Dependence of Galaxy Stellar Properties on the Primordial Spin Factor

To study if and how strongly the stellar properties of galactic subhalos depend on the single parameter initial condition, we utilize the particle snapshots and subhalo catalogs at $z=0$ from the 300 Mpc volume run (TNG300-1) of the IllustrisTNG suite of cosmological hydrodynamical simulations [45, 46, 47, 48, 49, 50], which have already been released to the general public²²2https://www.tng-project.org/data/. The IllustrisTNG 300-1 run, performed on a period box of side length $L_{\rm tot}\equiv 205\,h^{-1}{\rm Mpc}$ for the Planck $\Lambda$CDM cosmology [51], simulated the gravitational and hydrodynamical dynamics of $2500^{3}$ DM particles and equal number of baryonic cells with individual masses, $5.9$ and $1.1$, in unit of $10^{7}\,M_{\odot}$, respectively, by implementing the competent AREPO code [52].

The TNG 300-1 subhalo catalogs contain the substructures of friends-of-friends (FoF) groups identified via the SUBFIND algorithm [53], providing various information on their intrinsic and stellar properties like their comoving positions (${\bf x}_{c}$), total masses ($M_{\rm t}$), stellar masses ($M_{\star}$), stellar formation epochs ($a_{f\star}$), specific star formation rate (sSFR), photometric magnitudes at eight different wavebands ($U,\ B,\ V,\ K,\ g,\ r,\ i,\ z$), stellar metallicities ($Z_{\star}$) as well as the comoving positions and masses of each constituent particle ($\{{\bf x}_{\alpha},\ m_{\alpha}\})$, respectively. Here, the stellar formation epoch, $a_{f\star}$, is defined as the scale factor at the mean age of constituent stars. Among the central subhalos of the TNG FoF groups, only those that contain more than $100$ stellar particles are included in our main sample of the target galaxies [54]. Throughout this paper, we will consider six different stellar properties of each target galaxy, $R_{90\star}$, $R_{50\star}$, $a_{f\star}$, sSFR, $g-r$ colors, and $Z_{\star}$, where $R_{90\star}$ and $R_{50\star}$, denote two different stellar sizes determined as the radial distances which enclose $90\%$ and $50\%$ of $M_{\star}$, being called the stellar 90 and 50 percent-mass radii, respectively.

For each target galaxy in the main sample, we locate the initial positions, $\{{\bf q}_{\alpha}\}$, of its constituent DM and gas particles at the initial redshift $z_{i}=127$, and compute their center of mass, $\bar{\bf q}\equiv M^{-1}_{\rm t}\sum_{\alpha}m_{\alpha}{\bf q}_{\alpha}$ as the protogalactic site. Then, we determine the primordial spin factor, $\tau(\bar{\bf q})$, at each protogalactic site by taking the following procedure:

1. 1.

   Reconstruct the initial raw density contrast field, $\delta({\bf q})$, on $512^{3}$ grid points from the particle snapshot at $z_{i}$ via the cloud-in-cell algorithm.

2. 2.

   Compute the Fourier-space raw density contrast field, $\delta({\bf k})$, by using the fast Fourier transformation (FFT) code, where ${\bf k}$ is the Fourier-space wave vector.

3. 3.

   Compute the density hessian tensor smoothed with a Gaussian kernel on the scale of $R_{f}$ as $H_{ij}({\bf k})\equiv k_{i}k_{j}\delta({\bf k})\exp\left(-k^{2}R^{2}_{f}/2\right)$ where $k\equiv|{\bf k}|$.

4. 4.

   Compute the Gaussian filtered tidal tensor as $T_{ij}({\bf k})=H_{ij}({\bf k})/k^{2}$, which is equivalent to the initial potential hessian tensor.

5. 5.

   Conduct inverse Fourier transformations of the smoothed density hessian and tidal tensor fields into the real space counterparts, $H_{ij}({\bf q})$ and $T_{ij}({\bf q})$, respectively.

