Testing Modified Gravity at the Saddle
Point with LISA Pathfinder
Christian Trenkel
Astrium Ltd, Stevenage, UK
On behalf of the (growing) “LPF and MOND” team
Overview

Motivation

Modified Gravity inspired by MOND

LISA Pathfinder

Testing Modified Gravity with LISA Pathfinder

Summary and Discussion
Q2C5 Cologne 9-12 October 2012
Motivation

If we simply apply the gravitational laws to the observable
matter outside the Solar System, things don’t make sense, eg…
Instead of
GM v 2
1



v
r2
r
r

This may be for the following reasons:
 There are other constituents (Dark Matter, Dark Energy)
 Laws of gravity need to be modified
 …both of the above
Q2C5 Cologne 9-12 October 2012
Motivation

Direct search for Dark Matter particles underway (dedicated
underground searches, LHC and predecessors) for decades now
– so far no confirmed detection

Problem with most proposed Gravity modifications: almost by
definition they predict significant deviations from GR only in
extreme environments outside the Solar System – and are
therefore hard to test directly!

Direct experimental evidence (for anything – Dark Matter or
Modified Gravity!) would clearly be of great value

It becomes increasingly hard to explore new parameter space –
we should grab every opportunity!
Q2C5 Cologne 9-12 October 2012
Modified Gravity – inspired by MOND

Newtonian dynamics are modified when total gravitational acceleration
approaches a0 ≈ 10-10ms-2 (Milgrom 1983):
a
F  m  a
 a0 


with
Many forms possible for
 ( a / a0 )
 a 
a  a 0      1
 a0 
 a  a
a  a 0     
 a0  a0
Newtonian
“MONDian”
:
 ( a / a0 ) 
a / a0
1  a / a0 
 ( a / a0 ) 




Can also be seen as modification of Newton’s law of Gravity:
a N  a0
a grav  a0
Q2C5 Cologne 9-12 October 2012
1  a / a  
2 1/ 2
0
Automatically describes flat rotation curves – on the outskirts of
galaxies we have:
1/ 4
 GM
a2
GM
v4
F m  2 
 v  
2
a0
r
a0 r
 a0
a / a0
GM
 a0 a N
2
r
Modified Gravity – inspired by MOND
Purely phenomenological, non-relativistic formula with no
underlying theory
 Surprisingly successful in describing many (MANY!) galactic
rotation curves without Dark Matter:

Less successful on extragalactic scales
 Bullet cluster still needs Dark Matter – but less

Q2C5 Cologne 9-12 October 2012
Modified Gravity – inspired by MOND

MOND respectability increased when a relativistic theory (TeVeS)
was developed with non-relativistic MONDian limit (Bekenstein
2004)

Since TeVeS, (many) other theories with MONDian non-relativistic
limit exist…

Classification of theories according to non-relativistic MONDian
origin into three types (Magueijo & Mozaffari 2012) :
 Type I: The total gravitational potential is sum of Newtonian
potential and new scalar field:
 grav   N  
The new scalar field is solution of modified Poisson equation
   
    

  4a0
 
   G
 
 
Q2C5 Cologne 9-12 October 2012
Modified Gravity – inspired by MOND
 Type II: Total potential as for type I, but the source driving the scalar
field now depends on the Newtonian potential:
     2  N


    

   4  a0
4
 
2


 N 




 Type III: The total gravitational potential is a single field which
satisfies a non-linear Poisson equation:
   grav
   ~
  a0
 


   4G
grav




Q2C5 Cologne 9-12 October 2012
Modified Gravity – inspired by MOND
All these theories incorporate a free interpolating function
describing transition between MONDian and Newtonian regimes
 Approximate comparison of some proposed functions (Galianni et
al 2012):

1.00E+01
1.00E+00
Deviation from Newtonian v(gn/a0)-1
1.00E-01
TEVES-like
1.00E-02
1.00E-03
1.00E-04
1.00E-05
Linear
1.00E-06
1.00E-07
1.00E-08
1.00E-09
1.00E-10
1.00E-11
Quadratic
1.00E-12
1.00E-13
1.00E-02
1.00E+00
1.00E+02
1.00E+04
gn/a0
1.00E+06
Q2C5 Cologne 9-12 October 2012
1.00E+08
1.00E+10
Modified Gravity – inspired by MOND
Problem: galactic rotation curves only tell us something up to
≈10a0
 Significant solar system constraints only reach down to ≈105a0
 How can we access the acceleration regime in between?

