Evolution of Mass and Velocity Field in the Cosmic Web: Comparison Between Baryonic and Dark Matter

Three simulations in periodic cubic boxes of 25, 50, and 100 h⁻¹ Mpc were performed in the ΛCDM model with WMAP5 normalization, in which cosmological parameters such as Ω_m = 0.274, Ω_Λ = 0.726, h = 0.705, σ₈ = 0.812, Ω_b = 0.0456, and n_s = 0.96 (Komatsu et al. 2009) were adopted. We refer to the simulations as L025, L050, and L100, respectively, hereafter, with the numbers standing for the box size. The same random seeds and phases are used in generating the initial conditions of the three simulations. The velocity fields of baryonic matter in L025 and L100 have been investigated in ZF15. The simulations were run with the hybrid N-body/hydrodynamic cosmological code WIGEON, which employs the positivity-preserving Weighted Essentially Non-Oscillatory finite differences scheme to solve hydrodynamic equations, incorporating the standard particle-mesh method as the gravitational potential solver (Feng et al. 2004; Zhu et al. 2013). All the simulations were evolved from redshift z = 99 to z = 0 in a 1024³ grid with an equal number of dark matter particles. The space resolutions are 24.4 h⁻¹, 48.8 h⁻¹, and 97.7 h⁻¹ kpc, respectively. The corresponding particle mass resolutions are 1.30 × 10⁶, 1.04 × 10⁷, and 8.32 × 10⁷ M_⊙. A uniform UV background is switched on at z = 11.0. The radiative cooling and heating modules are implemented with a primordial composition (X = 0.76, Y = 0.24) following the method in Theuns et al. (1998). The processes of star formation, AGNs, and their feedback are not included.

To reduce the impact of discreteness, especially in underdense regions, the mass and velocity of dark matter particles are first assigned to a 512³ grid using the cloud-in-cell (CIC) algorithm. The resolution of the CIC grid, R_g, is two grid units of simulations. The resampled density and velocity fields on the CIC grids are then smoothed with Gaussian kernels of one, two, four, and eight CIC grid cells (denoted as smt01, smt02, smt04, and smt08 hereafter) using FFT. For instance, smoothing lengths R_s of 48.8, 97.6, 195, 390 h⁻¹ kpc are applied to L025. The effective smoothing length is given by ${R}_{\mathrm{eff}}=\sqrt{{R}_{g}^{2}+{R}_{s}^{2}}$. The same 512³ grid is also applied in resampling the density and velocity fields of gas, which are then smoothed with the same kernel as the dark matter. The peculiar velocity fields of both dark matter and baryonic matter are decomposed into three components: the curl-free divergence velocity ${{\boldsymbol{v}}}_{\mathrm{div}}$, the divergence-free curl velocity ${{\boldsymbol{v}}}_{\mathrm{curl}}$, and the uniform velocity ${{\boldsymbol{v}}}_{\mathrm{unif}}$ through the Helmholtz–Hodge decomposition (Sagaut & Cambon 2008),

$$\begin{eqnarray}&&{\boldsymbol{v}}={{\boldsymbol{v}}}_{\mathrm{curl}}+{{\boldsymbol{v}}}_{\mathrm{div}}+{{\boldsymbol{v}}}_{\mathrm{unif}},\end{eqnarray} \tag{ 1 }$$

where ${\rm{\nabla }}\times {{\boldsymbol{v}}}_{\mathrm{div}}=0$ and ${\rm{\nabla }}\cdot {{\boldsymbol{v}}}_{\mathrm{div}}={\rm{\nabla }}\cdot {\boldsymbol{v}};$ ${\rm{\nabla }}\cdot {{\boldsymbol{v}}}_{\mathrm{curl}}=0$ and ${\rm{\nabla }}\times {{\boldsymbol{v}}}_{\mathrm{curl}}={\rm{\nabla }}\times {\boldsymbol{v}}$. The uniform component ${{\boldsymbol{v}}}_{\mathrm{unif}}$ is both curl free and divergence free, which is actually negligible in the simulations and will not be discussed here.

The distributions of baryonic and dark matter at the resampled grid cells are classified into four categories of cosmic structures, i.e., voids, sheets/walls, filaments, and clusters/knots, using two identification schemes following Hahn et al. (2007b) and Forero-Romero et al. (2009). The former scheme is based on the density field and the latter is based on the peculiar velocity field of matter. We denote by ${\boldsymbol{r}}$ the comoving coordinates and ${\boldsymbol{x}}=a(t){\boldsymbol{r}}$ the proper coordinates, where a(t) is the cosmic expansion factor. Furthermore, the comoving peculiar velocity ${\boldsymbol{v}}({\boldsymbol{r}},t)=\dot{{\boldsymbol{r}}}$ is related to the physical velocity ${\boldsymbol{u}}$ by $a(t){\boldsymbol{v}}({\boldsymbol{r}},t)={\boldsymbol{u}}-\dot{a}(t){\boldsymbol{r}}$. The first method (d-web hereafter) identifies structures on the basis of the eigenvalues ${\lambda }_{t,1},{\lambda }_{t,2},{\lambda }_{t,3}$ of the tidal tensor, which is defined as the Hessian matrix of the rescaled peculiar gravitational potential ϕ,

