $t\ge t_0$ , the dissipation $\Pi=\nabla\Phi^{\mathsf T}\Gamma\nabla\Phi$ is non-negative, and $\int_{t_0}^ta^3Q\,dt'\ge0$ with $Q=-\dot\rho_v=n\dot\mu+\Pi$ .

(c) Unital exchange on a passive state, one step. Let matter be a finite-dimensional quantum system with energy operator $\hat E$ and state $\varrho$ , and let the exchange over a time step act as a unital completely positive trace-preserving map $\mathcal C$ on $\varrho$ ( $\mathcal C(\mathbb 1)=\mathbb 1$ ; random unitary mixtures and dephasing are examples) such that the exchange conserves the total energy of matter plus vacuum. If $\varrho$ is passive, that is diagonal in an energy eigenbasis with occupations non-increasing in energy (every thermal state at positive temperature is of this kind), then $\mathrm{Tr}[\hat E\,\mathcal C(\varrho)]\ge\mathrm{Tr}[\hat E\varrho]$ , and the vacuum loses the corresponding energy in that step.

In (a) the conclusion is (3) read through the hypothesis on the derivative. In (b) the distance of the state from a fixed equilibrium never increases (the dynamics is a contraction), so after a step in the source the mass cannot fall below its initial value, and a decreasing source is a positive superposition of steps. In (c) a unital map can only mix, and mixing raises the energy of a passive state. A unital map does not in general leave the state passive. With levels of energy $0,1,2$ and occupations $(0.5,0.3,0.2)$ , the map that exchanges the first and third levels raises the mean energy from $0.7$ to $1.3$ , and its repetition from the new state lowers it back to $0.7$ . A sequence of unital steps is therefore guaranteed a gain at each step if the state is passive at the start of each, for instance because the matter is held thermal between steps, and without that condition the example shows that a later step can reverse the gain.

Proof of (a). Apply (3) with $e_j$ evaluated at $\mu(\xi)$ : $G(\mu_2)\le G(\mu_1)$ for $G=e_2-e_1$ and $\mu_j=\mu(\xi_j)$ . By hypothesis $G'<0$ on $J$ , so $G$ is strictly decreasing there, and $G(\mu_2)\le G(\mu_1)$ forces $\mu_2\ge\mu_1$ . For dust, $G(\mu)=(n_2-n_1)\mu$ . For the exchange rate, total energy is $\rho_v+\rho_M=g+n\mu=\Phi_t(\xi(t))$ , the minimum of $\Phi_t$ , and it is continuous across a jump between two degenerate minima. With $na^3$ constant, $a^3\rho_M=a^3n\mu$ is non-decreasing by the first part, so $Q\ge0$ in the sense of measures. $\square$

Proof of (b). Put $y=K^{1/2}\xi$ , $c=K^{-1/2}b$ and $\mathcal A(t)=K^{1/2}\Gamma(t)K^{1/2}$ . The potential is $\Phi_t=\tfrac12y^{\mathsf T}y+n(t)\,c^{\mathsf T}y+\text{const}$ , so $\dot y=-\mathcal A(t)\,(y+n(t)c)$ and the instantaneous minimiser is $y_{\rm eq}(t)=-n(t)c$ . The mass is $\mu=\mu_0+c^{\mathsf T}y$ . Write $\theta(t)=n(t_0)-n(t)\ge0$ , non-decreasing, so that $y_{\rm eq}(t)=y_0+\theta(t)c$ with $y_0=y_{\rm eq}(t_0)=y(t_0)$ . For $u=y-y_0$ , $\dot u=-\mathcal A(u-\theta c)$ . Let $Y(t,t')$ be the propagator of $\dot z=-\mathcal Az$ from time $t'$ to time $t$ . Since $\tfrac{d}{dt}\lVert z\rVert^2=-2z^{\mathsf T}(\mathrm{sym}\,\mathcal A)z\le0$ and $\mathrm{sym}\,\mathcal A=K^{1/2}(\mathrm{sym}\,\Gamma)K^{1/2}\ge0$ , $\lVert Y\rVert\le1$ . For $\theta$ equal to a unit step at time $t'$ , the solution is $u(t)=(1-Y(t,t'))c$ , because $\partial_{t'} Y(t,t')=Y(t,t')\mathcal A(t')$ . A non-decreasing $\theta$ is a positive superposition of steps, $\theta(t)=\int_{t_0}^t d\theta(t')$ , so

c^{\mathsf T}u(t)=\int d\theta(t')\,\big[c^{\mathsf T}c-c^{\mathsf T}Y(t,t')c\big]\ge0

by the Cauchy–Schwarz inequality and $\lVert Y\rVert\le1$ . Hence $\mu(t)-\mu(t_0)=c^{\mathsf T}u\ge0$ . The rate at which the effective potential energy is dissipated is $-\nabla\Phi\cdot\dot\xi=\nabla\Phi^{\mathsf T}\Gamma\nabla\Phi=\Pi\ge0$ (only the symmetric part of $\Gamma$ contributes). Since $\Phi=g+n\mu$ , $-\dot\rho_v-n\dot\mu=-\nabla\Phi\cdot\dot\xi=\Pi$ , so $Q=-\dot\rho_v=n\dot\mu+\Pi$ . With $na^3$ constant, $\int_{t_0}^ta^3n\dot\mu\,dt'=a^3n\,(\mu(t)-\mu(t_0))\ge0$ , and $\int_{t_0}^ta^3\Pi\,dt'\ge0$ . $\square$

