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cosmological models obtained by dimensional reduction of 10-dimensional supergravity theories are an active field of research @xcite@xcite . since supergravity is the low energy limit of string theory , these models should provide a way towards understanding the quantum effects in cosmology . so far , there has been much activity in investigating models including fields of the ns - ns - sector . the discovery , that the conformal field theory of p - brane - like superstring backgrounds charged under rr - gauge fields is that of open strings with dirichlet boundary conditions @xcite , has made the rr - field contents of type ii string theories a very interesting subject for string cosmological models . recently , in a first study of cosmological models including rr - fields , it was found that inflating phases can occur in a certain range of parameters which label the solutions to the equations of motion @xcite . the ns - ns-3-form field strength had been set to zero in the model and the compactification ansatz was chosen to contain a number of maximally symmetric flat subspaces . the two branches of the solutions were separated and curvature singularities appeared in each branch .    in this paper we construct curved 4-dimensional cosmological models containing a ns - ns-3-form field strength as well as nontrivial type iib rr - fields . to construct the 4-dimensional model , we use a 5-brane - solution of type iib supergravity whose rr - sector consists of a 3-form and a scalar . by sl(2,r)-invariance of the action , which is realized as a sl(2,z)-subgroup in iib superstring theory , we generate 5-brane - solutions with nontrivial rr - fields and compactify the transverse coordinates on a 5-torus , giving a black hole in 5 dimensions . after generalizing the black hole solution to constant curvature solutions we compactify to 4 dimensions , obtaining isotropic frw - metrics together with nontrivial rr - fields . the outline of the paper is as follows : in section 2 we introduce the sl(2,r)-transformed 5-brane - solution of type iib supergravity and compactify it on a 5-torus . in section 3 we make one further compactification to 4 dimensions and obtain frw - models including nontrivial rr - fields in this way . in section 4 we investigate the influence of the rr - fields on the cosmological models with a detailed analysis of the inflationary phase . section 5 is devoted to the extremal limit of the model . our starting point is the bosonic part of the 10-dimensional sl(2,r)-invariant type iib supergravity action @xcite [ iib ] s_10=d^10x \ { e^-2 ( r + 4 ( ) ^2 - h_1 ^ 2 ) - ( ) ^2 - ( h_1 + h_2)^2 } . in this action the string frame metric @xmath2 , the dilaton @xmath3 and the three form field strength @xmath4 are the familiar fields contained in the ns - ns sector of all 10-dimensional string theories . the three form field strength @xmath5 and the scalar @xmath6 belong to the rr - sector . the bosonic field content of iib superstring theory also includes a selfdual five form field strength , which we have set to zero in order to be able to write a covariant action . a 5-brane - solution to this action is @xcite [ sol10 m ] ds_(10)^2= - dt^2 + + r^2 d_3 ^ 2 + _ ijdx^idx^j for the 10-dimensional line element in the string frame and [ sol10f ] = -(1-(r_-/r)^2 ) , h_1=q_3,h_2==0 , for the remaining background fields . the magnetic charge is @xmath7 and @xmath8 and @xmath9 are the line element and volume element of a 3-sphere , respectively , i.e d_3&=&d^2+^2(d^2+^2d^2 ) + _ 3&=&^2ddd . the rr - fields @xmath10 and @xmath6 are not excited in this solution . we use sl(2,r)-invariance of the action ( [ iib ] ) to generate nontrivial rr - fields , making the solution ( [ sol10m],[sol10f ] ) an intrinsic iib solution . a manifest sl(2,r)-invariant form of ( [ iib ] ) was given in @xcite and also used there to construct sl(2,r)-transformed 5-brane - solutions ( see also @xcite for black hole solutions of type ii stringtheory ) . we do not hodge - dualize the rr-3-form here , instead we construct a solution possessing two magnetic charges .    