THE EVOLUTION OF THE MASS–SIZE RELATION TO z = 3.5 FOR UV-BRIGHT GALAXIES AND SUBMILLIMETER GALAXIES IN THE GOODS-NORTH FIELD

Size determination in the observed near-infrared (rest-frame optical at high z) is more robust in tracing the distribution of stellar mass than the observed optical (rest-frame UV at high z) since the sizes at rest-frame UV can be strongly affected by dust and star formation. Therefore, we use K_s-band images from the Subaru MOIRCS Deep Survey (MODS) in the GOODS-North field (Kajisawa et al. 2006; Ouchi et al. 2007). The imaging observations are performed for the J and K_s bands over ∼112 arcmin² area of the GOODS-N field with a pixel scale of 0.12 arcsec. We used the images reduced by Bouwens et al. (2008). The resulting FWHM of the K_s images is ∼05. The deepest data cover ∼28 arcmin² reaching a 5σ of 25.4 AB mag in K_s band while the other regions are ∼1.3 mag shallower. More details of the images and data reduction can be found in Bouwens et al. (2008).

Sizes of galaxies were estimated by using the GALFIT code (Peng et al. 2002) in a similar procedure used by Trujillo et al. (2007) and Williams et al. (2010). For each galaxy, a square postage stamp of 120 pixels (144) around it was made, and we use a mask to exclude neighboring galaxies from the fit. A range of Sérsic (1968) profile models convolved with the point-spread function (PSF) of the image are fitted to the K_s images of galaxies. The convolved models for each object were compared with the galaxy surface brightness distribution and the best-fit model determined using minimized χ² of the fit. The PSF used by GALFIT was taken from the median of the unsaturated stars over the entire MOIRCS K_s images. We perform a test to check the PSF of the image by measuring sizes of a number of stars in the field listed in the Barger et al. (2008) catalog. The derived sizes of all the stars are found to be ≪0.02 pixel (effectively zero). Therefore, the PSF is a good approximation of a point source. We note that within our K_s-band images, there is a shallower region with seeing of 068. We followed a similar procedure with an appropriate PSF (which is made from the median of stars in this region) to measure sizes of galaxies in this region separately. We also verified that the derived sizes of the shallower region are reliable by its appropriate PSF.

We determine the circularized effective radius $r_{e} = a_{e} \sqrt{(1-\epsilon)}$, from the half-light radius along the semimajor axis a_e and ellipticity () as output by GALFIT. This removes the effects of ellipticity. We allowed the Sérsic index n (which measures the shape of the surface brightness profile of galaxy) to vary between 0.5 and 5 and the effective radius between 0.01 and 60 pixels (00012 and 72). Initial guesses for the effective radius r_e, ellipticity, and position angle were taken from the SExtractor catalog, and magnitude was taken from the original (Barger et al. 2008) catalog, and we set the Sérsic index to 2 initially.

Our results show that the median Sérsic index measured for all galaxies is 2.0 where 50% of measurements lies between 4.0 and 1.0. For galaxies with stellar masses between 10¹⁰ and 10¹¹M_☉, the median Sérsic index is 2.4 where 50% lies between 4.6 and 1.2.

Due to the possibility of color gradients, it is best to use the same rest-frame band for measuring sizes of galaxies at all redshifts. However, only deep K imaging is available, so there is a possibility of systematic effects with redshift. Franx et al. (2008) show that such systematic effects are small and will not significantly affect the results. As an approximation, we corrected sizes to the rest-frame g band using star-forming galaxies from studies of the GOODS-Chandra Deep Field-South (CDF-South; Franx et al. 2008). We derive the best linear fit to the median of the ratio of sizes in the K band and rest-frame g band as a function of redshift and apply this fit ($r_{e, k-\rm band}/r_{e, g-\rm band} = 0.15 z + 0.64$) to galaxies with z ≲ 2 in our sample.

To gauge the accuracy of the measured sizes and reliability of our results, we perform a realistic simulation. About 8500 Sérsic profiles were generated with uniformly distributed random parameters in the ranges of 19 < K_AB < 25, 01 < r_e < 2'', 0.5 < n < 5 and 0 < < 0.8. The mock galaxies were then convolved with the PSF of the image. Finally, we added these galaxies to 144 blank-sky postage stamps randomly taken from our K_s-band image. The structural parameters of the model galaxies were then measured in a manner identical to that used for the actual images.

