UG-07-01
Scaling cosmologies,
geodesic motion and pseudo-susy
Wissam Chemissany, André Ploegh and Thomas Van Riet
Centre for Theoretical Physics, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands
w.chemissany, a.r.ploegh, t.van.riet@rug.nl
Abstract
One-parameter solutions in supergravity carried by scalars and a metric trace out curves
on the scalar manifold. In ungauged supergravity these curves describe a geodesic mo-
tion. It is known that a geodesic motion sometimes occurs in the presence of a scalar
potential and for time-dependent solutions this can happen for scaling cosmologies. This
note contains a further study of such solutions in the context of pseudo-supersymmetry for
multi-field systems whose first-order equations we derive using a Bogomol’nyi-like method.
In particular we show that scaling solutions that are pseudo-BPS must describe geodesic
curves. Furthermore, we clarify how to solve the geodesic equations of motion when the
scalar manifold is a maximally non-compact coset such as occurs in maximal supergravity.
This relies upon a parametrization of the coset in the Borel gauge. We then illustrate this
with the cosmological solutions of higher-dimensional gravity compactified on a n-torus.
http://arxiv.org/abs/0704.1653v3
Contents
1 Preliminaries 2
2 (Pseudo-) supersymmetry 3
3 Multi-field scaling cosmologies 4
3.1 Pure kinetic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Potential-kinetic scaling solutions . . . . . . . . . . . . . . . . . . . . . . . 5
4 Geodesic curves and the Borel gauge 8
4.1 A solution-generating technique . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 An illustration from dimensional reduction . . . . . . . . . . . . . . . . . . 9
5 Discussion 10
A Curvatures 11
B The coset SL(N, IR)/ SO(N) 11
1 Preliminaries
We consider scalar fields Φi that parametrize a Riemannian manifold with metric Gij
coupled to gravity through the standard action
µν∂µΦ
j − V (Φ)
. (1)
We restrict to solutions with the following D-dimensional space-time metric
ds2D = g(y)
2ds2D−1 + ǫf(y)
2dy2 , ds2D−1 = (ηǫ)abdx
adxb , (2)
where ǫ = ±1 and ηǫ = diag(−ǫ, 1, . . . , 1). The case ǫ = −1 describes a flat FLRW-space-
time and ǫ = +1 a Minkowski-sliced domain wall (DW) space-time. The scalar fields that
source these space-times can only depend on the y-coordinate Φi = Φi(y). The function f
corresponds to the gauge freedom of reparameterizing the y-coordinate.
Of particular interest in this note are scaling comologies, which have received a great
deal of attention in the dark-energy literature, see [1] for a review and references. One def-
inition (amongst many) of scaling cosmologies is that they are solutions for which all terms
in the Friedmann equation have the same time dependence. For pure scalar cosmologies
this implies that
H2 ∼ V ∼ T ∼ τ−2 , (3)
where τ denotes cosmic time, H the Hubble parameter and T is the kinetic energy T =
GijΦ̇
iΦ̇j . These relations imply that the scale factor is power-law a(τ) ∼ τ p. In the
case of curved FLRW-universes we also demand that H ∼ k/a2, which is only possible for
p = 1. Interestingly, scaling solutions correspond to the FLRW-geometries that possess a
time-like conformal vectorfield ξ coming from the transformation
τ → eλτ , xa → e(1−p)λxa , (4)
where xa are the space-like cartesian coordinates1. In the forthcoming we reserve the indices
a, b, . . . to denote space-like coordinates when we consider cosmological space-times. Apart
from the intriguing cosmological properties of scaling solutions they are also interesting
for understanding the dynamics of a general cosmological solution since scaling solutions
are often critical points of an autonomous system of differential equations and therefore
correspond to attractors, repellors or saddle points [2]. Scaling cosmologies often appear in
supergravity theories (see for instance [3,4]) but, remarkably, they also appear by spatially
averaging inhomogeneous cosmologies in classical general relativity [5].
We will use two coordinate frames to describe scaling comologies
τ − frame : ds2 = −dτ 2 + τ 2p ds2D−1 , (5)
t− frame : ds2 = −e2t dt2 + e2ptds2D−1 . (6)
The first is the usual FLRW-coordinate system and the second can be obtained by the
substitution t = ln τ .
2 (Pseudo-) supersymmetry
If the scalar potential V (Φ) can be written in terms of another function W (Φ) as follows
V = ǫ
Gij∂iW∂jW − D−14(D−2)W
, (7)
then the action can be written as “a sum of squares” plus a boundary term when reduced
to one dimension:
dy fgD−1
(D−1)
4(D−2)
W − 2(D − 2) ġ
+Gij∂jW ||2
gD−1W − 2(D − 1)ġgD−2f−1
, (8)
where a dot denotes a derivative w.r.t. y. The term ||Φ̇i/f + Gij∂jW ||2 is a shorthand
notation and the square involves a contraction with the field metric Gij. It is clear that
the action is stationary under variations if the terms within brackets are zero2, leading to
the following first-order equations of motion
W = 2(D − 2)
+Gij∂jW = 0 . (9)
1For curved FLRW-space-times the space-like coordinates are invariant.
2For completeness we should have added the Gibbons-Hawking term [6] in the action which deletes
that part of the above boundary term that contains ġ.
For ǫ = +1 these equations are the standard Bogomol’nyi-Prasad-Sommerfield (BPS)
equations for domain walls that arise from demanding the susy-variation of the fermions to
vanish, which guarantees that the DW preserves a fraction of the total supersymmetry of
the theory. The function W is then the superpotential that appears in the susy-variation
rules and equation (7) with ǫ = +1 is natural for supergravity theories. It is clear that for
every W that obeys (7) we can find a corresponding DW-solution, and if W is not related
to the susy-variations we call the solutions fake supersymmetric [7].
For ǫ = −1 these equations are the generalization to arbitrary space-time dimension D
and field metric Gij of the framework found in references [8–11]. So here we generalized
and derived in a different way (some of) the results of [8–11] by showing that analogously
to DW’s we can write the Lagrangian as a sum of squares. We refer to these first-order
equations as pseudo-BPS equations and W is named the pseudo-superpotential because
of the immediate analogy with BPS domain walls in supergravity [10, 11]. For the case of
cosmologies there is no natural choice for W as cosmologies cannot be found by demanding
vanishing susy-variations of the fermions3.
In [11] it is proven that for all single-scalar cosmologies (and domain walls) a pseudo-
superpotential W exists such that the cosmology is pseudo-BPS and that one can give
a fermionic interpretation of the pseudo-BPS flow in terms of so-called pseudo-Killing
spinors. This does not necessarily carry over to multi-scalar solutions as was shown in [14].
Nonetheless, a multi-field solution can locally be seen as a single-field solution [15] because
locally we can redefine the scalar coordinates such that the curve Φ(y) is aligned with a
scalar axis and all other scalars are constant on this solution. A necessary condition for
the single-field pseudo-BPS flow to carry over (locally) to the multi-field system is that the
truncation down to a single scalar is consistent (this means that apart from the solution
one can put the other scalars always to zero) [14].
