Doppler Magnification Dipole in Cosmology

• Doppler Magnification Dipole is a kinematic lensing effect where galaxy peculiar velocities induce a dipolar pattern in observed sizes and brightness, distinct from gravitational lensing.
• It employs linear perturbation theory and cross-correlation techniques to connect galaxy density fluctuations with size/flux modulations, providing insights into large-scale velocity fields.
• Observational strategies using surveys like DESI and SKA leverage this effect to achieve high signal-to-noise detections and constrain key cosmological parameters while addressing systematics such as PSF noise.

The Doppler magnification dipole is a kinematic lensing effect whereby peculiar velocities of galaxies along the line of sight modulate their observed sizes, brightnesses, and hence the measured convergence field. Around overdensities or underdensities in redshift space, this effect induces a dipolar pattern in the cross-correlation between galaxy number density fluctuations and galaxy sizes/brightnesses. Its physical and theoretical origin, formulation in linear cosmological perturbation theory, observational strategies, systematics, and implications for cosmological inference have been formally developed and quantified in recent literature (Andrianomena et al., 2018, Bonvin et al., 2016, Ye et al., 22 Nov 2025, Coates et al., 2020, Bacon et al., 2014).

1. Physical Origin and Phenomenology

At fixed observed redshift $z$, the real-space comoving distance $r$ of a galaxy with a line-of-sight peculiar velocity $V \cdot n$ receives a shift $\Delta r = - V \cdot n / \mathcal{H}(z)$, where $\mathcal{H}=aH$ is the conformal Hubble rate. This velocity-induced displacement leads to a systematic modulation of apparent sizes and fluxes:

• For $V \cdot n < 0$ (moving away), $\Delta r > 0$: the galaxy's true distance is smaller than suggested by its redshift, resulting in apparent magnification—the source appears larger and brighter.
• For $V \cdot n > 0$ (moving toward), $\Delta r < 0$: the source is farther away than implied by its redshift—an apparent demagnification.

Around an overdensity, the near-side galaxies are mostly infalling ($V \cdot n > 0$) and demagnified; far-side galaxies are receding ($V \cdot n < 0$) and magnified. This establishes an antisymmetric, dipolar pattern in the cross-correlation of density and size/brightness fields. The signature reverses for underdensities.

Doppler magnification is distinct from gravitational lensing ($\kappa_g$), which arises from the integrated effect of foreground density fluctuations along the sightline and yields mostly even ($\ell=0,2,\dots$) multipoles. Doppler magnification ($\kappa_v$) depends only on the local velocity at the galaxy position and is dominant at low redshift ($z \lesssim 0.5$) (Andrianomena et al., 2018, Bonvin et al., 2016, Bacon et al., 2014, Ye et al., 22 Nov 2025).

2. Linear Theory and Formalism

2.1 Convergence Field

The observed convergence $\kappa$ from galaxy size/flux measurements has two leading terms:

$$\kappa(\mathbf{r}) \simeq \kappa_v(\mathbf{r}) = \left[ \frac{1}{\mathcal{H} r} - 1 \right] V(\mathbf{r}) \cdot n$$

where $V(\mathbf{r})$ is the peculiar velocity, $n$ is the line-of-sight direction, and $r$ is the comoving distance (Andrianomena et al., 2018, Coates et al., 2020, Bacon et al., 2014, Ye et al., 22 Nov 2025). The prefactor arises from the derivative of conformal distance with respect to $z$.