Proof of (c). Let $\varrho=\sum_jr_j\vert j\rangle\langle j\vert$ in the energy eigenbasis, with energies $E_1\le E_2\le\dots\le E_d$ and $r_1\ge r_2\ge\dots\ge r_d$ . Define $W_{ij}=\langle i\vert \mathcal C(\vert j\rangle\langle j\vert )\vert i\rangle\ge0$ . Trace preservation gives $\sum_iW_{ij}=1$ and unitality gives $\sum_jW_{ij}=\langle i\vert \mathcal C(\mathbb 1)\vert i\rangle=1$ , so $W$ is doubly stochastic. The energy after the step is $\sum_iE_iq_i$ with $q=Wr$ (the diagonal elements of $\mathcal C(\varrho)$ in the energy basis; off-diagonal elements do not contribute to the trace). Let $R_k=\sum_{i\le k}r_i$ and $R'_k=\sum_{i\le k}q_i$ . Then $R'_k=\sum_jr_jv_j$ with $v_j=\sum_{i\le k}W_{ij}\in[0,1]$ and $\sum_jv_j=\sum_{i\le k}\sum_jW_{ij}=k$ . Because $r$ is sorted in decreasing order, a linear form $\sum_jr_jv_j$ under the constraints $0\le v_j\le1$ , $\sum_jv_j=k$ is largest at $v=(1,\dots,1,0,\dots,0)$ , where it equals $R_k$ . Hence $R'_k\le R_k$ for every $k$ . Summation by parts gives $\sum_iE_iq_i=E_d-\sum_{k<d}(E_{k+1}-E_k)R'_k\ge E_d-\sum_{k<d}(E_{k+1}-E_k)R_k=\sum_iE_ir_i$ . This is passivity in the sense of [PW78], [Lenard]: no unital mixing lowers the energy of a passive state. $\square$

Global minimality is used only in (a), and the monotonicity of the source only through $\partial_\mu e_2<\partial_\mu e_1$ . In (b) the constancy of $K$ is used in the substitution $y=K^{1/2}\xi$ that turns the dynamics into a contraction, and starting on the equilibrium branch is used in $u(t_0)=0$ . The condition that the vacuum is non-propagating means that its energy is $g(\xi)$ only; with kinetic energy the pressure is no longer $-\rho_v$ and the reconstruction of Lemma 1 changes character.

Corollary 1. (a) Under Theorem 1(a) for dust, $Q\ge0$ (as a measure) at all times. The comoving dark-matter density $a^3\rho_M$ at recombination is then not larger than at any later time, in particular $\omega_c(\mathrm{rec})/\omega_c(0)\le1$ , where $\omega_c(t)=a(t)^3\rho_M(t)/a_0^3$ is the comoving dark-matter density expressed as a density today, and a reconstruction anchored at recombination returns $w_{\rm eff}\ge-1$ at all later times whenever $\rho_{\rm DE,eff}>0$ .

(b) Under Theorem 1(b), $a^3n\mu$ at any $t\ge t_0$ is not smaller than at the start $t_0$ of the relaxation, and the transfer weighted by $a^3$ and accumulated since $t_0$ is non-negative. A reconstruction anchored at $t_0$ in which the transferred energy is counted as dust-like then has $\Delta\ge0$ .

(c) Under Theorem 1(c), the matter does not lose energy in a step that starts from a passive state, and the vacuum does not gain it. If the exchange is a semigroup of unital maps and the state is passive at time $t$ , applying (c) to each short step shows that the right derivative of the matter energy at $t$ is non-negative, $\dot\rho_v\le0$ . Nothing follows for $\omega_c$ , since the added energy is not an $a^{-3}$ -diluting rest mass.

The first statement is Lemma 1 together with $Q\ge0$ . A fit that returns $w_{\rm eff}<-1$ at some time after an early anchor, or $\omega_c(\mathrm{rec})/\omega_c(0)>1$ , needs $Q<0$ over some interval, which Theorem 1(a) forbids. For (b) and (c) the theorem restricts the sign of the energy transfer in the senses stated, cumulative since $t_0$ in (b) and over one step from a passive state in (c), and it translates into the dark-matter density only to the extent that the transferred energy dilutes as $a^{-3}$ .

Remark 1. In (b) the transfer is non-negative only cumulatively. Take $d_\xi=2$ , $K=\mathbb 1$ , $c=(1,0)$ , $\Gamma=\begin{pmatrix}\epsilon&\varpi\\-\varpi&\epsilon\end{pmatrix}$ with $\epsilon,\varpi>0$ (symmetric part $\epsilon\mathbb 1$ ), and let $n$ be prescribed as a function of time that drops by one unit, from $n_0+1$ to $n_0$ , at $t=0$ , the limit of a drop that is fast compared with the relaxation time. A step in $n$ is not compatible with $na^3$ constant, which the cumulative statement of Theorem 1(b) uses, so the example concerns the instantaneous sign of $Q=n\dot\mu+\Pi$ only. The solution of $\dot u=-\Gamma(u-c)$ with $u(0)=0$ has $\mu(t)-\mu(0)=1-e^{-\epsilon t}\cos\varpi t\ge0$ and $\dot\mu=e^{-\epsilon t}(\epsilon\cos\varpi t+\varpi\sin\varpi t)$ , which is negative during part of each period. The dissipation is $\Pi=\epsilon e^{-2\epsilon t}$ , so after the drop

Q=n_0\,e^{-\epsilon t}\big(\epsilon\cos\varpi t+\varpi\sin\varpi t\big)+\epsilon e^{-2\epsilon t}.

At the minima of the oscillating factor this equals $e^{-\epsilon t}\big(\epsilon e^{-\epsilon t}-n_0\sqrt{\epsilon^2+\varpi^2}\big)$ , which is negative once $n_0\sqrt{\epsilon^2+\varpi^2}>\epsilon e^{-\epsilon t}$ , and in particular at the first minimum if $n_0\sqrt{\epsilon^2+\varpi^2}>\epsilon$ . The mass overshoots its new equilibrium value and returns, and energy flows from matter to the vacuum at those times, although $\mu(t)\ge\mu(0)$ throughout.

