Star Formation at High Redshift: Possible Theories and Wild Speculations

Vincent L. Fish

24 November 1998 Astronomy 208 – Problem Set 4

In this paper, we attempt to give a coherent theory of star formation at high redshift. Toward this aim, we outline a possible theory of star formation at z = 0 and some of the complications that arise from the extension of this theory to the high redshift case. Applying Occam’s razor, we take the minimal extensions and modifications to existing star formation theory. We first explore isolated star formation and then consider star formation in the context of the star’s environment.

Low-z star formation generally begins in giant molecular clouds (GMCs). These GMCs are of nonuniform density, implying the existence of overdense regions with respect to an average cloud density. These overdense regions are driven to successively higher densities by gravitational instability. The highest density regions become the cores which will eventually become the sites of the rest of the star formation phenomenon.

This portion of the theory encounters some difficulty in the high-redshift case. The formation of H_{2}, necessary for the existence of molecular clouds, occurs at low redshift primarily through dust-catalyzed mechanisms. Theories of Big Bang nucleosynthesis indicate that the only elements that existed before the epoch of star formation began were hydrogen, helium, and trace amounts of lithium and beryllium. Thus, before the first stars formed, there was no dust to catalyze H_{2} production. Molecular hydrogen can and did form directly in the gas phase, although this process occurs much more slowly. At z ≈ 60 it is estimated that the ratio H_{2}/ H was only a few parts per million (Lepp & Shull 1984). It is likely that giant molecular clouds did not exist at all at high redshift (Norman & Spaans 1997). The first cloud cores may have formed out of atomic clouds, then, rather than molecular clouds. More likely, molecular hydrogen formation occurred concurrently in these atomic clouds with their collapse, as higher densities increased the rate coefficients for H_{2} formation mechanisms. This may then have increased the characteristic timescale of star formation.

Once a core has formed, material falls inward and the density increases due to the effects of gravity. It is unlikely that this feature of the model was substantially different at high redshift. However, magnetic fields play a significant role in the determination of timescales of collapse, and little is known of magnetic fields on parsec-size length scales at high redshift, so let us quickly consider this problem. The magnetic field at the time of the electroweak phase transition was likely 10^{22} G on a length scale of roughly 10^{−8}H_{EW}^{−1} , where H_{EW}^{−1} is the Hubble scale at the electroweak transition (Joyce & Shaposhnikov 1997). This corresponds to a comoving scale at z = 0 of about 10^{−4} pc, although inverse cascade of magnetic energy to larger scales may have increased this length scale to as large as 1 Mpc (Field 1998). (Assuming ideal magnetohydrodynamics, the magnetic flux is frozen into the material, so the magnetic field strength is inversely proportional to the square of the scale factor of the universe, since expansions along field lines do not dilute the magnetic field.) Assuming that even a small amount of inverse cascade occurred is enough to guarantee the existence of microgauss magnetic fields on a length scale of 0.3 pc, the typical size of a Bok globule.

Ambipolar diffusion has long been recognized as playing an important role in determining the time scale of star formation (Mouschovias 1976a, b). The timescale for star formation implied by a gravitational collapse not slowed by magnetic effects is too short compared with the observed rate, but ion-neutral friction slows the collapse. The problem with the current theory is that it slows collapse too much. In recent years, attention has been directed toward modifications of the model that cause faster collapse than that predicted by ion-neutral friction alone. For instance, dissipation of magnetic turbulence within a dense core can lead to a pressure imbalance and therefore an unstable turbulent cooling flow that speeds collapse (Myers & Lazarian 1998). In fact, it has been suggested that the timescale for star formation is determined by the timescale for dissipation of magnetic turbulence (Nakano 1998). It is reasonable to wonder how ambipolar diffusion and turbulent dissipation might have differed in the epoch of earliest star formation, although the answers are not necessarily clear. Since there was no interstellar radiation field and hence very few ionizing ultraviolet photons, the ionization fraction of HI may have been substantially smaller at high z. This would have reduced the braking effect of ion-neutral friction on the gravitational collapse of overdense regions. Consideration of differences in turbulent dissipation require a detailed analysis involving understanding of magnetohydrodynamic turbulence and typical magnetic fields threading hydrogen clouds at high redshift. The latter is poorly known, as addressed previously, and even the former is not well understood, despite decades of research. We therefore do not speculate as to how turbulent dissipation may vary with redshift, but leave this up to future researchers to consider.

