Detecting Relativistic Doppler by Multi-Tracing a Single Galaxy Population

New data from ongoing galaxy surveys, such as the $Euclid$ satellite and the Dark Energy Spectroscopic Instrument (DESI), are expected to unveil physics on the largest scales of our universe. Dramatically affected by cosmic variance, these scales are of interest to large-scale structure studies as they exhibit relevant corrections due to general relativity (GR) in the $n$-point statistics of cosmological random fields. We focus on the relativistic, sample-dependent Doppler contribution to the observed clustering of galaxies, whose detection will further confirm the validity of GR in cosmological regimes. Sample- and scale-dependent, the Doppler term is more likely to be detected via cross-correlation measurements, where it acts as an imaginary correction to the power spectrum of fluctuations in galaxy number counts. We present a method allowing us to exploit multi-tracer benefits from a single data set, by subdividing a galaxy population into two sub-samples, according to galaxies' luminosity/magnitude. To overcome cosmic variance we rely on a multi-tracer approach, and to maximise the detectability of the relativistic Doppler contribution in the data, we optimise sample selection. As a result, we find the optimal split and forecast the relativistic Doppler detection significance for both a DESI-like Bright Galaxy Sample and a $Euclid$-like H$\alpha$ galaxy population.


Introduction
The standard cosmological model (ΛCDM) pictures a universe dominated by the so-called dark components: dark matter and dark energy, which together account for about 95% of the total energy contribution.Despite its success in reproducing many cosmological observations, it is challenged by several issues (Abdalla et al., 2022;Perivolaropoulos and Skara, 2022;Debono and Smoot, 2016).Therefore, it has been posed that those problems, as well as the dark components themselves, may be due to an incorrect use of the theory to fit experimental data.General Relativity (GR) provides us with a description of the gravitational interaction which has passed various stunning tests (Holberg, 2010;Shapiro et al., 1968;Ashby, 2002;Hulse and Taylor, 1975;Weisberg and Taylor, 2005;Nanograv Collaboration, 2023;LIGO Scientific Collaboration and Virgo Collaboration, 2016), and has been established as the theoretical framework depicting cosmic evolution-as though it were the game rules which the whole content of the universe has been following.However, GR is yet to be confirmed in cosmological regimes, characterised by huge distances and extremely weak gravitational fields.Since such conditions are common across Email addresses: federico.montano@unito.it(Federico Montano), stefano.camera@unito.it(Stefano Camera) the universe, the ΛCDM model in fact relies upon the assumption of GR correctly describing gravity even at those still unprobed scales.For this reason, there is a considerable interest in setting up tests that could either falsify or further confirm GR on cosmological scales.
The large-scale structure (LSS) of the universe can be assessed as the product of cosmic evolution.In particular, LSS statistical properties, being both predictable and measurable, offer us the opportunity to look for GR-induced effects.We usually map biased tracers, e.g.galaxies, in redshift space to track the underlying (dark) matter distribution, so that a variety of phenomena come into play as corrections between the two distributions.Among those, we find the well-known clustering term (Desjacques et al., 2018), the Redshift Space Distortions (Kaiser, 1987), and a plethora of relativistic effects predicted by GR (Yoo, 2010;Bonvin and Durrer, 2011;Challinor and Lewis, 2011), whose leading contribution is given by the relativistic Doppler term.A measurement of the latter could confirm the validity of GR on large scales, though it has not been achieved yet by the ongoing galaxy surveys.The Doppler contribution, being relevant on the very scales plagued by cosmic variance, calls for new observational campaigns (DESI Collaboration: Aghamousa et al., 2016;Amendola et al., 2013;Mainieri et al., 2024;Schlegel et al., 2022) and strategies to be detected.For the sake of completeness, other effects enter the computation of the ob-served spatial distribution of sources: mostly Primordial Non-Gaussianity (Martin et al., 2014;Bartolo et al., 2004) and Wide-angle effects (Castorina and White, 2018).Nevertheless, we focus on the Doppler contribution in this paper.
To probe LSS properties we study the two-point statistic in Fourier space-i.e. the galaxy power spectrum-and address the relevance of the relativistic effects.The Doppler term can be investigated via a multi-tracer approach, that is, analysing the signal coming from different tracers at once.On the one hand, it is known that multi-tracer power spectra are able to boost the relativistic contribution thanks to the study of the cross-correlations, which show a milder scale dependence of the Doppler term (Mc-Donald, 2009).On the other hand, it allows us to even overcome the low statistical sampling limitation at large scales (cosmic variance), as was firstly noted in Percival et al. (2004) (see also White et al., 2009;McDonald and Seljak, 2009;Seljak, 2009;Abramo and Leonard, 2013;Fonseca et al., 2015;Alonso and Ferreira, 2015).
