---
title: Varying Electron Mass in Cosmology
url: https://www.emergentmind.com/topics/varying-electron-mass-model
type: topic
---

# Varying Electron Mass in Cosmology

Searching arXiv for recent papers on varying electron mass cosmology and related constraints.
The varying electron mass model denotes a class of phenomenological and microphysical frameworks in which the electron mass differs from its laboratory value as a function of redshift, cosmic time, or environment. In cosmology it is commonly parameterized as \(m_e(z)\equiv m_{e,0}[1+\Delta_e(z)]\), while some recombination-era analyses use \(\mu\equiv m_e/m_{e,0}\) during the CMB epoch and set \(\mu=1\) today [2407.16845][2405.11869]. In high-redshift spectroscopy, by contrast, \(\mu\) often denotes the proton-to-electron mass ratio, so that if the proton mass is assumed fixed, \(\Delta\mu/\mu \simeq -\Delta m_e/m_e\) and a nonzero spectroscopic signal maps directly onto an electron-mass shift [1308.1330]. The model is studied both as a modification of pre-recombination microphysics that can shrink the sound horizon and as a target of constraints from big bang nucleosynthesis, quasar absorption systems, radio molecular absorbers, atomic clocks, galaxy clusters, and equivalence-principle tests [2407.16845].

## 1. Definitions, notation, and scope

A standard phenomenological parametrization writes
\[
m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,
\]
with \(m_{e,0}\) the present-day mass and \(\Delta_e(z)\) the fractional shift [2407.16845]. A simpler one-parameter version, used in several recombination analyses, assumes a constant rescaling during the CMB epoch,
\[
\mu\equiv \frac{m_e}{m_{e,0}}\,,
\]
followed by \(\mu\to 1\) at low redshift [2405.11869]. Big-bang nucleosynthesis studies often denote the same quantity by \(\chi\equiv m_e/m_{e,0}\) and treat it as constant over the BBN\(\to\)recombination era [2206.13209].

A recurrent source of confusion is the symbol \(\mu\). In molecular-absorption work, \(\mu\equiv m_p/m_e\), not \(m_e/m_{e,0}\). For absorbers at redshift \(z\), one defines
\[
\Delta\mu/\mu \equiv (\mu_z-\mu_0)/\mu_0\,,
\]
and, if \(m_p\) is fixed, obtains to first order
\[
\Delta\mu/\mu \simeq -\Delta m_e/m_e\,.
\]
This relation underlies the use of \(\mathrm{H}_2\), HD, ammonia, and methanol spectra as indirect probes of a varying electron mass [1308.1330].

The model class is broader than a single cosmological ansatz. The literature includes a bottom-up recombination-era modification of \(m_e\) [2407.16845], scalar-field realizations coupled to the electron Yukawa sector [2407.16845], dark-sector interaction models with \(m_e(\phi)=m_{e0}e^{\beta\phi}\) [2202.07714], density-dependent symmetron realizations [2201.03127], hyperlight-scalar constructions that also modulate \(\alpha\) [2405.10358], and curvature-dependent proposals in which effective fermion masses depend on the local Weyl tensor [1010.6233].

## 2. Recombination microphysics and the CMB mechanism

The central cosmological mechanism is that hydrogenic binding energies scale as \(E_B\propto m_e\alpha^2\), while the Thomson cross section scales as
\[
\sigma_T\propto m_e^{-2}\,.
\]
Increasing \(m_e\) therefore raises the redshift of recombination \(z_*\), shortens the sound horizon, and can require a larger \(H_0\) to preserve the observed acoustic angle \(\theta_s=r_s/D_A(z_*)\) [2407.16845][2405.11869].

In Saha equilibrium,
\[
\frac{x_e^2}{1-x_e}=\frac{n_e n_p}{n_H}\propto (m_eT)^{3/2}e^{-E_B/(k_BT)}\,,
\]
so that approximately \(z_*\propto m_e\) [2407.16845]. The sound horizon is
\[
r_s=\int_{z_*}^{\infty}\frac{c_s\,dz}{H(z)}\,,
\]
and the drag-epoch sound horizon satisfies
\[
r_d\equiv \int_{z_d}^{\infty}\frac{c_s(z)}{H(z)}\,dz\,.
\]
Because larger \(m_e\) shifts \(z_*\) and \(z_d\) upward, both \(r_s\) and \(r_d\) decrease [2405.11869].

