4. THE STAR FORMATION RELATION IN THE EARLY UNIVERSE

While direct identifications of galaxies giving rise to z > 2 DLAs have proven extremely challenging, critical insights into the star formation relation in the early Universe can still be gained from comparing the incidence of DLAs with the spatial distribution of star formation rate (SFR) per unit area uncovered in deep imaging data (Lanzetta et al 2002, Wolfe and Chen 2006). Specifically, the SFR per unit area (Σ_SFR) is correlated with the surface mass density of neutral gas (Σ_gas), following a Schmidt-Kennicutt relation in nearby galaxies (e.g., Schmidt 1959, Kennicutt 1998). The global star formation relation, Σ_SFR = 2.5 × 10⁻⁴ (Σ_gas / 1 M_⊙ pc⁻²)^1.4 M_⊙ yr⁻¹ kpc⁻² (dashed line in Fig. 5), is established using a sample of local spiral galaxies and nuclear starbursts (solid grey points in Fig. 5) over a broad range of Σ_gas, from Σ_gas ≈ 10 M_⊙ pc⁻² to Σ_gas ≈ 10⁴ M_⊙ pc⁻².

Figure 5. The global star formation relation observed in nearby galaxies and at high redshifts. The correlation between the SFR per unit area (ΣSFR) and the total surface gas mass density (Σgas), combining both atomic (H I) and molecular (H2) for nearby spiral and starburst galaxies are shown in small filled circles (Kennicutt 1998, Graciá-Carpio et al 2008, Leroy et al 2008), together with the best-fit Schmidt-Kennicutt relation shown by the dashed line (Kennicutt 1998). A reduced star formation efficiency is observed both in low surface brightness galaxies and in the outskirts of normal spirals, which are shown in grey star symbols and open triangles, respectively (Wyder et al 2009, Bigiel et al 2010). CO molecules have been detected in many massive starburst galaxies (Mstar > 2.5 × 1010 M⊙) at z = 1−3 (e.g., Baker et al 2004, Genzel et al 2010, Tacconi et al 2013), which occur at the high surface density regime of the global star formation relation (open squares). In contrast, searching for in situ star formation in DLAs has revealed a reduced star formation efficiency in this metal-deficient gas. Specifically, green points and orange shaded area represent the constraints obtained from comparing the sky coverage of low surface brightness emission with the incidence of DLAs (Wolfe and Chen 2006, Rafelski et al 2011, 2016). Cyan squares and red circles represent the limits inferred from imaging searches of galaxies associated with individual DLAs, and the cyan and red bars represent the limiting ΣSFR based on ensemble averages of the two samples (Fumagalli et al 2015). The level of star formation observed in high-N(H I) DLAs (green pentagons and orange shaded area) is comparable to what is seen in nearby low surface brightness galaxies and in the outskirts of normal spirals. See the main text for a detailed discussion

Empirical constraints for a Schmidt-Kennicutt relation at high redshifts require observations of the neutral gas content in star-forming galaxies. Although observations of individual galaxies in H I emission remain out of reach, the sample of z = 1−3 galaxies with resolved CO maps is rapidly growing (e.g., Baker et al 2004, Genzel et al 2010, Tacconi et al (2013)). The observed Σ_SFR versus Σ_molecular for the high-redshift CO detected sample is shown in open squares in Fig. 5, which occur at high surface densities of Σ_molecular ≳ 100 M_⊙ pc⁻². Considering only Σ_molecular is appropriate for these galaxies, because locally it has been shown that at this high surface density regime molecular gas dominates (e.g., Martin and Kennicutt 2001, Wong and Blitz 2002, Bigiel et al 2008). In contrast, DLAs probe neutral gas with N(H I) ranging from N(H I) = 2 × 10²⁰ cm⁻² to N(H I) ≈ 5 × 10²² cm⁻². The range in N(H I) corresponds to a range in surface mass density of atomic gas from Σ_atomic ≈ 2 M_⊙ pc⁻² to Σ_atomic ≳ 200 M_⊙ pc⁻², which is comparable to the global average of total neutral gas surface mass density in local disk galaxies (e.g., Fig. 5). Therefore, DLAs offer an important laboratory for investigating the star formation relation in the distant Universe, and direct constraints can be obtained from searches of in situ star formation in DLAs.

