Degrees of freedom of quadratic scalar-nonmetricity theory

TL;DR

This work analyzes the degrees of freedom in quadratic scalar-nonmetricity (QSN) theory, a scalar-tensor extension of quadratic nonmetricity, by combining an ADM-based perturbative approach with a nonperturbative Dirac–Bergmann constraint analysis in the coincident gauge. The authors classify models into 13 coefficient-cases, separating those with propagating tensor modes from those without, and study three representative cases (II, V, VI) in detail. They find Case II propagates 10 DOFs both perturbatively and nonperturbatively, while Cases V and VI yield 8 DOFs nonperturbatively but only 6 and 5 modes at linear order on FLRW backgrounds, respectively, signaling strong coupling on cosmological backgrounds. The results illuminate how linear perturbations can miss nonperturbative modes and clarify the DOF content of STEGR and f(Q) within the QSN framework, with implications for cosmology and gravitational wave phenomenology.

Abstract

We study the number of degrees of freedom (DOFs) in quadratic scalar-nonmetricity (QSN) theory, whose Lagrangian is the linear combination of five quadratic nonmetricity invariants with coefficients depending on a dynamical scalar field. Working in the coincident gauge, we perform the Arnowitt-Deser-Misner decomposition and classify QSN models into distinct cases according to the numbers of their primary constraints. For cases that are physically viable in the sense that both a consistent cosmological background and tensor gravitational waves exist, we count the number of degrees of freedom based on two approaches. First we investigate the linear cosmological perturbations around an FLRW background. Then we perform a Dirac-Bergmann Hamiltonian constraint analysis to count the number of DOFs at the nonperturbative level. We focus on three representative cases. In case II, both the perturbative and nonperturbative approaches yield the same result, which indicates that the theory propagates 10 degrees of freedom. In contrast, in cases V and VI, the Hamiltonian analysis yields 8 degrees of freedom, while only 6 and 5 modes are visible at linear order in perturbations, respectively. This indicates that additional modes are strongly coupled on cosmological backgrounds.