$$\begin{eqnarray}&&{T}_{\alpha \beta }=\displaystyle \frac{{\partial }^{2}\phi }{\partial {r}_{\alpha }\partial {r}_{\beta }},\end{eqnarray} \tag{ 2 }$$

where α, β = 1, 2, and 3 denotes the components in the three axes. The peculiar gravitational potential is rescaled by $4\pi G\bar{\rho }(t)$, and obeys ∇² ϕ = δ, where $\bar{\rho }(t)$ is the cosmic mean density of matter and $\delta =(\rho -\bar{\rho })/\bar{\rho }$ is the overdensity field. The mean density and overdensity of baryonic matter, ${\bar{\rho }}_{b}(t),{\delta }_{b}(t)$, and of dark matter, ${\bar{\rho }}_{d}(t),{\delta }_{d}(t)$, are used separately to identify webs in corresponding matter fields. The second method (v-web hereafter) depends on the eigenvalues ${\lambda }_{v,1},{\lambda }_{v,2},{\lambda }_{v,3}$ of the rescaled velocity shear tensor,

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\alpha \beta }=-\displaystyle \frac{1}{2H(z)}\left(\displaystyle \frac{\partial {v}_{\alpha }}{\partial {r}_{\beta }}+\displaystyle \frac{\partial {v}_{\beta }}{\partial {r}_{\alpha }}\right),\end{eqnarray} \tag{ 3 }$$

where H(z) is the Hubble parameter and is used as the rescaling factor. The derivatives of the potential and velocity are performed in Fourier space. We count the number of eigenvalues above λ_th at each CIC grid cell. A cell with a value of 3, 2, 1, or 0 is marked as a cluster, filament, sheet, or void, respectively. The threshold value hence is important for classification results. Forero-Romero et al. (2009) pointed out that λ_th could be estimated in principle, considering the association of the deformation tensor, collapse timescale, and age of universe as well. A precise estimate of λ_th, however, needs to carefully deal with the anisotropic collapse of structures and is not available thus far. Forero-Romero et al. (2009) relaxed the assumption λ_th = 0 used in Hahn et al. (2007b), by taking into account that λ_th is dependent on the collapse timescale, and recommended values of 0.2–0.4 for the d-web. Hoffman et al. (2012) used λ_th ∼ 0.44 for the v-web and a larger value, λ_th ∼ 0.7, for d-web to provide the best match to the visual impression. Here, we investigated the threshold values of 0.2–1.2 for d-web, and ${\lambda }_{v,\mathrm{th}}=0.2\mbox{--}0.4$ for v-web. λ_th is first kept constant in time, and then set to vary with redshift.

Before probing the volume and mass content of the cosmic web, we first provide a direct visual impression of the mass distribution, vorticity, and identified structures in this subsection. Figure 1 displays a projected three-dimensional rendering of the density distribution of baryonic and dark matter, and the vorticity of dark matter in the simulations L025 and L050 at z = 0. A sharp picture of the cosmic web is visualized in the baryonic density field of L025 (the rendering procedure is improved compared to Figure 1 in ZF15 to feature the structures). In comparison, the large-scale structures are less prominent in L050, probably because the lower spatial numerical resolution leads to lower density contrast between the structures in the simulation. It would take more effort to visually identify the large-scale structures in the density field of dark matter, especially in underdense regions. The structures in the vorticity field, $| \omega | =| {\rm{\nabla }}\times v|$, of dark matter, however, are smooth and prominent.

Zoom In Zoom Out Reset image size

The vorticity of dark matter resembles that of baryonic matter (see Figure 1 in Zhu & Feng 2015) in both spatial configuration and strength distribution. The curl motions have developed significantly in the regions surrounding filaments and clusters, showing more extended morphologies than the density field. The evolution of the cosmic web since z = 5.0 in L025 is demonstrated in Figure 2. Vague protosheets and sheets appear at early times. Small-scale filaments can also be found visually at z = 5, when most of them are tenuous. Prominent filaments emerge late, at a redshift between z = 2 and z = 1, mainly built from tenuous predecessors. Evident cluster regions form at around z = 0.5, in agreement with C14.

Zoom In Zoom Out Reset image size

The filaments and sheets identified by d-web at z = 3.0 and z = 0.0 in L025 with a constant λ_th = 0.6 over redshift and R_s = R_g are presented in Figure 3. d-web is capable of finding filaments with various lengths and radii, as well as sheets. The spatial distribution of filaments at z = 3 is significantly different form that at z = 0 on a small scale, while it is similar on a large scale to some extent. Longer smoothing length and varying λ_th at different redshifts may enhance the similarity, as in Bond et al. (2010b). The most prominent filaments are connecting to massive clusters at z = 0. However, most of the filaments at high redshifts do not connect to any clusters. They will coalesce into prominent ones at lower redshifts, e.g., around z = 1. The d-web classification has extracted the filaments from the voids and sheets embracing them, leaving empty tubes (Figures 2(b) and (d)). The sheets are fluffy without distinct boundaries at high redshifts, due to low density contrast with respect to the voids. The sheets in baryonic matter evolve to a much smoother and thinner shape at z = 0. The distribution of sheets on a larger scale at z = 3 is similar to that at z = 0 at a certain level; we will revisit this later with the slice view. The discreteness of cold dark matter particles leads to the identified filaments and sheets appearing relatively rough. The sheets suffer more severely from the undersampling in low-density regions, even after smoothing.

