OK, this is a bit longer then usual.
In addition to my work for Euclid, I have been busy over the past months finishing two papers that use the gravitational lensing effect to reconstruct the fluctuations in the matter density field -- also known as the matter power spectrum -- across time. The power spectrum is like a fingerprint of the growth of cosmological structure, reflecting how local over-densities collapse under their own gravity within an expanding universe, while under-densities become further thinned out as time passes. Think of ripples on a lake's surface: they can have different sizes and heights, all overlapping in one coherent structure. The power spectrum quantifies the typical amplitude of these waves for a given wavelength. A perfectly smooth lake has a vanishing power spectrum.
The structure of matter in space grows in detail depending on its composition. For instance, dark matter, which makes up the largest fraction of the matter mix, has no pressure to resist collapse. In contrast, the smaller fraction of ordinary atoms and molecules, known as baryonic matter, generates pressure when sufficiently compressed. Theoretical models for the power spectrum have been developed for various matter compositions and expansion rates and are commonly used in cosmological analyses. However, these models become less accurate at small spatial scales where the influence of baryons and potential deviations from a purely cold dark matter scenario are significant. Similarly, the impact of massive active galaxies torching their neighbourhoods with powerful matter jets remains unclear. Therefore, it is valuable to have a method to directly measure the power spectrum and how it changes with both spatial scale and time.
This is where the reconstruction technique described in our papers is particularly effective. The technique is detailed in one of my previous papers [1], although we use only the part of the technique for constraining the matter power spectrum, completely ignoring galaxy biasing. At its core, the technique exploits the dependence of lensing strength on geometry, which is inferred from the shapes of millions of distant galaxies across the sky. This is achieved through a 'tomographic cosmic-shear analysis', where galaxies are grouped by their distance, or cosmological redshift. In this lensing analysis, which looks back in distance and time, the evolving power spectrum is observed in projection. We then invert the correlations of the small galaxy shape distortions in the sky to recover the original matter power spectrum. Unlike other methods, this technique does not rely on theoretical assumptions about the shape or evolution of the power spectrum. Instead, it infers the power in bands for a number of selected spatial scales, averaged within specific time epochs. While it is anchored to a 'reference power spectrum' for numerical reasons, the measurement can deviate freely from it. The only additional inputs we use are the expected change in lensing effect strength with the distance of the background galaxies and a model for extra correlations that slightly bias the lensing correlations (through the intrinsic alignment of galaxies). While neither of these inputs is known precisely, they are well understood enough to be predicted with quantifiable uncertainty, which is then accounted for as extra uncertainty in our results. Another assumption is the reasonable smoothness of the power spectrum when moving from one spatial scale to the next, achieved by a 'Tikhonov filter'. This filter is our latest addition to the technique compared to [1] and considerably reduces the space of possible solutions in the reconstruction, thereby lowering the noise level in the results. Our tests with simulations have shown that this filter is essential for obtaining useful results with currently available data.
Our data were provided by the Kilo-Degree Survey (KiDS), which recently released its fifth-year (and final!) KiDS-Legacy catalogue of galaxies. The data in this catalogue have been meticulously processed and calibrated to meet modern standards for cosmological lensing analysis. We used this new data set, along with the preceding KiDS-1000 catalogue (which had larger calibration uncertainties and less survey area), to produce two new papers: a KiDS-1000 paper and the more recent KiDS-Legacy paper.
