Solar System Constraints on Gauss-Bonnet Mediated Dark Energy
Solar system constraints on
Gauss-Bonnet mediated dark energy
Luca Amendola1, Christos Charmousis2 and Stephen C Davis3
1 INAF/Osservatorio Astronomico di Roma, Viale Frascati 33, 00040 Monte Porzio
Catone (Roma), Italy
2 LPT, Université Paris–Sud, Bâtiment 210, 91405 Orsay CEDEX, France
3 Lorentz Institute, Postbus 9506, 2300 RA Leiden, The Netherlands
E-mail: amendola@mporzio.astro.it, Christos.Charmousis@th.u-psud.fr and
sdavis@lorentz.leidenuniv.nl
Abstract. Although the Gauss-Bonnet term is a topological invariant for general
relativity, it couples naturally to a quintessence scalar field, modifying gravity at solar
system scales. We determine the solar system constraints due to this term by evaluating
the post-Newtonian metric for a distributional source. We find a mass dependent, 1/r7
correction to the Newtonian potential, and also deviations from the Einstein gravity
prediction for light-bending. We constrain the parameters of the theory using planetary
orbits, the Cassini spacecraft data, and a laboratory test of Newton’s law, always
finding extremely tight bounds on the energy associated to the Gauss-Bonnet term.
We discuss the relevance of these constraints to late-time cosmological acceleration.
Keywords: dark energy theory, gravity, string theory and cosmology
1. Introduction
Supernovae measurements [1] indicate that our universe has entered a phase of late-
time acceleration. One can question the magnitude of the acceleration and its equation
of state, although given the concordance of different cosmological data, acceleration
seems a robust observation (although see [2] for criticisms). Commonly, in order to
explain this phenomenon one postulates the existence of a minute cosmological constant
Λ ∼ 10−12 eV4. This fits the data well and is the most economic explanation in terms of
parameter(s). However such a tiny value is extremely unnatural from a particle physics
point of view [3]. Given the theoretical problems of a cosmological constant, one hopes
that the intriguing phenomenon of acceleration is a window to new observable physics.
This could be in the matter sector, in the form of dark energy [4, 5], or in the gravity
sector, in the form of a large distance modification of Einstein gravity [6, 7, 8, 9].
Scalar field driven dark energy, or quintessence [4] is one of the most popular of the
former possibilities. However these models have important drawbacks, such as the
fine tuning of the mass of the quintessence field (which has to be smaller than the
actual Hubble parameter, H0 ∼ 10−33 eV), and stability of radiative corrections from
http://arxiv.org/abs/0704.0175v2
Solar system constraints on Gauss-Bonnet mediated dark energy 2
the matter sector [10] (see however [11]). Modified gravity models have the potential
to avoid these problems, and can give a more profound explanation of the acceleration.
However, these are far more difficult to obtain since Einstein’s theory is experimentally
well established [12], and the required modifications happen at very low (classical)
energy scales which are (supposed to be) theoretically well understood. Furthermore,
many apparently successful modified gravity models suffer from instabilities or are
incompatible with gravity experiments. For example the self-accelerating solutions of
DGP [8] suffer from perturbative ghosts [13], and f(R) gravity theories [9] can conflict
with solar system measurements and present instabilities [14].
In this paper we will consider observational constraints on a class of gravity theories
which feature both dark energy and modified gravity. Specifically, we will examine
solar system and laboratory constraints resulting from the response of gravity to a
quintessence-like scalar field, which couples to quadratic order curvature terms such
as the Gauss-Bonnet term. Such couplings arise naturally [15], and modify gravity at
local and cosmological scales [15, 16]. Although the Gauss-Bonnet invariant shares
many of the properties of the Einstein-Hilbert term, the resulting theory can have
substantially different features, see for example [17]. It is a promising candidate for
a consistent explanation of cosmological acceleration, but as we will show, can also
produce undesirable effects at solar system scales.
In particular, we will determine constraints from deviations in planetary orbits
around the sun, the frequency shift of signals from the Cassini probe, and table-
top experiments. In contrast to some previous efforts in the field [18], we will not
suppose a priori the order of the Gauss-Bonnet correction or the scalar field potential.