The results of these simulations are shown in Figures 1 and 2. The left panel of Figure 1 shows that the scatter in the recovered sizes increases with magnitude. The random uncertainties increase significantly after K_AB = 23. However, the systematic errors are very small (<5%) even at the faintest magnitudes. Moreover, the simulation shows that the systematic errors on recovery of the Sérsic index are also very small and random uncertainties increase above K = 23 (right panel of Figure 1). From this result, we limit the sample studied in this paper to galaxies with K ⩽ 23 mag. We note that this limit is for the deeper region in our K-band images. Running a separate simulation for the shallower region, we find that the derived sizes of galaxies with K < 22.7 mag are reliable. Hence, we restricted the sample in this region to this slightly brighter magnitude limit.

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We have also explored how the recovery of size depends on the size itself. As shown in Figure 2, the uncertainties in retrieving sizes depend on both magnitude and size. Systematic offsets at all magnitudes appear to be negligible except for r_e>1.8 arcsec (which corresponds to 14.4 kpc at z ∼ 1) at the faintest magnitudes; however, galaxies this large and faint have not been seen at any redshift.

The random uncertainties increase with size in all magnitude bins. For objects with K_AB ⩽ 20 mag (top panel), the random uncertainties in size recovery are <5%. However, for galaxies with 21 < K_AB < 22 the increasing of random uncertainties with sizes is significant. The increase in random uncertainties at larger sizes could be due to decreasing of surface brightness. However, since we are mainly concerned about the overall sample properties, the lack of systematic errors is more important.

In Figure 3, we show the size distributions for UV-bright galaxies between 10 < log(M_*/M_☉) < 11. The blue histogram shows the measured half-light radii of GALEX/LBGs at z ∼ 1 while the distribution of sizes measured for BM/BX galaxies (z ∼ 2) is shown as a green histogram. It can be seen from this plot that UV-bright galaxies at z ∼ 2 are smaller compared to similar galaxies at z ∼ 1. Specifically, the median half-light radius of BM/BX galaxies in this mass range is 2.68 ± 0.19 kpc significantly smaller than the median effective radii of GALEX/LBGs (4.42 ± 0.52 kpc). This means that sizes of UV-bright galaxies evolve by a median factor of 0.60 ± 0.08 between z ∼ 2 and z ∼ 1. It is harder to measure the evolution for z ∼ 3 to z ∼ 2, as we have four LBGs within mass range of 10 < log(M_*/M_☉) < 11 at z ∼ 3 in our sample. The median half-light radius of these four massive galaxies is 2.22 ± 0.61 kpc, 0.82 ± 0.23 smaller than z ∼ 2 BM/BX galaxies. We perform a power-law fit on the size evolution and find r_e ∝ (1 + z)^{−1.11±0.13} over the range 0.6 ≲ z ≲ 3.5.

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The size–redshift relation for star-forming galaxies with spectroscopic redshifts in the GOODS-North field is shown in Figure 4. Galaxies are split into the four different stellar mass bins shown in the figure. In each panel, the small gray symbols are normal star-forming galaxies with spectroscopic redshifts selected to have specific star formation rates log  sSFR> − 10. Color symbols represent our samples of UV-bright galaxies and SMGs. LBGs (2.7 < z < 3.5) are shown as dark-green triangles, and BM/BX galaxies (1.4 < z < 2.7) are plotted as green squares. GALEX/LBGs at relatively lower redshifts (0.6 < z < 1.4) are plotted in different panels as blue stars. The black symbols at z ∼ 0 are star-forming galaxies from Guo et al. (2009) from SDSS.

