Low-mass Population III Star Formation due to the HD Cooling Induced by Weak Lyman–Werner Radiation

Lyman–Werner (LW) radiation photodissociating molecular hydrogen (H2) influences the thermal and dynamical evolution of the Population III (Pop III) star-forming gas cloud. The effect of powerful LW radiation has been well investigated in the context of supermassive black hole formation in the early Universe. However, the average intensity in the early Universe is several orders of magnitude lower. For a comprehensive study, we investigate the effects of LW radiation at 18 different intensities, ranging from J LW/J 21 = 0 (no radiation) to 30 (H-cooling cloud), on the primordial star-forming gas cloud obtained from a three-dimensional cosmological simulation. The overall trend with increasing radiation intensity is a gradual increase in the gas cloud temperature, consistent with previous works. Due to the HD cooling, however, the dependence of gas cloud temperature on J LW deviates from the aforementioned increasing trend for a specific range of intensities (J LW/J 21 = 0.025–0.09). In HD-cooling clouds, the temperature remained below 200 K during 105 yr after the first formation of the high-density region, maintaining a low accretion rate. Finally, the HD-cooling clouds have only a low-mass dense core (above 108 cm−3) with about 1–16 M ⊙, inside of which a low-mass Pop III star with ≤0.8 M ⊙ (a so-called “surviving star”) could form. The upper limit of star formation efficiency Mcore/Mvir,gas significantly decreases from 10−3 to 10−5 as HD cooling becomes effective. This tendency indicates that, whereas the total gas mass in the host halo increases with the LW radiation intensity, the total Pop III stellar mass does not increase similarly.


Introduction
The first stars, so-called Population III (Pop III) stars, formed from the primordial (metal-free) gas in the early Universe at z = 30-20 inside dark matter (DM) minihalos with 10 5-6 M e (Tegmark et al. 1997).Previous theoretical and numerical works have uncovered the hierarchical formation process of Pop III stars, from the large-scale structure formation to the physical processes around the accreting protostar (see Klessen & Glover 2023 for a review).The critical physical parameter of Pop III stars with which to evaluate their role in the formation and evolution of the first galaxies, which is one of the main observational targets of the James Webb Space Telescope (JWST), is the initial mass function (IMF).
Unlike nearby star-forming gas clouds where dust and metals are present, primordial star-forming gas clouds collapse by releasing the internal energy via molecular hydrogen (H 2 ) radiative cooling (e.g., Abel et al. 2002;Bromm et al. 2002).The difference in the thermal evolution of gas clouds affects the stellar mass and the shape of the IMF.When the gas cloud becomes self-gravitationally unstable (Jeans unstable), the temperature of the primordial gas is around 200 K, whereas the temperature of the solar-metallicity gas drops to 10 K (e.g., Omukai et al. 2005).Since the accretion rate, estimated by dividing the Jeans mass by the freefall time, is proportional to 3/2 power of the temperature (M T 3 2  µ ), the high temperature of the primordial gas is responsible for the increased mass accretion rate onto protostars and hence the higher stellar mass (∼100 M e ) than the present-day case (∼1 M e ).
In the primordial gas, there are two other coolants besides H 2 , and the thermal evolution and stellar mass will vary depending on which coolant is dominant.One is the atomic hydrogen (H), which can act when H 2 formation is suppressed due to additional effects.The H-cooling clouds can only be cooled to 8000 K and form more massive Pop III stars than the H 2 -cooling clouds.Such a situation is intensively investigated in scenarios that consider the formation of supermassive stars (intermediate-mass black holes) in the early Universe as the origin of the high-z quasars (see Woods et al. 2019 andInayoshi et al. 2020 for reviews).Another is the hydrogen deuteride (HD), which becomes effective once the temperature of the primordial gas falls below 200 K, the condition under which HD formation becomes efficient.The HD-cooling clouds can be cooled near the cosmic microwave background (CMB) temperature floor T CMB = 2.73(1 + z) K, thus forming low-mass Pop III stars (e.g., Hosokawa et al. 2012).
For H/HD cooling to be effective, some external effects must be considered to change the gas cloud's physical properties.One of the representative effects is H 2 photodissociation due to a radiation field incident from outside the cloud (Omukai 2001;O'Shea & Norman 2008).If the external radiation intensity in Lyman-Werner (LW) bands is strong enough to photodissociate H 2 in the primordial gas cloud fully (e.g., J crit = 100-1000 in Sugimura et al. 2014), the gas cloud cools only with H cooling, leading to the formation of supermassive stars.Previous works investigated the supermassive star formation process to explore the formation scenario of seed black holes of high-z quasars while considering such powerful radiation fields (e.g., Regan & Downes 2018a;Dunn et al. 2018).
The supermassive star formation under such critical radiation intensity is, on the whole, an extremely low-incidence event, comparable to the observed presence of high-z quasars (∼Gpc −3 ; Bañados et al. 2016).The background radiation intensity during Pop III star formation is about a few orders of magnitude less than J crit (e.g., Agarwal et al. 2012).In other words, most cases are associated with weak radiation intensities for the entire Pop III star formation event.For a comprehensive understanding of the effect of external radiation on Pop III star formation and the IMF, we have to consider the effect of such weak radiation intensities.Recent studies have updated the Pop III star formation scenario inside the primordial cloud irradiated by external LW radiation by considering, for example, gas fragmentation and the formation of less massive objects (Regan & Downes 2018b;Suazo et al. 2019;Prole et al. 2023), and a lack of a correlation between the LW radiation intensity and host halo masses when considering the self-shielding effect (Skinner & Wise 2020).
The next generation of telescopes have begun to explore the early Universe, and a comprehensive understanding of the formation and evolutionary processes of the first galaxies is becoming increasingly important.We have performed Pop III star formation simulations in all possible ranges of radiation intensity, from a no-radiation field to J crit , with 18 different radiation intensities.We skip the calculation of regions with a density of n H = 10 8 cm −3 and above, and thus successfully calculate the accretion process from the halo to the high-density regions for 100,000 yr after the high-density region formation.We investigate how the various thermal evolutions of gas clouds, which appear according to the external radiation intensity used as a parameter, affect the accretion growth of dense cores that host Pop III stars inside.
Previously, Hirano et al. (2015) investigated the effect of external radiation on Pop III star formation using a series of cosmological simulations.While Hirano et al. (2015) examined models with only five different intensities, in this work we comprehensively study the intensity dependence of the first star formation for the range from the minimum (consistent with the no-radiation case) to the maximum (H-cooling cloud).Furthermore, we study the parameter range for HD-cooling clouds by performing additional calculations around the parameter range where HD cooling becomes effective.In addition, as a new point of study, we compute the long-term evolution of star-forming gas clouds in the presence of external radiation and analyze gas clouds that have undergone different thermal evolutions depending on the effective coolants (HD, H 2 , and H).
This paper is organized as follows.Section 2 summarizes the numerical methodology of the cosmological simulations.Section 3 shows the simulation results under different LW radiation intensities.Section 4 discusses why HD cooling is effective in gas clouds exposed to LW radiation and determines the Pop III stellar mass and star formation efficiency.Section 5 finally summarizes the main conclusions of this study.

