in recent years , relativistic effects have been vastly studied .a number of the studies in the literature have shown how to calculate the observed clustering overdensity of galaxies in a relativistic context , i.e. the relativistic galaxy overdensity  , revealing relativistic corrections to the simple newtonian calculation that is standard for low redshift , small area surveys .galaxy surveys with large volume coverage spanning scales nearly and larger than the hubble radius , reaching high redshifts require relativistic corrections to the observed overdensity .basically , these corrections are owing to the large - scale effects of ( i ) the gravitational potential , both local at the observed galaxies and also along the line of sight , and ( ii ) peculiar velocities , by the apparent motion of the galaxies relative to the observer ; together , these constitute the so - called `` relativistic effects '' .current optical surveys only cover low redshifts and sky area with dimensions which are smaller than the hubble radius ; thus , containing negligible relativistic effects .however , upcoming surveys in the optical and the radio / infrared , will extend to high redshifts and encompass large sky area covering distance scales equal and larger than the horizon providing deeper and richer information on the observable universe , but most importantly , on cosmological scales at the survey redshifts . on these scales ,the relativistic effects become significant .these effects can , e.g. provide a means for a sensitive probe of the nature of dark energy and modified gravity : by comparing the predictions from the relativistic corrections to the observation of the forthcoming surveys .there are two fundamental issues underlying the relativistic analysis : firstly , we need to correctly identify the galaxy overdensity that is observed on the past light cone ; secondly , we need to account for all the distortions arising from observing on the past light cone , including redshift and volume perturbations ( with all relativistic effects included ) .these relativistic effects appear in the power spectrum of matter in redshift space .however , all of these effects together can suitably be accounted for simply by using the fact that the physical ( observed ) number of the galaxies depends on two main quantities : the survey _ volume _ and the apparent _ flux _ of the sources .thus in this work , we rely on the dependence of the physical galaxy number on the survey volume and the apparent flux , to re - trace the known relativistic galaxy overdensity ( at first order perturbation ) .we recover all previously calculated terms , and yet uncover subtle , but crucial ( background ) parameters which are missed out by previous works .we start by re - calculating the relativistic galaxy overdensity while revealing the new parameters in sec .[ sec : rdp ] ; we relate our results to some previous works in sec .[ sec : cpr ] . in sec .[ sec : fvls ] we give an overview of specific ( flux- and volume - limited ) samples of galaxy surveys and , in sec . [ sec : tlslps ] we demonstrate the possible implications of the new parameters on the large - scale prediction of galaxy clustering , via the galaxy power spectrum , in the standard concordance model .we conclude in sec .[ sec : concl ] .( we leave out the specifics on measuring the new parameters as a future project ; here we focus on their possible cosmological implications . )in cosmological surveys , observers usually look at a certain volume of the sky through a solid angle in a given direction , at a redshift away . apart from the redshift , the solid angle and the intrinsic properties of the observed sources , surveys will generically also depend on cosmic magnification via the apparent flux ( or luminosity ) of the observed sources  .the observed flux of a source is implicitly ( de)amplified in an inhomogeneous universe .( hereafter , we use `` magnification '' to mean _ magnification of sources _ , by the amplification of their flux and/or angular size . )thus , the physical number of galaxies observed in an _ arbitrary _ survey will depend on two main quantities , which are , the observed volume , and the apparent flux of the sources i.e .  .  moreover, a generic sample of cosmic objects in the sky would inherently contain both `` magnified '' and `` unmagnified '' fractions , respectively ; where the magnified fraction is proportional to the magnification bias .however , the actual measurable quantity being the number of objects per unit solid angle per redshift would consist of all the events together , with no distinctions between these fractions .nevertheless , these fraction can each be isolated by observations , and measured separately , given that the unmagnified fraction is volume dependent , and the magnified fraction is flux dependent .thus , a given volume element and flux interval , within a specified redshift range and solid angle interval , will lead to galaxies : [ dn2 ] dn(,z ) & = & _ g(,z ) d(,z ) + _ _ f(,z ) df(,z ) & & _g(,z ) d_dz , where ( henceforth , a tilde denotes _ redshift space _ ) we note that the decomposition of the total differential , top line in , follows directly from differential calculus ( given that the observed sky volume is independent of the apparent flux from a source ) , and [ def : n_g ] _ g . |_f _ * = ^f_*_f_cutdf ( , f ) , where denotes a given ( maximum ) flux value ( to be specified by the experiment ) with apparent magnitude ; is the instrument threshold or cut - off flux with apparent magnitude , and ( varying with both volume and flux ) implicitly measures the _ flux _ or _ luminosity function _ ; with , [ def : n_f ] _ _ f .| _ _ * = ^_*__mind(,f ) , where denotes a given ( maximum ) volume at a specified distance , and is the initial volume at a minimum distance ( at the observer , or another arbitrary position ) . note that , the ( ensemble ) average number of galaxies per unit volume , with flux in the infinitesimal range from to , is ; similarly , gives the average number of galaxies per unit flux , with distance ( or position ) within the volume element from to .consequently , we have and as the _ effective _ ( or _ cumulative _ ) : galaxy number per unit volume , and galaxy number per unit flux , respectively.evaluated at a fixed flux the cumulative number of galaxies per unit volume , with flux within the finite range , and at a fixed volume the cumulative number of galaxies per unit flux in a given finite volume ( or with distance within ) . ] ( we notice that the number density is evaluated at a fixed flux , whereas , the number density is evaluated at a fixed volume . )surveys like the sdss and the lcrs , among others , employ two flux cut - offs : the instrument threshold ( below which no source is detected ; thus , being the lower bound ) , and a certain maximum ( beyond which no source is detected ; thus , being the upper bound ) .thus , only sources with apparent magnitude in a given closed interval in a certain sky volume , are measured ; usually , the given volume will also be demarcated by two distance ( lower and upper ) limits .note that by integrating over volume , the number density is independent of distance ; conversely , depends on distance .the quantity , given by , is the observed galaxy number per unit solid angle per redshift which is what observers measure .if the survey also pertains the apparent size of the sources , then the observed flux from a source will depend on the apparent size of that source , i.e.  ; whence , [ tildeng ] _g(,z ) = _ g(,z ) ( , z ) + _ _ f(,z ) ( , z ) _ _ x(,z ) , where is the volume per unit solid angle per redshift ; with being the flux change per size interval ( i.e. flux per unit size ) , and is the source size per unit solid angle per redshift .the redshift distribution of the galaxies is obtained by averaging over all solid angles : .therefore , the ( observed ) relativistic galaxy overdensity , is obtained by [ obzdelta ] ^obs_g(,z ) & & , + [ obzdelta2 ] & = & e__v(|z ) ( _ g(,z ) + _ _ v(,z))&+ & e__f(|z ) q(|z ) _ _ m(,z ) , where is the average galaxy distribution density in redshift space ( hereafter , overbars denote `` background '' quantities , averaged over all solid angles , and is the perturbation in the given quantity , with ) ; is the number density contrast , and is the volume density contrast . moreover , by using that for magnified sources the flux per unit source size in the background is proportional to the underlying magnification per unit source size , i.e.  , we have ; consequently , is the magnification contrast .  the parameters and are the fractional contributions by the change in volume and the change in flux , respectively , in the background sample i.e .if the survey volume and the apparent flux are variable , then the change in volume will contribute a certain fraction in the average distribution density , and similarly , the change in flux will also contribute a fraction together yield the identity [ fracsum ] e__v(|z ) + e__f(|z ) = 1 , where for fixed - volume surveys ( at fixed ) , and the entire background sample is generated by the variation in flux , with ; of which sample is ideal for pure magnification analysis , see e.g.  .( fixing the apparent flux and / or size leads to with ; thus , corresponding to a pure number - count analysis , see e.g.  . )these parameters , and , have never previously been taken into account ( in the literature ) in the calculation of the relativistic galaxy overdensity .the parameter , is the cosmic _ magnification bias _ , given by [ b_m ] q .|_|z = ( 2 + 2__x - 5__f ) , where we denote , and corresponds to a _ size bias _ ; with the parameter corresponding to a _ flux _ or _ luminosity bias _ , and is the apparent magnitude .moreover , we have and , which implies that the relativistic _ number - count overdensity _ and the relativistic _ magnification overdensity _ are , separately , observable are volume and flux dependent , respectively .( the methods given by e.g.  can be applied to measure . )thus , and are basically the magnified and the unmagnified parts , respectively , of the relativistic ( galaxy ) overdensity .moreover , for a given source , the surface brightness is a constant and is unaffected by cosmic amplification ( or magnification ) of the source .this implies that , hence and we get . however , note that if the survey does not pertain the apparent size of the source , i.e.  , we will have in : and thus , where becomes the change in flux per unit solid - angle change per redshift change .thus , and we get ; when , and , we have and . (note that , given we get , up to first order , where the factor quantifies the cosmic ( de)magnification see . )  thus , we rewrite the relativistic overdensity : [ obzdelta3 ] ^obs_g(,z ) = e__v(z ) ^obs_n(,z ) + e__f(z ) ^obs__m(,z ) , where the number - count overdensity and the magnification overdensity are given in appendix [ sec : dmd ] .note that the ( total ) relativistic overdensity is unique and physically defined : it does not correspond to any of the standard gauge - invariant definitions of overdensity but it is automatically gauge - invariant  .it should be pointed out that previous works  have by implication applied the division given by , i.e. the division of a galaxy sample into separate `` volume - dependent '' and `` flux - dependent '' parts .the only difference is that ( noting difference in notation ) , in the previous works , the two respective parts are first computed _ individually _ and later merely summed directly ; however , doing so is inconsistent .for example , in the work by jeong et al ( see also ) , the relativistic overdensity is computed in parts ; firstly , the pure number - count overdensity is computed ( see ( 40)(55 ) , by ) , accounting for the overdensity solely owing to the change in the observed volume . afterwards , by using that most real cosmic samples also depend on the apparent flux from the sources , the contribution solely owing to the variation in flux is then added ( see ( 60 ) , by ) , given by _g = _ g ( ) + q , i.e. given in the notation of , where we rightly identify and ; thus , ( mis)leading to .( it is worthy of note that , the results by bonvin and durrer  and by challinor and lewis , respectively , are both recovered by jeong et al . )  however , given that the physical number of sources implicitly depends on both the survey volume and the sources apparent flux , then by differential calculus we have the total number differential to be fully decomposed as given by ; thus , leading to and .this way i.e .the approach via we give a consistent procedure for including the _ true _ effect , , of cosmic magnification in the relativistic overdensity .thus , and hence , the entire analysis in sec .[ sec : rdp ] , generalize the relativistic overdensity for a _ generic _ survey , in which both the survey volume and the instrument flux are variable ( within given intervals ) .specifically , in redshift surveys ( see ) , as well as continuum surveys ( see ) , the instrument is fixed at a particular flux or intensity limit and , only above such limit can sources be detected .such surveys are called , `` threshold surveys , '' with the given limit being the instrument s `` threshold . ''essentially , implies that contrary to previous analyses ( see )the volume - dependent and the flux - dependent parts , respectively , of the generic relativistic overdensity do not merely add ; they are scaled down by their respective fractional contribution in the ( total ) redshift - space average distribution density .an ideal , generic survey would involve variable volume and flux , respectively .thus , as previously mentioned , the generated catalogue of the cosmic objects would inherently contain both `` magnified '' ( flux - dependent ) and``unmagnified '' ( volume - dependent ) fractions , respectively ; with the observationally measurable quantity , being the number of objects per unit solid angle per redshift , consisting of all the events together ( without any distinctions between these fractions ) .however , in practice , analyses are often done for specific samples ; usually , for either flux - limited or volume - limited samples .( these sample types are discussed subsequently . ) in order to construct or extract these samples , certain measurement limits are placed on the experiment .generally , in observational cosmology , the principal limiting parameters are usually the apparent flux and the observed ( geometric ) cosmic depth ; these invariably determine the source intrinsic brightness , i.e. luminosity . the flux is prescribed by an apparent magnitude ; the cosmic depth is given in units of distance , and the luminosity is prescribed by an absolute magnitude .basically , once limits , and , are placed on the apparent magnitude and the cosmic depth , respectively , the limit on the absolute magnitude is automatically determined ( via the magnitude - distance relation ) , by [ m_lim ] m_lim = m_lim - 25 - 5_10 ( d_lim ) , where , in general , _ needs not be _ the apparent magnitude of the threshold flux ( it may be any fixed value , depending on interest or the choice of investigation ) ; usually , is the luminosity distance ( measured in ) at a given maximum redshift . note that , in reality , may be affected by the well - known -correction , galactic extinction , and other notable effects especially if is large , i.e. at a high redshift thus , a more general expression for will need to incorporate these effects  .   in flux - limited samples ,the measuring instrument is operated on a fixed flux value serving as the instrument s threshold and only sources with flux strictly above this threshold are extracted or detected . the flux limit is fixed by the apparent magnitude limit , and any source which is intrinsically brighter than the absolute limit , i.e. any , is extracted : [ m_lim ] m = m_lim - 25 - 5_10 ( r ) , where we notice that the absolute magnitude is mainly a function of the ( radial ) distance only ; the various sources are accessed by varying the observed distance . however , the distance is in turn , a function of redshift , so that computing first requires knowing the actual redshift .therefore , the experiment needs to have the sufficient resolution and sensitivity to determine .this way , the individual sources are differentiated , and their actual number is counted .( the instrument picks out the individual sources . )thus the given survey is automatically a `` number - count '' survey . invariably , the survey is also a `` redshift '' survey , with a catalogue consisting of sky positions and luminosities of the sources . for the this type of catalogue , in particular ,usually we have ; with the source sample also termed , a `` magnitude - limited '' sample .note that , by having a flux threshold , it implies that sources at higher redshifts must have higher luminosities in order to be detected by the instrument , i.e. sources at larger must have lower .( otherwise , they lead to a flux which is lower than the given instrument threshold , and are therefore filtered out . ) in fact , once the magnitude limit is applied , the flux becomes irrelevant ; the instrument no longer measures the variation in flux it only registers the number of the individual sources which have flux above the ( instrument ) threshold .thus , the _ change in flux per unit solid - angle change per redshift change , is zero _( i.e. ) ; which leads to and , given , we have . then by , the given distribution is adequately described by ( appropriate for _ pure _ number - count analysis only , see e.g.  ) .moreover , it should be noted that in order to extract or screen for the required sources , the instrument is fixed at the limit ; however , in order to determine the number ( density ) of these sources , the integral in must be evaluated from up to ( noting that a lower apparent magnitude corresponds to a higher flux ) .consequently , the resulting distribution ( density ) depends on the sources distance , and hence , on the sky volume containing , at redshift .given , it implies that intrinsically faint sources will not be detected by the the instrument , unless they are close enough ; whereas , brighter sources are detected even at large distances away .thus the given sample is _ spatially not uniform _  .besides , the sample will inherit all the factors affecting , which are mostly redshift - dependent ( see the discussion below , on ) .an ideal volume - limited survey would be one done by observing a _ fixed patch _ of the sky at a given , over a given period , and the flux and/or size distribution on the given sky patch is mapped .thus , at the given redshift , the _ change in volume per unit change in solid angle per change in redshift , is zero _( i.e. ) ; consequently , and .( such a survey would be appropriate for _ pure _ magnification analysis . )similarly , given , the generated sample is adequately described by .however , volume - limited samples may also be extracted from flux - limited samples .this is usually done by ( pre)specifying a maximum value a fixed redshift the depth of the cosmic volume to be investigated and , similarly , picking out only sources that are intrinsically brighter than ; with luminosities specified by an absolute magnitude : [ d_lim ] m = m - 25 - 5_10 ( d_lim ) , where we note that the absolute magnitude is mainly a function of the apparent magnitude only , and hence , the flux ( density ) . here, the sources are accessed by varying the apparent magnitude , and the sources in the extracted sample will be contained in a fixed volume . obviously , the cosmic depth , which only needs to be less or equal to ( see sec .[ sec : rdp ] ) , is essentially the limiting radius of the sample where the given sources are at particular distances .we notice that , given , in volume - limited samples the flux limit is no longer present and all _ possible _ fluxes are mapped ( within the specified cosmic volume ) .moreover , the extracted sample is _ spatially uniform _, in the sense that the gradient of the number density vanishes , since the number density of the given sample does not vary with distance  .