3 Multi-field scaling cosmologies
Let us turn to scaling solutions in the framework of pseudo-supersymmetry and see how
geodesic motion arises. First we consider the rather trivial case with vanishing scalar
potential V and then in section 3.2 we add a scalar potential V . Pseudo-supersymmetry
is only discussed in the case of non-vanishing V .
3.1 Pure kinetic solutions
If there is no scalar potential the solutions trace out geodesics since after a change of
coordinates y → ỹ(y) via dỹ = fg1−Ddy, the scalar field action becomes
′iΦ′jdỹ,
where a prime means a derivative w.r.t. ỹ. This new action describes geodesic curves
with affine parameter ỹ. The affine velocity is constant by definition and positive since the
metric is positive definite
′iΦ′j = ||v||2 . (10)
3Star supergravity is an exception [12] and that seems related to pseudo-supersymmetry [13].
The Einstein equation is
Ryy = 12GijΦ̇
iΦ̇j =
||v||2
g2−2Df 2 , Rab = 0 . (11)
In the gauge f = 1 the solution is given by g = eC2(y +C1)
D−1 , with C1 and C2 arbitrary
integration constants, but with a shift of y we can always put C1 = 0 and C2 can always
be put to zero by re-scaling the space-like coordinates. In the case of a four-dimensional
cosmology the geometry is a power-law FLRW-solution with p = 1/3.
3.2 Potential-kinetic scaling solutions
In a recent paper of Tolley and Wesley an interesting interpretation was given to scaling
solutions [16], which we repeat here. The finite transformation (4) leaves the equations
of motion invariant if the action S scales with a constant factor, which is exactly what
happens for scaling solutions since all terms in the Lagrangian scale like τ−2. Under (4)
the metric scales like e2λgµν and in order for the action to scale as a whole we must have
V → e−2λV , T = 1
gττGijΦ̇
iΦ̇j → e−2λT . (12)
Equations (12) imply that GijΦ̇
iΦ̇j remains invariant from which one deduces that dΦ
must be a Killing vector. The curve that describes a scaling solution follows an isometry
of the scalar manifold. It depends on the parametrization whether the tangent vector Φ̇
itself is Killing. This happens for the parametrization in terms of t = ln τ since
= limλ→0
Φi(eλτ)− Φi(τ)
d ln τ
. (13)
Thus a scaling solution is associated with an invariance of the equations of motion for
a rescaling of cosmic time and is therefore associated with a conformal Killing vector on
space-time and a Killing vector on the scalar manifold.
Pseudo-supersymmetry comes into play when we check the geodesic equation of motion
∇Φ̇Φ̇i = Φ̇
j∇jΦ̇i = Φ̇j
∇(jΦ̇i) +∇[jΦ̇i]
, (14)
where we denote Φ̇i = GikΦ̇
k. Now we have that the symmetric part is zero if we
parametrize the curve with t = ln τ since scaling makes Φ̇ a Killing vector. We also have
that ∇[jΦ̇i] = 0 since the pseudo-BPS condition makes Φ̇ a curl-free flow Φ̇i = −f∂iW . To
check that the curl is indeed zero (when f 6= 1) one has to notice that in the parametriza-
tion of the curve in terms of t = ln τ the gauge is such that ġ/g is constant and that
f ∼ W−1. Since the curl is also zero we notice that the curve is a geodesic with ln τ as
affine parametrization4
∇Φ̇Φ̇
i = 0 = Φ̈i + ΓijkΦ̇
jΦ̇k . (15)
4 One could wonder whether the results works in two ways. Imagine that a scaling solution is a geodesic.
This then implies that ∇[jΦ̇i] = 0 and therefore the flow is locally a gradient flow Φ̇i = ∂i lnW ∼ f∂iW .
The link between scaling and geodesics was discovered by Karthauser and Saffin in [17],
but no conditions on the Lagrangian were given in [17] such that the relation scaling-
geodesic holds. An example of a scaling solution that is not a geodesic was given by
Sonner and Townsend in [18].
A more intuitive understanding of the origin of the geodesic motion for some scaling
cosmologies comes from the on-shell substitution V = (3p− 1) T in the Lagrangian to get
a new Lagrangian describing seemingly massless fields. Although this is rarely a consistent
procedure we believe that this is nonetheless related to the existence of geodesic scaling
solutions.
Single field
For single-field models the potential must be exponential V = Λeαφ in order to have scaling
solutions. The simplest pseudo-superpotential belonging to an exponential potential is
itself exponential
W = ±
2 . (16)
If we choose the plus sign the solution to the pseudo-BPS equation is
φ(τ) = − 2
ln τ + 1
ln[6−2α
] , g(τ) ∼ τ
α2 . (17)
The minus sign corresponds to the time reversed solution.
Multiple fields
For a general multi-field model a scaling solution with power-law scale factor τ p obeys
V = (3p− 1)T from which we derive the on-shell relation
Gij∂iW∂jW =
⇒ W = ±
8 p V
3p− 1
. (18)
In general the above expression for the superpotential W ∼
V does not hold off-shell,
unless the potential is a function of a specific kind:
Gij∂iV ∂jV
. (19)
Scalar potentials that obey (19) with the extra condition that p ≷ 1
↔ V ≷ 0 allow for
multi-field scaling solutions. For a given scalar potential that obeys (19) there probably
exist many pseudo-superpotentials W compatible with V but if we make the specific choice
8 p V/(3p− 1) then all pseudo-BPS solutions must be scaling and hence geodesic.
As a consistency check we substitute the first-order pseudo-BPS equations into the right-
hand-side of the following second-order equations of motion
Φ̈i + ΓijkΦ̇
kΦ̇j = −f 2Gij∂jV −
3 ˙(ln g)− ˙(ln f)
Φ̇i , (20)
and choose a gauge for which
W , (21)
then we indeed find an affine geodesic motion since the right hand side of (20) vanishes.
For some systems one first needs to perform a truncation in order to find the above
relation (19). A good example is the multi-field potential appearing in Assisted Inflation
V (Φ1, . . . ,Φn) =
, Gij = δij . (22)
The scaling solution of this system was proven to be the same as the single-exponential
scaling [20]. The reason is that one can perform an orthogonal transformation in field
space such that the form of the kinetic term is preserved but the scalar potential is given
V = eαϕ U(Φ1, . . . ,Φn−1) ,
. (23)
The scaling solution is such that Φ1, . . . ,Φn−1 are frozen in a stationary point of U and
therefore the system is truncated to a single-field system that obeys (19). The same was
proven for Generalized Assisted Inflation [21] in reference [22]. The scaling solution in the
original field coordinates reads Φi = Ai ln τ + Bi, which is clearly a straight line and thus
a geodesic.
