Varying Electron Mass in Cosmology

• Varying Electron Mass Model is a framework where the electron mass differs from its laboratory value as a function of redshift or cosmic environment.
• The model alters hydrogen binding energies and Thomson cross sections, shifting recombination redshift and sound horizon scales that impact H0 estimates.
• Observational constraints from CMB, BAO, SNe, BBN, and high-redshift spectroscopy rigorously test these variations against standard cosmology.

Searching arXiv for 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 (Schöneberg et al., 2024, Seto et al., 2024). 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 (Bagdonaite et al., 2013). 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 (Schöneberg et al., 2024).

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 (Schöneberg et al., 2024). 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 (Seto et al., 2024). Big-bang nucleosynthesis studies often denote the same quantity by $\mu\equiv m_e/m_{e,0}$0 and treat it as constant over the BBN$\mu\equiv m_e/m_{e,0}$1recombination era (Seto et al., 2022).

A recurrent source of confusion is the symbol $\mu\equiv m_e/m_{e,0}$2. In molecular-absorption work, $\mu\equiv m_e/m_{e,0}$3, not $\mu\equiv m_e/m_{e,0}$4. For absorbers at redshift $\mu\equiv m_e/m_{e,0}$5, one defines

$\mu\equiv m_e/m_{e,0}$6

and, if $\mu\equiv m_e/m_{e,0}$7 is fixed, obtains to first order

$\mu\equiv m_e/m_{e,0}$8

This relation underlies the use of $\mu\equiv m_e/m_{e,0}$9, HD, ammonia, and methanol spectra as indirect probes of a varying electron mass (Bagdonaite et al., 2013).

The model class is broader than a single cosmological ansatz. The literature includes a bottom-up recombination-era modification of $\mu=1$0 (Schöneberg et al., 2024), scalar-field realizations coupled to the electron Yukawa sector (Schöneberg et al., 2024), dark-sector interaction models with $\mu=1$1 (Hoshiya et al., 2022), density-dependent symmetron realizations (Solomon et al., 2022), hyperlight-scalar constructions that also modulate $\mu=1$2 (Baryakhtar et al., 2024), and curvature-dependent proposals in which effective fermion masses depend on the local Weyl tensor (Landau et al., 2010).

2. Recombination microphysics and the CMB mechanism

The central cosmological mechanism is that hydrogenic binding energies scale as $\mu=1$3, while the Thomson cross section scales as

$\mu=1$4

Increasing $\mu=1$5 therefore raises the redshift of recombination $\mu=1$6, shortens the sound horizon, and can require a larger $\mu=1$7 to preserve the observed acoustic angle $\mu=1$8 (Schöneberg et al., 2024, Seto et al., 2024).

In Saha equilibrium,

$\mu=1$9

so that approximately $\mu$0 (Schöneberg et al., 2024). The sound horizon is

$\mu$1

and the drag-epoch sound horizon satisfies

$\mu$2

Because larger $\mu$3 shifts $\mu$4 and $\mu$5 upward, both $\mu$6 and $\mu$7 decrease (Seto et al., 2024).

The model exhibits a pronounced parameter degeneracy. One analysis finds that the Thomson visibility is left invariant under

$\mu$8

creating a near-exact degeneracy with $\mu$9, with direction

$\Delta\mu/\mu \simeq -\Delta m_e/m_e$0

This is the technical reason a percent-level shift in $\Delta\mu/\mu \simeq -\Delta m_e/m_e$1 can translate into a nontrivial shift in the inferred Hubble constant while leaving the primary TT/TE/EE spectra nearly unchanged (Schöneberg et al., 2024).

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 $\Delta\mu/\mu \simeq -\Delta m_e/m_e$2 all follow their known $\Delta\mu/\mu \simeq -\Delta m_e/m_e$3-scalings (Seto et al., 2024, Wang et al., 26 Aug 2025). This suggests that the varying-$\Delta\mu/\mu \simeq -\Delta m_e/m_e$4 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 $\Delta\mu/\mu \simeq -\Delta m_e/m_e$5 at recombination is correlated with $\Delta\mu/\mu \simeq -\Delta m_e/m_e$6, but the quantitative preference depends on the dataset and on whether additional degrees of freedom such as $\Delta\mu/\mu \simeq -\Delta m_e/m_e$7, $\Delta\mu/\mu \simeq -\Delta m_e/m_e$8, or dynamical dark energy are varied simultaneously (Schöneberg et al., 2024).

The following representative constraints illustrate the current parameter space:

Analysis	Electron-mass parameter	Reported $\Delta\mu/\mu \simeq -\Delta m_e/m_e$9 or tension
P18+DESI	$m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,$0	$m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,$1, $m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,$2
P18+DESI+PantheonPLUS+$m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,$3	$m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,$4	$m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,$5, $m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,$6
$m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,$7DESI BAO	$m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,$8	$m_e(z)=m_{e,0}[1+\Delta_e(z)]\,,$9, $m_{e,0}$0
CMB+SNe+DESI DR2	$m_{e,0}$1	$m_{e,0}$2, tension $m_{e,0}$3

These entries are drawn from distinct analyses and use different parameterizations of the electron-mass shift (Schöneberg et al., 2024, Seto et al., 2024, Toda et al., 12 Apr 2025).

