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the accumulating astronomical evidence for an accelerating cosmic expansion has stimulated many investigations into the nature of the dark energy , or other possible deviant gravitational effects , which might be responsible for this unexpected dynamics ( for a review see , _ e.g. _ , @xcite ) . besides proposing to add some new ( and purely theoretical ) matter species into the energy budget of the universe , many investigators have also focused their attentions on modifying general relativity ( gr ) on the largest scales , so as to introduce significant modifications in the behaviour of gravity at late times when it is comparatively weak . one example of the latter sort is provided by the family of @xmath3 gravity models , which had also been considered before the discovery of cosmic acceleration ( see for example refs . @xcite ) with reference to alternative forms of inflation and the existence of singularities . in refs . @xcite , the authors discuss a specific model where the correction to gr is a polynomial function of the @xmath4 and @xmath5 quadratic curvature invariants ( here , @xmath6 and @xmath7 are respectively the ricci scalar , ricci tensor and riemann tensor calculated in the standard way from the physical metric @xmath8 ) and showed that there exist late - time accelerating attractors in friedmann cosmological solutions to the theory . barrow and clifton established the general existence conditions for de sitter , einstein static , gdel universes in theories where the lagrangian is an arbitrary function of these three invariants @xcite . when the ricci scalar @xmath9 in the einstein - hilbert action is replaced by some general functions of @xmath9 and @xmath10 , it becomes necessary to distinguish between two different variational approaches to deriving the field equations . in the metric approach , as in refs . @xcite , the metric components @xmath8 are the only variational quantities and the field equations are generally of fourth - order , which makes the theories phenomenologically richer but more stringently constrained in most cases . within the palatini variational approach , on the other hand , we treat the metric @xmath8 and the connection @xmath11 as independent variables and extremize the action with respect to both of them ; the resulting field equations are second order and easier to solve . the palatini @xmath3 gravity is also proposed as an alternative to dark energy in a series of works @xcite . there has since been growing interest in these modified gravity theories : for the local tests of the palatini and metric @xmath3 gravity models see @xcite ; for the cosmologies of these two classes of models see @xcite .    both approaches to modifying gravity are far from problem - free . in the metric @xmath3 gravity models , the theory is conformally related to standard gr plus a self - interacting scalar field @xcite , which generally introduces extra forces inconsistent with solar system tests @xcite . the palatini approach , on the other hand , generally leads to a large ( or even negative ) sound - speed - squared term in the growth equation of the non - relativistic matter perturbations on small scales . this induces effects on the cosmic microwave background ( cmb ) and the matter clustering power - spectra which deviate unacceptably from those which are observed @xcite . again , these examples highlight the difficulties encountered when trying to make modifications to standard gr which are compatible with observations . in this work we will focus on the ricci - squared gravity models within the palatini variational approach , which we also denote by the @xmath12 gravity . it turns out that the ricci tensor , @xmath13 and ricci scalar , @xmath14 , appearing in the field equations in the palatini approach are not the ones calculated from the physical metric , @xmath8 , ( we consider the metric @xmath8 as the physical one because it is this metric which the matter lagrangian density @xmath15 depends on and the energy - momentum conservation law holds with respect to ) as in gr , and we denote the gr equivalents by @xmath16 and @xmath9 respectively to distinguish them from the palatini quantities ) . such a modification of gravity has indeed been considered in @xcite and shown to give an accelerating cosmology . however , our work differs from @xcite in that we replace the ricci scalar in the gravitational action with @xmath17 rather than simply @xmath18 and we concentrate more on the cosmology at the first - order perturbation level , especially the late - time cold - dark - matter ( cdm ) density perturbation growth . we emphasize the similarity to the palatini @xmath19 gravity models also . our presentation is organized as follows . in sec . [ sect : equations ] we briefly introduce the model and outline the methods used to derive the background and first - order covariant and gauge - invariant ( cgi ) perturbed field equations . in sec .  [ sect : background ] , we present the modified friedmann equation and apply it to a specific family of theories with @xmath20 ; the constraints on the parameter space @xmath21 from cosmological data are also given . then , in sec .  [ sect : perturbations ] , we analyse the growth of cdm density perturbations at late cosmological times for this model . since this analysis shares some similarities with the palatini @xmath19 gravity models , we also present a similar discussion of the latter for comparison . our discussion and conclusions are presented in sec . [ sect : conclusions ] .    throughout this work our convention is chosen as @xmath22u^{c}=r_{ab\ d}^{\ \ c}u^{d},r_{ab}=r_{acb}^{\ \ \ c}$ ] where @xmath23 run over @xmath24 and @xmath25 ; the metric signature is @xmath26 and the universe is assumed to be spatially flat and filled with cdm and black body radiation . in this section we briefly introduce the main ingredients of @xmath12 gravity and outline the strategy for deriving the general perturbation equations that govern the dynamics of small inhomogeneities in the cosmological models that arise in this theory . we will start our discussion with the modified einstein - hilbert action in the present model , @xmath27,\end{aligned}\ ] ] in which @xmath28 with @xmath29 the newtonian gravitational constant . here , @xmath30 is given by @xmath31 and @xmath32 ; note that @xmath33 is a new and independent variable with respect to which we extremise the action , and is different from the christoffel symbol @xmath11 calculated using the metric @xmath8 . @xmath34 is assumed to be a symmetric tensor ( if it contains an antisymmetric part then the field equation will be spoiled as discussed in @xcite ) and @xmath8 could be used to raise or lower its indices . varying the action eq . ( [ eq:1 ] ) with respect to the metric @xmath8 ( note that @xmath35 as they are independent ) gives the modified einstein equations : @xmath36 & = & \kappa t_{ab}^{f}\ \\end{aligned}\ ] ] where @xmath37 with @xmath38 and @xmath39 is the energy - momentum tensor of the fluid matter ( cdm and radiation ) .    on the other hand , varying the action with respect to the new variable @xmath40 with the relation @xmath41 , one arrives at another field equation @xmath42 & = & 0,\end{aligned}\ ] ] where @xmath43 represents the covariant derivative compatible to @xmath44 ( the covariant derivative compatible to @xmath8 is denoted , as conventionally , by @xmath45 ) . just like in the palatini @xmath46 models , this equation implies some relation between the physical metric @xmath8 and the metric @xmath47 whose christoffel symbol is @xmath40 . however , because of the presence of the second term in the parentheses this relation is nontrivial and some further algebra will be needed to explicate it . before doing that , we will present some preliminary definitions and expressions , one of which is the notation of @xmath48 decomposition . the main idea of @xmath48 decomposition @xcite is to make spacetime splits of physical quantities with respect to the 4-velocity @xmath49 of an observer . the projection tensor @xmath50 is defined as @xmath51 and can be used to obtain covariant tensors perpendicular to @xmath49 . for example , the covariant spatial derivative @xmath52 of a tensor field @xmath53 is defined as @xmath54    the energy - momentum tensor and covariant derivative of the 4-velocity are decomposed respectively as @xmath55 in the above , @xmath56 is the projected symmetric trace - free ( pstf ) anisotropic stress , @xmath57 the heat flux vector , @xmath58 the isotropic pressure , @xmath59 the pstf shear tensor , @xmath60}$ ] , is the vorticity , @xmath61 ( @xmath62 is defined here as the mean expansion scale factor ) the volume expansion rate scalar , and @xmath63 is the fluid acceleration ; the overdots denote time derivatives expressed as @xmath64 , brackets mean antisymmetrisation , and parentheses symmetrization . the velocity normalization is chosen to be @xmath65 . the quantities @xmath66 are referred to as _ dynamical _ quantities and @xmath67 as _ kinematical _ quantities . note that the dynamical quantities can be obtained from the energy - momentum tensor @xmath68 through the relations @xmath69    decomposing the riemann tensor and making use the einstein equations , we could obtain , after linearization , the perturbed ( constraint and propagation ) equations @xcite . here , we shall not list all of them because most are irrelevant for the following discussion ; rather we will use the linearised raychaudhuri equation @xmath70 the linearised conservation equations for the energy density : @xmath71 and the linearised friedmann equation @xmath72    the above equations are derived and presented for standard general relativity , and so the @xmath73 variables describe imperfect fluid matter . for general modified gravity theories , such as those presented here , the modification to gr might be parameterized as an effective energy - momentum tensor . in this case the formalism of these equations is preserved and one just needs to replace @xmath74 by the total effective quantities of the same sort : @xmath75 @xcite .      