The scaling solutions of [14, 18] were constructed for an axion-dilaton system with an
exponential potential for the dilaton
(∂φ)2 − 1
eµφ(∂χ)2 − Λeαφ
. (24)
Clearly this two-field system obeys (19) and (one of) the pseudo-superpotential(s) is given
by (16). The pseudo-BPS scaling solution therefore has constant axion and is effectively
described by the dilaton in an exponential potential. Note that this solution indeed de-
scribes a geodesic on SL(2, IR)/ SO(2) with ln τ as affine parameter. All examples of scaling
solutions in the literature seem to occur for exponential potentials, however by performing
a SL(2, IR)-transformation on the Lagrangian (24) the kinetic term is unchanged and the
potential becomes a more complicated function of the axion and the dilaton. The same
scaling solution then trivially still exists (and (19) still holds) but the axion is not con-
stant in the new frame and instead the solution follows a more complicated geodesic on
SL(2, IR)/ SO(2).
However another scaling solution is given in [18] that is not geodesic and with varying
axion in the frame of the above action (24). This is an illustration of the above, since the
solution is not geodesic we know that there does not exists any other pseudo-superpotential
for which the varying axion solution is pseudo-BPS, consistent with what is shown in [14]
for that particular solution.
4 Geodesic curves and the Borel gauge
For the last example of the previous section the pseudo-BPS scaling solutions described
geodesics on the symmetric space SL(2, IR)/ SO(2). In this section we consider a general
class of symmetric spaces of which SL(2, IR)/ SO(2) is an example and they are known as
maximally non-compact cosets U/K. It seems that for this class of spaces the geodesic
equations of motion can be solved easily. The symmetry of the geodesic equations is the
symmetry of the scalar coset U/K. In the case of maximal supergravity the symmetry U is
a U-duality and is a maximal non-compact real slice of a complex semisimple group. The
isotropy group K is the maximal compact subgroup of U .
4.1 A solution-generating technique
In the Borel gauge the scalar fields are divided into r dilatons φI and (n − r) axions χα,
with r the rank of U and n the dimension of U/K (see for instance [23]). The dilatons
are related to the generators HI of the Cartan sub-algebra (CSA) and the axions to the
positive root generators Eα through the following expression for the coset representative
L in the Borel gauge
L = Παexp[χ
αEα]ΠIexp[−12φ
IHI ] . (25)
In this language the geodesic equation is
φ̈I + ΓIJKφ̇
J φ̇K + ΓIαJ χ̇
αφ̇J + ΓIαβχ̇
αχ̇β = 0 , (26)
χ̈α + ΓαJKφ̇
J φ̇K + ΓαβJ χ̇
βφ̇J + Γαβγχ̇
βχ̇γ = 0 . (27)
Since ΓIJK = 0 and Γ
JK = 0 at points for which χ
α = 0 a trivial solution is given by
φI = vI y , χα = 0 . (28)
How many other solutions are there? A first thing we notice is that every global U -
transformation Φ → Φ̃ brings us from one solution to another solution. Since U generically
mixes dilatons and axions we can construct solutions with non-trivial axions in this way.
We now prove that in this way all geodesics are obtained and this depends on the fact that
U is maximally non-compact with K the maximal compact subgroup of U .
Consider an arbitrary geodesic curve Φ(t) on U/K. The point Φ(0) can be mapped to
the origin L = 1 using a U -transformation, since we can identify Φ(0) with an element of U
and then we multiply the geodesic curve Φ(t) with Φ(0)−1, generating a new geodesic curve
Φ2(t) = Φ(0)
−1Φ(t) that goes through the origin. The origin is invariant under K-rotations
but the tangent space at the origin transforms under the adjoint of K. One can prove that
there always exists an element k ∈ K, such that AdjkΦ̇2(0) ∈ CSA [24]. Therefore χ̇α2 = 0
and this solution must be a straight line. So we started out with a general curve Φ(t) and
proved that the curve Φ3(t) = kΦ(0)
−1Φ(t) is a straight line.
4.2 An illustration from dimensional reduction
The metric Ansatz for the dimensional reduction of (4 + n)-dimensional Einstein-gravity
on the n-torus (Tn) is
ds24+n = e
2αϕds24 + e
2βϕMabdza ⊗ dzb , (29)
where
4(n+ 2)
, β = −
. (30)
The matrix M is a positive-definite symmetric n×n matrix with unit determinant, which
depends on the 4-dimensional coordinates, describing the moduli of Tn. The modulus
ϕ controls the overall volume and is named the breathing mode or radion field. Notice
that we already truncated the Kaluza–Klein vectors in the Ansatz. The reduction of the
Einstein–Hilbert term gives
−g{R− 1
(∂ϕ)2 + 1
Tr∂M∂M−1} . (31)
The scalars parametrize IR × SL(n, IR)/ SO(n) where ϕ belongs to the decoupled IR-part
and M is the SL(n, IR)/ SO(n) part.
If we take the four-dimensional part of space-time to be a flat FLRW-space then that
part of the metric will be power-law with p = 1/3 and the scalars follow a geodesic with
ln τ as an affine parameter. According to the solution-generating technique, the Ansatz for
the scalars is
ϕ = v0 ln τ + c0 , M = ΩDΩT , D = diag(e−
~βa·~φ) , (32)
with ~φ = ~v ln τ and ~β the weights of SL(n, IR) in the fundamental representation (see
appendix B for some explanations on the SL(n, IR)/ SO(n)-coset in this representation).
The diagonal matrix D represents the straight-line solution and Ω is an arbitrary SL(n, IR)-
matrix in the fundamental representation. Therefore M = ΩDΩT is the most general coset
matrix describing a geodesic curve.
The Friedmann equation implies that the affine velocity is restricted to be
v20 + ||v||2 = 43 , (33)
which is the only constraint coming from the 4-dimensional Einstein equation. If we sub-
stitute this solution in (29) and define new coordinates ~y = ~zΩ we find
ds24+n = −τ 2α v0dτ 2 + τ
+2αv0d~x23 +
~βa·~v+2β v0dy2a . (34)
This is similar to what is called a Kasner solution in general relativity (see for instance [25]).
Kasner solutions are a general class of time-dependent geometries that look like
ds2 = −τ 2p0dτ 2 +
τ 2padx2a . (35)
Kasner solutions solve the Einstein equations in vacuum if the following two conditions are
satisfied
p0 + 1 =
pa , (p0 + 1)
p2a . (36)
For the metric (34) these conditions are satisfied if the lower-dimensional Friedmann equa-
tion is satisfied. For this calculation one needs the properties of the weight-vectors ~βa
(given in appendix B) and the relation between α and β (30). We therefore conclude that
the general spatially flat FLRW-solution lifts up to the most general Kasner solution with
SO(3)-symmetry in 4 + n dimensions.
5 Discussion
In this note we have studied multi-field scaling solutions using a first-order formalism for
scalar cosmologies a.k.a. pseudo-supersymmetry. We derived these first-order equations
via a Bogomol’nyi-like method that was known to work for domain wall solutions as was
first shown in [26, 27]5 and we showed that it trivially extends to cosmological solutions.
