# Secular gravity gradients in non-dynamical Chern–Simons modified gravity for satellite gradiometry measurements

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## Abstract

With continuous advances in related technologies, precision tests of modern gravitational theories with orbiting gradiometers becomes feasible, which may naturally be incorporated into future satellite gravity missions. In this work, we derive, at the post-Newtonian level, the new secular gravity gradient signals from the non-dynamical Chern–Simons modified gravity for satellite gradiometry measurements, which may be exploited to improve the constraints on the mass scale \(M_{CS}\) or the corresponding length scale \({\dot{\theta }}\) of the theory with future missions. For orbiting superconducting gradiometers, a bound \(M_{CS}\ge 10^{-7}\ \mathrm{eV}\) and \({\dot{\theta }} \le 1\ \mathrm{m}\) could in principle be obtained, and for gradiometers with optical readout based on the similar technologies established in the LISA PathFinder mission, an even stronger bound \(M_{CS}\ge 10^{-6}\)–\(10^{-5}\ \mathrm{eV}\) and \({\dot{\theta }} \le 10^{-1}\)–\(10^{-2} \ \mathrm{m}\) might be expected.

## 1 Introduction

Among modifications of Einstein’s general relativity (GR), extensions to the Einstein–Hilbert action with second order curvature terms are of particular interest, which may arise from the full, but still lacking, quantum theory of gravity [1]. The Chern–Simons (CS) modified gravity [2, 3, 4, 5, 6] belongs to such extensions of GR, which has physical roots in particle physics and string theory. In particle physics, the CS modification or the Pontryagin term \(^{\star }RR\) is known to be related to the chiral current anomaly that caused by spacetime curvature [7, 8], which may also serve as a possible candidate that sources the baryon asymmetry of the Universe [9] through gravi-leptogenesis [10]. In string theory, the CS modification emerges as an anomaly-canceling term through the Green–Schwarz mechanism [11]. More interestingly, the CS extension to GR may provide us insights into the physics of possible parity-violations in gravitation, that includes effects like amplitude birefringent gravitational waves [5, 12, 13], different gravito-magnetic (GM) sectors [12, 13, 14], and etc.. Therefore, experimental tests of the CS modified gravity and the resulted constraints are of importance.

The CS modified gravity is considered as an effective theory, that the ultra-violet modifications to gravitation and their possible observable effects are to be studied in more fundamental and sophisticated theories such as string theory or loop quantum gravity. Theoretical studies had shown that in CS modified gravity the Lorentz symmetry can be satisfied [5, 15, 16], and, up to now, experimental constraints on CS gravity are mainly from astrophysical observations and Solar system tests. Of course, it is natural to expect that stronger bounds on CS gravity might come from particle physics because of the connections between these theories. In this work, we will focus on the tests, with orbiting gradiometers, of the non-dynamical formulation of the CS modified gravity [5], where the coupling field or the deformation parameter \(\theta \) is externally prescribed (one notices that the arbitrariness in \(\theta \) could not be completely removed in the dynamical formulation due to the different choices of the potential \(V(\theta )\)). The first constraint [14] on the time derivative of the coupling scalar \({\dot{\theta }}\) and the corresponding mass scale \(M_{CS}\sim 1/{\dot{\theta }}\) of the non-dynamical CS gravity was obtained based on the observations from the LAGEOS I, II [17, 18] and the Gravity Probe-B [19] missions, which had set \(M_{CS} \ge 10^{-13}\ \mathrm{eV}\) and \({\dot{\theta }} \le 10^{6} \ \mathrm{m}\), and a stronger bounds \(M_{CS}\ge 4.7\times 10^{-10}\ \mathrm{eV}\) and \({\dot{\theta }} \le 0.4\times 10^{3} \ \mathrm{m}\) (been revised in [20]) was obtained based on the data from double binary pulsars [21].