In principle, the Schmidt-Kennicutt relation can be rewritten in terms of N(H I) for pure atomic gas following

(5)

which is justified for regions probed by DLAs with a low molecular gas content (see Sect. 2 and Fig. 3d). For reference, the local Schmidt-Kennicutt relation has K = 2.5 × 10⁻⁴ M_⊙ yr⁻¹ kpc⁻², β = 1.4, and N₀ = 1.25 × 10²⁰ cm⁻² for a pure atomic hydrogen gas. Following Eq. (5), the N(H I) distribution function, f_{N(H I)} (e.g., Fig. 3a), can then be expressed in terms of the Σ_SFR distribution function, h(Σ_SFR), which is the projected proper area per dΣ_SFR interval per comoving volume (Lanzetta et al 2002). The Σ_SFR distribution function h(Σ_SFR) is related to f_{N(H I)} according to h(Σ_SFR) dΣ_SFR = (H₀ / c) f_{N(H I)} dN(H I).

This exercise immediately leads to two important observable quantities. First, the sky covering fraction (C_A) of star-forming regions in the redshift range, [z₁, z₂], with an observed SFR per unit area in the interval of Σ_SFR and Σ_SFR + dΣ_SFR is determined following

(6)

where c is the speed of light and H(z) is the Hubble expansion rate. Equation (6) is equivalent to f_{N(H I)} dN(H I) dX, where dX≡ (1 + z)² H₀ / H(z) dz is the comoving absorption pathlength. In addition, the first moment of h(Σ_SFR) leads to the comoving SFR density (Lanzetta et al 2002, Hopkins et al 2005),

(7)

Constraints on the star formation relation at high redshift, namely K and β in Eq. (5), can then be obtained by comparing f_{N(H I)}-inferred C_A and _* with results from searches of low surface brightness emission in deep galaxy survey data. Furthermore, estimates of missing light in low surface brightness regions can also be obtained using Eq. (7) (e.g., Lanzetta et al 2002, Rafelski et al 2011).

In practice, Eq. (5) is a correct representation only if disks are not well formed and a spherical symmetry applies to the DLAs. For randomly oriented disks, corrections for projection effects are necessary and detailed formalisms are presented in Wolfe and Chen (2006) and Rafelski et al (2011). In addition, the inferred surface brightness of in situ star formation in the DLA gas is extremely low after accounting for the cosmological surface brightness dimming. At z = 2−3, only DLAs at the highest-N(H I) end of f_{N(H I)} are expected to be visible in ultra-deep imaging data (cf. Lanzetta et al 2002, Wolfe and Chen 2006). For example, DLAs of N(H I) > 1.6 × 10²¹ cm⁻² at z ≈ 3 are expected to have V-band (corresponding roughly to rest-frame 1500 Å at z = 3) surface brightness µ_V ≲ 28.4 mag arcsec⁻², assuming the local Schmidt-Kennicutt relation. The expected low surface brightness of UV photons from young stars in high-redshift DLAs dictates the galaxy survey depth necessary to uncover star formation associated with the DLA gas. At N(H I) > 1.6 × 10²¹ cm⁻², roughly 3% of the sky (C_A ≈ 0.03) is expected to be covered by extended low surface brightness emission of µ_V ≲ 28.4 mag arcsec⁻². For comparison, the sky covering fraction of luminous starburst galaxies at z = 2−3 is less than 0.1%.

Available constraints for the star formation efficiency at z = 1−3 are shown in colour symbols in Fig. 5. Specifically, the Hubble Ultra Deep Field (HUDF; Beckwith et al 2006) V-band image offers sufficient depth for detecting objects of µ_V ≈ 28.4 mag arcsec⁻². Under the assumption that DLAs originate in regions distinct from known star-forming galaxies, an exhaustive search for extended low surface brightness emission in the HUDF has uncovered only a small number of these faint objects, far below the expectation from applying the local Schmidt-Kennicutt relation for DLAs of N(H I) > 1.6 × 10²¹ cm⁻² following Eq. (6). Consequently, matching the observed limit on * from these faint objects with expectations from Eq. (7) has led to the conclusion that the star formation efficiency in metal-deficient atomic gas is more than 10 × lower than expectations from the local Schmidt-Kennicutt relation (Wolfe and Chen 2006; green pentagons in Fig. 5).

