Early-universe constraints on the electron mass

Abstract: We investigate the impact of a nonstandard electron mass $m_e$ on early-Universe thermal history, focusing on neutrino decoupling and Big Bang Nucleosynthesis (BBN). In the standard cosmology, neutrino--electron interactions keep neutrinos in thermal contact with the electromagnetic plasma until shortly before $e<sup>\pm$ annihilation. Varying $m_e$ shifts the decoupling epoch and the entropy transfer from $e<sup>\pm$ annihilation, thereby modifying the neutrino energy density and the inferred effective number of relativistic species, $N_{\mathrm{eff}}$. Independently, during BBN the rates of charged-current weak processes, and hence the neutron-to-proton ratio, depend on $m_e$. By confronting BBN predictions for the primordial light-element abundances with observations and imposing cosmological constraints on $N_{\mathrm{eff}}$, we obtain a bound on $m_e$ in the early Universe of $m_e = 0.504<sup>{+0.007}_{-0.006}$ MeV or $m_e=0.510\pm0.007$ MeV ($1σ$), depending on the considered nuclear reaction network (NACRE II or PRIMAT, respectively). The allowed range is close to the present laboratory value at the level of 1.4\%, thus supporting the constancy of the electron mass over cosmological timescales.

• The paper constrains the electron mass during neutrino decoupling and BBN to 0.504^{+0.007}_{-0.006} MeV with NACRE II or 0.510 ± 0.007 MeV with PRIMAT, consistent with the laboratory value within about 1.4%.
• The analysis combines modified neutrino-decoupling and BBN calculations with N_eff, deuterium, and helium-4 data, finding that direct changes to weak rates, the neutron lifetime, and plasma thermodynamics dominate over expansion-rate effects.
• The results show that helium-4 provides the strongest constraint, while nuclear-reaction network differences remain comparable to the quoted uncertainty and future measurements could test the mild preference for a slightly lower electron mass.

Overview

This paper constrains the value of the electron mass $m_e$ in the early Universe, at the epoch of neutrino decoupling and Big Bang Nucleosynthesis (BBN), by combining two independent probes: the effective number of relativistic species, $N_{\rm eff}$, inferred from cosmological data, and the primordial light-element abundances. The central result is a bound on the electron mass at MeV temperatures of

$$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$

at $1\sigma$, i.e., agreement with the laboratory value $m_{e,0}=0.511$ MeV to within about 1.4%. This constitutes one of the most stringent constraints on $m_e$ available for any pre-recombination epoch.

The analysis proceeds in two stages: a neutrino-decoupling calculation yielding $N_{\rm eff}(m_e)$, and a modified BBN computation propagating $m_e$ through the weak sector, plasma thermodynamics, and expansion history.

Neutrino decoupling with a varying electron mass

The authors solve the coupled evolution equations for the comoving photon temperature $z = T_\gamma a$ and per-flavor comoving neutrino temperatures $z_{\nu_\alpha}$ through neutrino decoupling and $N_{\rm eff}$0 annihilation, using a modified version of the NUDEC_BSM code [Escudero's framework]. A technical refinement is the introduction of a fixed mass scale $N_{\rm eff}$1 in the comoving variable $N_{\rm eff}$2, so that variations in $N_{\rm eff}$3 do not redefine the independent variable; the mass dependence enters only through $N_{\rm eff}$4. Finite-temperature QED corrections to the electromagnetic plasma equation of state are included via the $N_{\rm eff}$5, $N_{\rm eff}$6 functions.

The calculation adopts three approximations, all justified at the precision targeted: thermal (undistorted) Fermi–Dirac neutrino spectra, no flavor oscillations, and vanishing chemical potentials. Spectral distortions shift the standard $N_{\rm eff}$7 at the $N_{\rm eff}$8 level and oscillations at a few $N_{\rm eff}$9, both below the sensitivity relevant here.

The qualitative behavior is as follows. For larger $$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$0 (e.g., 5 MeV), $$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$1 annihilation occurs earlier, while neutrinos remain efficiently coupled to the electromagnetic plasma; they then share the entropy release, reducing photon heating and enhancing $$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$2. For smaller $$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$3 (e.g., 0.1 MeV), annihilation happens after complete decoupling, and the outcome approaches the instantaneous-decoupling limit $$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$4. Consequently $$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$5 is a monotonically increasing function of $$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$6, saturating near 3.044 at low masses and rising toward the asymptote

$$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$7

in the tightly coupled limit. An important corollary is that small $$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$8 produces almost no change in $$m_e = 0.504^{+0.007}_{-0.006}\ \text{MeV} \quad (\text{NACRE II}), \qquad m_e = 0.510 \pm 0.007\ \text{MeV} \quad (\text{PRIMAT})$$9: once decoupling precedes annihilation, further reductions in $1\sigma$0 leave the entropy transfer essentially unchanged. The $1\sigma$1 observable therefore yields only a one-sided upper bound,

$1\sigma$2

This asymmetry — no meaningful lower bound from $1\sigma$3 — is an intrinsic limitation of this channel.