A review of the literature concludes that electron-mass variations allow significant easing of the Hubble tension, from the current $m_{e,0}$4 significance down to between $m_{e,0}$5 and $m_{e,0}$6 significance, depending on the precise model and data, and that CMB+BAO+SNe combinations prefer such variations at between $m_{e,0}$7 and $m_{e,0}$8 depending on the model and the data (Schöneberg et al., 2024). DESI BAO are especially important because they indicate a slightly longer sound horizon product $m_{e,0}$9 than some earlier BAO compilations, which in the varying-$\Delta_e(z)$0 scenario shifts the fit toward larger $\Delta_e(z)$1 (Seto et al., 2024).

The model’s performance is not uniform across extensions. In a joint analysis of $\Delta_e(z)$2, $\Delta_e(z)$3, and $\Delta_e(z)$4, the fit gives

$\Delta_e(z)$5

with no indication of spatial curvature deviating from flatness (Wang et al., 26 Aug 2025). By contrast, combining varying $\Delta_e(z)$6 with a sign-switching cosmological constant does not successfully solve the Hubble tension because the two sectors require opposite shifts in $\Delta_e(z)$7 to maintain the relevant CMB scales (Toda et al., 2024). Likewise, in 2025 DESI-era fits, $\Delta_e(z)$8 gives a larger upward shift in $\Delta_e(z)$9 than CPL+$\mu\equiv \frac{m_e}{m_{e,0}}\,,$0, because the late-time CPL freedom absorbs part of the degeneracy breaking that otherwise pushes $\mu\equiv \frac{m_e}{m_{e,0}}\,,$1 (Smith et al., 24 Oct 2025).

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

The BBN sector probes varying $\mu\equiv \frac{m_e}{m_{e,0}}\,,$2 through neutron–proton interconversion, neutron beta decay, and the thermodynamics of neutrino decoupling. One key scaling is

$\mu\equiv \frac{m_e}{m_{e,0}}\,,$3

so increasing $\mu\equiv \frac{m_e}{m_{e,0}}\,,$4 suppresses the weak rates, raises the helium mass fraction $\mu\equiv \frac{m_e}{m_{e,0}}\,,$5, and lowers the deuterium abundance D/H (Seto et al., 2022, Schöneberg et al., 2024).

A full BBN modification with an emulator yields a weak BBN-only $\mu\equiv \frac{m_e}{m_{e,0}}\,,$6 CL bound of

$\mu\equiv \frac{m_e}{m_{e,0}}\,,$7

which by itself does not compete with CMB+BAO (Schöneberg et al., 2024). Once primordial abundances are combined with CMB and BAO, however, the allowed window narrows substantially. One joint analysis reports

$\mu\equiv \frac{m_e}{m_{e,0}}\,,$8

with

$\mu\equiv \frac{m_e}{m_{e,0}}\,,$9

The same work concludes that an acceptable electron mass at the BBN time would be only approximately $\mu\to 1$0 greater than the current electron mass and that large $\mu\to 1$1 shifts as a resolution of the Hubble tension are effectively ruled out (Seto et al., 2022).

A more recent early-Universe treatment couples a neutrino-decoupling solver to an $\mu\to 1$2-aware BBN network and includes the induced shift in $\mu\to 1$3. Using D/H, $\mu\to 1$4, and Planck $\mu\to 1$5, it finds

$\mu\to 1$6

at $\mu\to 1$7, with the allowed range close to the present laboratory value at the level of $\mu\to 1$8 (Garramone et al., 5 Feb 2026). This constrains not only recombination-era phenomenology but also any model in which $\mu\to 1$9 departs appreciably from $\mu\equiv m_e/m_{e,0}$00 MeV during the MeV epoch.

5. Microphysical realizations and excluded variants

At the field-theory level, a minimal toy construction introduces a scalar $\mu\equiv m_e/m_{e,0}$01 with canonical kinetic term and potential $\mu\equiv m_e/m_{e,0}$02, coupled to the electron Yukawa interaction,

$\mu\equiv m_e/m_{e,0}$03

with $\mu\equiv m_e/m_{e,0}$04, so that $\mu\equiv m_e/m_{e,0}$05 (Schöneberg et al., 2024). 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,

$\mu\equiv m_e/m_{e,0}$06

In their fit to Planck, BAO, Pantheon, and SH0ES(R19), the coupling parameter satisfies $\mu\equiv m_e/m_{e,0}$07, corresponding to $\mu\equiv m_e/m_{e,0}$08, with $\mu\equiv m_e/m_{e,0}$09 and an inferred recombination-era shift

$\mu\equiv m_e/m_{e,0}$10

(Hoshiya et al., 2022).