2.2 Cross-Correlation and Dipole Multipole

Define the cross-correlation between galaxy-number fluctuations $\delta_g$ and convergence $\kappa$ as a function of separation $s$ and orientation $\mu = n \cdot \hat{s}$:

$$\xi_{g\kappa}(s, \mu) = \langle \delta_g(\mathbf{r})\kappa(\mathbf{r}+\mathbf{s}) \rangle$$

This is expanded as

$$\xi_{g\kappa}(s,\mu) = \sum_{\ell=0}^{\infty} \xi_\ell(s) P_\ell(\mu)$$

The dipole moment ($\ell=1$) is

$$\xi_1(s) = \frac{3}{2} \int_{-1}^{1} d\mu\, \mu\, \xi_{g\kappa}(s,\mu)$$

In GR, the analytic prediction in the flat-sky, linear approximation is (Andrianomena et al., 2018):

$$\xi_1(s) = \frac{1}{2\pi^2}\left[1 - \frac{1}{\mathcal{H} r}\right] (b + \frac{f}{3}) f \int_0^{\infty} dk\, k P_m(k,z) j_1(k s)$$

where $b$ is the galaxy bias, $f$ is the logarithmic growth rate, $P_m(k,z)$ is the matter power spectrum, and $j_1$ is the spherical Bessel function.

Generalized forms for modified gravity models involve scale-dependent growth $\mu(a,k)$ and velocity-growth kernels $\mathcal{G}(z,k)$, leading to more complex kernel integrals (Andrianomena et al., 2018, Coates et al., 2020).

3. Observational Measurement and Survey Forecasts

3.1 Estimator Construction

Given pixelized survey data, the estimator for the dipole is constructed by weighting galaxy pairs by the Legendre $P_1$ function of their orientation:

$$\hat{\xi}_1(d) = A^{-1}\sum_{ij} \delta_i \kappa_j P_1(\cos \beta_{ij})\, \delta_K(d_{ij} - d)$$

where $\delta_K$ is the Kronecker delta for a separation bin, and $A$ is the normalization (Andrianomena et al., 2018, Ye et al., 22 Nov 2025, Bonvin et al., 2016). The angle $\beta_{ij}$ is between the line of sight at $i$ and the separation vector $ij$.

3.2 Current and Future Survey Prospects

DESI Bright Galaxy Sample (BGS) and SKA2:

• DESI BGS: 14,000 deg$^2$, $0.05 < z \lesssim 0.4$, median $z \sim 0.2$, combining spectroscopic redshifts and multiband imaging.
• SKA Phase-2: 20,000 deg$^2$, $0 < z \lesssim 1.5$, focus on $z \lesssim 0.5$, HI spectroscopic redshifts coupled with imaging for sizes (Andrianomena et al., 2018, Ye et al., 22 Nov 2025).

For practical detectability, forecasts indicate cumulative dipole $S/N \sim 70$ for SKA2 with $\sigma_\kappa=0.3$ over $40 < d < 180$ Mpc/$h$ and $0.1 < z < 0.5$. For DESI with LSST Y1 imaging, recent simulations yield per-bin $S/N \approx 10$–13 for $\Delta z=0.1$ bins (Ye et al., 22 Nov 2025).

Measurement strategies employ the cross-correlation of spectroscopically selected galaxy positions from DESI with size/convergence information from LSST imaging, using size-fit algorithms and robust PSF modeling.

3.3 Covariance Analysis

The total estimator variance is decomposed into:

• Cosmic variance (of $\delta$ and $\kappa$)
• Intrinsic size scatter and shot noise ($\propto \sigma_\kappa^2$)
• Measurement error and limited galaxy number ($\propto \sigma_\kappa^2/n_{\rm gal}$)

Size intrinsic scatter ($\sigma_\kappa$) dominates over measurement error for current imaging quality. For LSST, simulations yield $\sigma_\kappa \sim 0.39$–0.48 per galaxy (Ye et al., 22 Nov 2025). Cosmic variance dominates at large scales.

4. Systematics and Theoretical Considerations

Systematic Effects:

• Size Measurement Noise: Dominant at current imaging depths; improvements below $\sigma_\kappa \sim 0.1$ would require new “standard-ruler” relations or advanced simulation calibration.
• PSF and Seeing: Can bias size measurements, mitigated by multi-band deconvolution and comparison to calibration samples.
• Selection Biases: Must account for size-dependent selection effects via completeness corrections and forward modeling.
• Lensing Contamination: For $z < 0.5$, gravitational lensing contributes $\lesssim$7% to the dipole (Bonvin et al., 2016). At higher $z$, joint fits to multiple multipoles can isolate the Doppler signal.
• Redshift Binning: The analytic formulae for the dipole depend on the redshift window. For flux-limited counts, the correction changes for narrow versus broad bins; intermediate cases interpolate smoothly as shown in simulations (Chen, 2018).