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As Figure 4 shows, sizes of UV-bright galaxies at a fixed stellar mass increase toward lower redshifts. There is also a trend that the size evolution may be stronger for the most massive galaxies. For example, UV-bright galaxies with stellar masses 9 < log(M_*/M_☉) < 10 evolve by a median factor of approximately 1.15 ± 0.25 from z ∼ 2 to z ∼ 0.8. However, galaxies with masses 10 < log(M_*/M_☉) < 10.5 grow by a median factor of approximately 1.82 ± 0.25. At higher stellar mass bins (10.5 < log(M_*/M_☉) < 11), this evolution is even stronger: a factor of 2.05 ± 1.26 over the same redshift range. However, selection effects may influence this trend (especially in the low-mass bin) and more complete samples are needed.

SMGs are also shown as pink diamonds in Figure 4; they are discussed further in Section 6.4. Galaxies which have X-ray luminosities >10⁴² erg s⁻¹ (i.e., potential AGN hosts) are marked with open circles. These galaxies have sizes similar to the others. Since AGN is point-like, one would expect galaxies with AGN emission to have smaller half-light radii. Their "normal" rest-frame optical sizes are thus perhaps not strongly influenced. Nonetheless, due to possible effects on the size and mass estimates, X-ray-detected galaxies should be considered low-confidence points.

We now investigate how mass–size distribution of the UV-bright galaxies compares to other galaxy populations. We further check if the relation between stellar mass and size of these galaxies exists to high redshift.

The stellar mass–size distributions of our samples are shown in Figure 5. We have divided our sample into three redshift bins: 0.5 < z < 1.5, 1.5 < z < 2.5, and 2.5 < z < 3.5. UV-bright galaxies are color coded in the same way as in Figure 4. Solid and dotted lines in each panel show the mass–size relation from Shen et al. (2003) for star-forming galaxies and early-type galaxies at z ∼ 0, respectively. The gray symbols are the K-selected sample of galaxies in CDF-South with stellar masses >10^9.8 M_☉ (Franx et al. 2008). Comparing our results with the galaxies from CDF-S allows us to see where our galaxies lie relative to a purely mass-selected sample. Brown solid lines in the middle of each panel show the median effective radii in narrow mass bins for all galaxies in CDF-S. As can be seen, at z ∼ 1 and z ∼ 2, UV-bright galaxies have larger effective radii compared to all galaxies at a given stellar mass. For galaxies between 10¹⁰ and 10¹¹M_☉, the median sizes differ by a factor of 1.89 ± 0.23 and 1.12 ± 0.09 at z ∼ 1 and 2, respectively. As the spectroscopic sample contains only four UV-bright galaxies in this mass range at z ∼ 3, the difference with CDF-S is not well constrained at this redshift (1.27 ± 0.36).

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We further investigate how the size–mass distribution of GOODS-N UV-bright galaxies compares to that of UV-bright galaxies in CDF-S. This would help to verify if the previously reported size–mass relation for a large photometric sample (Franx et al. 2008) is consistent with our spectroscopic sample. We define analogs of the z ∼ 1, 2, and 3 LBGs in CDF-S by their optical colors and magnitudes. Specifically, we selected the analogs of the z ∼ 1 GALEX/LBGs by requiring (B − V) < 0.6 and R ⩽ 23.8. This selection is based on the locus of the GOODS-N GALEX/LBGs in this color–magnitude plane. The analogs of BM/BX galaxies in CDF-S are similarly selected with R < 25.1 and (B − V) < 0.8 and z_phot = 1.5–2.5. We note that the R-band magnitude in both surveys is estimated by the average values of V- and I-band magnitudes. We verified that the spectroscopic samples are effectively selected by these criteria. In order to identify analogs of UV-bright objects in CDF-S at z ∼ 2.5–3.5, we find the location of LBGs in GOODS-N on the color–color (B − V) versus (V − I) diagram and apply the following criteria to the CDF-S galaxies: (1) 0.5 < (B − V) < 1.2 for (V − I) < 0.35; (2) (B − V)>3 × (V − I) + 0.5 and (B − V) < 1.2 for 0.35 < (V − I) < 0.5. We note that these selection criteria do not impose a "U-drop out" criteria, but do select galaxies with similar brightness and UV slope as the usual LBG methods.

