document: cosmological models with scalar field matter have been much studied in the context of inflation and , more recently , in the context of the late - time acceleration that is indicated by current astronomical observations ( see @xcite for a recent review ) . 
   one theoretical motivation for these studies is that scalar fields arise naturally from the compactification of higher - dimensional theories , such as string or m - theory . 
   however , the type of scalar field potential obtained in these compactifications is sufficiently restrictive that until recently it was considered to be difficult to get accelerating cosmologies in this way , although the existence of an accelerating phase in a hyperbolic ( @xmath13 ) universe obtained by compactification had been noted @xcite , and non - perturbative effects in m - theory have since been shown to allow unstable de sitter vacua @xcite .    in an earlier paper , we pointed out that compactification on a compact hyperbolic manifold with a time - dependent volume modulus naturally leads to a flat ( @xmath14 ) expanding universe that undergoes a transient period of accelerating expansion @xcite . 
   numerous subsequent studies have shown that such cosmological solutions are typical to all compactifications that involve compact hyperbolic spaces or non - vanishing @xmath15-form field strengths ( flux ) @xcite , and this was additionally confirmed in a systematic study @xcite . 
   furthermore , the transient acceleration in these models is easily understood @xcite in terms of the positive scalar field potential that both hyperbolic and flux compactifications produce in the effective , lower - dimensional , action . 
   this perspective also makes clear the generic nature of transient acceleration .    for 
   any realistic application one would want the lower - dimensional spacetime to be four - dimensional , but for theoretical studies it is useful to consider a general @xmath16-dimensional spacetime . assuming that we have gravity , described by a metric @xmath17 , coupled to @xmath18 scalar fields @xmath19 taking values in a riemannian target space with metric @xmath20 and with potential energy @xmath1 
   , the effective action must take the form @xmath21 where @xmath22 is the ( spacetime ) ricci scalar . 
   we are interested in solutions of the field equations of the action ( [ action ] ) for which the line element has the friedmann - lematre - robertson - walker ( flrw ) form for a homogeneous and isotropic spacetime . in standard 
   coordinates , @xmath23 where the function @xmath24 is the scale factor , and @xmath25 represents the @xmath26-dimensional spatial sections of constant curvature @xmath27 . 
   we normalise @xmath27 such that it may take values @xmath28 for a riemann tensor @xmath29 with metric @xmath30 on @xmath31 . 
   the scalar fields are taken to depend only on time , which is the only choice compatible with the symmetries of flrw spacetimes . 
   the universe is expanding if @xmath32 and accelerating if @xmath33 . 
   we need only discuss expanding cosmologies because contracting cosmologies are obtained by time - reversal .    in some simple cases , 
   the target space has a one - dimensional factor parametrised by a dilaton field @xmath34 , and the potential takes the form @xmath35 for some ` dilaton coupling ' constant @xmath36 and constant @xmath37 . 
   this model is of special interest , in part because of its amenability to analysis . 
   the special case for which the dilaton is the _ only _ scalar field was analysed many years ago ( for @xmath38 ) using the observation that , for an appropriate choice of time variable , cosmological solutions correspond to trajectories in the ` phase - plane ' parametrised by the first time - derivatives of the dilaton and the scale factor @xcite . 
   this method ( which has recently been extended to potentials that are the sum of two exponentials @xcite ) allows a simple visualisation of all possible cosmological trajectories . 
   moreover , all trajectories for flat cosmologies can be found explicitly @xcite ( see also @xcite ) , and a related method allows a visualisation of their global nature @xcite . it was noted in @xcite that there is both a ` critical ' and a ` hypercritical ' value of the dilaton coupling constant @xmath36 , at which the set of trajectories undergoes a qualitative change . in spacetime dimension 
   @xmath16 , these values are @xcite @xmath39 below the critical value ( @xmath40 ) there exists a late - time attractor universe undergoing accelerating expansion , whereas only transient acceleration is possible above it . 
   the hypercritical coupling ( @xmath41 ) separates ` steep ' exponential potentials ( which arise in flux compactifications ) and ` gentle ' exponential potentials ( which arise from hyperbolic compactification , in which case @xmath42 so the potential is still too steep to allow eternal acceleration ) . 
   one aim of this paper is to generalise this type of analysis to the multi - scalar case . for @xmath14 , 
   this has been done already for what could be called a ` multi - dilaton ' model @xcite , and for the multi - scalar model with exponential potential ( [ eq.exppot ] ) @xcite . here 
   we consider _ all _ cosmological trajectories ( arbitrary @xmath27 ) for an exponential potential of either sign , and for any spacetime dimension @xmath16 . 
   in particular , we find exact solutions for all flat cosmologies when @xmath43 , following the method used in @xcite for @xmath44 , and the exact phase - plane trajectories for all @xmath27 when @xmath45 . 
   a more ambitious aim of this paper is to determine what can be said about cosmologies with more general scalar potentials . what kind of model - independent behaviour can one expect , and how generic is the phenomenon of transient acceleration ? 
   exponential potentials are simple partly because of the power - law attractor cosmologies that they permit , but such simple solutions do not occur for other potentials so other methods are needed . in this paper 
   , we develop an alternative method of visualising cosmological solutions that applies to _ any _ potential , and we illustrate it by an application to exponential potentials . 
   as we explain , exponential potentials serve as ` reference potentials ' in determining the late - time behaviour of cosmologies arising in a large class of models with other potentials . 
   our starting point for the new formalism is the observation @xcite that flat cosmological solutions of gravity coupled to @xmath18 scalar fields with @xmath5 can be viewed as null geodesics in an ` augmented ' target space of dimension @xmath46 with a metric of lorentzian signature ( see @xcite for related results ) 
   . trajectories in this space corresponding to non - flat cosmologies are neither null nor geodesic . 