This first-order formalism allows a better understanding of the geodesic motion that comes
with a specific class of scaling solutions. One of the main results of this note is a proof that
shows that all pseudo-BPS cosmologies that are scaling solutions must be geodesic. This
complements to the discussion in [14] where the first example of a non-geodesic scaling
cosmology was shown to be non-pseudo-BPS. Moreover we gave constraints on multi-field
Lagrangians for which the pseudo-BPS cosmologies are geodesic scaling solutions.
Having illustrated the importance of geodesic motion in scalar cosmology, we tackled
the problem of solving the geodesic equations in the second part of this note. We showed
that the most general geodesic curve can be written down for maximally non-compact
coset spaces U/K. These coset spaces appear in all maximal and some less-extended
supergravities [29]. We used a solution-generating technique based on the symmetries
of the coset. We were able to prove that the most general solution is given by a U-
transformation on the “straight line”, (φI(t) = vIt, χα = 0) in the Borel gauge. We
illustrated this technique for the coset SL(n, IR)/ SO(n). Since SL(n, IR)/ SO(n) is also
the moduli space of the n-torus we applied it to find the cosmological solutions of higher-
dimensional gravity compactified on a n-torus. This exercise nicely illustrates why the
straight line is the generating solution since, from a higher-dimensional point of view,
all solutions that correspond to the non-straight line geodesics can be seen as coordinate
transformations of the solutions associated with the straight line. The oxidation of the
straight line solutions corresponds to the most general SO(3)-invariant Kasner solution of
(4 + n)-dimensional vacuum GR.
The same technique was used in [3] to find all geodesic scaling cosmologies of the
CSO-gaugings in maximal supergravity.
The solution-generating technique presented here should be considered complementary
to the “compensator method” developed by Fré et al in [30]. There the straight line
5See also [28].
also serves as a generating solution but instead of rigid U -transformations one uses local K
transformations that preserve the solvable gauge to generate new non-trivial solutions. This
technique is a nice illustration of the integrability of the second–order geodesic equations
of motion [31].
Acknowledgments
We are grateful to Dennis Westra for useful discussions and comments on the manuscript
and to Jan Rosseel for many useful discussions. This work is supported in part by the Eu-
ropean Communitys Human Potential Programme under contract MRTN-CT-2004-005104
in which the authors are associated to Utrecht University. The work of AP and TVR is
part of the research programme of the Stichting voor Fundamenteel Onderzoek der Materie
(FOM).
A Curvatures
For the metric Ansatz (2) the Ricci tensor is given by
Rab = −ǫ(ηǫ)ab
gġḟ
+ (D − 3) ġ
, Ryy = (D − 1)
− ( g̈
. (37)
B The coset SL(N, IR)/ SO(N)
Consider a general coset U/K. It is not difficult to construct a coset representative using
the Lie algebras U and K of U and K respectively. Since K is a subgroup of U we have the
decomposition U = K ⊕ F, with F the complement of K in U. For a given representation
of the algebra U we define a coset representative via L(y) = exp(yifi) where the fi form a
basis of F in some representation of U.
To derive the metric we define a Lie algebra valued one-form from the coset represen-
tative L(y) via
L−1dL ≡ E + Ω , (38)
where E takes values in F and Ω in K. We notice that L−1dL is invariant under left
multiplication with a y-independent element g ∈ U . Multiplying L from the right with
local elements k ∈ K results in
E → k−1E k , Ω → k−1Ω k + k−1dk . (39)
In supergravity the parameters yi are scalar fields that depend on the space-time coordi-
nates yi = φi(x). The one-form L−1dL can be written out in terms of coset-coordinate
one-forms dφi which themselves can be pulled back to space-time coordinate one-forms
dφi = ∂µφ
idxµ. Now we can write
L−1dL = Eµdx
µ + Ωµdx
µ . (40)
Under the φ-dependent K-transformations k(φ(x)) we have that Ωµ → k−1Ωµk + k−1∂µk
and Eµ → k−1Eµk. It is clear that Eµ is covariant under local K-transformations and
Ωµ transforms like a connection. Using this connection Ωµ we can make the following
K-covariant derivative on L and L−1
DµL = ∂µL− LΩµ , DµL−1 = ∂µL−1 + ΩµL−1 . (41)
To find a kinetic term for the scalars we notice that the object
Tr[DµLD
µL−1] = −Tr[EµEµ] , (42)
has all the right properties as it contains single derivatives on the scalars, it is a space-time
scalar, it is invariant under rigid U transformations and under local K-transformations.
Thus,
e−1Lscalar = −Tr[EµEµ] ≡ −12g(φ)ij∂µφ
i∂µφj . (43)
If SO(N) is the maximal compact subgroup of U and we work in the fundamental
representation, then the Lie algebra of SO(N) is the vector space of antisymmetric matrices,
L−1dL+ (L−1dL)T
, Ω =
L−1dL− (L−1dL)T
, (44)
and a calculation shows that
e−1Lscalar = −Tr[E2] = +14Tr[∂M∂M
−1] , (45)
where M is the SO(N)-invariant matrix M = LLT .
No we specify to U = SL(N, IR). In general SL(N, IR) has rank N − 1 and its maximal
compact subgroup is SO(N). There will therefore be N−1 dilaton fields φI and N(N−1)/2
axion fields χα. The Cartan generators are given in terms of the weights ~β of SL(N, IR) in
the fundamental representation
( ~H)ij = (~βi)δij . (46)
The weights can be taken to obey the following algebra
βiI = 0 ,
βiIβiJ = 2δIJ , ~βi · ~βj = 2δij −
. (47)
The first of these identities holds in all bases since it follows from the tracelessness of the
SL generators. The second and third identity can be seen as convenient normalizations of
the generators. The positive step operators Eij are all upper triangular and a handy basis
is that they have only one non-zero entry [Eij ]ij = 1. The negative step operators are the
transpose of the positive. The SO(N) algebra is spanned by the following combinations
(Eβ − E−β) . (48)
The action will generically look complicated but when all axions are set to zero L is diagonal
L = diag[ exp(−1
~βi · ~φ)] and the action becomes
Tr∂M∂M−1 = −1
βiJβiI)∂φ
I∂φJ = −1
δIJ∂φ
I∂φJ . (49)
This action describes N − 1 dilatons that parametrize the flat scalar manifold IRN−1.
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	Preliminaries
	(Pseudo-) supersymmetry
	Multi-field scaling cosmologies
	Pure kinetic solutions
	Potential-kinetic scaling solutions
	Geodesic curves and the Borel gauge
	A solution-generating technique
	An illustration from dimensional reduction
	Discussion
	Curvatures
	The coset SL(N,IR)/SO(N)

  One-parameter solutions in supergravity carried by scalars and a metric trace
out curves on the scalar manifold. In ungauged supergravity these curves
describe a geodesic motion. It is known that a geodesic motion sometimes occurs
in the presence of a scalar potential and for time-dependent solutions this can
happen for scaling cosmologies. This note contains a further study of such
solutions in the context of pseudo-supersymmetry for multi-field systems whose
first-order equations we derive using a Bogomol'nyi-like method. In particular
we show that scaling solutions that are pseudo-BPS must describe geodesic
curves. Furthermore, we clarify how to solve the geodesic equations of motion
when the scalar manifold is a maximally non-compact coset such as occurs in
maximal supergravity. This relies upon a parametrization of the coset in the
Borel gauge. We then illustrate this with the cosmological solutions of
higher-dimensional gravity compactified on a $n$-torus.