Methods
We perform three-dimensional cosmological N-body/hydrodynamical simulations of Pop III star formation under different external LW background radiation intensities.We use a hierarchical zoom-in technique to achieve a sufficiently high spatial resolution to follow the formation of the star-forming gas cloud within the dark matter (mini)halo.We adopt a stiff equation of state (EOS) technique to follow the long-term dynamical and thermal evolution of the star-forming gas cloud.
We generate a cosmological initial condition using the publicly available code MUSIC (Hahn & Abel 2011).The parent cosmological simulation has a volume of 1 h −1 comoving Mpc (cMpc) on a side.We insert a series of nested refinement regions, reaching the maximum refined region inside a volume of 0.15 h −1 cMpc on a side.The particle masses of dark matter and gas components in the maximally refined region are m DM = 12.4 M e and m gas = 2.3 M e , respectively.We adopt the standard Lambda cold dark matter (ΛCDM) cosmology with cosmological parameters matter density Ω m = 0.3086, baryon density Ω b = 0.04825, dark energy density Ω Λ = 0.6914 in units of the critical density, Hubble constant of h = 0.6777, normalization of the density fluctuation amplitude σ 8 = 0.8288, and primordial index n s = 0.961 (Planck Collaboration et al. 2014).The initial ionization fraction is x e = 2.737 × 10 −4 (Seager et al. 1999(Seager et al. , 2000;;Wong et al. 2008).
The cosmological simulations are performed by using the parallel N-body/smoothed particle hydrodynamics (SPH) code GADGET-2 (Springel 2005), suitably modified for the primordial star formation case (Hirano et al. 2018), including solving chemical rate equations for 14 primordial species (e − , H, H + , H − , He, He + , He ++ , H 2 , H 2 + , D, D + , HD, HD + , and HD − ; Yoshida et al. 2006Yoshida et al. , 2007)).We employ a hierarchical refinement technique to follow the gas cloud collapse with the refinement criterion that the local Jeans length is always resolved by 15 times the local smoothing length by progressively increasing the spatial resolution using the particle-splitting technique (Kitsionas & Whitworth 2002).The minimum mass of baryon particles becomes m M 8.05 10 gas,min 3 