On the other hand, independent observations of DLA galaxies at z = 2−3 have suggested that these absorbers are associated with typical star-forming galaxies at high redshifts. These include a comparable clustering amplitude of DLAs and these galaxies (e.g., Cooke et al 2006), the findings of a few DLA galaxies with mass and SFR comparable to luminous star-forming galaxies found in deep surveys (e.g., Møller et al 2002, , Christensen et al 2007), and detections of a DLA feature in the ISM of star-forming galaxies (e.g., Pettini et al 2002, Chen et al 2009, Dessauges-Zavadsky et al 2010). If DLAs originate in neutral gas around known star-forming galaxies, then these luminous star-forming galaxies should be more spatially extended than has been realized. Searches for low surface brightness emission in the outskirts of these galaxies based on stacked images have indeed uncovered extended low surface brightness emission out to more than twice the optical extent of a single image. However, repeating the exercise of computing the cumulative _* from Eq. (7) has led to a similar conclusion that the star formation efficiency is more than 10 × lower in metal-deficient atomic gas at z = 1−3 than expectations from the local Schmidt-Kennicutt relation (Rafelski et al 2011, Rafelski et al 2016). The results are shown as the orange shaded area in Fig. 5). In addition, the amount of missing light in the outskirts of these luminous star-forming galaxies is found to be ≈ 10% of what is observed in the core (Rafelski et al 2011).

At the same time, imaging searches of individual DLA galaxies have been conducted for ≈ 30 DLAs identified along QSO sightlines that have high-redshift LLS serving as a natural coronograph to block the background QSO glare, improving the imaging depth in areas immediate to the QSO sightline (Fumagalli et al 2015). These searches have yielded only null results, leading to upper limits on the underlying surface brightness of the DLA galaxies (cyan squares and red circles in Fig. 5). While the survey depth is not sufficient for detecting associated star-forming regions in most DLAs in the survey sample of Fumagalli et al (2015) based on the local Schmidt-Kennicutt relation, the ensemble average is beginning to place interesting limits (cyan and red arrows).

The lack of in situ star formation in DLAs may not be surprising given the low molecular gas content. In the local Universe, it is understood that the Schmidt-Kennicutt relation is driven primarily by molecular gas mass (Σ_molecular), while the surface density of atomic gas (Σ_atomic) “saturates” at ∼ 10 M_⊙ pc⁻² beyond which the gas transitions into the molecular phase (e.g., Martin and Kennicutt 2001, Wong and Blitz 2002, Bigiel et al 2008). As described in Sect. 2 and Fig. 3d, the transitional surface density from atomic to molecular is metallicity dependent. Therefore, the low star formation efficiency observed in DLA gas can be understood as a metallicity-dependent Schmidt-Kennicutt relation. This is qualitatively consistent with the observed low Σ_SFR in nearby low surface brightness galaxies (e.g., Wyder et al 2009; star symbols in Fig. 5) and in the outskirts of normal spirals (e.g., Bigiel et al 2010; open triangles in Fig. 5), where the ISM is found to be metal-poor (e.g., McGaugh 1994, Zaritsky et al 1994, Bresolin et al 2012). Numerical simulations incorporating a metallicity dependence in the H₂ production rate have also confirmed that the observed low star formation efficiency in DLAs can be reproduced in metal-poor gas (e.g., Gnedin and Kravtsov 2010).

A metallicity-dependent Schmidt-Kennicutt relation has wide-ranging implications in extragalactic research, from the physical origin of DLAs at high redshifts, to star formation and chemical enrichment histories in different environments, and to detailed properties of distant galaxies such as morphologies, sizes, and cold gas content. It is clear from Fig. 5 that there exists a significant gap in the gas surface densities, between Σ_gas ≈ 10 M_⊙ pc⁻² probed by these direct DLA galaxy searches and Σ_gas ≈ 100 M_⊙ pc⁻² probed by CO observations of high-redshift starburst systems (open squares in Fig. 5). Continuing efforts targeting high-N(H I) DLAs (and therefore high Σ_gas) at sufficient imaging depths are expected to place critical constraints on the star formation relation in low-metallicity environments at high redshifts. Similarly, spatially resolved maps of star formation and neutral gas at z > 1 to mean surface densities of Σ_SFR < 0.1 M_⊙ yr⁻¹ kpc⁻² and Σ_{atomic, molecular} ≈ 10−100 M_⊙ pc⁻² will bridge the gap of existing observations and offer invaluable insights into the star formation relation in different environments.