Interestingly, the GR contribution happens also to be dependent upon the specifications of the target sample, like the slope of the luminosity function.Since different galaxy populations have different power spectra, it is necessary to account for those specific aspects in order to estimate the presence of a Doppler signal in the data (Maartens et al., 2021).A kind of optimisation work can hence be carried out, with the aim of searching for a promising target to look at.For instance, low-z probes are expected to exhibit a significant Doppler effect, the lensing term being negligible.
Furthermore, it is possible to fulfil the multi-tracer requirements, namely, to consider at least two tracers simultaneously, with a single galaxy population if a luminosity cut is applied.Splitting a galaxy sample into a faint and a bright sub-sample, according to luminosity, enables us to make cross-correlation measurements within only one data set (Bonvin et al., 2014(Bonvin et al., , 2016;;Gaztanaga et al., 2017;Fonseca and Clarkson, 2021;Bonvin et al., 2023).The using of a multi-tracer approach, either within a single galaxy population or combining different tracers, has already been proven as an effective strategy to improve measurement uncertainties, both with mock and real data (Blake et al., 2013;Ross et al., 2014;Beutler et al., 2016;Marín et al., 2016;Wang et al., 2020;Zhao et al., 2021a,b;Mergulhão et al., 2022Mergulhão et al., , 2024)).Nonetheless, we focus on the relativistic signal in this analysis.
In fact, this paper comes out as a follow-up work of Montano and Camera (2024), which was focused on exploiting the benefit of cross-correlation measurements against auto-correlation ones.Now, we explore the potential of the multi-tracer technique and provide, for the first time, a comprehensive theoretical estimation of all the relevant quantities to study the impact of the Doppler signal in a full power spectrum analysis.In doing so, we use the faint-bright division strategy and adopt an information (or Fisher) matrix approach.We present forecasts regard-ing the probability of detecting the relativistic contribution within two samples: a DESI-like low redshift Bright Galaxy Sample (BGS), and a high-z Euclid -like Hα emitter population (Hahn et al., 2023;Euclid Collaboration et al., 2024a).
This paper is structured as follows: in Section 2 we define the multi-tracer power spectrum and its covariance; in Section 3 we describe our data sets, outlining the features of the luminosity cut technique as well as the studied galaxy populations.Then, we explain the information analysis formalism used in Section 4 and show the results in Section 5. Finally, we draw our conclusions in Section 6.

Multi-tracer power spectrum
Galaxy surveys map the extragalactic sky in some 'observed' space.In the traditional jargon of galaxy clustering, this is 'redshift space', to emphasise that the reconstruction of galaxy positions along the line of sight is affected by spurious peculiar velocity contributions to their cosmological redshift, resulting into an anisotropy in the signal between the radial direction and that transverse to it, globally referred to as 'redshift-space distortions' (RSDs).In fact, observed radial positions are not the only ones to differ from real galaxy locations.Most notably, observed transverse positions are affected by gravitational lensing distortions.To these two main perturbations, other subdominant ones are also present, arising in a full general relativistic treatment due to de-projection from observations on the past light-cone (Bonvin and Durrer, 2011;Yoo, 2010;Challinor and Lewis, 2011).Therefore, the number density contrast of galaxy counts is known to be affected by several terms.Many studies have been carried out about the derivation and physical meaning and impact of the various corrections (Jeong and Schmidt, 2015;Camera et al., 2015;Di Dio et al., 2016;Di Dio and Beutler, 2020;Beutler and Di Dio, 2020;Castorina and Di Dio, 2022;Martinelli et al., 2022;Foglieni et al., 2023).
In this work, we focus on the leading local contributions to the number density.The density contrast of fluctuations in galaxy number counts then reads where δ is the matter density contrast (in comovingsynchronous gauge), b is the (linear) galaxy bias, H is the conformal Hubble factor, v is the peculiar velocity field, and the subscript '∥' denotes the component of a vector along the line of sight (which is always oriented from the observer towards the source).The first term in Eq. ( 1) is a purely clustering term (Desjacques et al., 2018), the second is the RSD term (Kaiser, 1987), and the last one is the relativistic Doppler.The amplitude of the last term is with a prime denoting derivation with respect to conformal time, r the comoving radial distance, and Q and E, respectively, the magnification and evolution bias.It is worth emphasising that α is sample-dependent, due to the sample dependence of the two biases-just as the galaxy bias depends on the sample under consideration.For a luminosity-limited catalogue we have (3) where n(z; L > L c ) is the comoving (volumetric) number density of sources with a luminosity larger than a cut set at L c .