Zoom In Zoom Out Reset image size

Volume and mass content are the most explicit quantities to demonstrate the evolution of the cosmic web (Hahn et al. 2007b; Bond et al. 2010; C14). Table 1 summarizes the mass fraction and mean density of each cosmic web component identified by d-web in our simulations at z = 0 with λ_th = 0.6 and R_s = R_g. A lower fraction of baryonic matter is found to residing in clusters and filaments, which is more significant in L025 with the smallest box size. The mass fraction of gas in filaments is less than that of dark matter by 8% in its absolute value, while for both sheets and voids it is ∼3.6% higher. It implies that more baryonic matter is likely to inhabit the underdense or mildly overdense region at z = 0, which may be attributed to thermal pressure effect and nonthermal turbulent motions. The box size has minor effects on the results on these two global quantities. The mean density of baryon in each environment is usually close to that of dark matter, i.e., 1 + δ > 50 in clusters, and around ∼6, 0.8, and 0.2 in filaments, sheets, and voids respectively, in agreement with the results in recent simulation studies (e.g., Aragon-Calvo et al. 2010; C14). In L025, however, the mean density of baryon in clusters is nearly twice that of dark matter. In comparison, the excursion model in Shen et al. (2006) gives an approximation 36 and 6 times denser than the critical density for filaments and sheets. The median density of filaments of baryonic matter is also lower than the estimated density of WHIMs, i.e., ∼10–30 (Dave et al. 2001; Fang et al. 2010). It is probably because the mean density in the background, i.e., the voids, is only 0.2 times that of the cosmic mean. Meanwhile, more matter will reside in clusters and filaments when the simulation box gets smaller and the resolution is increased.

In Figure 4, we investigate the evolution of the volume and mass fraction in voids, sheets, filaments, and clusters since z = 5.0 in L025 with the d-web scheme. The smoothing length is fixed as R_s = R_g; meanwhile, three values of λ_th, i.e., 0.2, 0.4, and 0.6 are probed. The overall evolutionary histories inferred from the different values of λ_th are similar. The volume fraction of each component varies slightly with time, while for the evolution of the mass fraction, it is clearly more significant. The volume fraction of clusters is less than 1% throughout time, and hence is not plotted in this figure. The exact volume fractions in voids and sheets, as well as the mass fraction in voids at a given redshift, are sensitive to the value of λ_th. A lower λ_th would lead to more cells being identified as parts of filaments and sheets, instead of voids. For λ_th = 0.4 and 0.6, the voids occupy the largest volume share, and then sheets and filaments in turn. However, the volume fraction of sheets is larger than voids at z < 5 for λ_th = 0.2. Forero-Romero et al. (2009) showed that the volume fraction of sheets at z = 0 would be larger than voids for λ_th < 0.25, and a lower λ_th close to 0.0 would make the voids too small and isolated.

Zoom In Zoom Out Reset image size

In terms of the mass fraction, clusters grows gradually with time, while filaments demonstrate rapid growth. On the contrary, the mass content in voids keeps shrinking toward lower redshifts, descreasing more sharply for higher λ_th. The mass fraction in sheets varies slowly at 3 < z < 5 and decreases with time from z ∼ 3. At redshift z = 5, about 33%–45% of the mass resides in sheets, while about 18%–30% and 2%–3% have collapsed into filaments and clusters, respectively. Down to z = 0, more than half of the mass is found in the filaments. For threshold values between 0.2 and 0.6, the filaments become the primary collapsed structures, replacing sheets, as measured by the mass fraction at around z ∼ 2–3. The phenomenon that there is less baryonic matter residing in filaments and clusters occurred as early as z > 3. It is noticeable that the shortage of baryon has been largely remedied in clusters at low redshifts, while the relative difference remains around 10%–20% between z = 0 and z = 5 in filaments. The shortage of gas is slightly more apparent for higher λ_th.

The evolution of the volume and mass fractions in L025 with different Gaussian smoothing lengths for fixed λ_th = 0.4 is demonstrated in Figure 5. The effect of different values of R_s is more significant in the mass fraction than in the volume fraction. A larger R_s will lead to notably more mass in voids, but less mass in filaments. For sheets, however, the dependence of the mass fraction on R_s turns over before and after z ∼ 3. The redshift where filaments take over sheets as the primary structure decreases moderately for longer smoothing lengths, and is around z ∼ 2 for R_s = 195 h⁻¹ kpc. The discrepancy between the mass fraction of baryonic and dark matter in filaments and clusters is narrowed by increasing R_s. For L025, with the space resolution of 24.4 h⁻¹ kpc, the relative difference is reduced to ∼5%–10% while taking R_s = 195 h⁻¹ kpc.