   however , we will see that they are projections of null geodesics in a ` doubly - augmented ' target space with a metric of signature @xmath47 for @xmath13 and signature @xmath48 for @xmath49 . 
   it turns out that the extension of these results to @xmath50 is very simple . 
   flat cosmologies are again geodesics in the augmented target space but with respect to a conformally rescaled metric ; the geodesic is timelike if @xmath44 and spacelike if @xmath43 . 
   null geodesics are unaffected by the conformal rescaling and hence continue to correspond to ( @xmath14 ) cosmologies for @xmath5 . 
   the analysis of acceleration is also simple in this framework : within the lightcone is an ` acceleration cone ' , and if a geodesic enters its acceleration cone then the corresponding universe will accelerate . in the case of a flat target space with an exponential potential , the fixed point 
   geodesics are just straight lines , and the universe is accelerating if the straight line lies within the acceleration cone . 
   the further extension to @xmath51 cosmologies is achieved exactly as in the @xmath5 case . 
   cosmological trajectories are projections of geodesics in the doubly - augmented target space with respect to a conformally rescaled metric of signature @xmath47 for @xmath13 and @xmath48 for @xmath49 . 
   the geodesic is timelike if @xmath44 , null if @xmath5 and spacelike if @xmath43 . 
   the plan of this paper is as follows . 
   section [ sec.mscalar ] reviews , in our conventions , the equations of motion in multi - scalar cosmologies , and we derive a useful alternative criterion for the acceleration of the scale factor . in section [ sec.fixedpoints ] , we consider possible fixed points , concluding that these occur only for exponential potentials , and we determine the nature of these fixed points for arbitrary @xmath16 , @xmath27 and sign of the potential . in section [ sec.phase ] 
   we discuss the phase - plane trajectories and present some exact @xmath43 solutions . in section [ sec.manifold ] 
   we develop the interpretation of multi - scalar cosmologies as geodesic motion . in section [ sec.applications ] 
   we introduce the notion of the acceleration cone , and we show how the cosmologies arising in exponential - potential models fit into the new framework ; other potentials of current interest are also considered . 
   we summarise in section [ sec.discussion ] . 
   we begin by briefly reviewing the equations of motion for multi - scalar cosmologies that follow from the action ( [ action ] ) . 
   these equations will be given in two different forms , distinguished by the choice of time coordinate . 
   the second form will be useful to our analysis in the next section of possible ` fixed points ' , which correspond to power - law cosmologies in the first form .    to simplify the equations we define @xmath52 where @xmath53 is the levi - civit connection for the target space metric @xmath54 , and we introduce the following ` characteristic functions ' of the potential v : @xmath55 these are the components of a 1-form on the target space dual to a vector field @xmath56 . in a vector notation 
   , the scalar field equation can now be written as @xmath57 where @xmath58 is the hubble function . 
   the friedmann constraint is @xmath59 = \left(d-1\right)\alpha_c^{-2}k,\ ] ] where the norm @xmath60 ( and hence the inner product ) is the one induced by the target space metric : @xmath61 . 
   differentiating the friedmann constraint , and using the scalar field equation ( [ eq.sca ] ) , we deduce the ` acceleration equation ' @xmath62 is the ` critical ' constant given in ( [ exponents ] ) . 
   clearly , acceleration is possible only if @xmath44 . 
   note too that @xmath63 can not vanish unless either @xmath43 or @xmath64 ( or @xmath5 , @xmath14 and @xmath65 ) so only in these cases can an expanding universe recollapse .    for the following section , it will prove useful to rewrite the above equations in terms of a new time coordinate @xmath66 , defined as a function of @xmath67 by the relation @xmath68 note that we have allowed for the possibility that @xmath6 , although @xmath5 is excluded . 
   we will also set @xmath69 in terms of the variable @xmath70 , the condition @xmath71 for expansion is @xmath72 , while the condition @xmath73 for acceleration is @xmath74 where the overdot denotes differentiation with respect to @xmath75 .    in the new time 
   coordinate , the scalar field equation becomes @xmath76 and the friedmann constraint becomes @xmath77 = \left(d-1\right)\alpha_c^{-2 } k\,.\ ] ] the acceleration equation is now @xmath78 and the condition ( [ accel1 ] ) for acceleration is therefore equivalent to @xmath79 , and hence valid for all @xmath80 . 
   as expected , it can be satisfied only if @xmath44 . 
   we now wish to determine whether the system of equations ( [ newscalar ] , [ accelerationeq ] ) subject to the constraint ( [ newfried ] ) , admits any fixed point solutions for which @xmath81 we shall see that these conditions are consistent only for exponential potentials , for which the equations ( [ newscalar ] , [ accelerationeq ] ) become those of an autonomous dynamical system , and we determine the fixed points and their type for all @xmath16 , @xmath27 and for either sign of the potential . for this analysis , 
   it is convenient to introduce the quantity @xmath82 recall that @xmath83 is the ` hypercritical ' constant given in ( [ exponents ] ) .      given @xmath84 , equation ( [ newscalar ] ) yields @xmath85 where @xmath86 solves the quadratic equation @xmath87 given @xmath88 , equation ( [ accelerationeq ] ) yields @xmath89 \nonumber\\ & = & \left(d-2\right)c^2a^2 + { 1\over2}\left(d-1\right ) \left(d-2\right)\left[\dot\beta^2 - ca^2 \dot\beta \right ] \end{aligned}\ ] ] where we have used ( [ cterms ] ) to get to the second line . 
   using this equation to eliminate the @xmath90 term from ( [ quadc ] ) we arrive at the quadratic equation @xmath91 there are therefore two types of fixed point :    * @xmath92 . using this in ( [ quadc ] ) we deduce that @xmath93 and that @xmath94 this requires @xmath95 and @xmath96 it then follows that @xmath97 this is just the friedmann constraint for @xmath14 , so this type of fixed point occurs only for @xmath14 . 