Moving to Fourier space, Eq. (1) becomes where GR (k) corresponds to the redshift-space kernel at first order in perturbation theory; its Newtonian (a.k.a.standard) and relativistic terms are with f := −d ln δ/d ln(1 + z) the growth rate, µ the cosine between the wavevector k and the line of sight, and k = |k|.Since the relativistic contribution (6) to the kernel is proportional to k −1 , it mainly affects the largest scales, which are actually difficult to access due to cosmic variance.For this reason, the Doppler term has hitherto gone undetected, because of the lack of galaxy survey data on very large scales.Labelling tracers by X and Y , the power spectrum tells us about their correlation in Fourier space and reads with P (k) the (linear) matter power spectrum and tracerdependent quantities having subscripts X and Y .Interestingly, we have , where the asterisk denotes complex conjugation.If X = Y , i.e. in the case of auto-correlation, the imaginary term vanishes, thus the Doppler contribution becomes ∝ k −2 ; conversely, if X ̸ = Y the relativistic effects leads to a dipole in the imaginary part, proportional to k −1 (McDonald, 2009;Gaztanaga et al., 2017).In other words, b and α being different for the two tracers results in a cross-power spectrum which is more promising than the auto-correlation one and allows for a detection of the Doppler effect (Bonvin et al., 2023;Montano and Camera, 2024).
Being interested in the large scales, we rely on a simple Gaussian covariance matrix Γ.Therefore, we define the power spectra data vector as P = (P XX , P XY , P Y Y ) and consequently write the covariance associated with a measurement of such a multi-tracer galaxy power spectrum as where PXY = P XY + N XY , N XY is the noise related to a measurement of P XY and N k represents the number of independent modes available in the observed volume V .Technically, we compute this latter quantity by taking into account the width of z−, µ− and k−bins-which we dub ∆z, ∆µ and ∆k, respectively-as well as considering V (z, ∆z) as the volume of a spherical shell rescaled by the observed area of the sky f sky , so that We stress the fact that we find Eq.( 8) to be the correct and most general expression for a multi-tracer power spectrum in this form.When we work with imaginary contributions to the cross-correlation, we indeed need to deal with the complex nature of the off-diagonal terms and the above-mentioned relation P XY = P * Y X .As may be easily noticed, Γ is thus consistently Hermitian and nonnegative, as it must always be for a covariance matrix.Also, our Eq.( 8) coherently recovers both Eq.(2.5) of Barberi Squarotti et al. (2024) and Eq. ( 28) of Jolicoeur et al. (2023) in the case of a cross-power spectrum P XY being fully real (see also Karagiannis et al. (2024) for a more in-depth analysis on multi-tracer covariance).
Equation ( 1) considers, on top of the Kaiser RSD correction, only the dominant term due to GR.However, other effects are expected to play a role (Castorina and Di Dio, 2022;Foglieni et al., 2023).All those not included contributions, especially lensing and wide-angle effects Castorina and White (2018); Paul et al. (2023), may be added in future analyses.We neglect these terms now for simplicity, because they cannot be directly added in a full power spectrum study.In addition to that, the very scales at which the Doppler contribution becomes dominant are also of a concern for constraining Primordial Non-Gaussianity, which in fact show a similar k-dependence and therefore needs to be distinguished from the relativistic effect (Namikawa et al., 2011;Euclid Collaboration et al., 2022;Abramo and Bertacca, 2017;Viljoen et al., 2021a,b;Camera et al., 2015;Lorenz et al., 2018).Nonetheless, since our main purpose is to understand whether we will be able to measure the Doppler effect within only one galaxy population, still through a multi-tracer approach, we shall assume the validity of Eq. ( 1), despite its non-completeness.

Luminosity cut technique
Galaxy surveys typically provide us with flux/magnitude limited catalogues of redshift and angular position of sources.Such catalogues are essentially sets of galaxies, within given sky patches, that are observed with a flux density higher than the detector threshold.However, as was first proposed in Bonvin et al. (2014) (see also Bonvin et al. (2016); Gaztanaga et al. (2017) and the recent Bonvin et al. (2023)), nothing forbids us to split such a set into two sub-samples, according to luminosity, and then treat the two sub-selections as if they were two independent populations.Labelling with the subscript "c" the critical flux density (or magnitude)-i.e. the detector threshold-and with "s" the fixed division between the two sub-samples, we define a faint sample as the selection of all the sources whose observed brightness is within [F c , F s )-or analogously ∈ (m s , m c ]-and a bright sample which includes all the galaxies brighter than F s (or m s ).In line with the notation in Montano and Camera (2024), we have n T = n F + n B , where n X is the galaxy number density for the faint (X = F), bright (X = B), or the total-namely, the whole catalogue-(X = T) sample.