3. Where the theorem applies

Mass-coupled static fluids

Take $\rho=g(\xi)+n\mu(\xi)$ for dust whose mass per particle is $\mu(\xi)$ , with $\xi$ at a strict minimum at fixed $n$ : $\nabla g+n\nabla\mu=0$ with Hessian $\mathcal K=\nabla^2g+n\nabla^2\mu>0$ . This is the local version of Theorem 1(a), and it gives the rate of change of the mass.

Proposition 1 (Hessian identity). Along the branch of strict minima, $d\xi/dn=-\mathcal K^{-1}\nabla\mu$ , and

\frac{d\mu}{dn}=-\nabla\mu^{\mathsf T}\mathcal K^{-1}\nabla\mu\le0,\qquad \frac{d\rho_v}{dn}=n\,\nabla\mu^{\mathsf T}\mathcal K^{-1}\nabla\mu\ge0, \tag{4}

where $\mu$ and $\rho_v=g$ are evaluated on the branch. With $\chi=n\,\nabla\mu^{\mathsf T}\mathcal K^{-1}\nabla\mu/\mu\ge0$ , the mass changes as $d\ln \mu/d\ln a=3\chi\ge0$ , and the adiabatic sound speed squared of the tracking fluid is $c_s^2=-\chi\le0$ .

Proof. Differentiate $\nabla g+n\nabla\mu=0$ along the branch: $\mathcal K\,d\xi/dn+\nabla\mu=0$ . Then $d\mu/dn=\nabla\mu^{\mathsf T}d\xi/dn=-\nabla\mu^{\mathsf T}\mathcal K^{-1}\nabla\mu$ , and $d\rho_v/dn=\nabla g^{\mathsf T}d\xi/dn=-n\nabla\mu^{\mathsf T}d\xi/dn$ gives the second relation. With $na^3$ constant, $dn/d\ln a=-3n$ , so $d\ln\mu/d\ln a=-3n\,(d\mu/dn)/\mu=3\chi$ . The total energy density on the branch is $\rho_{\rm eq}(n)=g+n\mu$ with $\rho_{\rm eq}'=\mu$ (envelope theorem), the pressure is $p=n\rho_{\rm eq}'-\rho_{\rm eq}=-g$ , and $c_s^2=dp/d\rho=n\,(d\mu/dn)/\mu=-\chi$ . $\square$

The sign $c_s^2<0$ is the adiabatic instability of mass-varying dark sectors found in [AZK], [BFT 1], [BFT 2]. It does not enter the sign theorem, but it means that a fluid obtained from a non-propagating auxiliary variable has a gradient instability. For the global minimiser, the concavity of $\rho_{\rm eq}(n)=\min_\xi[g+n\mu]$ , a minimum of functions affine in $n$ , gives the monotonicity of $\mu$ at once, including across jumps.

Unimodular heating

In unimodular gravity the cosmological term is an integration function. Write the field equations as $G_{ab}+\Lambda g_{ab}=8\pi GT_{ab}$ with $\Lambda(x)$ determined by the Bianchi identity, $\nabla_b\Lambda=8\pi G\nabla^aT_{ab}$ [JPS]. In a homogeneous universe $u^b\nabla^aT_{ab}=-(\dot\rho_M+3H(\rho_M+p_M))=-Q$ , so $\dot\rho_{\rm vac}=-Q$ with $\rho_{\rm vac}=\Lambda/8\pi G$ , which is (1). Consider matter whose reduced dynamics is a Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generator, the most general form of a Markovian completely positive evolution of a density operator, $\dot\varrho=-\tfrac{i}{\hbar}[\hat h,\varrho]+\sum_k(L_k\varrho L_k^\dagger-\tfrac12\{L_k^\dagger L_k,\varrho\})$ with self-adjoint $L_k$ , as in spontaneous-collapse models.

Proposition 2 (heating by self-adjoint Lindblad operators). For self-adjoint $L_k$ the heating rate satisfies $\dot E=-\tfrac12\sum_k\langle[L_k,[L_k,\hat h]]\rangle$ . For $\hat h=\hat p^2/2m+V(\hat x)$ and $L_k=L_k(\hat x)$ a function of position, it equals $\dot E=\tfrac{\hbar^2}{2m}\sum_k\langle(\nabla L_k)^2\rangle\ge0$ in every state.

Proof. In the Heisenberg picture the dissipator acts on an operator $O$ as $\sum_k(L_kOL_k-\tfrac12L_k^2O-\tfrac12OL_k^2)=-\tfrac12\sum_k[L_k,[L_k,O]]$ , and the unitary part does not change the energy. For $O=\hat h$ only the kinetic term fails to commute with $L_k(\hat x)$ . From $[L,\hat p_j]=i\hbar\,\partial_jL$ one finds $[L,\hat p^2]=i\hbar(\nabla L\cdot\hat p+\hat p\cdot\nabla L)$ and $[L,[L,\hat p^2]]=-2\hbar^2(\nabla L)^2$ . Hence $[L,[L,\hat h]]=-(\hbar^2/m)(\nabla L)^2$ and the dissipator gives $(\hbar^2/2m)(\nabla L)^2$ , a positive operator. $\square$

In unimodular gravity (a model assumption) the heating is the energy $Q\ge0$ received by matter, so $\dot\rho_{\rm vac}\le0$ , and $w_{\rm vac}\ge-1$ if $\rho_{\rm vac}>0$ . The map generated by self-adjoint $L_k$ is unital, so in a finite-dimensional truncation Theorem 1(c) gives the same sign over a step that starts from a passive state, while the computation above holds in every state. The quantity $w_{\rm vac}$ of section 1 refers to the vacuum itself. The equation of state that an observer reconstructs is a different quantity, by Lemma 1: if the heated matter is counted as cold dark matter at a late anchor, the reconstruction has $w_{\rm eff}<-1$ at earlier times, as in [DCK]. The proposition therefore does not exclude a reconstruction that looks phantom, only a phantom vacuum energy.