6. 6.

   Find the orthonormal eigenvectors of $T_{ij}(\bar{\bf q})$ at each protogalactic site, and express $H_{ij}(\bar{\bf q})$ in the principal frame of $T_{ij}(\bar{\bf q})$ via a similarity transformation

7. 7.

   Compute the primordial spin factor, $\tau(r_{f})$, as

   $$\tau\equiv\left(\frac{H^{2}_{12}+H^{2}_{23}+H^{2}_{31}}{H^{2}_{11}+H^{2}_{22}+H^{2}_{33}}\right)^{1/2}\,.$$

   (2.1)

It is important to note the difference between eq. (2.1) and the original definition of $\tau$ given in our prior works [20, 21]. Basically, eq.(2.1) replaces the protogalaxy inertia tensor, $(I_{ij})$, by the density hessian matrix , $(H_{ij})$, on the ground that $(H_{ij})$ was numerically found to approximate $(I_{ij})$ quite well as long as $R_{f}$ is comparable to the protogalaxy Lagrangian size [10]. There are two main advantages of using $(H_{ij})$ instead of $(I_{ij})$ for $\tau$. First, the former can be uniquely defined unlike the latter, which depends sensitively on the subhalo identification algorithm, i.e., subhalo boundary. Second, it is in principle possible to statistically reconstruct the former from real observational data [31, 10, 27, 28], while the latter is not practically measurable.

The non-negligible value of $\tau$ is the sole initial condition for the first order generation of protogalaxy angular momenta [30]. From here on, this single parameter initial condition, $\tau$, will be called the primordial spin factor, and its mutual information with the six stellar properties of the TNG galaxies at $z=0$ will be determined for three different cases of the smoothing scale, $r_{f}\equiv R_{f}/(h^{-1}{\rm Mpc})=0.5$, $1$, and $2$, in sections 2.2-3.

To evaluate the mutual information (MI) between a galaxy stellar property (representatively say, $S$) and the primordial spin factor ($\tau$), we first divide two dimensional configuration space spanned by $S$ and $\tau$ into a total number of $N_{\rm pixel}$ pixels of small area. Let $N_{g}$ denote the total number of the selected target galaxies in the main sample. Let also $n(a,b)$ denote the number of those target galaxies whose values of $(S,\tau)$ belong to the $(a,b)$th pixel. The ${\rm MI}(S,\tau)$, can be computed as

$${\rm MI}(S,\tau)=\sum_{a=1}^{N_{a}}\sum_{b=1}^{N_{b}}P(S_{a},\tau_{b})\log\frac{P(S_{a},\tau_{b})}{P(S_{a})P(\tau_{b})}\,,$$

(2.2)

Here, $(S_{a},\tau_{b})$ represent the median of $(S,\tau)$ values in the $(a,b)$th pixel, while $P(S_{a},\tau_{b})\equiv n(a,b)/N_{\rm g}$, $P(S_{a})\equiv\sum_{b=1}^{N_{b}}\,n(a,b)/N_{\rm g}$, and $P(\tau_{b})\equiv\sum_{a=1}^{N_{\rm a}}\,n(a,b)/N_{\rm g}$, where $N_{a}$ and $N_{b}$ denote the total numbers of bins into which the ranges of $S$ and $\tau$ values are divided, satisfying $N_{a}N_{b}=N_{\rm pixel}$. A larger amount of ${\rm MI}(S,\tau)$ translates into the existence of a stronger connection between $S$ and $\tau$. To evaluate the statistical significance of MI values, we create $1000$ resamples of the target galaxies by randomly shuffling their $S$ and $\tau$ values, and compute the average and standard deviation over the $1000$ resamples for each property.