=
´-.This study considers the effect of external radiation in the LW bands, which photodissociates the important coolants, H 2 and HD molecules.We assume a blackbody spectrum of Pop II stars with the effective temperature of 10 4 K as the light source. 3The collapsing gas cloud increases its central density and finally becomes optically thick.After that, the opaque shell surrounds the further collapsing region, and the abundance of both coolants can recover because the external radiation is consumed by photodissociating the surrounding gas (the selfshielding mechanism).We adopt the self-shielding functions (Wolcott-Green & Haiman 2011;Wolcott-Green et al. 2011) for H 2 and HD molecules.For each SPH particle, we calculate the column densities along six directions (±X, ±Y, ±Z) to account for the directional dependence of the self-shielding effect (six-ray approximation; Yoshida et al. 2008).
We follow cosmological structure formation for redshift z = 99-39.Then, we restart the cosmological simulations by adding uniform LW radiations from z = 39. 4 The model parameter is the LW radiation intensity J LW that is the intensity at the LW bands normalized in units of J 21 = 10 −21 erg s −1 cm −2 Hz −1 sr −1 .We adopt 18 models with J LW /J 21 = 0-30 (Table 1).The model with the maximum intensity J LW /J 21 = 30 corresponds to the atomic-cooling halo (ACH) case.We specifically examine the parameter dependence in more detail by taking nine parameters within J LW / J 21 = 0.02-0.09.We calculate the gravitational collapse of the star-forming gas cloud until the gas number density first reaches n H = ρ/m H = 10 8 cm −3 .
Finally, we study the long-term evolution of the star-forming gas cloud.To accelerate the evolution, we adopt a stiff-EOS technique with a threshold density n th = 10 8 cm −3 above which the gravitational collapse is artificially prohibited (Hirano & Bromm 2017).This study assumes the dense region with n H n th (the "core") is the host site where the Pop III stars form in the interior.We analyze the formation and evolution of cores (mass and number) in simulations with different J LW /J 21 .We continue hydrodynamical simulations for 10 5 yr (the typical timescale of the Pop III star accretion phase; e.g., Figure 1 in Hirano & Bromm 2017) after the gas density first reaches n th = 10 8 cm −3 .For the case with J LW /J 21 = 30, we stop the simulation at 5 × 10 4 yr because of the enormous computational cost and because it is not the intermediate radiation intensity, which is the main objective of this study.

Results
Figure 1 overviews the thermal evolution of primordial starforming gas clouds irradiated by 18 different LW radiation intensities.The gas temperature increases with increasing radiation intensity J LW /J 21 due to the increase of the H 2 photodissociation rate.In particular, in the case of the highest intensity J LW /J 21 = 30, the self-shielding effect is not effective enough to maintain high H 2 photodissociation rates, resulting in an isothermally contracting H-cooling gas cloud at 8000 K.With the intermediate intensities J LW /J 21 = 0.025-0.09,by contrast, HD cooling works, and the gas temperature in the high-density region remained below 200 K until the end of the simulations.In HD-cooling clouds, the mass of the highdensity region decreases by two orders of magnitude compared to the others (Table 1), which could impact the Pop III stellar masses born in their interiors.The following subsections explain how temperature evolution varies qualitatively with radiation intensity.

Delay of the Halo Formation
First, we discuss the effects of different LW radiation intensities on the host halo properties.Columns 2-5 in Table 1 summarize the physical properties at the host halo scale.
As J LW increases from J LW /J 21 = 0 to 30, the formation epoch delays from z = 24.88 to 13.08 and host halo mass increases from M vir = 8.74 × 10 5 M e to 2.76 × 10 7 M e .Figure 2 shows the time evolution of the maximum density of the collapsing gas inside the host halo.As structure formation proceeds and the mass of the DM halo increases, its self-gravity causes gas to accumulate in the central region of the halo, where the DM density is maximum.At this time, the gas density increases due to gas contraction, and the gas temperature also increases adiabatically.When the gas density ( 2)

Note.
Column (1): radiation intensity at the LW bands in units of J 21 = 10 −21 erg s −1 cm −2 Hz −1 sr −1 .Column (2): redshift when the maximum gas number density first reaches 10 8 cm −3 (t th = 0 yr).Columns (3)-( 5): radius, total mass, and gas mass at the virial scale where the total matter density exceeds 200 times the cosmic average.Column (6): ratio of the collapse and freefall timescales averaged at n H = 10 3 -10 5 cm −3 .Columns (7) and (8): whether HD cooling is enabled in the gas cloud (whether the abundance ratio of H 2 and HD molecules overcomes the critical value