Instead we will calculate leading-order gravity corrections for each of them, and obtain
constraints on the relevant coupling constants (checking they fall within the validity of
our perturbative expansion). Hence our analysis will apply for large couplings, which
as we will see, are in accord with Gauss-Bonnet driven effective dark energy models. In
this way we will show such models generally produce significant deviations from general
relativity at local scales. We also include higher-order scalar field kinetic terms, although
for the solutions we consider they turn out to be subdominant.
In the next section we will present the theory in question and calculate the
corrections to a post-Newtonian metric for a distributional point mass source. In
section 3, we derive constraints from planetary motion, the Cassini probe, and a table-
top experiment. For the Cassini constraint, we have to explicitly derive the predicted
frequency shift for our theory, as it does not fall within the usual Parametrised Post-
Newtonian (PPN) analysis. We discuss the implications of our results in section 4.
2. Quadratic Curvature Gravity
We will consider a theory with the second-order gravitational Lagrangian
R− (∇φ)2 − 2V (φ)
Solar system constraints on Gauss-Bonnet mediated dark energy 3
ξ1(φ)LGB + ξ2(φ)Gµν∇µφ∇νφ+ ξ3(φ)(∇φ)2∇2φ+ ξ4(φ)(∇φ)4
, (1)
which includes the Gauss-Bonnet term LGB = R2 − 4RµνRµν + RµνρσRµνρσ. Note for
example that such a Lagrangian with given ξ’s arises naturally from higher dimensional
compactification of a pure gravitational theory [15]. On its own, in four dimensions, the
Gauss-Bonnet term does not contribute to the gravitational field equations. However
we emphasise that when coupled to a scalar field (as above), it has a non-trivial effect.
Throughout this paper we take the dimensionless couplings ξi and their derivatives
to be O(1). There is then only one scale for the higher curvature part of the action,
given by the parameter α, with dimensions of length squared. Similarly we assume that
all derivatives of the potential V are of O(V ), which in our conventions has dimensions
of inverse length squared. These two simplifying assumptions will hold for a wide range
of theories, including those in which ξi and V arise from a toroidal compactification of
a higher dimensional space [15]. On the other hand it is perfectly conceivable that they
do not apply for our universe, in which case the corresponding gravity theories will not
be covered by the analysis in this article.
Using the post-Newtonian limit, the metric for the solar system can be written [12]
ds2 = −(1− h00)(c dt)2 + (δij + hij)dxidxj +O(ǫ3/2) . (2)
with h00, hij = O(ǫ). The dimensionless parameter ǫ is the typical gravitational strength,
given by ǫ = Gm/(rc2) where m is the typical mass scale and r the typical length scale
(see below). For the solar system ǫ is at most 10−5, while for cosmology, or close to the
event horizon of a black hole, it is of order unity. The scale of planetary velocities v,
is of order ǫ1/2, and so the h0i components of the metric are O(ǫ
3/2), as are ∂th00 and
∂thij . In what follows, we will take φ = φ0 +O(ǫ). For the linearised approximation we
are using, we can adopt a post-Newtonian gauge in which the off-diagonal components
of hij are zero. We can then write
hij = −2Ψδij , h00 = −2Φ , (3)
and so c2Φ is the Newtonian potential.
In this paper we will consider the leading-order corrections in ǫ without assumptions
on the magnitude of V and α. To leading order in ǫ, the Einstein equations take the
nice compact form,
ρm − V − 2αξ′1D(Φ + Ψ, φ) + O(ǫ2, αǫ3/r2, V ǫr2) (4)
2ξ′1D(Ψ, φ) +
D(φ, φ)
+O(ǫ2, αǫ3/r2, V ǫr2) (5)
where primes denote ∂/∂φ, and V , ξ′1, etc. are evaluated at φ = φ0. The matter energy
density in the solar system is ρm, and G0 is its bare coupling strength (without quadratic
gravity corrections). Other components of the energy-momentum tensor are higher order
in ǫ. The scalar field equation is
∆φ = V ′ − α [4ξ′1D(Φ,Ψ) + ξ2D(Φ−Ψ, φ) + ξ3D(φ, φ)] + O(ǫ2, αǫ3/r2, V ǫr2) . (6)
Solar system constraints on Gauss-Bonnet mediated dark energy 4
We have defined the operators
X,ii , D(X, Y ) =
X,ijY,ij −∆X∆Y . (7)
with i, j = 1, 2, 3 where to leading order, the Gauss-Bonnet term is then LGB =
8D(Φ,Ψ). For standard Einstein gravity (V = α = 0), the solution of the above
equations is
Φ = Ψ = −Um , φ = φ0 , (8)
where
ρm(~x
′, t)
|~x− ~x′|
. (9)
We will now study solutions which are close to the post-Newtonian limit of general
relativity, and take
Φ = −Um + δΦ , Ψ = −Um + δΨ , φ = φ0 + δφ , (10)
where δφ, etc. are the leading-order α- and V -dependent corrections.
