[Paper Review] Modified gravity without new degrees of freedom

Laurent Freidel|ArXiv.org|Dec 17, 2008

Cosmology and Gravitation Theories19 references15 citations

TL;DR

This paper demonstrates that modified gravity theories proposed by Bengtsson and Krasnov—previously considered non-metric and with no apparent metric interpretation—can be reformulated as standard metric gravity coupled to an auxiliary symmetric matrix field of unit determinant. The key result is that these theories propagate only two degrees of freedom (a single graviton) because the auxiliary field is non-dynamical at quadratic order, despite the action being non-local after integrating it out.

ABSTRACT

We show that the new type of "non-metric" gravity theories introduced independently by Bengtsson and Krasnov can in fact be reexpressed explicitely as a metrical theory coupled to an auxiliary field. We unravel why such theories possess only one propagating graviton by looking at the quadratic perturbation around a fixed solution. And we give a general construction principle with a new class of example of such modified gravity theories still possessing only two propagating degrees of freedom.

Motivation & Objective

• To resolve the apparent paradox that non-metric gravity theories can have only two degrees of freedom despite lacking a standard metric formulation.
• To show that these theories can be recast as a standard metric theory coupled to an auxiliary field with unit determinant.
• To clarify why the auxiliary field does not propagate by analyzing quadratic perturbations around a fixed background.
• To provide a general construction principle for new modified gravity theories that preserve two degrees of freedom while modifying the Einstein-Hilbert action.
• To extend previous work by Carroll on scalar auxiliary fields to the case of symmetric tensor auxiliary fields, preserving the graviton count.

Proposed method

• Reformulate SU(2) BF theory in terms of a metric $ g_{\mu\nu} $ and a symmetric, unimodular matrix field $ h^{ij} $, using the Urbantke construction.
• Express the BF action in terms of $ g_{\mu\nu} $ and $ h^{ij} $, showing that $ h^{ij} $ acts as an auxiliary field with determinant one.
• Introduce a potential $ V(h^{ij}) $ for the auxiliary field to generate modified gravity theories, preserving the two-d.o.f. structure.
• Analyze small metric fluctuations $ h_{\mu\nu} $ around a background solution, and perform a non-local field redefinition to decouple the non-propagating mode.
• Demonstrate that the auxiliary field $ b_{ab} $ (related to $ h^{ij} $) satisfies purely algebraic equations of motion at quadratic order, confirming it does not propagate.
• Construct a general class of theories by coupling a symmetric tensor field $ \pi_{\mu\nu} $ to gravity via $ \sqrt{g} G_{\mu\nu} (g^{\mu\nu} + \pi^{\mu\nu}) $, with a Pauli-Fierz kinetic term and a potential $ V(\pi) $, ensuring only one propagating graviton.

Experimental results

Research questions

• RQ1 Can non-metric gravity theories with only two degrees of freedom be reformulated as standard metric theories with auxiliary fields?
• RQ2 Why do these modified gravity theories still propagate only two degrees of freedom despite the absence of a metric interpretation?
• RQ3 What is the role of the auxiliary field in ensuring the absence of additional propagating modes at the quadratic level?
• RQ4 How can one systematically construct new modified gravity theories that preserve the two-d.o.f. structure while modifying the Einstein-Hilbert action?
• RQ5 Can the mechanism of non-propagating auxiliary fields be generalized beyond scalar fields to tensor fields, and what are the implications for the effective action?

Key findings

• The modified gravity theories of Bengtsson and Krasnov can be explicitly rewritten as a metric theory coupled to an auxiliary symmetric matrix field $ h^{ij} $ of unit determinant.
• The auxiliary field $ h^{ij} $ is non-dynamical at quadratic order: its equations of motion are purely algebraic, meaning it does not propagate any degrees of freedom.
• After integrating out the auxiliary field, the effective action for the metric becomes non-local, with a kinetic term for the transverse-traceless graviton of the form $ h^{TT}_{\mu\nu} \frac{\Box}{1 + \frac{\Box}{M^2}} h^{TT\mu\nu} $.
• The modification to the Newtonian potential includes a contact term $ V(x) \sim \frac{1}{\alpha} \delta^4(x) $, significant only at ultraviolet scales.
• The theory's two-d.o.f. structure is protected by a topological symmetry in the underlying BF theory, which removes 8 of the 10 components of the auxiliary field.
• A general construction principle is established: coupling gravity to a symmetric tensor field $ \pi_{\mu\nu} $ with a Pauli-Fierz kinetic term and a potential $ V(\pi) $, followed by a field redefinition $ \hat{h}_{\mu\nu} = h_{\mu\nu} + \pi_{\mu\nu} $, results in a theory with only one propagating graviton.