BBN dependence on the electron mass

BBN predictions are obtained with a modified version of PRyMordial [Burns et al.], which retains refinements beyond the Born approximation: $1\sigma$4 QED radiative corrections, finite nucleon-mass effects and weak magnetism (relative rate shifts of order $1\sigma$5), and finite-temperature corrections needed for sub-percent accuracy. Crucially, the neutron lifetime is not imposed from laboratory measurement; it follows from the assumed $1\sigma$6 via

$1\sigma$7

so that increasing $1\sigma$8 suppresses charged-current phase space and lengthens $1\sigma$9, raising the neutron abundance at the onset of nucleosynthesis and hence the $m_{e,0}=0.511$0 yield.

A key diagnostic isolates the role of the expansion history: propagating only $m_{e,0}=0.511$1 into PRyMordial while keeping weak rates fixed produces shifts over $m_{e,0}=0.511$2 MeV of $m_{e,0}=0.511$3, but nearly flat responses in deuterium, helium-4, and lithium-7. The paper concludes that the dominant sensitivity of BBN abundances to $m_{e,0}=0.511$4 arises from direct microphysics — weak rates, neutron lifetime, plasma thermodynamics — rather than from the associated change in the expansion rate. This is a substantive finding, since many analyses attribute such effects primarily to expansion-rate modifications.

Results are computed with two nuclear reaction-rate compilations. They agree closely for helium-4 (which depends mainly on the neutron fraction) but differ systematically for deuterium, where PRIMAT predicts lower D/H due to its larger low-energy $m_{e,0}=0.511$5-factors for deuterium-destruction channels. Lithium-7 exhibits non-monotonic behavior with $m_{e,0}=0.511$6, reflecting interplay between the $m_{e,0}=0.511$7/$m_{e,0}=0.511$8 channels and neutron bottlenecks.

Constraints on $m_{e,0}=0.511$9

The statistical analysis is a $m_e$0 fit over $m_e$1 using deuterium, helium-4, and $m_e$2, with observational inputs

Observable	Value
$m_e$3	$m_e$4
$m_e$5	$m_e$6
$m_e$7	$m_e$8

Theoretical uncertainties on D/H and $m_e$9 follow the rate-induced estimates of Ref. Barenboim et al.; the $N_{\rm eff}(m_e)$0 theoretical uncertainty is neglected as subdominant. Notably, lithium is excluded from the fit because the well-known factor-of-three "lithium problem" would inject poorly controlled systematics — a concession that the constraint set is deliberately restricted to better-understood observables.

Individual and combined results are:

Observable	Network	$N_{\rm eff}(m_e)$1 best fit	$N_{\rm eff}(m_e)$2 range
$N_{\rm eff}(m_e)$3	—	$N_{\rm eff}(m_e)$4 MeV	$N_{\rm eff}(m_e)$5 MeV
D	NACRE II	$N_{\rm eff}(m_e)$6	$N_{\rm eff}(m_e)$7
D	PRIMAT	$N_{\rm eff}(m_e)$8	$N_{\rm eff}(m_e)$9
$m_e$0He	either	$m_e$1 / $m_e$2	$m_e$3
Combined	NACRE II	$m_e$4	$m_e$5
Combined	PRIMAT	$m_e$6	$m_e$7

Helium-4 dominates the combined error budget, owing chiefly to the high precision of the latest $m_e$8 determination ($m_e$9). Two points deserve emphasis. First, helium alone prefers $z = T_\gamma a$0 slightly below the laboratory value, with the deviation sitting at the edge of the $z = T_\gamma a$1 interval under NACRE II — a mild tension worth monitoring as helium measurements improve. Second, the choice of nuclear compilation shifts the combined central value between $z = T_\gamma a$2 and $z = T_\gamma a$3 MeV, i.e., the residual network systematics are comparable to the quoted statistical uncertainty. The paper treats the two compilations as complementary rather than adjudicating between them.

Limitations and open questions

Several assumptions bound the interpretation of these results. The neutrino-decoupling calculation neglects spectral distortions and oscillations, acceptable here but potentially relevant if future CMB experiments reduce $z = T_\gamma a$4 below current levels. The interaction rates entering the collision terms are evaluated with interpolation over precomputed exact-$z = T_\gamma a$5 tables, accurate but limited to the tabulated grid. The exclusion of lithium means the fit does not test whether any value of $z = T_\gamma a$6 could alleviate the lithium problem; the non-monotonic $z = T_\gamma a$7Li response suggests this is unlikely to be a simple resolution, but it is left unquantified. Finally, the analysis assumes zero lepton chemical potentials and standard expansion history; scenarios combining $z = T_\gamma a$8 variation with extra radiation or nonstandard baryon asymmetry would require a joint treatment. Whether future precision measurements of $z = T_\gamma a$9 and $z_{\nu_\alpha}$0 can resolve the mild sub-laboratory preference from helium, or discriminate more sharply between reaction networks, remains open.

Conclusion

By generalizing the NUDEC_BSM and PRyMordial codes to a free electron mass, this work establishes percent-level constraints on $z_{\nu_\alpha}$1 during the first minutes of cosmic evolution: $z_{\nu_\alpha}$2 MeV (NACRE II) or $z_{\nu_\alpha}$3 MeV (PRIMAT), consistent with the laboratory value at the level of 1.4%. The result demonstrates that the dominant lever arm of $z_{\nu_\alpha}$4 on BBN is the direct modification of weak-interaction rates and the neutron lifetime, not the induced change in expansion rate. These bounds complement existing recombination-era limits and extend tests of fundamental-constant constancy to epochs roughly fifteen orders of magnitude earlier in time than the CMB.