Note that the Laplacian carries a distribution and therefore we have to be careful
with the implementation of the D operator. We see that δφ is O(V, αǫ2), and so, to
leading order, we have
∆ δφ = V ′ − 4αξ′1D(Um, Um) . (11)
Having calculated δφ, we obtain
∆ δΦ = −V + 4αξ′1D(Um, δφ) (12)
∆ δΨ =
+ 2αξ′1D(Um, δφ) . (13)
In the case of a spherical distributional source ρm = mδ
(3)(x),
. (14)
In accordance to our estimations for ǫ the solar system Newtonian potentials are
Um . 10
−5, and the velocities satisfy v2 . Um. For planets we have Um . 10
(with the maximum attained by Mercury).
With the aid of the relation
D(r−n, r−m) = 2nm
n+m+ 2
∆r−(n+m+2) (15)
the above expressions evaluate, at leading order, to
φ = φ0 +
r2V ′
− 2ξ′1
α(G0m)
Φ = −G0m
− 64(ξ
α2(G0m)
Ψ = −G0m
− 32(ξ
α2(G0m)
. (18)
Solar system constraints on Gauss-Bonnet mediated dark energy 5
We find that there are now non-standard corrections to the Newtonian potential which
do not follow the usual parametrised expansion, in agreement with [19], but not [18]
(which uses different assumptions on the form of the theory). First of all note that the
Gauss-Bonnet coupling α couples to the running of the dark energy potential V ′, giving
a 1/r contribution to the modified Newtonian potential (17). We absorb this into the
gravitational coupling,
G = G0
. (19)
The corresponding term in (18) gives a constant contribution to the effective γ PPN
parameter. The r2V terms in (17), (18) are typical of a theory with a cosmological
constant, whereas the final, 1/r7 terms are the leading pure Gauss-Bonnet correction,
which is enhanced at small distances. If we take the usual expression for the PPN
parameter γ = Ψ/Φ, we see that it is r dependent. In using the Cassini constraint on γ
we must be careful to calculate the frequency shift from scratch.
For the above derivation we have assumed δφ ≪ Um, which implies V ≪ Um/r2
and α ≪ r2/Um. This will hold in the solar system if
V ≪ 10−36m−2 and α ≪
1023m2 (everywhere)
1029m2 (planets only)
in geometrised units. Note that strictly speaking there is also a lower bound on our
coupling constants, if the above analysis is to be valid. Indeed, if we were to find
corrections of order ǫ2 ∼ 10−14, then it would imply that higher-order corrections from
general relativity were just as important as the ones appearing in (17), (18).
3. Constraints
3.1. Planetary motion
Deviations from the usual Newtonian potential will affect planetary motions, which
provides a way of bounding them. This idea has been used to bound dark matter in the
solar system [20], and also the value of the cosmological constant [21]. We will apply the
same arguments to our theory. From the above gravitational potential (17), we obtain
the Newtonian acceleration
gacc(r) = −c2
64(αξ′1)
Gmeff
where rg ≡ Gm/c2 is gravitational radius of the mass m. The above expression gives
the effective mass meff felt by a body at distance r. If the test body is a planet with
semi-major axis a, we can use this formula at r ≈ a. Its mean motion n ≡
Gm/a3
will then be changed by δn = (n/2)(δmeff/m). By evaluating the statistical errors of
the mean motions of the planets, δn = −(3n/2)δa/a, we can derive a bound on δmeff
and hence our deviations from general relativity
δmeff
64(αξ′1)
. (22)
Solar system constraints on Gauss-Bonnet mediated dark energy 6
The values of a for the planets are determined using Kepler’s third law, with a constant
sun’s mass m⊙. Constraints on δΦ then follow from the errors δa, in the measure of a.
These can be found in [22], and are also listed in the appendix for convenience. Given
their different r-dependence, the two corrections to δmeff are unlikely to cancel. We will
therefore bound them separately, giving constraints on α and V .
