Effect of LISA Pathfinder spacecraft self-gravity on anomalous gravitational signals near the Sun-Earth saddle point predicted by quasilinear MOND

The possibility of sending the LISA Pathfinder spacecraft through the Sun-Earth saddle point following its nominal mission around L1 has now been studied for a few years. The principal motivation for doing so is to search for anomalous gravity gradients predicted by several alternative theories of gravity. In turn, these have originally been motivated by the dark matter problem, and predict deviations from General Relativity in regions of low acceleration. All signal estimates to date have ignored the presence of the spacecraft mass distribution and its self-gravity, on the basis that the gravitational field due to Sun and Earth is larger than that due to the spacecraft itself, for any realistic saddle point fly-by distances. In this paper, we show that at least for one of the theoretical frameworks, Quasilinear MOdified Newtonian Dynamics (QMOND), the presence of the local mass distribution cannot be ignored. Using simplified representations of the spacecraft mass distribution, we demonstrate that internal self-gravity, in particular internal gravitational gradients, can enhance the QMOND signals by more than 3 orders of magnitude. These preliminary results indicate that the parameter space accessible to LISA Pathfinder may be significantly larger than previously thought. We find further that the details of the matter distribution as well as of the trajectory can affect the expected signal shape, due to the coupling between internal and external gravitational fields and field gradients. We hope that this work will motivate a more comprehensive investigation of the effect, not just in QMOND, but also in the context of other theoretical frameworks.

Figure 1

Transition functions ($\nu (y)-1$) as a function of gravitational acceleration parameter $y=|g_N|/a_0$.

Figure 2

Evaluation of PDM density for the different transition functions as function of distance from center of sphere.

Figure 3

Discrete grid used to solve PDM density Eq. (1). PDM density is evaluated at black grid points. The transition function is evaluated at “intermediate” red grid points.

Figure 4

PDM density plots around Sun-Earth SP, in the absence of any other baryonic matter. Cubic grid of side length 1000 km with grid spacing 5 km. (a) TeVeS-like transition, (b) linear transition, (c) quadratic transition.

Figure 5

PDM densities along a trajectory parallel to the Sun-Earth line, missing the SP by 50 km. (a) our results, (b) results from [10].

Figure 6

Simple model used to investigate effect of grid size and resolution on local PDM density and differential acceleration.

Figure 7

First mass model: LISA Pathfinder test masses.

Figure 8

PDM distribution surrounding LPF TMs placed at saddle point. (a) TeVeS-like transition, (b) linear transition, (c) quadratic transition.

Figure 9

Logarithmic absolute PDM density distribution on $1{\mathrm{m}}^3$ grid surrounding LPF TMs placed at SP. (a) TeVeS-like transition, (b) linear transition, (c) quadratic transition.

Figure 10

Logarithmic PDM density distribution on ${(2\mathrm{m})}^3$ grid surrounding LPF TMs placed at SP. (a) TeVeS-like transition, (b) linear transition, (c) quadratic transition.

Figure 11

Absolute Newtonian gravitational acceleration and resulting PDM distribution for linear transition function at 2.5 km from the SP, along the Sun-Earth line in Sun direction.

Figure 12

Absolute Newtonian gravitational acceleration and resulting PDM distribution for linear transition function at 4.0 km from the SP, perpendicular to the Sun-Earth line.

Figure 13

Absolute Newtonian gravitational acceleration and resulting PDM distribution for linear transition function at 50 km from the SP, along the Sun-Earth line.

Figure 14

Reference trajectories past the SP used to compute effect of LPF self-gravity on anomalous differential acceleration between TMs.

Figure 15

Anomalous gravity gradients between LPF TMs on reference trajectories past the SP, for all three transition functionsfirst mass model.

Figure 16

Second mass model describing LISA Pathfinder test masses, optical bench, and internal balance masses.

Figure 17

PDM distribution surrounding LISA Pathfinder TMs placed at SP—second mass model. (a) TeVeS-like transition, (b) linear transition, (c) quadratic transition.

Figure 18

Anomalous gravity gradients between LPF TMs on reference trajectories past the SP, for all three transition functions—second mass model.

Figure 19

PDM density distribution for TeVeS-like transition function, at positions (a) $(x,z)=(0,10\mathrm{km})$ and (b) $(x,z)=(25\mathrm{km},0)$.

Figure 20

Absolute Newtonian gravitation across second mass model, at positions (a) $(x,z)=(0,10\mathrm{km})$ and (b) $(x,z)=(25\mathrm{km},0)$.

Figure 21

PDM density for linear transition function clustering inside and outside a single IBM, at positions (a) $(x,z)=(0,10\mathrm{km})$ and (b) $(x,z)=(25\mathrm{km},0)$.

Figure 22

Direct comparison of anomalous gravitational gradients predicted for the two mass models, for all three transition functions and both trajectories.