Given the existence of strong correlations between $S$ and $M_{\star}$ [55, 56, 57], we classify the main sample into low-, intermediate-, and high-mass galaxies corresponding to three stellar mass ranges of $9\leq m_{\star}<9.5$, $9.5\leq m_{\star}<10$ and $10\leq m_{\star}<11$ with $m_{\star}\equiv M_{\star}/(h^{-1}M_{\odot})$, respectively, and separately compute the mean amounts of ${\rm MI}(S,\tau)$ by eq. (2.2) in each $m_{\star}$-range. Figure 1 plots the amounts of MI (red histogram) from the original sample, and average MI over the shuffled resamples with one standard deviation errors (blue histogram) versus the six stellar properties in the three different ranges of $m_{\star}$ for the three different case of $r_{f}$. As can be seen, for the case of $r_{f}=0.5$, all of the six stellar properties exhibit statistically significant signals of MI in all of the three stellar mass ranges except for the stellar metallicities that show negligibly low MI for the case of high-mass galaxies. The amounts of MI increase as $m_{\star}$ increases and that the largest amount of MI is produced by $R_{90\star}$ in all of the three $m_{\star}$ ranges. The low and intermediate-mass galaxies exhibit an overall trend that the amounts and statistical significances of MI are lower for the case of larger $r_{f}$. Whereas, the high-mass galaxies yield the least significant amounts of MI on the scale of $r_{f}=1$ rather than on $r_{f}=2$, which implies that the initial density and tidal field defined on the larger scales ($r_{f}\geq 2$) also contribute to the establishments of the high-mass galaxy stellar properties.

The galaxy stellar properties are well known to be a function of the total mass, $M_{\rm t}$, of its host DM halo [36, 39]. Furthermore, the DM halos with different total masses were found to have different $\tau$ values [20], which implies that the MI signals displayed in figure 1 could be at least partially contributed by the differences in $M_{\rm t}$ among the galaxies belonging to different $S$-$\tau$ pixels. To nullify a possible effect of mass difference on ${\rm MI}(S,\tau)$, it is necessary to eliminate $M_{\rm t}$ differences depending on the $\tau$ values. Binning the ranges of $\log m_{\rm t}$ and $\tau$, we count the number of galaxies whose values of $m_{\rm t}$ fall in each bin. Then, we look for a $\tau$ bin which yields the lowest galaxy number (say, $n_{\rm min}$) at a given $m_{\rm t}$ bin. Selecting only $n_{\rm min}(\log m_{\rm t})$ galaxies from each of the $\tau$ bins, we end up having a controlled galaxy sample where the effect of mass differences among different pixels disappear. Using this controlled sample, we recalculate the MI values by following the same procedure described in section 2.2.

Figure 2 shows the same as figure 1 but from the $m_{\rm t}$-controlled sample. As can be seen, although the controlled sample tends to yield slightly lower amounts of MI between the galaxy stellar properties and primordial spin factors, the signals are still quite significant for the case of $r_{f}=0.5$, confirming that $m_{\rm t}$-difference among the galaxies in the uncontrolled sample contribute at most only partially to the MI values. Note that the stellar ages and colors of the high-mass galaxies and the two stellar sizes of the low and intermediate-mass galaxies from the $m_{\rm t}$ controlled sample yield even larger and more significant amounts of MI for the cases of $r_{f}=0.5$ and $2$, respectively, than those from the original uncontrolled sample. This result implies that the $m_{\rm t}$ dependences of these stellar properties in fact played the role of veiling their connections with the primordial spin factors.

Recalling that the galaxy stellar properties also exhibit strong variations with environmental density contrasts, $\delta_{\rm nl}$ [37, 38], we further control the target galaxy sample to have identical joint distributions of $(m_{\rm t},\delta_{\rm nl})$, where the local density contrast field $\delta_{\rm nl}$ is reconstructed by applying the cloud-in-cell method to the TNG particle snapshot at $z=0$ in the same manner used for the reconstruction of the initial density field. Figure 3 shows the same as figure 1 but from the $m_{\rm t}$ and $\delta_{\rm nl}$-controlled sample. As can be seen, no drastic change is witnessed after the $\delta_{\rm nl}$ differences are eliminated. In all of the three $m_{\star}$ ranges, the amounts of MI between $R_{90\star}$ and $\tau$ are substantially reduced for the case of $r_{f}=0.5$, while for the low-mass galaxies, the statistical significances of MI between $R_{90\star}$ and $\tau$ is enhanced for the case of $r_{f}=2$.

