Cosmic Magnification Angular Power Spectrum

• Cosmic magnification angular power spectrum is a statistical tool that measures brightness and size fluctuations of extragalactic sources induced by gravitational lensing and relativistic effects.
• It decomposes the signal into weak lensing, Doppler, ISW, time-delay, and potential terms, providing insights into the geometry and dynamics of the universe.
• The observable is computed via spherical-harmonic decomposition and integrated source kernels, enabling precise constraints on dark energy, modified gravity, and large-scale structure evolution.

Cosmic magnification angular power spectrum is a pivotal observable in modern cosmological surveys, encoding the impact of weak lensing and all relativistic corrections on the observed number counts of galaxies and other extragalactic sources across the sky. By quantifying the statistics of apparent brightness and size fluctuations induced by matter inhomogeneities, the angular power spectrum of cosmic magnification ($C_\ell^{\mu\mu}$) yields powerful constraints on both the geometry and dynamics of the late universe. This observable is sensitive to fundamental physics, including the growth of large-scale structure, dark energy, and modifications of general relativity, and it provides critical complementarity to galaxy clustering and cosmic shear probes.

1. Theoretical Foundations and Formalism

Cosmic magnification arises from gravitational lensing, which alters the flux and apparent number density of background sources owing to mass inhomogeneities along the line of sight. More generally, in a flux-limited survey, the observed magnification overdensity is defined as

$$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$

where $\mu$ is the magnification, and ${\cal Q}(z)$ is the magnification bias, determined by the slope of the source cumulative number counts at the limiting flux threshold. This field is sensitive not only to the standard lensing (convergence) term but also to relativistic corrections from large-scale gravitational potentials and peculiar velocities.

The total observed magnification fluctuation is a sum of five physically distinct contributions: $$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$ where:

• $\Delta^{\rm lens}_{\cal M}$: standard weak lensing (convergence) term;
• $\Delta^{\rm Dop}_{\cal M}$: Doppler term from line-of-sight velocities;
• $\Delta^{\rm ISW}_{\cal M}$: integrated Sachs–Wolfe (ISW) term from time-varying gravitational potentials;
• $\Delta^{\rm td}_{\cal M}$: Shapiro time-delay term;
• $\Delta^{\rm pot}_{\cal M}$: local gravitational potential at the source.

Each term is expressed via integrals along the past light cone, involving the Bardeen gravitational potentials ($$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$0, $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$1), the source peculiar velocity, and projection kernels dependent on cosmological distances, bias, and redshift.

The angular power spectrum $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$2 for redshift bins $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$3 and $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$4 is defined as the two-point correlation function in spherical harmonics: $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$5 where $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$6 are multipole coefficients of $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$7. In linear perturbation theory: $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$8 with $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$9 the transfer function, $\mu$0 the primordial potential spectrum, and $\mu$1 the integrated source kernel collecting all physical contributions (Duniya et al., 8 Jul 2025, Duniya, 2016, Duniya et al., 2023).

2. Physical Contributions and Explicit Decomposition

The decomposition of $\mu$2 is explicit in the line-of-sight formalism:

• Lensing term:

$\mu$3

• Doppler:

$\mu$4

where $\mu$5 is the velocity potential.

• ISW:

$\mu$6

with primes denoting derivatives with respect to conformal time.

• Time-delay:

$\mu$7

• Potential:

$\mu$8

This formalism is valid for all metric theories of gravity, with the relevant evolution equations for $\mu$9 and ${\cal Q}(z)$0 given by the specific model (GR, ${\cal Q}(z)$1, beyond-Horndeski, etc.) (Duniya et al., 2022, Duniya et al., 2023).

3. Angular Power Spectrum Calculations and Key Dependencies

The computation of ${\cal Q}(z)$2 requires integrating the projected source kernels, which encode the redshift evolution, scale dependence, and all relevant physical effects: ${\cal Q}(z)$3 where ${\cal Q}(z)$4 is the slope of cumulative counts, and ${\cal Q}(z)$5 the spherical Bessel function (Duniya et al., 8 Jul 2025).

The standard Limber approximation,

${\cal Q}(z)$6

is applicable on ${\cal Q}(z)$7, but exact, non-Limber integrals are required for the ultra-large-scale regime (${\cal Q}(z)$8). The kernel ${\cal Q}(z)$9 depends strongly on the width of the redshift bins and survey selection function. Wide bins enhance the integrated lensing contribution and amplify biases when magnification is neglected (Tanidis et al., 2019).

