Effects of short-distance modifications to general relativity in spinning binary systems

We investigate the possibility of testing short-distance modifications to general relativity via higher-curvature terms in the fundamental gravity Lagrangian by analyzing their impact on observations of spinning astronomical binary systems. By using effective field theory methods applied to the two-body problem, generic lower bounds on the short-distance scale accompanying high-curvature terms can be set. In particular, we extend known results by deriving spin-dependent effects in binding energy, radiation emission process, and spin precession equations in binary systems, which are the fundamental ingredients to observe spin-dependent effects in gravitational wave detections from compact binary coalescences and spin precession in double binary pulsars.

Figure 1

Diagrams representing the Newtonian potential (first on the left) and the leading corrections from the ${\mathrm{Riemann}}^4$ terms in the fundamental Lagrangian in Eq. (1). The last diagram must be supplemented with its mirror image under $1\leftrightarrow 2$.

Figure 2

Diagrams describing the LO radiative process from a spinless binary system. The first one must be supplemented by its mirror image under $1\leftrightarrow 2$. Green wavy lines represent radiation, and blue dotted lines are $\varphi$ longitudinal modes.

Figure 3

Diagrams representing the LO spin-orbit potential in GR. Diagrams must be supplemented with their $1\leftrightarrow 2$ mirror image. The dashed red line represent an $A_i$-polarized longitudinal mode, and blue dotted lines are $\varphi$ modes.

Figure 4

Diagrams determining (part of) the LO spin terms in gravitational radiation emission. The first one contributes at LO to the magnetic quadrupole, the remaining ones to the electric quadrupole. Additional contribution comes from the expression of the radiative multipoles in terms of the energy-momentum tensor; see Appendix pp1.

Figure 5

Diagrams representing the leading corrections to the nonspinning potential from ${\mathcal{C}}^2$ interactions (blue dotted lines represent $\varphi$ propagators). The cross diagram vanishes. The $a$ diagram must be complemented by its mirror image under particle exchange.

Figure 6

Diagrams representing the leading corrections to the linear-in-spin potential from ${\mathcal{C}}^2$ interactions. Blue dotted lines represent $\varphi$ propagators, red dashed lines for $A$, and green wavy lines for $\sigma$. All cross diagrams vanish, as well as the first (diagram with the field $A$ coupling to $S_1$) and third ($\sigma$ coupling to $S_1$) diagrams on the second line. Diagrams obtained by exchanging particles 1 and 2 should be added.

Figure 7

Leading ${\mathcal{C}}^2$ correction to the emission process for nonspinning sources. Blue dotted lines are longitudinal modes exchanged between the sources, and the green wavy line the radiative gravitational mode.

Figure 8

Diagrams representing the leading magnetic contribution to the radiation emission process from ${\mathcal{C}}^2$ interaction, generated by bulk terms $\sim ({\partial }_m{\partial }_n\varphi {)}^2\times {\partial }_i{\partial }_j{A}_k{R}^{0ijk}$. Blue dotted lines represent $\varphi$ propagators, red dashed lines for $A_i$, and green wavy lines for ${\sigma }_{ij}$.

Figure 9

Diagrams representing the leading electric contribution to the radiation emission process from ${\mathcal{C}}^2$ interaction, generated by bulk terms $\sim ({\partial }_m{\partial }_n\varphi {)}^2{\partial }_i{\partial }_j\varphi (R_{0i0j}+{\delta }^{kl}{R}_{ikjl})$. Blue dotted lines represent $\varphi$ propagators, and green wavy lines for $\sigma$.

Figure 10

Diagrams representing the leading corrections to the spinning potential from $\mathcal{C}\tilde{\mathcal{C}}$ interactions (blue dotted lines represent $\varphi$ propagators, red dashed lines $A_i$ ones) at 2PN order with respect to leading GR behavior. Only the first diagram is nonvanishing.

Figure 11

Diagrams giving the LO effect of $\mathcal{C}\tilde{\mathcal{C}}$ on the radiation emission for the nonspinning case. At leading order, diagram (a) contributes to the magnetic coupling and (b) and (c) to the electric part.

Figure 12

Diagrams giving the LO effect of $\mathcal{C}\tilde{\mathcal{C}}$ on the radiation emission for the spinning case.

Figure 13

Diagrams representing the correction to the spinning potential mediated by the ${\tilde{\mathcal{C}}}^2$ bulk interaction. Diagrams must be supplemented with their mirror images under $1\leftrightarrow 2$. The three diagrams in the first line vanish.

Figure 14

Diagrams representing the correction to the linear-in-spin electric radiation emission mediated by the ${\tilde{\mathcal{C}}}^2$ bulk interaction. Diagrams must be supplemented with their mirror images under $1\leftrightarrow 2$.

Figure 15

Diagrams representing the correction to the linear-in-spin magnetic radiation emission mediated by the ${\tilde{\mathcal{C}}}^2$ bulk interaction. Diagrams must be supplemented with their mirror images under $1\leftrightarrow 2$.