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recent proposals in theoretical physics have suggested that our four - dimensional universe could lie on a brane embedded in a higher - dimensional space - time  @xcite . one of the most studied is the randall  sundrum ( rs ) model  @xcite where our four - dimensional universe is a brane embedded in five - dimensional anti - de sitter space - time ( ads@xmath0 ) . indeed it is possible in this framework to have only one brane in a non - compact space and this is the scenario we will consider in this letter . this brane - world perspective may dramatically change our picture of the early universe . in this model , the friedmann equation is modified  @xcite to @xmath1 so that the hubble parameter @xmath2 at high energy - density . bulk gravity is strongly coupled to the dynamics on the brane at high energy , which changes the evolution of cosmological perturbations @xcite . thus , it is important to establish whether the evolution of the perturbations generated during inflation could be significantly different from standard four - dimensional slow - roll inflation . in a recent paper @xcite we calculated these effects for the simplest realization of slow - roll inflation on a brane . the purpose of this letter is to illustrate the effect of these brane - world corrections on the spectral index of density perturbations created during inflation in some specific models .    the simplest way to realize inflation in the rs model is to have a single scalar field , the inflaton , confined to the brane  @xcite and only gravity in the bulk . the amplitude of the resulting scalar curvature perturbations is given by @xmath3 where @xmath4 is the time derivative of the inflaton @xmath5 and @xmath6 is the inflaton fluctuation on a spatially flat hypersurface . if one makes the assumption that back - reaction due to metric perturbations in the bulk can be neglected , the quantum expectation value of the inflaton fluctuations on super - horizon scales in the de sitter space - time is @xmath7 . it has been shown that the curvature perturbation @xmath8 on comoving hypersurfaces is constant outside the horizon in this model  @xcite and , indeed , independently of the gravitational theory so long as energy - momentum is conserved  @xcite . one can then use eq . ( [ curvature0 ] ) as initial conditions to calculate observables like cosmic microwave background ( cmb ) temperature anisotropy . although the formula  ( [ curvature0 ] ) for the curvature perturbation is exactly the same as in the four - dimensional case , the relation between the hubble parameter , @xmath9 , and the scalar field potential , @xmath10 , is different  @xcite due to the modified friedmann equation  ( [ modfriedmann ] ) . this changes the prediction of the spectrum of scalar perturbations for a given potential and observational predictions of brane - world inflation have been made using this approach  @xcite . a crucial assumption is that back - reaction due to metric perturbations in the bulk can be neglected . in the extreme slow - roll limit this is necessarily correct because the coupling between inflaton fluctuations and metric perturbations vanishes ; however , this is not necessarily the case when slow - roll corrections are included in the calculation . previous work  @xcite has shown that such bulk effects can be subtle and interesting ( see also @xcite for other approaches ) . in particular , sub - horizon inflaton fluctuations on a brane excite an infinite ladder of kaluza klein modes of the bulk metric perturbations at first order in slow - roll parameters , and a naive slow - roll expansion breaks down in the high - energy regime once one takes into account the back - reaction of the bulk metric perturbations , as confirmed by direct numerical simulations  @xcite . however , an order - one correction to the behaviour of inflaton fluctuations on sub - horizon scales does not necessarily imply that the amplitude of the inflaton perturbations receives corrections of order one on large scales ; one must consistently quantise the coupled brane inflaton fluctuations and bulk metric perturbations . this requires a detailed analysis of the coupled brane - bulk system @xcite . the rs model is a brane - world model where the background spacetime is ads@xmath0 and matter fields are confined to a 3-brane . we will be considering the single - brane model first proposed in ref . @xcite . the brane - world inflation model under consideration has the action @xmath11 where @xmath12 is the trace of the extrinsic curvature of the boundary @xmath13 , the ads curvature scale is defined by @xmath14 and the tension @xmath15 is tuned to make the brane flat if it has no other energy density . the brane is our four - dimensional universe and , because we are interested in studying early - universe inflation , we will make the slow - roll approximation , where the induced metric is assumed to be de sitter but with a slowly changing hubble parameter . we will describe this in coordinates where the background line element has the form @xmath16\,,\ ] ] with the function @xmath17 given by @xmath18  @xcite . here , @xmath9 is the hubble parameter , which is constant in a pure de sitter universe , and the coordinate @xmath19 is conformal time on the brane . the brane is located at @xmath20 , which would be exactly constant for a pure de sitter universe . during inflation the universe is approximately de sitter in that the hubble parameter @xmath9 changes slowly as the field rolls slowly down the potential . this slow - rolling behaviour is usually expressed in terms of slow - roll parameters defined as @xmath21 where the overdot denotes derivatives with respect to proper time on the brane , not the conformal time @xmath19 . we will use the convention that @xmath19 is positive , so that early times corresponds to @xmath22 and late times to @xmath23 . the equations of motion for the scalar metric perturbations are most conveniently expressed in terms of a master variable , a technique first used in the study of brane - world perturbations in ref . we define a master variable @xmath24 , whose equation of motion is  @xcite @xmath25 where @xmath26 note that eq . ( [ eom : chi ] ) is separable . the other equations are simplified by defining a new variable , @xmath27 , describing the scalar field and metric perturbations on the brane , and given by @xmath28 in terms of the field @xmath24 evaluated on the brane and the curvature perturbation @xmath29 on comoving hypersurfaces . this new variable obeys the following equation of motion  @xcite @xmath30 and the junction condition at the brane is @xmath31_{_{\rm b } } = \kappa_5 ^ 2 \dot\phi h \tau \xi \,.\ ] ] here we have neglected all terms suppressed by the slow - roll parameters except for the terms responsible for the coupling between @xmath27 and @xmath24 , i.e. , the terms that induce the mixing between the brane and bulk modes . it is useful to define the time - independent wronskian of two states @xmath32 and @xmath33 by  @xcite @xmath34 which is conserved .    at late times , @xmath35 , a growing bound state solution is  @xcite @xmath36 where @xmath37 is the overall amplitude , while @xmath38 can be determined through @xmath37 . a decaying bound state is given by  @xcite @xmath39\ , , \nonumber\end{aligned}\ ] ] where @xmath40 is linearly related to @xmath41 . at early times ( and high energies ) , @xmath42 , the growing and decaying solutions both take the form  @xcite @xmath43 the growing and decaying solutions have a different amplitude and phase , with the phase difference being approximately equal to @xmath44 . we see that the amplitude of @xmath45 increases as @xmath46 while the amplitude of @xmath27 tends to a constant at large @xmath19 , becoming increasingly dominated by higher modes and more closely bound to the brane . the early - time solution ( [ earlytimesolution ] ) differs from the solution for the scalar field in 4d at leading order , where @xmath27 satsifies the equation of motion @xmath47 and has solutions at late times given by @xmath48 where @xmath49 , and at early times by @xmath50 where @xmath51 and @xmath52 for standard 4d inflation . one might expect that the @xmath53 difference between the early - time solutions would lead to a significant difference to the amplitude of quantum perturbations generated during inflation . in fact this does not happen . we now illustrate how the quantum effects can be calculated and use the result to determine the slow - roll corrections to the spectral indices of scalar perturbations for some monomial potentials . in ref .  @xcite we calculated the growing and decaying mode solutions ; these were found to differ significantly from the solution in the standard 4d inflationary paradigm . however , the amplitude of large - scale perturbations from the quantum vacuum state in the early universe is only affected at the first order in the slow - roll parameters . we follow the method developed in refs . @xcite to relate the amplitude of the growing mode solution to the creation and annihilation operators , which is summarized here .    in principle , this method involves the construction of properly normalised positive- and negative - frequency solutions at early times and the evaluation of the inner product ( wronskian ) between these solutions and the bound states . however , there is a short - cut : all we need are the values of the real coefficients of the positive and negative frequency parts . we first explain this short - cut by making use of the familiar four - dimensional example , and then proceed to the brane - world model . let us begin with the standard four - dimensional case . the wronskian is defined as @xmath54 we are interested in the quantum field that behaves at late times as @xmath55 , where @xmath56 is a time - independent quantum operator . this field is quantised on small scales where we can expand @xmath57 in terms of negative- and positive - frequency modes , @xmath58 where @xmath59 and @xmath60 are annihilation and creation operators , respectively , which define the vacuum @xmath61 . note that because @xmath19 decreases as proper time increases , the negative- and positive - frequency functions are @xmath62 and @xmath63 . the mode functions should be normalised as @xmath64 to ensure the canonical commutational relation between @xmath65 and its conjugate momentum . we use the constancy of the wronskian to express @xmath56 in terms of @xmath59 and @xmath60 . at late times , @xmath66 where the numerator can be calculated using the early - time solutions and the denominator using the late - time solutions ( [ late4dxi ] ) as @xmath67 up to the first order in slow - roll parameters . we expand the growing mode and the decaying mode solutions at early times by @xmath68 this allows us to evaluate the expectation value of the square of the operator @xmath56 as @xmath69 which fully characterizes the gaussian random field . thus , the problem reduces to finding @xmath70 , the expansion coefficient for the decaying mode solution . the positive- and negative - frequency modes have the form @xmath71 , where the normalisation factor @xmath72 is not needed here , and @xmath73 is some phase ( which is irrelevant in the four - dimensional case ) . comparing the two expressions ( [ xi4d ] ) and ( [ xisplit ] ) , and using the wronskian @xmath74 , @xmath75 is determined to be @xmath76 thus , the quantity @xmath75 entering ( [ fin4d ] ) is expressed in terms of the ratio @xmath77 of the amplitudes and difference @xmath78 of the phases of the decaying and growing modes evolved back in time . these are evaluated analytically as @xmath79 and @xmath80 , giving @xmath81 the quantity that is related to observables is the comoving curvature perturbation @xmath29 , which is defined as @xmath82 \left ( \frac{\phi_{,\tau}}{h } { \cal r}_c \right).\ ] ] thus , one finds the power spectrum of the curvature perturbation to be @xmath83 \left.\frac{h^2}{2\pi\dot\phi}\right|_{k = ah } \,,\ ] ] where @xmath84 and @xmath85 , using the fact that conformal time is @xmath86 up to the first order in slow - roll parameters . the reason for preferring this method for dealing with the brane - world case is that one does not need to know the properly - normalised positive- and negative - frequency modes at early times , merely the quantities @xmath77 and @xmath87 . for the brane - world a similar calculation presented in our previous work  @xcite allow us to include the effect of interactions between the brane and bulk . the wronskian method , using the formula in eq . ( [ wronskian ] ) , allows us to deal with complicated early - time behaviour . the expectation value of the square of the operator @xmath56 is obtained as @xmath88 \,,\ ] ] where @xmath89 is determined by numerical solution . the left panel of fig .  1 shows the dependence of k on the energy scale of the inflation @xmath90 . at low energies , @xmath91 , the correction @xmath12 is negligible and we recover the standard vacuum ( [ zz ] ) . in the high - energy limit , @xmath92 , the correction can be approximated as a linear function of @xmath93 , which we write as @xmath94 . the right panel of fig . 1 shows @xmath95 in the high energy limit . this modifies eq . ( [ powerspectrum ] ) for the power spectrum by changing the coefficient @xmath96 to @xmath97 while @xmath98 . one of the strengths of the slow - roll inflation paradigm is that the perturbations generated by quantum fluctuations have a power spectrum that is almost scale - invariant . in 4d inflation the first - order correction to the tilt , @xmath99 , is well known to be @xmath100 . the second - order correction was calculated by stewart and lyth  @xcite . this was extended to high energy inflation in the rs model in ref . @xcite , including the modified effect of the friedmann equation ( [ modfriedmann ] ) in eq . ( [ powerspectrum ] ) but without including bulk gravity effects on the initial vacuum state . this gives @xmath101 where @xmath102 . the actual values of @xmath93 and @xmath103 for any given potential will change . alternative slow - roll parameters @xmath104 and @xmath105 are defined in terms of the potential by @xmath106 for brane - world inflation in the high - energy limit and @xmath107 for standard 4d inflation . these are related to the slow - roll parameters given in eq.([slowrolldef ] ) by @xmath108 and @xmath109 . note that the different friedmann relation means that the field , @xmath5 will have a different value at horizon crossing . the calculations presented in the preceding section show that we can determine the spectral index to second order for brane - worlds , including bulk gravity effects at high energies as well as the modified friedmann equation , by using the formula given in eq . ( [ powerspectrum ] ) , replacing @xmath96 with @xmath110 as given in eq . ( [ modifiedc1 ] ) . tables  [ table : monomialpotentials1 ] and  [ table : monomialpotentials2 ] show the slow - roll parameters and spectral indices for some monomial potentials . the differences at first order arise from the modified evolution on the brane , not from the brane - bulk mixing . the second - order corrections in brane - worlds have a contribution from the brane - bulk mixing  @xcite in addition to the familiar stewart  lyth correction , and these are of the same order of magnitude . for both the standard and brane - world inflation models the second - order correction is less than 1 part in @xmath111 so the first - order result should be sufficient for observational cosmology . .showing the values of the slow - roll parameters @xmath104 and @xmath105 for three monomial potentials @xmath112 and 50 e - folds of inflation . [ cols="^,^,^,^,^,^ " , ]     in the standard inflation paradigm slow - roll corrections change the dynamics on sub - horizon scales by introducing an effective mass that is negligible at early times ( though this is not negligible on super - horizon scales ) . the problem thus