To be as scrupulous and conservative as possible in detecting a signal of net $\tau$-dependences of the galaxy stellar properties, we control the galaxy sample even more strictly to have identical joint distributions of $(m_{\rm t},\delta_{\rm nl},q)$ among different $(S,\tau)$ pixels, where $q$ is the environmental shear [58] on which the galaxy properties were shown to depend [40, 41, 44], defined as [43]:

$$q\equiv\bigg{\{}\frac{1}{2}\left[(\varrho_{1}-\varrho_{2})^{2}+(\varrho_{2}-\varrho_{3})^{2}+(\varrho_{3}-\varrho_{1})^{2}\right]\bigg{\}}^{1/2}\,,$$

(2.3)

where $\{\varrho_{1},\ \varrho_{2},\ \varrho_{3}\}$ are three eigenvalues of the local tidal field measured at $z=0$. The local tidal field is reconstructed for the determination of $q$ from $\delta_{\rm nl}$ in the same manner used to construct $(T_{ij})$ from $\delta$, under the assumption that the magnitude of vorticity is negligible on the scale $r_{f}\geq 0.5$. Table 1 lists the numbers of the target galaxies included in the controlled samples for the three cases of $r_{f}$ in the three $m_{\star}$-ranges. As can be read, the sizes of the controlled samples depend on $r_{f}$, since the $\tau$ values for a galaxy varies with $r_{f}$.

Figure 4 shows the same as figure 1 but from the $m_{\rm t}$, $\delta_{\rm nl}$ and $q$-controlled sample. As can be seen, the strictly controlled sample still yield significant signals of MI between the galaxy stellar properties and primordial spin factors, despite that possible effects of $M_{\rm t}$, $\delta_{\rm nl}$ and $q$ are all eliminated. It is interesting to notice that it varies with $m_{\star}$ which stellar property among the six has the largest amounts of ${\rm MI}(\tau)$ and how many stellar properties yield statistically significant MI signals. The overall trends are that for the case of $r_{f}=0.5$, the more stellar properties yield stronger signals, as $m_{\star}$ increases, while $R_{90\star}$ is $m_{\star}$-independent signals of MI. Another interesting aspect of the results shown in figure 4 is that the significant signals of ${\rm MI}(\tau)$ seem to be contained in the two stellar sizes of the low and intermediate-mass galaxies, and in $g-r$ colors of the high-mass galaxies for the case of $r_{f}=1$ and $2$, respectively. The results shown in figure 4 compellingly demonstrates that the stellar properties of present galaxies have significant net $\tau$ dependences, independent of total mass and environments, whose establishments should be ascribed to the multi-scale influences of initial density and tidal fields at the protogalactic stages.

Now that among the 90 percent-mass radius, $R_{90\star}$, is found to yield the most robust signals of ${\rm MI}(\tau)$ in all of the three stellar mass ranges, it should be worth investigating what probability density distribution this stellar property follows. Given that the galaxy stellar sizes are strongly correlated with the halo spin parameter, $\lambda$ [8], and that the probability density function of $\lambda$ was found to be approximated much better by a $\Gamma$-distribution with two characteristic parameters [20] rather than by the conventional log-normal model [59], we speculate that $R_{90\star}$ may also be well modeled by a $\Gamma$ distribution. To verify this speculation, we determine the distribution of $R_{90\star}$ by taking the following steps. Divide the range of $R_{90\star}$ into multiple differential intervals of equal size, $\Delta_{90\star}$. Count the numbers, $n_{\rm g}$, of the galaxies whose values of $R_{90\star}$ fall in each interval, and then determine the probability density, $p(R_{90\star})$, at each differential interval as $n_{\rm g}/(N_{\rm g}\Delta_{90\star})$.