( note that , by , the given source distribution depends only on the flux ; the position or distance of the selected sources is irrelevant . )we investigate the three - dimensional ( 3d ) power spectrum associated with , with the main goal of showing how the parameters and affect the predictions of the large scale structure without using any approximations , e.g. the distant - source approximation ( ) , or the limber s approxiamtion  .we adopt the zero anisotropic stress metric , given by [ metric ] ds^2 = a^2 , i.e. with . on very large ( linear ) scales , the matter and the galaxy rest frames coincide ( given the homogeneity and isotropy of these scales ) , and we can set the matter and the galaxy velocities to be equal : , i.e. galaxies follow trajectories that are parallel to that of the underlying matter ( equivalently , the galaxies are comoving with the underlying matter ) ; hence there is essentially no momentum ( ex)change between the two frames to cause any bias between the velocities .however , on small ( non - linear ) scales , the implicit assumption of the matter - galaxy velocity equality may fail ; implying velocity biasing , which may need to be taken into account .given , the observed 3d galaxy power spectrum is [ p_obs ] p^obs_g(k,,z ) & = & e__v(z)^2 p^obs_n(k,,z ) & & + 2e__v(z ) p^obs_n_m(k,,z ) & & + ^2 p^obs__m(k,,z ) , where we used , with and being the number - count and the magnification power spectra , respectively ( see appendix [ sec:3dps ] ) ; the cross power spectrum between and , is given by [ pnm ] p^obs_n_m = 2q p_m , where the parameters in the square brackets are given in appendices [ sec : dmd ] and [ sec:3dps ] , accordingly ; with being the cross power spectrum ( not shown ) between the matter comoving overdensity and the gravitational potential .  to probe the galaxy power spectrum ,we assume the standard late - time universe : dominated by a cosmological constant and cold dark matter ( cdm ) ; considering mainly the very large ( linear ) scales .we use the present - epoch matter density parameter , and the hubble constant , for all numerical computations .furthermore , we adopt a galaxy bias , which by , the evolution bias .( note that throughout this work , for simplicity , and are assumed to be constants ; however , a proper modelling of these parameters as functions of be crucial for proper observational analysis . )  ) linear power spectra at , with galaxy bias and magnification bias .the number - count power spectrum ( red line ) has much higher power relative to the magnification power spectrum ( blue line)on sub - horizon scales .the _ black lines _ show the full observed power spectrum , for the chosen parameters : , ( solid ) ; , ( dashed ) ; ( dot - dashed ) , and , ( dotted ) .the _ magenta line _ shows the full observed power spectrum with , i.e. which corresponds to previous works in the literature .the black vertical line shows the hubble scale . ]  [ cols="^,^ " , ]   in fig .[ fig:1 ] , we show the plots of the radial linear power spectra ( i.e. with ) given by , and , respectively at the source redshift , with magnification bias .we see that is bounded by on all scales , as prescribed by the parameters : and .( the transverse power spectra , i.e. with , follow the same behaviour except that they are relatively lower in power ; thus we do not show those . ) obviously , we see the effect of and , they regulate the amplitude of on all scales , by dictating the relative contributions by and , respectively .moreover , the plots reveal that mostly dominates over ( which falls sharply towards smaller scales , i.e. increasing ) .we also plot the total power spectrum , with , which corresponds to the analysis by previous works in the literature . on horizon scales, we see that for all values of and obeying .this suggests that the relativistic effects , although already known to be small at low redshifts , may actually be lower than previously predicted for a small magnification bias .  in fig .[ fig:2 ] , we show the plots of the radial power spectra , at ( top panels ) and at ( bottom panels ) ; for magnification biases ( left panels ) and ( right panels ) , respectively .the number - count power spectrum and the magnification power spectrum are shown , accordingly .we also show the total observed power spectrum , and the associated _ standard _ power spectrum ( see appendix [ sec : stdpk])which is the power spectrum of the sum of the comoving galaxy overdensity , the redshift distortion and the net weak lensing magnification . furthermore , the total power spectrum and the standard power spectrum for the case where ( i.e. as in previous works ) , are also shown .we see that , well within sub - horizon scales , for both values of : at both and .in essence , for small , we have for all scales and redshifts .in other words , if and are not taken into account , it could lead to an incorrect prediction of the large - scale clustering of galaxies ; particularly , it could lead to a false boost of the horizon - scale power and hence , an overestimation of relativistic effects in the observed galaxy power spectrum .however , for increasing redshift ( i.e. from top to bottom panels ) , we see a gradual growth in power ( for a fixed ) on horizon scales , indicating large - scale growth in relativistic effects with increasing .moreover , at , starts to dominate over on super - horizon scales .thus the magnification effect increases with increasing redshift . for small , we have the reverse of the scenarios for ; while the behaviours of and remain unchanged throughout ( they remain linear on horizon scales ) .  , and galaxy bias ; with magnification bias ( top panel ) and ( bottom panel ) .line notations are as in fig .[ fig:2].,title="fig : " ] + , and galaxy bias ; with magnification bias ( top panel ) and ( bottom panel ) .line notations are as in fig .[ fig:2].,title="fig : " ]  in fig .[ fig:3 ] we show ( for completeness ) plots of the power spectra for very large .the plots show that , increasing the value of only results in enhancing the large - scale amplitude of the magnification power spectrum , with on hubble scales ; consequently , for any values of and we get ( with the highest horizon - scale value of occurring for ) . by comparing figs .[ fig:2 ] and [ fig:3 ] , it implies that for small magnification bias ( ) the relativistic effects , on super - horizon scales , are lower than previously predicted ( by the analysis of previous works ) .moreover , a value of e.g.  , implies that of the observed sample occur mainly owing to the magnification of the flux ( and/or angular size ) of faint background sources ; with the remaining ( ) occurring solely as a result of the variation , and/or distortion , in the survey volume .basically , of the observed sources have been magnified into the sample , and have a number density which varies only with flux ; the remaining of the sources are intrinsically bright enough to be detected , as the survey volume is varied ( and these sources have a number density which depends only on volume or distance ) .moreover , notice that throughout this work we used , which is reasonable for linear , gaussian perturbations . however , taking non - gaussianity into account may lead to the transformation ( via the relativistic magnification contrast ) , where contains the non - gaussianity ( and scale ) dependence ; thus , changing the shape of with respect to spatial scales .in general , the analysis in this work corresponds to an arbitrary survey .however , in practice , analyses are often done for specific samples : flux - limited , or volume - limited .thus , care must be taken to use the appropriate relativistic overdensity , in the large - scale analysis .a clear , straightforward reanalysis of the ( observed ) relativistic overdensity of cosmological surveys has been given .the calculation was based on the simple fact that the physical number of sources depends mainly on the survey volume and the ( source ) apparent flux ; consequently , giving a consistent approach for incorporating the effect of the cosmic magnification of sources , in the relativistic galaxy overdensity . along withalready known terms , new background parameters have been uncovered , and the possible consequence of these parameters in the observed galaxy power spectrum was demonstrated in the standard , universe .the new parameters separately quantify , ( i ) the fraction of apparently faint background sources , which have been magnified into the observed sample ( by the amplification of their flux and/or angular size ) , and ( ii ) the fraction of the background sources which are apparently bright enough to be observed .these parameters dictate the contributions from the magnified ( flux - dependent ) and the unmagnified ( volume - dependent ) parts , respectively , of the relativistic overdensity ; thereby regulating the amplitude of the observed ( total ) power spectrum .evidently , contrary to previous analyses , the magnified and the unmagnified parts of the relativistic overdensity , in a generic survey , do not merely add ; they are scaled down by and by , respectively .  for small positive magnification bias ,the magnification effect remains mostly sub - dominant at low redshifts ; however , this effect gradually becomes dominant on horizon scales , at high redshifts .moreover , for small positive magnification bias , the relativistic effects on horizon scales may be lower than previously predicted ( in the literature ) in the observed power spectrum . in other words , if these new parameters are not taken into account if they are each set to unity , could lead to a false boost of the horizon - scale power , and hence an overestimation of the relativistic effects , in the observed galaxy power spectrum .nevertheless , the results showed a gradual growth of the relativistic effects on horizon scales , for increasing redshift ( as already known ) ; the magnification effect also becoming prominent at higher .( a negative magnification bias gives mostly the reverse of the scenario for a positive bias . )  in general , for a generic survey , the parameters and can alter the predictions of galaxy clustering on all scales , and consequently , the effect of the relativistic corrections on horizon scales .however , care must be taken to use the relativistic overdensity appropriately , e.g. when considering pure number - count surveys ( , ) or pure magnification surveys ( , ) .the proper underlying conditions and/or regimes of approximation need to be observed .