in eq . ( [ eq:4 ] ) , we see that @xmath76 is a symmetric @xmath77 tensor density of weight 1 , and so we can introduce a new metric @xmath47 by means of the following relation @xmath78 where the levi - civita connection of the metric @xmath47 is just @xmath40 , as we referred to above .    to go further , we need to express @xmath34 explicitly . this is easy to do in principle , because eq . ( [ eq:3 ] ) is just an algebraic equation for @xmath34 . to see this , let us write the symmetric tensor @xmath34 in a general way as @xmath79 where @xmath80 is the 4-velocity of the observer referred to above . substituting eqs . ( [ eq:6 ] , [ eq:13 ] ) into eq . ( [ eq:3 ] ) , we get @xmath81   \nonumber \\ & & -\frac{1}{2}\left ( \delta + 3\xi + f\right ) u_{a}u_{b}-\frac{1}{2}\left ( \delta + 3\xi + f\right ) h_{ab }   \nonumber \\ & = & \kappa \left(\rho^{f}u_{a}u_{b}-p^{f}h_{ab } + 2u_{(a}q_{b)}^{f } + \pi_{ab}^{f}\right ) ,   \nonumber\end{aligned}\ ] ] which leads to the following four equations : @xmath82 \upsilon _ { a } & = & \kappa q_{a}^{f } , \\ \label{eq:17 } ( 1 + 4f\xi ) \sigma _ { ab } & = & \kappa \pi _ { ab}^{f},\end{aligned}\ ] ] where @xmath83 are functions of @xmath84 . thus given the specified form of @xmath85 and the values of @xmath86 , the quantities @xmath87 can be obtained from eqs . ( [ eq:14 ] , [ eq:15 ] ) , at least numerically . then , @xmath88 and @xmath89 can also be calculated from eqs . ( [ eq:16 ] , [ eq:17 ] ) provided the values of @xmath90 and @xmath91 are given . note that @xmath88 and @xmath92 are nonzero only at first order in perturbation . taking the time derivatives of eqs . ( [ eq:14 ] , [ eq:15 ] ) , and using the background values of @xmath87 , we could easily obtain @xmath93 and @xmath94 by solving the two linear algebraic equations . similarly , @xmath95 and @xmath96 could be worked out ( here , @xmath52 is the spatial derivative ) . in what follows , we shall assume that @xmath97 and their derivatives have been calculated . the next step is to find out the relation between @xmath8 and @xmath47 . we could rewrite eq . ( [ eq:12 ] ) as @xmath98 taking the determinants of both sides and equating we get @xmath99 with @xmath100    thus , we conclude from eq . ( [ eq:18 ] ) that @xmath101 obviously , @xmath102 and @xmath103 need to be evaluated respectively . for @xmath102 , we have @xmath104\nonumber.\end{aligned}\ ] ] to calculate this , let us write @xmath105 $ ] and @xmath106 where @xmath107 run over @xmath108 , @xmath109 and @xmath110 are first order metric variables of which @xmath110 is traceless , @xmath111 is the spatial component of @xmath80 , and @xmath112 is the metric of 3 dimensional flat space . as a result @xmath113 and @xmath114 . from these expressions the components of @xmath115 can also be obtained and one can substitute all these quantities into the above equation to get @xmath116 . since @xmath88 and @xmath117 are only of first order and because @xmath117 is traceless , it is then not difficult to see that up to first order ( note that the facts @xmath118 and @xmath119 indicate that @xmath120 and @xmath121 ) @xmath122    for @xmath123 , we know that it is symmetric as the inverse matrix of a symmetric matrix , and so could be written as @xmath124 using @xmath125 and @xmath126 it is then easy to obtain , to first order , that @xmath127    as a result , we have now the relations between the two metrics @xmath47 and @xmath8 and their inverses as @xmath128 where @xmath129.\end{aligned}\ ] ]    a discussion of how the two ricci tensors @xmath130 and @xmath131 are related to one another is given in the appendix , with the help of which the einstein equation eq . ( [ eq:3 ] ) can be rewritten as @xmath132 where @xmath133 and ( see the appendix for a definition of the tensor @xmath134 ) @xmath135    with the aid of eqs .  ( [ eq:8 ] , [ eq:33 ] , [ eq:34 ] , [ eq:35 ] ) one could identify @xmath136 and @xmath137 and express them in terms of @xmath138 and @xmath139 which are functions of @xmath140 and @xmath141 ( c.f . ( [ eq:14 ] , [ eq:15 ] , [ eq:28 ] , [ eq:29 ] ) ) , and @xmath142 which are also functions of @xmath143 ( c.f . , ( [ eq:16 ] , [ eq:17 ] ) ) . however , from eqs . ( [ eq:26 ] - [ eq:31 ] , [ eq : a6 ] , [ eq : a7 ] ) one can see that this process will involve a lot of calculation . in the present work we will not perform detailed numerical calculations of the perturbation equations of the @xmath12 model ; instead , in the next two sections of the paper we will :    \1 . study the background evolutions of general ricci - squared gravity models . as an example , we will consider a specific family of theories with @xmath1 , and constrain