Relativistic corrections become dominant on the largest angular scales ($$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$0), with the Doppler term especially significant at low redshift (Duniya, 2016, Duniya et al., 8 Jul 2025, Grimm et al., 2020). Tomographic binning suppresses the amplitude of these corrections, with Euclid-style bins ($$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$1) reducing Doppler contributions (e.g., $$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$2 at $$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$3 for $$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$4) (Grimm et al., 2020).

4. Sensitivity to Cosmological Models and Fundamental Physics

The cosmic magnification angular power spectrum encodes a range of physical information:

• Dark Energy and Modified Gravity: The signal is sensitive to the growth of structure and to the temporal–spatial evolution of metric potentials. In quintessence models, relativistic terms amplify the difference between $$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$5CDM and $$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$6CDM on large scales, especially for $$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$7, where deviations in $$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$8 can reach $$\Delta_{\cal M}^{\rm obs} = \Delta^{\rm lens}_{\cal M} + \Delta^{\rm Dop}_{\cal M} + \Delta^{\rm ISW}_{\cal M} + \Delta^{\rm td}_{\cal M} + \Delta^{\rm pot}_{\cal M},$$9 at $\Delta^{\rm lens}_{\cal M}$0 (Duniya et al., 8 Jul 2025). In $\Delta^{\rm lens}_{\cal M}$1 gravity, the signature includes a distinct grouping of $\Delta^{\rm lens}_{\cal M}$2 between integer and decimal $\Delta^{\rm lens}_{\cal M}$3 exponents in the Hu–Sawicki model, with sign changes in the relativistic correction at $\Delta^{\rm lens}_{\cal M}$4 (Duniya et al., 2022). Beyond-Horndeski models introduce scale- and time-dependent modifications via four $\Delta^{\rm lens}_{\cal M}$5-functions, directly affecting the magnification spectrum on large and small scales (Duniya et al., 2023).
• Interacting Dark Energy: IDE suppresses matter and velocity perturbations on large scales, causing a uniform decrease in the low-$\Delta^{\rm lens}_{\cal M}$6 $\Delta^{\rm lens}_{\cal M}$7 amplitude, especially for low redshifts, with effects up to $\Delta^{\rm lens}_{\cal M}$8 for large couplings; at $\Delta^{\rm lens}_{\cal M}$9 these effects diminish (Duniya et al., 2022, Duniya, 2016).
• GR Effects ("Ultra-large-scale Relativistic Corrections"): Magnification probes general-relativistic terms (Doppler, ISW, time-delay, gravitational potential) that are inaccessible to standard number-count clustering. The amplitude of these terms may exceed that of standard weak lensing on large scales at low $\Delta^{\rm Dop}_{\cal M}$0.

The table below summarizes these dependencies:

Model	Scale/Redshift Sensitivity	Main Effect in $\Delta^{\rm Dop}_{\cal M}$1
$\Delta^{\rm Dop}_{\cal M}$2CDM	$\Delta^{\rm Dop}_{\cal M}$3, $\Delta^{\rm Dop}_{\cal M}$4	Relativistic suppression
Quintessence	$\Delta^{\rm Dop}_{\cal M}$5, $\Delta^{\rm Dop}_{\cal M}$6	Enhanced difference ($\Delta^{\rm Dop}_{\cal M}$7)
$\Delta^{\rm Dop}_{\cal M}$8 (Hu–Sawicki)	all $\Delta^{\rm Dop}_{\cal M}$9 ($\Delta^{\rm ISW}_{\cal M}$0 at $\Delta^{\rm ISW}_{\cal M}$1)	Sign flip around $\Delta^{\rm ISW}_{\cal M}$2
Beyond-Horndeski	$\Delta^{\rm ISW}_{\cal M}$3 at $\Delta^{\rm ISW}_{\cal M}$4	Boosted relativistic signal
IDE	$\Delta^{\rm ISW}_{\cal M}$5, $\Delta^{\rm ISW}_{\cal M}$6	Uniform suppression/boost

5. Observational Strategies and Survey Considerations

Precision measurement of $\Delta^{\rm ISW}_{\cal M}$7 requires careful design of redshift binning, flux selection, and sky coverage:

• Wide redshift bins amplify the lensing kernel and sensitivity to magnification; however, they increase the risk of systematic biases if magnification is neglected (Tanidis et al., 2019).
• Flux-limited samples with steep number-counts (large $\Delta^{\rm ISW}_{\cal M}$8) maximize magnification bias and hence the signal-to-noise in $\Delta^{\rm ISW}_{\cal M}$9 (Duniya et al., 8 Jul 2025).
• Low-redshift, large-scale surveys can directly probe relativistic and Doppler terms, with the total relativistic signal surpassing cosmic variance at $\Delta^{\rm td}_{\cal M}$0 for $\Delta^{\rm td}_{\cal M}$1 (Duniya et al., 2023, Duniya et al., 8 Jul 2025). For ISW, time-delay, and potential terms, $\Delta^{\rm td}_{\cal M}$2 is required.
• Multi-tracer techniques can be deployed to beat cosmic variance, especially necessary for detecting relativistic corrections at $\Delta^{\rm td}_{\cal M}$3 (Duniya et al., 2022, Duniya et al., 8 Jul 2025).
• Tomographic cross-spectra between widely separated bins can isolate lensing-magnification from density fluctuations and are crucial for robust cosmological constraints (Tanidis et al., 2019).