+ * acknowledgements : * thanks to roy maartens and chris clarkson for meaningful discussions , and daniele bertacca for useful comments .this work was carried out with financial support from ( i ) the government of canada s international development research centre ( idrc ) , and within the framework of the aims research for africa project , and ( ii ) the south african square kilometre array project and the south african national research foundation .in this section we give the explicit forms of the perturbation terms in , i.e. the density distortion , the volume distortion and the magnification distortion .( throughout this work , flat space is assumed with most of the calculations being drawn from the rigorous work by , and relevant references therein . )we adopt a general metric , given by [ conf : metric]ds^2 & = & a()^2 -(1 + 2 ) d^2 + 2b_|i d dx^i & & + dx^i dx^j , where is the scale factor , and is the conformal time ; , and we have used [ bkd : metrictens ] |g_00 = -1,|g_i0 = = |g_0j,|g_ij = _ ij , and given the 4-scalars , , and : [ pertmetriccomps ] _ 00 = -2 , _i0 = b_i , _ ij = -2(d_ij - e_ij ) , where and is a traceless transverse tensor , such that it has no contribution to the term , , in the diagonal plane .( note that , , and for a scalar . )the -velocities of a particle moving in , are [ vels ] u^&= & a^-1(1 - , v_|i ) , u_&= & a(-1 - , v_|i+b_|i ) , where is the velocity potential ( a scalar ) .  given , we define the following potentials [ invphi ] & & -h - , + [ invpsi ] & & d + ^2e + h , + [ invvel ] v & & v + e , which are all gauge - invariant , where ; is the comoving hubble parameter , and a prime denotes derivative with respect to conformal time .here we give the expression for the distortion arising in the galaxy number density . in order to compute , which is in redshift space , we need to relate it to the distortion in real space .thus by taking a gauge transformation from real to redshift space : [ delta : g2 ] _g(*n*,z ) = _ g(*n*,z ) - ( * n*,z ) , where the background number density remains the same in both real and redshift spaces , respectively .we have used that , to first order , .we have [ delta : g3 ] |n_g(|z ) = , where we used , and the linear galaxy _ evolution bias _ is given by [ b_e ] b_e .|_|z , with being the background _galaxy number per unit volume .a photon which propagates along the direction from a source will be seen by an observer , located elsewhere in the universe with fundamental -velocity , to be moving with an energy under the direction .the redshift of the propagating photon , in moving from to is given by ( see  for details ) [ zexprssn ] 1+z & = & ( 1+|z ) , where is the background comoving distance at , is a scalar metric potential , and .the parameters , and are given by . a prime ( henceforth ) denotes derivative with respect to conformal time . hence given and ,the density distortion becomes [ delta : g4 ] _g(*n*,z ) & = & _ ( * n*,z ) + 3(+ ) ( * n*,z ) + 3*n*(*n*,z)&+ & 3 ^ 0_|r_sd|r ( + )(*n*,z ) -b_e(|z ) , where is the overdensity in flat slicing ; terms outside integrals are the relative values between and .( note that the term in has been missed by . )thus , we obtain [ flatcurvdelta ] _ = _g + ( 3-b_e)hv -3 , where is the galaxy comoving overdensity . the galaxy and the matter comoving overdensities ,respectively , are related by ( see e.g.  for a derivation)where is the linear _ galaxy bias _ .the ( gaussian ) galaxy bias relates to the evolution bias , by [ b_g ] b = -h ( 3 -b_e ) = 1 - b_e , where the second equality follows from the standard continuity equation .the distortion that arises in the observed volume of galaxy redshift surveys , i.e. the volume density contrast , is given by ( see  for details ) [ obsvolpert ] __ v & = & - 4*n * + ^|r_s_0d|r ( |r - |r__s ) ^2_(+ ) & & + 3^|r_s_0 d|r ( + ) + ^|r_s_0 d|r ( + ) & & + ( + ) & & + - 2(+ ) , where the squared gradient ( with ) is the laplacian on the surface transverse to the line of sight , and is an affine parameter along the photon trajectory : ( i.e.  being a scalar ) . by using that galaxies ( hence , pressureless matter )move along geodesics , we have the euler equation for the line - of - sight velocity as [ euler ] * n* + h * n * - _ r = 0 .thus given , , and , the ( observed ) number - count overdensity is [ eq : delta_n ] ^obs_n(*n*,z ) & = & _ g(*n*,z ) - _ r v_(*n*,z)&+ & ^|r_s_0d|r ( |r - |r__s ) ^2_(+ ) ( * n*,z)&+ & ( 3 - b_e)hv(*n*,z ) + (*n*,z)&- & ( b_e -1 - - ) ( * n*,z ) & - & 2(*n*,z ) + ^|r_s_0d|r ( + ) ( * n*,z ) & + & ( b_e - - ) , where , and are functions of ; with being the line - of - sight component of the velocity , as seen by .the first two lines give the `` standard '' term of the relativistic number - count overdensity : consisting of the comoving galaxy overdensity , the ( kaiser ) redshift distortion term and the weak ( gravitational ) lensing term , respectively .the standard term constitutes the most dominant contributions ( especially at low ) .the rest of the lines together constitute the `` relativistic corrections '' ; with the third , fourth and fifth lines giving the ( peculiar velocity and gravitational ) potential effect and the time delay ( integral term ) in the photon signals . in the last two lines , the first and the second terms in the square brackets correspond to the doppler and the integrated sachs - wolfe ( isw ) effects , respectively .similarly , a lengthy but straightforward calculation ( see  for details ) yields the magnification distortion ; hence the ( observed ) relativistic magnification overdensity becomes [ magdens ] ^obs__m(,z ) & = & q(|z)^|r__s _ 0 d|r(|r__s - |r ) ^2_(+ ) ( , z ) & + & 2q(|z)(,z ) - ^|r__s_0 d|r ( + ) & + & 2q(|z )( 1 - ) , where the first line gives the weak lensing term , and the remaining lines give the relativistic effects .thus , by combining and we get the explicit form of the ( observed ) relativistic galaxy overdensity taking note of .  in general , note that the overdensity solely measures the `` magnification of sources''owing mainly to the magnification of the sources fluxes by : weak lensing , doppler effect , isw effect , time delay and ( gravitational ) potential difference ( see  for concise statements on these effects , pertaining the magnification of sources ) .similarly , the redshift distortions , weak lensing , doppler effect , isw effect , time delay and potential difference effects also combine to induce distortions in the survey volume ( see ) which then increases ( or decreases ) the number of counted sources resulting in the `` ( de)amplification of the number - count overdensity '' .moreover , notice that there are still some residual lensing / magnification effects left in ; however , these effects are mainly owing to the inherent volume distortion , and not owing to dependence on flux / luminosity . +we compute the various 3d ( linear ) power spectra of the observed overdensities in and in , repectively ( in fourier space ) .   consider the galaxy relativistic number - count overdensity , then we can transform it in terms of the matter comoving overdensity in fourier space by [ dn_1 ] ^obs_n = _ m , + where is a parameter which measures the ( kaiser ) redshift distortions ( in the concordance model , becomes the standard matter _linear growth rate_as shown by  ) ; , with and being the wave vector and the wavenumber , respectively ; is a dimensionless parameter .thus given , [ pn_1 ] & = & ( b + f^2)^2 + 2(b + f^2 ) & & + ^2 + ^2 , where is the dm linear power spectrum ( see  ) , and  [ dn_2 ] a & & ( 3-b_e)f - x^2 , + [ dn_3 ] b & & ^|r_s_0d + ( b_e - - ) ^|r_s_0d , + [ dn_4 ] c & & -(b_e - - ) f + ^|r_s_0d ( -|r__s ) k ,  where the variable is the invariant gravitational potential evaluated at a background comoving distance an arbitrary position between and .here we consider the relativistic magnification overdensity of magnified galaxies .we transform the magnification overdensity in terms of the gauge - invariant gravitational potential : [ dm_1 ] ^obs__m = 2q , and consequently , we obtain the associated magnification linear power spectrum given by [ pm_1 ] p^obs__m = 4q^2 ( d^2 + ^2 e^2 ) p _ , with being the linear power spectrum of the gravitational potential , and [ dm_2 ] d & & 2 - - 2(1 - ) ^|r_s_0d & & + ^|r_s_0d , + [ dm_3 ] e & & x(1 - ) - ^|r_s_0d ( -|r__s ) k , where we note that in , similarly in , the main contribution of the lensing source - term in fourier space only adds to the transverse power spectrum ( with ) , since the radial ( line - of - sight ) power spectrum ( with ) has this term to be zero .moreover , for the transverse power spectrum : the kaiser redshift distortion term together with all the complex terms i.e . the parameters , and eliminated .given the parameters and , here we give the `` standard '' overdensity which is made up by the comoving ( galaxy ) overdensity , the redshift distortion and the total lensing terms by ( in real space ) [ delta_std1 ] ^std_g & & ( 1 - q_eff ) ^|r_s_0d ( - |r__s ) ^2 _ ( + ) & & + e__v ( _ g + _ r v _ ) , where used , and ; defines an `` effective '' magnification bias . in fourier space , [ delta_std2 ] ^std_g =_ m , where and are as given in ; given the metric , & & - ^|r_s_0d ( 1-^2 ) ( - |r__s ) k^2 , + r & & ^|r_s_0d ( - |r__s ) k .+ consequently , we have [ pk_std ] & = & e^2__v ( b + f^2)^2 + 2e__v ( 1 - q_eff ) ( b + f^2 ) & & + ( 1 - q_eff)^2 + ( 1 - q_eff)^2 ^2 , + which is the associated standard power spectrum .

Abstract: the main goal of galaxy surveys is to map the distribution of the galaxies , for the purpose of understanding the properties of this distribution and its implications for the content and evolution of the universe . however , in order to realise the potential of these surveys , we need to ensure that we are using a correct analysis : a relativistic analysis , which has been extensively studied recently . in this work , the known relativistic overdensity of galaxy surveys is re - examined . subtle , but crucial parameters which appear to have been missed by previous works are uncovered . the possible implication of these parameters on the observed galaxy power spectrum is demonstrated , in the standard concordance model . the results show that these parameters can alter the predictions of galaxy clustering on all scales ; thus , the effects of relativistic corrections ( on hubble scales ) .