Zoom In Zoom Out Reset image size

Figure 6 presents the evolution of fractions in all three simulations with λ_th = 0.4. Results from L025 with ${R}_{s}=2{R}_{g},4{R}_{g}$ are compared to results from L050 and L100 with R_s = R_g. Both the volume and mass fractions of dark matter in different simulations are very close, once the smoothing lengths are equal. Since we have used the same random seeds and phases to produce the initial conditions, it is obviously not beyond our expectations. In addition, except for slight differences, the deficiency of baryonic matter in filaments and clusters can be seen in all simulations. In terms of the mass fraction, baryonic matter is less than dark matter by ∼5%–15% relatively in filaments and clusters at an effective smoothing scale of R_eff = 0.29 h⁻¹ Mpc in L100.

Zoom In Zoom Out Reset image size

For a short summary, a nice bit of mass has fallen into sheets and protosheets, as well as into filaments, at z = 5. In terms of mass, sheets have been dominant over filaments since z > 2–3 for the d-web web classification scheme with λ_th = 0.2–0.6. The fast increase of the mass fraction in filaments, with decreasing redshifts, is not in conflict with the expectations from the structure growth history in the LCDM universe. Actually, filaments are structures experiencing two-dimensional collapse. The rapid growth of mass in filaments since z ∼ 3 should come from the further collapsing of sheets and accretion from nearby underdense environments as well.

We further investigate the impact of the web classification scheme, by looking into the results of v-web. Figure 7 shows the mass fraction of dark matter in the cosmic web identified by v-web in L025, and the corresponding ratios of baryon to dark matter. ${R}_{s}=2{R}_{g}$ has been used to identify v-web to reduce the noise in undersampled regions. ${\lambda }_{v,\mathrm{th}}=0.2\mbox{--}0.4$ are used, which are smaller than the ${\lambda }_{t,\mathrm{th}}$ in d-web, similar to Hoffman et al. (2012). Clearly, the overall evolution is consistent with d-web. However, relatively more grid cells are identified as sheets and clusters at low redshifts, rather than the voids and filaments identified in d-web. The mass fraction found in filaments (clusters) is moderately lower (higher) for v-web. Moreover, relative to d-web, the time since the filaments surpass sheets in mass fraction is postponed to around z ∼ 2. The deficiency in baryonic matter remains at ∼5%–20%, except for filaments at z = 0. Our results with the two different classification methods are generally consistent with each other.

Zoom In Zoom Out Reset image size

Figure 8 gives the probability distributions of the density of baryonic and dark matter in each cosmic web environment. The overtaking of sheets by voids, filaments by sheets, and clusters by filaments by cell number occurs at around δ ∼ −0.6, 1.0, and 200.0. Each component covers a large density range. The density range in filaments, especially, almost covers the entire density range in the simulation sample. The cluster has the highest peak density, and then the filament, sheet, and void in a decreasing sequence, which is consistent with theoretical expectations and previous studies. Remarkable changes in time are observed regarding the density distribution function in filaments. In particular, the single-peak distribution at high redshifts gradually changes to a double peak distribution at z = 0, while a long tail at the high density end is well-developed at z = 1.

Zoom In Zoom Out Reset image size

Moreover, a plateau appears in the density probability distribution function in sheets, and the covering density range expands progressively with time. The peak value decreases from slightly larger than the cosmic mean (δ > 0) at z = 3 to smaller than the cosmic mean, i.e., δ < 0 at z = 0. The probability distribution in voids has a single-peak pattern the entire time, while the width at half peak grows gradually from high redshifts to low redshifts. The density of cells in voids decreases moderately with time; the peak value is ∼0.30 at z = 3, and about ∼0.06 at the present epoch. A long tail at higher density has also developed in clusters at z = 1, which is where the pdf of the clusters mainly change. The density range of each cosmic morphology environment in L025 is slightly wider than that in L050 and L100, while the probability peak is shallower. As we have argued in the earlier paragraph, the higher resolution in L025 is helpful to develop more refined structures and sharper density contrast in the nonlinear regimes, i.e., resolving higher density peaks and lower density floors.

