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We review the current status of knowledge concerning the early phases of star formation during cosmic dawn. This includes the first generations of stars forming in the lowest mass dark matter halos in which cooling and condensation of gas with primordial composition is possible at very high redshift ( z > 20 ), namely metal-free Population III stars, and the first generation of massive black holes forming at such early epochs, the so-called black hole seeds. The formation of black hole seeds as end states of the collapse of Population III stars, or via direct collapse scenarios, is discussed. In particular, special emphasis is given to the physics of supermassive stars as potential precursors of direct collapse black holes, in light of recent results of stellar evolution models, and of numerical simulations of the early stages of galaxy formation. Furthermore, we discuss the role of the cosmic radiation produced by the early generation of stars and black holes at high redshift in the process of reionization.

We obtain two equations (following from two different approaches) for the density profile in a self-gravitating polytropic cylindrically symmetric and rotating turbulent gas disk. The adopted physical picture is appropriate to describe the conditions near to the cloud core where the equation of state of the gas changes from isothermal (in the outer cloud layers) to one of \"hard polytrope\", and the symmetry changes from spherical to cylindrical. On the assumption of steady state, as the accreting matter passes through all spatial scales, we show that the total energy per unit mass is an invariant with respect to the fluid flow. The obtained equation describes the balance of the kinetic, thermal and gravitational energy of a fluid element. We also introduce a method for approximating density profile solutions (in a power-law form), leading to the emergence of three different regimes. We apply, as well, dynamical analysis of the motion of a fluid element. Only one of the regimes is in accordance with the two approaches (energy and force balance). It corresponds to a density profile of a slope -2, polytropic exponent 3/2, and sub-Keplerian rotation of the disk, when the gravity is balanced by the thermal pressure. It also matches with some observations and numerical works and, in particular, leads to a second power-law tail (of a slope approx. -1) of the density distribution function in dense, self-gravitating cloud regions.

The density structure of the interstellar medium determines where stars form and release energy, momentum and heavy elements, driving galaxy evolution 1 – 4 . Density variations are seeded and amplified by gas motion, but the exact nature of this motion is unknown across spatial scales and galactic environments 5 . Although dense star-forming gas probably emerges from a combination of instabilities 6 , 7 , convergent flows 8 and turbulence 9 , establishing the precise origin is challenging because it requires gas motion to be quantified over many orders of magnitude in spatial scale. Here we measure 10 – 12 the motion of molecular gas in the Milky Way and in nearby galaxy NGC 4321, assembling observations that span a spatial dynamic range 10 −1 –10 3  pc. We detect ubiquitous velocity fluctuations across all spatial scales and galactic environments. Statistical analysis of these fluctuations indicates how star-forming gas is assembled. We discover oscillatory gas flows with wavelengths ranging from 0.3–400 pc. These flows are coupled to regularly spaced density enhancements that probably form via gravitational instabilities 13 , 14 . We also identify stochastic and scale-free velocity and density fluctuations, consistent with the structure generated in turbulent flows 9 . Our results demonstrate that the structure of the interstellar medium cannot be considered in isolation. Instead, its formation and evolution are controlled by nested, interdependent flows of matter covering many orders of magnitude in spatial scale. Statistical analysis of velocity fluctuations in the interstellar medium (ISM) of the Milky Way and NGC 4321 show that the motion of molecular gas over scales ranging from 0.1 to 1,000 pc is similar, and consistent with that generated by a combination of gravity and turbulence. ISM structure at one scale is therefore linked to structure at other scales.

The infrared-radio correlation (IRRC) of star-forming galaxies can be used to estimate their star formation rate (SFR) based on the radio continuum luminosity at MHz-GHz frequencies. For its application in future deep radio surveys, it is crucial to know whether the IRRC persists at high redshift z. Previous works have reported that the 1.4 GHz IRRC correlation of star-forming galaxies is nearly z-invariant up to z=4, but depends strongly on stellar mass M. This should be taken into account for SFR calibrations based on radio luminosity. To understand the physical cause behind the M dependence of the IRRC and its properties at higher z, we constructed a phenomenological model for galactic radio emission. Our model is based on a dynamo-generated magnetic field and a steady-state cosmic ray population. It includes a number of free parameters as well as observed scaling relations. We find that the resulting spread of the infrared-to-radio luminosity ratio, q(z, M), with respect to M is mostly determined by the scaling of the galactic radius with M, while the absolute value of the q(z, M) curves decreases with more efficient conversion of supernova energy to magnetic fields and cosmic rays. Decreasing the slope of the cosmic ray injection spectrum, aCR, results in higher radio luminosity, decreasing the absolute values of the q(z, M) curves. Our model reproduces the observed dependence of the IRRC on M and z when the efficiency of supernova-driven turbulence is 5%, 10% of the kinetic energy is converted into magnetic energy, and aCR = 3. For galaxies with intermediate to high M (10^9.5-10^11 M_sun), our model results in an IRRC that is nearly independent of z. For galaxies with lower M (M=10^8.5 M_sun), we find that the IR-to-radio flux ratio increases with increasing redshift. This matches the observational data in that mass bin which currently, however, only extends to z~1.5.

