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dark energy , discovered via the observed luminosity distances out to high-@xmath5 type ia supernovae @xcite , is the most mysterious element in physics today ( see * ? ? ? * for a recent review ) . as dark energy primarily affects the expansion rate of the universe , one can gain information on the nature of dark energy by measuring the cosmological distances as well as the expansion rates of the universe accurately . while the cosmic microwave background ( cmb ) and the type ia supernovae can be used for measuring the angular diameter distance out to @xmath6 and the luminosity distances out to @xmath7 , respectively , the power spectrum of matter distribution in the universe can be used to measure the angular diameter distances _ as well as _ the expansion rates of the universe out to a wider range of redshifts .    _ two _ length scales are encoded in the matter power spectrum , @xmath8 ( see , e.g. , * ? ? ? * ) :    * the comoving hubble horizon size at the matter - radiation equality , @xmath9 $ ] . * the comoving sound horizon size at the so - called drag epoch at which baryons were released from photons , @xmath10 , where @xmath11 $ ] is the sound speed of photon - baryon fluid . the former determines the overall shape of the power spectrum of dark matter including the location of the peak of @xmath8 at @xmath12 , whereas the latter determines the location of the baryonic features called the baryon acoustic oscillations ( baos ) . these length scales can be predicted from the 5-year data of the wilkinson microwave anisotropy probe ( wmap ) @xcite , the wmap 5-year data yield @xmath13 , @xmath14 , @xmath15 , and @xmath16 . ] : @xmath17 and @xmath18 these lengths can be used as the `` standard rulers , '' which give us the angular diameter distances as well as the expansion rates of the universe @xcite . we , as observers who measure the angular and redshift distribution of galaxies , can measure _ four _ distance ratios given by @xmath19 where @xmath20 is the proper ( i.e. , not comoving ) angular diameter distance . we measure @xmath21 and @xmath22 by comparing the predicted lengths with the corresponding observed lengths perpendicular to the line of sight , and @xmath23 and @xmath24 from the lengths parallel to the line of sight . , and that perpendicular to the line of sight , @xmath25 , i.e. , @xmath26 . the angular observables , @xmath27 and @xmath28 , are measured from @xmath25 , while the line - of - sight observables , @xmath29 and @xmath30 , are measured from @xmath31 . ] the baos have been detected in the current galaxy redshift survey data from the sloan digital sky survey ( sdss ) and the two - degree field galaxy redshift survey ( 2dfgrs ) @xcite . however , the current data are not yet sensitive enough to yield @xmath20 and @xmath32 separately @xcite ; thus , one can only determine a combined distance scale ratio from the spherically averaged power spectrum . since two spatial dimensions are available on the sky and one dimension is available along the line of sight , one can measure @xmath33^{1/3 }   = \frac{r_s(z_{drag})}{[(1+z)^2d_a^2(z)c / h(z)]^{1/3}}.\ ] ] @xcite have measured this quantity at @xmath34 from the sdss luminous red galaxies ( lrg ) , and @xcite have extended their analysis to include more data from the sdss lrg , as well as the sdss main galaxy samples and the 2dfgrs galaxies at @xmath35 . @xcite have combined these measurements with the cmb distance ratios determined from the wmap 5-year data , the `` wmap distance priors , '' to obtain the constraints on dark energy properties . the analysis performed in @xcite is a proto - type of what one can do in the future . it is clear that we can gain more information if we can measure @xmath20 and @xmath32 simultaneously at various redshifts . therefore , in the future we should be able to perform a much more sensitive test of dark energy properties by combining @xmath20 and @xmath32 from the future galaxy survey data , and the cmb distance priors from the future cmb experiments such as planck . moreover , the baos capture only a part of information encoded in the shape of @xmath8 . one would miss another baryonic feature , the silk damping scale , by only measuring baos . a more serious drawback is that one would miss the other prominent standard ruler , @xmath36 , completely , by only measuring baos . nevertheless , there is one major advantage of using baos : the phases ( not the amplitude ) of baos are less sensitive to the distortion of the shape of @xmath8 due to non - linear matter clustering , non - linear galaxy bias , or non - linear redshift space distortion @xcite . as a result , many studies have focused on developing various ways to extract the distance information from baos . most of the previous work focused only on extracting the baos from the spherically averaged @xmath8 ( which gives @xmath2 ) ( e.g. , * ? ? ? * ) . @xcite have studied the monopole and quadrupole moments in the galaxy power spectrum and their implications for determinations of the dark energy equation of state parameter , @xmath37 , and concluded that even in the worst case scenario ( i.e. , absence of the baos feature on the observed power spectrum ) , galaxy survey can still provide useful limits on @xmath37 from a combination of the monopole and quadrupole power spectra . recently , @xcite have explored an extraction of the quadrupole moment of the two - dimensional power spectrum , @xmath38 , which gives a different distance combination , @xmath39 .    