the allowed @xmath144 parameter space using data sets supernovae ( sne ) luminosity distances and the cmb shift parameter . present a simple argument to show that this class of modified gravity theory , like those arising in the palatini @xmath19 theory , generally possesses a scale - dependent effective sound - speed - squared term which affects the growth of cdm density perturbations and thus influences the matter power spectrum @xcite on small scales . in order to analyse the cosmological background evolution we can neglect the @xmath90 and @xmath91 terms , and hence the quantities @xmath145 . as a result , the equations are greatly simplified . we are interested in the modified friedmann equation in the present model . from eqs . ( [ eq:8 ] , [ eq:11 ] , [ eq:33 ] ) we have @xmath146 in which @xmath147 is the hubble expansion rate and @xmath148 is expressed as @xmath149 with @xmath150 and @xmath151 given in eqs . ( [ eq:34 ] , [ eq:35 ] ) . after a lengthy calculation , and using eq . ( [ eq:9 ] ) to eliminate the term @xmath152 which appears in eq . ( [ eq:37 ] ) , we obtain the following simple result @xmath153^{2 } & = & \frac{1}{6}(\delta -3\omega \xi ) .\end{aligned}\ ] ] there are two interesting points regarding eq . ( [ eq:38 ] ) . firstly , we see that only @xmath154 , and not its time derivatives @xmath155 or @xmath156 enter the equation . secondly , the second - order derivative of @xmath157 does not appear either ; to see the consequence of this , note that since @xmath158 ( with @xmath159 for matter and @xmath160 for radiation ) , the dependence of @xmath161 on @xmath141 could be expressed in terms of @xmath162 , _ e.g. _ , in radiation - dominated era @xmath163/3 and in matter - dominated era @xmath164 , and so we have @xmath165 . consequently , eq .  ( [ eq:38 ] ) has the form @xmath166 where @xmath167 is a complicated function of @xmath140 and @xmath141 ( at late times @xmath168 and it becomes a function of @xmath140 alone ) .     model . for clearness we only show the cases of @xmath169 ( the dashed curve ) and @xmath170 ( @xmath171 , the solid curve ) . the other parameters defining the cosmology are @xmath172 and @xmath173 . clearly an accelerating phase following the standard matter - dominated era can be realized within this model . note that the redshift range @xmath174 corresponds to the future of the present epoch , where the universe evolves into a de sitter stage . ]    as we discussed above , knowing @xmath140 and @xmath141 means that we know @xmath175 and @xmath154 . thus , given a specific form for @xmath176 eq . ( [ eq:38 ] ) completely determines the background cosmological evolution of the model . as a particular example let us consider the case of @xmath177 where @xmath178 and @xmath179 are the model parameters . note that here @xmath180 is always non - negative , and @xmath181 corresponds to a picking the standard @xmath171 cosmology of gr .    for convenience , we shall define the following dimensionless quantities @xmath182 then eqs .  ( [ eq:14 ] , [ eq:15 ] ) could be rewritten as @xmath183 where @xmath184 and than eq . ( [ eq:38 ] ) reduces to @xmath185^{2}\frac{h^{2}}{h_{0}^{2}}=\frac{1}{6}\left ( \tilde{\delta}-3\omega \tilde{\xi}\right)\end{aligned}\ ] ] where @xmath186 . in this paper we will set @xmath173 , so today we have @xmath187 and there are 3 equations ( eqs . ( [ eq:42 ] , [ eq:43 ] , [ eq:46 ] ) ) for the 5 parameters @xmath188 and @xmath189 . therefore , we are able to express all the other quantities in terms of @xmath179 and @xmath190 , which can therefore be treated as the two independent degrees of freedom of our model . note that @xmath191 is a constant , and once evaluated at the present day , it could be used all through the cosmic history , which helps determine @xmath192 at arbitrary times .    in figure  [ fig : figure1 ] we have plotted the effective equation of state , defined by @xmath193 ( where a star - superscript denotes the derivative with respect to @xmath194 ) , as a function of the redshift . the values of @xmath179 are indicated beside the curves . at early times , when the @xmath12 corrections are negligible , the models all mimic the @xmath195 evolution of @xmath196 , and the same thing happens in the future . this is because during this era the matter ( relativistic and non - relativistic ) is greatly diluted so that the right - hand sides of eqs . ( [ eq:42 ] , [ eq:43 ] ) both vanish ; consequently , we can solve them to show that @xmath197 and so @xmath12 is also constant . the deviation from @xmath171 occurs mainly at intermediate times , that is , in the recent past and future . we now use the observational data on the background cosmology to constrain the parameter space ( in the @xmath198 plane ) of the present model . for this we jointly use the 157 measurements on sne luminosity distance in the gold data sets of riess _ et  al_. @xcite and the cmb shift ( r ) parameter . the sne