Modified hydrogen recombination rate due to non-Hubble redshift effects

The gist The accumulation of observational data in recent years has led to a peculiar situation in which measurements of the cosmological parameters made by various collaborations and using different methods yield significantly different values, which this work addresses by modifying the models used to obtain these values

Introduction and Motivation

The paper investigates a mechanism that can introduce corrections to the rate at which Lyman-alpha photons escape the resonance line, specifically focusing on interactions of Lyman-alpha line photons with moving neutral hydrogen atoms as a source of such a correction The study revisits the standard approach to the recombination calculation and finds that this modification augments the standard ΛCDM model, slightly relaxing tensions between CMB and DESI BAO data concerning H0 and ns, while shifting ns toward values preferred by simple attractor models of inflation

Effects of Lyman-alpha Absorption and Re-emission

The initial idea considered was that special-relativistic corrections to the energy exchange between Lyman-alpha resonance photons and moving hydrogen atoms can make photons leave the resonance line faster than in the standard recombination approach due to redshift from cosmic expansion The process is modeled as H0 + γ → H∗ → H′0 + γ′, where the energy of the resulting photon p′γ (in the laboratory reference frame) differs from that of the initial photon pγ

The energy change in a single interaction can be expressed as p′γpγ ≈ 1−VH cos α − hνmH+V 2H (1), where VH is ∼ 10−5 and hνmH is ∼ 10−8 The criterion for a photon to leave the resonance line in one interaction is given by the inequality VH(1 + sin θ cos(φ + α)) − hνmH(1 − sin θ cos φ) < 0 (3), which leads to a geometrical probability Pθ,φ = hνmH/VH 1− cos α squared (4)

Energy Loss Estimation

The time necessary to cross the resonance line from the right border to the left is estimated as ∆tX = ∆να/vX, where ∆να ≈ 10−4 eV/h is the effective Doppler-broadened resonance line width The line exit rate vSR can be estimated using Eq. (7), which involves a complex integral over the angle α and includes factors like n1s/n, Γ, and q 1− V 2H 1− cos α squared For small values of VH, the integral over the angle α tends to unity

Modified Effective Three-Level Hydrogen Atom Equations

The standard approach is expressed in terms of a three-level system (Eq. 17), but this is modified by introducing a new parameter vα describing the additional rate of photons leaving the resonance This leads to a modification of Eq. (10) to ddt nαn = νa n 1a da dt n(ν+) − n(ν−) − vSRn/n(ν−) + R nn (15), where VSR = vSR/na The resulting modification to the electron concentration equation is given by Eq. (18), which shows that the main difference lies in the coefficient KSR < K

Numerical Results and Conclusions

An MCMC analysis using a modified version of CLASS, letting VSR vary freely, constrained log10 VSR to be between −15.1 ± 1.3 and −14.1 Cosmologies with a non-zero VSR provide a moderately improved fit compared to ΛCDM, driving down the value of the scalar spectrum index ns and inducing a slight upward shift in H0 with a corresponding downward shift in omegam The paper concludes that using CMB data alone cannot definitively establish a preference for or against a non-zero VSR, constraining it to VSR < 1.54 × 10−14s −1 at the 95% confidence level The findings suggest that non-Hubble redshift effects that remove Lyman-alpha photons from the resonance line could play a significant role in explaining observed tensions between cosmological measurements, and therefore should be studied in more detail with extended datasets

Acknowledgments The authors thank A.V. Shepelev and D.A. Solovyov for valuable discussions on this topic

Appendix Calculations Appendix In this appendix, the calculation of the change in photon momentum in a 1s → 2p → 1s process is performed using three input parameters The final change in the photon energy is given by Eq. (26), which describes p′γpγ

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1. No-Hubble Redshift Correction Integration: The improved AI system can incorporate a mechanism to modifying the models used to obtain these values by treating Lyman-alpha photon interactions with moving neutral hydrogen atoms as a source of correction, leading to modified cosmological parameter constraints like prefers slightly higher values of H0 and smaller values of omegam and S8 compared to the standard ΛCDM model.

2. Modified Recombination Solver: The system can utilize a modified version of the Boltzmann solver CLASS [8] to perform MCMC analysis, allowing it to sample cosmological parameters including the new parameter VSR, which is treated as a new model parameter describing an additional rate of photons leaving the resonance.

3. Parameter Space Exploration with VSR: The improved system can perform comprehensive MCMC analyses that allow for the variation of log10 VSR to obtain constraints on cosmological parameters, as demonstrated by Table 1 showing constraints on log10 VSR −15.1 ± 1.3 −-14.1.

4. Tension Resolution Capability: The system can be configured to analyze how the modification augments the standard ΛCDM model, slightly relaxing the tension between the CMB and DESI BAO [4] in terms of H0 and shifting the value of ns toward the range preferred by simple attractor models of inflation.

The accumulation of observational data in recent years has led to a peculiar situation in which measurements of the cosmological parameters made by various collaborations and using different methods yield significantly different values. A solution to this problem may be found by modifying the models used to obtain these values. In this work, we revisit the standard approach to the recombination calculation and investigate a mechanism that can introduce corrections to the rate at which Lyman-alpha photons escape the resonance line. We find that interactions of Lyman-alpha line photons with moving neutral hydrogen atoms can become a source of such a correction. An analysis of CMB data for the modified model favors the existence of a non-Hubble redshift effect and prefers slightly higher values of H 0 and smaller values of Ω m and S 8 compared to the standard Λ CDM model, as well as a smaller value of the scalar spectrum index n s.