in this paper , we shall develop a method for extracting @xmath0 and @xmath1 simultaneously from the two - dimensional baos . since we do not use spherical averaging or truncate the legendre expansion of baos at arbitrary orders , our method uses more information than most of the previous methods . to our knowledge , the full two - dimensional extraction of @xmath0 and @xmath1 from baos has been explored only by @xcite . this paper is organized as follows . in [ sec : fitex ] we give a brief account of the original one - dimensional `` fit - and - extract '' ( fitex ) method , which was developed by @xcite for extracting baos from a spherically averaged one - dimensional @xmath8 . we then extend this method to the two - dimensional fitex-2d method by including the full two - dimensional information without spherical averaging . in [ sec : montecarlo ] we extract @xmath0 and @xmath1 from simulated noisy data using the fitex-2d method , and show that the fitex-2d yields unbiased estimates of @xmath0 and @xmath1 . in [ sec : millennium ] we repeat the same analysis for a more realistic simulation , using the millennium simulation @xcite . in   [ sec : darkenergy ] we propagate errors in @xmath32 and @xmath20 to those in the dark energy equation of state with the parametrization of @xmath40 . we conclude in   [ sec : conclusion ] . throughout this paper we shall use the cosmological parameters given by @xmath41 , @xmath42 , @xmath43 , @xmath44 , and @xmath45 @xcite , which are the maximum likelihood values inferred from the wmap 5-year data @xcite combined with the current bao data @xcite and type ia supernova data @xcite . we develop a method for extracting @xmath0 and @xmath1 simultaneously from the two - dimensional baos without spherical averaging . our method builds upon the existing `` fit - and - extract '' ( fitex ) method developed by @xcite for extracting @xmath2 from a spherically averaged , one - dimensional @xmath8 . the fitex method extracts baos by fitting and removing the non - oscillatory part of @xmath8 , which leaves only the oscillatory component , i.e. , baos . @xcite model the non - oscillatory , smooth part by the following functional form : @xmath46 ^ 2 k^{n_s } , \label{eq : psmooth1d}\ ] ] where @xmath47 is the primordial tilt , while @xmath48 , @xmath49 , @xmath50 , @xmath51 , and @xmath52 are free parameters . @xcite have shown that this function is flexible enough to fit out the smooth part of the spherically averaged @xmath8 measured from the hubble volume simulation @xcite . they have tested the fitex method particularly for a large scale , @xmath53 , at high redshifts , @xmath54 , that are relevant to the hobby eberly dark energy experiment ( hetdex ; * ? ? ? we make a simple extension of the one - dimensional fitex method by including angular dependence . we model the two - dimensional smooth power spectrum by @xmath55 , \label{eq : psmooth2d}\end{aligned}\ ] ] where @xmath56 is the cosine of the angle @xmath57 between @xmath58 and the line of sight , i.e. , @xmath59 and @xmath60 . therefore , @xmath61 and @xmath62 for @xmath63 and @xmath64 , respectively .    here , @xmath65 is the legendre polynomials : @xmath66 the odd multipoles must vanish by symmetry . one may include @xmath67 if necessary , but we find it sufficient to include the terms only up to @xmath68 .    finally , @xmath69 is given by the 6th - order polynomials with only even powers of @xmath70 : @xmath71 where all of @xmath72 s are varied simultaneously for each @xmath73 . the odd powers must vanish because they are not analytic in @xmath58 @xcite . we include the terms only up to @xmath74 , as we include the multipoles up to @xmath68 . if , for instance , @xmath75 is included , then @xmath76 may also be included for consistency .    aside from the primordial tilt , @xmath47 , the fitex-2d contains 17 free parameters ( 5 for @xmath77 plus @xmath78 for the angular dependence ) . while it may sound like many , the number of data points available on the _ two - dimensional _ power spectrum is usually much larger , and thus our fit is well behaved .    it may be instructive to use the conventional model for the redshift space power spectrum to show what these parameters are supposed to capture . the leading order angular distortion is given by the so - called kaiser effect , which arises from coherent converging velocity flow toward the linear overdensity region @xcite . the linear kaiser power spectrum is given by @xmath79p^{linear}(k),\end{aligned}\ ] ] where @xmath80 is a @xmath70-independent function that depends on the linear galaxy bias , @xmath81 , and the cosmological parameters ( mainly @xmath82 ) via @xmath83 where @xmath84 is the growth factor of linear density fluctuations . we therefore find @xmath85 and the other terms are zero . another example is the so - called finger - of - god ( fog ) effect , which arises from random motion within virialized halos . when the distribution of the pairwise peculiar velocity within a halo is given by an exponential distribution with the velocity dispersion @xmath86 @xcite , one finds @xcite @xmath87 while the fog yields many terms when expanded into the legendre polynomials , it is still a good approximation to truncate the expansion at @xmath68 if @xmath70 is sufficiently smaller than @xmath88 . note that the fog effect yields terms in the form of powers of @xmath89 ; thus , it makes sense to use the same number for the maximum power of @xmath70 ( see eq . ( [ eq : gl ] ) ) and the maximum multipole ( see eq . ( [ eq : psmooth2d ] ) ) of the fitex-2d fitting function .    in general , neither of these two expressions are adequate . the linear kaiser formula is valid only on very large scales , while the exponential fog formula is valid only on very small scales . at the intermediate scales we find more complicated expressions from , e.g. , the 3rd - order perturbation theory @xcite . to account for these complications we have included @xmath70-dependent coefficients for the legendre polynomials .    