The wavelet analysis technique is a powerful tool and is widely used in broad disciplines of engineering, technology, and sciences. In this work, we present a novel scheme of constructing continuous wavelet functions, in which the wavelet functions are obtained by taking the first derivative of smoothing functions with respect to the scale parameter. Due to this wavelet constructing scheme, the inverse transforms are only one-dimensional integrations with respect to the scale parameter, and hence the continuous wavelet transforms (CWTs) constructed in this way are more ready to use than the usual scheme. We then apply the Gaussian-derived wavelet constructed by our scheme to computations of the density power spectrum for dark matter, the velocity power spectrum and the kinetic energy spectrum for baryonic fluid. These computations exhibit the convenience and strength of the CWTs. The transforms are very easy to perform, and we believe that the simplicity of our wavelet scheme will make CWTs very useful in practice.

The spatial distribution between dark matter and baryonic matter of the Universe is biased or deviates from each other. In this work, by comparing the results derived from IllustrisTNG and WIGEON simulations, we find that many results obtained from TNG are similar to those from WIGEON data, but differences between the two simulations do exist. For the ratio of density power spectrum between dark matter and baryonic matter, as scales become smaller and smaller, the power spectra for baryons are increasingly suppressed for WIGEON simulations; while for TNG simulations, the suppression stops at $k=15-20\, {h {\rm Mpc}^{-1}}$, and the power spectrum ratios increase when $k\gt 20\, {h {\rm Mpc}^{-1}}$. The suppression of power ratio for WIGEON is also redshift-dependent. From z = 1 to z = 0, the power ratio decreases from about 70 per cent to less than 50 per cent at $k=8\, {h {\rm Mpc}^{-1}}$. For TNG simulation, the suppression of power ratio is enhanced with decreasing redshifts in the scale range $k\gt 4\, {h {\rm Mpc}^{-1}}$, but is nearly unchanged with redshifts in $k\lt 4\, {h {\rm Mpc}^{-1}}$. These results indicate that turbulent heating can also have the consequence to suppress the power ratio between baryons and dark matter. Regarding the power suppression for TNG simulations as the norm, the power suppression by turbulence for WIGEON simulations is roughly estimated to be 45 per cent at $k=2\, {h {\rm Mpc}^{-1}}$, and gradually increases to 69 per cent at $k=8\, {h {\rm Mpc}^{-1}}$, indicating the impact of turbulence on the cosmic baryons are more significant on small scales.

Continuous wavelet analysis has been increasingly employed in various fields of science and engineering due to its remarkable ability to maintain optimal resolution in both space and scale. Here, we introduce wavelet-based statistics, including the wavelet power spectrum, wavelet cross correlation, and wavelet bicoherence, to analyze the large-scale clustering of matter. For this purpose, we perform wavelet transforms on the density distribution obtained from the one-dimensional Zel’dovich approximation and then measure the wavelet power spectra and wavelet bicoherences of this density distribution. Our results suggest that the wavelet power spectrum and wavelet bicoherence can identify the effects of local environments on the clustering at different scales. Moreover, we apply the statistics based on the three-dimensional isotropic wavelet to the IllustrisTNG simulation at z = 0, and investigate the environmental dependence of the matter clustering. We find that the clustering strength of the total matter increases with increasing local density except on the largest scales. Besides, we notice that the gas traces dark matter better than stars on large scales in all environments. On small scales, the cross correlation between the dark matter and gas first decreases and then increases with increasing density. This is related to the impacts of the active galactic nucleus feedback on the matter distribution, which also varies with the density environment in a similar trend to the cross correlation between dark matter and gas. Our findings are qualitatively consistent with previous studies on matter clustering.

In this work, we propose new statistical tools that are capable of characterizing the simultaneous dependence of dark matter and gas clustering on the scale and the density environment, and these are the environment-dependent wavelet power spectrum (env-WPS), the environment-dependent bias function (env-bias), and the environment-dependent wavelet cross-correlation function (env-WCC). These statistics are applied to the dark matter and baryonic gas density fields of the TNG100-1 simulation at redshifts of z=3.0-0.0, and to Illustris-1 and SIMBA at z = 0. The measurements of the env-WPSs suggest that the clustering strengths of both the dark matter and the gas increase with increasing density, while that of a Gaussian field shows no density dependence. By measuring the env-bias and env-WCC, we find that they vary significantly with the environment, scale, and redshift. A noteworthy feature is that at z = 0.0, the gas is less biased in denser environments of Δ ≳ 10 around 3 h Mpc−1, due to the gas reaccretion caused by the decreased AGN feedback strength at lower redshifts. We also find that the gas correlates more tightly with the dark matter in both the most dense and underdense environments than in other environments at all epochs. Even at z = 0, the env-WCC is greater than 0.9 in Δ ≳ 200 and Δ ≲ 0.1 at scales of k ≲ 10 h Mpc−1. In summary, our results support the local density environment having a non-negligible impact on the deviations between dark matter and gas distributions up to large scales.