The model exhibits a pronounced parameter degeneracy. One analysis finds that the Thomson visibility is left invariant under
\[
(\Omega_b h^2,\Omega_m h^2,m_e)\rightarrow (\lambda,\lambda,\lambda)\,,
\]
creating a near-exact degeneracy with \(H_0\), with direction
\[
\Delta_{\Omega_b h^2}=\Delta_{\Omega_m h^2}=\Delta_{z_*}=\Delta_{m_e},\qquad
\Delta_h\simeq 3.23\,\Delta_{m_e}\,.
\]
This is the technical reason a percent-level shift in \(m_e\) can translate into a nontrivial shift in the inferred Hubble constant while leaving the primary TT/TE/EE spectra nearly unchanged [2407.16845].

At the level of recombination codes, published analyses modify RECFAST or HYREC-2 so that binding energies, photoionization cross sections, case-B recombination coefficients, two-photon decay rates, and \(\sigma_T\) all follow their known \(m_e\)-scalings [2405.11869][2508.19081]. This suggests that the varying-\(m_e\) model is not a single observable rescaling but a coordinated deformation of the full recombination network.

## 3. Cosmological fits, DESI-era constraints, and the Hubble tension

Recent likelihood analyses consistently use Planck CMB data together with BAO and supernova samples. The broad pattern is that \(m_e>m_{e,0}\) at recombination is correlated with \(H_0>H_0^{\Lambda{\rm CDM}}\), but the quantitative preference depends on the dataset and on whether additional degrees of freedom such as \(\Omega_K\), \(A_{\rm lens}\), or dynamical dark energy are varied simultaneously [2407.16845].

The following representative constraints illustrate the current parameter space:

| Analysis | Electron-mass parameter | Reported \(H_0\) or tension |
|---|---:|---:|
| P18+DESI | \(\Delta_e=0.0121\pm0.0063\) | \(H_0=70.03\pm1.06\), \(GT\simeq2.0\sigma\) |
| P18+DESI+PantheonPLUS+\(\Omega_k\) | \(\Delta_e=0.0188\pm0.0052\) | \(H_0=71.61\pm1.00\), \(GT\simeq1.0\sigma\) |
| \(\mathcal D+\)DESI BAO | \(\mu=1.0092\pm0.0055\) | \(H_0=69.44\pm0.84\), \(T_{H_0}=2.89\sigma\) |
| CMB+SNe+DESI DR2 | \(\mu_e=1.0101\pm0.0046\) | \(H_0=69.44\pm0.67\), tension \(3.12\sigma\) |

These entries are drawn from distinct analyses and use different parameterizations of the electron-mass shift [2407.16845][2405.11869][2504.09136].

A review of the literature concludes that electron-mass variations allow significant easing of the Hubble tension, from the current \(\sim5\sigma\) significance down to between \(3.4\sigma\) and \(1.0\sigma\) significance, depending on the precise model and data, and that CMB+BAO+SNe combinations prefer such variations at between \(2\sigma\) and \(3.6\sigma\) depending on the model and the data [2407.16845]. DESI BAO are especially important because they indicate a slightly longer sound horizon product \(r_d h\) than some earlier BAO compilations, which in the varying-\(m_e\) scenario shifts the fit toward larger \(H_0\) [2405.11869].

The model’s performance is not uniform across extensions. In a joint analysis of \(\Delta m_e/m_e\), \(\Omega_K\), and \(A_{\rm lens}\), the fit gives
\[
\Delta m_e/m_e = 0.0109^{+0.0066}_{-0.0062},\quad
H_0 = 69.61^{+0.60}_{-0.55}\ {\rm km\,s^{-1}\,Mpc^{-1}},\quad
S_8=0.808\pm0.012,
\]
with no indication of spatial curvature deviating from flatness [2508.19081]. By contrast, combining varying \(m_e\) with a sign-switching cosmological constant does not successfully solve the Hubble tension because the two sectors require opposite shifts in \(\Omega_m\) to maintain the relevant CMB scales [2407.01173]. Likewise, in 2025 DESI-era fits, \(\Lambda{\rm CDM}+m_e\) gives a larger upward shift in \(H_0\) than CPL+\(m_e\), because the late-time CPL freedom absorbs part of the degeneracy breaking that otherwise pushes \(m_e/m_{e,0}>1\) [2510.21931].

## 4. Big-bang nucleosynthesis, neutrino decoupling, and early-Universe bounds

The BBN sector probes varying \(m_e\) through neutron–proton interconversion, neutron beta decay, and the thermodynamics of neutrino decoupling. One key scaling is
\[
\Gamma_{n\to p\,e\,\nu}\sim m_e^5\,f[(m_n-m_p)/m_e]\,,
\]
so increasing \(m_e\) suppresses the weak rates, raises the helium mass fraction \(Y_p\), and lowers the deuterium abundance D/H [2206.13209][2407.16845].