   * @xmath98 . using this in ( [ quadc ] ) 
   we deduce that @xmath99 it follows that this type of fixed point can occur only for @xmath44 , and that @xmath100 using this in the friedmann constraint we deduce that @xmath101 we may exclude the possibility that @xmath102 because this yields a @xmath14 fixed point of the type already considered , so @xmath51 and we need @xmath64 for @xmath40 and @xmath103 for @xmath104 . note that @xmath105 for this type of fixed point , so the fixed point cosmology is neither accelerating nor decelerating . 
   the above results are in complete analogy with those of @xcite for the one - scalar case . to see why 
   , it should first be appreciated that the fixed point conditions have been derived on the assumption that @xmath106 is constant , and that @xmath107 is covariantly constant . 
   it then follows from ( [ cterms ] , [ quadc ] ) that @xmath56 must be covariantly constant too . 
   however , for any space that admits a non - zero covariantly constant vector , there exist coordinates in which this vector is _ constant _ , not just covariantly constant . for coordinates 
   in which @xmath56 is constant the potential takes the form @xmath108 which is of the form ( [ eq.exppot ] ) with @xmath109 . 
   moreover , for any target space on which this potential is globally defined , one can find new coordinates such that @xmath110 where @xmath111 is a metric on a ` reduced ' target space with @xmath112 coordinates @xmath113 . in these target space 
   coordinates , the scalar field equations are @xmath114 and the friedmann constraint is @xmath115   - \left(d-1\right)\dot\beta^2 =    k\left(d-1\right)e^{2\left(a\varphi-\beta\right ) } |v_0|^{-1}\ , .\ ] ] these imply the acceleration equation for @xmath116 . if we suppose that the reduced target space is flat then the scalar field equations , together with the acceleration equation , define an autonomous dynamical system : @xmath117 - \dot\beta^2 + a\dot\beta \dot\varphi\ , , \nonumber\\ \ddot\chi & = & a\dot\varphi\ ,   \dot\chi - \left(d-1\right)\dot\beta\ , \dot\chi\ , . 
   \end{aligned}\ ] ] note that we may consistently set @xmath118 to recover the equations of the one - scalar model . as there are no fixed points with @xmath119 , the fixed points of the multi - scalar model are the same as those of the one - scalar model . 
   if the reduced target space is not flat then a connection term must be added to the last equation in ( [ autonomous ] ) . 
   as this introduces dependence on @xmath113 , the equations no longer define an autonomous dynamical system in just @xmath46 variables . 
   it is not clear how much difference this makes to the results : note that the connection term has no effect on the stability of fixed points with @xmath118 because it is quadratic in @xmath120 . 
   however , we will suppose for the analysis to follow that the reduced target space _ is _ flat . we may then assume with no essential loss of generality that it is also one - dimensional ( i.e. @xmath121 ) so we have an autonomous dynamical system for just three variables . 
   we linearise about a fixed point with @xmath122 and @xmath123 by writing @xmath124 the linearised equations for @xmath125 take the form @xmath126 , where @xmath127 is the @xmath128 matrix @xmath129 the eigenvalues of this matrix determine the nature of the fixed point . 
   we consider the two types of fixed point in turn :    * @xmath14 . in this case 
   @xmath130 the eigenvalues of @xmath127 are @xmath131 for an expanding universe we must take the top sign ( corresponding , for @xmath44 , to the upper branch of the @xmath14 hyperboloid ) . 
   then we have a stable node ( all eigenvalues real and negative ) for @xmath40 . 
   for @xmath132 we have a saddle , with one real - negative and two real - positive eigenvalues ; the instability due to the positive eigenvalues is what leads to the ` flatness problem ' of standard big - bang cosmology . 
   + for @xmath43 we may have @xmath133 , in which case the fixed point is an unstable node ( all eigenvalues real and positive ) . 
   all @xmath13 trajectories start at this unstable fixed point and end at the stable fixed point on the other branch of the @xmath14 hyperboloid ; as @xmath134 at this other fixed point , all @xmath13 universes recollapse to a big crunch singularity . 
   however , for @xmath135 there is a long quasi - static period on some of these trajectories . 
   * @xmath51 . in this case 
   @xmath136 the eigenvalues of @xmath127 are @xmath137 , \qquad \mp 2a/\alpha_c\,,\ ] ] where @xmath138 for one eigenvalue and @xmath139 for another . 
   take the top sign again . for @xmath140 
   all eigenvalues are real but one is positive and the others negative , so we have a saddle point . for @xmath42 
   all eigenvalues are real and negative provided _ either _ @xmath141 _ or _ ( if @xmath142 ) @xmath143 otherwise ( i.e. @xmath142 and @xmath144 ) we have one real negative eigenvalue and two complex eigenvalues with negative real parts , and hence a stable ` spiral - node ' ( a spiral in the @xmath145 plane and a node in the @xmath146 direction ) . note that @xmath147 where the equality occurs for @xmath148 . 