Many experimental quantities show a power-law distribution \\(p(x)\\propto x^{-\\alpha}\\). In astrophysics, examples are: size distribution of dust grains or luminosity function of galaxies. Such distributions are characterized by the exponent \\(\\alpha\\) and by the extremes \\(x_\\text{min}\\) \\(x_\\text{max}\\) where the distribution extends. There are no mathematical tools that derive the three unknowns at the same time. In general, one estimates a set of \\(\\alpha\\) corresponding to different guesses of \\(x_\\text{min}\\) \\(x_\\text{max}\\). Then, the best set of values describing the observed data is selected a posteriori. In this paper, we present a tool that finds contextually the three parameters based on simple assumptions on how the observed values \\(x_i\\) populate the unknown range between \\(x_\\text{min}\\) and \\(x_\\text{max}\\) for a given \\(\\alpha\\). Our tool, freely downloadable, finds the best values through a non-linear least-squares fit. We compare our technique with the maximum likelihood estimators for power-law distributions, both truncated and not. Through simulated data, we show for each method the reliability of the computed parameters as a function of the number \\(N\\) of data in the sample. We then apply our method to observed data to derive: i) the slope of the core mass function in the Perseus star-forming region, finding two power-law distributions: \\(\\alpha=2.576\\) between \\(1.06\\,M_{\\sun}\\) and \\(3.35\\,M_{\\sun}\\), \\(\\alpha=3.39\\) between \\(3.48\\,M_{\\sun}\\) and \\(33.4\\,M_{\\sun}\\); ii) the slope of the \\(\\gamma\\)-ray spectrum of the blazar J0011.4+0057, extracted from the Fermi-LAT archive. For the latter case, we derive \\(\\alpha=2.89\\) between 1,484~MeV and 28.7~GeV; then we derive the time-resolved slopes using subsets 200 photons each.

In the present work we apply virial analysis to the model of self-gravitating turbulent cloud ensembles introduced by Donkov \\& Stefanov in two previous papers, clarifying some aspects of turbulence and extending the model to account not only for supersonic flows but for trans- and subsonic ones as well. Make use of the Eulerian virial theorem at an arbitrary scale, far from the cloud core, we derive an equation for the density profile and solve it in approximate way. The result confirms the solution \\(\\varrho(\\ell)=\\ell^{-2}\\) found in the previous papers. This solution corresponds to three possible configurations for the energy balance. For trans- or subsonic flows, we obtain a balance between the gravitational and thermal energy (Case 1) or between the gravitational, turbulent and thermal energies (Case 2) while for supersonic flows, the possible balance is between the gravitational and turbulent energy (Case 3). In Cases 1 and 2 the energy of the fluid element can be negative or zero end thus the solution is dynamically stable and shall be long lived. In Case 3 the energy of the fluid element is positive or zero, i.e., the solution is unstable or at best marginally bound. At scales near the core, one cannot neglect the second derivative of the moment of inertia of the gas, which prevents derivation of an analytic equation for the density profile. However, we obtain that gas near the core is not virialized and its state is marginally bound since the energy of the fluid element vanishes.