A full BBN modification with an emulator yields a weak BBN-only \(68\%\) CL bound of
\[
|\Delta_e|\lesssim 5\%\,,
\]
which by itself does not compete with CMB+BAO [2407.16845]. Once primordial abundances are combined with CMB and BAO, however, the allowed window narrows substantially. One joint analysis reports
\[
\chi = \frac{m_e({\rm BBN})}{m_{e0}} = 1.0028 \pm 0.0064,\qquad
H_0 = 68.0 \pm 1.1\ {\rm km/s/Mpc},
\]
with
\[
Y_p = 0.24705^{+0.00035}_{-0.00043},\qquad
10^5\,{\rm D/H} = 2.510 \pm 0.029\,.
\]
The same work concludes that an acceptable electron mass at the BBN time would be only approximately \(1\%\) greater than the current electron mass and that large \(m_e\) shifts as a resolution of the Hubble tension are effectively ruled out [2206.13209].

A more recent early-Universe treatment couples a neutrino-decoupling solver to an \(m_e\)-aware BBN network and includes the induced shift in \(N_{\rm eff}\). Using D/H, \(Y_P\), and Planck \(N_{\rm eff}\), it finds
\[
m_e = 0.504^{+0.007}_{-0.006}\ {\rm MeV}\quad {\rm (NACRE\,II)},\qquad
m_e=0.510\pm0.007\ {\rm MeV}\quad {\rm (PRIMAT)}
\]
at \(1\sigma\), with the allowed range close to the present laboratory value at the level of \(1.4\%\) [2602.05720]. This constrains not only recombination-era phenomenology but also any model in which \(m_e\) departs appreciably from \(0.511\) MeV during the MeV epoch.

## 5. Microphysical realizations and excluded variants

At the field-theory level, a minimal toy construction introduces a scalar \(\phi\) with canonical kinetic term and potential \(V(\phi)\), coupled to the electron Yukawa interaction,
\[
\mathcal L \supset -\frac12(\partial\phi)^2 - V(\phi) + y_e(\phi)\,\bar e_L H e_R + {\rm h.c.}\,,
\]
with \(y_e(\phi)/y_{e,0}=\exp[\zeta(\phi-\phi_0)]\), so that \(m_e(\phi)=y_e(\phi)\,v/\sqrt2\) [2407.16845]. This framework is purely phenomenological, but it motivates several explicit realizations.

Hoshiya and Toda study a dark-sector interaction model in which both electrons and dark matter couple conformally to the same rolling scalar,
\[
f_e(\phi)=f_{DM}(\phi)=\exp(\beta\phi)\,,
\qquad
m_e(\phi)=m_{e0}e^{\beta\phi}\,.
\]
In their fit to Planck, BAO, Pantheon, and SH0ES(R19), the coupling parameter satisfies \(\delta=-1.4^{+1.1}_{-1.1}\times10^{-3}\), corresponding to \(\beta=0.029^{+0.011}_{-0.011}\), with \(H_0=69.9^{+1.6}_{-1.5}\ {\rm km/s/Mpc}\) and an inferred recombination-era shift
\[
\frac{m_e(z_*)-m_{e0}}{m_{e0}}\approx (0.9\pm0.7)\%
\]
[2202.07714].

An environment-dependent realization couples the electron to a symmetron scalar. In the Einstein frame,
\[
V(\phi)=-\frac12\mu^2\phi^2+\frac14\lambda\phi^4,\qquad
A(\phi)=1+\frac{\phi^2}{2M^2},\qquad
\mathcal L_{\rm Yukawa}=-g_s\bar\psi\psi\phi\,,
\]
and the effective potential in matter density \(\rho\) is
\[
V_{\rm eff}(\phi)\simeq \frac12\Bigl(\frac{\rho}{M^2}-\mu^2\Bigr)\phi^2+\frac14\lambda\phi^4\,.
\]
This yields a density-dependent vacuum expectation value and hence a density-dependent effective electron mass \(m_e(\rho)\) [2201.03127]. The paper proposes this as a mechanism by which \(m_e\) could differ between recombination and the present local environment.

A broader extension introduces a hyperlight scalar that modulates both \(m_e\) and \(\alpha\). With
\[
V(\phi)=\frac12 m_\phi^2\phi^2,
\]
and couplings such as
\[
\Delta m_e/m_e = d_e\,\phi/M_{\rm Pl}
\quad {\rm or}\quad
\Delta m_e/m_e \simeq \frac12 d_e^{(2)}(\phi/M_{\rm Pl})^2,
\]
the explored mass range is
\[
10^{-32}\ {\rm eV}\lesssim m_\phi \lesssim 10^{-28}\ {\rm eV},
\]
and the scalar can compose up to a percent of the present dark matter density while shifting early-time constants [2405.10358].