The strongest bound on the combination ξ′1α comes from Mercury, with
. 1.8× 10−12 . (23)
Neglecting the cosmological constant term, and using a ≈ 5.8×107 km and rg ≈ 1.5 km,
we find
|ξ′1α| .
(3a5δa)
≈ 3.8× 1022m2 . (24)
We see that this is within range of validity (20) for our perturbative treatment of gravity.
In cosmology, the density fraction corresponding to the Gauss-Bonnet term is [15]
ΩGB = 4ξ
. (25)
If this is to play the role of dark energy in our universe, it needs to take, along with the
contribution of the potential, a value around 0.7 at cosmological length scales (and for
redshift z ∼ 1).
If we wish to accurately apply the bound on α (24) to cosmological scales, details of
the dynamical evolution of φ will be required. These will depend on the form of V and
the ξi, and are expected to involve complex numerical analysis, all of which is beyond
the scope of this work. Here we will instead assume that the cosmological value of φ
is also φ0, which, while crude, will allow us to estimate the significance of the above
result. Given the hierarchy between cosmological and solar system scales it is natural
to question this assumption but we will make it here, and discuss it in more detail in
the concluding section.
Making the further, and less controversial, assumption that dφ/dt ≈ H , we obtain
a very stringent constraint on ΩGB:
|ΩGB| ≈ 4|ξ′1α|H20 . 8.8× 10−30 . (26)
Hence we see that solar system constraints on Gauss-Bonnet fraction of the dark energy
are potentially very significant, despite the fact that the Gauss-bonnet term is quadratic
in curvature.
Since we are assuming that all the ξi are of the same order, the above bound
also applies to the dark energy fractions arising from the final three terms in (1).
Clearly there are effective dark energy models for which the analysis leading to the
above bound (26) does not apply. However any successful model will require a huge
variation of ξ1 between local and cosmological scales, or a very substantial violation of
one of our other assumptions.
Solar system constraints on Gauss-Bonnet mediated dark energy 7
For comparison, we apply similar arguments to obtain a constraint on the potential.
The strongest bound comes from the motion of Mars [21], and is
|V | .
9rgδa
≈ 1.2× 10−40m−2 . (27)
This suggests ΩV = V/(3H
0) . 7.3×1011, which is vastly weaker than the corresponding
cosmological constraint (ΩV . 1). Hence planetary orbits tell us little of significance
about dark energy arising from a potential, in sharp contrast to the situation for Gauss-
Bonnet dark energy.
3.2. Cassini spacecraft
The most stringent constraint on the PPN parameter γ was obtained from the Cassini
spacecraft in 2002 while on its way to Saturn. The signals between the spacecraft
and the earth pass close to the sun, whose gravitational field produces a time delay.
The smallest value of r on the light ray’s path defines the impact parameter b. A
small impact parameter maximises the light delay. During that year’s superior solar
conjunction the spacecraft was re = 8.43AU = 1.26 × 1012m away from the sun, and
the impact parameter dropped as low as bmin = 1.6R⊙. A PPN analysis of the system
produced the strong constraint
δγ ≡ γ − 1 = (2.1± 2.3)× 10−5 . (28)
Given that our theory is not PPN we have to undertake the calculation from scratch.
The above constraint comes from considering a round trip, in which the light travels
from earth, grazes the sun’s ‘surface’, reaches the spacecraft, and then returns by the
same route. We take the path of the photon to be the straight line between the earth
and the spacecraft, ~x = (x, b, 0) with x varying from −xe to x⊕. For a round trip (there
and back), the additional time delay for a light ray due to the gravitational field of the
sun is then
c∆t = 2
h00(r) + hxx(r)
dx = −2
(Φ + Ψ)|r=√x2+b2 dx . (29)
For the solution (17) and (18), this evaluates to
c∆t = 4rg
1− 2αξ
a3⊕ + r
+ b2(a⊕ + re)
1024(αξ′1)
, (30)
where we have assumed x⊕ ≈ a⊕ ≫ b, and similarly for the spacecraft.
Rather than directly measure ∆t, the Cassini experiment actually found the
frequency shift in the signal [23]
ygr =
. (31)
The results obtained were
ygr = −
10−5 s
(2 + δγ) . (32)
Solar system constraints on Gauss-Bonnet mediated dark energy 8
If gravitation were to be described by the standard PPN formalism, then δγ would be
the possible deviation of the PPN parameter γ from the general relativity value of 1.