Figure 5 plots $p(R_{90\star})$ for the three cases of $m_{\star}$ range, revealing that $p(R_{90\star})$ has a long tail in the large size section ($R_{90\star}>R_{90\star,c}$), but drops rapidly in the opposite section ($R_{90\star}\leq R_{90\star,c}$) with size threshold in the range of $15\leq R_{90\star,c}/{\rm kpc}\leq 20$. Comparing this numerically determined $p(R_{90\star})$ to a $\Gamma$ distribution, we find that unlike the case of $p(\lambda)$, a single $\Gamma$ distribution fails to match simultaneously the stellar size distribution $p(R_{90\star})$ in the whole range of $R_{90\star}$. Instead, a separate treatment of two ranges, $R_{90\star}\leq R_{90\star,c}$ and $R_{90\star}>R_{90\star,c}$, in fitting $p(R_{90\star})$ to a single $\Gamma$ distribution turns out to work well, yielding two different sets of characteristic parameters. In other words, the stellar size distribution, $p(R_{90\star})$, turns out to have two different $\Gamma$ modes, and this fitting result leads us to discover that the following bimodal Gamma distribution describes quite well $p(R_{90\star})$ in the whole range of $R_{90\star}$:

	$\displaystyle p(R_{90\star})$	$\displaystyle=$	$\displaystyle\xi\times\Gamma\left(R_{90\star};k_{1},\theta_{1}\right)+(1-\xi)\times\Gamma\left(R_{90\star};k_{2},\theta_{2}\right)\,,$		(2.4)
	$\displaystyle\Gamma\left(R_{90\star};k_{i},\theta_{i}\right)$	$\displaystyle=$	$\displaystyle\frac{1}{f(k_{i})\theta^{k_{i}}_{i}}R_{90\star}^{k_{i}-1}\exp\left(-\frac{R_{90\star}}{\theta_{i}}\right)\,,\,\,{\rm for}\,\,i\in\{1,2\}$		(2.5)
	$\displaystyle f(k_{i})$	$\displaystyle=$	$\displaystyle\int_{0}^{\infty}t^{k_{i}-1}e^{-t}\,dt\,.$		(2.6)

Here, $\xi$ and $1-\xi$ represent the fractions of the first and second $\Gamma$ distributions with characteristic parameters $(k_{1},\theta_{1})$ and $(k_{2},\theta_{2})$, respectively.

Employing the $\chi^{2}$-statistics³³3Technically, the python package, ${\rm``scipy.optimize.curve\_fit"}$ is utilized for the computation of the best-fit parameters and associated errors for the two $\Gamma$ distributions., we fit the numerically obtained $p(R_{90\star})$ to eqs. (2.4)-(2.6) by adjusting the values of $(k_{1},\ \theta_{1})$ and $(k_{2},\ \theta_{2})$ as well as $\xi$. Figure 5 compares the best-fit bimodal $\Gamma$ distributions (thick solid lines) with the numerical results (red filled circles), confirming its validity in all of the three $m_{\star}$-ranges. Table 2 lists the best-fit parameter values for the three cases of $m_{\star}$ ranges. To physically understand this bimodal nature of $p(R_{90\star})$, recall the previous result that the probability density function of $\tau$ was also found to follow a single $\Gamma$-distribution whose best-fit parameters $(k,\ \theta)$ turned out to be scale dependent [21]. In other words, if $\tau$ is measured from the initial tidal fields smoothed on a different scale, then $p(\tau)$ is described by a single $\Gamma$ distribution with a different set of two parameters. Given this previous finding, we interpret the bimodal nature of $p(R_{90\star})$ as an evidence supporting that the galaxy stellar sizes $R_{90\star}$ are affected by multi-scale initial conditions.