Remark 2 (momentum diffusion on the mass shell). The diffusion of massive particles in energy-momentum from spacetime discreteness is the model class of [DPS]. The momentum of a particle of mass $m$ lies on the mass shell $\Sigma_m\cong\mathbb H^3$ , with energy $E=m\cosh\zeta$ in terms of the rapidity $\zeta$ , and the Lorentz-invariant diffusion is $\partial_\tau f=\kappa\nabla^2_\Sigma f$ with $\tau$ the proper time of the particle. The Laplacian acts on a radial function as $f''+2\coth\zeta\,f'$ , which for $f=\cosh\zeta$ is $3\cosh\zeta$ . Hence $d\langle E\rangle/d\tau=3\kappa\langle E\rangle>0$ , assuming that boundary terms vanish. The most general Lorentz-invariant massless diffusion in [DPS] contains a drift in the energy of the particle in addition to the diffusion, and the sign of the drift is not fixed by Lorentz invariance. Energy diffusion from matter into the discrete structure is the scenario of [PS]; in unimodular gravity it gives $Q<0$ and $\dot\rho_{\rm vac}>0$ , the direction opposite to Proposition 2, so it lies outside the proposition and is not excluded by the theorems. A drift that removes energy from the particle lies outside as well.

What the theorem requires of the vacuum

The sign depends on the vacuum having pressure $-\rho_v$ , and a scalar field with a kinetic term can reverse it. As a derivation within the model, take dust with mass $m(\phi)=m_0e^{\beta\phi}$ (with $\phi$ in Planck units) and a free scalar field with $V=0$ . The matter action gives $\rho_c=n\,m(\phi)$ and the field equation $\ddot\phi+3H\dot\phi=-\beta\rho_c$ . If $\dot\phi=0$ initially, then $(a^3\dot\phi)^\cdot=-\beta a^3\rho_c$ gives $\beta\dot\phi=-(\beta^2/a^3)\int a^3\rho_c\,dt<0$ , so $\dot m/m=\beta\dot\phi<0$ for either sign of $\beta$ : the particle mass falls with time and energy flows from matter into the kinetic energy of the field. The effective potential $nm_0e^{\beta\phi}$ has no minimum, so (a) does not apply, and the field has pressure equal to its energy density, so the vacuum is not of the type in section 1. Models of this class (coupled quintessence [Amendola]) that start away from the tracking branch, or that have no minimiser, are outside Theorem 1.

4. A field that tracks a minimum makes the electron lighter in the past

Fits of CMB and baryon acoustic oscillation (BAO) data in which the electron mass at recombination is free return central values above the laboratory one. Hydrogen recombines when the photon temperature reaches a fixed fraction of the binding energy, which is proportional to $m_e$ ; a heavier electron makes recombination earlier and shrinks the sound horizon. Write $\delta\equiv m_e(z_*)/m_e^{\rm lab}-1$ . The published values are in the table.

data	$\delta$ (%)
Planck + ACT DR6 + CMB lensing + DESI DR1 BAO [ACT DR6]	$0.96\pm0.60$
the same + type Ia supernovae [ACT DR6]	$0.63\pm0.56$
ACT DR6 + Planck ( $\ell<1000$ ) + lensing + DESI DR2 BAO + supernovae [Toda–Seto]	$0.78\pm0.47$
CMB + DESI DR2 BAO [Weiner]	$0.76\pm0.48$

These analyses treat $m_e$ at recombination as a free parameter with no field dynamics. Here a scalar $\phi$ on which $m_e$ depends, with bare potential $V(\phi)$ , supplies the dynamics. The vacuum of section 1 is replaced by a scalar with a kinetic term whose homogeneous value is assumed to follow the minimum of its effective potential (hypothesis (T1) below); this is a model assumption, and it holds when the field mass is large compared with the Hubble rate, so that the kinetic energy and the oscillations about the minimum are negligible.

Example 1. Take $m_e=m_0e^{\beta\phi/M_P}$ , with the reduced Planck mass $M_P=(8\pi G)^{-1/2}$ , and $V=\tfrac12m_\phi^2\phi^2$ with $m_\phi\gg H$ . Non-relativistic electrons of number density $n$ add the energy density $n\,m_e(\phi)$ , so $U(\phi)=\tfrac12m_\phi^2\phi^2+n\,m_0e^{\beta\phi/M_P}$ . At the minimum $m_\phi^2\phi=-(\beta/M_P)\,n\,m_e$ , hence $\phi$ has the sign opposite to $\beta$ and $\ln(m_e/m_0)=\beta\phi/M_P=-\beta^2n\,m_e/(m_\phi^2M_P^2)<0.$ The sign of the coupling drops out because it enters squared, and the denser the gas, the lighter the electron.

The field moves so as to lower the energy of the matter it is immersed in, and matter of given composition has lower energy when its constituents are lighter. The general statement needs the derivative of the energy of a relativistic plasma with respect to a mass.

For one species of mass $M$ with $N_{\rm int}$ internal states in kinetic equilibrium at temperature $T$ , define $\int_k\equiv N_{\rm int}\int d^3k/(2\pi)^3$ , $E=\sqrt{k^2+M^2}$ , and occupation numbers $f$ . Then $\rho=\int_kEf$ and $p=\int_k\frac{k^2}{3E}f$ (summed over particles and antiparticles when both are present), and we call $n_{\rm sc}=\int_k(M/E)f=(\rho-3p)/M$ the scalar density of the species, using $E-k^2/E=M^2/E$ . It is positive wherever the species is present, and $n_{\rm sc}\to n$ for non-relativistic particles. For fermions it is the expectation value of $\bar\psi\psi$ , and (5) below is the Feynman–Hellmann relation: the derivative of the energy with respect to a mass is the expectation value of the operator that the mass multiplies.