luminosity distance is expressed as @xmath199 where @xmath200 . the measurements supply the extinction - corrected distance modulus @xmath201 ( with @xmath202 in units of mega - parsecs ) and its uncertainty , @xmath203 , for individual sne , so that the standard @xmath204 minimization , defined by @xmath205 ^{2}}{\sigma _ { i}^{2}}\end{aligned}\ ] ] is easy to implement , where @xmath206 is the theoretically predicted distance modulus . as @xmath207 appears only as it does in eq . ( [ eq:47 ] ) , we could marginalize over it by integrating the probability density @xmath208 for all values of @xmath207 . for the cmb r - parameter , defined as @xmath209 we adopt the observational value @xmath210 at @xmath211 from @xcite . note that this does not depend on the specified value of @xmath207 . our constraining result is shown in figure  [ fig : figure2 ] , where we have shown the 68% and 95% confidence regions respectively . the constrained intervals are roughly @xmath212 and @xmath213 at the 95% confidence level , with the best fitting values being @xmath214 with @xmath215 . also note that the concordance @xmath216 model ( the white star ) lies within the 68% confidence region of our constraints .     and @xmath217 in the present model from joint sne and cmb shift parameter data sets . the grey and light grey regions represent the 68.3% and 95.4% confidence contours respectively . the white circle ( @xmath218 ) is the best - fitting parameter of our model , and the star is the concordance @xmath216 model . ] thus , we see that the background cosmological data is able to constrain @xmath219 to be of order @xmath220 . in the next section we will briefly investigate the possible constraint from the growth of dark - matter density perturbations , and show that this may provide a potentially more stringent restriction on @xmath221 . however , considering that this latter limit depends on the properties of the dark matter , our background constraints given in this section are less model dependent . in this section we study the effects of the @xmath12 corrections to gr on the cdm density perturbation growth . we start by recalling the case of @xmath19 gravity because it shares some similarities with the @xmath12 one , while being technically simpler than the latter , and because a similar analysis for the former is still missing from refs . @xcite ( see however @xcite for a slightly different treatment ) .      recall that in our simplified model the universe is filled with cdm and radiation , and at later times the radiation energy density is negligible , so @xmath222 .    taking the spatial derivative of the raychaudhuri equation eq . ( [ eq:9 ] ) ( with the @xmath223 there being replaced by @xmath224 ) , and working in the cdm frame ( where the observer is comoving with cdm particles and thus @xmath225 ) @xcite , we have @xmath226 where @xmath227 is the cdm density perturbation contrast that is defined through @xmath228 , and @xmath229 is the hubble expansion rate with respect to conformal time ( note that a prime denotes the conformal time derivative , and a dot the cosmic comoving proper - time derivative ) ; @xmath230 are respectively the harmonic expansion coefficients for @xmath231 and @xmath232 ( defined via @xmath233 and @xmath234 @xcite ) . clearly we need to know about @xmath235 and @xmath236 which arise from the @xmath12 modifications to gr ( c.f . ( 33 ) ) .    in the palatini @xmath19 model , in which the ricci scalar @xmath14 in the gravitational action is replaced with @xmath237 , eqs . ( [ eq:26 ] , [ eq:27 ] ) still hold , but with ( see for example @xcite ) @xmath238then , with the help of the calculations in the appendix , it is straightforward to show that @xmath239where @xmath240 , and the modified einstein equation @xcite , @xmath241 can be rewritten as @xmath242 in which the effective total energy - momentum tensor is given by @xmath243using eq . ( [ eq:8 ] ) , we can now identify @xmath244   \nonumber \\ & & -\frac{3\dot{\lambda}^{2}}{2\lambda^{2}}+\frac{\ddot{\lambda}}{\lambda } + % \frac{1}{2\lambda } \box \lambda , \\ \label{eq:55 } \kappa p^{\mathrm{tot } } & = & \frac{1}{\lambda } \kappa p^{f}-\frac{1}{2\lambda } ( \mathbf{r}+f )   \nonumber \\ & & + \frac{1}{2}\left [ \mathbf{r}+\frac{3}{\lambda } \box \lambda -\frac{3}{2\lambda ^{2}}\nabla ^{a}\lambda \nabla _ { a}\lambda \right ]   \nonumber \\ & & -\frac{1}{3\lambda } ( \theta \dot{\lambda}+\hat{\nabla}^{2}\lambda)-\frac{1}{2\lambda } \box \lambda .