in figure  [ fig1 ] and [ fig2 ] we show the performance of @xmath90 . in figure  [ fig1 ] we show a simple analytical model . ] for the non - linear galaxy power spectrum in redshift space given by @xmath91\\ & & \times\frac1{1+f^2k^2\mu^2\tilde{\sigma}_v^2 } , \label{eq : pkred}\end{aligned}\ ] ] where @xmath92 , @xmath93 , and @xmath94 are the density - density , density - velocity , and velocity - velocity power spectra computed from the 3rd - order perturbation theory , and they are given by eq . ( 63 ) , ( 64 ) , and ( 65 ) in @xcite , respectively . this form is similar to eq . ( 71 ) of @xcite , but we have replaced @xmath95 and @xmath96 in his formula by @xmath97 and @xmath98 , respectively , where @xmath99 is the 1-d peculiar velocity dispersion with an empirical fudge factor of @xmath100 calibrated off our simulations presented in @xcite . here , @xmath86 is given by @xmath101    we chose @xmath102 and @xmath103 . the contour of power spectrum is anisotropic in fig . [ fig1 ] due to the redshift space distortion ; however , we recover isotropy after subtracting the best - fitting @xmath90 from the anisotropic data ( see fig .  [ fig2 ] ) . we see that the baos have been extracted successfully , with isotropy of the oscillation phases recovered well . in   [ sec : err_fitex2d ] we show how well we can estimate @xmath0 and @xmath1 from the two - dimensional baos extracted from noisy data using the fitex-2d method . in [ sec : err_full ] we compare the bao results to the accuracy one would obtain from the full modeling of @xmath38 , including the overall shape . in other words , for the former ( baos ) we only use @xmath28 and @xmath30 for measuring @xmath0 and @xmath1 , while for the latter ( full modeling ) we can use @xmath28 , @xmath30 , @xmath27 , @xmath104 , as well as the silk damping scale for measuring @xmath0 and @xmath1 , provided that non - linear effects ( non - linear matter clustering , non - linear redshift space distortion , and non - linear bias ) are under control . note that the treatment of non - linear effects in this section is too simple to be realistic . for a more realistic treatment we shall use the galaxy power spectrum from the millennium simulation @xcite in   [ sec : millennium ] .      to estimate errors in @xmath0 and @xmath1 from the fitex-2d method , we use simple monte carlo simulations .    for the underlying spectrum we use the same data as shown in fig . [ fig1 ] , which includes a simplified modeling of non - linear matter clustering and non - linear redshift space distortion as given by eq . ( [ eq : pkred ] ) . as for the galaxy bias , we use a linear bias with @xmath103 .    once the underlying spectrum is specified , it is straightforward to compute the errors in @xmath105 , @xmath106 , provided that the distribution of @xmath105 is a gaussian . we use the standard formula that includes sampling variance as well as shot noise ( see , e.g. , * ? ? ? * ) @xmath107 where @xmath108 is the number density of galaxies , @xmath109 is the survey volume , @xmath110 and @xmath111 are the fundamental wavenumbers , i.e. , the resolution in @xmath25 and @xmath31 . we take these to be @xmath112 . we use @xmath106 from eq . ( [ eq : sigp ] ) to calculate the r.m.s . error in @xmath105 , and generate 1000 monte carlo realizations . we then apply the fitex-2d method to remove the smooth component from each realization to extract baos . for each realization , we measure @xmath0 and @xmath1 simultaneously by fitting the phases of extracted two dimensional baos to those of the reference baos extracted from either ( i ) the linear power spectrum , or ( ii ) the non - linear power spectrum given by eq . ( [ eq : pkred ] ) , with known @xmath113 and @xmath114 . ( later we find that using the linear spectrum as the reference bao yields the biased estimates of @xmath113 and @xmath114 . ) we use a simplex downhill method for @xmath115-minimization in the two - dimensional parameter space . the number of free parameters for this analysis is two , i.e. , @xmath0 and @xmath1 , and we do not include the amplitude in the fit . we have checked that including the amplitude does not change the results very much , as the amplitude and the phases of baos are nearly uncorrelated ( see appendix  [ sec : cc_no_beta ] for more details ) . this is true in both real and redshift space . when we apply fitex-2d to the simulated data , we perform a fit out to @xmath116 . we choose the survey parameters , @xmath109 , @xmath5 , and @xmath108 , such that they roughly match those expected for the hobby - eberly dark energy experiment ( hetdex ) @xcite : @xmath117 , and @xmath118 with the sky coverage of 420  deg@xmath119 , which yields @xmath120 . emitting galaxies between @xmath118 over 420  deg@xmath119 in 3 years of observations on the hobby - eberly telescope . ] we find that , when the phases extracted by fitex-2d are compared with the reference baos extracted from the linear power spectrum , the best - fitting values of @xmath0 and @xmath1 averaged over 1000 simulations disagree with the underlying , `` true '' values by 0.05% and 0.63% for @xmath0 and @xmath1 , respectively , due to the phase shift of baos caused by non - linearities ( including non - linear redshift space distortion ) . this result extends the previous study by @xcite , who studied a spherically averaged 1-d power spectrum and found that the bias was less than 1% in @xmath121 .    on the other hand , when the phases are compared with the reference baos extracted from the _ non - linear _ power spectrum ( eq . ( [ eq : pkred ] ) ) , the best - fitting values of @xmath0 and @xmath1 agree with the true values to well within the monte carlo sampling error ; thus , we confirm that the fitex-2d method yields unbiased estimates of @xmath0 and @xmath1 .    in figure  [ fig3 ] we show the projected error ellipses on @xmath0 and @xmath1 from the baos extracted with the fitex-2d ( larger , dotted contours ; same in all four panels ) . we find 1.8% and 2.5% errors on @xmath0 and @xmath1 , respectively , with the cross - correlation coefficient of @xmath122 , from the monte carlo simulations . for the same survey parameters , the bao fisher matrix proposed by @xcite yields 1.5% and 2.5% errors on @xmath0 and @xmath1 , respectively , with @xmath123 . therefore , we conclude that the fitex-2d method yields the results that nearly saturate the fisher matrix bound , i.e. , it is nearly an optimal method in a sense that it can yield the smallest errorbars one can obtain with the bao - only analysis .      