Much of the rest of the theory of single-star formation in isolation seems not to be strongly sensitive to redshift. After material starts to fall inward onto the core, a central condensation forms. Surrounding gas falls in toward the core, increasing the mass of the core. The kinetic energy of the infalling material is converted to heat, some of which is dissipated through accretion shocks. These shocks, if intense enough, can dissociate H_{2} molecules. When the central condensation grows to a critical size, heat can no longer be radiated away fast enough to keep its temperature from increasing. In the Milky Way ISM, this critical mass is approximately 0.01 solar masses. At high redshift, the “interstellar” medium would be devoid of metals and therefore dust. The opacity of the medium should be lower, so the heat that is radiated from the central condensation should not be self-absorbed as strongly by the surrounding gas cloud, and possibly not even by the outer layers of the core. Assuming that the emission from the core is primarily thermal, which is a reasonable assumption given the high density of material, radiation should be more effective at cooling at high redshift than in the z = 0 ISM. Cores at high redshift may then have been able to attain a higher mass before heating up.

As the core starts to grow, it heats up. At low redshift, the core becomes surrounded by a thick envelope of dust. In the epoch of initial star formation, this would not, of course, have existed, due to the nonexistence of dust. The infalling material has angular momentum, and as the material is pulled inward by gravity, it encounters a point at which the centrifugal effects of rotation balance the gravitational force inward. Accreting material gathers in a disk around the protostar until angular momentum can be lost. Some of the material can be ejected in streams perpendicular to the plane of rotation. The outflow carries angular momentum away from the disk, reducing the total angular momentum in the disk, allowing material at the inside of the disk to accrete onto the protostar below. The mechanism behind this is not entirely clear, although popular theories suggest that a magnetocentrifugal wind is responsible. Magnetic reconnection is important as a driving force in theories behind this wind. Petschek and Shu, among others, have proposed models involving an X-wind (Shu 1992). In these models, magnetic fields of opposite parity are forced close to each other at a characteristic radius, where reconnection occurs. Material that follows the field lines near this point may be ejected perpendicular to the disk. These theories are likely also applicable to circumstellar disks at high redshift, up to uncertainties in the strength of the magnetic field.

Eventually, the accretion of material onto the core ends. The reservoir of material to be accreted can be depleted. The outflow can become strong enough to balance the inflow of material into the accretion disk. Radiation pressure from the star can disperse the material that would accrete. Whatever the mechanism that causes it may be, accretion stops at some point, and the mass of the future star is reached. Eventually, as the collapse continues, the temperature and density in the center of the protostar become high enough to allow fusion to occur. The first product to be fused is deuterium, since its fusion point is the lowest of all materials in the star. At high redshift, there was substantially more deuterium, since deuterium is easy to destroy due to its small binding energy but hard to synthesize. Thus, the deuterium-burning stage in high-redshift stars may have been longer than for stars created today. (As a side note, brown dwarfs at high redshift would have been brighter and luminous for a longer timescale for precisely the same reason.)

The newly-born star then enters the T Tauri stage. In normal models of low-redshift pre-main-sequence stars, the T Tauri object is surrounded by an envelope of dust and gas. The T Tauri star evolves along the Hayashi track and becomes visible beyond a “birthline” in the H-R Diagram, where the dust dissipates enough that the luminosity of the star inside is visible (Hansen & Kawaler 1994). At high redshift, the nonexistence of dust would imply a significantly lower envelope opacity, so the birthline would be shifted farther up along the Hayashi track.

So far the discussion has been limited to the birth of stars in isolation from the rest of their surroundings. This is a good starting place for the discussion, but ignores many important facets of the problem that are highly relevant to star formation at large z. Let us now turn to examining star formation in the context of cosmology.