Lemma 2 (mass derivative of the thermodynamic potentials). For an ideal gas let $\mathcal F(M;T,\mathcal N)$ be the Helmholtz free energy density at fixed temperature and fixed net number density $\mathcal N$ (particle minus antiparticle number for a gas that contains both), and $\mathcal E(M;s,\mathcal N)$ the energy density at fixed entropy density $s$ and $\mathcal N$ . Then

\left.\frac{\partial\mathcal F}{\partial M}\right\vert_{T,\mathcal N}=\left.\frac{\partial\mathcal E}{\partial M}\right\vert_{s,\mathcal N}=n_{\rm sc}=\frac{\rho-3p}{M}>0. \tag{5}

For a mixture whose masses depend on $m_e$ , $\partial\mathcal F/\partial m_e=\partial\mathcal E/\partial m_e=\sum_in_{{\rm sc},i}\,dM_i/dm_e$ , at fixed composition and at chemical equilibrium.

Proof. The grand potential density is $-p(M;T,\mu_c)$ with $p=\pm N_{\rm int}T\int\frac{d^3k}{(2\pi)^3}\ln(1\pm e^{-(E-\mu_c)/T})$ for the particle species, plus the antiparticle term for $-\mu_c$ . Differentiating under the integral with $\partial E/\partial M=M/E$ gives $\partial p/\partial M\vert _{T,\mu_c}=-n_{\rm sc}$ . An integration by parts in $k$ turns the logarithm into the kinetic pressure $\int_kk^2f/3E$ . The free energy is $\mathcal F=-p+\mu_c\mathcal N$ with $\mu_c$ fixed by $\mathcal N$ ; since $\partial\mathcal F/\partial\mu_c\vert _{M}=-\partial p/\partial\mu_c+\mathcal N=0$ , the derivative at fixed $\mathcal N$ equals that of $-p$ at fixed $\mu_c$ , which is $n_{\rm sc}$ . The energy density satisfies $d\mathcal E=T\,ds+\mu_c\,d\mathcal N+\nu\,dM$ with $\nu=\partial\mathcal E/\partial M\vert_{s,\mathcal N}$ , and $\mathcal F=\mathcal E-Ts$ gives $d\mathcal F=-s\,dT+\mu_c\,d\mathcal N+\nu\,dM$ , so $\nu=\partial\mathcal F/\partial M\vert_{T,\mathcal N}$ . For a mixture the chain rule gives the sum, and at chemical equilibrium the composition extremises the energy at fixed entropy, so by the envelope theorem its variation does not contribute. $\square$

In the cosmological plasma the species that depend on $m_e$ are free electrons and positrons, with $dM/dm_e=1$ , and hydrogen and helium with bound electrons. A bound level of binding energy $B\propto m_e$ gives an atom the mass $M=M_{\rm nucleus}+m_e-B$ and $dM/dm_e=1-B/m_e$ , which is $1-2.7\times10^{-5}$ for ground-state hydrogen ( $B=13.6$ eV). Coulomb corrections to the ideal-gas thermodynamics are neglected. We call $\nu\equiv\partial\mathcal E/\partial m_e=\sum_in_{{\rm sc},i}\,dM_i/dm_e$ the electron scalar density; by Lemma 2 it is positive wherever electrons are present.

Let state $j$ of the plasma be specified by its entropy density, its conserved number densities and its composition (ionisation fractions in equilibrium or frozen out). For dust the energy at fixed particle number, $n\mu$ , is also the free energy. For a plasma the two derivatives with respect to a mass agree by Lemma 2, so the free energy at fixed temperature and the energy at fixed entropy give the same stationarity condition. Cosmological evolution conserves entropy per comoving volume, so the energy density $e_j(m)$ of state $j$ at fixed entropy and numbers, with masses evaluated at electron mass $m$ , is the function the field responds to, and $U_j(\phi)=V(\phi)+e_j(m_e(\phi))$ . Its derivative $\nu_j(m)=e_j'(m)$ is the electron scalar density of state $j$ . A homogeneous field at rest is stationary where $U_j'(\phi)=0$ , since by Lemma 2 the derivative of $e_j$ reproduces the source term $-\sum_in_{{\rm sc},i}\partial M_i/\partial\phi$ of the field equation, and the minima of $U_j$ are the stable positions.

(T1) Tracking: in each state $j$ compared, $\phi$ is at a global minimiser of $U_j$ .

(T2) Standard Model sources only: apart from $V$ , the $\phi$ -dependence of $U_j$ comes from Standard Model species whose masses are non-decreasing functions of $m_e$ . A field that changes only $m_e$ qualifies, and so does a Higgs portal provided every mass it changes increases with the Higgs vacuum value.

(T3) More matter in the past: on an interval of $m$ containing the two values of $m_e$ compared, $\nu_1(m)>\nu_2(m)$ for the earlier state 1 and the later state 2.

Theorem 2 (electron mass sign). Let state 1 be earlier and state 2 later, and let $\phi_1,\phi_2$ satisfy (T1) in the two states. Under (T2) and (T3), $m_e(\phi_1)\le m_e(\phi_2)$ . In particular, for the recombination plasma (state 1) and a laboratory environment (state 2) whose electron scalar density, counting bound electrons, is smaller than that of the recombination plasma over the interval of (T3), $\delta\le0$ .

Proof. This is Theorem 1(a) with $X$ the set of field values, $g=V$ , $\mu(\xi)=m_e(\phi)$ and $e_j$ as a function of $m_e$ . Under (T2) the difference $e_2-e_1$ depends on $\phi$ only through $m_e$ , and its derivative is $\nu_2-\nu_1<0$ on the interval of (T3), by Lemma 2. The conclusion of (a) is $m_e(\phi_2)\ge m_e(\phi_1)$ . For $\delta$ , $1+\delta=m_e(\phi_1)/m_e(\phi_2)\le1$ . $\square$

The proof uses neither the shape of $V$ nor the size of the coupling, and it holds for a field with several components. The input is Lemma 2, which fixes the sign of $\partial\mathcal E/\partial m_e$ , and (T3), the comparison of the electron scalar densities of the two states. A weaker reading of tracking, a continuous branch of local minima, gives the same sign.