morphological patterning , such as animal coat markings , may be caused by a chemical field  .the turing model now has strong experimental support ( * ? ? ?* for a review ) ,  .this mechanism has some limitations , and so we continue to investigate variations of the model  .we consider the standard reaction - diffusion model with the addition of convection  .the more general form is a system with advection , as in  .our motivation is theoretical , to see the effect of convection on the robustness of the pattern formation .we consider that a weak amount of convection , or a similar effect , may be present _ in vivo_. this may soften the standard requirements for the formation of pattern , increasing the applicability of the model .  there may be a direct biological application , such as the formation of the vertebral precursors ( somitogenesis ) .the organism growth could produce such a convective effect  .however , convection - induced patterning is not generally supposed for somitogenesis  .our system does not fit the standard turing analysis since our reaction functions are already unstable .however , we have shown previously that an unstable function may produce a turing pattern  .furthermore , we have shown that , in the presence of convection , we can see similar behaviour irrespective of the stability of the fixed point  .experimentally , convection is introduced from the boundaries  .the effect of boundary conditions is also likely to be relevant _ in vivo_. the boundary can have a significant effect on the pattern formed  .however , we follow the suggestion of cross and hohenberg to examine the system in a boundary - free environment first  .  in numerical simulations we makeform an initial point disturbance .pattern is formed , spreading outwards from its initiation point .the disturbance is oscillatory and complex .our focus here is the way in which the pattern propagates .the speed of propagation has been given for a two - species system with no convection  and one with equal convection  .a related theoretical study considers the relationship between the onset of the instability and the longer - term behaviour of the non - equilibrium state  . herethe simplest reaction - diffusion system with oscillatory kinetics was considered , and complex behaviours were found .limit cycles are inherent in these oscillatory models , since an unstable spiral at the steady state must be bounded  . in this paper , we seek to exploit the limit cycle to aid our analysis . to this end, we select the lambda - omega reaction functions .these are chosen to produce the simplest limit cycle possible : the unit circle .the reaction functions chosen in a full model correspond to actual reaction mechanisms .examples of these include the belousov - zhabotinsky reaction , and brusselator - type systems such as the chlorine dioxide - iodine - malonic acid reaction .the schnakenberg model can be considered an intermediate step towards these functions . for an initial investigation ,we have chosen simple , related functions .the intention is to extend the analysis to more realistic models .secondarily , the results in this simple case may form a useful basis for comparison to more complex systems .we transform the problem so that the oscillations are removed , or at least reduced .this brings the onset of the pattern into much sharper relief .the two aspects the pattern , and the initiation of the pattern are clearly distinguished .thus this onset of pattern is effectively a phase transition .the transformation is to convert the dependent variables into polar form .we then consider that the formation of the limit cycle approximates a travelling wave , and so we employ a fisher solution to the problem  .the relative sizes of the parameters affects our analysis : we consider various situations .we carry out numerical experiments to discover which of these estimates are valid .we suppose that some form of chemical mechanism is the underlying basis for biological pattern formation . if the chemicals are well - mixed , then the law of mass action valid , and an ode is an appropriate model :     we consider reaction functions chosen for their simplicity in the form of the limit cycle produced: this is a simple form of the - class of functions . herethe steady state is the origin and the limit cycle is the unit circle .there is no parameter to determine the stability : the steady state is unstable and the limit cycle stable .this behaviour is clear in the phase plane ( see figure [ fig : ode ] ) .the circled points on the diagram show the start of the phase plane trajectories .the phase curves spiral out from the steady state to join the limit cycle .the trajectories starting far away from the steady state spiral into the limit cycle .these functions are chemically unrealistic , as they stand .however , we can relate these functions to the schnakenberg reactions , chosen to be the simplest chemical form which can produce a limit cycle .this mechanism was proposed theoretically , but has been used as a model for actual reaction mechanisms  .the key reaction there is the cubic autocatalytic one , . using the law of mass action , this produces the term .the reverse step gives .these types of terms form the core of the - functions , the cubic terms .we consider a new coordinate system for the dependent variables as a polar form as follows: by differentiating these identities , we transform the first order ode  into: the equation clearly decouples and is resolvable .the remaining single - species ode in has steady states at .the negative state is unrealistic because must be positive , zero is unstable and one is stable .the point is our limit cycle : the unit circle .furthermore , , time and so progression around the limit cycle is constant .this type of analysis is often used in similar cases , such as the analysis of travelling wave trains  .biological pattern formation is by definition spatially differentiated .there are many models for this .one is that chemicals diffuse within an organism , then cells respond differently dependent on the concentration of one of these chemicals . if the concentration of the chemicals has formed a pattern , then this is reflected by the cells .  for an initial analysis , we choose the simplest case : one spatial dimension and some diffusion . by diffusionwe refer to the averaged gross effect of random motion of the chemicals ; with passive movement this is equivalent to brownian motion . in the case of general reaction functions , this system is the one proposed by turing  .we add a convective term to the system .this can be considered a disturbance to the system , as an investigation of stability . for a small amount of convection, this could correspond to axial growth  .however , for convection to be significant in axial growth we would require extremely slow diffusion of the chemicals to fit the model .the general system is then as follows : we consider different convection on each species ( ) .this effect could be created in a chemical flow reactor , with one of the reactants held fixed in a packed bed  . for biological applications ,one of the chemicals may be held within a cell , while the other flows freely .if both chemicals flow freely , it is possible that their movement may be hampered by obstacles such as the extra - cellular matrix . in this case , a larger molecule may be affected more strongly than a smaller one , and different convection speeds may result for each chemical . this concept parallels that of different diffusion rates , as in the turing model .we consider that .if this is not the case , we make the transformation , giving a positive value for with no change to the form of .we can remove one of the convective terms by a simple change of coordinates : with and as in  and , and positive constants . from the original system  we have .if turns out to be negative , we can reverse the sign as we did for and .equivalently , we choose the coordinate change previously , and set . in the case where the convection is the same on both species ( ) , then both convection terms are removed ( ) .this recovers the basic turing model .we have reaction functions that have limit - cycle behaviour and convection , which is known to drive instability .the appearance of pattern is then to be expected , although the form might be more difficult to predict .satnoianu , merkin and scott  studied a similar system to  previously , with schnakenberg reaction functions in place of the - ones .they found that periodic behaviour is emergent in the system over a broad parameter range .   with .the reactants are initially at steady state : , with a small disturbance at .the boundaries are held at zero derivative : , .,width=403 ]  in the numerical experiment , we start at the steady state , except for a small disturbance at .we try to simulate a boundless environment to this end we find zero derivative boundary conditions the most effective .the initial disturbance propagates and becomes pronounced , forming a regular pattern with aligned oscillations .the propagation is linear , forming a v - shape .the convective effect is to skew the pattern to the right ( figure [ fig : pattern ] ) . given the parameters in the figure ( ) , the direction of the internal oscillations is distinct from the angle of propagation of the pattern .the emergence of this pattern is the main focus of our study .before we start any specific analysis , we rescale the spatial variable to remove one of the parameters .we choose .this yields the system : where .following suite , we expect , but we go one step further in defining .this produces a symmetric first spatial derivative on both equations , with the difference between the two diffusion rates .we wish to again convert into the polar form . during the conversion, we see that there is grouping of the terms parameterised by and : + r(1-r^2 ) \ , \nonumber\\   \theta_t & = & \theta_{yy } + 2\frac{r_y}{r}\theta_y  + \frac{u}{r^2}[\bar{\varepsilon}v_{yy } - \bar{\gamma}v_y ] + 1 \ .\end{aligned}\ ] ] we complete the transformation :   transitions sharply from to .this is a travelling wave , propagating outwards .there are minor ripples in the established solution : the limit cycle settled on by the pde is not the unit circle , but is close . the oscillation is at an angle to the travelling wave front .there is a difference in behaviour between the left and right sides of the pattern : the angle of alignment and the frequency of the pattern is different on either side . there is also a clear centre to the propagation , roughly at .this is a numerical solution of  using nag d03pcf , plotting species with , .the reactants are initially at steady state : , with a small disturbance at .the boundaries are held at zero derivative : , .,width=403 ]  if we translate our numerical results into this polar form , the onset of the pattern becomes very clearly demarked : the new coordinate transitions sharply from to .the primary behaviour is that of a travelling wave , propagating outwards .the internal behaviour of the pattern , the steady oscillation , is reduced to a secondary effect .there are minor ripples in the established solution : the limit cycle settled on by the pde is not the unit circle , but is close .there is a difference in behaviour between the left and right sides of the pattern : the angle of alignment and the frequency of the pattern is different on either side ( figure [ fig : polar ] ) .next we examine this travelling wave analytically .we consider the very simplest situation : equal diffusion on both chemical species ( ) combined with no convection of ( =0 ) .we make a simplifying assumption for an initial analysis : , .the problem reduces to then decouples , as for the ode .the solution for is , to within an arbitrary constant .the remaining equation in is of the form similar to the fisher equation and should therefore yield a propagating wave solution .this analysis is covered in greater detail in  .we look for a solution of the form which gives we carry out a phase plane analysis in the phase plane .we find that gives a stable point at , which suggests the solution we are looking for .the point , which corresponds to our limit cycle , is always a saddle point .we are looking for a phase plane trajectory that leaves this saddle point and goes to the zero steady state : this will be our travelling wave solution .  small wave speed , , gives a spiral in the phase plane , which would imply at some point on the trajectory . from the definition of know this to be impossible and thus this is unrealistic for a travelling wave solution . for have a node and the trajectory discussed above , leaving from the saddle point will head directly to the node , remaining in the fourth quadrant of the phase plane and therefore realistic .this trajectory equates to the travelling wave that we are looking for .we expect that the least speed wave will be achieved so we predict a wave of speed . converting this back into our unscaled system, we have to the right .there is also a solution to the left : . in the original system, we have .we now apply the method for the full system . herewe may have different diffusion on the two species ( ) or some convection ( ) .we again make the assumption , .we introduce the travelling wave coordinate and look for a solution .we also linearise the system , dropping the term : using the same approach as before , we find . converting this back to the unscaled coordinate system we have .we see that and remain linked by the term .these parameters occur on the differential equation , and .  