Appendix

To second order the metric (3.10) can be rewritten as gµν =

−1−2ϕ−2ϕ² aj+ 2ϕaj

ai+ 2ϕai δij−2ϕ(δij +ςij) + 2ϕ²δij +ςij −aiaj

, (3.48) whereγij ≡δij +ςij (exact). It is also useful to have the form of the inverse metric

g^µν =

−e^−2ϕ(1−e^4ϕγ^ija_ia_j) e^2ϕa^j e^2ϕaⁱ e^2φγ^ij

. (3.49)

To second order one has g^µν ≃

−1 + 2ϕ−2ϕ²+δijaiaj aj+ 2ϕaj−ςjkak

a_i+ 2ϕa_i−ς_ika_k δ_ij + 2ϕδ_ij −ς_ij + 2ϕ(ϕδ_ij −ς_ij)

. (3.50)

100 3.6 Appendix The relevant integrals for computing Feynman diagrams like the one represented in fig. 3.4 (see for instance [124] and [117]) are

Z d³k

The integral relevant for fig. 3.5 is Z d³k

and to reconstruct the metric out of the effective EMT we used Z d³q

The relevant integrals for computing Feynman diagrams like the one represented in fig. 3.6 are (see again [124])

Z d^2+ǫk Other useful formulas to anti-Fourier transform the string effective EMT at next-to-leading order, are where a disk of radius q_ǫ around the origin has been cut out of the integral. Moreover

Z 2π 0

e^ixcosθdθ= 2πJ0(x), (3.58)

0

effective EMT the following integral Z

The effective EMT (3.44) in coordinate space is T_ij^(I)(x) = −4

where r denotes the distance to the string in the transverse two-dimensional space.

HereC^′ denotes theq-independent part of the quantity defined in text in (3.39).

To explicitly check conservation in the Fourier space of the string effective EMT (3.44), let us write down the conservation of the EMT in q-space, keeping only the components transverse to the string world-sheet:

∂^aTab(q) = which has an extra piece with respect to (3.47). Let us restrict for simplicity to the total derivative term and let us fix the index b = 2. To make sense of the integral we have to integrate over a region Ω obtained by cutting out of the plane the the two regions r < rǫ and r > R, and we will finally (but after taking the other limits first) let r_ǫ →0 andR → ∞.

By changing coordinates fromy, z toρ, θ according toy =rcosθ,z =rsinθand using the Green-Gauss theorem one obtains

− π

The first integral is clearly vanishing in the limit R→ ∞. Expanding the exponential in the second integral, taking the limit rǫ → 0 and finally plugging this result into (3.61), one has

102 3.6 Appendix

Extracting the three- and four-graviton vertices from binary pulsars and coalescing binaries

In Sec. 1.1 I have discussed the confrontation of scalar-tensor theories with GR by means of tests of relativistic gravity. This confrontation followed the studies of DEF and is summarized in Fig. 1.8, where future GW detections seem not to be competitive with binary pulsar observations already at hand. Surprisingly, even the detection of a neutron star-black hole system with the most sensitive space-based interferometer LISA is found to be less constraining than the electromagnetic observation of the same system performed by means of pulsar timing. In the course of presenting these conclusions by DEF I have pointed out how they strongly depend on the assumptions made. Notably, DEF parametrized a certain class of scalar-tensor theories by considering the couplings of matter to the scalar field only up to second order in perturbation. Within this choice, DEF confronted different tests of gravity in the radiative regime only to the lowest orders probed by binary pulsars, i.e. up to terms of O(v/c)⁵ in the dynamics.

On the contrary, GW detections will probe gravity up to O(v/c)¹²! Therefore, there is the necessity to envisage testing frameworks for gravity in the dynamical regime probed by GWs. In Sec. 1.4, I have described one such test where, extending the PPN approach to the radiative regime, one applies a phenomenological attitude towards the phasing coefficients of the GW formula. Another example of extended PPN tests is the one I present in the present chapter. This is a study that I have conducted with my supervisor and the rest of his group at the University of Geneva: inspired by the NRGR method of ref. [6], that I have described in Chap. 2, we constrained GR non-linearities from a field-theoretical perspective [125]. Here I report our publication, while in the following chapter I discuss more details in the context of a follow-up of our work.