This luminosity cut immediately leads to the need to model the linear, magnification and evolution biases for the sub-sample, as they are different from those of the total population now.Specifically, we must describe the biases of the faint population (b F , Q F and E F ) properly, considering that it is defined by both a lower and an upper flux limit.Conversely, the bright selection does not require any peculiar treatment because it can be thought of as a total catalogue with a more conservative luminosity threshold, which can be described by the usual recipes present in the literature.Therefore, we write the linear bias of the faint population as following Ferramacho et al. (2014).Given the number densities n T , n F and n B , b F only depends on the cumulative biases of the total and bright samples (b T and b B ) that in turn depends on F c (or m c ) and F s (or m s ), respectively.Concerning the magnification bias Q F , it has to be derived by calculating the variation of the number of faint galaxies due to luminosity perturbation at both brightness limits (Bonvin et al., 2023).This results in a redefinition of the bias in a way that takes into account also the logarithmic derivative of the galaxy number density with respect to the upper luminosity cut: On the other hand, since a redshift partial derivative is involved in the computation of the evolution bias, it is straightforward to rephrase it in the case of a faint sample as

Galaxy populations
The relativistic Doppler contribution is sampledependent, therefore the estimation of its detection probability cannot help but focus on specific targets.In this work, we consider the low redshift probe of a Bright Galaxy Sample and a high-z Hα emission line galaxy population.
We take as the first case study the population of the bright near galaxies and model it after the Bright Galaxy Sample (BGS) of the Dark Energy Spectroscopic Instrument (Schlafly et al., 2023;Hahn et al., 2023).The DESI BGS will target more than 10 million galaxies at z < 0.6 within a sky area of about 14 000 deg 2 (see also DESI Collaboration: Aghamousa et al., 2016;DESI Collaboration: Adame et al., 2024;Krolewski et al., 2024).Following the procedure illustrated in Appendix A.2 of Montano and Camera (2024), we choose to model this galaxy population according to Smith et al. (2024) and fix the critical (maximum) r-band magnitude for our analysis to m c = 20.175.Briefly, we adopt the fit parameters given in Smith et al. (2024) to build an analytical, luminosity-dependent Halo Occupation Distribution from which we consistently recover the galaxy number density and the values of the linear, magnification and evolution biases.In doing so, we assume the number of central galaxies within a dark matter halo to be modelled as an error function depending upon the halo mass, whose shape is given by the absolute magnitude of the galaxies themselves (Zheng et al., 2005).It is worth stressing that having a model which is able to describe the number of detected objects as a function of magnitude is crucial to our purpose: since we aim to work with a luminosity cut we indeed need to estimate the densities and the biases of the two magnitude-limited sub-samples.We work, in line with what is usually done in the case of the BGS, in a r-magnitude-based framework, and use the K correction as in Jelic-Cizmek et al. (2021).
Further, we focus on a high redshift emission line galaxy population which mimics the Hα target sample of the Euclid satellite (Laureijs et al., 2011;Amendola et al., 2013Amendola et al., , 2018;;Euclid Collaboration et al., 2024a).Euclid has recently started its scientific survey mapping up to 30 million Hα galaxies in 0.9 ≲ z ≲ 1.8 over almost 15 000 deg parametrise the linear bias by means of the phenomenological formula presented in Pan et al. (2020), which can be employed in describing Euclid -like Hα targets.For the sake of completeness, we point out that there is another luminosity function model, dubbed Model 1, widely present in the literature (see e.g.Maartens et al., 2020Maartens et al., , 2021;;Montano and Camera, 2024).However, we shall not focus on that because its analytical formulation features a discontinuity at z = 1.3 in E which may lead to unphysical outcomes.
Figure 1 shows the galaxy number density for both galaxy populations after a flux cut between a faint and a bright sub-sample is applied.Within the redshift range z ∈ [0.05, 0.55] we display the BGS number densities referred to a magnitude split of m s = 19; whereas in z ∈ [0.9, 1.8] Hα curves, with F s = 2.8 × 10 −16 erg cm −2 s −1 , are shown.A factor of 50 amplifies the latter for a visualisation preference.Further, we leave in Appendix A the fitting function that can be used to describe number densities and biases with the luminosity cut applied here.The choice of the division between the two sub-populations may seem arbitrary at this point, but it shall be clarified in Section 5.

Information analysis
In order to quantify the presence of a relativistic Doppler signal in the data, we carry out an information matrix analysis in several parameter spaces, which we shall discuss in the following section.We start defining the auto-and cross-power spectra data vector as the vector that includes the three correlations, i.e.