The mass distribution of dark matter in the cosmic web at z = 0 revealed by d-web and v-web in this work is basically in agreement with the results obtained from the dark matter-only simulation in C14. The characteristic density in each environment presented here (see Figure 8) is also consistent with C14. However, our results with the volume and mass fractions, and their evolution in voids and filaments at z > 0, show different behaviors in comparison with C14 in several aspects. First, in contrast to C14, the volume fraction in voids/sheets is decreasing/increasing with time in our simulation samples. This is partly because the d-web (also v-web) method tends to identify regions with underdeveloped density and velocity fluctuations as voids. Namely, none of the three eigenvalues is larger than λ_th, due to lack of significant collapsing activity. For a constant threshold value, the volume and mass fractions in voids would be higher while going to higher redshifts. Second, the mass fraction in filaments changes slowly toward z = 3.8 in C14, which is significantly different from our results, even in L100, which has a comparable box size. A larger/smaller share of mass has been found in voids/filaments in our samples, becoming more apparent as the redshift increases. The margin between filaments and sheets in terms of mass fraction at z > 0.5 is decreasing going to higher redshifts in C14, albeit in a much slower pace compared to our results. The epoch when sheets surpass filaments in C14 is expected to be much earlier than z ∼ 4. In addition, the probability distribution functions in our identified web are more widespread.

The key factor leading to these differences is the cosmic web identification method. The NEXUS+ scheme used in C14 is a multiscale morphological analysis tool implemented with filtering of the density field in logarithmic space, and is different from d-web and v-web, which have a single-scale filter. Cautun et al. (2013) have shown that the NEXUS+ scheme was successful in capturing filaments and walls with a wide range of sizes. C14 further showed that the mass fraction is insensitive to the identification tracer, i.e., either density field or tidal tensor, as well as velocity divergence, etc., in their NEXUS scheme. However, it has been justified only at z = 0, and its validity is not clear when going to high redshifts. On the other hand, more insightful investigations is needed to set a time-dependent λ_th that is physically meaningful for the d-web and v-web schemes. Though d-web with a fixed λ_th could identify both filaments and walls with different sizes at low and high redshifts in our simulation samples, it would be interesting to implement a time-varying λ_th in our scheme and justify whether it is able to reduce the difference between our statistics and C14.

Figure 9 shows the evolution of the volume and mass fractions of dark matter in L025 with ${\lambda }_{\mathrm{th}}(z)={\lambda }_{\mathrm{th}}(0)/(1+z)$. Generally, d-web provides a better match than v-web. Taking a fixed threshold value of λ_th(0) = 1.2 with ${R}_{s}=2{R}_{g}$ in the d-web scheme, the volume fractions of all structures and the mass fraction of filaments and voids at z = 0 are found to be almost identical to those in C14. The mass fractions in sheets and clusters, however, still shows differences of about 5% when comparing with C14. At z > 0, once a time-dependent λ_th(z) is applied, the consistency of the overall evolution trend between d-web and C14 in both mass and volume fractions is somewhat improved, whereas there still exist distinct differences in the absolute value of the volume and mass fractions. The d-web scheme predicts relatively higher mass fractions in sheets and clusters, but lower fractions in filaments at high redshifts, in contrast to C14.

Zoom In Zoom Out Reset image size

The remaining discrepancy can hardly be reduced by adjusting either ${\lambda }_{\mathrm{th},0}$ or smoothing length. The numerical results with an alternative monotonic λ_th(z), ${\lambda }_{\mathrm{th}}(0)\ast {H}_{0}/H(z)$, are given in Figure 10 and show slight changes. The differences in volume and mass fractions between d-web and C14 are apparent, especially at high redshifts. It should result from the different kernels we have used in the identification of cosmic web components. In addition, the d-web scheme with time-varying λ_th also predicts a turnover redshift of z ∼ 2–3, at which the universe goes through a transition from being sheet dominated to filament dominated, in agreement with results presented in the last subsection where a fixed λ_th was used.

Zoom In Zoom Out Reset image size

Before studying the properties of velocity in the cosmic web, we first take a look at the divergence and vorticity field of baryonic and dark matter. Figure 11 displays the spatial distributions of divergence and projected vorticity ω with ${R}_{s}=2{R}_{g}$ in a slice at z = 0; both have been rescaled to enhance the contrast. Specifically, ${sgn}{(\omega )(| \omega | t)}^{1/4}$ and ${sgn}{({\rm{\nabla }}\cdot {\boldsymbol{v}})(| {\rm{\nabla }}\cdot {\boldsymbol{v}}| t)}^{1/4}$ are plotted, where t is the cosmic time. The structure in the baryonic matter looks smoother than that in the dark matter. For the divergence, it makes a transition from positive to negative, usually at the boundaries between voids and overdense structures, which enclose its minima (negative) in mildly overdense regions. It is just what we expect from simple physics: the collapsing process will slow down and even cease in the central region of collapsed structures, where the relaxing process takes place. The vorticity is well-developed within multistreaming regions. Evident signs of quadripolar multistreaming regions, which was first demonstrated by Laigle et al. (2015), can be found in baryonic matter. Furthermore, more complex multistreams can be located at the cross-sections of prominent structures. The pattern in dark matter is somewhat vague in the overdense region and shows relatively more small-scale structures.