UG-07-01
Scaling cosmologies,
geodesic motion and pseudo-susy
Wissam Chemissany, André Ploegh and Thomas Van Riet
Centre for Theoretical Physics, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands
w.chemissany, a.r.ploegh, t.van.riet@rug.nl
Abstract
One-parameter solutions in supergravity carried by scalars and a metric trace out curves
on the scalar manifold. In ungauged supergravity these curves describe a geodesic mo-
tion. It is known that a geodesic motion sometimes occurs in the presence of a scalar
potential and for time-dependent solutions this can happen for scaling cosmologies. This
note contains a further study of such solutions in the context of pseudo-supersymmetry for
multi-field systems whose first-order equations we derive using a Bogomol’nyi-like method.
In particular we show that scaling solutions that are pseudo-BPS must describe geodesic
curves. Furthermore, we clarify how to solve the geodesic equations of motion when the
scalar manifold is a maximally non-compact coset such as occurs in maximal supergravity.
This relies upon a parametrization of the coset in the Borel gauge. We then illustrate this
with the cosmological solutions of higher-dimensional gravity compactified on a n-torus.
http://arxiv.org/abs/0704.1653v3
Contents
1 Preliminaries 2
2 (Pseudo-) supersymmetry 3
3 Multi-field scaling cosmologies 4
3.1 Pure kinetic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Potential-kinetic scaling solutions . . . . . . . . . . . . . . . . . . . . . . . 5
4 Geodesic curves and the Borel gauge 8
4.1 A solution-generating technique . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 An illustration from dimensional reduction . . . . . . . . . . . . . . . . . . 9
5 Discussion 10
A Curvatures 11
B The coset SL(N, IR)/ SO(N) 11
1 Preliminaries
We consider scalar fields Φi that parametrize a Riemannian manifold with metric Gij
coupled to gravity through the standard action
µν∂µΦ
j − V (Φ)
. (1)
We restrict to solutions with the following D-dimensional space-time metric
ds2D = g(y)
2ds2D−1 + ǫf(y)
2dy2 , ds2D−1 = (ηǫ)abdx
adxb , (2)
where ǫ = ±1 and ηǫ = diag(−ǫ, 1, . . . , 1). The case ǫ = −1 describes a flat FLRW-space-
time and ǫ = +1 a Minkowski-sliced domain wall (DW) space-time. The scalar fields that
source these space-times can only depend on the y-coordinate Φi = Φi(y). The function f
corresponds to the gauge freedom of reparameterizing the y-coordinate.
Of particular interest in this note are scaling comologies, which have received a great
deal of attention in the dark-energy literature, see [1] for a review and references. One def-
inition (amongst many) of scaling cosmologies is that they are solutions for which all terms
in the Friedmann equation have the same time dependence. For pure scalar cosmologies
this implies that
H2 ∼ V ∼ T ∼ τ−2 , (3)
where τ denotes cosmic time, H the Hubble parameter and T is the kinetic energy T =
GijΦ̇
iΦ̇j . These relations imply that the scale factor is power-law a(τ) ∼ τ p. In the
case of curved FLRW-universes we also demand that H ∼ k/a2, which is only possible for
p = 1. Interestingly, scaling solutions correspond to the FLRW-geometries that possess a
time-like conformal vectorfield ξ coming from the transformation
τ → eλτ , xa → e(1−p)λxa , (4)
where xa are the space-like cartesian coordinates1. In the forthcoming we reserve the indices
a, b, . . . to denote space-like coordinates when we consider cosmological space-times. Apart
from the intriguing cosmological properties of scaling solutions they are also interesting
for understanding the dynamics of a general cosmological solution since scaling solutions
are often critical points of an autonomous system of differential equations and therefore
correspond to attractors, repellors or saddle points [2]. Scaling cosmologies often appear in
supergravity theories (see for instance [3,4]) but, remarkably, they also appear by spatially
averaging inhomogeneous cosmologies in classical general relativity [5].
We will use two coordinate frames to describe scaling comologies
τ − frame : ds2 = −dτ 2 + τ 2p ds2D−1 , (5)
t− frame : ds2 = −e2t dt2 + e2ptds2D−1 . (6)
The first is the usual FLRW-coordinate system and the second can be obtained by the
substitution t = ln τ .
2 (Pseudo-) supersymmetry
If the scalar potential V (Φ) can be written in terms of another function W (Φ) as follows
V = ǫ
Gij∂iW∂jW − D−14(D−2)W
, (7)
then the action can be written as “a sum of squares” plus a boundary term when reduced
to one dimension:
dy fgD−1
(D−1)
4(D−2)
W − 2(D − 2) ġ
+Gij∂jW ||2
gD−1W − 2(D − 1)ġgD−2f−1
, (8)
where a dot denotes a derivative w.r.t. y. The term ||Φ̇i/f + Gij∂jW ||2 is a shorthand
notation and the square involves a contraction with the field metric Gij. It is clear that
the action is stationary under variations if the terms within brackets are zero2, leading to
the following first-order equations of motion
W = 2(D − 2)
+Gij∂jW = 0 . (9)
1For curved FLRW-space-times the space-like coordinates are invariant.
2For completeness we should have added the Gibbons-Hawking term [6] in the action which deletes
that part of the above boundary term that contains ġ.
For ǫ = +1 these equations are the standard Bogomol’nyi-Prasad-Sommerfield (BPS)
equations for domain walls that arise from demanding the susy-variation of the fermions to
vanish, which guarantees that the DW preserves a fraction of the total supersymmetry of
the theory. The function W is then the superpotential that appears in the susy-variation
rules and equation (7) with ǫ = +1 is natural for supergravity theories. It is clear that for
every W that obeys (7) we can find a corresponding DW-solution, and if W is not related
to the susy-variations we call the solutions fake supersymmetric [7].
For ǫ = −1 these equations are the generalization to arbitrary space-time dimension D
and field metric Gij of the framework found in references [8–11]. So here we generalized
and derived in a different way (some of) the results of [8–11] by showing that analogously
to DW’s we can write the Lagrangian as a sum of squares. We refer to these first-order
equations as pseudo-BPS equations and W is named the pseudo-superpotential because
of the immediate analogy with BPS domain walls in supergravity [10, 11]. For the case of
cosmologies there is no natural choice for W as cosmologies cannot be found by demanding
vanishing susy-variations of the fermions3.
In [11] it is proven that for all single-scalar cosmologies (and domain walls) a pseudo-
superpotential W exists such that the cosmology is pseudo-BPS and that one can give
a fermionic interpretation of the pseudo-BPS flow in terms of so-called pseudo-Killing
spinors. This does not necessarily carry over to multi-scalar solutions as was shown in [14].
