Coda Lens: Simulating CMB Lensing in Coupled DE Models

• Coda Lens is a simulation approach that creates CMB lensing maps by ray-tracing N-body outputs in cosmologies with coupled dark energy and cold dark matter.
• It employs full-sky HEALPix projections and the Born approximation to capture nonlinear structure formation and minute lensing features.
• Comparative analyses reveal distinct lensing signatures between ΛCDM, standard cDE, and bouncing cDE models, offering robust tests of cosmological theories.

Coda Lens refers to the rigorous simulation and analysis of Cosmic Microwave Background (CMB) weak lensing effects—specifically deflection-angle and lensing-potential maps—produced by tracing CMB photons through N-body simulated cosmic structures in models that incorporate coupled dark energy (DE) and cold dark matter (CDM) cosmologies. This approach leverages high-fidelity, large-volume N-body simulations (notably the COupled Dark Energy Cosmological Simulations, CoDECS) to generate full-sky lensing predictions that are sensitive to both the background cosmological evolution and the dynamics of structure formation, particularly under nonstandard DE scenarios where DE directly couples to CDM. The Coda Lens methodology makes it possible to discern subtle signatures of DE interaction in lensing observables, providing robust means to test and constrain cosmological models beyond the standard ΛCDM paradigm.

1. Calculation of Lensing Maps from N-Body Simulations

Coda Lens simulations generate lensing potential and deflection-angle maps via “ray-tracing” through the evolving three-dimensional gravitational potential from CoDECS N-body snapshots, capturing the non-linear cosmic web’s influence on photon propagation. The procedure consists of:

• Extracting dense grids of the gravitational potential from numerous simulation snapshots, spanning finely sampled redshifts.
• Stacking these grids along the observer’s past light cone, with simulation boxes arranged in spherical shells; each box is randomly rotated and translated per shell to minimize repetition and ensure accurate sampling of large-scale modes.
• Computing the lensing observables under the Born approximation, integrating potentials along undeflected light rays:

$$\Psi(\hat{n}) = -2 \int_0^{r^*} \frac{r^* - r}{r^* r} \frac{\Phi(r\hat{n}; \eta_0 - r)}{c^2} \, dr \tag{1}$$

where $\Psi(\hat{n})$ is the projected lensing potential, $r^*$ is the comoving distance to the last scattering surface, $\Phi$ is the gravitational potential, $\eta_0$ is present conformal time, and $c$ is the speed of light.

The deflection angle is given by:

$$\alpha(\hat{n}) = -2 \int_0^{r^*} \frac{r^* - r}{r^* r} \nabla_{\hat{n}}\left[ \frac{\Phi(r\hat{n}; \eta_0 - r)}{c^2} \right] dr \tag{2}$$

• Lensing potential and its gradient are spatially interpolated over the pre-computed potential grids.
• Projection onto the full sky is achieved using the HEALPix pixelization scheme (Nside = 2048, $\sim$1.72 arcmin angular resolution), capturing both arcminute-scale and degree-scale features. The underlying simulation grid spacing ($\simeq 400$ kpc/h) ensures high-fidelity structure on all relevant angular scales.

This methodology enables the construction of lensing maps that reliably encode both the geometry and physical properties of large-scale structure as influenced by varying DE-CDM coupling.

2. Coupled Dark Energy Cosmologies and Their Role in Lensing

Coda Lens simulations are distinguished by their inclusion of coupled dark energy (cDE) models, where the dark energy component is represented by a scalar field $\phi$ engaging in both self-interaction (e.g., exponential or SUGRA potentials) and explicit coupling to CDM. The direct DE-CDM coupling modifies the evolution of matter perturbations through two principal mechanisms:

• The emergence of a “fifth force,” acting as an additional attractive force among CDM particles, thereby enhancing the effective gravitational interaction.
• The introduction of a coupling-dependent friction (drag) term in the momentum equation for CDM, whose sign and magnitude depend on properties of the scalar field (notably $\dot\phi\beta_c$).

These effects are imprinted on the lensing signal, since the lensing potential integrates contributions from the time-varying gravitational field and expansion geometry along the photon path. Distinctions in the growth factor $D_+$ and Hubble parameter $H(a)$ caused by DE-CDM interactions affect both linear and nonlinear evolution of cosmic structure, resulting in measurable differences in the lensing observables—even in models sharing identical present-day normalization in linear density fluctuations (e.g., the same $\sigma_8$).

3. Comparative Outcomes for ΛCDM and Coupled DE Models

Three principal cosmological models are investigated in the Coda Lens framework:

• Standard ΛCDM (serving as a reference model),
• EXP003: a canonical cDE model with exponential potential and constant positive coupling ($\beta_c = +0.15$),
• SUGRA003: a “bouncing” cDE scenario characterized by a SUGRA potential and constant negative coupling ($\beta_c = -0.15$).