Theoretical Extensions:

The Doppler magnification dipole responds directly to changes in cosmological parameters, especially the amplitude and shape of the matter power spectrum, bias, and growth rate. Numerical simulations validate the linear theory up to few-percent discrepancies at high $z$, with larger deviations at low $z$ due to non-linear velocity dispersion (Coates et al., 2020).

5. Complementarity, Cosmological Constraints, and Survey Futures

5.1 Relationship to Other Probes

• Redshift-Space Distortions (RSD): RSD is sensitive to velocity gradients ($\partial V$), while the Doppler dipole probes the velocity field ($V$) itself, yielding increased sensitivity to large-scale (low-$k$) modes (Andrianomena et al., 2018, Bonvin et al., 2016, Bacon et al., 2014).
• Galaxy Bias: The cross-correlation dipole has different dependence on bias and growth, $f(b + f/3)$, helping break degeneracies present in RSD-only analyses.
• Cosmic Shear: Doppler magnification affects sizes, not shapes, thus is immune to shear systematics and offers an orthogonal probe (Bonvin et al., 2016).

Combining these measurements can facilitate robust constraints on cosmological geometry, growth, and gravitational physics, including joint Fisher analyses.

5.2 Cosmological Parameter Forecasts

For SKA2 ($\sigma_\kappa=0.3$), the Doppler dipole places competitive constraints on modified gravity parameters: $E_{11} \lesssim 0.03$ (scale-independent $\mu$ models), $B_0 \lesssim 5.7 \times 10^{-6}$ ($f(R)$ gravity) (Andrianomena et al., 2018), improving over Planck+BAO+RSD results. For DESI/LSST, detection should be at $>10\sigma$ per redshift bin, enabling direct measurement of the growth rate $f(z)$ and constraints on $\Lambda$CDM and extensions (Ye et al., 22 Nov 2025, Bonvin et al., 2016, Coates et al., 2020).

5.3 Large-scale Relativistic Features

At very large scales or high redshift, relativistic corrections (Doppler, ISW, time-delay) have pronounced effects on the magnification dipole. The Doppler term dominates the GR corrections to the magnification power spectrum at low $z$, enabling direct tests of dark energy and its interactions (Duniya, 2016).

6. Extension to CMB and Kinematic Dipoles

The analogous kinematic modulation (“Doppler magnification dipole”) arises in the CMB through our peculiar motion relative to the CMB rest frame, coupling dipole modulation with off-diagonal harmonics in $a_{\ell m}$. The effect is isolated by optimal quadratic estimators for $\beta = v/c$, with experimental systematics from intensity-to-temperature conversion, noise, and masking. Planck 2018 detects the Doppler coupling at $\simeq2.6\sigma$; future CMB-S4 will improve precision but cosmic variance remains a limiting factor (Ferreira et al., 2021).

7. Summary and Prospects

The Doppler magnification dipole is a theoretically well-defined, kinematic lensing observable that robustly probes the velocity field of large-scale structure. Its measurement in wide-field, high-precision spectroscopic plus imaging surveys such as DESI, SKA, and LSST is feasible at high $S/N$ in multiple independent redshift bins. Doppler magnification dipole analyses provide stringent, systematics-independent constraints on cosmological parameters, modified gravity, and relativistic effects. They are complementary to RSD and cosmic shear, and offer unique access to ultra-large scales and velocity fields. Accurate modeling of bin width effects, systematics, and relativistic corrections is essential for reliable scientific inference (Andrianomena et al., 2018, Bonvin et al., 2016, Ye et al., 22 Nov 2025, Coates et al., 2020, Bacon et al., 2014, Duniya, 2016, Chen, 2018, Ferreira et al., 2021).