Nonetheless, a multi-field solution can locally be seen as a single-field solution [15] because
locally we can redefine the scalar coordinates such that the curve Φ(y) is aligned with a
scalar axis and all other scalars are constant on this solution. A necessary condition for
the single-field pseudo-BPS flow to carry over (locally) to the multi-field system is that the
truncation down to a single scalar is consistent (this means that apart from the solution
one can put the other scalars always to zero) [14].
3 Multi-field scaling cosmologies
Let us turn to scaling solutions in the framework of pseudo-supersymmetry and see how
geodesic motion arises. First we consider the rather trivial case with vanishing scalar
potential V and then in section 3.2 we add a scalar potential V . Pseudo-supersymmetry
is only discussed in the case of non-vanishing V .
3.1 Pure kinetic solutions
If there is no scalar potential the solutions trace out geodesics since after a change of
coordinates y → ỹ(y) via dỹ = fg1−Ddy, the scalar field action becomes
′iΦ′jdỹ,
where a prime means a derivative w.r.t. ỹ. This new action describes geodesic curves
with affine parameter ỹ. The affine velocity is constant by definition and positive since the
metric is positive definite
′iΦ′j = ||v||2 . (10)
3Star supergravity is an exception [12] and that seems related to pseudo-supersymmetry [13].
The Einstein equation is
Ryy = 12GijΦ̇
iΦ̇j =
||v||2
g2−2Df 2 , Rab = 0 . (11)
In the gauge f = 1 the solution is given by g = eC2(y +C1)
D−1 , with C1 and C2 arbitrary
integration constants, but with a shift of y we can always put C1 = 0 and C2 can always
be put to zero by re-scaling the space-like coordinates. In the case of a four-dimensional
cosmology the geometry is a power-law FLRW-solution with p = 1/3.
3.2 Potential-kinetic scaling solutions
In a recent paper of Tolley and Wesley an interesting interpretation was given to scaling
solutions [16], which we repeat here. The finite transformation (4) leaves the equations
of motion invariant if the action S scales with a constant factor, which is exactly what
happens for scaling solutions since all terms in the Lagrangian scale like τ−2. Under (4)
the metric scales like e2λgµν and in order for the action to scale as a whole we must have
V → e−2λV , T = 1
gττGijΦ̇
iΦ̇j → e−2λT . (12)
Equations (12) imply that GijΦ̇
iΦ̇j remains invariant from which one deduces that dΦ
must be a Killing vector. The curve that describes a scaling solution follows an isometry
of the scalar manifold. It depends on the parametrization whether the tangent vector Φ̇
itself is Killing. This happens for the parametrization in terms of t = ln τ since
= limλ→0
Φi(eλτ)− Φi(τ)
d ln τ
. (13)
Thus a scaling solution is associated with an invariance of the equations of motion for
a rescaling of cosmic time and is therefore associated with a conformal Killing vector on
space-time and a Killing vector on the scalar manifold.
Pseudo-supersymmetry comes into play when we check the geodesic equation of motion
∇Φ̇Φ̇i = Φ̇
j∇jΦ̇i = Φ̇j
∇(jΦ̇i) +∇[jΦ̇i]
, (14)
where we denote Φ̇i = GikΦ̇
k. Now we have that the symmetric part is zero if we
parametrize the curve with t = ln τ since scaling makes Φ̇ a Killing vector. We also have
that ∇[jΦ̇i] = 0 since the pseudo-BPS condition makes Φ̇ a curl-free flow Φ̇i = −f∂iW . To
check that the curl is indeed zero (when f 6= 1) one has to notice that in the parametriza-
tion of the curve in terms of t = ln τ the gauge is such that ġ/g is constant and that
f ∼ W−1. Since the curl is also zero we notice that the curve is a geodesic with ln τ as
affine parametrization4
∇Φ̇Φ̇
i = 0 = Φ̈i + ΓijkΦ̇
jΦ̇k . (15)
4 One could wonder whether the results works in two ways. Imagine that a scaling solution is a geodesic.
This then implies that ∇[jΦ̇i] = 0 and therefore the flow is locally a gradient flow Φ̇i = ∂i lnW ∼ f∂iW .
The link between scaling and geodesics was discovered by Karthauser and Saffin in [17],
but no conditions on the Lagrangian were given in [17] such that the relation scaling-
geodesic holds. An example of a scaling solution that is not a geodesic was given by
Sonner and Townsend in [18].
A more intuitive understanding of the origin of the geodesic motion for some scaling
cosmologies comes from the on-shell substitution V = (3p− 1) T in the Lagrangian to get
a new Lagrangian describing seemingly massless fields. Although this is rarely a consistent
procedure we believe that this is nonetheless related to the existence of geodesic scaling
solutions.
Single field
For single-field models the potential must be exponential V = Λeαφ in order to have scaling
solutions. The simplest pseudo-superpotential belonging to an exponential potential is
itself exponential
W = ±
2 . (16)
If we choose the plus sign the solution to the pseudo-BPS equation is
φ(τ) = − 2
ln τ + 1
ln[6−2α
] , g(τ) ∼ τ
α2 . (17)
The minus sign corresponds to the time reversed solution.
Multiple fields
For a general multi-field model a scaling solution with power-law scale factor τ p obeys
V = (3p− 1)T from which we derive the on-shell relation
Gij∂iW∂jW =
⇒ W = ±
8 p V
3p− 1
. (18)
In general the above expression for the superpotential W ∼
V does not hold off-shell,
unless the potential is a function of a specific kind:
Gij∂iV ∂jV
. (19)
Scalar potentials that obey (19) with the extra condition that p ≷ 1
↔ V ≷ 0 allow for
multi-field scaling solutions. For a given scalar potential that obeys (19) there probably
exist many pseudo-superpotentials W compatible with V but if we make the specific choice
8 p V/(3p− 1) then all pseudo-BPS solutions must be scaling and hence geodesic.
As a consistency check we substitute the first-order pseudo-BPS equations into the right-
hand-side of the following second-order equations of motion
Φ̈i + ΓijkΦ̇
kΦ̇j = −f 2Gij∂jV −
3 ˙(ln g)− ˙(ln f)
Φ̇i , (20)
and choose a gauge for which
W , (21)
then we indeed find an affine geodesic motion since the right hand side of (20) vanishes.
For some systems one first needs to perform a truncation in order to find the above
relation (19). A good example is the multi-field potential appearing in Assisted Inflation
V (Φ1, . . . ,Φn) =
, Gij = δij . (22)
The scaling solution of this system was proven to be the same as the single-exponential
scaling [20]. The reason is that one can perform an orthogonal transformation in field
space such that the form of the kinetic term is preserved but the scalar potential is given
V = eαϕ U(Φ1, . . . ,Φn−1) ,
. (23)
The scaling solution is such that Φ1, . . . ,Φn−1 are frozen in a stationary point of U and
therefore the system is truncated to a single-field system that obeys (19). The same was
proven for Generalized Assisted Inflation [21] in reference [22]. The scaling solution in the
original field coordinates reads Φi = Ai ln τ + Bi, which is clearly a straight line and thus
a geodesic.