Ignoring magnification leads to significant bias in cosmological parameters. For example, in deep radio continuum surveys, neglecting magnification can bias $\Delta^{\rm td}_{\cal M}$4 upward by $\Delta^{\rm td}_{\cal M}$5 and $\Delta^{\rm td}_{\cal M}$6 downward by $\Delta^{\rm td}_{\cal M}$7 for wide bins (Tanidis et al., 2019). Additionally, omitting magnification increases parameter degeneracies, especially between galaxy bias and $\Delta^{\rm td}_{\cal M}$8, and degrades constraints on extensions to $\Delta^{\rm td}_{\cal M}$9CDM.

6. Practical Computation and Numerical Results

The calculation of $\Delta^{\rm pot}_{\cal M}$0 incorporates transfer functions and source kernels assembled from linear perturbation theory with model-dependent evolution of the metric potentials and the velocity field. For $\Delta^{\rm pot}_{\cal M}$1CDM (with vanishing anisotropic stress, $\Delta^{\rm pot}_{\cal M}$2), Planck-normalized parameters are typically adopted (Duniya et al., 8 Jul 2025). The relative impact of different contributions is quantified by computing the ratio $\Delta^{\rm pot}_{\cal M}$3, with Doppler and total relativistic terms exceeding $\Delta^{\rm pot}_{\cal M}$4 by orders of magnitude at low $\Delta^{\rm pot}_{\cal M}$5 and $\Delta^{\rm pot}_{\cal M}$6 (e.g., at $\Delta^{\rm pot}_{\cal M}$7, enhancement by $\Delta^{\rm pot}_{\cal M}$8 at $\Delta^{\rm pot}_{\cal M}$9; at $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$00, $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$01 boost) (Duniya, 2016).

In interacting and modified gravity models, the evolution equations of the scalar metric potentials are explicitly altered, requiring numerical integration of the system to obtain $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$02 and the fractional deviations from $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$03CDM or among gravity models. For $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$04 gravity, a characteristic feature is the grouping of angular power spectra according to the parity of the Hu–Sawicki exponent $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$05, observable as a few-percent level separation at $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$06 (Duniya et al., 2022).

Detectability is limited by cosmic variance, characterized by

$$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$07

where $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$08 is the observed sky fraction. For typical wide-area surveys ($$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$09), detection thresholds of $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$10 on $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$11 at $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$12 are achievable (Duniya et al., 8 Jul 2025). At higher $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$13, multi-tracer methods and cross-correlations provide a path to overcoming the cosmic variance limit.

7. Implications, Limitations, and Future Prospects

Cosmic magnification angular power spectrum constitutes a major probe of both the background and perturbation-level properties of the universe. Its multi-component structure enables simultaneous sensitivity to geometry, growth, and the ultra-large-scale relativistic regime, including the signatures of interacting dark energy, quintessence, $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$14 gravity, and beyond-Horndeski models (Duniya et al., 8 Jul 2025, Duniya et al., 2022, Duniya et al., 2023, Duniya, 2016). Its most distinctive leverage is on large angular scales at low redshift, where relativistic corrections are significant and may in principle be disentangled from cosmic variance via multi-tracer analyses and cross-correlations.

Neglecting magnification systematically biases cosmological inference, particularly in deep, wide-bin surveys (Tanidis et al., 2019). Modeling relativistic corrections is required for robust cosmological and fundamental physics constraints as survey areas and sensitivities expand (Euclid, SKA, LSST, etc.).

The ability of $$\Delta_{\cal M}^{\rm obs}(\mathbf{n},z) = {\cal Q}(z) \left[\mu(\mathbf{n},z) - 1\right],$$15 to distinguish among dark energy and modified gravity models depends critically on survey design, redshift coverage, control of systematic effects, and the ultimate reach in cosmic variance–limited regimes. The angular power spectrum of cosmic magnification, with its full relativistic decomposition and redshift tomography, will remain a foundational observable in high-precision cosmology.