   thus , for any @xmath16 the @xmath51 fixed point is an unstable saddle if @xmath40 , and is stable for @xmath42 but can be either a node or a spiral - node . for @xmath141 
   it is always a node , but for @xmath148 it becomes a spiral - node at @xmath41 and for @xmath149 it becomes a spiral - node at @xmath150 .    in the supergravity context 
   we are restricted to @xmath151 , but there is no scalar potential possible in @xmath152 and the only possible potential in @xmath153 is a positive exponential ( with @xmath154 so there is no fixed - point solution ) . 
   we now have all the information needed to determine the qualitative behaviour of all phase - space trajectories . 
   we begin with the one - scalar model , obtained by setting @xmath155 . 
   the trajectories for @xmath44 are sketched in @xcite but , for the reader s convenience , we give a brief summary in words . 
   all @xmath14 trajectories begin , after a big bang singularity , with a period of kinetic energy domination with effectively vanishing potential . 
   for @xmath133 they end up approaching a similar late - time phase , but for @xmath156 they approach the @xmath14 fixed - point solution , which is accelerating if @xmath40 and decelerating if @xmath42 . for @xmath40 , all @xmath51 trajectories sufficiently close to a @xmath14 trajectory also approach the @xmath14 fixed point ; in fact , all @xmath13 trajectories have this property . 
   the behaviour of a @xmath49 trajectory is determined by its relation to the separatrix of the unstable @xmath49 fixed point ( which is the einstein static universe for @xmath157 ) ; those on one side approach the @xmath14 fixed point while those on the other side represent universes that recollapse to a big crunch . 
   for @xmath42 _ all _ @xmath49 universes recollapse to a big crunch whereas all @xmath13 universes now approach a zero - acceleration milne universe ( the @xmath14 fixed point now being a saddle point ) . for @xmath144 
   the @xmath13 fixed point is an attractor spiral so that all @xmath13 trajectories spiral around a point of zero - acceleration ; as observed in @xcite , this leads to eternal oscillation between acceleration and deceleration . 
   for @xmath43 the @xmath14 trajectories were studied in @xcite but there seems to be no complete study of all trajectories for this case . 
   we will therefore present some results of the one - scalar model for @xmath43 before moving on to discuss modifications that arise in the multi - scalar case . 
   it is well known that anti - de sitter space is a solution of einstein s equations with a negative cosmological constant . 
   this space must correspond to one of the cosmological trajectories in a model with @xmath45 ( which implies that @xmath157 ) . 
   set @xmath158 to get the equations @xmath159\ ] ] where @xmath160 these equations imply that @xmath161 = cx^{2/(d-1)}\ ] ] for constant @xmath162 . 
   the curves with @xmath163 are the @xmath49 trajectories and those with @xmath164 are the @xmath13 trajectories . 
   the two branches of the @xmath165 curve , a hyperbola , are the @xmath14 trajectories . 
   the ads solution corresponds to @xmath166 , and hence @xmath167 ; for this special case we have @xmath168\ ] ] which has the solution @xmath169 , and hence @xmath170 the friedmann constraint implies that @xmath13 , as expected , and @xmath171,\ ] ] and hence @xmath172 this is anti - de sitter space . 
   in contrast to de sitter space , which can be realised ( for @xmath173 ) as a flrw universe for _ any _ @xmath27 ( in particular as the @xmath14 fixed - point solution ) , anti - de sitter space is realised as an flrw cosmology only for @xmath13 , and it does not correspond to a fixed point cosmology ( at least in the sense of this paper ) . 
   we now turn to the @xmath14 trajectories for arbitrary @xmath36 . in this case 
   we have to solve @xmath174 subject to the constraint @xmath175 the constraint is solved by setting @xmath176 and the equation of motion becomes @xmath177.\ ] ] this equation is immediately integrated if @xmath41 . 
   having found @xmath178 we integrate ( [ xydef ] ) to deduce that @xmath179^{1\over 2\left(d-1\right)}\,,\ ] ] for integration constant @xmath180 ( we have set to zero the other integration constant as it can be absorbed into @xmath37 ) . 
   if @xmath93 then we introduce the new variable @xmath181 the two branches of the @xmath14 hyperbola correspond to positive and negative @xmath182 ; we take @xmath183 . 
   we now consider separately the cases @xmath133 and @xmath156 :    * @xmath156 . in this case 
   the equation for @xmath182 is @xmath184,\ ] ] where we recall that @xmath185 is the function of @xmath36 given in ( [ omegadef ] ) . 
   integration of ( [ xydef ] ) now leads to @xmath186^{{\alpha_c^2\over    2\alpha_h\omega^2}}\ , . 
   \end{aligned}\ ] ] * @xmath154 . in this case 
   the equation for @xmath182 is @xmath187 there is a fixed point at @xmath188 , so we must distinguish between @xmath189 and @xmath190 . 
   one finds that @xmath191 for @xmath189 integration of ( [ xydef ] ) yields @xmath192^{\alpha_c^2\over    2\alpha_h\omega^2}\ , . 
   \end{aligned}\ ] ] for @xmath190 integration of ( [ xydef ] ) yields @xmath193^{\alpha_c^2\over    2\alpha_h\omega^2}\ , . 