The scaling solutions of [14, 18] were constructed for an axion-dilaton system with an
exponential potential for the dilaton
(∂φ)2 − 1
eµφ(∂χ)2 − Λeαφ
. (24)
Clearly this two-field system obeys (19) and (one of) the pseudo-superpotential(s) is given
by (16). The pseudo-BPS scaling solution therefore has constant axion and is effectively
described by the dilaton in an exponential potential. Note that this solution indeed de-
scribes a geodesic on SL(2, IR)/ SO(2) with ln τ as affine parameter. All examples of scaling
solutions in the literature seem to occur for exponential potentials, however by performing
a SL(2, IR)-transformation on the Lagrangian (24) the kinetic term is unchanged and the
potential becomes a more complicated function of the axion and the dilaton. The same
scaling solution then trivially still exists (and (19) still holds) but the axion is not con-
stant in the new frame and instead the solution follows a more complicated geodesic on
SL(2, IR)/ SO(2).
However another scaling solution is given in [18] that is not geodesic and with varying
axion in the frame of the above action (24). This is an illustration of the above, since the
solution is not geodesic we know that there does not exists any other pseudo-superpotential
for which the varying axion solution is pseudo-BPS, consistent with what is shown in [14]
for that particular solution.
4 Geodesic curves and the Borel gauge
For the last example of the previous section the pseudo-BPS scaling solutions described
geodesics on the symmetric space SL(2, IR)/ SO(2). In this section we consider a general
class of symmetric spaces of which SL(2, IR)/ SO(2) is an example and they are known as
maximally non-compact cosets U/K. It seems that for this class of spaces the geodesic
equations of motion can be solved easily. The symmetry of the geodesic equations is the
symmetry of the scalar coset U/K. In the case of maximal supergravity the symmetry U is
a U-duality and is a maximal non-compact real slice of a complex semisimple group. The
isotropy group K is the maximal compact subgroup of U .
4.1 A solution-generating technique
In the Borel gauge the scalar fields are divided into r dilatons φI and (n − r) axions χα,
with r the rank of U and n the dimension of U/K (see for instance [23]). The dilatons
are related to the generators HI of the Cartan sub-algebra (CSA) and the axions to the
positive root generators Eα through the following expression for the coset representative
L in the Borel gauge
L = Παexp[χ
αEα]ΠIexp[−12φ
IHI ] . (25)
In this language the geodesic equation is
φ̈I + ΓIJKφ̇
J φ̇K + ΓIαJ χ̇
αφ̇J + ΓIαβχ̇
αχ̇β = 0 , (26)
χ̈α + ΓαJKφ̇
J φ̇K + ΓαβJ χ̇
βφ̇J + Γαβγχ̇
βχ̇γ = 0 . (27)
Since ΓIJK = 0 and Γ
JK = 0 at points for which χ
α = 0 a trivial solution is given by
φI = vI y , χα = 0 . (28)
How many other solutions are there? A first thing we notice is that every global U -
transformation Φ → Φ̃ brings us from one solution to another solution. Since U generically
mixes dilatons and axions we can construct solutions with non-trivial axions in this way.
We now prove that in this way all geodesics are obtained and this depends on the fact that
U is maximally non-compact with K the maximal compact subgroup of U .
Consider an arbitrary geodesic curve Φ(t) on U/K. The point Φ(0) can be mapped to
the origin L = 1 using a U -transformation, since we can identify Φ(0) with an element of U
and then we multiply the geodesic curve Φ(t) with Φ(0)−1, generating a new geodesic curve
Φ2(t) = Φ(0)
−1Φ(t) that goes through the origin. The origin is invariant under K-rotations
but the tangent space at the origin transforms under the adjoint of K. One can prove that
there always exists an element k ∈ K, such that AdjkΦ̇2(0) ∈ CSA [24]. Therefore χ̇α2 = 0
and this solution must be a straight line. So we started out with a general curve Φ(t) and
proved that the curve Φ3(t) = kΦ(0)
−1Φ(t) is a straight line.
4.2 An illustration from dimensional reduction
The metric Ansatz for the dimensional reduction of (4 + n)-dimensional Einstein-gravity
on the n-torus (Tn) is
ds24+n = e
2αϕds24 + e
2βϕMabdza ⊗ dzb , (29)
where
4(n+ 2)
, β = −
. (30)
The matrix M is a positive-definite symmetric n×n matrix with unit determinant, which
depends on the 4-dimensional coordinates, describing the moduli of Tn. The modulus
ϕ controls the overall volume and is named the breathing mode or radion field. Notice
that we already truncated the Kaluza–Klein vectors in the Ansatz. The reduction of the
Einstein–Hilbert term gives
−g{R− 1
(∂ϕ)2 + 1
Tr∂M∂M−1} . (31)
The scalars parametrize IR × SL(n, IR)/ SO(n) where ϕ belongs to the decoupled IR-part
and M is the SL(n, IR)/ SO(n) part.
If we take the four-dimensional part of space-time to be a flat FLRW-space then that
part of the metric will be power-law with p = 1/3 and the scalars follow a geodesic with
ln τ as an affine parameter. According to the solution-generating technique, the Ansatz for
the scalars is
ϕ = v0 ln τ + c0 , M = ΩDΩT , D = diag(e−
~βa·~φ) , (32)
with ~φ = ~v ln τ and ~β the weights of SL(n, IR) in the fundamental representation (see
appendix B for some explanations on the SL(n, IR)/ SO(n)-coset in this representation).
The diagonal matrix D represents the straight-line solution and Ω is an arbitrary SL(n, IR)-
matrix in the fundamental representation. Therefore M = ΩDΩT is the most general coset
matrix describing a geodesic curve.
The Friedmann equation implies that the affine velocity is restricted to be
v20 + ||v||2 = 43 , (33)
which is the only constraint coming from the 4-dimensional Einstein equation. If we sub-
stitute this solution in (29) and define new coordinates ~y = ~zΩ we find
ds24+n = −τ 2α v0dτ 2 + τ
+2αv0d~x23 +
~βa·~v+2β v0dy2a . (34)
This is similar to what is called a Kasner solution in general relativity (see for instance [25]).
Kasner solutions are a general class of time-dependent geometries that look like
ds2 = −τ 2p0dτ 2 +
τ 2padx2a . (35)
Kasner solutions solve the Einstein equations in vacuum if the following two conditions are
satisfied
p0 + 1 =
pa , (p0 + 1)
p2a . (36)
For the metric (34) these conditions are satisfied if the lower-dimensional Friedmann equa-
tion is satisfied. For this calculation one needs the properties of the weight-vectors ~βa
(given in appendix B) and the relation between α and β (30). We therefore conclude that
the general spatially flat FLRW-solution lifts up to the most general Kasner solution with
SO(3)-symmetry in 4 + n dimensions.