Not all realizations are viable. Landau et al. consider a Weyl-tensor model in which effective fermion masses depend on
\[
W^2\equiv W_{abcd}W^{abcd},
\qquad
m_i^{\rm eff}(x)=m_i\Bigl[1+\frac{\xi_i}{M_P^4}W^2(x)\Bigr].
\]
To explain the reported Milky-Way ammonia anomaly, the required coupling is \(\sim10^{36}\ {\rm m^4}\), whereas modern torsion-balance Eötvös experiments constrain the relevant parameters to \(\lesssim10^{16}\ {\rm m^4}\). The discrepancy is twenty orders of magnitude or more, and the authors conclude that the model is not viable [1010.6233].

## 6. Spectroscopic and low-redshift observational constraints

Post-recombination bounds are dominated by spectroscopy. For molecular hydrogen absorbers, the leading-order wavelength shift of transition \(i\) is written as
\[
\lambda_i = \lambda_i^0(1+z_{\rm abs})[1+K_i(\Delta\mu/\mu)]\,,
\]
or equivalently \(\Delta v_i/c = K_i(\Delta\mu/\mu)\), where the sensitivity coefficients for \(\mathrm H_2\) span roughly \(-0.02\lesssim K_i\lesssim +0.05\) [1308.1330]. Comprehensive fits with VPFIT model all relevant \(\mathrm H_2\), HD, Lyman-\(\alpha\) forest, and metal-line absorption simultaneously and then introduce \(\Delta\mu/\mu\) as a global free parameter [1308.1330][1611.05191].

Three representative high-redshift measurements anchor the current spectroscopic literature. Toward QSO B0642–5038 at \(z_{\rm abs}=2.659\), a long-range distortion correction shifts the result to
\[
\Delta\mu/\mu = (12.7 \pm 4.5_{\rm stat} \pm 4.2_{\rm sys})\times10^{-6},
\]
corresponding, under fixed \(m_p\), to \(\Delta m_e/m_e\simeq -12.7\times10^{-6}\) [1308.1330]. Toward Q1232+082 at \(z_{\rm abs}=2.33771\), the final result is
\[
\Delta\mu/\mu = (19 \pm 9_{\rm stat} \pm 5_{\rm syst})\times10^{-6},
\]
using 106 H\(_2\)/HD transitions [1611.05191]. Toward J1443+2724 at \(z_{\rm abs}=4.224\), the corrected limit is
\[
\Delta\mu/\mu = (-9.5\pm5.4_{\rm stat}\pm5.3_{\rm sys})\times10^{-6},
\]
which probes the first \(10\%\) of cosmic time [1501.05533].

At the population level, the optical \(\mathrm H_2\) program finds no large deviation from constancy. One paper reports that seven high-redshift \(\mathrm H_2\) sightlines converge on \(|\Delta\mu/\mu|<1\times10^{-5}\), with weighted mean \((4.0\pm1.8)\times10^{-6}\) [1308.1330], while another quotes \(\langle\Delta\mu/\mu\rangle=(3.4\pm1.6)\times10^{-6}\) from eleven absorbers in the range \(z=2.0\)–\(4.2\) [1611.05191]. Radio molecular absorbers at \(z\approx0.7\)–\(0.9\) constrain \(|\Delta\mu/\mu|\lesssim {\rm few}\times10^{-7}\), and laboratory clock comparisons limit \(|d\mu/dt|/\mu\lesssim10^{-16}\ {\rm yr}^{-1}\) [1308.1330].

Low-redshift, non-spectroscopic tests are much weaker but probe distinct systematics. Using galaxy-cluster gas mass fractions together with Type Ia supernova distances, one analysis finds
\[
\Delta\mu/\mu = 0.0177 \pm 0.098,
\]
that is, no variation within \(1\sigma\), and then maps this null result into constraints on a lepton-specific 2HDM parameter space [2407.08921].

A common misconception is that null quasar or clock bounds automatically exclude all recombination-era varying-\(m_e\) models. The current literature does not support that statement in general. Post-recombination constraints are at the \(10^{-6}\)–\(10^{-5}\) level for quasar \(\Delta\mu/\mu\) and at the local-drift level for clocks, but the review literature emphasizes that such probes can be insensitive to an abrupt dark-ages transition [2407.16845]. This suggests that the central open question is not whether \(m_e\) can vary arbitrarily—it cannot—but whether a microphysical model can produce an approximately percent-level shift around recombination while remaining compatible with BBN, spectroscopy, laboratory bounds, and equivalence-principle tests.

Source: https://www.emergentmind.com/topics/varying-electron-mass-model