in the original system we have .again , the parameters for each species and are linked by the functions and respectively .  this general result for the wavespeedsgives a dependence on the angle variable .we consider some general numerical experiments to assess the behaviour of the system .what we find is two distinct types of behaviour .we refine our wavespeed estimates in light of these particular cases .   for parameters close to those of the simple system , namely small convection ( ) and near - equal diffusion ( ) , we expect the behaviour to remain similar to that of the simple system .previously we have seen ode - like and turing - type behaviour in a convection - free system  . in both cases , we see that the internal angle of the pattern ( the group direction ) is not in line with the wavefront . as such , the oscillation is perceptible at the wavefront .this suggests that an average value approximation of may be an appropriate estimate .in this case we approximate with its average value : .this yields . in the unscaled coordinate system we have .if we consider the limit as the amount of diffusion tends to zero , we see that the centreline of the propagation is at .in this situation , the convection does not increase the separation of the left and right wavefronts ; the spread of the propagation is solely due to diffusion . in the original systemthe incident fronts are at .  for dimensional reasons ,it is appropriate to write .then the diffusive contribution to the wavespeed becomes , which is reminiscent of pythagorean form .more simply , the term is additive and the wavespeed is proportional to the diffusion of each species .as the strength of convection increases , we see an alignment of the incident wave with the internal group direction .this means that the value of is approximately constant at the onset of the wave .we investigate the effect of this on the wavespeed estimate .we consider the range of values possible for the right wavespeed with a fixed value for .since has a range of  ] .  on the left we have a similar result : $ ]is the range once .we note that for , and for , .we propose that the maximum spread of the instability is achieved .that is , the wavespeeds are maximised within their given possible ranges . for this , the left wavespeed must reach its lower limit , and the right its highest .this gives and .then the onset angle to the left is expected to be approximately , and to the right of the right wave the angle should be roughly .  .the oscillations are aligned with the travelling waves . to the left of the left travelling wave ( the light , solid line ) ,the angle is roughly , and to the right of the right wave ( the dark , dashed line ) the angle is approximately .there is a smooth transition between the left and the right of the pattern .the oscillations at the centre line ( ; medium , dot - dashed line ) are roughly perpendicular to the centre line itself .this is a numerical solution of  using nag d03pcf , plotting species with , .the reactants are initially at steady state : , with a small disturbance at .the boundaries are held at zero derivative : , .,width=403 ]  we see this is the case in figure [ fig : polar_angle ] . there is a smooth transition between the left and the right of the pattern .the oscillations at the centre line ( approximately ) are roughly perpendicular to the centre line itself .  in the unscaled coordinate system , for strong convection, we have and . in the original system , this is and for . in the other case , there is a reversal and and .this analytical prediction was made from observing that the internal waves become aligned with the onset of the instability .however , when we make the assumption that the spread of the propagation is maximised , it follows that the onset angle become constant .furthermore , the suggestion of as a transition value between behaviours occurs naturally from the algebra .having carried out an analysis together with some initial numerical investigation , we now conduct a more comprehensive experiment .we vary our parameters over a wide range and measure the incident wavespeeds in each case .the results confirm our analyses , as we can see in figure [ fig : ef05fig5 ] .  for the left wavespeed the difference between the low and high parameter estimateis pronounced , so the transitional behaviour is noticable .there is less difference between the estimates on the right , and the transition is close to the intersection of the estimates .this seems quite remarkable , that the behavioural transition occurs where the estimates intersect . as a result , there is no discernable area of transitional behaviour .  in both casesthe low parameter behaviour endures longer for high , compared to low .this suggests that the transition , caused primarily by the convection , , is held in check more strongly by the diffusion on , where the convection is applied .there is a concept of stability in systems such as these , which considers whether the system at a particular spatial point , once destabilised , will return to the steady state .if so , the system is classified as convectively unstable .if the chemical species do not return to stasis then the system is deemed absolutely unstable .  in line with our analysis, we define absolute instability as , with equivalence giving convective instability . combining this with our results, we see that for small parameters we require for convective instability in the unscaled system .this inequality is unlikely to be satisfied whilst keeping within the small parameter regime .however , if we consider the original system , the condition is , which is quite possible to satisfy while still remaining in the ` small parameter ' regime ) .for example , gives , and then satisfies the condition .  for large parameters we always have absolute instability in the unscaled system .referring to the observed data ( figure [ fig : ef05fig5 ] ) , we see the left and right wavespeeds always have different signs .thus , we can classify this system as only exhibiting absolutely unstable behaviour .  however , if we consider the original system  we find that or will give convective instability rather than absolute instability .we see that , in the original system , we can always shift the stability from absolute to convective with sufficient convection applied to the system . in the flow coordinate system , the stability is essentially absolute .however , when we convert to this system we make an arbitrary choice to remove the parameter p rather than , for example , q. then the distinction between absolute and convective instability becomes similarly arbitrary .at the start of our analysis , in section [ sec : pattern ] , we chose a new coordinate system for the purpose of eliminating one of the convective terms .this reduction in the number of parameters proved useful in the ensuing work .however , the choice of coordinates , as we observed previously , was arbitrary .now we consider an alternative coordinate system , one centred on the convection . fromwe choose .this gives the system as : where .we can recalculate the apparent wavespeeds from this perspective .for our ` small parameter ' regime we have . herewe see that we have aligned our system centrally with the flow .  in our large parameter case , we first consider .this ensures .now the left wavespeed is , and the right wavespeed is .  in the other case , we define : now the wavespeeds are , and the right wavespeed is .  consider the right wavespeed . in each caseit is a function of only one of the diffusion parameters .we examine the equation for the relevant species . in both cases ,we find a negative convection term for that species .this suggests that negative convection drives the onset of instability .  for the left wavespeedthe connection is with the positive convection .however , in this direction is decreasing .from this perspective , the signs of the convective terms change . in this sensethen , the connection is the same .if we understand that the negative convection drives the instability wavefront , then we can consider only the associated species .for example , looking for the right wavespeed in , we see that is the relevant species .we make the assumption that the other species is much smaller than , and so disregard it. then we can carry out a travelling wave analysis for this single species .the result is exactly the same as we found in our polar - form analysis , with a much simpler approach .we can confirm this behaviour by observing that the initiation angles and correspond to and only , respectively .we can go one step further in unravelling the puzzle .if we simply separate the species perhaps we could call this a super - linearisation we can produce two wavespeeds , one from each species .in our example , for the right wavespeed we have the possibility of from the equation and from .then we apply the ` maximisation ' principle suggested earlier . for large , we can be sure that the wavespeed given from the equation will maximise the propagation .thus we can find the wavespeeds by applying two simple concepts to the problem .the behaviour for small convection relies on some more complex way of combining the two species .the wavespeed in this case is dependent on the strength of diffusion of both the species .therefore the polar - form approach given previously retains some merit .the addition of convection to the reaction - diffusion model has produced many patterns  .advection , being the broader definition , has potential to produce even more .aspects of the pattern have been observed , such as the different behaviour to the right and the left of the free - forming pattern  .previously analyses have considered a variety of models , most restricted in different ways . no movement for one species  , equal diffusion and equal convection  and fixed diffusion - convection ratio have all been considered . here, we have considered the most general diffusion and convection possible , as in more recent work  .much previous work was devoted to the discussion of ` flow and diffusion distributed structures '  .this is spatial pattern with no variation in time .our reduction of the system to two parameters allows for a clear and concise analysis of the system .however , after our change to convective coordinates , the meaning of ` time - constant ' is lost .in fact , in our reduced system , a time - constant solution is only possible for the trivial case , .the standard analysis to date has been a linear existence analysis for pattern , involving dispersion relations .this follows the turing derivation , but is difficult for this more complex model . in our analysis we have employed a different approach .we have not directly considered the internal behaviour of the pattern , but instead looked for approximate descriptions of the propagation speed of the pattern .this information has already been found for other versions of this system ( less general diffusion or convection , more complex reactions )  .  to understand the understand a systemwell , it is best to isolate influences  .this is why we consider the pattern forming away from the spatial boundaries . in this instancewe also have symmetrical , parameterless reactions , giving the most elementary form possible .previous work has shown boundary effects mixed in with other behaviour  , and that the boundary can control the existence of pattern  .our analysis is the natural one given the system and its behaviour .we have combined two simple techniques to reduce a complex problem to a more manageable one .we consider only the ( apparently ) linear aspect .our analysis predicted the speed of the incident travelling waves over the full parameter range . in combination with the numerical investigation ,we demonstrate two different behaviours of the system .there is a distinct transition between them , with the transition region indicated by the analysis .our work also confirms the validity of the fisher - type analysis in this two - species system .furthermore , we give some insight to the different behaviour to right and left of the pattern .  having successfully analysed this simple - system we must attempt a similar analysis on a more complex system .the next step is to consider a more realistic set of reaction functions , such as the schnakenberg ones . in this case , we already know that first - order effects will come into play  .this should lead to a clearer understanding of the turing phenomenon .we would like to acknowledge support from nih grant number r01gm076692 .any opinions , findings , conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the nih or the united states government .the authors are grateful to two anonymous reviewers for their helpful comments .

Abstract: we consider a diffusion model with limit cycle reaction functions . in an unbounded domain , diffusion spreads pattern outwards from the source . convection adds instability to the reaction - diffusion system . the result of this instability is a readiness to create pattern . we choose the lambda - omega reaction functions for their simple limit cycle . we carry out a transformation of the dependent variables into polar form . from this we consider the initiation of pattern to approximate a travelling wave . we carry out numerical experiments to test our analysis . these confirm the premise of the analysis , that the initiation can be modelled by a travelling wave . furthermore , the analysis produces a good estimate of the numerical results . most significantly , we confirm that the pattern consists of two different types .