Clearly there were no stars before a certain redshift. Stars did not exist at the time of the Big Bang, so the first star was created at some later time. Therefore, it is wholly reasonable to assume that no stars existed at all before a certain redshift. What this particular redshift was is, of course, a matter of great uncertainty. From the observational perspective, it is very difficult to observe objects even at z = 5, and the objects thus observed are incredibly luminous. Seeing a single star or even an entire star-forming region at such a great distance is currently not feasible with existing technology and probably will be unfeasible for a long time. Furthermore, even if such technology existed, it would be difficult to make a statement about the nonexistence of stars at redshifts higher than a critical z, since that would involve obtaining conclusive evidence that there are no more stars discovered as the limits to our detection get pushed back farther into the recesses of the past. Each incremental advancement of the horizon corresponds to an increasingly more difficult advancement in technology, which at some point exceeds either physical limit or economic justification. Practically, observations can set the lower bound on the redshift of formation of the first stars, but only theory can set the upper bound.

One possible upper bound on the redshift of first star formation might come from the cosmic microwave background (CMB). The CMB as we see it now formed at z ≈ 10^{3} at the time of recombination. At the time, the temperature of the background was several thousand Kelvin. Star formation could not possibly have occurred while there were still sufficient numbers of CMB ultraviolet photons to excite neutral hydrogen from the n = 1 to n = 2 state. More importantly, though, star formation would likely not occur until the temperature of the CMB fell to roughly 30 K, a typical temperature for giant molecular clouds and the cores they contain. If the temperature of the cloud is too high, it is thermally supported against collapse. Since the temperature of the CMB scales inversely with the scale factor of the universe, that places a reasonable upper limit on the redshift of first star formation at z ≈ 10.

Noticeably absent from the discussion so far is the role of galaxy formation in the process of star formation. The question of whether galaxies formed before, concurrently with, or after the first stars formed is a completely open question at this point and is tied to much more difficult issues such as the nature of dark matter and the cosmology of the universe. Observations indicate that 90% of stars now form in OB associations in spiral arms of galaxies (Lada 1998). Regardless of whether or not galaxies existed at the time of first star formation, it seems unlikely that sufficient structure existed in early galaxies to support the spiral density waves that are so important in z = 0 star formation. More likely, high-redshift stars form in isolated clouds or cloud complexes that are not as affected by large-scale galactic dynamics.

Another problem to be addressed is the complete lack of observed population III stars. While stars with relatively low metallicities are observed, especially in globular clusters and elliptical galaxies, no star with zero (or very near zero) metallicity has ever been observed. A star of roughly 1 solar mass has a lifetime of approximately 10^{10} years, which is nearly the time since the first stars were formed. This implies that in initial star formation, the initial mass function (IMF) must have had a distinctly different form from that which is observed today. For most observed systems, the IMF follows a power-law form with an exponent of nearly 2.5 for the high-mass end of the spectrum. Although it was once thought that this power law is valid all the way down to the hydrogen-burning limit (roughly 0.08 solar masses, depending on metallicity), recent evidence indicates that the Salpeter IMF has a break at a mass slightly less than 1.0 solar masses (Lada 1998). While the functional form of the IMF at lower masses is not yet well-determined, it is still clear that star formation today proceeds in such a way as to produce many more stars of low mass than of high mass. Furthermore, the IMF appears to be robust as seen from one star-forming region to another. This makes all the more perplexing the fact that no population III stars are observed. Something substantial must have been different at high redshift to produce an IMF that is lacking in low-mass stars.

A possible answer to this is inspired, ironically, from zero-redshift star formation in giant molecular clouds. Within a GMC it is observed that star formation tends to occur in smaller clusters within the cloud, rather than in individual, isolated sites. Although several stars form simultaneously, one star is likely to begin fusion before the others. The energy from fusion eventually propagates through the star and is released at the photosphere in the form of infrared, optical, and ultraviolet photons. The ultraviolet photons ionize the surrounding gas, forming HII regions. The temperature of these regions increases, which leads to a pressure imbalance, causing the entire region to expand. The temperature difference between the HII region and the surrounding molecular cloud is high enough that the HII region can expand as though it were in vacuo. These expanding HII regions can be strong enough to disperse the surrounding gas in the molecular cloud.

