[Paper Review] Which Fundamental Constants for CMB and BAO?

J. Rich|arXiv (Cornell University)|Mar 20, 2015

Cosmology and Gravitation Theories33 references3 citations

TL;DR

This paper demonstrates that cosmic microwave background (CMB) anisotropy spectra depend only on dimensionless combinations of fundamental constants, such as $\alpha^2 m_e/m_p$, $m_p/m_\chi$, and $Gm_\chi m_p/\hbar c$. By analyzing the CMB using a three-scale model, it shows that pre-recombination values of these combinations are consistent with present-day values at ~15% precision, offering model-independent constraints on time variations of fundamental constants when combined with low-redshift BAO and H₀ measurements.

ABSTRACT

We use the three-scale framework of Hu et al. to show how the Cosmic Microwave Background anisotropy spectrum depends on the fundamental constants. As expected, the spectrum depends only on \emph{dimensionless} combinations of the constants, and we emphasize the points that make this generally true for cosmological observables. Our analysis suggests that the CMB spectrum shape is mostly determined by $α^2m_e/m_p$ and the proton-CDM-particle mass ratio, $m_p/\mchi$. The distance to the last-scattering surface depends on $Gm_p\mchi/\hbar c$, so published CMB observational limits on time variations of the constants implicitly assume the time-independence of this quantity, as well as assuming a flat-\lcdm~cosmological model. On the other hand, low-redshift BAO, $H_0$ and baryon-mass-fraction measurements can be combined with the \emph{shape} of the CMB spectrum to give information that is largely independent of these assumptions. In particular we show that the pre-recombination values of $G\mchi^2/\hbar c$, $m_p/\mchi$ and $α^2m_e/m_p$ are equal to their present values at a precision of $\sim15\%$.

Motivation & Objective

• To clarify which fundamental constants govern the CMB anisotropy power spectrum and why only dimensionless combinations are physically meaningful.
• To resolve confusion in prior studies that report limits on time variations of dimensioned constants like $m_e$ or $G$, which are not directly measurable.
• To demonstrate that CMB observables depend only on dimensionless ratios, even when the underlying constants vary over time.
• To derive model-independent constraints on time variations of $Gm_\chi^2/\hbar c$, $m_p/m_\chi$, and $\alpha^2 m_e/m_p$ by combining CMB data with low-redshift BAO, $H_0$, and baryon-fraction measurements.

Proposed method

• Uses the three-scale framework of Hu et al. to analytically derive the dependence of CMB anisotropy features on fundamental constants.
• Identifies the key dimensionless combinations governing the CMB spectrum: $\alpha^2 m_e/m_p$, $m_p/m_\chi$, and $Gm_\chi m_p/\hbar c$, which determine the angular scale, peak height, and distance to last-scattering surface.
• Applies redshifting arguments to show that the inverse-length dimension of CMB features is transferred to the present-day CMB temperature $T_0$, making all observables depend only on dimensionless ratios.
• Combines CMB-derived dimensionless parameters with low-redshift measurements of $H_0$, BAO, and baryon mass fraction to break degeneracies and constrain time variations.
• Assesses the robustness of limits by comparing results from different observational probes: distance ladder, galaxy cluster dynamics, and cosmological deceleration.
• Emphasizes that only dimensionless combinations are physically measurable, even in cosmology, due to the dependence of SI standards on fundamental constants.

Experimental results

Research questions

• RQ1 Which combinations of fundamental constants determine the shape of the CMB anisotropy power spectrum?
• RQ2 Why are limits on time variations of dimensioned constants like $m_e$ or $G$ physically problematic, and what are the correct dimensionless alternatives?
• RQ3 How can low-redshift BAO, $H_0$, and baryon-fraction measurements be combined with CMB data to yield model-independent constraints on early-universe constant variations?
• RQ4 To what precision do the pre-recombination values of $Gm_\chi^2/\hbar c$, $m_p/m_\chi$, and $\alpha^2 m_e/m_p$ agree with their present values?
• RQ5 Why is the CMB distance to the last-scattering surface dependent on a dimensionless combination involving $Gm_\chi m_p/\hbar c$ and $T_0$?

Key findings

• The CMB anisotropy spectrum depends only on dimensionless combinations of fundamental constants, such as $\alpha^2 m_e/m_p$, $m_p/m_\chi$, and $Gm_\chi m_p/\hbar c$, confirming that only such combinations are physically meaningful.
• The shape of the CMB spectrum is primarily determined by $\alpha^2 m_e/m_p$ and $m_p/m_\chi$, while the distance to the last-scattering surface depends on $Gm_\chi m_p/\hbar c$.
• Published CMB limits on constant variations implicitly assume the time independence of $Gm_\chi m_p/\hbar c$ and a flat $\Lambda$CDM model, which introduces model dependence.
• When combined with low-redshift BAO, $H_0$, and baryon-fraction data, the CMB spectrum yields constraints on pre-recombination values of $Gm_\chi^2/\hbar c$, $m_p/m_\chi$, and $\alpha^2 m_e/m_p$ that are independent of these assumptions.
• The pre-recombination values of $Gm_\chi^2/\hbar c$, $m_p/m_\chi$, and $\alpha^2 m_e/m_p$ are consistent with their present values at a precision of approximately 15%.
• Limits on $m_e$ variation in prior studies are better interpreted as limits on $Gm_e^2/\hbar c$, but this is physically questionable due to the negligible role of electron gravity; the correct physical variables are the dimensionless combinations.