The formation of the most massive quasars observed at high redshifts requires extreme inflows of gas down to the length scales of the central compact object. Here, we estimate the maximum inflow rate allowed by gravity down to the surface of supermassive stars, the possible progenitors of these supermassive black holes. We use the continuity equation and the assumption of free-fall to derive maximum allowed inflow rates for various density profiles. We apply our approach to the mass-radius relation of rapidly accreting supermassive stars to estimate an upper limit to the accretion rates allowed during the formation of these objects. We find that the maximum allowed rate \\(\\dot M_{\\rm max}\\) is given uniquely by the compactness of the accretor. For the compactness of rapidly accreting supermassive stars, \\(\\dot M_{\\rm max}\\) is related to the stellar mass \\(M\\) by a power-law \\(\\dot M_{\\rm max}\\propto M^{3/4}\\). The rates of atomically cooled halos (0.1 -- 10 M\\(_\\odot\\) yr\\(^{-1}\\)) are allowed as soon as \\(M\\gtrsim1\\) M\\(_\\odot\\). The largest rates expected in galaxy mergers (\\(10^4-10^5\\) M\\(_\\odot\\) yr\\(^{-1}\\)) become accessible once the accretor is supermassive (\\(M\\gtrsim10^4\\) M\\(_\\odot\\)). These results suggest that supermassive stars can accrete up to masses \\(>10^6\\) M\\(_\\odot\\) before they collapse via the general-relativistic instability. At such masses, the collapse is expected to lead to the direct formation of a supermassive black hole even within metal-rich gas, resulting in a black hole seed that is significantly heavier than in conventional direct collapse models for atomic cooling halos.

We present a comparison of the Milky Way's star formation rate (SFR) surface density (\\(\\Sigma_{\\rm SFR}\\)) obtained with two independent state-of-the-art observational methods. The first method infers \\(\\Sigma_{\\rm SFR}\\) from observations of the dust thermal emission from interstellar dust grains in far-infrared wavelengths registered in the Herschel infrared Galactic Plane Survey (Hi-GAL). The second method determines \\(\\Sigma_{\\rm SFR}\\) by modeling the current population of O-, B-, and A-type stars in a 6 kpc \\(\\times\\) 6 kpc area around the Sun. We find an agreement between the two methods within a factor of two for the mean SFRs and the SFR surface density profiles. Given the broad differences between the observational techniques and the independent assumptions in the methods for computing the SFRs, this agreement constitutes a significant advance in our understanding of the star formation of our Galaxy and implies that the local SFR has been roughly constant over the past 10\\,Myr.

Much of what we know about molecular clouds, and by extension star formation, comes from molecular line observations. Interpreting these correctly requires knowledge of the underlying molecular abundances. Simulations of molecular clouds typically only model species that are important for the gas thermodynamics, which tend to be poor tracers of the denser material where stars form. We construct a framework for post-processing these simulations with a full time-dependent chemical network, allowing us to model the behaviour of observationally-important species not present in the reduced network used for the thermodynamics. We use this to investigate the chemical evolution of molecular gas under realistic physical conditions. We find that molecules can be divided into those which reach peak abundances at moderate densities (\\(10^3 \\, {\\rm cm^{-3}}\\)) and decline sharply thereafter (such as CO and HCN), and those which peak at higher densities and then remain roughly constant (e.g. NH\\(_3\\), N\\(_2\\)H\\(^+\\)). Evolving the chemistry with physical properties held constant at their final values results in a significant overestimation of gas-phase abundances for all molecules, and does not capture the drastic variations in abundance caused by different evolutionary histories. The dynamical evolution of molecular gas cannot be neglected when modelling its chemistry.

We identify stellar structures in the PHANGS sample of 74 nearby galaxies and construct morphological masks of sub-galactic environments based on Spitzer 3.6 micron images. At the simplest level, we distinguish centres, bars, spiral arms, interarm and discs without strong spirals. Slightly more sophisticated masks include rings and lenses, publicly released but not explicitly used in this paper. We examine trends using PHANGS-ALMA CO(2-1) intensity maps and tracers of star formation. The interarm regions and discs without strong spirals dominate in area, whereas molecular gas and star formation are quite evenly distributed among the five basic environments. We reproduce the molecular Kennicutt-Schmidt relation with a slope compatible with unity within the uncertainties, without significant slope differences among environments. In contrast to early studies, we find that bars are not always deserts devoid of gas and star formation, but instead they show large diversity. Similarly, spiral arms do not account for most of the gas and star formation in disc galaxies, and they do not have shorter depletion times than the interarm regions. Spiral arms accumulate gas and star formation, without systematically boosting the star formation efficiency. Centres harbour remarkably high surface densities and on average shorter depletion times than other environments. Centres of barred galaxies show higher surface densities and wider distributions compared to the outer disc; yet, depletion times are similar to unbarred galaxies, suggesting highly intermittent periods of star formation when bars episodically drive gas inflow, without enhancing the central star formation efficiency permanently. In conclusion, we provide quantitative evidence that stellar structures in galaxies strongly affect the organisation of molecular gas and star formation, but their impact on star formation efficiency is more subtle.