Then, from a purely theoretical perspective, given a parameter set θ = {θ 1 • • • θ n }, which the power spectra depend upon, with {θ α } the n so far unspecified parameters, we write the information matrix in a general way.It reads where the sum runs over all the configurations (µ m , k n ) and parenthesis around indexes indicate the non-trivial symmetrisation which ensures I = {I αβ } to be itself real.Indeed, it is worth noting that such symmetrisation is always required, as I is defined as the Hessian matrix of a Gaussian likelihood, but usually not made explicit when working with real power spectra.In each redshift bin, the marginal errors on the parameters {θ α } are σ θα = (I −1 ) αα (z i ).These values provide us with an estimation of the uncertainty of a measurement of θ(z i ) due to statistical noise and parameter degeneracies.On the other hand, if we look at the entire z-range we obtain the cumulative marginal errors by taking the diagonal elements of the inverse of the total information matrix I = i I(z i ).
Throughout this paper, we assume a standard ΛCDM Planck Collaboration et al. (2020) cosmology in presenting our forecasts.Furthermore, we pick f sky = 0.36, corresponding to a survey observing the majority of the extragalactic sky.Such an assumption is common in literature and enables us to describe a Euclid -like and a DESI-like survey at once, even though the nominal sky coverage is 14000 deg 2 (f sky = 0.339) for the Euclid satellite (Euclid Collaboration et al., 2024a,b) and for DESI (DESI Collaboration: Adame et al., 2024) 1 .This value is required for the computation of the probed volume and hence limits the largest measurable scale, which in turn enters our analysis as the smallest wavenumber considered, k min .Since we seek to detect an effect which is dominant on large scales, k min plays a crucial role: we fix it directly from the volume by i.e. the wavenumber associated with the length scale of the ith redshift bin.On the other hand, the largest wavenumber is expected to not dramatically affect our results.We thus make the conservative choice to fix it to the scale at which the non-linear growth of structure becomes dominant at z = 0, i.e. k max = 0.2 h Mpc −1 , although there are recipes to describe it as a redshift dependent quantity and to push to mildly non-linear scales.Equation ( 14) shows that we must define z-, µ-and k-bins in order to estimate the presence of a relativistic signal in the data.Therefore, we use 10 µ-bins in the range µ ∈ [−1, 1] and 30 log-spaced bins in k with k min ≤ k ≤ k max .As far as redshift bins are concerned, we divide the BGS and Hα targets into 3 and 4 z-bins, respectively.In our study this results in a redshift bin width of ∆z ≈ 0.17 within the range 0.05 ≤ z ≤ 0.55 for the BGS (Bonvin et al., 2023) and ∆z = 0.225 within 0.9 ≤ z ≤ 1.8 for the Hα (Model 3 -given) sample (Euclid Collaboration: Blanchard et al., 2020).We highlight that the width of the redshift bins constrains the probed volume and thus the k min value.Such an aspect is reflected in the choice of considering thicker ∆z than the usual ∆z ≈ 0.1.

Results
In this Section, we present the results of our information matrix analysis, obtained by exploring three different sets of parameters.We focus on the significance of the relativistic effects within the data, as a unique signal which we forecast across the whole redshift range.Afterwards, we investigate the possibility of disentangling the amplitude of the Doppler effect α X (z) for both the faint and the bright sub-sample, individually.

Relativistic Doppler detection
Firstly, we estimate the probability of detecting the relativistic Doppler effect by considering three dummy N , A K and A D -as our parameter set θ(z i ).Those variables, whose fiducial values are all fixed to unity, act as amplitudes of the three leading contributions to the galaxy power spectrum: the real-space clustering signal, usually referred to as Newtonian (A N ); the linear Kaiser Redshift Space Distortion term (A K ) and the Doppler (A D ).We then rephrase Eqs. ( 5) and (6) to and obtain the expression of the power spectra with the amplitudes.Thus, we set up our information matrix approach with the parameter set to compute the marginal error, cumulative on the entire redshift range, associated with a measurement of the Doppler amplitude.Since we are interested in finding a tailored sample to detect the relativistic effect, we focus this section on the dependence of σ AD upon the splitting flux/magnitude.For this reason, we decide to fold together all the z-bins, by considering the cumulative marginal errors instead of the differential ones, and study whether the curve σ AD (F s )/σ AD (m s ) shows a minimum.This would mean that there exists a sort of optimal division between the faint and bright sub-samples.
