[Paper Review] Linear stability of the new relativistic theory of modified Newtonian dynamics

Constantinos Skordis, Tom Złośnik|arXiv (Cornell University)|Sep 29, 2021

Cosmology and Gravitation Theories38 references4 citations

TL;DR

This paper proposes a relativistic extension of Modified Newtonian Dynamics (MOND) that reduces to MOND in weak-field galactic regimes and to ΛCDM-like cosmology in the large-scale universe. Using second-order perturbation theory on a Minkowski background, it demonstrates linear stability by showing healthy massive vector and scalar modes with positive Hamiltonians, while identifying a non-propagating mode with potential instability at low wavenumbers (k < k* ≲ 1 Mpc⁻¹), relevant only on cosmological scales.

ABSTRACT

We have recently proposed a simple relativistic theory which reduces to Modified Newtonian Dynamics (MOND) for the weak-field quasistatic situations applied to galaxies, and to cosmological behaviour as in the $\Lambda$CDM model, yielding a realistic cosmology in line with observations. A key requirement of any such model is that Minkowski space is stable against linear perturbations. We expand the theory action to 2nd order in perturbations on a Minkowski background and show that it leads to healthy dispersion relations involving propagating massive modes in the vector and the scalar sector. We use Hamiltonian methods to eliminate constraints present, demonstrate that the massive modes have positive Hamiltonian and show that a non-propagating mode with a linear time dependence may have negative Hamiltonian for wavenumbers $k< k_*$ and positive otherwise. The scale $k_*$ is estimated to be around $\lesssim \mathrm{Mpc}^{-1}$ so that the low momenta instability may only play a role on cosmological scales.

Motivation & Objective

• To develop a relativistic theory that unifies MOND's success in galactic dynamics with a cosmologically viable ΛCDM-like behavior.
• To ensure Minkowski spacetime is stable under linear perturbations, a fundamental requirement for any viable relativistic gravity model.
• To analyze the Hamiltonian structure of the theory and verify the positivity of energy for propagating modes.
• To investigate the stability of non-propagating modes and assess the potential for cosmological-scale instabilities.

Proposed method

• Expand the theory's action to second order in perturbations around a Minkowski background.
• Apply Hamiltonian formalism to identify and eliminate constraints in the phase space formulation.
• Diagnose the dispersion relations of vector and scalar modes to confirm propagation and stability.
• Compute the Hamiltonian for each mode, verifying positivity for massive modes.
• Identify a non-propagating mode with linear time dependence and evaluate its Hamiltonian as a function of wavenumber k.
• Estimate the critical wavenumber k* ≲ 1 Mpc⁻¹ to assess the domain of potential instability.

Experimental results

Research questions

• RQ1 Does the proposed relativistic MOND theory exhibit linear stability on a Minkowski background?
• RQ2 Are the massive vector and scalar modes in the theory propagating with positive energy?
• RQ3 What is the behavior of the non-propagating mode with linear time dependence in terms of Hamiltonian sign?
• RQ4 At what wavenumber scale k* does the instability of the non-propagating mode become relevant?
• RQ5 Is the potential instability confined to cosmological scales, given the estimated value of k*?

Key findings

• The theory supports two propagating massive modes in the vector and scalar sectors, both with positive-definite Hamiltonians, indicating stability.
• The dispersion relations for the massive modes are healthy, showing no tachyonic or ghost-like behavior.
• A non-propagating mode with linear time dependence has a negative Hamiltonian for wavenumbers below k* ≲ 1 Mpc⁻¹.
• For wavenumbers above k*, the Hamiltonian of the non-propagating mode becomes positive, suggesting stability on smaller scales.
• The critical scale k* is estimated to be less than or equal to approximately 1 Mpc⁻¹, implying that any instability is confined to cosmological length scales.
• The overall theory is linearly stable on astrophysical scales, with potential issues only at very large cosmological wavelengths.