\end{aligned}\]]thus @xmath245 in which @xmath246 represent the terms _ not _ involving second order ( time and spatial ) derivatives . the reason why we keep only two second derivative terms explicitly on the right - hand side of eq . ( [ eq:56 ] ) is that , after taking the spatial covariant derivative , the first term contributes a @xmath247 piece to eq . ( [ eq:50 ] ) while the second term contributes a @xmath248 piece . none of the remaining terms in @xmath246 contribute these two pieces to eq . ( [ eq:50 ] ) . to be more explicit , recall that @xmath249 in the model , so @xmath250 where we have used eq . ( [ eq:10 ] ) to background order . as a result , @xmath251and eq . ( [ eq:50 ] ) can be recast into the form @xmath252 \delta _ { \mathrm{cdm}}^{\prime \prime } + [ \cdots ] \delta _ { \mathrm{cdm}}^{\prime }   \nonumber \\ + \left [ \cdots + \frac{\dot{\lambda}}{6\lambda h}k^{2}\right ] \delta_{\mathrm{cdm } } & = & 0   \nonumber\end{aligned}\]]which , after rearrangement , gives @xmath253\delta_{\mathrm{cdm}}^{\prime }   \nonumber \\ + \left[\cdots + \frac{\dot{\lambda}}{3(2\lambda h+\dot{\lambda})}k^{2}\right ] \delta_{\mathrm{cdm } } & = & 0.\end{aligned}\]]here , @xmath246 denotes complicated terms that are determined completely by the background evolutions of the model , and are unimportant for our analysis here . what is essential in eq . ( [ eq:58 ] ) is that it tells us that , as long as the quantity @xmath254 does not vanish , in general there will appear an effective sound - speed - squared term for the growth of matter density perturbations . depending on the sign of @xmath254 , this sound - speed - squared term could be either positive or negative , in both cases the small - scale density perturbation growth becomes extremely scale - dependent , altering the shape of the matter power spectra significantly @xcite . notice that the terms in @xmath255 $ ] could also modify the evolution of density contrasts differently as compared with the prediction in standard general relativity , but in a scale - independent manner , and at small scales their effects are subdominant . one more comment on the modified friedmann equation in the palatini gravity models is appropriate . using eq . ( [ eq:11 ] ) with @xmath256 replaced by the @xmath257 given in eq . ( [ eq:54 ] ) , we can see that only the @xmath258 terms are involved and the @xmath259 terms cancel ( we only consider terms to background order here ) . since @xmath260 we see that @xmath261 and could be moved to the left - hand side of eq .  ( 11 ) ; the remaining terms on the right - hand side are also functions of @xmath140 only and so eq .  ( [ eq:39 ] ) is also realized . to be more explicit , the modified friedmann equation in palatini @xmath19 gravity is @xcite @xmath262 ^{2 } & = & \frac{1}{6\lambda}\left [ \kappa ( \rho^{f}+3p^{f})-(\mathbf{r}+f)\right ]   \nonumber\end{aligned}\ ] ] which can be shown to be just eq . ( [ eq:38 ] ) if @xmath263 there , as expected , because the metrics then take the same form ( of course , the definitions of @xmath161 are different in the two cases ) . now consider the @xmath12 gravity model . as discussed in the last section , the detailed forms of @xmath264 are very complicated , but fortunately we need not evaluate the full formulae explicitly . our experience of the simpler theory described in the last subsection shows that what is most relevant for our analysis are the second order ( time and space ) derivative terms ( note that the term @xmath265 , if exists , can also contribute to @xmath266 because @xmath267 contains @xmath268 where @xmath269 is the harmonic expansion coefficient of @xmath270 via @xmath271 , and because @xmath272 ; however it turns out no @xmath265 terms appears in @xmath273 ) , which are straightforward to identify .    we shall formally repeat the procedure of the last subsection . note that the quantities @xmath274 are determined by @xmath275 which in our case are due to the radiation matter species . at late times the radiation energy density is negligible so that , to a good approximation , @xmath275 and thus @xmath274 , vanish . as a result , eqs . ( [ eq:26 ] , [ eq:27 ] ) become @xmath276with @xmath277 defined in eqs . ( [ eq:28 ] , [ eq:29 ] ) . meanwhile , since @xmath278 we have @xmath168 at late times ; @xmath279 , and hence @xmath161 and @xmath280 become functions of @xmath281 only , _ i.e. _ , @xmath282 , @xmath283 . the analysis in eq .  ( [ eq:57 ] ) then also applies to @xmath161 and @xmath154 here , so @xmath284    now , from eq . ( [ eq:33 ] ) we obtain @xmath285where the relation @xmath286 and eq . ( [ eq:8 ] ) are used ; @xmath151 is given in eq . ( [ eq:34 ] ) . note that @xmath287 does not appear in this formula , and what we need to evaluate is just the collection of second - order derivative terms in @xmath288 . after some manipulation we obtain a similar expression to eq . ( [ eq:56 ] ) for the palatini @xmath19 model : @xmath289 the following analysis then completely parallels that for the palatini @xmath19 model and the cdm density perturbation growth equation can be shown to be ( like eq . ( [ eq:58 ] ) ) @xmath290\delta _ { \mathrm{cdm}}^{\prime }   \nonumber \\ + \left[\cdots + \frac{(\lambda \omega ) ^{\cdot } } { 3(2\lambda h+\dot{\lambda})}k^{2}\right ] \delta _ { \mathrm{cdm } } & = & 0.