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Integrated Sachs-Wolfe maps from the Gower Street $w$ CDM simulations

This paper presents a validated pipeline using the Gower Street N-body simulations to generate full-sky Integrated Sachs-Wolfe maps for 791 $w$ CDM cosmologies, demonstrating excellent agreement with linear theory and revealing that quintessence models produce higher ISW amplitudes than phantom models.

Original authors: Mina Ghodsi Yengejeh, András Kovács, István Szapudi, István Csabai

Published 2026-05-13

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CC BY 4.0

Original authors: Mina Ghodsi Yengejeh, András Kovács, István Szapudi, István Csabai

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Listening to the Echo of the Universe's Expansion

Imagine the Universe as a giant, expanding balloon. Inside this balloon, there are clumps of matter (like galaxies) and empty spaces (voids). These clumps create "gravity wells"—like deep bowls in a landscape.

The Integrated Sachs-Wolfe (ISW) effect is like a sound echo traveling through this landscape. As light from the Big Bang (the Cosmic Microwave Background, or CMB) travels across the Universe to reach our telescopes, it has to roll down into these gravity wells and then climb back out.

In a normal, steady world: If the landscape didn't change while the light was traveling, the energy the light gains rolling down would be exactly lost when it climbs back up. The net result is zero.
In our expanding Universe: The Universe is stretching (accelerating) because of Dark Energy. This stretching changes the shape of the gravity wells while the light is passing through them. By the time the light climbs out, the well has changed shape, so the light doesn't get its energy back perfectly. It ends up slightly hotter or cooler. This tiny temperature change is the ISW signal.

The Problem: We Need Better Maps

Scientists want to measure this tiny temperature change to understand Dark Energy (the mysterious force pushing the Universe apart). However, the signal is incredibly weak and hard to find. To find it, we need to compare the CMB temperature map with maps of where galaxies are.

To do this accurately, we need to know exactly what the signal should look like under different theories of Dark Energy. Until now, most computer simulations only modeled the simplest version of Dark Energy (called $\Lambda$ CDM). But what if Dark Energy is more complex? What if it changes over time?

The Solution: The "Gower Street" Simulation Suite

The authors of this paper created a new, powerful tool to solve this. They used a massive collection of 791 computer simulations called the Gower Street (GS) suite.

Think of these simulations as 791 different "what-if" universes.

In some universes, Dark Energy is very strong and pushes apart violently (called "phantom" models).
In others, it's weaker and behaves differently (called "quintessence" models).
In the standard one, it behaves exactly as we currently think it does.

The team took these simulations and built a new pipeline (a set of computer instructions) to generate full-sky maps of what the ISW temperature signal would look like in each of these 791 different universes.

What They Did (The Method)

Upgraded the Software: They took an existing software tool (called pyGenISW) that could only handle the standard universe and upgraded it to handle all these weird, alternative universes.
Traced the Light: They simulated light rays traveling through these virtual universes, calculating how the gravity wells changed as the universe expanded.
Created Maps: They turned these calculations into colorful maps showing where the sky would be slightly hotter or cooler for each type of Dark Energy.

What They Found (The Results)

The team tested their new maps to make sure they were accurate.

The Check: They compared their computer-generated maps against strict mathematical formulas. The results matched perfectly, proving their new tool works reliably.
The Comparison: They looked at the three specific examples mentioned in the paper:
- The "Phantom" Universe (Sim 127): Dark Energy is super strong ( $w = -1.79$ ). The gravity wells decay very slowly. The resulting ISW signal on the map is weak (faint colors).
- The "Standard" Universe (Sim 742): Dark Energy is normal ( $w = -1$ ). The signal is medium strength.
- The "Quintessence" Universe (Sim 401): Dark Energy is weaker/different ( $w = -0.34$ ). The gravity wells decay faster. The resulting ISW signal is strong (bright, vivid colors).

Key Finding: The maps showed that universes with "Quintessence" Dark Energy produce a much louder "echo" (stronger ISW signal) than "Phantom" universes. This happens because the gravitational potential (the depth of the gravity wells) decays faster in those models.

Why This Matters

The paper concludes that they have successfully built a reliable toolkit for scientists.

Before, if a scientist wanted to study Dark Energy using the ISW effect, they were mostly stuck with the "Standard" model.
Now, they have a library of 791 different maps. If future telescopes (like Euclid or DESI) find a signal that doesn't match the Standard model, scientists can look through this library to see which "what-if" universe matches the new data.

In short: The authors built a "simulator" that lets us see what the Universe's temperature map would look like if Dark Energy behaved in 791 different ways. They proved the simulator works, and they found that different types of Dark Energy leave very distinct "fingerprints" on the sky.

Technical Summary: Integrated Sachs–Wolfe maps from the Gower Street wCDM simulations

Problem and Motivation
The late-time linear Integrated Sachs–Wolfe (ISW) effect serves as a direct probe of cosmic acceleration and the nature of dark energy. However, detecting these weak secondary temperature anisotropies in the Cosmic Microwave Background (CMB) requires accurate theoretical predictions across a broad range of cosmological models. While previous analyses have largely focused on the standard $\Lambda$ CDM model, the nature of dark energy remains unknown, with theoretical alternatives ranging from phantom ( $w < -1$ ) to quintessence ( $w > -1$ ) models. Existing tools, such as the pyGenISW package, were previously limited to $\Lambda$ CDM cosmologies. To address the need for robust theoretical modeling across a wider dark-energy parameter space, this work aims to generate full-sky ISW maps for a suite of $w$ CDM cosmologies, enabling analyses that go beyond the standard model.