=
. Column (11): mass accretion rate at the Jeans radius when t th = 0 yr.We end simulations at t th = 10 5 yr, except for the J LW /J 21 = 30 model, for which we stopped the calculation at t th = 5 × 10 4 yr.The horizontal lines distinguish the five ways gas clouds are classified according to their thermal evolution (R1-R5; see Section 3.2).
reaches about 10 cm −3 , the high pressure of the hot gas stops the gas contraction.The subsequent collapse requires either (1) radiative cooling, which releases the adiabatic compression heating and reduces the gas pressure, or (2) halo mass growth until the halo's gravity can compress the hot gas.In models with J LW /J 21 > 0, external LW radiation photodissociates H 2 molecules, the main driver of radiative cooling in the primordial gas cloud, and prevents gas contraction.For gas contraction to resume under the influence of external radiation, it is necessary to wait until the gas column density increases due to the increase in mass of the gas that has stopped contracting over time and the self-shielding factor tied to photodissociation becomes sufficiently high.When the selfshielding factor is sufficiently high, H 2 formation begins inside the gas cloud, and radiative cooling resumes the contraction of the gas cloud.
The LW radiation intensities, which determine the thermal evolution of the collapsing clouds (Figure 1), also affect the structure of the collapsing gas clouds inside host halos (Figure 3).The radius of the halo increases from R vir = 126 to 708 pc with the LW radiation intensity (Table 1).In a 1 kpc square region (top panels of Figure 3), all halos show a spherical density distribution, which is the gas cloud's structure during the increasing temperature phase due to adiabatic compression.The 100 pc square region (middle) shows the gas density distribution inside the halo.The three left panels show a structure that deviates from spherical symmetry due to the rapid temperature drop caused by H 2 cooling.By contrast, the two right panels show a broadened spherically symmetric structure due to the strong LW radiation inhibiting H 2 formation.At the 10 pc scale (bottom), the J LW /J 21 = 0.1 model shows a large filament, where the gas temperature decreases at 100 cm −3 (Figure 1), and the structure of collapsing gas changes.By contrast, the cloud of the J LW /J 21 = 30 model isothermally collapses due to H cooling (∼8000 K) and thus retains a spherically symmetric structure up to this scale.Although it is spherically symmetric at the DM halo scale, the structure of the dense gas cloud varies from spherically symmetric to filamentary due to different temperature evolution in its interior.

Various Thermal Evolutions of the Gas Cloud
The thermal evolution of gas clouds does not vary monotonically with increasing LW radiation intensity (Figure 1).We categorize 18 models into three types depending on the dominant chemical species contributing to the radiative cooling of the gas cloud: (a) HD-, (b) H 2 -, and (c) H-cooling clouds.The HD-cooling cloud is cooled by deuterium hydrogen (HD), provided that the ratio of HD to H 2 is above  Following the above definition, we have classified 18 models in five ranges with J LW /J 21 as a parameter: (R1) HD-cooling clouds for J LW /J 21 = 0, (R2) H 2 for J LW /J 21 = 0.003-0.02,(R3) HD for J LW /J 21 = 0.025-0.09,(R4) H 2 for J LW /J 21 = 0.1-10, and (R5) H for J LW /J 21 = 30.
Note that these classifications are based on results at the end of the simulations (t th = 10 5 yr).
To identify differences in the thermal evolution of the gas clouds with the main coolant, Figure 4 divides the 18 models into three portions according to the range of J LW /J 21 .The left panels of Figure 4 show R1 and R2 models.For the gas cloud without external radiation (R1; J LW /J 21 = 0), the gas temperature became lower than ∼200 K, where HD cooling could be efficient.The abundance ratio certainly exceeds the critical value for HD cooling when the gas number density is above 10 6 cm −3 .We classify the R1 model as an HD-cooling cloud.
By contrast, R2 models (J LW /J 21 = 0.003-0.02)show a relatively high-temperature evolution.H 2 photodissociation is effective in the low-density region (n H < 10 3 cm −3 ), reducing the H 2 abundance, but as density increases H 2 photodissociation becomes ineffective due to the self-shielding effect.The thermal evolution of the gas cloud is driven by H 2 cooling, so the temperature of the gas cloud does not fall below 200 K.We classify R2 models as H 2 -cooling clouds.
The central panels of Figure 4 show the models with intermediate ("weak") LW radiation (R3; J LW /J 21 = 0.025-0.090).The abundance ratio is higher than for models in other panels and above the critical value required for HD cooling.As a result, the gas temperature remains below 200 K until the end of the simulations.We classify R3 models as HDcooling clouds.While Hirano et al. (2015) showed that HD cooling becomes effective under a certain model and radiation intensity, this result shows for the first time that HD-cooling clouds appear under a certain range of external radiation intensity.We will discuss why HD cooling is enabled for R3 models in Section 4.1.
The right panels of Figure 4 show the remaining models irradiated by higher intensities.In these cases, the high photodissociation rate counteracts the self-shielding effect, so the gas clouds have a low H 2 abundance up to a high density.R4 models (J LW /J 21 = 0.1-10) show the thermal evolution on the high-temperature side.H 2 formation does not proceed until the density increase causes the star formation rate to exceed the photodissociation rate and the gas cloud experiences a significant temperature drop.This temperature gap produces the filamentary structure shown in Figure 3 (see also Section 3 in Hirano et al. 2023).Eventually, H 2 cooling becomes effective in these gas clouds.We classify R4 models as H 2cooling clouds.
The R5 model (J LW /J 21 = 30) shows the isothermal collapse at 8000 K.The H 2 photodissociation rate is sufficiently high to We classify the R5 model as the H-cooling cloud.