to calculate the errors in @xmath0 and @xmath1 expected from the full modeling of the two - dimensional galaxy power spectrum , @xmath124 , we use the fisher matrix given by ( see , e.g. , * ? ? ? * ; * ? ? ? * )    @xmath125    where @xmath126 for @xmath127 , 2, ... ,7 , respectively , @xmath128 , where @xmath48 is the overall amplitude of the power spectrum , @xmath98 is the linear redshift distortion parameter , @xmath129 is the calibrated 1-d velocity dispersion ( see eq . ( [ eq : pkred ] ) ) , and @xmath47 and @xmath130 describe the shape of the initial ( primordial ) power spectrum : @xmath131    here , the weight function , @xmath132 , is one half of the so - called `` effective volume , '' @xmath133 v_{survey } \equiv \frac12v_{eff}(k,\mu ) . \label{eq : weight_fm}\ ] ] the effective volume is equal to the actual survey volume , @xmath109 , in the sampling variance dominated regime , @xmath134 , whereas it is small in the shot - noise dominated regime , @xmath135 . in figure  [ fig4 ] we show @xmath136 for @xmath117 and @xmath103 as a function of @xmath5 . the factor of @xmath137 accounts for symmetry in @xmath138 . the derivatives of @xmath139 with respect to @xmath140 are calculated and given in the appendix  [ sec : derivatives ] . cccccccc @xmath141 ( % ) & 0.279 & 0.877 & 0.317 & 0.282 & 0.479 & 0.416 & 1.100 + @xmath142 ( % ) & 0.437 & 0.786 & 1.124 & 0.801 & 0.509 & 0.539 & 1.134 + @xmath143 & 0.382 & @xmath144 & @xmath145 & 0.082 & @xmath146 & @xmath147 & 0.038 + @xmath148 ( % ) & 0.187 & 0.762 & 0.317 & 0.259 & 0.386 & 0.363 & 0.775 + & @xmath98 , @xmath129 & @xmath149 , @xmath129 & @xmath149 , @xmath98 , @xmath129 & @xmath149 , @xmath47 , @xmath130 & @xmath149 , @xmath98 , @xmath47 & @xmath149 , @xmath98 , @xmath129 , + & & & & & @xmath130 & @xmath47 , @xmath130 + @xmath141 ( % ) & 0.327 & 0.891 & 1.101 & 1.089 & 1.233 & 1.250 + @xmath142 ( % ) & 1.457 & 1.101 & 1.468 & 0.984 & 1.362 & 1.530 + @xmath143 & @xmath150 & @xmath151 & 0.005 & @xmath152 & @xmath153 & @xmath154 + @xmath148 ( % ) & 0.322 & 0.869 & 0.879 & 0.974 & 1.000 & 1.014 [ tb : fisher ]    unlike for baos , which are insensitive to the parameters that affect the overall shape , for the full modeling we need to make sure that we take into account potential degeneracy between @xmath0 and @xmath1 and any other parameters that affect the overall shape . in this paper we include @xmath149 , @xmath98 , @xmath155 , @xmath47 , and @xmath130 . ( we shall comment on the effects of non - linear bias in   [ sec : caveat ] ) . we study the effects of marginalization over various parameter combinations by taking the submatrix , @xmath156 , of the full @xmath157 matrix with the index , @xmath158 , of @xmath140 running from 1 to 7 , such that the submatrix includes the matrix components of desired parameters to be marginalized . in other words , the parameters that are not included in the submatrix are fixed and not marginalized over . then , we compute the marginalized errors in @xmath159 and @xmath160 as @xmath161    to simplify the analysis , we fix all the other cosmological parameters , such as @xmath162 , @xmath163 , etc . these cosmological parameters will be determined by the future cmb mission , planck , accurately , and therefore it is a good approximation to simply fix them , and vary only @xmath159 , @xmath160 , @xmath48 , @xmath98 , @xmath129 , @xmath47 and @xmath130 . the fiducial value for the bias is set to @xmath103 and @xmath164 is computed from the fiducial cosmological model . for the following reason : while @xmath96 can be calculated from the cosmological parameters assuming the validity of general relativity , one might choose to let @xmath96 free and use it for testing the validity of general relativity . in this paper we chose to assume the validity of general relativity , but one can extend our analysis to let @xmath96 free in a straightforward manner . ] we expect that the analysis of the bispectrum ( fourier transform of three - point function ) will give a precise determination of @xmath81 ( as well as non - linear bias parameters such as @xmath165 ) @xcite , and therefore it is also a good approximation to simply fix it . however , we also explore a more conservative case where we do not know what @xmath81 is , i.e. , we marginalize over the overall amplitude as well as @xmath98 . in the future work we also plan to investigate the effect of marginalization over @xmath165 , using a joint analysis of the power spectrum and bispectrum . therefore , our calculation presented here will provide the lower limit to the errors in @xmath159 and @xmath160 expected from the full modeling of the power spectrum measured in a survey like hetdex . we use the same survey parameters that we have used in   [ sec : err_fitex2d ] , and we integrate eq . ( [ eq : fisher_fm ] ) up to @xmath116 .    in figure  [ fig3 ] we show the resulting error ellipses from the full modeling , in the smaller , solid contours , with four choices of marginalization . ( we present the results from more choices of marginalization in table  [ tb : fisher ] . ) first , for all choices of marginalization we find that the sizes of the errors in both @xmath0 and @xmath1 are substantially smaller than those from the bao - only analysis with the fitex-2d . for example , determinations of both @xmath0 and @xmath1 are improved by more than a factor of two in the case of the amplitude marginalization . this is expected , as we are able to use more information encoded in the power spectrum ; namely , the hubble horizon at the matter - radiation equality epoch and the silk damping scale . second , @xmath0 and @xmath1 are _ anti - correlated _ for the amplitude marginalization , with the cross - correlation coefficient of @xmath166 ( see top - left panel of fig .  [ fig3 ] ) , as opposed to a positive correlation seen in the bao - only analysis . this is due to the marginalization over the overall amplitude : if we fixed the overall normalization , then we would still find a positive correlation between @xmath0 and @xmath1 with @xmath167 . the origin of the negative correlation is the so - called alcock - paczynski ( ap ) test @xcite : when the redshift space distortion is known perfectly well , the departure of the power spectrum in redshift space from isotropy , i.e. , dependence of @xmath38 on @xmath168 , can be used to determine @xmath39 , resulting in @xmath169 for a power - law power spectrum . the contributions from departures of @xmath8 from a pure power - law , i.e. , the existence of `` standard rulers , '' such as baos , the hubble horizon at the matter - radiation equality and the silk damping scale , make @xmath170 bigger than @xmath171 . ( see appendix  [ sec : cc ] for more details . ) when @xmath149 and @xmath98 are marginalized over simultaneously , the correlation between @xmath0 and @xmath1 nearly disappears : the ap test no longer works when we marginalize over the linear redshift space distortion . we find @xmath172 ( see top - right panel of fig .  [ fig3 ] ) for this case . when @xmath149 is marginalized over while the other parameters ( @xmath98 , @xmath173 , @xmath47 , and @xmath130 ) are held fixed , we find 0.88% and 0.79% errors on @xmath0 and @xmath1 , respectively , with @xmath166 . the more parameters we marginalize over , the greater the cross - correlation coefficient between @xmath0 and @xmath1 as well as the errors on @xmath0 and @xmath1 become . note that the increase in the errors does not necessarily imply the decrease in the statistical power in constraining dark energy properties : since the cross - correlation coefficient is also reduced , the error in the combined 1-d distance scale , @xmath174 , is much less affected by the marginalization ( see table  [ tb : fisher ] ) . the error in @xmath175 has been computed as @xcite : @xmath176    finally , the errors in @xmath0 , @xmath1 , and @xmath174 for various choices of marginalization are : ( @xmath177 ) = ( 0.88% , 0.79% , 0.76% ) , ( 1.10% , 1.13% , 0.78% ) , ( 1.10% , 1.47% , 0.88% ) , and ( 1.25% , 1.53% , 1.01% ) for the marginalization over @xmath149 , @xmath149 and @xmath98 , @xmath149 , @xmath98 and @xmath129 , and @xmath149 , @xmath98 , @xmath129 , @xmath47 and @xmath130 , respectively ( see table  [ tb : fisher ] for more comprehensive list ) . this result should be compared with that from the bao - only analysis : ( @xmath177)=(1.76% , 2.47% , 1.08% ) . it is clear that the full analysis , even with a generous set of marginalization choices , beats the bao - only analysis with a significant gain in the distance determination accuracies . our analysis presented in   [ sec : err_full ] is too simplistic and optimistic , as it ignores any systematic errors due to our lack of understanding of the effects of various non - linearities in the power spectrum .    among the three major non - linearities , non - linear matter clustering is under control , at least for high redshifts , i.e. , @xmath178 , as one can model non - linear evolution of matter fluctuations almost exactly by the 3rd - order perturbation theory @xcite . while the nominal 3rd - order perturbation theory breaks down at lower redshifts , @xmath179 , there have been a number of studies aiming at improving upon our ability to compute @xmath8 at @xmath179 or even lower redshifts @xcite . therefore , it is quite possible that the non - linear matter clustering will be fully under control in the near future , at least for the scales that are relevant to the bao scales , i.e. , @xmath180 .    in a separate paper @xcite , we show that non - linear galaxy biasing is also under control in the weakly non - linear regime . one can use the perturbation theory approach combined with the local bias assumption @xcite to model the galaxy power spectrum with non - linear bias . the most problematic one is the non - linear redshift space distortion . our understanding of non - linear redshift space distortion , especially the finger - of - god ( fog ) effect , is limited @xcite . therefore , whether one can achieve the accuracy of @xmath0 and @xmath1 ( @xmath1 in particular ) reported in fig . [ fig3 ] depends crucially on our ability to correct for the fog effect . this is work in progress . note that the marginalization over @xmath173 should capture some of the increase in the errors in distance scales due to our ignorance of fog . how realistic is our result for the determinations of @xmath0 and @xmath1 from the bao phases using the fitex-2d method ? since our monte carlo simulations used in   [ sec : err_fitex2d ] are too simple , in this section we test the fitex-2d method further by using the millennium simulation @xcite . we use the millennium galaxy catalogue , generated by the semi - analytical galaxy formation code @xcite . we have measured the two - dimensional power spectrum of galaxies in redshift space from the millennium simulation , and applied the fitex-2d method to remove the smooth component . we then find the best - fitting @xmath0 and @xmath1 from the bao phases extracted from the fitex-2d . again , we use the data up to @xmath116 .    in figure  [ fig5 ] we show the result . the best - fitting value that we find from the millennium simulation corresponds to one point at the center of the contours . we find the errors from the monte carlo simulations that we described in   [ sec : err_fitex2d ] with the survey parameters replaced by those of the millennium simulation : @xmath181 , @xmath182 , and @xmath183 . ( there are 17,238,935 galaxies in the millennium simulation at @xmath183 . ) for the theoretical power spectrum that we use for generating monte carlo simulations , we use the best - fitting power spectrum for the galaxy catalogue of the millennium simulation found in @xcite . since the volume of the millennium simulation is @xmath184 times as small as that would be surveyed by hetdex , the uncertainties in @xmath0 and @xmath1 are larger for the millennium simulation . ( compare fig . [ fig5 ] with the larger contours of fig . [ fig3 ] . ) we find 5.1% and 6.8% errors on @xmath0 and @xmath1 , respectively , with the cross - correlation coefficient of @xmath185 , from the monte