Zoom In Zoom Out Reset image size

In order to manifest the growth of divergence in the evolutionary background of the cosmic web, we show the distribution of density, divergence, and web classification of baryonic matter in the same slice from z = 3 to z = 1.0 in Figure 12. Voids, sheets, filaments, and clusters are represented by dark blue, light blue, yellow, and red, respectively. The collapsed structures are mainly sheets and filaments in this slice. Numerous sheets with varying sizes have formed by z = 3. At the same epoch, the associated divergence has developed substantially. Many of the sheets at early time have disturbed boundaries as the three-dimensional rendering has illustrated in Figure 3. Afterwards, small-size sheets would either consolidate with neighbors of similar scale or be absorbed by their large-size kin in time. The right column of Figure 12 shows that the spatial distribution of sheets and voids on a large scale has been set up by z = 3 and experienced slight change at a later time, indicating that the outline of the cosmic web may have formed at z > 3, confirming our interpretation of Figure 3. The divergence is gradually enhanced around the prominent sheets, while the configuration on a large scale is slightly changed. We note that at z = 1.0, a portion of the cells identified as sheets has positive divergence, about $({\rm{\nabla }}\cdot {\boldsymbol{v}})t\lt 0.1$. These cells are lying adjacent to prominent sheets and filaments, where the potential tidal field should have a complex structure. Cells with one eigenvalue large than λ_th and two negative eigenvalues might be classified as sheets by d-web, i.e., collapsing in one dimension while expanding in the other two dimensions. The region belonging to filaments is much less than sheets at z = 3, in good agreement with our volume and mass fraction statistics presented in the last section. Afterwards, filaments keep emerging and growing, usually in the cross region between multisheets. In comparison with Figure 11, the divergence mildly evolved between z = 1 and z = 0 in sheets. Significant changes only take place in filaments having large cross-sections.

Zoom In Zoom Out Reset image size

Figure 13 demonstrates the rapid growth of vorticity of gas in the same slice since z = 2.0. Weak curl motions would develop in the sandwich layer of sheets that is well confined in the normal direction and is the place where the first shell-crossing happens. Quadripolar multistreaming regions tend to arise at the intersections of multisheets, i.e., the seeds of filaments. Moderate vorticity has developed in a few filamentary regions at z = 2.0. The vorticity is boosted significantly when matter from multiflows keeps collapsing and swirling around filaments. The baroclinicity produced in strong curved shocks surrounding filaments should be the driving factor for gas (Zhu et al. 2013). The sign of the quadripolar multiflow in isolated filaments became more evident at z = 1.0 and evolved moderately afterwards, in sync with the growth of filaments. Even more complex patterns of vorticity are found in the cross-sections of prominent filaments. The curl motions in the progenitor of the prominent filaments at the position (x = 0.50, y = 0.90) were already more complex than a quadripolar pattern at z = 1.0. Matter should have been accreted into filaments along multisheets, as well as directly from nearby voids. The rapid growth of vorticity is in sync with the dramatic rise of filaments at z < 2–3.

Zoom In Zoom Out Reset image size

In Figure 14, we plot the probability distribution function (pdf) of vorticity rescaled by the cosmic time, i.e., $| \omega | t$ measured in L025 at z = 0 with different smoothing lengths. The vorticities of both baryonic and dark matter are found to decrease with increasing smoothing lengths, in agreement with Libeskind et al. (2014). Vortical motions are effectively boosted in the nonlinear regime, but a large smoothing length would reduce the nonlinearity. The vorticity of baryonic matter exhibits a stronger heavy-tailed distribution pattern than dark matter, suggesting that the vorticity of baryonic matter can be developed more effectively in the highly nonlinear regimes. The pdf of the vorticity in baryonic matter at z = 0 is likely to be described by a log-normal distribution (Zhu et al. 2010). We also compare the distribution in each cosmic environment in the top right plot of Figure 14. Consistent with ZF15, the vorticities are triggered and pumped up in sheets, and especially in filaments, and then dissipated in clusters for both baryonic and dark matter. The vorticity of dark matter in the voids is relatively higher than that in baryonic matter, as displayed in Figure 11. Actually, it might be due to the undersampling issue of dark matter particles in voids. Moreover, the growth of the vorticity since z = 3 can be clearly seen in the bottom left plot of Figure 14. During the epoch between z = 2 and z = 1, the vortical component of the velocity field experiences the largest gains. Meanwhile, the difference between the vorticities of baryonic and dark matter widened. The bottom right plot in Figure 14 compares the pdf of the vorticity of baryonic matter at z = 0 in three simulations. Obviously, as the cosmic vorticity is a typical nonlinear feature emerging from gravitational collapse, higher spatial resolution is capable of resolving higher nonlinearity, and hence the global vorticity of L025-smt01 is the highest. On the other hand, for the same smoothing length, a larger simulation box could host more collapsed structures and have relatively more cells with very large vorticities, i.e., a longer tail in the pdf. For instance, L050-smt01 has more cells with $| \omega | t\gt 40$ than L025-smt02.