Mass of the Dense Core
The Pop III IMF is the critical parameter to model the early Universe.This study could not directly determine the stellar mass because we artificially restrict the cloud collapse by the stiff-EOS technique.However, we can discuss the upper limit of stellar mass by analyzing the mass of the high-density region, the core mass M core at n H 10 8 cm −3 , inside of which Pop III star(s) form.
First, all models have only one high-density core at the end of simulations.Figure 5 plots radial profiles for all models.During the 10 5 yr accretion phase, all gas clouds do not fragment regardless of increased gas mass inside the host halo from M vir,gas = 1.29 × 10 5 M e to 4.27 × 10 6 M e with increasing LW radiation intensities from J LW /J 21 = 0 to 30.Due to the increase in the ratio of the total gas mass to the Jeans mass, the massive gas cloud may be more likely to fragment, although no apparent Jeans-scale fragmentation occurs within the scope of .Thermal properties of the gas cloud as a function of the gas number density: temperature, H 2 abundance, the ratio of H 2 and HD abundances, and the ratio of the collapse and freefall timescales t col /t ff (from top to bottom panels).The top three rows are density-averaged values calculated from data at t th = 10 5 yr, and the bottom row is the time evolution for the maximum density particle during the cloud collapse until t th = 0 yr.Line colors represent the model parameter radiation intensities J LW /J 21 .We classify the models into three groups using the intensity range with which the HD cooling becomes effective as the boundary: (left) J LW /J 21 = 0-0.02,(center) J LW / J 21 = 0.025-0.09,and (right) J LW /J 21 = 0.1-30.On the left panels, HD cooling is enabled exceptionally on models without LW radiation (J LW /J 21 = 0).the current simulations.This is the same for HD-cooling clouds, which should more easily fragment due to the decreased Jeans mass (Ripamonti 2007).
Then, we analyze the core mass for each model and summarize in Figure 6(b).The core mass, the upper limit of stellar mass, does not increase monotonically with external radiation intensity.Furthermore, the core mass exhibits a characteristic behavior for classifications R1-R5: (a) HD cooling: The core masses are smaller by 1-2 orders of magnitude, 1-16 M e .This is because the HD cooling enabled the gas to cool further from 200 K and decrease the mass accretion rate as M T 3 2  µ .This is a new path of enabling HD cooling by external LW radiation, resulting in low-mass Pop III star formation.If the final mass of the star is less than 0.8 M e , it has a lifetime exceeding the age of the Universe (a so-called "surviving star").(b) H 2 cooling: The core masses are about 10 3 M e , independent of J LW .This is in a typical mass range for the top-heavy Pop III IMF model.

