Case for testing modified Newtonian dynamics using LISA pathfinder

We quantify the potential for testing modified Newtonian dynamics (MOND) with Laser Interferometer Space Antenna pathfinder, should a saddle point fly-by be incorporated into the mission. We forecast the expected signal-to-noise ratio for a variety of instrument noise models and trajectories past the saddle. For standard theoretical parameters, the signal-to-noise ratio reaches middle to high double figures even with modest assumptions about instrument performance and saddle approach. Obvious concerns, like systematics arising from Laser Interferometer Space Antenna pathfinder self-gravity, or the Newtonian background, are examined and shown not to be a problem. We also investigate the impact of a negative observational result upon the free function determining the theory. We demonstrate that, if Newton’s gravitational constant is constrained to not be renormalized by more than a few percent, only contrived MONDian free functions would survive a negative result. There are exceptions, e.g. free functions not asymptoting to 1 in the Newtonian limit, but rather diverging or asymptoting to zero (depending on their mother relativistic MONDian theory). Finally, we scan the structure of all proposed relativistic MONDian theories, and classify them with regards to their nonrelativistic limit, finding three broad cases (with a few subcases depending on the form of the free function). It is appears that only the Einstein-Aether formulation, and the subcases where the free function does not asymptote to 1 in other theories, would survive a negative result without resorting to “designer” free functions.

Figure 1

Log plot of ratio between the MONDian and Newtonian forces, $F_{\varphi }/F_N$, against $z=(k/4\pi )|F_{\varphi }|/a_0$ (bottom axis) and $F_N/a_0$ (top axis). So that $F_N\sim F_{\varphi }$ when $F_{\varphi }\sim a_0$ (and so $z=\kappa /4\pi$; also $F_N\sim a_0$) and at the same time have $F_{\varphi }/F_N\sim \kappa /4\pi \ll 1$ in the Newtonian regime ($z\gg 1$, $F_N\to \infty$), we must trigger MONDian behavior in $\varphi$ at accelerations much larger than $a_0$ (when $z\sim 1$).

Figure 2

The transverse MOND stress signal $S_{yy}$ along the lines $y=25$, 100, and 400 km (top to bottom), for the Sun-Earth saddle, taking the effect of the Moon into account. The different lines represent different lunar phases: new Moon (thick, black, solid line), full Moon (thick, black, dashed line) and with the Moon appears 18° away from the Sun toward positive $y$ (thin, black, solid line). We also show, for the $y=25\mathrm{km}$ case, the Newtonian stresses (grey) rescaled by $\kappa /4\pi$ (see text for discussion).

Figure 3

The ASD of the MOND tidal stress signal for a trajectory with $b=50\mathrm{km}$ and $v=1.5\mathrm{km}{\mathrm{s}}^{-1}$, compared to the ASD of the basic noise model described in the text, assuming a baseline of $1.5\times {10}^{-14}{\mathrm{s}}^{-2}/\sqrt{\mathrm{Hz}}$. This scenario generates a SNR of 28.

Figure 4

Signal-to-noise ratio contours, for various impact parameters up to 600 km and base noise ASD. We set the spacecraft velocity at $1.5\mathrm{km}{\mathrm{s}}^{-1}$. Calamitous assumptions would still lead to SNR of 5. More optimistic ones ($b$ around 50 km, noise halfway up the scale) would lead to SNRs easily around 50.

Figure 5

Plot of SNR against satellite velocity for an impact parameter of 50 km and a baseline noise of $1.5\times {10}^{-14}{\mathrm{s}}^{-2}/\sqrt{\mathrm{Hz}}$. We note a broad peak around $v=2\mathrm{km}{\mathrm{s}}^{-1}$. Higher speeds shift the signal to higher temporal frequencies; however, the rough speeds of all trajectories in the Earth-Moon system are already optimal, given the noise properties of the instrument.

Figure 6

This figure replots Fig. 3, adding on the best- and worst-case scenarios for more realistic noise models, as at the time of writing. We have assumed a trajectory with the geometry described in the main text, with impact parameter of $b=50\mathrm{km}$ and velocity $v=1.5\mathrm{km}{\mathrm{s}}^{-1}$. We have also plotted the contribution of $\varphi$ to the Newtonian background.

Figure 7

The SNR for the improved noise models (best- and worst-case scenario) assuming $v=1.5\mathrm{km}{\mathrm{s}}^{-1}$ for a variety of impact parameters $b$.

Figure 8

ASD plot of the MONDian and Newtonian signal (multiplied by $\kappa /4\pi$), as compared to the noise ASD. We consider the effect of subtracting the Newtonian component in $\varphi$. This only affects very small and very large frequencies.

Figure 9

This plot illustrates the systematic effects that might result from an incorrect Newtonian subtraction. We consider the transverse tidal stresses felt in trajectories with impact parameters $b=100$, 500, 1000 km. We then subtract the DC constant Newtonian tidal stress contributions from $\varphi$ (top) and its full contribution (bottom). As we can see, an imperfect subtraction produces a spurious ramp in the stress.

Figure 10

Signal-to-noise ratio contours, for various impact parameters up to 600 km and base noise ASD, using the same templates as in Fig. 4 but with an exponential falloff in the model-dependent region $r>r_0$. As we can see, for impact parameters $b>400\mathrm{km}$ the SNR drops more sharply, but nothing changes very much for $b<400\mathrm{km}$.

Figure 11

Log plot of ratio between the MONDian and Newtonian forces, $F_{\varphi }/F_N$, against $z=(k/4\pi )|F_{\varphi }|/a_0$ (bottom axis) and $F_N/a_0$ (top axis). So that $F_N\sim F_{\varphi }$ when $F_{\varphi }\sim a_0$ (and so $z=\kappa /4\pi$; also $F_N\sim a_0$) and at the same time have $F_{\varphi }/F_N\sim \kappa /4\pi \ll 1$ in the Newtonian regime ($z\gg 1$, $F_N\to \infty$), we must trigger MONDian behavior in $\varphi$ at accelerations much larger than $a_0$. However, by allowing a sharper intermediate power law in $\mu$, the trigger acceleration $a_N^{\mathrm{trig}}$ may be smaller (in this illustration by a factor of 10).

Figure 12

The size of the MOND bubble as a function of the intermediate power $n$. It is easy to collapse to bubble by an order of magnitude (say to around 20 km) with $n\sim 2$. However, to make the bubble much smaller (say, on the order of a few kilometers), very dramatic intermediate powers would be required.

Figure 13

Contours of the power $n$ needed to obtain $\mathrm{SNR}=1$, for different noise levels and impact parameters up to $b=600\mathrm{km}$. For $n\ne 1$, the function is “unnatural.” We see that as soon as we plunge deep into the MOND bubble, a rather unnatural designer $\mu$ becomes necessary to accommodate a negative result.

Figure 14

Log-log plot of ratio between the MONDian and Newtonian forces, $F_{\varphi }/F_N$, against $z=(k/4\pi )|F_{\varphi }|/a_0$ for functions (55), with $n=0$, 1, 2. For $n=0$, we realize the divergent function (30) and as we can see there are only two regimes, corresponding to two power laws, with the ratio never flattening to a constant. For all other cases, we have a 3-piece function, with a falloff to the Newtonian regime which depends crucially on $n$.

Figure 15

Contours of the power $n$ for models (55) required in order to obtain $\mathrm{SNR}=1$, for different noise levels and impact parameters up to $b=600\mathrm{km}$. These are to be seen as an upper bound on parameter $n$, should a negative result from a LPF saddle experiment be found, for a given experimental setup.