Zoom In Zoom Out Reset image size

The value of velocity can directly characterize the flows corresponding to divergence and vorticity. The mean divergence velocities ${{\boldsymbol{v}}}_{\mathrm{div}}$ as a function of density in each web component in L050 since z = 3.0 are shown Figure 15. The magnitude of divergence velocity in each structure was set up by z = 3 and changed mildly since then, mainly between z = 3 and z = 1. It is not beyond our expectation, as the outline of the sheets and voids on a large scale is found, in previous paragraphs, to already be in place by z = 3.0. In addition, the divergence velocity of matter evolves slowly with respect to the density in each environment until δ ∼ 100, during the formation of cosmic web. At high overdensity regimes, the small number of grid cells leads to significant dispersion. The level of divergence motions in sheets and filaments is marginally higher than in voids and clusters, consistent with the divergence distribution given in Figure 12. For a particular box size, e.g., L050, the magnitude is nearly the same in the four morphology environments, around 100 km s⁻¹. Notable differences between baryonic matter and dark matter can only be found in high density regions, δ > 30, where the dispersion due to the limited number of cells becomes evident. The distribution of ${{\boldsymbol{v}}}_{\mathrm{div}}$ in the filaments of all three simulations is also compared in Figure 15. The largest mean divergence velocity is found in L100, and can be about twice the value in L025, because more massive structures can form in a larger simulation box and hence result in a higher bulk velocity.

Zoom In Zoom Out Reset image size

Similarly, Figure 16 gives the distribution and evolution of the mean curl velocity, v_curl, in the comic web. The curl motions are effectively developed when the matter flows into the central layer of sheets and into filaments, and then experiences dissipation when flowing into clusters, basically confirming our previous result in ZF15 and the slice view in Figure 12. The resampling and smooth processes in this work may have partly blurred the velocity fields at the boundaries between sheets and filaments for δ > 1, resulting in the fast growth of curl motions in sheets with respect to density, compared to ZF15. On the other hand, Figure 8 shows that most of the cells with 1 < δ < ∼100 belong to filaments. Hence, the curl velocity in the cosmic web is mainly contributed by filaments, which conforms with the visual evolution presented in Figure 13.

Zoom In Zoom Out Reset image size

The magnitude of v_curl has increased rapidly for δ ≤ 20–30 since z = 3. At z = 0, the curl velocity is ∼10% of the divergence velocity at δ ∼ 0, then increases to ∼30% at δ ∼ 3–5, and eventually becomes comparable to the divergence velocity at δ ∼ 20–30. This evolution pattern is basically consistent with that in Libeskind et al. (2014), which shows that the vorticity of dark matter grows rapidly as ω ∝ ρ for ρ < 10, and turns to ω ∝ ρ^1/3 for ρ > 10. In filaments, the curl velocities are pumped up to ∼100 km s⁻¹ for δ ≳ 30 at z = 0. The curl velocity of baryonic matter is moderately higher than that of dark matter in sheets and filaments within the overdensity range 1–30. The effect of box size on the curl velocity is the same as the effect on divergence velocity. A valley at δ ∼ 0 is presented in the sheets and filaments for both modes of motion, which is more evident for v_curl at high redshifts. The valley is observed in all three simulations. A possible explanation is that a fraction of matter with δ ∼ 0 in the sheets and filaments have not experienced shell-crossing or curved shocks for dark and baryonic matter, respectively; the fraction decreases with time because of continuous anisotropic gravitational collapse.

The power spectrum of velocity at different times reflects the spatial correlation of motions associated with the evolving cosmic web. In Figure 17, we plot the compensated power spectrum ${k}^{2}P(k)$ of the total, curl, and divergence velocities. For the sake of clarity, the comoving velocity rather than the proper velocity is used in this figure to separate lines at different redshifts. On large scales, namely, in the linear regime, the power spectrum is dominated by the divergence component, in agreement with theoretical expectation and a previous study (Zheng et al. 2013). The dominance of irrotational velocity can be extended to the mildly nonlinear regime, i.e., around a few megaparsecs. Actually, the power spectrum of the divergence velocity has been well-developed at >2 Mpc by z = 3. Combined with the exploration on mass and velocity distribution given in the preceding sections, we conclude that the outline of the cosmic web above ∼2–3 Mpc has been formed by z = 3, when the sheets served as the frame. The power spectrum of the curl velocity increases rapidly from z = 3.0, in accordance with the growth of curl motions that largely developed in filament components. At z = 0.0, the contribution from the curl motion reaches 10% at a scale of ∼5 Mpc, and can exceed the divergence velocity at scales smaller than ∼2 Mpc for the baryonic matter. The power spectrum of both modes drops dramatically at a few times the resampling smoothing length.

Zoom In Zoom Out Reset image size

The relative contribution of curl velocity in dark matter is weaker than baryonic matter, remaining subordinate to irrotational velocity in the highly nonlinear regime. Pichon & Bernardeau (1999) predicted that the curl motion will become significant at scales ∼3–4 Mpc after the shell-crossing in large-scale caustics. The predicted scale is in good agreement with our simulations. Using a set of cold dark-matter-only simulations, Zheng et al. (2013) demonstrated that the power spectrum of curl motions would equal to the divergence motions at k ∼ 10 h/Mpc, comparable with our result in L100. They also expect that the contribution of curl velocity would be twice that of divergence at sufficiently small scales, which is hard to verify for dark matter due to numerical factors related to the discrete particle, limited resolution, and smooth processes. For the baryonic matter in our simulations, the ratio of the contribution of the curl velocity to divergence velocity reaches a peak value of ∼1.5 at the scales ∼1.3, 0.8, and 0.4 Mpc in L100, L050, and L025, respectively, i.e., close to the expectation in Zheng et al. (2013).