The comparison of size–mass distributions for UV-bright galaxies from both fields is shown in Figure 6. In each panel, the color symbols are the median log effective radii of UV-bright galaxies in narrow mass bins, with the blue and green symbols representing UV-bright galaxies in GOODS-N and the brown symbols are the ones in CDF-S. The error bars show the 1σ dispersion. UV-bright samples from both fields follow consistent size–mass relation, further confirming that UV-bright galaxies are on average larger than overall galaxies at a fixed stellar mass. We note that at high redshift (z ∼ 2.5–3.5) the comparison is weak because of the limited number LBGs in GOODS-N.

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The UV-bright galaxies show a weak stellar mass–size relation at z = 1 and z = 2 in Figure 5. To quantify this relation, we fitted a power-law function of the form r_e ∝ M^α to the individual UV-bright galaxies in each redshift bin. The fitting results are plotted as color dashed-dotted lines in Figure 6. At z ∼ 1, the size of GALEX/LBGs in GOODS-N scales with stellar mass as r_e ∝ M^0.19±0.05 and the BM/BX galaxies at z ∼ 2 have r_e ∝ M^0.30±0.06. The uncertainties were estimated using bootstrap resampling. The results are comparable to the relation for late-type galaxies at z ∼ 0 from Shen et al. (2003). They found r_e ∝ M^0.15 for low-mass (log(M) < 10.6) late-type galaxies and steeper relation for high-mass late-type galaxies (r_e ∝ M^0.4). In the highest redshift bin (z ∼ 3), we use UV-bright galaxies from CDF-S to find the best fit (α = 0.32 ±  0.06). Table 2 lists the best-fit power-law parameter to the mass–size relation of UV-bright galaxies. The slopes of the mass–size relation at different redshift bins are consistent and there is no significant evolution. Our results confirm the persistence of the size–mass relation for star-forming galaxies up to high redshift.

It is worth checking whether or not other star-forming populations at high redshift have the same size–mass relation. Therefore, we compare star-forming BzK galaxies (sBzK) with BM/BX galaxies. The size distribution as a function of stellar mass is shown in Figure 7 with the sBzK galaxies plotted as gray filled circles. The medians of log effective radii of sBzK galaxies are overplotted as red squares. BM/BX galaxies and SMGs are marked with the green and pink circles, respectively, and the green dashed-dotted line shows the mass–size relation for BM/BX galaxies. As one can see, the mass–size relation for sBzK galaxies is comparable to the one for BM/BX galaxies. The average offset of median half-light radii of sBzK galaxies to the BM/BX mass–size relation is 0.037 dex. The effective radii of sBzK galaxies scale with stellar mass as r_e ∝ M^0.31±0.09 close to the size–mass relation for UV-bright galaxies. This similarity could be due to the significant overlap between the UV-selected galaxies and sBzK galaxies (e.g., Reddy et al. 2005). We note that stellar masses can have significant uncertainties depending on the methods used to calculate them. We verify that by using the stellar mass estimates from Reddy et al. (2006) for UV-bright galaxies, the size–mass distribution of these galaxies covers the same region in the size–mass plane (see, e.g., the middle panel of Figure 5). Hence, this shows that our results are robust against using different estimates of stellar masses.

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One of the surprising results of Franx et al. (2008) was the tight correlation between stellar mass surface density (M_*/r²_e) and color of galaxies to z ∼ 3. They showed that bluer galaxies have lower surface densities than red quiescent ones. They indicated that the color of galaxies correlates more fundamentally with the stellar mass surface density than the mass.

We show the tight relation between color and stellar mass surface density for both star-forming and quiescent galaxies with M_☉ ∼ 10¹⁰–10¹¹ in Figure 8. In the left panel, the observed color (V₆₀₆ − z₈₅₀) is plotted versus the stellar mass surface density for galaxies at redshift ∼1. The black and gray symbols are galaxies from GOODS-N and CDF-S fields, respectively. The blue stars and brown circles indicate the GALEX (LBGs) and their analogs from CDF-S, respectively. In the right panel, the relation between observed (z₈₅₀–K) and surface density for galaxies at higher redshift ∼2 is illustrated, with BM/BX galaxies in GOODS-N shown by green squares and their analogs in CDF-S as brown circles. As can be seen in this figure, the blue star-forming galaxies have lower surface density than red galaxies at both z ∼ 1 and z ∼ 2, though the correlation between color and surface density is more tight at z ∼ 1 than z ∼ 2. In general, using a sample of galaxies with spectroscopic redshifts confirms that the color of galaxies tightly correlates with surface density and this relation holds out to high redshifts.