Secondly, we extend our analysis by including an additional set of nuisance parameters, on top of the three fictitious amplitudes.Specifically, we account for possible departures from Poissonian errors and add to θ a number of 3 × (number of z−bins) noise parameters, one per redshift bin and per power spectrum type (P F F , P F B and P BB ) considered (see e.g.Casas-Miranda et al., 2002;Hamaus et al., 2010;Paech et al., 2017;Ginzburg and Desjacques, 2020;Euclid Collaboration et al., 2024c).Therefore we take into account where the fiducial value of N i XY = 0 and the superscript (i), running over all the z-bins, means that we have an independent additional parameter for each bin.In doing so, we substitute back in Eq. ( 13) and then build an information matrix whose dimension is [3 × (1 + number of z−bins)] 2 .
BGS results are depicted in Fig. 2, where we plot the cumulative marginal error associated with a measurement of the relativistic Doppler A D , as a function of the splitting magnitude.Being the fiducial value A D = 1, we notice that the multi-tracer technique allows us to reach a highsignificance detection, with σ AD ≪ 1.Also, the tiny discrepancy between the (solid) curve that includes the nuisance parameters and that (dashed line) which does not,  demonstrate the robustness of our analysis and allows us to argue about the behaviour of the BGS sample itself.Finding σ AD (m s ) has a minimum at roughly m s = 19.0,tells us it is possible to enhance the Doppler contribution within BGS data with the luminosity cut strategy.However, our purely analytical calculations might be nonaccurate when m s ≲ 18.5 because of the number density of the bright subsample n F → 0 in the highestz bin.For this reason, the accuracy for σ AD (m s ≲ 18.5) is probably optimistic.Analogously, Fig. 3 shows the σ AD (F s ) forecasts for the Hα galaxy population.Again, results for both with-and without-noise cases are displayed; nevertheless, they almost overlap in our graph, being their relative difference |σ AD (F s ) (w/o noise) − σ AD (F s ) (w/ noise) |/σ AD (F s ) (w/ noise) < 1%.As expected, we find the marginal error on A D marginalised over the nuisance parameters to be bigger than σ AD (F s ) (w/o noise) .Both curves reach the 3 σ detection level (σ AD ≃ 0.3) around F s = 2.8 × 10 −16 erg cm −2 s −1 .Hα galaxies allow for a detection of the relativistic effect with a lower significance than that given by the BGS due to the higher impact shown by the Doppler term at low redshift (Beutler and Di Dio, 2020).
Moreover, if we look at the amplitudes of the clustering and Kaiser terms, namely A N and A K , we notice that they are almost unaffected by the splitting flux/magnitude value.As a consequence, within the entire F s (or m s ) interval tested, they present roughly constant marginal errors, with discrepancies with respect to the average values well below 10% (see Table 1 for details).The detection of the dominant terms of the power spectrum could indeed be affected by the noise alone, but it appears to be given by the number density of the total sample in a multi-tracer analysis, in which we merge information from both faint BGS Hα Table 1: Cumulative marginal errors on the Newtonian (σ A N ) and Kaiser (σ A K ) contributions for both BGS and Hα targets.Superscripts refer to the presence/absence of noise parameters in the information matrix; whereas the reported values are expressed in the format (average value) ± (percentage variation), with the mean and variation being computed over the whole splitting flux/magnitude interval.Note that the uncertainty on A N and A K are hardly affected by the luminosity cut.
and bright sub-samples.Finally, it is worth noting that the results we present in this paper definitely look better than those obtained by focusing only on the faint-bright cross-correlation power spectrum, as can be appreciated if one compares Figs. 2 and 3 with Fig. 5 of Montano and Camera (2024).That is due to the capability of the multi-tracer approach of putting all the information coming from both auto-and cross-power spectra together to get tighter constraints.This happens because, unlike the only cross-correlation case, a multi-tracer analysis is able to somehow overcome the cosmic variance limit, that always affects single spectra measurements (Fonseca et al., 2015).
To draw the reader's attention on how our cumulative marginal error turn into constrains within each redshift bin, we show in Fig. 4 the uncertainties in each redshift bin, σ AD (z i ). 2 We fix the luminosity cut to the best performing values, i.e. m s = 19.0 and F s = 2.8 × 10 −16 erg cm −2 s −1 for the BGS and Hα sample, respectively.Leaving aside the three auto-correlations (blue, orange, and green candles) that are unable to provide us with a detection of the relativistic contribution (see Appendix B for insights on the differences between those cases), we can appreciate how the marginal error improves for the cross-power spectrum (red candles) because of the different scaling of the Doppler (Montano and Camera, 2024;Bonvin et al., 2023).On top of that, the gain given by the multi-tracer measurements (purple candles) overcoming cosmic variance is remarkable.Thereby, Fig. 4 demonstrates the importance of combining sample optimisation and multi-tracer analyses.