The longer paper [2] provides a detailed look at the algorithm and initial tests, applying the methodology to the KiDS-1000 data. The primary finding that the power spectrum is compatible with structure growth dominated by dark matter after redshift z~1 (approximately the last 8 billion years, give or take 2/3 of the age of the Universe) is not surprising, as it aligns with previous studies. These studies also support our finding that the power spectrum has an amplitude about 10% lower than what is predicted from Cosmic Microwave Background (CMB) studies, which probe a much earlier phase of cosmic evolution. We therefore, too, see the notorious, so-called S8-tension. Similarly, when we compare the CMB prediction for the late-time power spectrum to our measurement, we find that, first, on large scales KiDS-1000 could fit the CMB prediction but, second, on smaller scales (the size of galaxy clusters or smaller), the fit only works if the amplitude is suppressed by up to ~35%. This suggests a suppression effect at small scales could potentially resolve the discrepancy between CMB and lensing results. However, the KiDS-1000 data on its own would prefer a solution with both an overall lower amplitude and no additional suppression or boost (with an upper limit of around 20%).
The KiDS-1000 data become more intriguing when examining potential structure growth driven exclusively by dark matter. As mentioned, when averaged over the past 8 billion years, a low-amplitude (growing) reference model can be chosen to fit the true matter power spectrum without significant deviations. To study changes in these deviations over different periods, we divided the past 8 billion years into three separate epochs and averaged the deviations for each. If there were no changes or suppression, the results would be consistent across all three epochs, and all would align with a 10% lower amplitude. However, our findings show a different story: (i) the most recent epoch (last ~3 billion years, z<0.3) fits the reference model well; (ii) the middle epoch (~3--6 billion years ago, 0.3<z<0.6) clearly shows a preference for less power at smaller scales (down by up to ~50%), while still agreeing with the amplitude at large scales; (iii) the earliest epoch (~6--8 billion years ago, 0.6<z<~1) even disagrees on the large scales, settling on approximately 2.5 times more power. Therefore, there is an inconsistency with the standard cosmology reference. While the variations between the epochs average out over the entire lookback time, they are clearly present during separate epochs. Although the evidence is not definitive (with a significance of roughly 3 sigma), it provides interesting insights into the KiDS-1000 data and possibly the S8-tension. In the paper, we discuss various possible interpretations, from systematic errors in the data or intrinsic alignment assumptions to non-standard cosmologies. However, my personal guess is that non-standard cosmologies would struggle to explain the strongly varying deviations from the standard reference model.
In our most recent paper [3], we have updated the measurements using the latest, improved Legacy-KiDS data which no longer show the S8-tension with CMB results (after a blinded analysis) [5]. The improvements in the data -- volume, reduction, and calibration -- are excellent news because they provide a higher signal-to-noise ratio for our measurements and a greater accuracy in the input data for our reconstruction algorithm. Our concise letter, which presents these findings, repeats the analyses from paper [2] and [4] (which adopt a generic double power-law for the power spectrum or the regularised deprojection technique). We found that the inconsistencies we previously saw in the three-epoch results from the KiDS-1000 data have disappeared, likely due to increased redshift accuracy. All three epochs now agree with the dark-matter-only reference on large scales within a standard cosmology. The reference model, which was again chosen to fit the data, has a higher amplitude that is entirely consistent with the CMB prediction. To our excitement, the average deviations from the dark-matter-only reference over the past 8 billion years show consistency on large scales but also a significant (2.8 sigma) suppression of structure growth by about 30% on small scales. Crucially, this suppression is derived from lensing data alone, without external constraints from the CMB or other cosmological probes. This suppression is also weakly visible in two of the separate epochs (the most recent and the middle) but not in the most distant one, where KiDS-1000 previously indicated a much higher amplitude. This could be a weak indication that the suppression has been increasing over time. If these findings are supported by other surveys, the S8 tension between lensing and CMB results would be resolved. It would also mean that the evolution of the matter power spectrum before and after redshift z~1 can be reconciled within a standard cosmological model, provided there is an additional suppression of structure growth at scales of galaxy clusters or smaller.
The ongoing Euclid Wide Survey will provide an excellent opportunity to test our findings, as one of its key focuses is to produce data for tomographic lensing analyses.
[2] Simon et al., 2025, A&A, 698, A217
[3] Broxterman and Simon et al., 2025, A&A, submitted