following @xcite , the action of the sl(2,r)-matrix @xmath11 on the background fields is  & = & + e^- & = & + h_1 & = & dh_1 + h_2 & = & -bh_1 . this transformation clearly includes s - duality for the choice @xmath12 , which interchanges the two 3-forms and inverts the string coupling @xmath13 . in detail , the transformed scalar fields are [ sl2sol10f ] &= & + &=&-(1-(r_-/r)^2)+ ( d^2+c^2(1-(r_-/r)^2 ) ) and the string frame line element changes to [ sl2sol10 m ] ds^2_(10)=ds_(10)^2 . the einstein frame metric is not changed by the sl(2,r ) transformation . we compactify this solution to five dimensions by wrapping the five transverse coordinates of the 5-brane over a 5-torus , i.e.  we make the ansatz for the string frame metric ds^2=_dx^dx^+ e^a _ ijdx^i dx^j , where @xmath14 is a 5-dimensional lorentz metric , @xmath15 are coordinates on the 5-torus and @xmath16 depends on the @xmath17 only . we neglect the moduli coming from compactifying the three forms as well as vector fields coming from the metric . the effective 5-dimensional action is [ 5dstring ] s_5&=&d^5x \{e^- ( + ( ) ^2 - ( ) ^2 - h_1 ^ 2 ) - + & & - e^ ( ( ) ^2 + ( h_1+h_2)^2 ) } , where we have used the rescaled modulus field @xmath18 and the 5-dimensional dilaton @xmath19 is given by @xmath20 . from the transformed 5-brane - solution ( [ sl2sol10f ] , [ sl2sol10 m ] ) we get a 5-dimensional solution [ sl2sol5string ] ds_5 ^ 2&= & + h_1&=&dq_3,h_2=-bq_3 + & = & + & = & ( d^2 + c^2(1-(r_-/r)^2 ) ) + & = & . for @xmath21 the identity transformation , i.e.  vanishing rr - fields @xmath10 and @xmath6 , this is just the 5-dimensional black hole solution of @xcite , written in the string frame . in these equations @xmath22 and @xmath23 are the values of the parameter @xmath24 at the outer and inner horizon , respectively . the metric has a curvature singularity at @xmath25 and also at the spacelike surface @xmath26 whereas @xmath22 is a regular event horizon . this is a black hole solution for @xmath27 only , in the other case the solution describes a naked singularity . a particular interesting solution is the extremal limit at @xmath28 . we will discuss the non - extremal and extremal cases in a cosmological context in the next sections . following @xcite , where this solution was generalized to constant curvature spaces , we can produce a 4-dimensional cosmological solution from this black hole by kaluza - klein compactification of the time coordinate and re - interpretation of the radius coordinate as 4-dimensional time . in order to get the right signature after compactification we must change signature from @xmath29 to @xmath30 in the 5-dimensional solution . the generalization of ( [ sl2sol5string ] ) to constant curvature spaces is given by [ sol5gen ] ds_5 ^ 2&= & + h_1&=&dq_k , h_2=-bq_k , q=2t_+t_- + & = & + & = & ( d^2 + c^2(1-(t_-/t)^2 ) ) + & = & , where @xmath31 labels the constant curvature three - spaces with line element and volume element d_k&=&d^2+()^2 ( d^2+^2d^2 ) , + _ k&=&()^2 ddd . it should be emphasized that the time coordinate in this solution is the radius coordinate of the 5-dimensional black hole and the signature is different from usual black hole physics . we reduce to 4 dimensions using the ansatz ds_5 ^ 2=y^2dy^2+ds_4 ^ 2 where the @xmath32-modulus @xmath33 turns out to be y^2(t ) = . by introducing @xmath34 we arrive at the 4-dimensional action [ cosstring ] s_4&=&d^4x\ { e^-2(r+4()^2-()^2 - ( d)^2 -h_1 ^ 2 ) - + & & -e^+(()^2 + ( h_1+h_2)^2 ) } . the 4-dimensional dilaton @xmath35 is given by @xmath36 . the 5-dimensional black hole ( [ sol5gen ] ) reduces to the 4-dimensional frw - solution [ frwstring ] ds_4 ^ 2&= & ( -+ t^2 d_k^2 ) + h_1&=&dq_k , h_2=-bq_k , q=2t_+t_- + & = & ( ) + & = & ( ) + & = & ( d^2+c^2(1-t_-^2/t^2 ) ) + & = & for the special choice of sl(2,r)-parameters @xmath37 , the rr - fields @xmath10 and @xmath6 and the @xmath38-modulus field @xmath39 are zero and we recover the cosmological models described in @xcite . these models have been studied in detail there , both in the string and einstein frame . since the sl(2,r)-transformation does not affect the einstein metric , we concentrate on the string frame in the following . let us briefly recall the properties of the model in the pure ns - ns - case , i.e.  without contributions from the rr - fields . the solution describes a closed @xmath40 , open @xmath41 , or flat @xmath42 cosmology respectively . in the string frame , for @xmath43 , the three space oscillates ) . ] between the extrema @xmath44 . for @xmath45 the solution is not oscillating , the