1.00E+01
1.00E+00
Deviation from Newtonian v(gn/a0)-1
1.00E-01
1.00E-02
1.00E-03
1.00E-04
?
1.00E-05
1.00E-06
1.00E-07
1.00E-08
1.00E-09
1.00E-10
1.00E-11
1.00E-12
1.00E-13
1.00E-02
1.00E+00
1.00E+02
1.00E+04
gn/a0
1.00E+06
Q2C5 Cologne 9-12 October 2012
1.00E+08
1.00E+10
Modified Gravity – inspired by MOND
A priori, direct tests of these theories using a spacecraft within the
Solar System appear poor:
Pioneer
Mercury
Earth
Saturn
Neptune
1.E-01
1.E-02
1.E-03
1.E-04
a [ms-2]

1.E-05
108
1.E-06
1.E-07
1.E-08
1.E-09
1.E-10
1.E-11
0.1
1
10
100
Distance from Sun [AU]
Q2C5 Cologne 9-12 October 2012
1000
10000
Modified Gravity – inspired by MOND
But: gravitational Saddle Points provide MONDian “habitats”
(Bekenstein & Magueijo 2006)
 Anomalous gravitational accelerations also result in
anomalous gravity gradients
 TeVeS predicts MOND gravity gradients ≥10-13s-2 within
elliptical bubble around SP – for certain interpolation
function

Sun
1532km
259000km
Earth
766km
Q2C5 Cologne 9-12 October 2012
LISA Pathfinder

Key LPF characteristics relevant for our
purposes:
 Mission deep into implementation
phase – launch a few years from now
 Instrument on board forms a highly
sensitive gravity gradiometer
 Nominal mission Lissajous orbit
around L1
 Micropropulsion system on board
with limited thrust and total dV
…more details later as required…
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

TeVeS and other theories of modified gravity inspired by MOND
predict potentially measureable anomalous gravity gradients in
macroscopic regions around gravitational Saddle Points

In a few years, LISA Pathfinder will be in orbit around L1, carrying
the most sensitive gravity gradiometer ever on-board, with peak
sensitivity in the mHz frequency band.
→ Obvious question: Can we put the two together?
Can we use LISA Pathfinder to look for MONDian effects?
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

Need to establish that:
(1) We can fly LISA Pathfinder through a gravitational Saddle
Points (SPs) in the Sun-Earth-Moon System, such that the
low acceleration regime can be explored
(2) LISA Pathfinder gradiometer has the sensitivity for a
meaningful test of at least some predicted anomalous
MONDian gradients
… while using LPF “as built” – no interference with nominal
mission allowed – this test would be done during mission
extension
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

In the Sun-Earth-Moon system, there are two SPs that could
potentially be targeted:
SP Trade-off:
→ Focus has been on Sun-Earth SP
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder
LPF will lie nominally on a stable manifold whilst orbiting the
Earth-Sun L1 point
 Only a small manoeuvre is needed to reach an unstable
manifold
 This allows an option to return towards Earth and access the
gravitational saddle point between Sun and Earth

1.5mio km
Nominal Orbit
around L1
Earth
Sun
Saddle Point
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder
Search for suitable trajectories - assumptions and constraints:

dV manoeuvres up to 2m/s have been considered:
 compatible estimated cold gas control authority (≈ 4-5m/s)
following nominal mission
 reasonable timescales for manoeuvres, including single thruster
failure

Consider both Rockot and VEGA launch options

Proof of principle at this stage – exact trajectory can only be chosen
once nominal mission is underway
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