From (30) we obtain
ygr = −
b2V (a⊕ + re)
1536(αξ′1)
4αξ′1V
. (33)
Requiring that the corrections are within the errors (28) of (32), implies
|ξ′1α| .
. 1.6× 1020m2 . (34)
This suggests the dark energy bound
|ΩGB| . 3.6× 10−32 , (35)
although obtaining this bound from solar system data requires major assumptions about
the cosmological behaviour of φ, as we will point out in section 4.
The data obtained by the spacecraft were actually for a range of impact parameters
b, but we have just used the most conservative value b = bmin = 1.6R⊙. The above
constraint is even stronger than (24), which was obtained for planetary motion. This
is because the experiment involved smaller r, and so the possible Gauss-Bonnet effects
were larger.
Taking the above expression for ygr (33) at face value, we can also constrain the
potential to be |V | . 10−22m2 and the cross-term |αξ′1V ′| . 10−5. However these are
of little interest as they are much weaker than the planetary motion constraints (24),
(27), and also the former is far outside the range of validity (20) of our analysis.
3.3. A table-top experiment
Laboratory experiments can also be used to obtain bounds on deviations from Newton’s
law. For illustration we will consider the table-top experiment described in [24]. It
consists of a 60 cm copper bar, suspended at its midpoint by a tungsten wire. Two 7.3 kg
masses are placed on carts far (105 cm) from the bar, and another mass of m ≈ 43 g is
placed near (5 cm) to the side of bar. Moving the masses to the opposite sides of the bar
changes in the torque felt by it. The experiment measures the torques N105 and −N5
produced respectively by the far and near masses. The masses and distances are chosen
so that the two torques roughly cancel. The ratio R = N105/N5 is then determined, and
compared with the theoretical value. The deviation from the Newtonian result is
Rexpt
RNewton
− 1 = (1.2± 7)× 10−4 . (36)
In fact, to help reduce errors, additional measurements were taken. To account for the
gravitational field of the carts that the far masses sit on, the experiment was repeated
with only the carts and a m′ ≈ 3 g near mass. The measured torque was then subtracted
from the result for the loaded carts.
The Gauss-Bonnet corrections to the Newton potential (17) will alter the torques
produced by all four masses, as well as the carts. Furthermore, since δΦ is non-linear
Solar system constraints on Gauss-Bonnet mediated dark energy 9
in mass, there will be further corrections coming from cross terms. The expressions
derived in section 2 are just for the gravitational field of a single mass, and so will not
fully describe the above table-top experiment. However, we find that the contribution
from the mass m will dominate the other corrections, and so we can get a good estimate
of the Gauss-Bonnet contribution to the ratio R by just considering m.
The torque experienced by the copper bar, due to a point mass at ~X = (X, Y, Z) is
d3x (~x ∧ ~F )z = ρCu
yX − xY
r=| ~X−~x|
, (37)
where ρCu is the bar’s density. A full list of parameters for the experiment is given
in table I of [24]. The bar’s dimensions are 60 cm × 1.5 cm × 0.65 cm. Working in
coordinates with the origin at the centre of the bar, the mass m = 43.58 g is at
~X = (24.42,−4.77,−0.03) cm. Treating m as a point mass, Newtonian gravity implies
a torque of N5 ≈ (8.2 cm2)GmρCu is produced. The Gauss-Bonnet correction is
δN5 = ρCu
64G3m3(αξ′1)
yX − xY
| ~X − ~x|9
≈ −(0.025 cm−4)
(Gm)3(αξ′1)
. (38)
To be consistent with the bound (36), we require δN5/N5 to be within the range of δR.
This implies
|αξ′1| . (18 cm3)
. 1.3× 1022m2 , (39)
which is comparable to the planetary constraint (24). Extrapolating it to cosmological
scales gives
|ΩGB| . 3.1× 10−30 . (40)
There are of course many more recent laboratory tests of gravity, and we expect
that stronger constraints can be obtained from them. Table-top experiments frequently
involve multiple gravitational sources, or gravitational fields which cannot reasonably be
treated as point masses. A more detailed calculation than the one presented in section 2
will then be required. For example, the gravitational field inside a sphere or cylinder
will not receive corrections of the form (17), and so any experiment involving a test
mass moving in such a field requires a different analysis.