extension of the corresponding three spaces is bounded from below at @xmath46 and tends to infinity as @xmath47 . the dilaton @xmath35 is always divergent at @xmath46 , corresponding to the minimal extension of the three spaces . for @xmath43 , the dilaton is also divergent at @xmath48 , while for @xmath45 , where the time region has no upper bound , it tends to zero for @xmath47 . the @xmath32-modulus @xmath49 is divergent at @xmath46 , indicating decompactification to 5 dimensions at this points . for @xmath43 , the @xmath32-radius becomes zero at the maximum extension of three - space . in the other cases @xmath45 , the modulus tends to a constant value as @xmath50 .    in the einstein frame , the curvature scalar is singular at @xmath46 in all three cases and also at @xmath48 in the case @xmath43 . the extension of three - space shrinks to zero at these points , indicating a big - crunch or big - bang . let us now study the qualitative changes caused by adding nontrivial rr - fields to the solution described above . as can be seen from the dilaton expression in ( [ frwstring ] ) , in the case @xmath51 one divergence cancels leaving the expression for the dilaton _ d=0= ( ) . for @xmath43 , there is still a divergence at @xmath48 , but the divergence at @xmath46 has disappeared in all cases @xmath52 . the case @xmath51 corresponds to zero ns - ns - charge and a constant rr - scalar @xmath53 . we want to see how the scale factors of the frw - models behave in this special case . for this we change to a coordinate system which is usually used to study frw - cosmologies ds^2=-d^2+a^2()d_k^2 . the comoving time coordinate @xmath54 is defined by a solution to the differential equation [ tau ] d^2=dt^2 , t(=0)=t_- and the scale factor @xmath55 is [ scalefactor ] a^2()=t^2 ( ) . for @xmath51 the scale factor @xmath55 goes to zero at @xmath56 . calculating @xmath57 we find the two different behaviors _ 0=\ {    r@:l 0 & d= 0 + & d=0    . so by switching off the ns - ns-3-form @xmath58 completely , we produce a singularity in the string frame scale factor , whereas a finite value of the ns - ns - charge smoothes out this singularity . at the same time we find that the @xmath38-compactification radius is zero for @xmath51 at @xmath46 , whereas the @xmath32-radius is still divergent there . the dilaton @xmath35 takes the minimum value at @xmath46 for @xmath51 and it diverges there for @xmath59 . the magnetic charge of the ns - ns-3-form prevents the universe from collapsing , as was already recognized in @xcite . without ns - ns - charge , the scale factor exhibits a singularity regardless of the existence of rr - charge . the qualitative behavior of the dilaton - singularities does not change when rr - fields are added to a non - zero ns - ns - field strength . the minimum extension of the universe at @xmath46 is @xmath60 and for @xmath43 the maximum extension is given by @xmath61 . however , the duration and effectivity of a possible initial inflationary phase depend on the parameters c and d as well as on @xmath62 and @xmath63 , as can be seen from the equations ( [ tau ] ) and ( [ scalefactor ] ) . we will discuss this dependence in detail now .      in the presence of a ns - ns - charge the string frame scale factor of the cosmology is free of singularities , although in the einstein frame the familiar curvature singularity close to the time @xmath64 appears . it is interesting to investigate how this feature is affected by adding rr - fields and whether a phenomenologically interesting inflationary phase can occur . from a phenomenological point of view one can argue that the einstein frame is the preferred coordinate system . on the other hand one can treat both frames on equal footing , as has been done in recent years in numerous analyses where the transition of a superinflationary pre - big bang phase into a a frw type cosmology was worked out in the string frame @xcite@xcite .    in the following we will examine in more detail the effect of rr - fields on the string frame scale factor close to the time @xmath64 , where the influence of these fields are most apparent . it was pointed out in @xcite that in the asymptotic limit @xmath65 the expansion scales like @xmath66 and from the discussion in the previous section we know that the s - dual model contains a singularity in the scale factor at this point . one can easily imagine that close to @xmath56 a short phase of superluminal expansion might occur in the sense of power - law inflation with @xmath67 and @xmath68 sufficiently large to ensure an expansion that could be of phenomenological interest .    