Cosmic Dipoles from Large-Scale Structure Surveys

TL;DR

This work develops a gauge-aware, relativistic framework to quantify cosmic dipoles across CMB, supernova luminosity distances, and galaxy surveys within $\Lambda$CDM using Planck parameters and Boltzmann codes. It shows that the observer’s conformal Newtonian velocity principally drives the dipoles, while intrinsic velocities of sources decay with redshift, making a universal common rest frame an approximation rather than a reality. The analysis clarifies misconceptions about the intrinsic dipole and the Ellis–Baldwin test, and demonstrates that while local density fluctuations can contribute at low redshift, they are largely suppressed when integrating over realistic redshift distributions. These results illuminate the source of the observed tensions between CMB- and LSS-derived dipoles and provide a practical, extendable calculation scheme for testing cosmological models beyond $\Lambda$CDM.

Abstract

Large-scale structure surveys can be used to measure the dipole in the cosmic microwave background (CMB), in the luminosity distances inferred from type-Ia supernova observations, and in the spatial distribution of galaxies and quasars. The measurements of these cosmic dipoles appear to be mutually inconsistent, even though they are expected to indicate the common observer velocity. This observational tension may represent a significant challenge to the standard model of cosmology. Here we study in detail what contributes to the cosmic dipoles from CMB, supernova, and galaxy survey in the standard $Λ$CDM model, though our theoretical model can be applied beyond the standard model. While measurements of the cosmic dipoles yield the relative velocities between the source samples and the observer velocity, the motion of the observer is the dominant contribution in the conformal Newtonian gauge, and the intrinsic velocities of the samples fall steeply with increasing redshift of the sources. Hence the cosmic dipoles of CMB, type-Ia supernovae, and galaxies should be aligned but can have different amplitudes. We also clarify several misconceptions that are commonly found in the literature.

Figures (3)

• Figure 1: Transfer functions $\mathcal{T}_{\delta p}$ for the individual contributions to the dipole power spectra in Eqs. \ref{['CMBdipole']}, \ref{['TFSN']}, \ref{['TFLSS']}. Green curves show the transfer functions at the decoupling epoch $z_\star$, and they oscillate on small scales, while the other curves show the transfer functions at $z=0$. The gravitational potential changes little over time ($\psi_\star$, $\psi_o$), but the growth of the velocity potential is significant. None of the contributions to the transfer function is comparable to the matter density fluctuation $\delta_v$ in the comoving gauge (dot-dashed). Two short vertical lines indicate the peak positions $x=2.08$ and $x=3.87$ for $j_1(x)$ and $j_1'(x)$ with $x=k\bar{r}_\star$, beyond which the transfer functions are suppressed by the spherical Bessel function, except the contribution of the observer motion (solid) without the spherical Bessel function. Note that only the individual transfer functions are plotted here, not including the spherical Bessel function contribution. At lower redshift, the peak positions are shifted to higher wave numbers.
• Figure 2: Dimensionless dipole power spectra from the individual contributions in Eqs. \ref{['CMBdipole']}, \ref{['TFSN']}, \ref{['TFLSS']}. In the dipole transfer functions ${\cal T}_1(k)$, each transfer function $\mathcal{T}_{\delta p}(k,\eta_z)$ for the individual contributions $\delta p$ is multiplied by spherical Bessel functions, and only this product is considered for each curve to compute the dipole power at each redshift $z$. From the top to bottom, three curves represent the matter density fluctuation $\delta_v$ (solid), the redshift-space distortion (dot dashed), the observer motion $k\mathcal{T}_{v_m^{\rm cN}}(\eta_{\bar{\rm o}})/3\mathcal{H}\bar{r}_z$ in Eq. \ref{['TFSN']} (dashed). In the middle, five green curves show the lensing convergence $\kappa$ (solid), the velocity $kv_m^{\rm cN}$ (dashed), the gravitational potential $\psi$ (dotted), the velocity potential $\mathcal{H}_zv_m^{{\rm cN}}$ (short dashed), and the line-of-sight integration of $\psi/\bar{r}_z$ (dot dashed). In the bottom, two blue curves show the time derivative of gravitational potential $\psi'/\mathcal{H}_z$ (dotted) and the line-of-sight integration of $\psi'$ (dashed). The full dipole power needs to be computed with the full transfer function, which is the sum of all the individual components and integrated over the redshift range, before squared and integrated in Fourier space. All these contributions to the intrinsic velocity of the sources fall as the source redshift increases, while the observer velocity $\bm{v}^{\rm cN}_{\rm o}$ in the conformal Newtonian gauge remains constant. Two short horizontal lines represent the mean dipole power of CMB (solid: $C_1^{\rm CMB}=4.5\times10^{-6}$) from our fiducial cosmological model and the dipole power from the Planck measurements (dashed: $C_1^{\rm Planck}= 2.1\times 10^{-6}$), which to a good approximation represent the observer velocity $\bm{v}^{\rm cN}_{\rm o}$ in the conformal Newton gauge.
• Figure 3: Dimensionless dipole power spectra from the matter density fluctuations integrated over various redshift distributions. Solid curve shows the dipole power with the distribution in Eq. \ref{['2MASS']} as a function of the mean redshift $z$. The redshift distribution of the CatWise sample is similar to the solid curve with mean redshift $z=1$. Dotted, dashed, and dot-dashed curves show the dipole power with Gaussian redshift distribution in Eq. \ref{['GAUSS']}, and the variance for the Gaussian distributions is set $\sigma_z=0.05$, 0.1, and 0.2, respectively. In comparison, the green solid curve shows the dipole power from the matter density fluctuations $C_1^\delta$ at each redshift slice shown in Figure \ref{['Fig:dipole']}. The density contributions are greatly reduced in amplitude, once averaged over the redshift distributions. Three horizontal lines at $z=1$ indicate the observed dipole power $C_1^{\rm obs}$ from the CatWise survey, the dipole power $C_1^{\rm EB}$ expected from the Ellis-Baldwin formula for the survey, and the shot-noise power. The dipole power from the Planck measurements is shown as the gray horizontal line. Note that the matter density contributions in this plot should be multiplied by the bias factor $b\approx2$ and the Ellis-Baldwin coefficient ${\cal M}\approx 6$, and hence the dipole power $C_1\propto b^2{\cal M}^2$ is boosted by $\approx140$ for various curves to be compared to the observed dipole power $C_1^{\rm obs}$.