reduces to quantizing a massless scalar field in de sitter space . by contrast , in the brane - world model it is essential to use the slow - roll approximation even to formulate the problem in a way where the bulk wave equation is separable . in the brane - world model we have shown that the coupling to bulk metric perturbations can not be ignored in the equations of motion . indeed , we have shown that there are order - unity differences between the classical solutions without coupling and with slow - roll induced coupling . however , the change in the amplitude of quantum - generated perturbations is at next - to - leading order  @xcite because there is still no mixing at leading order between positive and negative frequencies when scales observable today crossed the horizon , so the bogoliubov coefficients receive no corrections at leading order . the amplitude of perturbations generated is also subject to the usual slow - roll corrections on super - horizon scales . thus , the correction we calculate is in addition to the usual stewart  lyth correction  @xcite and we find it to be of the same order of magnitude . our results also show that the ratio of tensor - to - scalar perturbation amplitudes are not influenced by brane - bulk interactions at leading order in slow - roll . it is not possible for us to calculate the next - order corrections without also deriving the slow - roll correction to tensor amplitudes . this correction has not yet been calculated because it requires one to go beyond the approximation of a separable wave equation in the bulk for a de sitter brane . from an end - user s perspective our results are significant in establishing rigorously the validity of assumptions made in earlier brane - world literature . the most significant difference between the prediction for the spectral indices in the brane - world at high energy and in standard inflation with the same potential arises at first order in slow - roll from the modified evolution of the hubble parameter and scalar field , leading to different values of the slow - roll parameters when the number of e - foldings of inflation is fixed  @xcite . it is remarkable that the predictions from inflation theories should be so robust that this result hold in spite of the leading - order change to the solutions of the classical equations of motion . the authors are supported by stfc . a.  cardoso , t.  hiramatsu , k.  koyama and s.  s.  seahra , arxiv:0705.1685 [ astro - ph ] and references therein . k.  koyama , a.  mennim , v.  a.  rubakov , d.  wands and t.  hiramatsu , [ arxiv : hep - th/0701241 ] . r.  maartens , d.  wands , b.  a.  bassett and i.  heard , phys . d * 62 * , 041301 ( 2000 ) [ arxiv : hep - ph/9912464 ] . d.  langlois , r.  maartens , m.  sasaki and d.  wands , phys . d * 63 * , 084009 ( 2001 ) [ arxiv : hep - th/0012044 ] . d.  wands , k.  a.  malik , d.  h.  lyth and a.  r.  liddle , phys . d * 62 * , 043527 ( 2000 ) [ arxiv : astro - ph/0003278 ] . a.  liddle and a.  n.  taylor , phys . d * 65 * , 041301 ( 2002 ) ; a.  r.  liddle and a.  j.  smith , phys . d * 68 * ( 2003 ) 061301 ; s.  tsujikawa and a.  r.  liddle , jcap * 0403 * ( 2004 ) 001 . d.  seery and a.  taylor , phys . d * 71 * , 063508 ( 2005 ) [ arxiv : astro - ph/0309512 ] . e.  ramrez and a.  r.  liddle phys .  rev . d * 69 * 083522 ( 2004 )    k.  koyama , d.  langlois , r.  maartens and d.  wands , jcap * 0411 * , 002 ( 2004 ) [ arxiv : hep - th/0408222 ] . k.  koyama , a.  mennim and d.  wands , phys . d * 72 * ( 2005 ) 064001 [ arxiv : hep - th/0504201 ] . k.  koyama , s.  mizuno and d.  wands , jcap * 0508 * ( 2005 ) 009 [ arxiv : hep - th/0506102 ] . c.  de rham , phys . d * 71 * , 024015 ( 2005 ) [ arxiv : hep - th/0411021 ] ; c.  de rham and s.  watson , arxiv : hep - th/0702048 . t.  hiramatsu and k.  koyama , arxiv : hep - th/0607068 . k.  koyama , a.  mennim and d.  wands , phys . d * 72 * , 064001 ( 2005 ) [ arxiv : hep - th/0504201 ] ; a.  cardoso , k.  koyama , a.  mennim , s.  s.  seahra and d.  wands , phys . d * 75 * , 084002 ( 2007 ) [ arxiv : hep - th/0612202 ] . s. mukohyama , phys . d62 : 084015 ( 2000 ) [ arxiv : hep - th/0004067 ]      d.  s.  gorbunov , v.  a.  rubakov and s.  m.  sibiryakov , jhep * 0110 * , 015 ( 2001 ) [ arxiv : hep - th/0108017 ] ; t.  kobayashi and t.  tanaka , phys . d * 71 * , 124028 ( 2005 ) [ arxiv : hep - th/0505065 ] ; t.  kobayashi and t.  tanaka , phys . d * 73 * , 044005 ( 2006 ) [ arxiv : hep - th/0511186 ] ; t.  kobayashi , phys . d * 73 * , 124031 ( 2006 ) [ arxiv : hep - th/0602168 ] . m.  v.  libanov and v.  a.  rubakov , phys . d * 72 * , 123503 ( 2005 ) [ arxiv : hep - ph/0509148 ] .
we quantify the slow - roll corrections to primordial density perturbations arising from inflation driven by a four - dimensional scalar field with a monomial potential in a five - dimensional non - compact bulk spacetime . 
 although the difference between the classical brane - world solutions and standard four - dimensional solutions is large at early times , the change to the amplitude at late times of perturbations generated from quantum fluctuations is first - order in slow - roll parameters , leading to second - order slow - roll corrections to the spectral index . 
 this confirms that the leading - order effects are correctly given by previous work in the literature .