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Since GOODS-N includes 14 SMGs with redshifts, we compare their masses and sizes to the UV-selected galaxies at z ∼ 2. As Figure 4 illustrates, the optical rest-frame sizes of the SMGs (pink diamonds) and UV-selected galaxies of similar mass are comparable. The median half-light radius of all SMGs is 2.90 ± 0.45 kpc which is in good agreement with the median optical rest-frame sizes of the BM/BX galaxies over the whole mass range (2.68 ± 0.25 kpc). The median effective radius of SMGs with stellar masses of 10¹⁰–10¹¹ is 2.65 ± 0.56 which is similar to the BM/BX of at the same mass range (2.68 ± 0.19). The errors are computed using bootstrap resampling.

We further compare the size distributions of SMGs and BM/BX galaxies in Figure 9. In the top panel, the histograms show the size distribution of BM/BX galaxies and SMGs. In the bottom panel, the normalized cumulative distribution functions are shown for both samples. As this plot shows, the size distributions of SMGs are comparable with z ∼ 2 UV-bright galaxies and the SMGs follow the same size distribution as BM/BX galaxies in the optical rest-frame with SMGs being slightly larger. According to a K-S test, the probability that two distributions being the same is 72%, suggesting no significant difference between the rest-frame optical sizes of the SMGs and UV-bright galaxies.

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We study the size evolution of galaxies with spectroscopic redshifts between z ∼ 0.5and3.5. Our results are summarized in Figure 10. As this plot illustrates, at each epoch, the median sizes of UV-bright galaxies (color squares) with stellar masses 10 < log(M_*/M_☉) < 11 are larger (by a median factor of 0.45 ± 0.09) than quiescent galaxies in a similar mass range selected from CDF-S (red circles). The red triangle is the median sizes of quiescent galaxies studied in (van Dokkum et al. 2008b, VD08 hereafter) with a median stellar mass of 1.7 × 10¹¹M_☉ and median redshift of 2.3. Table 3 lists the median half-light radii for UV-bright and quiescent galaxies seen in Figure 10. The growth of UV-bright galaxies with time is also illustrated in this figure. The dashed line shows that UV-bright galaxies scaled up their sizes toward lower redshifts as (1 + z)^{−1.11±0.13}.

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Previous studies (Franx et al. 2008; Kauffmann et al. 2003) have reported that there is a correlation between color and stellar mass surface density of galaxies at both low and high redshifts. However, the results based on photometric redshifts can be uncertain especially for star-forming galaxies. By means of our large sample of galaxies with secure redshifts, we have confirmed that the tight correlation between color and stellar mass surface density of galaxies that holds out to high redshifts; as the stellar mass densities of blue star-forming galaxies are smaller than those of red quiescent ones. This verifies that the galaxies with higher specific star formation rates are larger than the ones with lower specific star formation rates.

We have also explored the stellar mass–size distribution for galaxies in our sample. We have confirmed that there is a relation between stellar mass and size for UV-bright galaxies at 0.5 < z < 3.5. The relations are consistent for both spectroscopic and photometric redshift samples. Our results verify that there is no significant evolution of the slope of the size–mass relation to redshift ∼3.5. The evolution of the relation from z = 1 to z = 0 is not well established, as it is not straightforward to select LBGs at z ∼ 0. Some authors find evolution (e.g., Franx et al. 2008; Williams et al. 2010) in the stellar mass–size relation; however, Barden et al. (2005) have reported that the stellar mass–size relation for disk galaxies remains constant from z ∼ 1 to the present. Selection effects likely play a role. Comparing to the theoretical work, the size evolution predicted by Somerville et al. (2008) is somewhat stronger than the evolution we measured here.