as a measure to describe an accelerated expansion of the universe we use the quantity @xmath69 . here @xmath70 is the hubble parameter @xmath71 , a dot means the derivative with respect to @xmath54 and the last part of the equation holds if the expansion is of power law type , @xmath67 . in the general case we are not able to find analytical expressions for the quantity @xmath72 , because equation ( [ tau ] ) can not be integrated directly . instead we will give numerical solutions for a certain choice of parameters in fig . it is shown how the acceleration changes with transformations that lie within a so(2,r)-subgroup of sl(2,r ) , parametrized by the rotation angle @xmath73 . an increasing rotation angle corresponds to an enlargement of the fields of the rr - sector and simultaneously a reduction of the ns - ns-3-form field strength . we observe that the initial acceleration is very much enhanced by exchanging ns - ns - charge by rr - fields but a subsequent decelerating phase becomes also gradually more important and thus taking off much of what was gained in the accelerating phase . at the s - dual point we encounter again the transition from a non - singular to the singular universe and @xmath74 . the number of e - foldings @xmath75 produced during the expansion increases only slightly by enlarging the rr - fields and remains of the order of one for almost all values of @xmath73 . since @xmath75 must exceed a certain limit as a prerequisite for a viable inflationary scenario , none of the cosmologies fulfills this requirement . an expression for the number of e - foldings produced between the time @xmath76 and @xmath77 can easily be derived from ( [ scalefactor ] ) ( t;c)= ( ) + ( d+ ( 1-()^2 ) ) . since the string frame metric is only influenced by the sl(2,r)-parameters @xmath78 and @xmath79 , we study the case where for a fixed value of d ( fixed ns - ns - charge ) the parameter @xmath78 is increased . for the specific choice @xmath80 and @xmath78 a free parameter , it becomes apparent that the rr - fields enhances the number of e - folds only logarithmically . however , for @xmath62 being very close to zero , @xmath75 can become sufficiently large for having cosmological significance . upon expanding @xmath55 to second order in @xmath54 it can also be seen how the singular case is recovered , a()=t_- - ( k- ( ) ^2 ) ( 1 + ) ^2 + o ( ^4 ) . the pre - factor of the quadratic term gets large for small @xmath62 and @xmath79 and for large @xmath78 . also , the behavior of @xmath72 with this choice of parameters is qualitatively the same as shown in fig .  1 for the so(2 ) transformed models , as well as for all other sets that have been investigated . moreover , curved spatial sections ( @xmath81 ) have only little influence on the qualitative picture . it was shown in @xcite that for vanishing rr - fields the 5-dimensional solution ( [ sol5gen ] ) has an extremal limit @xmath82 , where the theory corresponds to a wzw model ) part . for @xmath83 , the 2d part corresponds to a @xmath84 coset model and for @xmath85 it is trivial . ] and is therefore exact to all orders in @xmath86 . whereas for p - brane superstring backgrounds charged under rr gauge fields the corresponding conformal field theory is known to be that of open strings with dirichlet boundary conditions @xcite , the higher order contributions to the tree level equations of motions for non - p - brane like backgrounds still lack an explicit sigma model description @xcite . for this the extremal limit described above may not remain exact after performing an sl(2,r ) transformation . the corresponding 4d solution , valid for arbitrary @xmath87 , is ds^2&=&t_-^2 ( -d^2+d^2_k ) + & = & ( ( ) ^2 ) + & = & - ( ) + & = & ( d^2+c^2 ( ) ^2 ) + & = & + h_1&=&2d t_-^2_k , h_2=-2b t_-^2_k in this solution , @xmath88 is the conformal time defined by the transformation [ conftime ] t^2=t^2_-+(t_+^2-kt_-^2)()^2 and a possible constant part of the 5-dimensional dilaton in ( [ sol5gen ] ) has to be adjusted @xcite _ 0 _ 0 + ( t_+^2-kt_-^2 ) in order to be able to perform the extremal limit . for