Proposition 3 (local tracking). Let $\lambda\in[0,1]$ label a continuous sequence of states from today ( $\lambda=0$ ) back to an earlier epoch, and let $\phi(\lambda)$ be a $C^1$ branch of non-degenerate local minima of $U_\lambda$ for a single-component field. Under (T2), if $\nu(m;\lambda)$ does not decrease with $\lambda$ at fixed $m$ , then $m_e(\phi(\lambda))$ is non-increasing in $\lambda$ .

Proof. Differentiate $U'_\lambda(\phi(\lambda))=0$ along the branch: $U''_\lambda\phi'+\partial_\lambda U'_\lambda=0$ . By Lemma 2 and (T2), $U'_\lambda=V'+\nu(m_e(\phi);\lambda)\,m_e'(\phi)$ , so $\partial_\lambda U'_\lambda=m_e'\partial_\lambda\nu$ and $dm_e/d\lambda=m_e'\phi'=-(m_e')^2\partial_\lambda\nu/U''_\lambda\le0$ . $\square$

The branch can end where $U''_\lambda\to0$ , after which the field jumps; Theorem 2 covers jumps between global minima.

For a field that changes only $m_e$ , (T3) holds between recombination and any later state of the universe at its mean density. After electron–positron annihilation the net electron number is conserved, so the number density scales as $(1+z)^3$ and the scalar density is $n_{\rm sc}=n\langle m_e/E\rangle\approx n(1-3T/2m_e)$ . At recombination $T=0.26$ eV and $T/m_e\approx5\times10^{-7}$ . (T3) needs only that the correction $3T/2m$ be below $1-(1+z)^{-3}$ , which holds for every $m$ well above $T$ , where the expansion is accurate. So (T3) holds analytically on that range, which contains $m_e$ and values of $m_e$ many orders of magnitude away from it. With $\Omega_bh^2=0.02237$ [Planck VI] and helium fraction $Y_p=0.245$ (so $1-Y_p/2$ electrons per baryon), the mean electron density is $n_e(z_*=1090)\approx2.9\times10^2\ {\rm cm}^{-3}$ at last scattering and $2.2\times10^{-7}\,{\rm cm}^{-3}$ today.

Before and during annihilation the pairs add to the scalar density, and a field sourced by the pairs responds to their scalar density $(\rho-3p)/M$ of Lemma 2. For an ideal electron–positron gas with the baryon-to-photon ratio $\eta_b=273.9\,\Omega_bh^2\times10^{-10}=6.1\times10^{-10}$ and the same helium fraction, the scalar density at $m=m_e$ increases monotonically with temperature from $T_0=2.7255$ K to 10 MeV. This establishes (T3) in the annihilation era at $m=m_e$ only, which is the input to Proposition 3 and to the pair case of Corollary 2.

Which environment is today

The theorem compares tracked values in two homogeneous states. The laboratory value is the tracked value for whatever environment the field responds to there. Theorem 2 gives $\delta\le0$ provided that environment has an electron scalar density below $n_e(z_*)\approx290\ {\rm cm}^{-3}$ . The environment to which a tracking field responds is set by its range $1/m_\phi$ . A field whose range is of the order of a parsec or more, so that the Earth and the Sun contribute little to the density averaged over it, responds to the interstellar gas, a few nucleons per cubic centimetre near the Sun, and the condition holds. A field of range below the size of the Earth responds to the density of the Earth or of the laboratory, as screened fields of chameleon [KW] or symmetron [HK] type do, and the condition fails; intermediate ranges respond to the average density over the range. Applied to pairs of environments of similar composition, the theorem orders them by electron scalar density: $m_e$ in any environment less dense than the recombination plasma is at least as large as at recombination.

Corollary 2. Assume (T1) to (T3), with (T3) holding on an interval that contains the tracked values of $m_e$ in the two states, and that the laboratory value of $m_e$ is set by an environment less dense than the recombination plasma. Then $\delta\le0$ , for an arbitrary potential and an arbitrary sign and number of couplings.

After annihilation, (T3) holds for every $m$ well above $T(z_*)\approx0.26$ eV (shown above), so the interval contains any tracked value of $m_e$ in that range. In the annihilation era (T3) is shown only at the single value $m=m_e$ . A field sourced by the scalar density of the electron–positron plasma (the pair case) then falls under the corollary if the two tracked values of $m_e$ lie in a neighbourhood of $m_e$ on which, by continuity of $\nu$ in $m$ , the inequality persists; the size of that neighbourhood is not quantified here. For a Higgs portal under the proviso in (T2), (T3) is an assumption. Potentials of chameleon or symmetron form are covered if the screening is not active at the density of the laboratory. Screened fields whose laboratory value is set by the density of the Earth or of the apparatus do not satisfy the hypothesis on the environment, and are treated in section 6, under tracking of the local density.

The central values in the table lie between $1.1$ and $1.7$ standard deviations above $\delta=0$ ( $0.96/0.60=1.6$ , $0.63/0.56=1.1$ , $0.78/0.47=1.7$ , $0.76/0.48=1.6$ ). Under the hypotheses of Theorem 2 the sign $\delta>0$ is excluded, so the theorem is in tension with the central values at a level that is not decisive; the fits themselves impose no field dynamics.

5. A field that does not track: the excursion bound

A light field that is frozen at early times and released later can move in either direction, so Theorem 2 does not apply. The obstruction is then different: the field must still arrive at today’s value, and if its coupling to electrons is the same along the way, that value is constrained by tests of the equivalence principle.