our setting is the _ stochastic block model _  a random graph model in which a set of vertices is randomly partitioned into _ blocks _ and then , conditioned on the partition , existence of edges between all pairs of vertices are independent bernoulli trials with parameters determined by the block membership of the pair .( the model details are specified in section [ c ] . )the realized partition of the vertices is not observed , nor are the bernoulli trial parameters known .however , the realized vertex adjacencies ( edges ) are observed , and the main inferential task is to estimate the partition of the vertices , using the realized adjacencies as a guide .such an estimate will be called consistent if and when , in considering a sequence of realizations for with common model parameters , it happens almost surely that the fraction of misassigned vertices converges to zero as .rohe et al . proved the consistency of a block estimator that is based on spectral partitioning applied to the normalized laplacian , and sussman et al . extended this to prove the consistency of a block estimator that is based on spectral partitioning applied to the adjacency matrix .importantly , both of these procedures assume that and the rank of are known ( where ^{k \times k} ] satisfies , and the _ communication probability matrix _^{k \times k} ] which are all parameters of the model , and there are corresponding random adjacency matrices such that for each _ modality _ and for each pair of vertices that are distinct , is with probability independently of the other pairs of vertices but possibly with dependence across the modalities .as above , for model identifiability purposes we assume that , for each distinct , there exists an such that or .also , it is assumed that we know positive integers which are upper bounds on respectively .we will also describe next in section [ b ] how the adjacency - spectral partitioning procedure of that section can be modified for this more complicated scenario so that theorem [ a ] will still hold for it .the adjacency - spectral partitioning procedure that we work with is given as follows :  first , take the realized adjacency matrix , and compute a singular value decomposition ( \sigma \oplus \sigma_r ) [ v|v_r]^t ] and  ] into at most clusters using the minimum least squares criterion , as follows : if it is known that the rows of are pairwise not equal , then compute which minimizes over all matrices such that there are at most distinct - valued rows in  , otherwise , if it is known that the columns of are pairwise not equal , then compute which minimizes over all matrices such that there are at most distinct - valued rows in , otherwise compute which minimizes \|_f ] for , , , and such that  ] are each real - orthogonal matrices and is a diagonal matrix with its diagonals non - increasingly ordered , then define and and then , according as the rows of all are known to be distinct - valued , the columns of are known to be distinct - valued , or neither , compute which minimizes \|_f ] or \|_f ] , and if then )^2\geq c \right ] \leq 2e^{-\frac{2c}{m}} ] , the nonnegative matrix has all of its entries bounded by , thus all of its row sums bounded by , and thus its spectral radius is bounded by , ie we have as desired .+ next , for all , let random variable denote the number of vertices in block .the nonzero eigenvalues of are the same as the nonzero eigenvalues of . by the definition of and the law of large numbers , almost always for each , thus we express as the sum of two positive semidefinite matrices and obtain that the minimum eigenvalue of is at least .similarly the minimum eigenvalue of is at least .the minimum eigenvalue of a product of positive semidefinite matrices is at least the product of their minimum eigenvalues , thus the minimum eigenvalue of ( which is equal to ) is at least , as desired .  it almost always holds that , it always holds that , and it almost always holds that .[ cc ]  * proof : * by lemma [ aa ] and weyl s lemma ( e.g. , see ) , we obtain that for all it almost always holds that .for all , the singular value of is zero , thus almost always .lemma [ bb ] can in fact be strengthened to show that there is an such that almost always , hence , thus we have almost always that , as desired .showing that is done the same way that was shown in lemma [ bb ] . + it is worth noting that a consequence of corollary [ cc ] is that , for any chosen real number such that , the random variable which counts the number of which are greater than is a consistent estimator for ( is almost always equal to ) .our goal in this article is to show a robustness result , that overestimating " with in the adjacency - spectral partitioning procedure does not ruin the consistency of the procedure .+ recall from section [ b ] the singular value decomposition ( \sigma \oplus \sigma_r ) [ v|v_r]^t ] ,  ] and  ] is an orthogonal matrix .in particular , note that the columns of together with the columns of form an orthonormal basis for the eigenspace associated with eigenvalue in the matrix .  there almost always exists a real orthogonal matrix such that \ q -  \ [ u_c | u_r ] \\|_f \leq \frac{\sqrt{6}}{\alpha^2 \gamma^2 } \cdot \sqrt { \frac{\log n } { n}} ] .then \ - \ [ x_c | x_r ] \ \|_f \leq \frac{3^{1/8}6^{1/2}}{\alpha^2 \gamma^2 } \cdot n^{-1/8 } \log^{5/8}n ] with and then using corollary [ cc ] and the definitions of and . always it holds that .[ hh ]  * proof : * it is clear ( with the matrix from corollary [ gg ] ) that \q ] has distinct - valued rows , and the triangle inequality , we have that   -x  \|_f + \| \tilde{{{\mathcal x}}}_c \|_f.\end{aligned}\ ] ] then , by two uses of the triangle inequality and then equation ( [ aaa ] ) , we have \|_f & \leq & \| { \mathcal c } - [ \tilde{{{\mathcal x}}}_\ell | \tilde{{{\mathcal x}}}_c ] \|_f + \| \tilde{{{\mathcal x}}}_c \|_f \\ & \leq &  \| { \mathcal c}-x \|_f + \| x - [ \tilde{{{\mathcal x}}}_\ell | \tilde{{{\mathcal x}}}_c ] \|_f + \| \tilde{{{\mathcal x}}}_c \|_f \\ & \leq & 2 \cdot  \|  [ \tilde{{{\mathcal x}}}_\ell | \tilde{{{\mathcal x}}}_c ]  -x  \|_f + 2 \cdot \| \tilde{{{\mathcal x}}}_c \|_f\end{aligned}\ ] ] which , by corollary [ ee ] , corollary [ gg ] , and note [ hh ] , is almost always bounded by ^{1/2 } + 2 r^{1/2 }  \ 3^{1/8 } n^{3/8}\log^{1/8}n,\end{aligned}\ ] ] which is almost always bounded by for any fixed .thus line ( [ bbb ] ) is shown .now , it easily follows from line ( [ bbb ] ) that \nonumber  \\ \mbox { { \it with euclidean norm at least } } \frac{\alpha \beta \gamma}{3 } \mbox { { \it is almost always less than } } n^\epsilon ;  \label{ccc}\end{aligned}\ ] ] indeed , if this was not true , then \|_f \geq \sqrt{n^\epsilon \left ( \frac{\alpha \beta \gamma}{3 } \right ) ^2} ] ; by lemma [ ff ] , these balls are almost always disjoint .the number of vertices which the block membership function assigns to each block is almost always at least , thus ( by line ( [ ccc ] ) and the pigeonhole principle ) almost always each ball contains exactly one of the distinct - valued rows of . and, for any fixed , the number of misassignments from is thus almost always less than .theorem [ a ] is now proven in the scenario where there is a single modality of communication and it is known that the rows of are pairwise nonequal .  in the general case where there are multiple modalities of communication and/or the rows of not known to be pairwise nonequal , then the above proof holds mutatis mutandis ( affecting relevant bounds by at most a constant factor ) ; in place of use or  ] or  ] and in place of  ] or  ] , or  ] , as appropriate , and similar kinds of adjustments .in this section we provide a consistent estimator for the number of blocks , if indeed is not known .( the only assumption used is our basic underlying assumption that an upper bound is known for . )  to simplify the notation , in this section we assume that there is only one modality of communication and we also assume that it is known that the rows of are distinct - valued .these simplifying assumptions do not affect the results we obtain , and the analysis can be easily generalized to the general case in the same manner as was done at the end of section [ f ] .  in the adjacency - spectral partitioning procedure from section [ b ] , recall that one of the steps was to compute which minimized over all matrices such that there are at most distinct - valued rows in .then the block assignment function was defined as partitioning the vertices into blocks according to equal - valued corresponding rows in  .let us now generalize the procedure of section [ b ] .suppose that , for any fixed positive integer , we instead compute which minimizes over all matrices such that there are at most distinct - valued rows in .then the block assignment function is defined as partitioning the vertices into parts ( some possibly empty ) according to equal - valued corresponding rows in .we shall call this adjusted procedure the adjacency - spectral partitioning procedure from section [ b ] with parts . "[ fff ] let real number such that be chosen and fixed .for the adjacency - spectral procedure from section [ b ] with parts , if then almost always , and if then almost always .* proof : * using equation ( [ aaa ] ) , corollary [ ee ] , corollary [ gg ] , and note [ hh ] in the manner used to prove line ( [ bbb ] ) , we obtain that almost always -x\|_f \leq n^\xi ] . by lemma [ ff ] ,these balls are almost always disjoint and , in fact , their centers are almost always at least distance one from the other . by the pigeonhole principle, there is at least one ball that contains none of the distinct - valued rows of .together with the fact that each block almost always has more than vertices , we obtain almost always that \|_f \geq \sqrt{\alpha n \left ( \frac{\alpha \beta \gamma}{3 } \right ) ^2 } ] . + let real number such that be chosen and fixed .define the random variable to be the least positive integer such that for the adjacency - spectral procedure from section [ b ] with parts it happens that . by theorem [ fff ], we have the following consistency result for .  [ ggg ] almost always .the adjacency - spectral partitioning procedure of section [ b ] assumed that an integer was known such that , but it also assumed that the number of blocks was known .we next extend the adjacency - spectral partitioning procedure of section [ b ] ( we call it the extended adjacency - spectral partitioning procedure " ) so that it only has the assumption that an integer is known such that , and it is not assumed that is known .the procedure is as follows :  let real number such that be chosen and fixed .successively for , do the spectral partitioning procedure of section [ b ] with parts until it happens that , then return the from the last successive iteration ( i.e. the iteration where ) .  [ iii ] with the extended adjacency - spectral partitioning procedure , for any fixed , the number of misassignments is almost always less than .