   \end{aligned}\ ] ]      in the multi - scalar case , the ` spectator ' fields @xmath113 have an effect on the scale factor due to the @xmath194 term in the acceleration equation , and in the friedmann constraint ( [ friedchi ] ) . 
   phase space trajectories with non - zero @xmath120 will depend on the reduced target space metric but , as long as there are no new fixed points , one may expect the qualitative behaviour to be independent of this metric . 
   let us therefore continue to suppose that the target space ( and hence the reduced target space ) is flat . in this case , all @xmath14 trajectories lie in a hyperboloid in phase space that separates the @xmath13 and @xmath49 trajectories . 
   this is true for either sign of @xmath195 but we will now suppose that @xmath44 ; in this case , we see from ( [ eq.accpot ] ) that acceleration occurs whenever the phase - plane trajectory enters the region with hyperspherical boundary @xmath196 . 
   this was observed in @xcite in the context of an analysis of @xmath14 trajectories , but it is also valid for non - zero @xmath27 . 
   the only exponential models with eternally accelerating cosmological solutions are those for which a fixed point lies inside or on the sphere of acceleration , as happens when @xmath140 . 
   for all other values of @xmath197 , any acceleration can be at most transient , and only a subset of the trajectories undergo even transient acceleration ; these are the trajectories that pass through the sphere of acceleration . 
   the more scalar fields there are , the more freedom there is to avoid the sphere of acceleration , so transient acceleration in multi - scalar models is less generic than it is in the single - scalar model . to make this statement 
   quantitative would require an understanding , along the lines of @xcite , of what the appropriate measure might be on the space of trajectories . 
   the most significant fact about a @xmath14 fixed point is whether it lies inside or outside the sphere of acceleration . 
   figure [ fig.k0psdia ] illustrates a few examples in a model with two scalar fields and a positive exponential , scalar potential . 
   we now develop a method to describe the generic evolution of multi - scalar cosmologies . 
   it will be convenient to define a new independent variable @xmath198 by @xmath199 although we will still use @xmath70 where this is more convenient . 
   the friedmann constraint can now be written as @xmath200 and the scalar field equation as @xmath201 these two equations imply the acceleration equation @xmath202 = 0\,.\ ] ] using the friedmann constraint to eliminate @xmath203 from the scalar and acceleration equations we have @xmath204 & = &   k\,\alpha_c^{-2}\left(d-1\right)e^{-2\beta}\ ,   \mathbf{a}\ , , \nonumber\\ \partial_t^2\gamma + \alpha_h \left|\partial_t{{\bm{\phi}}}\right|^2   & = & k\,\alpha_c^{-2}\alpha_h e^{-2\beta}\ , . 
   \label{accgamma}\end{aligned}\ ] ] before considering the general case ( arbitrary @xmath27 and arbitrary @xmath195 ) , we first discuss the special cases of @xmath5 ( but arbitrary @xmath27 ) and @xmath14 ( but arbitrary @xmath195 ) . 
   we may consider the @xmath205 variables @xmath206 to be maps from the cosmological trajectory to an ` augmented ' target space . in this notation , 
   the friedmann constraint for @xmath5 can be written as @xmath207 where @xmath208 is a _ lorentz - signature _ metric on the augmented target space . using the friedmann constraint to eliminate the @xmath209 term in the second equation in ( [ accgamma ] ) yields @xmath210 if this is taken together with the scalar field equation then the two combine into the single ( albeit coordinate - dependent ) equation @xmath211 for @xmath14 this is the equation for a geodesic in a non - affine parametrisation . in terms of the the new time variable @xmath212 defined by @xmath213 we have @xmath214 for @xmath14 
   this is the equation of an affinely parametrised geodesic . 
   although the cosmological trajectory in the augmented target space is not a geodesic for @xmath51 , it can be viewed as the projection of a geodesic in a ` doubly - augmented ' target space of dimension @xmath215 that is foliated by hypersurfaces isometric to the augmented target space . 
   let @xmath216 be the @xmath215 coordinates . 
   we take the metric to be @xmath217 where @xmath218 
   as we now have one more variable we also need another equation . 
   we take this to be the ` projection equation ' @xmath219 with this choice , the friedmann constraint becomes @xmath220 and the combined scalar field and acceleration equations ( [ knonzero ] ) are equivalent to the single geodesic equation @xmath221 thus , cosmological trajectories for @xmath5 are null geodesics in the doubly - augmented target space . 
   note that the signature of this space is lorentzian if @xmath103 but non - lorentzian ( with two ` times ' ) if @xmath64 . 
   because @xmath222 is not constant , the motion is not restricted to a single hypersurface of constant @xmath222 and this accounts for the fact that the projection of the motion onto one such hypersurface is not geodesic . 
   this argument fails for @xmath14 , for which the projected motion _ is _ geodesic , because the metric on the doubly - augmented target space is degenerate when @xmath14 .      the friedmann constraint for @xmath14 , but arbitrary @xmath195 , can be written as @xmath223 thus , the cosmological trajectory in this space is timelike for @xmath44 , null for @xmath5 and spacelike for @xmath43 . 
   as we have seen , it is a null _ geodesic _ when @xmath5 . 
   when @xmath50 the cosmological trajectories are no longer geodesic with respect to the metric @xmath224 , but they _ are _ geodesics with respect to the conformally - rescaled metric @xmath225 where the conformal factor is @xmath226 in fact , one finds that the equations ( [ basiceqs ] ) are equivalent , for @xmath14 , to the equation @xmath227 where @xmath228 is the levi - civit connection for @xmath229 , and when @xmath5 as @xmath230 is undefined in this case . ] 
   @xmath231 this is the equation for a geodesic in a non - affine parametrisation ; noting that @xmath232 we deduce that the geodesics are affinely parametrised by a new time - coordinate @xmath233 for which @xmath234 in other words , @xmath235 the friedmann constraint now takes the form @xmath236 notice that @xmath233 differs from @xmath212 as defined ( for @xmath5 ) by ( [ tprime ] ) . 
   in fact , @xmath237 this difference occurs because the affine parameter of a null geodesic is affected by a conformal rescaling of the metric , and the metric we are now considering is the conformally rescaled one @xmath238 ; clearly a null curve that is geodesic with respect to @xmath239 is also geodesic with respect to any other conformally equivalent metric , such as @xmath238 , but the affine parametrisation will differ in general . 