HD-cooling Clouds Driven by the Slow Collapse
Our simulations show that HD cooling is enabled in the primordial star-forming cloud exposed by the external LW radiation with intermediate intensities J LW /J 21 = 0.025-0.09(R3 models), leading to a significant reduction in the core mass that limits the upper limit of the Pop III stellar masses.Because HD molecules form abundantly in the ionized primordial gas, Nakauchi et al. (2014) examined the condition of efficient HD cooling in the relic HII region of Pop III stars irradiated by weak background LW radiation (see also Johnson & Aykutalp 2019).This work presents the condition of efficient HD cooling in the primordial cloud without ionization by some phenomenon.In this subsection, we use the collapse timescale to explain the direct cause of HD cooling's effectiveness and show why HD cooling was ineffective in other models with lower and higher intensities (R2 and R4 models).
The cooling rate of HD exceeds that of H 2 when the abundance ratio exceeds f HD /f H2 = 10 −3 .There are two main processes for the HD formation: The reaction rate coefficients for these reactions differ significantly from those for the reverse reaction at high temperatures.The second reaction produces more HD at low temperatures 200 K, however, which is the typical lower limit by H 2 cooling (Galli & Palla 2002).There is a dilemma: for HD cooling, which can cool the gas cloud more, to be effective, the gas cloud's temperature must fall to just below the lower limit of the coolable temperature by H 2 cooling.One solution has been proposed as a scenario in which the gas cloud collapses slowly over time, allowing the gas to lose thermal energy through a prolonged cooling time, thus achieving the low temperatures required for efficient HD formation (Ripamonti 2007).In the internal energy change during a given density increase, the amount of energy gain due to the adiabatic compression is constant, while the amount of energy loss due to the radiative cooling increases with the prolonged collapse time.We confirm the same dependence on the collapse time in the density-temperature diagram (top row of Figure 4) as the faster decline of gas temperature during n H = 10 2 -10 4 cm −3 in HD-cooling models (R3 models) than in H 2 -cooling models (R2 and R4 models).Hirano et al. (2014) investigated the effect of different collapse timescales on the thermal evolution of primordial gas clouds from one-zone simulations, showing that HD cooling is effective if the contraction timescale is about 3 times higher than the freefall timescale (see also Gurian et al. 2023).We follow their argument and calculate the collapse timescale to investigate the cause of HD-cooling clouds.We define the collapse time normalized by the freefall time, t col /t ff , as a collapse degree of the star-forming clouds.The collapse time is the time it takes for the maximum gas density of the contracting gas cloud to increase by a factor of 10 1/4 as and the freefall time is ´-- We calculate the normalized collapse timescale for each collapsing cloud (bottom row of Figure 4).Then, we determine an average value between densities n H = 10 3 -105 cm −3 at which HD formation and cooling must be effective (Table 1 and Figure 6(a)).Figure 6(a) shows that HD-cooling clouds (R3 models; cyan region) experience slow contractions, t col /t ff = 4-11, whereas H 2 -cooling clouds experience relatively fast contractions, t col / t ff = 2-5. 5The collapse timescale for HD-cooling clouds is consistent with one examined in the previous work, about 10 or more (Figure 24 of Hirano et al. 2014) for primordial gas clouds with no external radiation field (J LW /J 21 = 0).Even under the LW radiation field, a sufficiently slow contraction of the gas cloud can promote HD formation.
Finally, we discuss why HD cooling only became effective for the intermediate LW radiation intensities.This is due to the balance between the decrease of radiative cooling efficiency caused by H 2 photodissociation and the increase of self-gravity caused by the increased mass of the gas cloud with increasing radiation intensity.First, the photodissociation rate is also low when the radiation intensity is lower (R2 models).The H 2 abundance recovers to the same degree as for the case without radiation (R1 model) due to self-shielding effects (Figure 4).When H 2 cooling becomes effective, the gas cloud contracts while losing internal energy through H 2 radiative cooling.At this time, if this energy loss rate is high, the gas cloud collapse can accelerate.The resultant rapid contraction speed is maintained until the cloud reaches the loitering phase, leading to heating.HD cooling cannot be effective in these cases.
By contrast, with higher intensities (R4 models), H 2 molecules are sufficiently photodissociated and H 2 cooling is inefficient.Since the energetic LW photon photodissociates H 2 molecules, the thermal energy of the collapsing gas cloud cannot be released by H 2 radiative cooling and the cloud collapse stops.During the stagnation phase, the halo mass increases due to the accretion and merging of substructures, and the mass of the gas cloud that has stopped contracting (n H = 10 cm −3 ) also increases (see the dependence of the enclosed mass at R ∼ 10 pc on the radiation intensity in Figure 5).When the H 2 column density becomes sufficiently high that the H 2 photodissociating photons cannot reach the interior of the gas cloud (the self-shielding effect), H 2 cooling becomes effective, and the gas cloud restarts the collapse again.At this point, the self-gravity of the gas cloud is sufficiently strong that the density increase continues while the temperature (and pressure) of the gas cloud remains high (see Figure 4).Then, HD cooling cannot be effective in this case, either.
At intermediate intensities (R3 models), the cloud collapses over time with a reduction in cooling efficiency due to H 2 photodissociation, and its self-gravity is not too strong.Therefore, the cloud can collapse slowly, and HD cooling is activated.

Temporal HD-cooling Clouds
In some models, HD cooling that became effective during the collapse phase (t th 0 yr) became ineffective during the accretion phase (t th 0 yr).We distinguish these models as temporal HD-cooling gas clouds (Y-N combination in columns 7 and 8 of Table 1 and crosses in Figure 6).Figure 7 shows the thermal properties of one of the temporal HD-cooling clouds (J LW /J 21 = 0.1) at three different epochs.During the contraction phase (t th 0 yr), the abundance ratio of HD and H 2 exceeds the critical value and the gas temperature falls below 200 K in the high-density region.However, during the accretion phase (t th 0 yr), the HD cooling becomes inefficient.This model is classified as an H 2 -cooling cloud.
In simulation studies of star formation, stopping and analyzing simulations at the contraction stage is common practice due to computational costs.However, simulations during the accretion phase are necessary to identify gas clouds in which HD cooling becomes effective.