Significant progress has been made over the last few years, in three
phases:
(1)
Single dV manoeuvre
 understanding search space
 difference in launchers
 types of trajectories
 typical transfer times and SP flyby distances
(2)
Double and multiple dV manoeuvres – minimising flyby distances
(3)
Search for trajectories including double SP flybys

In parallel, the navigation issue has been studied
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

Results from phase (1) – single dV manoeuvre:
 Many possible trajectories exist to take LPF from L1 to the SP – many
“needles in the haystack”
 Chaotic search space for single manoeuvre if no subsequent
corrections are applied
 LGA could potentially be used as additional manoeuvre
 Transfer time to reach the SP from L1 is typically 1 - 1.5 years with
Rockot, and around 1 year with VEGA
 Typical SP flyby distances of 100-1000kms
VEGA examples:
348days, 2333km
Q2C5 Cologne 9-12 October 2012
512days, 130km
Testing Modified Gravity with LISA Pathfinder

Results from phase (2) – multiple manoeuvres

Add second manoeuvre only at the apogees, starting with the
best solutions from the single-manoeuvre search
One
manoeuvre
strategy
Two
manoeuvres
strategy
Rockot
1635
Rockot
LGA
VEGA LGA
fast
VEGA 130
Total DV
0.3225 m/s
0.8673 m/s
0.2301 m/s
-1.232 m/s
Flyby
distance
1635 km
396 km
2333 km
130 km
DV1
0.3225 m/s
0.8673 m/s
0.2301 m/s
-1.232 m/s
DV2
1.4 m/s
1.8 m/s
1.87 m/s
0.05 m/s
Total DV
1.7225 m/s
2.6673 m/s
2.1001 m/s
1.282 m/s
Flyby
distance
242 km
253 km
355 km
72 km
Additional manoeuvres (keeping total dV manageable) show:
SP flyby distance can be reduced to “zero”

Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder


Results from phase (3) – double SP crossings:

Promising examples have been found

More work required
a: Launch, 24/2/2013.
inclination = 57.6°
perigee altitude: 322 km
b: Libration orbit, 73 days after
launch

 c: Exiting libration orbit, 258 days
after launch. The spacecraft has
spent 185 days around L1.
d: Reaching the SP for the first
time, 543 days after launch (285 days
after escaping from L1).

e: Reaching the SP for the second
time, 582 days after launch (39 days
after the first passage).

Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

The issue of navigation – just as critical:
 Can LPF be navigated along a nominal trajectory, given the
limitations of the micropropulsion system and navigation
errors?
 Results so far: the navigation issue appears manageable
 Ground contact with SC required every few days – affects
operational costs
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder
Gravitational accelerations accessible to LPF during SP flyby:

Given that
 LPF speed through SP region ≈1.5km/s (basically free-fall)
 SP flyby distance will be dominated by SC tracking errors –
estimated of order 1-10km depending on ground station(s)

We conclude that
 Newtonian gradients around SP are ≥ 2x10-11s-2. For a bestcase miss distance of 1km, ag ≥ 2x10-8ms-2
 Even if LPF flies through the SP exactly, it will spend, at most,
≈6s in an environment with ag ≤ 1x10-7ms-2
 LPF will spend at least 300s in ag ≤ 1x10-5ms-2
 Spacecraft self-gravity (< 10-8ms-2 ) is not an issue (more later)
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

Experiencing ag ≈ 1x10-7ms-2 to 1x10-5ms-2 at 1AU from the Sun is
not too bad:
Pioneer
Mercury
Earth
Saturn
Neptune
1.E-01
1.E-02
1.E-03
a [ms-2]
1.E-04
1.E-05
1.E-06
1.E-07
1.E-08
1.E-09
1.E-10
1.E-11
0.1
1
10
100
1000
Distance from Sun [AU]