Key results:

• EXP003 (standard cDE): The lensing potential maps and power spectra demonstrate an increased amplitude relative to ΛCDM, attributable to both the enhanced growth rate of CDM density perturbations and modified nonlinear collapse due to the fifth force and coupling-induced friction. This produces excess lensing power, most notably manifest on nonlinear ($\ell > 1000$) scales.
• SUGRA003 (bouncing cDE): Counter-intuitively, despite similar $\sigma_8$ normalization at $z=0$, the integrated lensing power is reduced by approximately 10% compared to ΛCDM. This reduction arises from the interaction of background quantities (such as $\Omega_M(a)$ and $H(a)$) and the evolving growth factor, quantified by:

$$\frac{\mathcal{P}_\Psi}{\mathcal{P}_{\Psi,\Lambda}} = \left(\frac{\Omega_M(a)}{\Omega_{M,\Lambda}(a)}\right)^2 \left(\frac{H(a)}{H_\Lambda(a)}\right)^4 \frac{D_+(a)}{D_{+,\Lambda}(a)} \tag{3}$$

In SUGRA003, the lensing observable is suppressed due to the competition between increased fluctuation amplitude at intermediate redshift and a modified expansion history, culminating in a net reduction in lensing efficiency.

A table summarizing key model distinctions:

Model	DE Potential & Coupling $\beta_c$	Lensing Power (vs. ΛCDM)
ΛCDM	None	0
EXP003	Exponential	+0.15
SUGRA003	SUGRA	–0.15

Normalization to a common $\sigma_8$ does not erase these differences, as nonlinear dynamics tied to DE-CDM interaction remain significant.

4. Influence on Structure Formation and Lensing Signal

The lensing signal’s dependence on the structure formation history is a central feature of Coda Lens analysis:

• In EXP003, the fifth force accelerates the linear growth of CDM density contrasts ($D_+$), resulting in increased abundance and deeper gravitational potential wells (halos). The additional friction term modifies the formation and merging of non-linear structures, leading to further amplification of the lensing power.
• Lens potential is the integrated result of all gravitational potentials along the line of sight. Thus, the enhanced halo population and altered nonlinear evolution drive a rise in the lensing potential power spectrum, prominently where $\ell > 200$ and nonlinearities dominate.
• When comparing simulation results to semi-analytical predictions utilizing Halofit corrections, N-body approaches are shown to more fully capture small-scale features essential for accurate lensing statistics.

In SUGRA003, although the growth of fluctuations is initially enhanced at intermediate redshifts, the combined effect of the DE-CDM coupling and background expansion yields a lower cumulative lensing efficiency.

5. Distinction Between Baryons and CDM in Lensing Observables

The CoDECS simulations evolve baryon particles as collisionless matter; crucially, baryons remain uncoupled to the DE scalar field, and therefore do not experience the fifth force or friction term affecting CDM particles:

• While both trace the underlying large-scale structure, the amplitude of their contributions to the lensing signal diverge. Baryons systematically contribute only about one-third of the total deflection-angle root mean square.
• In SUGRA003, this leads to a distinctive effect: the CDM lensing amplitude is suppressed relative to ΛCDM, while baryons exhibit slightly higher power. The decoupling of baryons from DE severs the link to the lensing suppression affecting CDM, resulting in divergent trends for the two components in the bouncing scenario. This is especially pronounced in SUGRA003, where baryons sustain or enhance their gravitational collapse while the CDM signal is reduced.

6. Applications and Implications for Cosmological Inference

The Coda Lens approach underscores the sensitivity of CMB lensing to both the expansion history and the growth of cosmic structure. Its salient implications include:

• Providing discriminants between ΛCDM, standard cDE, and bouncing cDE scenarios, even allowing for degeneracy in linear normalization parameters (such as matching $\sigma_8$).
• Enabling upcoming high-resolution CMB lensing experiments (for instance, successors to Planck or Euclid’s cross-correlation studies) to place stronger empirical constraints on the nature and dynamics of dark energy via lensing observables.
• Validating advanced ray-tracing and map-generation techniques against both semi-analytic theory and N-body realizations—techniques that are critical for reliable lensing statistics on small angular scales.
• Allowing for decomposition and analysis of the respective contributions from coupled CDM and uncoupled baryons; this capability offers a route to breaking degeneracies among competing DE models in future observational data, particularly in cases where coupled and uncoupled matter manifest opposite trends.

7. Summary

Coda Lens defines a comprehensive simulation and analysis methodology for probing the gravitational lensing of the CMB in cosmological models with coupled dark energy and cold dark matter. By deploying ray-tracing algorithms through N-body simulation outputs and explicitly computing lensing integrals, this approach establishes that CDM-DE interactions—encoded as constant coupling parameters—produce measurable modifications to the CMB lensing potential and power spectrum. Standard cDE models enhance the lensing signal due to more vigorous growth and clustering, while bouncing cDE models can exhibit suppression relative to ΛCDM, owing to complex interplay between perturbation growth and background expansion rates. The separate evolution of baryons and CDM, particularly in coupled scenarios, generates observable effects that could help distinguish among dark energy models when analyzed via high-resolution CMB lensing data. Coda Lens thus offers a robust framework for leveraging simulated lensing maps as precision tools in cosmological model selection and parameter constraint.