* proof : * indeed , almost always the last value of ( which is ) is equal to by theorem [ ggg ] , and then almost always the number of misassignments is less than by theorem [ a ] .in section [ e ] we provided the consistent estimator for the number of blocks .it was based on theorem [ fff ] , which contrasted for the adjacency - spectral procedure from section [ b ] with parts what would happen when versus when . in this sectionwe are interested in contrasting for the adjacency - spectral procedure from section [ b ] with parts what would happen when versus when .this yields another consistent estimator for .  for the adjacency - spectral procedure from section [ b ] with parts ,the at - most distinct - valued rows of will be called the _ centroids _ , the _ centroid separation _ will refer to the minimum euclidean distance between all pairs of distinct centroids , and the _ minimum part size _ will refer to the least cardinality of the parts as partitioned by ; in particular , if one of the parts is empty then the minimum part size is zero , whereas the centroid separation would still be positive .[ mmm ] for the adjacency - spectral procedure from section [ b ] with parts , if then almost always the minimum part size is greater than and the centroid separation is at least .let and be any fixed real numbers . if then almost always it will * not * hold that the minimum part size is greater than and the centroid separation is at least . [ ddd ]  * proof : * as we did in section [ f ] , consider balls of radius about the distinct - valued rows of  ] ( these rows are the centers of the balls ) .since this centroid appears as a row of more than times , this would imply that \|_f \geq \sqrt{\vartheta n \left ( \frac{\zeta}{3 } \right ) ^2} ] ( where such that is fixed and denotes what would have been if we instead did the adjacency - spectral procedure from section [ b ] with parts instead of parts ) , which gives us the desired contradiction. + with theorem [ mmm ] we obtain another consistent estimator for .however , we would need to assume that positive real numbers and are known that satisfy and . assuming that such and are indeed known, we can define the random variable to be the greatest positive integer among the values ( note that is an upper bound on ) such that for the adjacency - spectral procedure from section [ b ] with parts the minimum part size is greater than and the centroid separation is at least . by theorem [ mmm ]we immediately obtain the following consistency result for .almost always .in order to define , lower bounds on and need to be known in addition to an upper bound on that needs to be known .this contrasts with , for which we only need to assume that an upper bound on is known .( because requires fewer assumptions , the extended adjacency - spectral partitioning procedure in section [ hhh ] utilizes and not . )nonetheless , it is useful to be aware of how the adjacency - spectral procedure from section [ b ] with parts changes in behavior when becomes greater than how it changes in behavior when becomes less than . andwhen lower bounds on and are also known then , in practice for a single value of , we can check for in order to have more confidence that their common value is indeed .as an illustration , consider the stochastic block model with parameters \quad m=\left [ \begin{array}{ccc }  .205 & .045 & .150 \\  .045 & .205 & .150 \\  .150 & .150 & .180 \end{array } \right ] , \ ] ] ( in particular , there is only one modality of communication ) and suppose edges are undirected . hererank ,  for each of the values and for each number of vertices , we generated monte carlo replications of this stochastic block model and to each of these realizations we applied the adjacency - spectral partitioning procedure of section [ b ] using as the upper bound on rank ( which , in the case of , is purposely incorrect for illustration purposes ) assuming that we know .note that rather than finding the actual minimum of , we use the -means algorithm which approximates this minimum .the five curves in figure [ qqq ] correspond to     respectively , and they plot the mean fraction of misassignments ( the number of misassigned vertices divided by the total number of vertices  , such fractions averaged over the monte carlo replicates ) along the -axis , against the value of along the -axis .note that when the performance of the adjacency - spectral partitioning is excellent ( in fact , the number of misassignments becomes effectively zero as gets to ) .indeed , even when and ( which is substantially greater than rank ) the adjacency - spectral partitioning partitioning performs very well . however , when , which is not an upper bound on rank ( violating our one assumption in this article ) , the misassignment rate of adjacency - spectral partitioning is almost as bad as chance .next we will consider the estimator for proposed in section [ e ] . recall that this estimator is defined as where is the matrix of centroids associated with each vertex , the adjacency spectral clustering procedure in section [ b ] is done with parts , and is fixed .we now consider stochastic block model parameters with stronger differences between blocks to illustrate the effectiveness of the estimator .in particular we let so that . for each we generated monte carlo replications of this stochastic block model . to each of these realizations we performed the adjacency spectral clustering procedure using ( figure [ fig : kest ] , left panel ) and ( figure [ fig : kest ] , right panel ) as our upper bound but this time assuming is not known .we used and computed the statistic .figure [ fig : kest ] shows the mean and standard deviation of this test statistic over the monte carlo replicates for each , and .the results demonstrate that for , is a good estimate when when we choose close to .on the other hand for smaller values of , our estimator will select too few blocks regardless of the choice of .interestingly , choosing close to , always equals the true number of blocks when we let , suggesting that this estimator has interesting behavior as a function of .note that for larger values of , will tend to be smaller , and for smaller values of , will tend to be larger .   using the parameters in line  for and .the unmarked dash line shows . ]our simulation experiment for estimating demonstrates that good performance is possible for moderate under certain parameter selections .this buttresses the theoretical and practical interest , as this estimator may serve as a stepping stone for the development of other more effective estimators .indeed , bounds shown in suggest that it may be possible to allow to be as small as using different proof methods .these methods in terms of the operator norm are an important area for further investigation when considering spectral techniques for inference on random graphs .  note additionally that for our first simulation , we used -means rather than minimizing since the latter is computationally unfeasible .this , together with fast methods to compute the singular value decomposition , indicates that this method can be used even on quite large graphs . for even larger graphs , there are also techniques to approximate the singular value decomposition that should be considered in future work .  further extensions of this work can be made in various directions .rohe et al . and others allow for the number of blocks to grow .we believe that this method could be extended to this scenario , though careful analysis is necessary to show that the estimator for the number of blocks is still consistent .another avenue is the problem of missing data , in the form of missing edges ; results for this setting follow immediately provided that the edges are missing uniformly at random .this is because the observed graph will still be a stochastic block model with the same block structure .other forms of missing data are deserving of further study .sparse graphs are also of interest and this work can likely be extended to the case of moderately sparse graphs , for example with minimum degree , without significant additional machinery .another form of missing data is that since we consider graphs with no self - loops , the diagonal of the adjacency matrix are all zeros .marchette et al . and scheinerman and tucker both suggest methods to impute the diagonals , and this has been show to improve inference in practice .this is related to one final point to mention : is it better to do spectral partitioning on the adjacency matrix ( as we do here in this article ) or on the laplacian ( to be used in place of the adjacency matrix in our procedure of this article ) ?there does nt currently seem to be a clear answer ; for some choices of stochastic block model parameters it seems empirically that the adjacency matrix gives fewer misassignments than the laplacian , and for other choices of parameters the laplacian seems to be better .a determination of exact criterion ( on the stochastic block model parameters ) for which the adjacency matrix is better than the laplacian and vice versa deserves attention in future work .but the analysis that we used here to reduce the required knowledge of the model parameters and to show robustness in the procedure will hopefully serve as an impetus to achieve formal results for spectral partitioning in the nonparametric setting for which the block model assumptions do nt hold . +* acknowledgements : * this work ( all authors ) is partially supported by national security science and engineering faculty fellowship ( nsseff ) grant number n00244 - 069 - 1 - 0031 , air force office of scientific research ( afosr ) , and johns hopkins university human language technology center of excellence ( jhu hlt coe ) .we also thank the editors and the anonymous referees for their valuables comments and critiques that greatly improved this work .bickel and a. chen , a nonparametric view of network models and newman - girvan and other modularities , _ proceedings of the national academy of sciences of the united states of america _ * 106 * ( 2009 ) .k. chaudhuri , f. chung , a. tsiatas , spectral clustering of graphs with general degrees in the extended planted partition model , _ journal of machine learning research : workshop and conference proceedings _ , ( 2012 ) pages 123

Abstract: for random graphs distributed according to a stochastic block model , we consider the inferential task of partioning vertices into blocks using spectral techniques . spectral partioning using the normalized laplacian and the adjacency matrix have both been shown to be consistent as the number of vertices tend to infinity . importantly , both procedures require that the number of blocks and the rank of the communication probability matrix are known , even as the rest of the parameters may be unknown . in this article , we prove that the ( suitably modified ) adjacency - spectral partitioning procedure , requiring only an upper bound on the rank of the communication probability matrix , is consistent . indeed , this result demonstrates a robustness to model mis - specification ; an overestimate of the rank may impose a moderate performance penalty , but the procedure is still consistent . furthermore , we extend this procedure to the setting where adjacencies may have multiple modalities and we allow for either directed or undirected graphs .