Equivalent to travelling out to between 25 and 250 AU!
Q2C5 Cologne 9-12 October 2012
10000
Testing Modified Gravity with LISA Pathfinder
So by flying LPF through the Sun-Earth SP, we can access about
half the acceleration gap, between 103 and 105a0 – but to what
(integrated) sensitivity, compared to predicted signals…?
1.00E+01
1.00E+00
1.00E-01
Deviation from Newtonian v(gn/a0)-1

1.00E-02
1.00E-03
1.00E-04
1.00E-05
1.00E-06
1.00E-07
1.00E-08
1.00E-09
1.00E-10
1.00E-11
1.00E-12
1.00E-13
1.00E-02
1.00E+00
1.00E+02
1.00E+04
gn/a0
1.00E+06
Q2C5 Cologne 9-12 October 2012
1.00E+08
1.00E+10
Testing Modified Gravity with LISA Pathfinder

Signal prediction of anomalous MOND gravity gradients in TeVeS
(Bevis et al 2010):
 Numerical method used to calculate anomalous gradients at grid
points of cubic volume around SP
 A typical LPF trajectory is then propagated through the volume and
the anomalous gradients are extracted at each point:
 LPF speed through SP region – 1.5km/s – is used to convert
spatial into temporal gradient variations
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

Results
 Anomalous MOND gradients as a function of flyby distance:
0km
50km
100km
400km
MOND signal ≈ 500s - 1000s long → ~ mHz (!!)
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

Total external gravity gradient seen by LPF for 50km flyby distance
Newtonian only
Newtonian + MOND

Smooth Newtonian background is predictable and can be subtracted
or filtered out
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

Comparison of predicted TEVES signal to LPF differential
acceleration performance requirement (divided by Test Mass
separation to give gradiometer performance):
→ If LPF “only” meets its requirements, TEVES MONDian gradient
detection does not look good!
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

Extensive test campaigns on flight hardware show that the
currently predicted LPF performance is substantially better:
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

Even the above estimate could be considered conservative and
“worst case”. The actual LPF performance could be as good as

We won’t really know until LPF flies!
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

SNRs can be calculated in analogy with gravitational wave
detection. Assuming stationary instrument noise, the following
SNR plot is obtained (Magueijo):
→ SNRs between
15 and 60
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

What about theories other than TeVeS?

Following Magueijo & Mozzafari (2012):

Type I and some type II theories with “designer transition
functions” can be severely constrained

Type III theories can in principle avoid detection by making
the transition steep enough
Q2C5 Cologne 9-12 October 2012
Testing Modified Gravity with LISA Pathfinder

Example of signals and noise for linear transition function
(Galianni et al (2012)):

SP flyby distance of 10km required for unity SNR (detection),
flyby distance of 1km required for SNR ≈ 2
Q2C5 Cologne 9-12 October 2012
Summary and Discussion

It seems clear that a direct test of MONDian behaviour is possible
at acceleration levels 103 – 105a0, by flying LPF through the SunEarth SP region once the nominal LPF mission is completed

MONDian gradients predicted by TeVeS would either be detected
and even measured in detail, or ruled out conclusively

MONDian gradients predicted by other theories are smaller –
although some could be constrained, it is clear that some
variants could avoid detection

LPF presents us with the rare chance explore the low
acceleration regime around the SP, and to subject some
alternative gravitational theories to a direct experimental test – at
the relatively minor cost of extended operations

A positive detection would represent a major breakthrough in
fundamental physics
Q2C5 Cologne 9-12 October 2012
Summary and Discussion

Recent developments & on-going work:

The baseline LPF micropropulsion system is now cold gas.
Sufficient propellant to take LPF through the SP in an
extension is available

Mission analysis and trajectory design: ESOC have been
contacted to consolidate trajectory work – but (small) funding
required…

Differential self-gravity of LPF spacecraft could potentially
generate local anomalous gradients (external field effect) –
currently under investigation

The “LPF and MOND team” plans to submit a proposal to ESA
for a LPF mission extension to test modified gravity at the
saddle point – we think there is enough value for money!
Q2C5 Cologne 9-12 October 2012