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the flow equations approach , pioneered by hoffman and turner @xcite and by kinney @xcite ( see also refs .
@xcite ) , is a means of generating large numbers of inflation models via a random process , permitting for example that ensemble to be compared with observational results as done by the wilkinson microwave anisotropy probe team @xcite .
it has however been pointed out by liddle @xcite that the flow equations algorithm is actually rather insensitive to the equations governing inflationary dynamics . in particular , the friedmann equation is not needed to determine the flow equation trajectories in the space of slow - roll parameters ; its only role is to measure the amount of expansion taking place along those trajectories .    in this
_ brief report _
, we make an explicit study of the effect of modifying the friedmann equation by considering the flow equations in the simplest braneworld inflation scenario , based on the randall  sundrum type ii model @xcite .
the flow equations are a set of differential equations linking a set of slow - roll parameters defined from the hubble parameter @xmath0 . in the standard cosmology , following the notation of kinney @xcite , the parameters can be defined as @xmath1 where primes are derivatives with respect to the scalar field @xmath2 . using the relation @xmath3 where we define the number of @xmath4-foldings @xmath5 as _ decreasing _ with increasing time ,
yields the flow equations @xmath6(^\ell\lambda_{{\rm h}})+^{\ell+1}\!\!\ !      \lambda_{{\rm h } } \ ; ; \ ; \ell \ge2 \ , , \nonumber\end{aligned}\ ] ] where @xmath7 is a convenient definition .    as pointed out in ref .
@xcite , these equations actually have limited dynamical input from inflation , since the above derivation has been made without reference to the friedmann equation . indeed ,
if written in the form @xmath8 they are a set of identities true for any function @xmath9 , and the reparametrization to @xmath10 modifies only the measure along the trajectories , not the trajectories themselves . in that light
it seems surprising that they can say much about inflation at all , but it turns out that the flow equations can be viewed as a ( rather complicated ) algorithm for generating functions @xmath11 which have a suitable form to be interpreted as inflationary models @xcite .
@xcite implied that the flow equation predictions ought to be little changed by moving to the braneworld , although this modifies the friedmann equation .
however this statement needs explicit justification , because although the trajectories are unaffected , there are several changes which affect the predictions : the measure of @xmath4-foldings along the trajectories changes , the endpoints of the trajectories change , and the equations relating the slow - roll parameters to the observables change . in this short article
we investigate these effects , restricting ourselves to the high - energy regime of the randall  sundrum type ii model @xcite .
we follow the notation of ref .  @xcite . in the randall  sundrum type ii braneworld model @xcite , the friedmann equation in the high - energy regime
can be written as @xmath12 where @xmath13 is the five - dimensional planck mass , related to the brane tension @xmath14 by @xmath15 .
the scalar wave equation is unchanged , and a useful expression is @xmath16    although they could be used , the standard cosmology definitions of the slow - roll parameters , eqs .
( [ e : eps ] ) and ( [ e : srdef ] ) , are actually not very convenient in the braneworld scenario , in particular because @xmath17 is no longer the condition to end inflation . instead , following ref .
@xcite , we define new slow - roll parameters relevant to the high - energy regime as follows : @xmath18 by analogy to the standard cosmology case , we define the higher - order slow - roll parameters as @xmath19 where one can identify @xmath20 .    from eq .
( [ eq : phi ] ) , the relation between @xmath2 and @xmath5 , measuring the length along the trajectories , is changed to @xmath21 from these definitions , and taking the convention we find a set of flow equations @xmath22\,;\\ \frac{d(^{\ell}\lambda_{\rm h})}{d\phi } & = & \frac{1}{\epsilon_{\rm h } } \frac{h'}{h}\left\ { \left[(\ell-1)\eta_{\rm h}-(\ell+1)\epsilon_{\rm h}\right](^{\ell}\lambda_{\rm h } ) \right . \quad
\\ & &   \quad \quad \left .
+ ^{(\ell+1)}\lambda_{\rm h}\right\}\ , ; \quad \ell\ge 2 \ , , \nonumber\end{aligned}\ ] ] which leads to the following set of flow equations for the braneworld @xmath23(^{\ell}\lambda_{\rm h } ) \\ & & \quad \quad + ^{(\ell+1)}\lambda_{\rm h } ; \,\ , \ell\ge 2 \ , , \nonumber\end{aligned}\ ] ] where @xmath24 .
note that the braneworld slow - roll parameters were purposefully defined @xcite so that the end of inflation is at @xmath25 , and so that standard expression for the spectral index @xmath26 still applies ; however the expression for the tensor - to - scalar ratio @xmath27 is modified from the usual @xmath28 . in common use ,
the other one being 16 times this one .
we use this convention to follow kinney @xcite . ] the observable quantities at first - order are given by @xcite @xmath29    following ref .
@xcite , we can analyze the fixed points where all the derivatives vanish , for which the conditions are @xmath30 as in the standard cosmology , these correspond to @xmath31 and are stable for @xmath32 .
the @xmath33 branch is stable for integration backwards in time .
we solve the flow equations following the method of kinney @xcite , both for the standard cosmology where we verify his results and for the system of braneworld flow equations written above .
we consider 60,000 initial conditions drawn from the ranges kinney uses ( taking the same ranges also for the braneworld slow - roll parameters ) .
the results are shown in fig .
[ fig ] , with observables computed to first - order in slow - roll .        for both models ,
the majority of the points lie effectively on the @xmath34 axis , and in addition we see in each case the now - familiar swathe of points following a tight diagonal locus to large values of @xmath27 and @xmath35 @xcite , plus some other scattered points .
the two distributions are extremely similar , though the swathe for the braneworld is slightly below that of the standard cosmology .
we conclude therefore that the change in the dynamical equations does not significantly affect the distribution of points in the space of observables .
we have modified the flow equations approach to implement it in the high - energy regime of the randall  sundrum type ii braneworld cosmology .
although the flow equation trajectories are independent of the dynamical equation driving inflation , changes do occur because the measure of length ( i.e.  the number of @xmath4-foldings ) along the trajectories changes , because the point corresponding to the end of inflation changes , and because the formulae giving the observables change . nevertheless , we have shown that those effects are small and that the distribution of observables predicted by the braneworld flow equations is very similar to that of the standard cosmology .
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we generalize the flow equations approach to inflationary model building to the randall  sundrum type ii braneworld scenario . as the flow equations are quite insensitive to the expansion dynamics 
 , we find results similar to , though not identical to , those found in the standard cosmology .