\end{aligned}\ ] ] again the @xmath291 @xmath292 $ ] in eq . ( [ eq:63 ] ) denotes the terms which are completely determined by the background evolution and are not of interests to us here . thus we see that , similar to the case of palatini @xmath19 gravity model , a scale - dependent sound - speed - squared term also appears in the palatini @xmath12 gravity model , whose sign depends on @xmath293 . note that in the case of @xmath263 the metric @xmath47 ( eq .  ( [ eq:59 ] , [ eq:60 ] ) ) has the same form as that in the palatini @xmath19 model , and eq . ( 63 ) reduces to eq .  ( 58 ) as expected .    in the general @xmath12 gravity models it is possible that @xmath294 , thus the scale - dependent effective sound - speed - squared term influences the matter perturbation growth and could alter the shape of the predicted matter power spectrum . our previous knowledge derived from refs . @xcite suggests that this effect might allow observational data to place stringent constraints on the parameter space of these theories . in fact , we can give a rough estimate of how stringent the constraint can be . consider first eq .  ( [ eq:58 ] ) for the palatini @xmath19 model : since on small scales the @xmath295 term dominates the other terms in front of @xmath227 ( the @xmath296 terms ) , it is obvious that the magnitude of the quantity @xmath297 determines the deviation from @xmath195 results . the observational constraint on the model parameter @xmath179 ( recall that @xmath298 ) , as shown in refs . @xcite , is @xmath299 ; we take @xmath300 and @xmath301 for illustrative purposes , and find that @xmath302 for the relevant redshift range of @xmath303 . in the analogous case of palatini @xmath12 gravity , eq .  ( [ eq:63 ] ) , we have @xmath304 as the dominant term instead of @xmath305 , and so the constraint should be @xmath306 in the same redshift range . again , taking @xmath301 , for the theories with @xmath20 we find that to satisfy the above constraint @xmath307 must also be limited to be @xmath308 or even smaller . thus , we conclude that the palatini @xmath12 models may be constrained by the observational data on the matter power spectrum just as stringently as are the palatini @xmath19 models . yet , we should note that a more exact quantitative constraint can only be obtained by exploiting a full parameter - space search as done in @xcite , and that the above conclusion depends on the assumption that the cdm particles have vanishing anisotropic stress . if , in contrast , the dark matter particles admit an anisotropic stress , in a manner similar to that prescribed in @xcite , then the effective sound - speed - squared terms might be canceled and leave no significant traces . we considered a general class of modified gravity models where the ricci scalar in the gravitational action of gr is replaced by a function @xmath17 and the field equations are derived using the palatini variational approach , _ i.e. _ , treating the metric @xmath8 and connection @xmath40 as independent variables so that the action is varied with respect to both of them . the strategy for deriving the cosmological equations at both the background ( zero - order ) and the first - order perturbation levels is outlined . the main step in this process is to determine the metric , @xmath47 , whose levi - civita connection is @xmath309 , relate it to the physical metric @xmath8 and thereby fix the relation between @xmath34 and @xmath16 . then , the correction to gr is treated as a new effective energy - momentum tensor while the field equations take the same form as in gr . the formulae laid down here might be useful for the numerical implementations of such modified gravity models . we also investigated in detail a power - law correction to the usual einstein - hilbert action given by @xmath20 . we used the snia luminosity distance and cmb shift parameter data to constrain its ( independent ) two - parameter space @xmath310 . we found that at 95% confidence level @xmath311 $ ] and @xmath312 $ ] . a slightly positive value of @xmath179 ( @xmath313 ) is preferred by the data used however , the standard @xmath171 model ( equivalent to @xmath181 ) with @xmath314 is still within the 68% confidence contour . hence , although the best fit to the data is a non - null leading - order correction to einstein gravity , the current data exhibits no significant preference over the concordance @xmath171 model of gr . the late - time growth of matter density perturbations in general palatini @xmath315 gravity models was also studied . it was shown that the equations governing this class of models look very similar to that in the palatini @xmath19 models . in particular , there exists a scale - dependent effective sound - speed - squared term in the perturbation growth equation which may be either positive or negative , depending on the background evolution in both models . in the @xmath19 case it is well known that these terms can lead to strong scale - dependence of the matter power spectrum , which is highly constrained by observational data @xcite , and we expect a similar feature to exist in the @xmath12 case . we estimate that this will produce a strong observational bound of @xmath316 unless some exotic properties are added to the dark matter candidate @xcite .    