   in contrast , when @xmath50 the cosmological trajectories in the augmented target space are geodesics with respect to a unique metric ( up to homothety ) in the class of metrics that are conformally equivalent to @xmath239 , and this metric is @xmath238 .      the steps that led to ( [ affinegeo ] ) for @xmath14 lead , for general @xmath27 , to the equation @xmath240 
   although this is not the equation of a geodesic when @xmath51 , it is the projection of a geodesic in the doubly - augmented target space with respect to the conformally rescaled metric @xmath241 where the conformal factor @xmath242 is as given in ( [ conformalfac ] ) , and @xmath243 is as given in ( [ doubleaug ] ) . from ( [ phistar ] ) and ( [ ttprime ] ) we deduce that @xmath244 given this , one then finds that ( [ general ] ) is equivalent to the geodesic equation @xmath245 and that the friedmann constraint is @xmath246 thus 
   , the general cosmological trajectory is a geodesic in the doubly - augmented target space , one that is timelike for @xmath44 , null for @xmath5 and spacelike for @xmath43 . 
   we now consider some applications of the formalism just developed . as we are particularly interested in accelerating cosmologies 
   we first consider the implications of the acceleration condition . 
   we will then see how to interpret the fixed point cosmologies that arise for exponential potentials . 
   this turns out to be useful when considering the asymptotic behaviour of more general potentials . 
   we first show how the condition for acceleration ( [ eq.accpot ] ) acquires a geometrical meaning in the new framework . 
   we will now assume that @xmath44 , as acceleration can not otherwise occur , and before proceeding we note that the friedmann constraint ( [ newfried ] ) can be written as @xmath247    we now introduce the following ` acceleration ' metric on the augmented target space , @xmath248 and the corresponding conformally rescaled metric @xmath249 where the conformal factor is the same as before . 
   noting that @xmath250 one finds that @xmath251 using the friedmann constraint in the form ( [ friedform ] ) to eliminate the @xmath252 term , we deduce that @xmath253 +   { k\over 2\alpha_c^{2}v}\ , e^{-2\beta}\,.\ ] ] recalling that acceleration occurs when @xmath254 , we see that the condition for acceleration is equivalent , for @xmath14 , to @xmath255 geometrically , this states that a universe corresponding to a timelike geodesic trajectory is accelerating when its tangent vector lies within a subcone of the lightcone defined by the acceleration metric on the augmented target space . 
   as one might expect , an analogous result holds for @xmath51 in terms of the doubly - augmented target space . in this case , the acceleration metric is @xmath256 . 
   \end{aligned}\ ] ] using the projection equation ( [ phistarhat ] ) one finds that @xmath257 , \ ] ] where @xmath258 is the conformally rescaled acceleration metric . 
   the condition for acceleration is therefore equivalent to @xmath259 this states that a universe is accelerating when the tangent to its geodesic trajectory in the doubly - augmented target space lies with an acceleration subcone of the lightcone . 
   we now discuss how the cosmologies arising for an exponential potential of the form ( [ exppot2 ] ) fit into the new geometrical framework . for reasons given earlier we assume in this case that the target space is flat , in which case the augmented target space metric @xmath239 is also flat , and hence @xmath238 is conformally flat , with conformal factor @xmath260.\ ] ]    our first task is to determine the trajectories in the augmented target space that are associated with the fixed point solutions . 
   consider first the @xmath14 fixed point . 
   this solution has the property that the linear function @xmath261 is time - independent : @xmath262 . in other words , @xmath263 is a vector tangent to the hyperplanes of constant @xmath264 . for @xmath265 
   there is only one such direction , which is determined by the gradient of @xmath242 . 
   thus , @xmath266 for @xmath265 . 
   in fact , this remains true for arbitrary @xmath18 as a direct calculation shows : @xmath267 a fixed point solution for @xmath51 has the property that the linear function @xmath268 is time - independent . as surfaces of constant @xmath269 are hyperplanes , a @xmath51 fixed point trajectory is _ also _ a straight line , as a direct computation confirms : @xmath270 thus , fixed point solutions are ( particular ) straight - line trajectories in the augmented target space . 
   in addition , as follows from our earlier results , the @xmath14 straight - line trajectories are geodesics with respect to the metric @xmath271 , whereas this is not true for @xmath51 . 
   this difference can be understood as follows : as we have assumed a flat target space , the metric @xmath239 is flat and _ all _ straight lines are geodesics with respect to it . 
   however , the relevant metric is @xmath238 , and the only geodesic of @xmath239 that is also a geodesic of @xmath238 is the line of steepest descent of the function @xmath242 ; as we have seen , this is precisely the direction of the @xmath14 fixed point trajectory . 
   note that the line determined by ( [ direction.nonzerok ] ) is a generator of the acceleration cone ( as expected from the zero - acceleration property of the @xmath51 fixed point ) . 
   more generally , if a straight line trajectory lies within the acceleration cone then it corresponds to an eternally accelerating universe , and if it lies outside the acceleration cone ( @xmath42 ) then it corresponds to an eternally decelerating universe . in the latter case 
   it might also lie outside the ` lightcone ' ( @xmath133 ) , in which case the geodesic corresponds to the @xmath14 fixed point for @xmath43 . 
   note that a change in the value of the constant vector @xmath56 effects a lorentz transformation of the @xmath14 straight - line trajectory in the augmented target space . 