An environment-dependent realization couples the electron to a symmetron scalar. In the Einstein frame,

$\mu\equiv m_e/m_{e,0}$11

and the effective potential in matter density $\mu\equiv m_e/m_{e,0}$12 is

$\mu\equiv m_e/m_{e,0}$13

This yields a density-dependent vacuum expectation value and hence a density-dependent effective electron mass $\mu\equiv m_e/m_{e,0}$14 (Solomon et al., 2022). The paper proposes this as a mechanism by which $\mu\equiv m_e/m_{e,0}$15 could differ between recombination and the present local environment.

A broader extension introduces a hyperlight scalar that modulates both $\mu\equiv m_e/m_{e,0}$16 and $\mu\equiv m_e/m_{e,0}$17. With

$\mu\equiv m_e/m_{e,0}$18

and couplings such as

$\mu\equiv m_e/m_{e,0}$19

the explored mass range is

$\mu\equiv m_e/m_{e,0}$20

and the scalar can compose up to a percent of the present dark matter density while shifting early-time constants (Baryakhtar et al., 2024).

Not all realizations are viable. Landau et al. consider a Weyl-tensor model in which effective fermion masses depend on

$\mu\equiv m_e/m_{e,0}$21

To explain the reported Milky-Way ammonia anomaly, the required coupling is $\mu\equiv m_e/m_{e,0}$22, whereas modern torsion-balance Eötvös experiments constrain the relevant parameters to $\mu\equiv m_e/m_{e,0}$23. The discrepancy is twenty orders of magnitude or more, and the authors conclude that the model is not viable (Landau et al., 2010).

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 $\mu\equiv m_e/m_{e,0}$24 is written as

$\mu\equiv m_e/m_{e,0}$25

or equivalently $\mu\equiv m_e/m_{e,0}$26, where the sensitivity coefficients for $\mu\equiv m_e/m_{e,0}$27 span roughly $\mu\equiv m_e/m_{e,0}$28 (Bagdonaite et al., 2013). Comprehensive fits with VPFIT model all relevant $\mu\equiv m_e/m_{e,0}$29, HD, Lyman-$\mu\equiv m_e/m_{e,0}$30 forest, and metal-line absorption simultaneously and then introduce $\mu\equiv m_e/m_{e,0}$31 as a global free parameter (Bagdonaite et al., 2013, Daprà et al., 2016).

Three representative high-redshift measurements anchor the current spectroscopic literature. Toward QSO B0642–5038 at $\mu\equiv m_e/m_{e,0}$32, a long-range distortion correction shifts the result to

$\mu\equiv m_e/m_{e,0}$33

corresponding, under fixed $\mu\equiv m_e/m_{e,0}$34, to $\mu\equiv m_e/m_{e,0}$35 (Bagdonaite et al., 2013). Toward Q1232+082 at $\mu\equiv m_e/m_{e,0}$36, the final result is

$\mu\equiv m_e/m_{e,0}$37

using 106 H$\mu\equiv m_e/m_{e,0}$38/HD transitions (Daprà et al., 2016). Toward J1443+2724 at $\mu\equiv m_e/m_{e,0}$39, the corrected limit is

$\mu\equiv m_e/m_{e,0}$40

which probes the first $\mu\equiv m_e/m_{e,0}$41 of cosmic time (Bagdonaite et al., 2015).

At the population level, the optical $\mu\equiv m_e/m_{e,0}$42 program finds no large deviation from constancy. One paper reports that seven high-redshift $\mu\equiv m_e/m_{e,0}$43 sightlines converge on $\mu\equiv m_e/m_{e,0}$44, with weighted mean $\mu\equiv m_e/m_{e,0}$45 (Bagdonaite et al., 2013), while another quotes $\mu\equiv m_e/m_{e,0}$46 from eleven absorbers in the range $\mu\equiv m_e/m_{e,0}$47–$\mu\equiv m_e/m_{e,0}$48 (Daprà et al., 2016). Radio molecular absorbers at $\mu\equiv m_e/m_{e,0}$49–$\mu\equiv m_e/m_{e,0}$50 constrain $\mu\equiv m_e/m_{e,0}$51, and laboratory clock comparisons limit $\mu\equiv m_e/m_{e,0}$52 (Bagdonaite et al., 2013).

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

$\mu\equiv m_e/m_{e,0}$53

that is, no variation within $\mu\equiv m_e/m_{e,0}$54, and then maps this null result into constraints on a lepton-specific 2HDM parameter space (Albuquerque et al., 2024).

A common misconception is that null quasar or clock bounds automatically exclude all recombination-era varying-$\mu\equiv m_e/m_{e,0}$55 models. The current literature does not support that statement in general. Post-recombination constraints are at the $\mu\equiv m_e/m_{e,0}$56–$\mu\equiv m_e/m_{e,0}$57 level for quasar $\mu\equiv m_e/m_{e,0}$58 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 (Schöneberg et al., 2024). This suggests that the central open question is not whether $\mu\equiv m_e/m_{e,0}$59 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.