Acceleration Explains Galactic Curves and Cosmic Expansion

Scientists are investigating how particle spectra experience additional broadening caused by both local, short-term acceleration and the background of deSitter spacetime. M. J. Luo, working independently, details this effect and proposes it offers a straightforward explanation for the acceleration interpolation relation needed to modify particle kinematics within the Modified Inertial interpretation of Modified Newtonian Dynamics (MOND). This research represents a significant step towards unifying our understanding of accelerated universal expansion and anomalies observed in galactic rotation curves, extending the well-known Unruh effect, typically associated with long-term, uniform acceleration, to encompass scenarios involving local, non-uniform acceleration. Importantly, this modified kinematic interpretation of MOND requires an extension of the equivalence principle, moving beyond first-order moments to consider fluctuations at the level of second-order moments, and the paper discusses the implications for reference frames and gravity.

Scientists have discovered a subtle connection between the expansion of the universe and the behaviour of galaxies. Their work suggests a unified explanation for these phenomena, potentially removing the need to invoke mysterious dark matter. This breakthrough links acceleration and spacetime geometry, offering a simpler picture of the cosmos. This effect can be regarded as a generalisation of the thermal blackbody spectrum generated by the Unruh effect, which arises from long-time uniform acceleration in a flat background, to the scenario of local short-time non-uniform acceleration.

This effect offers a unified framework for understanding the accelerated expansion of the universe and the anomalies in galactic rotation curves or radial acceleration. This manifests in several ways. Firstly, the interpolation function used in MOND to describe the transition from Newtonian dynamics to the deep-MOND regime cannot be derived from first principles.

Secondly, the interpolation function relies on the ratio between the system’s acceleration and a critical acceleration constant, and since acceleration is coordinate-dependent, MOND is not a covariant theory. In attempts like Bimetric MOND, Milgrom has sought to construct a concept analogous to a covariant acceleration vector by treating acceleration as the acceleration of the geometry of spacetime relative to an auxiliary geometry.

The behaviour of MOND can be interpreted in two ways: modifications of the gravitational theory or modifications of Newton’s second law (Modified Inertial). Modified Inertial approaches can preserve the elegance of covariant gravitational theories, and the concept of acceleration appears more naturally in kinematics than in metric-based gravitational theories.

Milgrom proposed an explanation for modified inertial behaviour based on the interaction between the Unruh temperature effect and the temperature effect of the background de Sitter spacetime. These coincidences have led to speculation that the influence of the deSitter cosmological background on locally accelerated systems may be the cause of MOND-like modifications.

However, these speculations still lack a solid physical foundation and mathematical proof. Known effects of deSitter background expansion on local gravitational systems seem negligible, and the static solution of Einstein’s gravity with a cosmological constant fails to produce the gravitational potential required by MOND. A more complete theory must first generalise the Unruh effect to non-flat, non-long-time non-uniform acceleration scenarios.

Second, it must explain how changes in the quantum spectrum of test particles reflect their classical equations of motion. Although MOND still faces unresolved issues, it serves as a phenomenological touchstone for testing new ideas. In previous work, the Gabor transform method, a form of Fourier transform with a short-time window, was employed to extract spectral information from a monochromatic wave function undergoing local short-time acceleration.

This revealed a slight Gaussian broadening of the spectrum, directly proportional to the square of the instantaneous acceleration. This short-time acceleration-induced broadening can be regarded as a local generalisation of the Unruh effect. This effect can also explain the influence of local non-uniform acceleration on the kinematics of test particles through the quantum equivalence principle.

It provides a more robust explanation for the kinematic modifications to acceleration in a deSitter background proposed by Milgrom. This paper derives the effect of spectral broadening due to local short-time acceleration using a general coordinate transformation method. The analysis was extended to a deSitter background to compute corrections to particle actions and the equivalent comoving background acceleration.

This parallel analysis enabled a direct comparison between local acceleration and the large-scale acceleration inherent in deSitter geometry. The structure of this paper details the calculation of corrections to the action and broadening effects, the computation of corrections in a deSitter background, and the derivation of the acceleration interpolation relation for MOND. The research begins with a general relativistic point particle action and accounts for acceleration through a coordinate transformation, specifically xμ → xμ + aμδs², where aμ represents the non-relativistic acceleration and δs denotes a very short proper time interval.

Gabor transforms and coordinate transformations reveal spectral broadening from short-time acceleration

A detailed analysis of particle spectra formed the core of this work, employing the Gabor transform method to meticulously extract spectral information. This technique was applied to monochromatic wave functions subjected to local, short-time acceleration within a precisely defined time window, allowing for the observation of subtle spectral changes.

The Gabor transform’s ability to resolve time-frequency characteristics proved crucial in identifying the broadening of spectra induced by acceleration, beyond the inherent Gaussian broadening expected in standard quantum mechanics. Following initial observations, a more rigorous general coordinate transformation method was implemented to calculate corrections to the action and the resulting broadening effects on test particles.

Specifically, particle coordinates were transformed via the equation xμ → xμ + aμδs². This approach allowed for a precise determination of how acceleration alters the particle’s trajectory and, consequently, its spectral characteristics. To further validate the findings, calculations were extended to a deSitter background. The same coordinate transformation method was used to compute corrections to particle actions within this expanding spacetime, alongside the determination of equivalent comoving background acceleration. This parallel analysis enabled a direct comparison between the effects of local acceleration and the large-scale acceleration inherent in deSitter geometry.

Cosmological expansion and galactic dynamics unified through spectral line broadening

Spectral line broadening attributable to local, short-time acceleration combined with a deSitter spacetime background reveals an extra broadening quantified as approximately −1/4q0H2 0, where H0 represents the Hubble constant. The research shows that this broadening is indistinguishable from that caused by general acceleration, suggesting a unified framework for understanding both cosmological expansion and galactic dynamics.