Physical parameters
Let us now move to estimate the uncertainty associated with a measurement of the following parameter set in each redshift bin (labelled by i), 2 Figure 4 can be seen as the information matrix complement of Fig. 2 in Montano and Camera (2024), although there the detection significance of the Doppler contribution is depicted and the multipower spectrum is not yet considered.: Differential marginal error associated with a measurement of A D in each redshift bin, in the case of an optimal split between the two sub-samples (namely, {mc, mc} = {20.175,19.0} for the BGS and {Fc, Fs} = {2.0,2.8} × 10 −16 erg cm −2 s −1 for Hα galaxies).Blue, orange, green, and red candles respectively depict the cases of total, faint, and bright auto-power spectra and the faint-bright cross-power spectrum, whilst purple candles show the multi-tracer case.
where the presence of σ 8 is given by expressing the observable power spectra as with, again, X = Y yielding the auto-correlation case.This choice for θ addresses the problem from a more physical point of view.In this context we mimic a measurement of the two relativistic Doppler amplitudes in each z-bin.
In other words, we take into account the difference between the biases of the faint and bright populations and no longer consider only one, general, amplitude.We thus fix the fiducial values of the parameters ( 19) to the theoretical predictions and estimate their uncertainties, focusing on the relativistic terms.We use once again the results shown above to select the best luminosity cut for both targets: i.e. m s = 19.0 for BGS and F s = 2.8 × 10 −16 erg cm −2 s −1 for Hα emitters. 3This choice allows us to study the relevance of the optimised Doppler signal in each sample.In fact, it is reasonable to assume that the best split found in Section 5.1 holds with the parameter set in Eq. ( 19) too.Since we are going to focus on α F and α B -which together enclose the relativistic contribution-we expect their detection significance to peak in the same way A D does.Illustratively, Fig. 5 shows our results of both BGS (lowz, left-hand side of the panel) and Hα population (high-z, right-hand side).The Doppler amplitude α X (z) is plotted against redshift using a symmetrical logarithmic scale (with linear range −10 < α < 10).Grey curves represent the predictions for both sub-samples, whilst coloured data points (faint sample in orange, bright sample in green) are the measurement forecasts.Each data point is associated with an error that is given by the study of the information matrix, namely σ α X (z i ).We find in general large error bars, meaning that it is much harder to constrain α F (z) and α B (z) in each redshift bin separately than assess the total presence of relativistic Doppler in the data.Moreover, we observe that the Doppler amplitude is wellconstrained within the first two BGS redshift bins, whereas this is not the case with the third z-bin, due to the number density falling off at z ≳ 0.35.At high redshift, the difference between the faint and bright sub-samples is hardly appreciable, meaning that the bias corrections of Eqs. ( 11) and ( 12) induced by the luminosity cut are more relevant for the BGS.This once again confirms the sampledependent nature of the relativistic Doppler effect and, in turn, the need for a search for tailored galaxy samples in the efforts to detect this effect.

Discussion and conclusions
A detection of an effect due to GR coming from galaxy clustering measurements might further confirm the validity of Einstein's theory of gravity in the very cosmological regime where it has not been tested yet.Several corrections affect the relation between the (underlying) dark matter density contrast and that of galaxies (Yoo, 2010;Bonvin and Durrer, 2011;Challinor and Lewis, 2011).Among them, the so-called relativistic Doppler term plays a dominant role with respect to the other contributions, especially at low redshift where lensing is subdominant (Castorina and Di Dio, 2022).Its relevance is given by the amplitude parameter (α), which behaves differently for different galaxy populations due to the presence of magnification and evolution biases.Moreover, the Doppler scaling ∝ k −1 in ∆(k) makes it important on large scales in the auto-correlation galaxy power spectrum and relevant even at intermediate scales in the cross-correlation case.The main issues to be addressed thus are: accessing the largest scales plagued by low statistical sampling due to the lack of Fourier modes, and finding a promising target.
In the search for an optimal galaxy sample to achieve a detection of the relativistic Doppler term, we have split a galaxy population according to luminosity (Bonvin et al., 2014), and then theoretically estimated the significance of the relativistic signal in a multi-tracer power spectrum analysis (Percival et al., 2004).To exploit multi-tracer benefits out of a single data set we have combined the autocorrelation of the two (faint and bright) sub-samples and their cross-power spectrum.We have presented forecasts regarding a low-z DESI-like BGS and a high-z Euclid -like Hα target.In doing so, we have coherently retrieved all the relevant quantities thanks to phenomenological relations calibrated on real data (Fonseca and Camera, 2020;Montano and Camera, 2024).
Our information analysis approach has shown that we can tune the division between the faint and bright samples in order to maximise the impact of the relativistic effect.Considering the entire redshift range, i.e. the cumulative marginal error on the Doppler contribution σ AD , we have found the following best selections: • m s = 19.0(with m s = 20.175)for BGS; • F s = 2.8 × 10 −16 erg cm −2 s −1 (with F c = 2.0 × 10 −16 erg cm −2 s −1 ) for Hα emitters.
