Detecting Local and Integrated Relativistic Effects by Multi-Tracing a Single Galaxy Population in Harmonic Space

TL;DR

This work tackles the challenge of detecting relativistic effects in galaxy clustering on cosmological scales by employing a two-tracer (faint-bright) split within a single galaxy population and analyzing the angular power spectrum $C_{ell}^{ij}$. By constructing a full multi-tracer tomographic covariance and comparing a complete GR model against a standard Newtonian-like model, the authors forecast the detectability of the leading Doppler effect and the sub-dominant $(H/k)^2$ terms across three representative survey targets (BGS, ELG, LBG) over $z\lesssim5$. They find that the Doppler term can reach ~3σ significance with optimally chosen faint fractions and binning, particularly for BGS at low redshift and LBGs at high redshift, while the higher-order corrections remain undetectable under realistic survey conditions. The results highlight the potential of harmonic-space multi-tracer analyses to test GR on cosmological scales and inform survey design, while also indicating that accessing the full suite of GR effects may require alternative datasets or more sub-samples. The study thus provides a practical pathway to validate GR with large-scale structure while outlining the limitations of the current method for probing sub-dominant relativistic corrections.

Abstract

Measuring relativistic effects on cosmological scales would provide further confirmation of the validity of general relativity in the still poorly tested condition of weak gravity. Despite their relevance, relativistic imprints in the distribution of galaxies on large scales have so far eluded detection, mainly because they are stronger on the largest cosmic scales, which are plagued by cosmic variance. Expanding on previous works, we study galaxy clustering by subdividing a galaxy population into two sub-samples -- bright and faint -- and we here focus on their two-point correlation function in harmonic space, i.e. via the angular power spectrum. Thanks to such a split in magnitude and by exploiting the multi-tracer technique, we are able to boost the impact of the relativistic contributions. We first focus on the leading relativistic contribution given by the Doppler effect and show that, with a carefully tailored luminosity cut, it can be detected. Then, we look at the sub-dominant effects predicted by general relativity and quantify how their statistical significance, as yet undetectable, varies with redshift binning and survey specifications. As case studies, we consider in our forecasts a bright galaxy sample at low redshift, an H$α$ emission-line galaxy survey at intermediate redshifts, and high-redshift Lyman-break galaxies at high redshift.

Figures (3)

• Figure 1: Harmonic-space power spectrum for faint-bright cross-correlation in the case of BGS. All spectra are plotted in the range $\ell \in [2,300]$. Solid curves correspond to the inclusion of all the contributions, whereas dashed ones correspond lack Doppler and the other GR terms. By $z_i$ we indicate the $i$th redshift bin, so that the $z_i \times z_i$ curve is the auto-correlation power spectrum in the $i$th bin, i.e. $C^{ii}_\ell$.
• Figure 2: Percentage change between the multi-tracer faint-bright tomographic matrix that includes all GR effects (i.e. the Doppler term is taken to be part of ${\sf C_\ell^{\rm (ext)}}$ here) and the one that does not. Three fixed multipoles are considered: $\ell=\{2,\,20,\,200\}$, from left to right. In line with \ref{['fig:Cl']}, we use the BGS specs, with $3$ redshift bins.
• Figure 3: Cumulative statistical significance associated with a detection of the relativistic contributions as a function of the fraction of faint galaxies. Low-$z$ DESI-like BGS curves in red, intermediate-$z$Euclid-like H$\alpha$ in green, and high-$z$ MegaMapper LBGs in blue. Different binning choices are shown via the line-style code, increasing the number of redshift bins going from the solid lines, through dotted lines, to dashed lines. Top:$\varDelta\chi^2$ for a model with all the GR effects against a null-hypothesis of a 'standard' angular power spectrum, i.e. Doppler included in ${{\sf C}_\ell^{\rm (ext)}}$. Bottom:$\varDelta\chi^2$ of the GR corrections $\mathcal{O}(\mathcal{H}^2/k^2)$, i.e. Doppler in ${{\sf C}_\ell^{\rm (std)}}$. The statistical significance of BGS is enhanced by a factor of $10$ to enhance readability.