Methodology
The authors extended the open-source pyGenISW package to handle $w$ CDM models and massive neutrinos, moving beyond its previous $\Lambda$ CDM-only limitation. The methodology relies on the Gower Street (GS) simulation suite, which consists of 791 full-sky, dark-matter-only N-body simulations run with pkdgrav3. These simulations span a flat $w$ CDM cosmology with dark energy equation-of-state parameters $-1.79 \lesssim w \lesssim -0.34$ , varying seven cosmological parameters including $\Omega_m$ , $\sigma_8$ , $w$ , and neutrino mass $m_\nu$ .

The ISW map construction pipeline involves the following steps:

Input Data: The pipeline utilizes light-cone catalogs from the GS simulations, converting raw particle counts into density contrast maps for each slice up to redshift $z \simeq 1.0$ .
Theoretical Framework: To compute the necessary expansion and growth histories, the authors implemented three complementary methods for calculating the linear growth factor ( $D(z)$ $D (z)$ ) and growth rate ( $f(z)$ $f (z)$ ):
- An Ordinary Differential Equation (ODE) approach (fiducial), solving the standard perturbation equation.
- A CAMB-based approach, extracting growth factors from the linear matter power spectrum.
- An analytic approximation using the growth-index formalism.
  The ODE approach was validated against the pyCCL library and found to be in excellent agreement.
Map Reconstruction: Using the extended pyGenISW, the pipeline performs a discrete spherical Bessel transform (SBT) on the density contrast maps. This involves projecting the density fields onto a basis of spherical Bessel functions adapted to the finite radial range of the light cone, applying a Neumann boundary condition. The ISW temperature anisotropies are then reconstructed by integrating the time evolution of the gravitational potential along the line of sight.
Validation: The pipeline was validated by comparing measured angular power spectra (ISW auto-correlation, galaxy auto-correlation, and ISW-galaxy cross-correlation) against theoretical predictions from TheoryCL and pyCCL.

Key Contributions

Code Extension: The authors successfully extended pyGenISW to support $w$ CDM cosmologies and massive neutrinos, making the code and resulting ISW data publicly available.
Simulation Suite Utilization: They generated full-sky ISW maps for 777 valid simulations out of the 791 available in the Gower Street suite, covering a wide range of dark energy equations of state.
Validation of Growth Modeling: The study confirmed that the ODE-based approach for calculating growth factors yields results consistent with pyCCL across the relevant redshift range ( $0 \le z \le 1$ ) for all tested $w$ CDM models.

Results

Power Spectrum Agreement: The measured ISW angular power spectra ( $C_{\ell}^{ISW-ISW}$ ) and ISW-density cross-correlations ( $C_{\ell}^{g-ISW}$ ) show excellent agreement with linear theory predictions (computed via pyCCL and TheoryCL) in the multipole range $2 \le \ell \le 200$ . Discrepancies at $\ell < 10$ are attributed to the finite simulation box size, and deficits at $\ell \ge 100$ are linked to resolution effects and the breakdown of linear theory assumptions.
Model Dependence: The ISW signal strength varies significantly with the dark energy equation of state. Quintessence-like models ( $w > -1$ , e.g., Sim 401 with $w \approx -0.34$ ) exhibit higher ISW amplitudes and broader temperature fluctuation distributions compared to phantom models ( $w < -1$ , e.g., Sim 127 with $w \approx -1.79$ ). This is consistent with the enhanced late-time decay of gravitational potentials in quintessence scenarios.
Redshift Evolution: While models with different $w$ values are nearly indistinguishable at very low redshifts ( $z < 0.2$ ) due to similar present-day potential decay rates (controlled primarily by $\Omega_m$ ), clear distinctions emerge at intermediate redshifts ( $0.3 \lesssim z \lesssim 1.0$ ) where the ISW kernel is most sensitive to the growth history.
Density Extrema: Analysis of density extrema (voids and superclusters) reveals that the ISW signal is enhanced in these regions. The quintessence model produces the largest temperature shifts ( $\Delta T_{ISW} \sim \pm 15 \, \mu\text{K}$ ), while the phantom model shows weaker signals ( $\Delta T_{ISW} \sim \pm 5 \, \mu\text{K}$ ).
Power Ratios: By computing the ratio of ISW power spectra relative to a fiducial $\Lambda$ CDM model ( $A_{ISW}^{Tg}$ ), the authors found that models with $w > -1$ and low $\Omega_m$ can produce amplitudes consistent with some observational claims of excess signals (e.g., $A_{ISW} \approx 5$ ), though no model in this suite explains the reported sign-reversal of the ISW signal at specific redshifts.

Significance and Claims
The paper concludes that the consistency between the generated $w$ CDM ISW maps and theoretical predictions confirms the robustness of the extended methodology. The authors assert that this framework establishes a reliable tool for theoretical and observational ISW-Large Scale Structure (LSS) analyses. Specifically, the maps are intended to support:

Covariance calculations for next-generation surveys.
Various map-based statistics, including stacking analyses and simulation-based inference.
Theoretical predictions for the redshift-dependent ISW imprint across a broad parameter space from phantom to quintessence domains.

The work does not claim to resolve existing observational tensions (such as the sign-reversal of the ISW signal) but rather provides the necessary theoretical infrastructure to model ISW signals across diverse cosmologies, facilitating future comparisons with data from surveys like Euclid, DESI, and Rubin-LSST.