Restriction on the Stellar Mass
This study obtains the core mass, an upper limit of the total stellar masses formed inside the core.A part of the dense region collapses and becomes Pop III star(s) because we artificially prevent structure formation above a specific density using the stiff-EOS technique to achieve long computation times.In addition, some physical processes occurring inside the core could cause differences between the core mass and stellar mass: a decrease by circumstellar disk fragmentation (e.g., Hirano & Bromm 2017;Susa 2019;Sugimura et al. 2020Sugimura et al. , 2023) ) and an increase by protostellar radiative feedback from high accretion rates (e.g., Hosokawa et al. 2011;Hirano et al. 2014;Hosokawa et al. 2016).
The physical properties at the early phase of the star formation could determine the accretion history of the Pop III star.Some studies constructed the correlation function to estimate the Pop III stellar mass without detailed, long-term simulations to avoid the huge computational costs (Hirano et al. 2014(Hirano et al. , 2015;;Gurian et al. 2023;Toyouchi et al. 2023).Figure 8 compares the obtained core masses and estimated stellar masses using functions in Hirano et al. (2015) and Toyouchi et al. (2023).The core masses of H 2 -cooling clouds are consistent with the estimated stellar masses.By contrast, the core masses for HD-cooling clouds are lower than the estimated stellar masses.This is consistent with Hirano et al. (2015), who constructed different fitting functions for H 2 -and HD-cooling clouds.

Restriction on the Star Formation Efficiency
When modeling the effect of Pop III stars on the formation and evolution of the first galaxies, the star formation efficiency, which determines how many Pop III stars form in a halo, is another important parameter in addition to the stellar mass, which determines the life and death of each Pop III star.For example, numerical simulations and semianalytical modeling often adopt the Pop III star formation efficiency of the gas mass to stellar mass in the halo, M star /M vir,gas , as a parameter.Similar to the upper limits on stellar mass shown in Section 4.3, we evaluate an upper limit on the star formation efficiency as  f M M M M III core vir,gas star vir,gas

=
, where M vir,gas is the total mass of gas within the virial radius. 6igure 6(c) shows the dependence of f III on J LW (see also Table 1).The star formation efficiency decreases with radiation intensity, f III ∼ 10 −2 → 10 −4 .This trend is because, with radiation intensity, the total gas mass of the halo scale increases (Table 1) while the core mass does not vary significantly (Figure 6(b)).In addition, the star formation efficiency declines for HD-cooling gas clouds, f III ∼ 10 −5 , due to the decrease of the core mass.Furthermore, all models in this study form only one core, so the cloud-scale fragmentation does not contribute to increasing the total mass.
This study finds that the star formation efficiency decreases with increasing radiation intensity and, even more significantly, in a specific parameter range where HD cooling is effective.In the early stage of the formation of the first galaxies, it is necessary to consider the decrease of the Pop III star formation efficiency with external LW radiation intensity.This is in direct contrast to the conventional interpretation that the H 2 photodissociation process increases the accretion rate by increasing the gas cloud's temperature, stellar mass, and star formation efficiency.Although this study does not directly calculate the accretion process onto protostars, such a trend has been identified in the core mass, which is the upper mass limit of the accreting gas mass.

Conclusions
This study examines the dependence of the Pop III star formation inside primordial gas clouds irradiated by external LW radiation with 18 different intensities J LW /J 21 = 0-30.We summarize the results below.
First, HD cooling becomes effective for models with weak intensities J LW /J 21 = 0.025-0.09and keeps the gas temperature below 200 K above a density of 10 4 cm −3 during the first 10 5 yr of the protostellar accretion phase.In HD-cooling clouds, the core mass, defined as the total gas mass of the region where n H 10 8 cm −3 , are M M 1 16 core -