vanishing rr - fields , we recover again the solution given in @xcite . in this limit , the geometry is simply @xmath89 for all three cases ( see figure 2a ) , where @xmath90 stands for a sphere , pseudo - sphere or flat space respectively . adding rr - fields to the solution , the universe behaves different for different values of @xmath87 . for @xmath91 , the conformal factor is oscillating between @xmath92 and @xmath93 . in the other cases the universe approaches the minimum size @xmath92 for @xmath94 and then expands for ever . especially for the s - dual case , where the ns - ns - field @xmath58 is switched off , the corresponding solutions exhibit again singularities ( see figures 2b - d ) . 2a    2b     +    2c    2d we constructed 4-dimensional frw - cosmologies from 5-dimensional black hole solutions following @xcite . our models contain nontrivial rr - fields , a 3-form and a scalar , which are generated by sl(2,r ) transformations of a black 5-brane solution of type iib supergravity . the effect of this rr - fields on the resulting cosmologies have been studied . for the s - dual model , where the ns - ns - charge is replaced by the corresponding rr - charge , the dilaton singularity at the minimum extension of spatial 3-space disappears , but the model still decompactifies to 5 dimensions there . also , other than the ns - ns - field strength , the rr - field strength can not prevent the universe from collapsing , and we find a singularity in the string frame scale factor , which is absent when the ns - ns - charge is different from zero . it was further shown that the influence of rr - fields on the cosmic expansion results in a large increase of the initial accelerating phase . it is however followed by a subsequent deceleration phase that increases in a similar way . for that reason the number of e - folds produced during the combined phases increases only slightly , almost logarithmically , with the rr - fields being turned on . thus , such models do not seem to be suitable for an explanation of spatial homogeneity and isotropy , especially that of the cmb radiation , based on inflationary cosmology . however , for a small fraction of parameter space , the superluminal expansion of the physical space is effective enough to solve some of the problems inflationary cosmology was invented for . we examined also the extremal limit of the model , for which the 5-dimensional black hole is on the verge of becoming a naked singularity . since we generalized the 5-dimensional solution to a constant curvature solution , it is not clear whether the successful description of extremal black holes as being composed of non - interacting branes , anti - branes and momentum @xcite can be applied to this extremal limit . the effect of rr - fields added to the extremal limit of the pure ns - ns case is to render dynamics to the steady universe . for the s - dual case we find again singularities in the string frame scale factor .    during writing this paper @xcite appeared , where also cosmological models from toroidally compactification of type ii theories are discussed . we would like to thank s.frste and r.dick for useful discussions . this work was supported by the `` sonderforschungsbereich 375 - 95 fr astro - teilchenphysik der deutschen forschungsgemeinschaft '' and the european commission tmr programs erbfmrx - ct96 - 0045 and erbfmrx - ct96 - 0090 . the work of s.s .  is supported by deutsche forschungsgemeinschaft . 99 a.a.tseytlin , int . j. mod d 1 ( 1992 ) 223 i.antoniadis , j.rizos , k.tamvakis , nucl . b 415 ( 1994 ) 497 e.j.copeland , a.lahiri , d.wands , phys . rev . d 51 ( 1995 ) 1569 r.easther , k.maeda , hep - 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we construct 4-dimensional cosmological frw  models by compactifying a black 5-brane solution of type iib supergravity , which carries both magnetic ns - ns - charge and rr - charge . 
 the influence of nontrivial rr - fields on the dynamics of the cosmological models is investigated . 
 = -1 cm -0.3 cm .2 cm    lmu - tpw 96 - 27 + tum - hep-258/96 + sfb-375/122 + hep - th/9610166 + october 1996 +     r. poppe@xmath0 , s. schwager@xmath1    .5 cm _ @xmath0 physik department , technische universitt mnchen + 85748 garching , germany _ .2 cm _ 
 @xmath1 sektion physik , universitt mnchen + theresienstr . 
 37 , 80333 mnchen , germany _