Write $\beta_e\equiv M_P\,d\ln m_e/d\phi$ today. If $\phi$ is light enough to mediate a force over the size of the Earth, a body $\alpha$ of mass $M_\alpha$ containing $N_\alpha$ electrons has scalar charge $q_\alpha=\partial M_\alpha/\partial\phi=\beta_eN_\alpha m_e/M_P$ , when only the electron rest mass depends on $\phi$ . Scalar exchange between bodies $\alpha$ and $\alpha'$ at distance $r$ gives an attractive force $q_\alpha q_{\alpha'}/(4\pi r^2)$ , gravity gives $M_\alpha M_{\alpha'}/(8\pi M_P^2r^2)$ , and the ratio is $2\beta_e^2\gamma_\alpha\gamma_{\alpha'}$ with $\gamma_\alpha=N_\alpha m_e/M_\alpha\approx(m_e/m_u)(Z/A)_\alpha$ , where $Z$ is the atomic number and $A$ the mass number. The Eötvös parameter for two test masses falling towards the Earth ( $E$ ) is

\eta_{\rm EP}\simeq2\beta_e^2\gamma_E(\gamma_{\rm Ti}-\gamma_{\rm Pt}). \tag{6}

The MICROSCOPE experiment measured $\eta_{\rm EP}({\rm Ti,Pt})=[-1.5\pm2.3\,({\rm stat})\pm1.5\,({\rm syst})]\times10^{-15}$ [MICROSCOPE]. The errors combine in quadrature to $2.75\times10^{-15}$ , so the 2 sigma range of $\lvert\eta_{\rm EP}\rvert$ extends to $1.5\times10^{-15}+2\times2.75\times10^{-15}=7.0\times10^{-15}$ . With $m_e/m_u=5.486\times10^{-4}$ , $(Z/A)_{\rm Ti}=22/47.867=0.4596$ , $(Z/A)_{\rm Pt}=78/195.084=0.3998$ and a bulk-Earth value $(Z/A)_E\approx0.487$ (an assumed input), (6) gives $\lvert\beta_e\rvert<6.3\times10^{-4}$ at this level. Varying $(Z/A)_E$ between $0.48$ and $0.50$ changes the bound by less than 2% of its value; the bound scales as $\sqrt{\eta_{\max}}$ .

Suppose now that $\lvert d\ln m_e/d\phi\rvert$ never exceeds a value $\beta_{\max}/M_P$ along the path from last scattering to today; the identification $\beta_{\max}=6.3\times10^{-4}$ is the assumption that the coupling does not grow towards the past. Then $\ln(1+\delta)\le\beta_{\max}\lvert\Delta\phi\rvert/M_P$ . A field cannot move arbitrarily far, because its kinetic energy density $\dot\phi^2/2$ is a share $\Omega_{\rm kin}=\dot\phi^2/(6H^2M_P^2)$ of the total. For a flat or negatively curved universe in which every non-kinetic energy density is non-negative, the Friedmann equation gives $\Omega_{\rm kin}\le1$ . With $x=\ln a$ as the time variable, $d\phi/dx=\pm\sqrt{6\Omega_{\rm kin}}\,M_P$ , and over the $x_*=\ln(1+z_*)\approx\ln1091=6.99$ e-folds from last scattering to today the Cauchy–Schwarz inequality gives

\frac{\lvert\Delta\phi\rvert}{M_P}\le\int_0^{x_*}\sqrt{6\Omega_{\rm kin}}\,dx\le x_*\sqrt{6\langle\Omega_{\rm kin}\rangle}=17.1\sqrt{\langle\Omega_{\rm kin}\rangle}, \tag{7}

with $\langle\Omega_{\rm kin}\rangle$ the average of $\Omega_{\rm kin}$ over e-folds.

Theorem 3 (excursion bound). Suppose $m_e$ depends on one field $\phi$ with $\lvert d\ln m_e/d\phi\rvert\le\beta_{\max}/M_P$ along the path from last scattering to today, that the universe is flat or negatively curved, and that all non-kinetic energy densities, including $V(\phi)$ , are non-negative. Then $\ln(1+\delta)\le\sqrt6\,x_*\,\beta_{\max}\sqrt{\langle\Omega_{\rm kin}\rangle}$ . Taking $\beta_{\max}=6.3\times10^{-4}$ , the MICROSCOPE 2 sigma bound on the present coupling of a field light enough to mediate a force over the size of the Earth, so that the hypothesis on the path says that the coupling was never larger in the past, this is $\delta\lesssim1.1\%\times\sqrt{\langle\Omega_{\rm kin}\rangle}$ .

The proof is the chain of the two inequalities above. The table shows what the central values of the fits would require at $\beta_{\max}=6.3\times10^{-4}$ : the required excursion is $\ln(1+\delta)/\beta_{\max}$ and the required $\langle\Omega_{\rm kin}\rangle$ is $(\lvert\Delta\phi\rvert/17.1\,M_P)^2$ .

$\delta$ (%)	required $\lvert\Delta\phi\rvert/M_P$	required $\langle\Omega_{\rm kin}\rangle$
0.96	15.2	0.78
0.78	12.3	0.52
0.76	12.0	0.49
0.63	10.0	0.34

A required $\langle\Omega_{\rm kin}\rangle$ above 1 is impossible under the hypotheses of Theorem 3, so a value of $\delta$ above $1.1\%$ is excluded under those hypotheses with $\beta_{\max}=6.3\times10^{-4}$ . The central values in the table need the field’s kinetic energy to carry between $34\%$ and $78\%$ of the total energy density on average over the e-folds since recombination, in an interval that the CMB and BAO describe as dominated by matter. As an illustration, if the kinetic energy never exceeded the share of a cosmological constant, $\Omega_{\rm kin}\le\Omega_\Lambda(z)$ with $\Omega_m=0.3$ in a $\Lambda$ CDM history, the middle expression of (7) gives $\lvert\Delta\phi\rvert\le1.98M_P$ and $\delta\le0.12\%$ .

6. Cases outside the hypotheses

Fields that start off the tracking branch, or have no minimiser

The rolling field of section 3 pushes the particle mass down for either sign of the coupling, so that $\omega_c(\mathrm{rec})/\omega_c(0)>1$ is produced by a coupled scalar with kinetic energy [Amendola]. The vacuum is then not of the type in Theorem 1, since $w\ne-1$ .