103

104 4.1 Introduction

PHYSICAL REVIEW D 80, 124035 (2009)

Extracting the three- and four-graviton vertices from binary pulsars and coalescing binaries

Umberto Cannella, Stefano Foffa, Michele Maggiore, Hillary Sanctuary and Riccardo Sturani

abstract

Using a formulation of the post-Newtonian expansion in terms of Feynman graphs, we discuss how various tests of General Relativity (GR) can be translated into meas-urement of the three- and four-graviton vertices. In problems involving only the con-servative dynamics of a system, a deviation of the three-graviton vertex from the GR prediction is equivalent, to lowest order, to the introduction of the parameter βPPN in the parametrized post-Newtonian formalism, and its strongest bound comes from lunar laser ranging, which measures it at the 0.02% level. Deviation of the three-graviton vertex from the GR prediction, however, also affects the radiative sector of the theory.

We show that the timing of the Hulse-Taylor binary pulsar provides a bound on the deviation of the three-graviton vertex from the GR prediction at the 0.1% level. For coalescing binaries at interferometers we find that, because of degeneracies with other parameters in the template such as mass and spin, the effects of modified three- and four-graviton vertices is just to induce an error in the determination of these paramet-ers and, at least in the restricted PN approximation, it is not possible to use coalescing binaries for constraining deviations of the vertices from the GR prediction.

DOI: 10.1103/PhysRevD.80.124035 PACSnumbers: 04.30.–w, 04.80.Cc, 04.80.Nn

4.1 Introduction

Binary pulsars, such as the Hulse-Taylor [13] and the double pulsar [14], are won-derful laboratories for testing General Relativity (GR). They have given the first ex-perimental confirmation of the existence of gravitational radiation [126, 60], provide stringent tests of GR and allow for comparison with alternative theories of gravity, such as scalar-tensor theories [127, 51, 128, 129, 37, 36, 11, 33, 12, 34] (see [130, 19, 90]

for reviews). Another very sensitive probe of the non-linearities of GR is given by the gravitational wave (GW) emission during the last stages of the coalescence of compact

ising signals for GW interferometers such as LIGO [38] and Virgo [39], especially in their advanced stage, and for the space interferometer LISA [40]. Various investigations have been devoted to the possibility of using the observation of coalescing binaries at GW interferometers to probe non-linear aspects of GR [37, 36, 103, 131, 76, 132, 133, 15, 16].

Compact binary systems probe both the radiative sector of the theory, through the emission of gravitational radiation, and the non-linearities intrinsic to GR which are already present in the conservative part of the Lagrangian. In a field-theoretical language, these non-linearities can be traced to the non-Abelian vertices of the theory, such as the three- and four-graviton vertices. It is therefore natural to ask whether from binary pulsars or from future observations of coalescing binaries at interferometers one can extract a measurement of these vertices, much in the same spirit in which the triple and quartic gauge boson couplings have been measured at LEP2 and at the Tevatron [134, 135, 136, 137].

In this paper we tackle this question. The organization of the paper is as follows.

In Section 4.2 we discuss how to “tag” the contribution of the three- and four-graviton vertices to various observables in a consistent and gauge-invariant manner, and we com-pare it with other approaches, such as the parametrized post-Newtonian (PPN) formal-ism [19]. In particular, we find that the introduction of a modified three-graviton vertex corresponds – in the conservative sector of the theory and at first Post-Newtonian or-der (1PN) – to the introduction of a value for the PPN parameter β_PPN different from the value βPPN = 1 of GR. However, a modified three-graviton vertex also affects the radiative sector of the theory, which is not the case for the PPN parameter βPPN. We also discuss subtle issues related to the possible breaking of gauge invariance which takes place when one modifies the vertices of the theory. In Section 4.3 we present our computations with modified vertices, and in Section 4.4 we compare these computa-tions with experimental results obtained from the timing of binary pulsars and with what can be expected from the detection of gravitational waves (GWs) at ground-based interferometers or with the space interferometer LISA. Section 4.5 contains our conclusions.