clumps remains low mass, <1 M⊙, unlike the case of Z ≲10−6 Z⊙. Due to the continuous infall of the low-mass clumps, the accretion is intermittent, but there is no significant quiescent period in its rate. This behavior of the accretion is similar to the results of the calculation using the barotropic relation in Chapter 2. In such a case, radiative feedback does not work, and SMS formation is relatively easy due to short quiescent periods. Therefore, we expect that the SMS formation is possible at Z ≃10−5 Z⊙.
(iii) Z ≃10−4 Z⊙ case
In this metallicity case, we have found that the rapid temperature rise due to the H2 formation heating leads to a long quiescent period corresponding to the KH timescale for a protostar mass of104 M⊙. If a similar decrease in the accretion rate occurs in the SMS formation, the ionizing feedback may turn on before the mass of the protostar reaches that of the SMS. However, calculations using a one-zone model of the protostellar collapse show that the temperature increase in the case of SMS formation is more moderate than in the case of ordinary-mass star formation.
Therefore, we need to perform hydrodynamic simulations to examine whether the halting of accretion by the H2 formation heating occurs in the SMS formation.
To summarize, we conclude that SMSs could form in the case at least Z ≲10−5 Z⊙. This relaxes the conventional condition for the SMS formation. In the case ofZ ≃10−4 Z⊙, we need further studies by performing numerical simulations with solving the energy equation.
4.3 Future prospects
We have found that circumstellar disks are gravitationally unstable in low-metallicity (Z ≲10−1 Z⊙) environments, and the accretion becomes intermittent due to the clump infall to the central star (Chapter3). From this fact, we conjecture that the intermittent accretion also occurs and affects the SMS formation from a slightly metal-enriched gas.
The existing calculation of the SMS formation in such an environment uses the barotropic equation of state derived from a one-zone model of the protostellar collapse (Chon &
Omukai 2020). As we showed in Chapter2, however, the disk evolution in the calculation using the barotropic relation can be different from that in the calculation solving the energy equation. This discrepancy causes different length of the quiescent period in the intermittent accretion. Therefore, we should perform hydrodynamic simulations by solving the energy equation to verify the SMS formation from a slightly metal-enriched gas. Specifically, we will perform calculations similar to those presented in Chapters 2
and 3, but for metal-enriched SMS formation by updating the thermal and chemical code.
In our study, we have followed the stellar growth until its mass reaches 3×104M⊙ by two-dimensional hydrodynamic simulations (Chapter 2). At the end of the calculation, since the accretion onto the protostar is still on going at a high rate, further stellar growth is expected thereafter. In order to study the fate of the growing star, we need to perform simulations including the effect of radiative feedback from the star for the entire stellar lifetime (∼Myr).
In our study, we used a two-dimensional hydrodynamic code, which allows us to follow the accurate thermal evolution with a higher resolution than previous studies.
However, this code may overestimate the density increase due to collisions of spiral arms because vertically extended structures are confined to the disk plane. Therefore, performing three-dimensional simulations with a high enough resolution is required in the future. Our goal in the future is to carry out long-term, high resolution and three-dimensional radiation-hydrodynamic simulations to investigate the final outcome of the SMS formation.
We have shown that SMSs are likely to be formed in the early universe. In order to prove the adequacy of SMSs as an origin of SMBHs, the number of SMSs need to account for that of SMBHs observed in the high-redshift universe. The number density of SMSs has been estimated based on results of hydrodynamic simulations (Agarwal et al. 2014; Chon et al. 2016; Habouzit et al. 2016; Wise et al. 2019) and analytical studies (Dijkstra et al. 2014), to be in the range 10−9 ... 10−2 Mpc−3. The reason for the variation in the estimated number density comes from the uncertainty of the SMS formation condition. The intensity of the external far-UV radiation required to form SMSs still varies by two orders of magnitude (e.g.,Shang et al. 2010;Sugimura et al. 2014;
Wolcott-Green et al. 2017). This variation is caused due to differences in the adopted radiation spectrum, gas geometry, and velocity field. In addition, three-dimensional simulations tend to be required higher intensity values than those using one-zone models because the turbulence affects the temperature evolution. Furthermore, the effects of the streaming velocity between baryons and dark matter (Tanaka & Omukai 2014; Hirano et al. 2017) and dynamical heating (Yoshida et al. 2003;Wise et al. 2019) would relax the required far-UV intensity. In addition, if we consider also the SMS formation in slightly metal-enriched environments, the number of SMSs is expected to increase significantly.
A possible solution to determine the SMS formation condition is to perform a large number of hydrodynamic simulations with various models for different far-UV spectrum and intensity, and metallicity. Our two-dimensional code may be useful for this purpose, since its computational cost is less than that of three-dimensional code, and is suitable for performing a large number of calculations.
4.3 Future prospects | 81 Future infrared telescopes (Euclid, Roman Telescope, and JWST) will detect more SMBHs at higher redshifts (z > 7.5) than those currently detected. The information on distant SMBHs, such as their redshift and masse, is expected to give constraints on their formation models. In addition, the possibility of direct detection of SMSs or their remnant BHs has been claimed. If a gas accretes onto an SMS with mass of>105 M⊙
at a high rate (>0.1 M⊙ yr−1), the stellar radiation becomes bright in the infrared and would be observed by JWST up to redshiftz = 20 (Hosokawa et al. 2013;Surace et al.
2018). In addition, if stars formed in the disk around a remnant BH falls into the BH, the tidal disruption event occurs. The afterglow associated with the tidal disruption event would be observable byJWST (Kashiyama & Inayoshi 2016). Recently, numerical studies, including our own, suggest the possibility of SMS binary formation (Chon et al.
2018; Latif et al. 2020;Patrick et al. 2020). If binary SMSs leave behind binary BHs as a result of their collapse and they merge within the cosmic time, gravitational waves emitted from their merger event would be observable by the next generation space-based detectorLISA. As described above, future instruments may provide more information on SMBHs in the early universe than ever before. In order to utilize obtained information, we need better theoretical understanding of the seed BH formation.
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