=
, which are more than two orders of magnitude smaller than in other models where HD cooling is ineffective.Suppose a small core forms a low-mass Pop III star with less than 0.8 M e .In that case, the resultant star becomes an important "fossil" in the early Universe because its lifetime exceeds the cosmic age, and low-mass Pop III stars can survive in the Milky Way galaxy until today.Previous studies have suggested the formation of Pop III "survivors" from circumstellar disk fragmentation (e.g., Clark et al. 2011;Greif et al. 2011;Stacy et al. 2016;Wollenberg et al. 2020).Without considering the small-scale phenomena, we present a novel formation path of low-mass Pop III stars from cloud-scale physics.
Second, HD cooling decreases the star formation efficiency f M M III core vir,gas = from 10 −3 to 10 −5 .The total gas mass in the halo scale increased with increasing radiation intensity, while the total core mass in the high-density regions remained almost flat (Figure 6(b)).The core mass is independent of the radiation intensity in each of HD-/H 2 -/H-cooling clouds.Furthermore, regardless of the halo mass, only one high-density core is formed in all models, and we identify no cloud-scale fragmentation.
This paper examines the effect of widely different LW radiation strengths on only one Pop III star formation site.To generalize the effects of the weak radiation field identified in this study and determine the typical range of radiation intensity over which HD cooling is effective, we are planning parameter survey simulations for many Pop III star-forming regions.In addition, although this study assumed a relatively soft spectrum energy distribution (SED) with a temperature of 10 4 K, recent JWST observations suggest significantly bluer SEDs for highredshift galaxies (e.g., Furtak et al. 2023).Since the chemical reaction rate varies with the SED shape even for the same intensity, future studies need to investigate both dependencies.

Figure 1 .
Figure1.Averaged gas temperature as a function of the gas number density of primordial star-forming gas clouds at t th = 10 5 yr.The color of the lines represents the LW radiation intensity, J LW /J 21 , the model parameter of this study.The gray dashed line represents 200 K, the temperature floor above which H 2 cooling is effective.

Figure 2 .
Figure 2. Time evolution of the maximum density of the collapsing gas cloud as a function of redshift.The dashed line indicates 200 times the cosmic average density, 200 n H,univ , which defines the virial scale.

Figure 3 .
Figure3.Gas density distribution around the collapsing center of the primordial gas cloud for the cases with J LW /J 21 = 0, 0.01, 0.03, 0.10, and 30 when t th = 0 yr.The box sizes are 1 kpc (top panels), 100 pc (middle), and 10 pc (bottom) on a side, respectively.

Figure 4
Figure 4. Thermal properties of the gas cloud as a function of the gas number density: temperature, H 2 abundance, the ratio of H 2 and HD abundances, and the ratio of the collapse and freefall timescales t col /t ff (from top to bottom panels).The top three rows are density-averaged values calculated from data at t th = 10 5 yr, and the bottom row is the time evolution for the maximum density particle during the cloud collapse until t th = 0 yr.Line colors represent the model parameter radiation intensities J LW /J 21 .We classify the models into three groups using the intensity range with which the HD cooling becomes effective as the boundary: (left) J LW /J 21 = 0-0.02,(center) J LW / J 21 = 0.025-0.09,and (right) J LW /J 21 = 0.1-30.On the left panels, HD cooling is enabled exceptionally on models without LW radiation (J LW /J 21 = 0).
(c) H cooling: The core has a large mass of 7.1 × 10 4 M e .It is a possible site to form an intermediate-mass black hole with 10 4−5 M e , a possible seed for high-z quasars (supermassive black holes).

Figure 5 .
Figure5.Radial profiles of (panel (a)) gas number density, (b) enclosed gas mass, and (c) gas accretion rate at t th = 10 5 yr.We classify models into three groups based on parameter J LW /J 21 as in Figure4.The colors represent the model parameter J LW /J 21 = 0-30.The dashed lines in the top panels show n th = 10 8 cm −3 , which is the maximum resolution density of our simulation.The dashed lines in the bottom panels show two critical values of the mass accretion rates, 4 × 10 −3 and 4.7 × 10 −2 M e yr −1 , for the protostellar radiative feedback.

Figure 6 .
Figure 6.J LW /J 21 dependence of (panel (a)) the normalized collapse timescale, (b) core mass, and (c) star formation efficiency.The cyan region represents R3 models where the HD cooling becomes effective (J LW /J 21 = 0.025-0.09).The symbols indicate whether HD cooling is effective at two epochs (t th = 0 and 10 5 yr), shown in columns 7-8 of Table 1: circles, crosses, and squares mean Y-Y, Y-N, and N-N, respectively.

Figure 8 .
Figure 8. Dependence of core masses on the mass accretion rate at the Jeans radius at t th = 0 yr.The filled and open symbols distinguish whether HD cooling is effective at t th = 10 5 yr: without HD cooling (filled symbols) and with HD cooling (open).Two lines show functions to estimate the Pop III stellar mass as a function of the mass accretion rate shown in Hirano et al. (2015, solid line) and Toyouchi et al. (2023, dashed).

Table 1
Model Parameters and Simulation Results e ) t th = 0 yr t th = 10 5 yr (M e ) ( M e yr −1 )