4. Discussion
We have shown that significant constraints on Gauss-Bonnet gravity can be derived from
both solar system measurements and table-top laboratory experiments (note that further
constraints arise when imposing theoretical constraints like absence of superluminal or
ghost modes, see [25]). The fact that the corrections to Einstein gravity are second
order in curvature suggests they will automatically be small. However this does not
take into account the fact that the dimensionfull coupling of the Gauss-Bonnet term
must be large if it is to have any hope of producing effective dark energy. Additional
constraints will come from the perihelion precession of Mercury, although the linearised
Solar system constraints on Gauss-Bonnet mediated dark energy 10
analysis we have used is inadequate to determine this, and higher-order (in ǫ) effects
will need to be calculated.
Performing an extrapolation of our results to cosmological scales suggests that the
density fraction ΩGB will be far too small to explain the accelerated expansion of our
universe. This agrees with the conclusions of [19]. Hence if Gauss-Bonnet gravity is to
be a viable dark energy candidate, one needs to find a loophole in the above arguments.
This is not too difficult, and we will now turn to this question.
In particular, we have assumed no spatial or temporal evolution of the field φ
between cosmological and solar system scales, even though the supernova measurements
correspond to a higher redshift and a far different typical distance scale. A varying φ
would of course imply that different values of ξi, and their derivatives, would be perceived
by supernovas and the planets. It is interesting to note that the size of the bound we have
found (26) is of order the square of the ratio of the solar system and the cosmological
horizon scales, s = (1AUH0)
2 ∼ 10−30. Therefore one could reasonably argue that the
small number appearing in (26) could in fact be due to the hierarchy scale, s, rather
than a very stringent constraint on ΩGB. This could perhaps be concretely realised with
something similar to the chameleon effect [26] giving some constraint on the running of
the quintessence theory. One other possibility is that the baryons (which make up the
solar system) and dark matter (which is dominant at cosmological scales) have different
couplings to φ [27]. Again, this would alter the relation between local and cosmological
constraints.
Alternatively, it may be that our assumptions on the form of the theory should
be changed. The scalar field could be coupled directly to the Einstein-Hilbert term, as
in Brans-Dicke gravity. Additionally, the couplings ξi and their derivatives could be of
different orders. The same could be true of the potential. In particular, if φ were to have
a significant mass, this would suppress the quadratic curvature effects, as they operate
via the scalar field. This would be similar to the situation in scalar-tensor gravity with
a potential, where the strong constraints on the theory can be avoided by giving the
scalar a large mass (which, however, would inhibit acceleration).
Finally, the behaviour of the scalar field could be radically different. We took it to
be O(ǫ), like the metric perturbations. However since our constraints are on the metric,
and not φ, this need not be true. Furthermore, since the theory is quadratic, there may
well be alternative solutions of the field equations, and not just the one we studied.
Hence to obtain a viable Gauss-Bonnet dark energy model, which is compatible
with solar system constraints, at least one of the above assumptions must be broken.
For many of the above ideas the higher-order scalar kinetic terms will play a significant
role. This then opens up the possibility that the higher-gravity corrections will cancel
each other, further weakening the constraints. We hope to address some of these issues
in the near future.
Solar system constraints on Gauss-Bonnet mediated dark energy 11
Acknowledgments
CC thanks Martin Bucher, Gilles Esposito-Farese and Lorenzo Sorbo for discussions.
SCD thanks the Netherlands Organisation for Scientific Research (NWO) for financial
support.
Appendix
For the benefit of readers without an astronomical background, we list relevant solar
system parameters. The values for δa come from table 4 of [22]. We take the Hubble
constant to be H0 = 70 kms
−1 Mpc−1.
R⊙ = 6.96× 108m
r⊙g ≡ Gm⊙/c2 = 1477m
H0/c = 7.566× 10−27m−1
1AU ≡ a⊕ = 149597870691m
G = 6.6742× 10−11m3 s−2 kg−1
c = 299792458m s−1
m⊙ = 1.989× 1030 kg
name a (109m) δa (m)
Mercury 57.9 0.105
Venus 108 0.329
Earth 149 0.146
Mars 228 0.657
Jupiter 778 639
Saturn 1433 4.22× 103
Uranus 2872 3.85× 104
Neptune 4495 4.79× 105
Pluto 5870 3.46× 106
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http://arxiv.org/abs/astro-ph/9908023
	Introduction
	Quadratic Curvature Gravity 
	Constraints
	Planetary motion
	Cassini spacecraft
	A table-top experiment
	Discussion