Theoretical studies of testing relativistic gravitational theories with orbiting gravity gradiometers in space were firstly carried out in 1980s [22, 23, 24], and such measurement schemes could be naturally incorporated into future satellite gradiometry missions or missions that carrying high sensitive gradiometer as one of the key payloads. For the baseline design of high sensitive gravity gradiometers in micro-gravity or zero-g environment in space, such as electrostatic or superconducting ones, one generally has pairs of proof masses aligned along each of the measurement axes, and a combinations of strategies of proof mass disturbances isolation, proof mass position sensing and control is employed, see [25, 26, 27, 28] for reviews. The proof masses are generally enclosed within sensor cages or housings, vacuum maintenances and other shielding devices, and, with such setup, fluctuations forces subjected to proof masses are to be reduced or isolated as much as possible. The relative motions or accelerations between the “free-falling” proof masses (with respect to certain noise level) in space will give rise to measurements of the tidal matrix from spacetime curvature \(R_{0i0}^{\ \ \ \ j}\) along certain orbits (for Newtonian limits, \(R_{0i0}^{\ \ \ \ j}\) reduces to \(\partial _i\partial _j U\) with *U* the Newtonian gravitational potential). For electrostatic and superconducting gradiometers, the difference between the compensating forces that restoring the proof masses to their nominal positions can be used as the direct readouts of the tidal accelerations. The GOCE satellite, launched in March 2009, carried an electrostatic gravity gradiometer containing six proof masses in three pairs to map out the geopotential of Earth, whose sensitive had reached \(10\ \mathrm{mE/Hz}^{1/2}\) in the frequency band of \(5\sim 100\ \mathrm{mHz}\) [27]. With the continuous advances, the multi-axis superconducting gravity gradiometer under the development could reach the sensitivity about \(10^{-2}\ \mathrm{mE/Hz}^{1/2}\) in the band around \(1\ \mathrm{mHz}\) in space [25, 29]. As an alternative optical readout method, the relative motions between proof masses as integrations of tidal accelerations can also be precisely measured by onboard laser interferometers^{1}. The LISA PathFinder (LPF) mission [30, 31], which can be view as a demonstration of an one dimensional optical gradiometer with the resolution of the onboard laser interferometer better than \(9\ \mathrm{pm}/\sqrt{\mathrm{Hz}}\) in the *mHz* band, had even reached the noise floor of \(10^{-3}\)–\(10^{-4} \ \mathrm{mE/Hz}^{1/2}\). Tests of relativistic gravitational theories including the CS modified gravity with satellite gradiometry now becomes more and more feasible, and for CS gravity a preliminary measurement scheme had been studied in [32, 33]. It was firstly noticed by Mashhoon and Theiss [22, 34, 35, 36] that along orbit motions relativistic secular tidal effects that growing with time may exist (known as the Mashhoon–Theiss anomaly), which would greatly improve the measurement accuracy of relativistic tidal components. Recently, the physical mechanism behind such secular tidal effects had been studied and explained in [37, 38]. From the post-Newtonian (PN) point of view, the difference between the relativistic precession of the local free-falling frame (or the parallel transported measurement axes) and the orbit plane with respect to the sidereal frame will produce modulations of Newtonian tidal forces along certain axes and then gives rise to periodic secular tidal signals [37, 39]. Back to the tests of the CS modified gravity, in the far field expansion of the non-dynamical theory, the CS gravity will add modifications to the GM sector of GR [12, 13], and therefore will give rise to new secular tidal effects that could be read out precisely along certain measurement axes of an orbiting gradiometer. In this work, we derive, at the PN level, the new secular tidal tensor from the non-dynamical CS modified gravity under the local Earth pointing frame along a relativistic polar and nearly circular orbit. For (possible) future experiments, we give the estimations of the bound on the characteristic mass scale \(M_{CS}\) that could be drawn from such a measurement scheme.

## 2 Models and settings

*M*and angular momentum \(\mathbf {J}\). The geocentric inertial coordinates system \(\{t,\ x^i\}\) is defined as follows, that one of its bases \(\frac{\partial }{\partial x^3}\) is parallel to the direction of \(\mathbf {J}\) and the coordinate time

*t*is measured in asymptotically flat regions. For an orbiting proof mass or satellite, we have the PN order relations

## 3 Reference orbit and local tetrad

*a*denotes the orbit radius, \(\varPsi =\omega \tau \) the true anomaly, \(\omega \) is the mean angular frequency with respect to the proper time \(\tau \) along the orbit and \(\varOmega \) is the longitude of ascending node with initial value \(\varOmega (0)=0\). The precession rate of the node \({\dot{\varOmega }}={\dot{\varOmega }}_{GR}+{\dot{\varOmega }}_{CS}\), where the Lense–Thirring precession rate \({\dot{\varOmega }}_{GR}=\frac{2GJ}{a^3}\) [42] and the correction from the non-dynamical CS modified gravity had been worked out in [14] as

*R*is the averaged radius of Earth, \(j_l(x)\) and \(y_l(x)\) are spherical Bessel functions of the first and second kind, and \(\chi \) was a new PN parameter introduced in [12, 13] which can be related to the mass scale as \(\chi =\frac{4}{a M_{CS}}\). For polar circular orbit, the GM force in CS modified gravity generated by spherical sources [14] will also change slightly the orbital eccentricity to \(e\sim \chi {\mathscr {O}}(\epsilon ^2)\), which is too small to be relevant to secular tidal effects at the PN level. Therefore, in this work the small eccentricity is ignored, and its effects together with other orbital perturbations in satellite gradiometry, such as those from geopotential harmonics, can be found in [27, 43].

## 4 Secular gradient observables

## 5 Conclusions

*N*in \(s^{CS}\) will be \(3.5\times 10^4\), and the secular signal will reach about \(4.3\chi \ \mathrm{mE}\). With proper data analysis methods employed, the total signal-to-noise ratio can be further amplified by a factor of the square root of the total cycles \(\sqrt{N}\). Therefore, for superconducting gradiometers with potential sensitivity better than \(10^{-2}\ \mathrm{mE}/\sqrt{\mathrm{Hz}}\) in low frequency band near \(0.1\ \mathrm{mHz}\) [29], a rather strong constraint on the CS mass scale of the non-dynamical theory may in principle be obtained as

## Footnotes

- 1.
Updated progresses and references of the research of optical gradiometer supported by the geo-Q research project can be found in http://www.geoq.uni-hannover.de/a07.html. http://www.geoq.uni-hannover.de/b07.html

## Notes

### Acknowledgements

The authors thank Professor Nicolás Yunes for discussions. The NSFC Grands no. 11571342, Natural Science Basic Research Plan in Shaanxi Province of China no. 2017JQ1028 and National Key R&D Program of China no. 2017YFC0602202 are acknowledged. This work is also supported by the State Key Laboratory of applied optics and the State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences.

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