Active media

A negative $\Gamma$ or a concave $g+F$ (population inversion, a negative temperature) reverses the sign. In one dimension with $K=1$ , $\Gamma=-\gamma<0$ and a drop of $n$ by $\theta$ at $t_0$ , the equation $\dot y=\gamma(y-y_{\rm eq})$ gives $\mu(t)-\mu(t_0)=\theta c^2(1-e^{\gamma(t-t_0)})<0$ . The generalised second law and the source of the free energy then have to be specified. Matter creation with a chemical potential [CCL], non-unital maps, and reservoirs at negative temperature are likewise outside (c).

Hysteresis

A field that follows a local minimum to a spinodal and then jumps is outside the global-minimum argument. In numerical examples of one-dimensional potentials with the structure of Theorem 1(a), a jump along the local-minimum branch can lower $\mu$ , while the net change of $\mu$ from the start to the end of the history is non-negative.

Quadratic couplings

For $m_e=m_0(1+\phi^2/2M_e^2)$ with $\phi=0$ today there is no linear coupling, $\beta_e=0$ , and (6) is empty. A displacement from zero cannot come from a Standard Model source, since any displacement raises $m_e$ , which Theorem 2 forbids. A positive $\delta=\phi_i^2/2M_e^2$ requires a field held away from zero by Hubble friction that later oscillates about zero. This class is studied in [BSW 1], [BSW 2] and [GBY]. The oscillating field is matter-like at late times, and its residual oscillation modulates $m_e$ in time, which atomic clocks constrain.

Fields also sourced by dark matter

If $\phi$ also changes the mass $m_X$ of a non-relativistic dark-matter particle, (T2) fails but inequality (3) still holds under (T1). Take $e_j=n_{e,j}m_e+n_{X,j}m_X$ with state 1 at recombination and state 2 today, and let both densities scale as $(1+z)^3$ , so that $n_{\cdot,1}-n_{\cdot,2}=n_{\cdot,0}[(1+z_*)^3-1]$ is common to the two terms. Then (3) reduces to $n_{e,0}(m_e^{\rm lab}-m_e^{\rm rec})+n_{X,0}(m_X^{\rm lab}-m_X^{\rm rec})\ge0$ , and with $m_e^{\rm rec}=(1+\delta)m_e^{\rm lab}$ it reads

\frac{m_X^{\rm lab}-m_X^{\rm rec}}{m_X}\ge\delta\,\frac{\rho_{e,0}}{\rho_{X,0}},\qquad \frac{\rho_{e,0}}{\rho_{X,0}}=(1-Y_p/2)\frac{m_e}{m_p}\frac{\Omega_bh^2}{\Omega_ch^2}=8.9\times10^{-5}\quad(\Omega_ch^2=0.120).

A heavier electron at recombination then requires the dark matter to have been lighter by a fraction of at least $8.9\times10^{-5}\delta$ , that is $7\times10^{-7}$ at $\delta=0.78\%$ . A field of this kind whose coupling saturates at high density would produce a step in $m_e$ .

Fields that track the local density today

Theorem 2 then compares the recombination plasma with a far denser environment and allows $\delta>0$ . Spectroscopy of absorbing gas measures $m_p/m_e$ in other environments, and it constrains this route in two steps. If the gas has an electron scalar density below that of the recombination plasma, Theorem 2 gives $m_e^{\rm gas}\ge m_e(z_*)=(1+\delta)m_e^{\rm lab}$ , so a limit on $\lvert\Delta(m_p/m_e)/(m_p/m_e)\rvert$ between gas and laboratory, attributed to $m_e$ alone, limits $\delta$ by the same amount. Methanol absorption in the lens towards PKS1830−211 at $z=0.88582$ limits the fractional change to $4\times10^{-7}$ at $2\sigma$ [Kanekar], and ammonia and cyanopolyyne lines in Galactic molecular cores limit it to $2\times10^{-8}$ at $3\sigma$ [Levshakov], both far below $\delta\sim10^{-2}$ . The step applies only if the absorbing gas has an electron density below about $290\ {\rm cm}^{-3}$ counting bound electrons. Molecular clouds are typically denser, and then the theorem gives only $m_e\le m_e(z_*)$ there and no limit on $\delta$ .

7. Open points: non-quadratic relaxation, hysteresis and fits with field dynamics

Theorem 1(a) covers global tracking for every potential, and Theorem 1(b) a quadratic vacuum energy with constant Hessian. For a relaxation $\dot\xi=-\Gamma(\xi,t)\nabla\Phi_t$ with a non-quadratic $\Phi_t$ the substitution that makes the flow a contraction is unavailable, and the cumulative sign of the transfer is open. The same holds for hysteresis, where the numerical examples of section 6 show a non-negative net change of $\mu$ , and for a heavy propagating auxiliary field that tracks the minimum adiabatically, where the kinetic energy of the field enters. For unital exchange, the gain in one step is shown from a passive state; an iterated statement needs the state to return to the passive set between steps.

For the electron mass, (T3) is established for an ideal plasma, without Coulomb corrections and without the full recombination history. A bound on the size of the neighbourhood of $m_e$ in which it persists during annihilation would complete the pair case of Corollary 2. A path along which $d\ln m_e/d\phi$ grows from a small value today interpolates between Theorem 3 and the quadratic class and is covered by neither. The values of $\delta$ in the table come from fits that vary $m_e$ by hand at recombination. A field model changes $m_e$ at other epochs, in particular during nucleosynthesis, and adds its own energy density, so confronting a model of section 6 with the data requires a fit that includes this dynamics. Such a fit would give the coupling required of the rolling field and its cost in CMB fits, the CMB and nucleosynthesis constraints on a field also sourced by dark matter, and the atomic-clock limits on the oscillating field of the quadratic class. The fifth-force and equivalence-principle limits on a field that tracks the local density form a separate question.

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