carlo simulations . these errors are larger than those from hetdex monte - carlo simulation by a factor of two ( rather than @xmath186 ) as the shot noise on the power spectrum of the millennium simulation is much smaller than that of hetdex simulation . the best - fitting values of @xmath0 and @xmath1 are well within 68% c.l . region , which indicates that the fitex-2d is able to yield unbiased estimates of the bao phases from the millennium simulation . these results indicate that the fitex-2d method that we have developed in this paper can be used for extracting the baos and measuring @xmath0 and @xmath1 safely from the real data . it would be interesting to apply the fitex-2d method to the two - dimensional power spectrum measured from the sdss lrg samples @xcite , and extract @xmath0 and @xmath1 from them . in   [ sec : montecarlo ] and   [ sec : millennium ] , we have estimated errors in @xmath0 and @xmath1 from two different approaches , i.e. , the bao fitting using the fitex-2d method and the full modeling . in this section , we propagate errors in @xmath0 and @xmath1 to those in the dark energy equation of state parameters . we parametrize @xmath187 using the linear model , @xmath40 @xcite . we propagate the errors in @xmath0 and @xmath1 to those in @xmath3 and @xmath4 by @xmath188 where @xmath189 is the fisher matrix for the dark energy parameters , @xmath190 is the fisher matrix for @xmath0 and @xmath1 , @xmath191 for @xmath127 and 2 , and @xmath192 for @xmath193 and 2 . partial derivatives of @xmath0 and @xmath1 with respect to @xmath3 and @xmath4 are given by @xmath194f(z^{\prime})g(z^{\prime})^{-3/2}dz^{\prime } \over\int_0^z g(z^{\prime})^{-1/2}dz^{\prime}},\\ { \partial \ln{h } \over \partial w_0}&=&{3\over2}\omega_\lambda \ln(1+z){f(z)\over g(z)},\\ { \partial \ln{h } \over \partial w_a}&=&{3\over2}\omega_\lambda \left[\ln(1+z)-\frac{z}{1+z}\right]{f(z)\over g(z)},\end{aligned}\ ] ] where @xmath162 and @xmath195 are given by @xmath196    figure [ fig6 ] shows the derivatives as a function of @xmath5 between @xmath197 in two different cosmological models , ( @xmath3 , @xmath4)=(@xmath198 , @xmath199 ) and ( @xmath200 , @xmath201 ) . the former is the @xmath202cdm model , while the latter resembles the maximum likelihood values of @xmath3 and @xmath4 from the wmap+bao+sn+bbn @xcite . the derivatives are similar for these cosmological models , and therefore we use the @xmath202cdm model as the fiducial model for computing the derivatives . we add the distance information from cmb as @xmath203 where we assume that the cmb experiment yields 1% determination of the angular diameter distance out to @xmath204 , i.e. , we use @xmath205    we are interested in how the bao - only analysis compares with the full modeling . in fig . [ fig7 ] we show the projected error contours on @xmath3 and @xmath4 calculated from the bao - only analysis with the fitex-2d and those from the full analysis at four redshift bins : @xmath206 , @xmath207 , @xmath208 , and @xmath209 . the survey area and the number of galaxies are 420  @xmath210 and @xmath211 for all redshift bins . from the bao - only analysis we find @xmath212 , @xmath213 , @xmath214 , and @xmath215 , whereas from the full modeling we find @xmath216 , @xmath217 , @xmath218 , and @xmath219 , for @xmath206 , @xmath207 , @xmath208 , and @xmath209 , respectively .    we therefore conclude that the full analysis yields much better constraints on @xmath3 and @xmath4 than the bao - only analysis . in this paper we have developed a method , called the fitex-2d method , to extract the two - dimensional phases of baos from galaxy power spectra in redshift space . our model builds on and extends the existing one - dimensional algorithm , called fitex , developed by @xcite . our method removes the smooth , non - oscillating component from the observed galaxy power spectrum in redshift space . the fitting function consists of the smooth one - dimensional spectrum that depends only on @xmath70 , @xmath77 given by eq . ( [ eq : psmooth1d ] ) , multiplied by the angle - dependent function expanded in the legendre polynomials with even multipoles . the coefficients of the legendre polynomials contain even powers of @xmath70 . the resulting function , given by eq . ( [ eq : psmooth2d ] ) , is able to capture the non - oscillating part of the galaxy power spectrum well . we have tested the fitex-2d method using the analytical model without any noise , the monte carlo realizations with noise expected from the hetdex experiment @xcite , and the galaxy catalogue created from the millennium simulation @xcite . in all cases the fitex-2d method yields unbiased estimates of the angular diameter distance , @xmath0 , and the expansion rate , @xmath1 . however , the baos capture only a part of distance information encoded in the galaxy power spectrum . to exploit the distance information , especially the equality scale , @xmath220 , we have explored the constraints on @xmath0 and @xmath1 from the full modeling of the galaxy power spectrum in redshift space . provided that three key non - linearities ( non - linear matter clustering , non - linear galaxy bias , and non - linear redshift space distortion ) are under control , we find that the full modeling yields the constraints that are better than the bao - only analysis by more than a factor of two both in @xmath0 and @xmath1 , and the dark energy parameters such as @xmath3 and @xmath4 .    while the effects of non - linear matter clustering @xcite and non - linear galaxy bias @xcite are being understood in the weakly non - linear regime that is relevant to the future galaxy surveys , the effects of non - linear redshift space distortion are poorly understood . while the fitex-2d method that we have developed in this paper are useful for obtaining _ robust _ constraints on @xmath0 and @xmath1 , hence the dark energy properties , one must understand non - linear redshift space distortion to fully exploit the full information content of the galaxy power spectrum in redshift space . we would then be able to reduce the errors in @xmath0 and @xmath1 by more than a factor of two . we thank the anonymous referee for illuminating comments on the correlation coefficients , which motivated our doing more thorough analysis . this material is based in part upon work supported by the texas advanced research program under grant no . 