Star-forming galaxies at high redshifts can be identified by means of different methods of sample selection and can be assigned as different populations of galaxies (e.g., submillimeter or sBzK) and therefore studied separately. However, many of these galaxy populations may overlap, hence it is worth checking out if they have comparable properties. We, therefore, compare sizes of UV-bright galaxies to other types of star-forming galaxy populations at high redshift, i.e., sBzK galaxies and SMGs. We show that these two populations have comparable sizes to the UV-bright galaxies. This suggests that, in general, star-forming galaxies have larger sizes than quiescent ones without regard to their type. However, sample selection might alter the results. We return to this possibility later, in Section 7.3.

The strong growth of both star-forming and quiescent galaxies has been shown by several authors (Trujillo et al. 2006b; Franx et al. 2008; Williams et al. 2010; Toft et al. 2007). The size growth of star-forming galaxies studied here going as (1 + z)^α, where α = −1.11 ± 0.13, agrees well compared to previous studies where the size evolves with α ∼ −1. For example, Bouwens et al. (2006) found α = −1.1 ± 0.3 for a sample of luminosity-selected dropout galaxies between z ∼ 6 and z ∼ 2.5, while Dahlen et al. (2007) found α = −1.10 ±  0.07 for luminosity-selected disk galaxies over the range z ∼ 1.1–2.2. Williams et al. (2010) also found a similar slope for star-forming galaxies in UDS at z ∼ 0.5–2.

The similarity between sizes of UV-bright galaxies and star-forming BzK galaxies is also shown in our study. The two populations are also compared by Overzier et al. (2010). They showed that rest-frame optical sizes of sBzK galaxies are somewhat larger than UV-selected galaxies. Since BzK galaxies are K-selected and thus tend to represent massive galaxies, it is not surprising that they appear larger. Hence, the sizes must be corrected for differing stellar masses for an accurate comparison to be made.

Our results also show that the SMGs have a median half-light radius of 2.90 ± 0.45 kpc comparable to the rest-frame optical sizes of the BM/BX galaxies. The result is in agreement with a recent study by Swinbank et al. (2010) where they use deep HST I and H-band imaging and show that the rest-frame optical sizes of the SMGs and UV-bright galaxies in GOODS-N are comparable. The typical half-light radius of the SMGs in the H-band (rest-frame optical) measured by Swinbank et al. (2010) is 2.8 ± 0.4 which is consistent with our measurements (see also Targett et al. 2010 for more comparison.). Almaini et al. (2005) also compared sizes of SMGs with LBGs at rest-frame optical and found no clear difference between sizes of the SMGs and LBGs. It is worth noting that SMGs have larger median stellar masses compared to BM/BX galaxies (also mentioned in Swinbank et al. 2010). This suggests higher stellar mass densities for SMGs compared to UV bright galaxies. However, the similarity in sizes between these two galaxy populations at high z does not by itself allow conclusive connections to be drawn between the two populations. The SMGs that are faint or lacking emission lines could be missing from our spectroscopic sample and they may have dramatically different mass or size properties. Therefore, selection effect could bias these results. Spectroscopic observation in near-IR might help to solve this problem; e.g., Kriek et al. (2009a) find a median size of 2.8 kpc for a small sample of near-IR spectroscopically confirmed star-forming galaxies at z ∼ 2.3. Indeed, our measured size evolution is consistent with their spectroscopic sample; however, near-IR spectroscopy over a large redshift range is needed to definitively quantify the size evolution.

With this large spectroscopic sample of high-redshift galaxies, we have removed an important source of uncertainty in the stellar mass–size relation and verified the results from larger samples based on photometric redshifts. However, the spectroscopic catalog we used in this study consists of many different spectroscopic surveys which made it inhomogeneous. Galaxies in this catalog are mostly relatively unobscured star-forming galaxies with emission lines, comparing these to quiescent galaxies can still be a problem because spectroscopy of high-redshift red galaxies is difficult. Therefore, using a larger and homogeneous sample of high-redshift galaxies with secure redshift will allow greatly improved constraints on their evolution. New ground- and space-based IR multi-object spectrographs (WFC3 grism, MMIRS, MOSFIRE) will allow investigations of more complete samples of galaxies and study the correlation between galaxy properties at higher redshifts. Dust-obscured starbursts are also likely to be missed in our UV-bright sample, hence NIR spectroscopy will also permit studies of this important population.