In the nonlinear regime, the power spectrum of velocity approximates a power law within a particular scale range for both baryonic and dark matter. For the baryonic matter, the upper end is about the same, ∼3 Mpc in three simulations. The lower end varies with the resolution, extended to ∼0.3 Mpc in L025. The growth of the upper end of the scale is in sync with the evolution of vorticity in filaments presented in Figure 8. The upper end of the scale range for dark matter is smaller than that for baryonic matter. In the highly nonlinear regime, the anisotropic collapsing of gas would develop numerous strong curved shocks, mostly associated with filaments, and hence a supersonic intermittent velocity field exhibiting P(k) ∝ k⁻² law in the power spectrum (e.g., Porter et al. 1992; Kritsuk et al. 2007; Zhu et al. 2013). After the shell-crossing in caustics, the curl motions of dark matter in filaments may have developed similar intermittent behavior.

The growth of the magnitude and power spectrum of the curl velocity indicates a close connection between matter flows and cosmic web. Given that the vorticity in filaments may play a very important role in building the spins of halo and galaxy (e.g., Laigle et al. 2015), we carry a further check on the correlations between vorticity and filaments through the study of velocity structure functions and fractal spatial dimensions. The velocity structure functions can give a more insightful view of the coherence of curl motions in sheets and filaments. In ZF15, we found that the velocity structure functions of curl motions of baryonic matter can be described by the intermittent model proposed by She & Leveque (1994) (see also Dubrulle 1994) in the supersonic-scale range. The velocity structure function reads as

$$\begin{eqnarray}&&{S}_{p}(r)=\langle \delta {v}_{r}^{p}\rangle =\langle | v(x)-v(x+r){| }^{p}\rangle \ \propto \ {r}^{{\zeta }_{p}},\end{eqnarray} \tag{ 4 }$$

where p = 1, 2, 3, .... She & Leveque (1994) and Dubrulle (1994) suggested that the relative scaling exponents Z_p would follow a universal scaling law, i.e.,

$$\begin{eqnarray}&&{Z}_{p}=\displaystyle \frac{{\zeta }_{p}}{{\zeta }_{3}}=(1-{\rm{\Delta }})\displaystyle \frac{p}{3}+\displaystyle \frac{{\rm{\Delta }}}{1-\beta }(1-{\beta }^{p/3}),\end{eqnarray} \tag{ 5 }$$

where β and Δ are related to the hierarchy properties of intermittent structures (more details can be found in Dubrulle 1994 and ZF15).

We apply the same analysis to the resampled curl velocity fields of both baryonic and cold dark matter in all three simulations. Figure 18 shows the longitudinal and transverse structure functions, ${S}_{p}^{l}$ and ${S}_{p}^{t}$, at z = 2 and z = 0 in L050. In agreement with ZF15, the structure functions are broken into two distinctive sections, where the breaking occurs at the scale that marks the turnover in the compensated power spectrum of the curl velocity. The breaking scale is almost the same for baryonic and dark matter. In other words, the curl motions show significant coherence below the breaking scale denoted by k_b, strongly confirming the results from the power spectrum study.

Zoom In Zoom Out Reset image size

The fractal dimensions of the most singular structures, derived from the curl velocity structure functions, are given in Figure 19, where d^l and d^t are from ${S}_{p}^{l}$ and ${S}_{p}^{t}$, respectively. According to the SL model, the most singular structures of the vortical motions have dimensions of 1.85–2.15 and 1.75–2.05 for baryonic and dark matter, respectively. The d^l and d^t in dark matter are smaller than the values in filaments in C14. On the other hand, the fractal dimensions of filaments and sheets can be measured using the box-counting method (Mandelbrot 1983). Figure 20 presents the results for dark matter in L025; the values are slightly larger than those for baryonic matter in ZF15. A transition of the mass distribution at the scale around k_b is also observed in the filaments. The fractal dimensions of curl motions is close to the fractal dimensions of filaments below k_b. In other words, the velocity power spectrum, structure functions, and fractal dimension of mass in filaments show an analogous transition over k_b, and the value of the fractal dimensions of curl motions and mass in filaments is similar. Combining the growth history of curl motions revealed in Section 4.2 and in ZF15, we note that the vortical kinetic energy is primarily associated with filaments for both baryonic and dark matter. The statistical properties investigated in our simulations indicate that the curl motions of cosmic matter is mainly driven by the formation of filaments, usually from further collapse within sheets. This result is consistent with that of Laigle et al. (2015), in which they claimed that vorticity in large-scale structures tends to be confined to, and predominantly aligned with, filaments.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size