   such a transformation can take any timelike line into any other timelike line , and any spacelike line into any other spacelike line , but it can not take a spacelike line to a timelike one or vice versa ( as expected from the fact that this requires a change of sign of the potential ) . also , it can not take a timelike or spacelike line into a null line , a fact that is consistent with the absence of a fixed - point solution for @xmath41 . 
   generic geodesics , corresponding to generic @xmath14 cosmologies , are not straight lines in the extended target space , but they still have a simple description . the regime where the solutions are dominated by kinetic energy is given by geodesics with tangent vectors having large @xmath272 . 
   this means that the generic geodesics start out in null directions , as could be anticipated from the fact that @xmath14 cosmologies are null geodesics when @xmath5 . 
   let us now follow the subsequent evolution for the @xmath44 case . as the potential becomes more important , the geodesics bend into the timelike cone , ultimately approaching the timelike straight line corresponding to the fixed point solution if @xmath156 . on the other hand , 
   if @xmath273 , then the geodesic ultimately approaches a null straight - line geodesic . 
   for the single - scalar case , this behaviour is shown in figure [ fig.k0geodesics ] . 
   cosmologies accelerate precisely when their corresponding geodesics bend their tangent vectors into the acceleration cone .    for @xmath51 , 
   cosmological trajectories are no longer geodesics in the augmented target space but they are projections of geodesics in the doubly - augmented target space . the vector tangent to the geodesic corresponding to a @xmath51 fixed point is @xmath274 where we have used ( [ phistarhat ] ) to get the last entry . note that this is _ not _ a constant @xmath215-vector ( even allowing for the fact that @xmath275 is constant at the fixed point ) so the fixed point solution is not a straight line in the doubly - augmented target space , as was to be expected because its metric is neither flat nor conformally flat . however , using ( [ fixedcurve ] ) , and then ( [ keq ] ) , one finds that @xmath276 = 0\,.\ ] ] this again confirms the zero acceleration of the @xmath51 fixed point cosmologies .      the asymptotic ( late - time ) behaviour of a large class of cosmologies in models with rather general potentials can be determined by comparison with the asymptotic behaviour of cosmologies in models with exponential potentials . 
   to demonstrate this , we discuss the case of flat cosmologies with positive potentials . in this case 
   , the scalar fields must approach @xmath277 at late times , where some components of the constant @xmath18-vector @xmath278 may be infinite . 
   the characteristic functions of the potential in this limit are also constant : @xmath279 the absolute value @xmath280 determines the late - time behaviour , by comparison with the critical and hypercritical exponents : if @xmath281 , then there will be late - time acceleration , with the trajectory approaching an approximate accelerating attractor ; obvious examples are models for which @xmath195 has a strictly positive lower bound , so most models of interest will be those for which @xmath195 either tends to zero or has a minimum at zero . for @xmath282 , there is late - time deceleration corresponding to the existence of an approximate late - time decelerating attractor . 
   the asymptotic geodesics in both these cases are timelike . in contrast , for @xmath283 the timelike geodesic asymptotes a null ( and hence decelerating ) trajectory .    to illustrate the above observations , we consider the following single - scalar examples .    * inverse power law potentials : @xmath284 these potentials have been much studied in quintessence models ( see e.g. @xcite ) . 
   they also arise from dynamical breaking of supersymmetry @xcite . with @xmath285 , 
   one finds @xmath286 . 
   this implies late - time acceleration , _ even though the potential tends to zero at late - times_. * that any other value of @xmath287 may occur is illustrated by the potential @xmath288 for which @xmath289 if we suppose that @xmath290 at late times . for such cosmologies the asymptotic behaviour is the same as it would be in the exponential potential model with dilaton coupling constant @xmath36 . 
   * it may happen that at late times the scalar fields become trapped near a minimum of the potential . 
   near such a minimum , which we may assume to occur at the origin of field space , the potential takes the form @xmath291 for integer @xmath292 . 
   for positive @xmath293 one finds @xmath294 which tends to zero ( i.e. @xmath286 ) as @xmath295 for any @xmath296 . 
   this implies late - time acceleration , due to the effective cosmological constant @xmath293 . 
   however , we may now consider the limit @xmath297 , in which case @xmath298 , implying that the late - time trajectory is null , and hence decelerating . 
   in this paper we have developed a geometrical method for the classification and visualisation of homogeneous isotropic cosmologies in multi - scalar models with an arbitrary scalar potential @xmath195 . 
   the method involves two steps . in the first step , 
   the target space parametrised by the @xmath18 scalar fields is augmented to a larger @xmath205-dimensional space of lorentzian signature , the logarithm of the scale factor playing the role of time . 
   this is reminiscent of the role of the scale factor in mini - superspace models of quantum cosmology , especially if one views the scalar fields of our model as moduli - fields for extra dimensions . in this case 
   , the lorentzian metric on the augmented target space is the one induced from the wheeler - dewitt metric on the space of higher - dimensional metrics . 
   this was the point of view adopted in @xcite , where it was also observed that when @xmath5 all flat ( @xmath14 ) cosmologies are null geodesics in this metric ( an observation that goes back to work of dewitt on kasner metrics @xcite ) . 
   when @xmath50 , flat cosmologies again correspond to trajectories in the augmented target space but they are neither null nor geodesic 
   . however , as we have shown , _ all _ flat cosmologies are geodesics with respect to a _ conformally rescaled _ metric on the augmented target space , the conformal factor depending both on the scale factor and the potential . 
   the conformal rescaling has no effect on null geodesics , of course , so flat cosmologies are still null geodesics whenever @xmath5 , but now the trajectories of flat cosmologies are timelike geodesics when @xmath44 and spacelike geodesics when @xmath43 . 