003658 - 0005 - 2006 . ek acknowledges support from an alfred p. sloan research fellowship . the millennium simulation databases used in this paper and the web application providing online access to them were constructed as part of the activities of the german astrophysical virtual observatory . in this appendix we describe what we have implemented in our fisher matrix code , which is publicly available as a part of `` cosmology routine library ( crl ) , '' developed by one of the authors ( ek ) . this code includes the non - linear matter power spectrum in both real and redshift space , as well as marginalization over the amplitude , the linear redshift space distortion , the velocity dispersion of fingers - of - god , the primordial tilt and running index . in the future release we plan to include non - linear galaxy bias and primordial non - gaussianity . a simple , approximate formula of the fisher matrix for galaxy survey is given by ( e.g. , * ? ? ? * ) @xmath221 where @xmath124 is the galaxy survey power spectrum calculated theoretically as a function of parameters , @xmath140 are the parameters to be extracted from the data , and @xmath132 is a function given by @xmath133 v_{survey } \label{eq : weight}\ ] ] here , @xmath108 and @xmath109 are the number density of galaxies and the volume of survey , respectively .    in linear theory , @xmath124 is given by @xmath222 where @xmath81 is the scale independent linear bias factor , @xmath223 is the linear matter power spectrum , and @xmath224 describes the linear redshift space distortion effect ( kaiser effect ) : @xmath225 where @xmath84 is the growth factor of the linear density fluctuations , and @xmath226 is the scale factor .      to calculate the logarithmic derivatives of @xmath8 in eq . ( [ eq : fisher ] ) , let us write down the non - linear galaxy power spectrum ( with linear bias ) as ( eq .  ( [ eq : pkred ] ) ) : @xmath227 \times\frac1{1+f^2k^2\mu^2\tilde{\sigma}_v^2}.\end{aligned}\ ] ] we compute the derivatives with respect to the following seven parameters : the angular diameter distance , @xmath0 , the hubble expansion rate , @xmath1 , the overall amplitude of the galaxy power spectrum , @xmath48 , the linear redshift space distortion factor , @xmath80 , the velocity dispersion with an empirically calibrated fudge factor , @xmath129 , the tilt of the primordial power spectrum , @xmath47 , and the running index , @xmath130 ( @xmath228}$ ] ) . we choose the convention such that @xmath229        we compute the derivatives with respect to @xmath159 and @xmath160 in a two step process . first , we write @xmath231 where @xmath232 finally , we need to know the `` effective spectral index '' , @xmath233 , given by @xmath234 or explicitly @xmath235 where @xmath236      in this subsection we explore the behaviour of the cross - correlation coefficient between @xmath0 and @xmath1 in various cases . in particular we focus on the effect of the marginalization over the overall amplitude with (   [ sec_cc_beta ] ) and without (   [ sec : cc_no_beta ] ) the additional marginalization over the redshift space distortion . let us evaluate the fisher matrices in the limit that the redshift space distortion is absent , i.e. , @xmath237 . in this limit , the weighting function in eq . ( [ eq : fisher ] ) and the effective spectral index in eq . ( [ eq : n_eff ] ) become independent of @xmath56 , i.e. , @xmath238 and @xmath239 . we obtain @xmath240 ^ 2w(k ) \int^1_0d\mu ( 1-\mu^2)^2\\ f_{12}&=&\int^{k_{max}}_{k_{min}}\frac{k^2dk}{2\pi^2}[n_{eff}(k)]^2w(k ) \int^1_0d\mu ( 1-\mu^2)(-\mu^2)\\ f_{13}&=&\int^{k_{max}}_{k_{min}}\frac{k^2dk}{2\pi^2}n_{eff}(k)w(k ) \int^1_0d\mu ( 1-\mu^2)\\ f_{22}&=&\int^{k_{max}}_{k_{min}}\frac{k^2dk}{2\pi^2}[n_{eff}(k)]^2w(k ) \int^1_0d\mu ( -\mu^2)^2\\ f_{23}&=&\int^{k_{max}}_{k_{min}}\frac{k^2dk}{2\pi^2}n_{eff}(k)w(k ) \int^1_0d\mu ( -\mu^2)\\ f_{33}&=&\int^{k_{max}}_{k_{min}}\frac{k^2dk}{2\pi^2}w(k ) \int^1_0d\mu\end{aligned}\ ] ] now , in order to understand the effect of the structure of @xmath241 , let us assume that the galaxy power spectrum is a pure power - law , i.e. , @xmath242 and @xmath243 is the independent of @xmath70 . in this limit , we obtain        the marginalized errors of parameters and the correlation coefficients are computed from the inverse of the fisher matrix . however , one can show that the matrix given in eq . ( [ eq:3by3fisher ] ) is singular . in other words , @xmath0 and @xmath1 are completely degenerate with the amplitude for a power - law power spectrum . this result shows that only the departure of the power spectrum from a pure power - law , i.e. , the existence of characteristic scales , can break the degeneracy between @xmath0 and @xmath1 , and @xmath48 . these scales are often called the `` standard rulers . ''    to understand the structure of the fisher matrix in eq . ( [ eq:3by3fisher ] ) better , let us add small perturbations , @xmath246 , to the diagonal elements , and invert the matrix . the result is @xmath247 we find that the correlation coefficient between @xmath0 and @xmath1 is @xmath248 as @xmath249 . therefore , @xmath159 and @xmath160 are totally anti - correlated , which implies that , although we can not determine @xmath159 and @xmath160 simultaneously , we can determine @xmath250 , even for a power - law power spectrum . and @xmath251 for @xmath252 and @xmath253 respectively . ] this is known as the alcock - paczyski ( ap ) test @xcite . there is a special case in which the covariance between @xmath48 and @xmath0 or @xmath1 may be ignored . one may imagine the situation where @xmath254 depends upon @xmath70 such that @xmath48 is uncorrelated with @xmath0 or @xmath1 . for example , if @xmath254 oscillates about zero , then @xmath255 would be small compared with @xmath256 ^ 2w(k)$ ] or @xmath257 . therefore , @xmath258 and @xmath259 may be ignored , making @xmath48 de - correlated with @xmath0 and @xmath1 . in this case , the fisher matrix is a 2-by-2 matrix given by @xmath260 the inverse of this matrix is then @xmath261 the correlation coefficient between @xmath0 and @xmath1 is thus given by @xmath262 this result has been derived by @xcite , and justifies the use of baos as a way to measure @xmath0 and @xmath1 with a correlation coefficient of @xmath263 .      * when the information is dominated by baos , the correlation coefficient between @xmath0 and @xmath1 is @xmath264 . the amplitude of the baos contributes little to the errors on @xmath0 and @xmath1 , as the amplitude information is de - correlated with @xmath0 and @xmath1 . * when the information is dominated by the ap test , @xmath265 . * in reality , as we have shown in this paper , baos contribute less than the overall shape of the power spectrum . also , the shape of the power spectrum is not exactly a power - law . as a result , the correlation coefficient from the full analysis is usually negative ( or small positive ) , but always greater than @xmath171 ( see table  [ tb : fisher ] ) . next , let us consider the case where the redshift space distortion can not be ignored . in this case , we see from eq . ( [ eq : weight ] ) and eq . ( [ eq : n_eff_exp ] ) that the weighting function , @xmath132 , and the effective spectral index , @xmath233 , are no longer independent of @xmath56 . the analytical treatment is also possible for this case , although the results are too complicated to be useful . we therefore report on the numerical results .    here , we choose the survey parameters given in [ sec : err_full ] with the non - linear power spectrum of eq . ( [ eq : pkred ] ) . the results from the numerical calculations of the fisher matrix are given in table [ tb : fisher ] . we find that the marginalization over the amplitude information , and that over the amplitude and the shape of the primordial power spectrum ( i.e. , @xmath47 and @xmath130 ) give the cross - correlation close to @xmath171 ; thus , one relies on the ap test . the marginalization over the amplitude and the linear redshift space distortion ( i.e. , @xmath98 ) drive the cross - correlation towards zero , as the ap test no longer works when the linear redshift space distortion is marginalized over . however , in both cases the errors in the combined 1-d distance scale , @xmath174 , are about the same . in other words , while one changes the orientation of the ellipse , the area is approximately preserved .    in summary , when the amplitude information is marginalized over , the information is mostly coming from the dependence of @xmath38 on @xmath168 , which yields a constraint on @xmath39 via the ap test , while when both the amplitude and the linear redshift space distortion are marginalized , the most information is coming from the standard rulers , which can constrain @xmath0 and @xmath1 separately , driving the cross - correlation towards zero .    finally , in fig . [ fig8 ] we show how different choices of marginalization over parameters influence the error contours of @xmath3 and @xmath4 : @xmath267 , @xmath268 , @xmath269 , and @xmath270 , for the cases of no marginalization , marginalization over @xmath149 , marginalization over @xmath149 , @xmath98 and @xmath129 , and marginalization over @xmath149 , @xmath98 , @xmath129 , @xmath47 and @xmath130 respectively .        * linear power spectrum with the linear redshift space distortion ( kaiser effect ) , * non - linear power spectrum from the 3rd - order perturbation theory with the linear redshift space distortion , * non - linear power spectrum from the 3rd - order perturbation theory with the non - linear redshift space distortion given by eq . ( 71 ) of @xcite , or * non - linear power spectrum from the 3rd - order perturbation theory with the non - linear redshift space distortion given by eq . ( [ eq : pkred ] ) . a given galaxy survey can be sliced up into multiple redshift bins . after entering the survey area in units of @xmath271 , one is asked to enter the following parameters at each redshift bin : the redshift range ( @xmath272 ) , the number of galaxies in units of millions in the bin , @xmath81 , @xmath273 in units of @xmath274 , and the redshift error in units of km / s . the linear power spectrum at @xmath275 has been precomputed using the camb code @xcite for the maximum likelihood parameters given in table  1 of @xcite (  wmap+bao+sn  ) . the ingredients of the non - linear power spectra , @xmath276 , @xmath277 , and @xmath278 , have been precomputed from the linear spectrum at @xmath275 . these spectra are then evolved to a specified redshift by the appropriate growth factor obtained by solving the differential equation given in eq . ( 76 ) of @xcite .
we present a method for extracting the angular diameter distances , @xmath0 , and the expansion rates , @xmath1 , of the universe from the _ two - dimensional _ baryon acoustic oscillations ( bao ) in the galaxy power spectrum . 
 our method builds upon the existing algorithm called the `` fit - and - extract '' ( fitex ) method , which allows one to extract only @xmath2 from a spherically averaged one - dimensional power spectrum . 
 we develop the fitex-2d method , an extension of the fitex method , to include the two - dimensional information , which allows us to extract @xmath0 and @xmath1 simultaneously . 
 we test the fitex-2d method using the millennium simulation as well as simplified monte carlo simulations with a bigger volume . 
 the baos , however , contain only a limited amount of information . 
 we show that the full modeling , including the overall shape of the power spectrum , yields much better determinations of @xmath0 and @xmath1 , hence the dark energy equation of state parameters such as @xmath3 and @xmath4 , than the bao - only analysis by more than a factor of two , provided that non - linear effects are under control .