5 Discussion
In this note we have studied multi-field scaling solutions using a first-order formalism for
scalar cosmologies a.k.a. pseudo-supersymmetry. We derived these first-order equations
via a Bogomol’nyi-like method that was known to work for domain wall solutions as was
first shown in [26, 27]5 and we showed that it trivially extends to cosmological solutions.
This first-order formalism allows a better understanding of the geodesic motion that comes
with a specific class of scaling solutions. One of the main results of this note is a proof that
shows that all pseudo-BPS cosmologies that are scaling solutions must be geodesic. This
complements to the discussion in [14] where the first example of a non-geodesic scaling
cosmology was shown to be non-pseudo-BPS. Moreover we gave constraints on multi-field
Lagrangians for which the pseudo-BPS cosmologies are geodesic scaling solutions.
Having illustrated the importance of geodesic motion in scalar cosmology, we tackled
the problem of solving the geodesic equations in the second part of this note. We showed
that the most general geodesic curve can be written down for maximally non-compact
coset spaces U/K. These coset spaces appear in all maximal and some less-extended
supergravities [29]. We used a solution-generating technique based on the symmetries
of the coset. We were able to prove that the most general solution is given by a U-
transformation on the “straight line”, (φI(t) = vIt, χα = 0) in the Borel gauge. We
illustrated this technique for the coset SL(n, IR)/ SO(n). Since SL(n, IR)/ SO(n) is also
the moduli space of the n-torus we applied it to find the cosmological solutions of higher-
dimensional gravity compactified on a n-torus. This exercise nicely illustrates why the
straight line is the generating solution since, from a higher-dimensional point of view,
all solutions that correspond to the non-straight line geodesics can be seen as coordinate
transformations of the solutions associated with the straight line. The oxidation of the
straight line solutions corresponds to the most general SO(3)-invariant Kasner solution of
(4 + n)-dimensional vacuum GR.
The same technique was used in [3] to find all geodesic scaling cosmologies of the
CSO-gaugings in maximal supergravity.
The solution-generating technique presented here should be considered complementary
to the “compensator method” developed by Fré et al in [30]. There the straight line
5See also [28].
also serves as a generating solution but instead of rigid U -transformations one uses local K
transformations that preserve the solvable gauge to generate new non-trivial solutions. This
technique is a nice illustration of the integrability of the second–order geodesic equations
of motion [31].
Acknowledgments
We are grateful to Dennis Westra for useful discussions and comments on the manuscript
and to Jan Rosseel for many useful discussions. This work is supported in part by the Eu-
ropean Communitys Human Potential Programme under contract MRTN-CT-2004-005104
in which the authors are associated to Utrecht University. The work of AP and TVR is
part of the research programme of the Stichting voor Fundamenteel Onderzoek der Materie
(FOM).
A Curvatures
For the metric Ansatz (2) the Ricci tensor is given by
Rab = −ǫ(ηǫ)ab
gġḟ
+ (D − 3) ġ
, Ryy = (D − 1)
− ( g̈
. (37)
B The coset SL(N, IR)/ SO(N)
Consider a general coset U/K. It is not difficult to construct a coset representative using
the Lie algebras U and K of U and K respectively. Since K is a subgroup of U we have the
decomposition U = K ⊕ F, with F the complement of K in U. For a given representation
of the algebra U we define a coset representative via L(y) = exp(yifi) where the fi form a
basis of F in some representation of U.
To derive the metric we define a Lie algebra valued one-form from the coset represen-
tative L(y) via
L−1dL ≡ E + Ω , (38)
where E takes values in F and Ω in K. We notice that L−1dL is invariant under left
multiplication with a y-independent element g ∈ U . Multiplying L from the right with
local elements k ∈ K results in
E → k−1E k , Ω → k−1Ω k + k−1dk . (39)
In supergravity the parameters yi are scalar fields that depend on the space-time coordi-
nates yi = φi(x). The one-form L−1dL can be written out in terms of coset-coordinate
one-forms dφi which themselves can be pulled back to space-time coordinate one-forms
dφi = ∂µφ
idxµ. Now we can write
L−1dL = Eµdx
µ + Ωµdx
µ . (40)
Under the φ-dependent K-transformations k(φ(x)) we have that Ωµ → k−1Ωµk + k−1∂µk
and Eµ → k−1Eµk. It is clear that Eµ is covariant under local K-transformations and
Ωµ transforms like a connection. Using this connection Ωµ we can make the following
K-covariant derivative on L and L−1
DµL = ∂µL− LΩµ , DµL−1 = ∂µL−1 + ΩµL−1 . (41)
To find a kinetic term for the scalars we notice that the object
Tr[DµLD
µL−1] = −Tr[EµEµ] , (42)
has all the right properties as it contains single derivatives on the scalars, it is a space-time
scalar, it is invariant under rigid U transformations and under local K-transformations.
Thus,
e−1Lscalar = −Tr[EµEµ] ≡ −12g(φ)ij∂µφ
i∂µφj . (43)
If SO(N) is the maximal compact subgroup of U and we work in the fundamental
representation, then the Lie algebra of SO(N) is the vector space of antisymmetric matrices,
L−1dL+ (L−1dL)T
, Ω =
L−1dL− (L−1dL)T
, (44)
and a calculation shows that
e−1Lscalar = −Tr[E2] = +14Tr[∂M∂M
−1] , (45)
where M is the SO(N)-invariant matrix M = LLT .
No we specify to U = SL(N, IR). In general SL(N, IR) has rank N − 1 and its maximal
compact subgroup is SO(N). There will therefore be N−1 dilaton fields φI and N(N−1)/2
axion fields χα. The Cartan generators are given in terms of the weights ~β of SL(N, IR) in
the fundamental representation
( ~H)ij = (~βi)δij . (46)
The weights can be taken to obey the following algebra
βiI = 0 ,
βiIβiJ = 2δIJ , ~βi · ~βj = 2δij −
. (47)
The first of these identities holds in all bases since it follows from the tracelessness of the
SL generators. The second and third identity can be seen as convenient normalizations of
the generators. The positive step operators Eij are all upper triangular and a handy basis
is that they have only one non-zero entry [Eij ]ij = 1. The negative step operators are the
transpose of the positive. The SO(N) algebra is spanned by the following combinations
(Eβ − E−β) . (48)
The action will generically look complicated but when all axions are set to zero L is diagonal
L = diag[ exp(−1
~βi · ~φ)] and the action becomes
Tr∂M∂M−1 = −1
βiJβiI)∂φ
I∂φJ = −1
δIJ∂φ
I∂φJ . (49)
This action describes N − 1 dilatons that parametrize the flat scalar manifold IRN−1.
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	Preliminaries
	(Pseudo-) supersymmetry
	Multi-field scaling cosmologies
	Pure kinetic solutions
	Potential-kinetic scaling solutions
	Geodesic curves and the Borel gauge
	A solution-generating technique
	An illustration from dimensional reduction
	Discussion
	Curvatures
	The coset SL(N,IR)/SO(N)