as the final remark , we give a brief comment about the static and spherically - symmetric solutions of the present model . the analogue for the palatini @xmath19 model was considered in @xcite and the authors found that the exterior spherically - symmetric vacuum solutions are unique . here we just want to point out the @xmath317 model also shares this feature . in fact , in the vacuum where @xmath318 it is easy to show that for our model @xmath319 are uniquely determined by eqs . ( [ eq:14 ] , [ eq:15 ] ) and so is @xmath34 . the full consideration of a static system also needs the interior solution and its matching the exterior , which is beyond the scope of this paper and will be left for further investigation . in this appendix we present the relation between @xmath130 and @xmath131 if the two metrics @xmath47 and @xmath320 satisfy the following relations @xmath321where @xmath161 is a scalar function and @xmath322 symmetric tensors . firstly , the requirement @xmath323 implies that @xmath324 then , with some algebra , and using eq . ( [ eq : a4 ] ) , we can easily show that ( here a comma denotes the ordinary derivative ) @xmath325   \nonumber \\ & = & \gamma _ { bc}^{a}(g_{ab})+\gamma _ { bc}^{a},\end{aligned}\]]where the difference between @xmath40 and @xmath326 , denoted @xmath327 is defined by @xmath328   \nonumber \\ & & + \frac{1}{2}\left [ \zeta _ { b}^{a}\nabla _ { c}\lambda + \zeta_{c}^{a}\nabla_{b}\lambda -g_{bc}\zeta ^{ad}\nabla _ { d}\lambda \right ]   \nonumber \\ & & + \frac{1}{2\lambda } \left [ \nabla _ { b}\xi _ { c}^{a}+\nabla_{c}\xi_{b}^{a}-\nabla ^{a}\xi _ { bc}\right ]   \nonumber \\ & & + \frac{1}{2}\zeta^{ad}[\nabla_{c}\xi_{bd}+\nabla_{b}\xi_{cd}-\nabla_{d}\xi_{bc}]\end{aligned}\]]and is a true tensor , as expected . from the definition in eq . ( [ eq:2 ] ) we can thus derive that @xmath329 note that @xmath34 differs from @xmath16 by a rank-2 symmetric tensor . we thank andrzej borowiec , john miritzis , thomas sotiriou and the referee for helpful discussions . bl is supported by the overseas research studentship , cambridge overseas trust and the department of applied mathematics and theoretical physics ( damtp ) at the university of cambridge . dfm acknowledges the a. humboldt foundation .                          , * * , ( ) ; and , * * , ( 2005 ) ; and , * * , ( 2007 ) ; , and ( 2006 ) , ; and ( 2006 ) , ; , * * , ( ) ; , * * , ( ) ; , * * , ( ) ; ( 2007 ) , [ arxiv : astro - ph/0701662 ] ; and ( 2007 ) , arxiv : 0705.1158 [ astro - ph ] ; , , and , * * , ( ) . arxiv : 0704.1733 [ astro - ph ] ; and ( 2007 ) , arxiv : 0706.1378 [ hep - th ] ; , and , * * , ( 2006 ) ; , and , arxiv : astro - ph/0610854 ; , and , arxiv:0709.0046 [ gr - qc ] ; , and , arxiv:0709.0891 [ gr - qc ] ; and ( ) , arxiv:0708.0432 [ hep - th ] .    , and , * * , ( ) ; , , and , * * , ( ) ; , * * , ( ) ;  , and ( 2007),[arxiv : astro - ph/0701479 ] ; , and , * * , ( ) ; , and ( 2007 ) , arxiv : 0705.0232 [ gr - qc ] ; , * * , ( 2006 ) ; , * * , ( 2007 ) ; and , * * , ( 2007 ) ; , * * , ( 2006 ) ; and , * * , ( 2006 ) .    , * * , ( ) ; , and ( 2006 ) , ; and , * * , ( 2006 ) ; , , and , * * , ( 2007 ) ; ( 2007 ) , arxiv : gr - qc/0703044 ; and ( 2007 ) , arxiv : 0704.0680 [ astro - ph ] ; and , * * , ( 2005 ) ; and , * * , ( 2006 ) ; and ( 2007 ) , arxiv : gr - qc/0702098 ; , * * , ( 2006 ) ; , , and , *  * , ( ) . 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we consider the cosmology of the ricci - tensor - squared gravity in the palatini variational approach . 
 the gravitational action of standard general relativity is modified by adding a function @xmath0 to the einstein - hilbert action , and the palatini variation is used to derive the field equations 
 . a general method of obtaining the background and first - order covariant and gauge - invariant perturbation equations is outlined . as an example , we consider the cosmological constraints on such theories arising from the supernova type ia and cosmic microwave background observations . 
 we find that the best fit to the data is a non - null leading - order correction to einstein gravity , but the current data exhibit no significant preference over the concordance model . 
 the growth of non - relativistic matter density perturbations at late times is also analyzed , and we find that a scale - dependent ( positive or negative ) sound - speed - squared term generally appears in the growth equation for small - scale density perturbations . 
 we also estimate the observational bound imposed by the matter power spectrum for the model with @xmath1 to be roughly @xmath2 so long as the dark matter does not possess compensating anisotropic stresses .