Interestingly, both cases roughly correspond to a 50% faint-50% bright division.This outcome, regarding BGS, is not in agreement with the optimal 10% faint-90% bright division obtained by Bonvin et al. (2023); however, we point out that such a difference might be given by the flattening of their number density at high m c values, which we have not seen in our HOD study.The BGS has turned out to be more promising than the high-z Hα population, exhibiting a minimum σ AD ≃ 0.005, to be compared with the minimum Hα value σ AD ≃ 0.3.This result may be due to both the larger number density of sources in the BGS and the α's specific features.Furthermore, we have checked the robustness of the forecasts presented by introducing various nuisance parameters within our information matrix and verifying that they do not have any significant impact on our findings.Also, we have used the most promising samples to make forecasts on the measurement of the Doppler amplitude in the two sub-samples.As expected on the basis of our previous results, α seems to be better constrained in the case of the BGS, particularly in the low-redshift bins, even though the error bars are such that we cannot forecast a clear joint detection of α B and α F yet.To study the Doppler presence within the full galaxy power spectrum the computation of the corresponding covariance matrix has been required.The covariance treatment presented has been carried out for the first time taking into account the relativistic contribution being imaginary.Since P XY (k) = P * Y X (k), a Hermitian covariance matrix indeed allows us to work with complex data vectors.
To conclude, thanks to the improved sensitivity and the enhanced volume probed, the current galaxy surveys (Euclid Collaboration et al., 2024a;DESI Collaboration: Adame et al., 2024) are likely to provide us with a first detection of a relativistic signature on the large scales of the cosmic structures.As those effects are sample-dependent, accurate sample optimisation works will have a paramount relevance for the target selection (see e.g.Bonvin et al. (2023) and Sobral-Blanco et al. (2024)).However, in order to obtain more accurate forecasts on the probability of detecting a GR-driven term, rather than focusing on the relevance of the relativistic Doppler itself, future studies shall include other contributions.In particular, it has been shown that the wide-angle effect should be considered as they are in fact not negligible with respect to the Doppler (Paul et al., 2023;Castorina and White, 2018;Noorikuhani and Scoccimarro, 2023;Jolicoeur et al., 2024).Additionally, Primordial Non-Gaussianity should be included (Abramo and Bertacca, 2017;Camera et al., 2015;Bruni et al., 2012), as well as the other local and integrated effects in the full relativistic correction to the galaxy number density in redshift space (Foglieni et al., 2023;Castorina and Di Dio, 2022;Beutler and Di Dio, 2020).Extensions of this study are going to focus on the analysis of the faint-bright multi-tracer power spectrum in harmonic space (Novara et al., 2024) and assess the reliability of the luminosity cut technique using simulated data.

Figure 2 :
Figure2: Cumulative marginal error associated with a measurement of the Doppler amplitude A D , within the entire redshift range, as a function of the splitting magnitude for the BGS.Solid(dashed) line refers to the analysis which includes(does not include) nuisance parameters, on top of the amplitudes A N , A K and A D .Magnitude limit: mc = 20.175.

Figure 3 :
Figure 3: Same as Fig. 2 but in the case of the Euclid-like Hα sample: marginal error on A D varying the splitting flux Fs.Since a flux-based description is used, the faint end is on the left-hand side now (where the critical flux is Fc = 2.0 × 10 −16 erg cm −2 s −1 ).
Figure4: Differential marginal error associated with a measurement of A D in each redshift bin, in the case of an optimal split between the two sub-samples (namely, {mc, mc} = {20.175,19.0} for the BGS and {Fc, Fs} = {2.0,2.8} × 10 −16 erg cm −2 s −1 for Hα galaxies).Blue, orange, green, and red candles respectively depict the cases of total, faint, and bright auto-power spectra and the faint-bright cross-power spectrum, whilst purple candles show the multi-tracer case.

Figure 5 :
Figure5: Doppler amplitude α(z), for the faint (orange) and bright (green) subsample in the best-split scenario.Results for the BGS are depicted on the left (0.05 ≤ z ≤ 0.55) while Hα curves lie on the right (0.9 ≤ z ≤ 1.8).Coloured data points and error bars show our forecast about a measurement of the Doppler amplitude, i.e. they are given by α F ± σα F and α B ± σα B in each redshift bin; on the other hand, grey lines draw the theoretical predictions.Let us caution that on y-axis a symmetric logarithmic scale is adopted, with linear range α ∈ [−10, 10], to facilitate visualising.