   this is true not only for a potential of fixed sign but also for one that changes sign ; in this case the tangent to the geodesic becomes null just as the scalar fields take values for which @xmath5 . 
   the second step , which is needed for non - flat ( @xmath299 ) cosmologies , is to further enlarge the target space to a ` doubly - augmented ' target space foliated by copies of the augmented target space . 
   one can choose the metric on this space , of signature @xmath47 for @xmath13 and @xmath48 for @xmath300 and degenerate for @xmath14 , such that geodesics yield ( on projection ) cosmological trajectories for _ any _ @xmath27 when projected onto a given hypersurface , and the geodesics are again timelike , null or spacelike according to whether @xmath195 is positive , zero or negative . 
   this general construction thus includes all the previous ones as special cases . 
   accelerating cosmologies have a simple interpretation in this new framework . within the lightcone ( defined by the metric with respect to which the trajectories are geodesics ) 
   there is an acceleration subcone . 
   a given trajectory corresponds to an accelerating universe whenever its tangent vector lies within the acceleration cone . in particular 
   , a flat cosmology undergoes acceleration whenever its geodesic trajectory bends into the acceleration cone within the lightcone of the augmented target space . 
   this can happen only if the trajectory is timelike , which it will be if the potential is positive . 
   we have also presented in this paper a complete treatment ( complementing many previous studies ) of cosmological trajectories , for any @xmath27 and any spacetime dimension @xmath16 , for the special case of simple exponential potentials . 
   these potentials are characterised by a sign ( the potential may be positive or negative ) and a ( dilaton ) coupling constant @xmath36 ( the magnitude of an @xmath18-vector coupling constant @xmath56 ) . as has long been appreciated @xcite , cosmological solutions in such models 
   correspond , for an appropriate choice of time parameter , to trajectories of an autonomous dynamical system in a ` phase space ' parametrised by the time - derivatives of the scale factor and scalar fields . 
   the qualitative features of these trajectories ( which should not be confused with trajectories in the ` augmented target space ' parametrized by the fields themselves ) are determined by the position and nature of any fixed points , which are of two types . 
   there is always a @xmath14 fixed point unless @xmath41 ( the ` hypercritical ' value of @xmath36 defined by this property ) but it occurs only for @xmath44 if @xmath133 and only for @xmath43 if @xmath133 . 
   there is also a @xmath51 fixed point if @xmath44 , which coincides with the @xmath14 fixed point when @xmath102 , where @xmath301 is the ` critical ' value of @xmath36 ( defined as the value at which fixed point cosmologies have zero acceleration ) . 
   these fixed point cosmologies have a very simple interpretation as trajectories in the augmented target space : they are straight lines . in the @xmath14 case 
   these lines are also geodesics . for @xmath43 
   these geodesic lines lie outside the lightcone . for @xmath44 
   they lie inside the lightcone , but may lie either inside or outside the acceleration cone . for models obtained by ( classical ) compactification from a higher dimensional theory without a scalar field potential , the @xmath14 
   geodesic line always lies outside the acceleration cone , so only transient acceleration is possible in these models . for a single scalar field 
   the transient acceleration is generic in the sense that it is a feature of all @xmath14 trajectories when @xmath133 and of some when @xmath156 . 
   in contrast , in the multi - scalar case , there are trajectories that correspond to eternally decelerating universes even when @xmath133 , so transient acceleration is less generic for more than one scalar field . 
   although exponential potentials are of limited phenomenological value , they are important as ` reference potentials ' in determining the asymptotic behaviour of cosmologies in models with other potentials , such as inverse power potentials . essentially , any potential that falls to zero slower than the critical exponential potential ( as do inverse power potentials ) will lead to late - time eternal acceleration . 
   any potential that falls to zero faster than the hypercritical exponential potential will have a late - time behaviour that is approximately that of a model with zero potential . 
   of course , there is no evidence for the existence of cosmological scalar fields , but there _ is _ evidence that the expansion of the universe is accelerating and hence for dark energy . 
   whatever produces this energy , it seems reasonable to suppose that it can be modeled by scalar fields . 
   if so , it is possible that future observations may be interpreted as telling us something about the potential energy of these fields . in view of our current complete ignorance of what this potential 
   might be , we have tried , as much as possible , to understand generic properties , and we hope that the methods developed here will be of further use in this respect . 
   the interpretation of solutions of gravitational theories as geodesics in a suitable metric space has a considerable history of which we were mostly unaware at the time of writing this paper . 
   the geodesic interpretation of @xmath14 cosmologies that we have presented is closely related to the maupertuis - jacobi principle of classical mechanics . 
   relatively recent work on this topic includes @xcite 
   . a particle mechanics formulation of the geodesic interpretation described here , including our proposed extension to @xmath51 , has recently been found , and applied to a model in which cosmological singularities correspond to horizons in the augmented target space @xcite .
summary: for gravity coupled to @xmath0 scalar fields , with arbitrary potential @xmath1 , it is shown that all flat ( homogeneous and isotropic ) cosmologies correspond to geodesics in an @xmath2-dimensional ` augmented ' target space of lorentzian signature @xmath3 , timelike if @xmath4 , null if @xmath5 and spacelike if @xmath6 . 
   accelerating cosmologies correspond to timelike geodesics that lie within an ` acceleration subcone ' of the ` lightcone ' . 
   non - flat ( @xmath7 ) cosmologies are shown to evolve as projections of geodesic motion in a space of dimension @xmath8 , of signature @xmath9 for @xmath10 and signature @xmath11 for @xmath12 . 
   this formalism is illustrated by cosmological solutions of models with an exponential potential , which are comprehensively analysed ; the late - time behaviour for other potentials of current interest is deduced by comparison .