Implications of the Holst term in a $f\mathbf{(}R\mathbf{)}$ theory with torsion

We analyze a modified $f(R)$ theory of gravity in the Palatini formulation, when a Holst term endowed with a dynamical Immirzi field is included. We study the basic features of the model, especially in view of eliminating the torsion field via the Immirzi field and the scalar-tensor degrees of freedom of the $f(R)$ model. The main task of this study is the investigation of the morphology of the gravitational wave polarization when their coupling to a circle of test particles is considered. We first observe that the dynamics of the scalar mode of the $f(R)$ Lagrangian is frozen out, since its first order term identically vanishes. This allows a detailed characterization of the linearized theory, which outlines the emergence of a modified Newtonian potential in the static limit, and when time independence is relaxed a standard gravitational wave plus the scalar wave associated to the Immirzi field. Investigating the effect of the coupling of this scalar-tensor wave on a circle of test particles, we arrive to define two effective gravitational polarizations, corresponding to an equivalent phenomenological wave, whose morphology is anomalous with respect the standard case of general relativity. In fact, the particle circle suffers modifications as it was subjected to modified plus and cross modes, whose specific features depend on the model free parameters and are, in principle, detectable via a data analysis procedure.

Figure 1

The deformation induced by the polarization ${\beta }_B$ for $\omega t=\pi /2$ (left) and $\omega t=3\pi /2$ (right). The dotted circle represents the unperturbed test masses, whereas with the solid lines is shown the breathing mode. For the sake of clarity the effects due to $\delta \beta$ are magnified with respect to the actual dynamics.

Figure 2

The deformation induced by the polarization ${\beta }_R$ for $\omega t=\pi /2$ (left) and $\omega t=3\pi /2$ (right). The dotted circle represents the unperturbed test masses, whereas with the solid lines the ${\beta }_R$ mode is shown (in $\omega t=\pi ,2\pi$ we just recover the initial rest position); the rotation angle $\delta$ is also shown. For the sake of clarity the effects due to $\delta \beta$ are magnified with respect to the actual dynamics.

Figure 3

Effective plus polarization for ${\epsilon }_B\ll 1$, in $\omega t=\pi /2$ (left) and $\omega t=3\pi /2$ (right). The dotted circle represents the unperturbed test masses, whereas the dashed and solid lines are the standard and modified polarization, respectively. For the sake of clarity the effects due to $\delta \beta$ are magnified with respect to the actual dynamics.

Figure 4

Effective cross polarization for ${\epsilon }_R\ll 1$, in $\omega t=\pi /2$ (left) and $\omega t=3\pi /2$ (right). The dotted circle represents the unperturbed test masses, whereas the dashed and solid lines are the standard and modified polarization, respectively. For the sake of clarity the effects due to $\delta \beta$ are magnified with respect to the actual dynamics.