Even though giant molecular clouds may not have existed at the time of first star formation, this scenario may still be applicable, and may explain the lack of observed population III stars. Consider a cluster in which star formation is occurring at high redshift. As stated in more detail in the first part of this paper, conditions at high redshift were probably conducive to the formation of higher-mass stars and to the formation of high-mass stars on a shorter timescale (relative to low-mass stars) than is observed today. If this is indeed true, it is possible that in a typical star-forming cluster one or more very high-mass stars reached the stage of fusion quickly, while lower-mass protostars were still in earlier stages of evolution. These stars would produce many ultraviolet photons. In fact, if their average masses were only a small fraction greater than the average mass of a high-mass star formed first in a cluster today, the ultraviolet photon flux would have been significantly higher, due to the steepness of the mass-luminosity relation. Furthermore, without dust in high-redshift atomic and molecular clouds to absorb these photons and reradiate them at longer wavelengths, the ambient interstellar radiation field could have been very strong, indeed. The ionization of the cloud may have occurred rapidly and efficiently enough to interfere with the formation of low-mass stars, which would have still been in early stages of evoluation. Thus, the high-mass stars may effectively have prevented the formation of low-mass stars in the same cluster. Furthermore, the highestmass stars only survive as stars for a few million years, a timescale which is comparable to the timescale of (low-mass) star formation. Thus, in addition to ionizing and dispersing the medium from which low-mass stars were trying to form, high-mass stars may have dispersed the medium even more by their winds and catastrophic explosions upon their demise.^{1}

Winds and supernova explosions did eventually dump metals into the surrounding medium, but initially molecular clouds and the interstellar medium were dust-free. As the first stars ignited, ultraviolet photons not only ionized the clouds from which the stars formed, but also the interstellar medium. Ultraviolet photons in the interstellar radiation field may have reached other clusters were star formation had not occurred or been completed, disrupting the process through photoionization. In fact, cosmologists who trace the ionization history of the universe theorize that the ionized fraction of hydrogen, which fell slowly after recombination, rose again sharply after the star-formation epoch began (Ostriker & Gnedin 1996). This may have inhibited future star formation for some time. As long as the interstellar radiation field contained Lyman-α photons or many photons of higher energy, atomic hydrogen would be ionized. Fortunately, redshifting caused a gradual decay of photon energies. Some Lyman-α radiation was also lost by reprocessing by atomic hydrogen in two-photon (forbidden) 2s → 1s transitions. ^{2} While star formation is a continual process today, it is quite possible that after the population III stars formed, there was a long period of low or no star formation as hydrogen clouds recovered from photoionization. If so, this would make observation of population III stars all the more difficult, since the universe would look dark and uninteresting at a lower redshift than the redshift of first star formation. This would be quite a discouragement to observers hoping to look back to the epoch of first star formation!

1The more reasonable theory might suggest that these winds and explosions would induce gas compressions that would trigger further star formation, not inhibit it. However, our assumption in this paper is that the clouds are somewhat precarious to begin with, especially with photoionization, such that winds and shocks do more toward dispersing the cloud than inducing star formation. This is one of those “wild speculations” referred to in the title!

2A similar phenomenon is observed in the radiation from high-redhsift galaxies and quasars. As the radiation from such sources travels to us, it gets redshifted. If, for instance, photons of higher energy are redshifted to the Lyman α frequency at the time of passage through a hydrogen cloud, absorption takes place. When we observe the radiation, then, we see a Lyman-α forest of absorption lines caused by gas clouds between the source and us.

We might also wonder whether the first stars ended up in bound clusters or whether the clustering associated with formation was a transient phenomenon. A direct application of the virial theorem suggests that a cluster will not be bound if less than half of the material is converted into stars. A more careful analysis, allowing for adiabatic adjustment of the gas and stars in a cluster, suggests that in a typical GMC the cluster can remain bound if the star formation efficiency is as low as 30% (Lada 1998). The theory outlined in this paper seems to indicate a low star formation efficiency, since much of the material that would coalesce to become low-mass stars is blown away before it can condense. Furthermore, the first stars would have been type O stars, which seem not to form in clusters that remain bound. Thus, it seems unlikely that the first star clusters remained bound after star formation ceased.^{3}

In this paper, we have discussed several aspects of the problem of star formation at high redshift. We have predicted several differences in the isolated star-formation process between high and low redshift. We have also generalized this theory to include star formation in the context of an environment of clouds, clusters, galaxies, and radiation. The theories put forth concerning star formation at high redshift are intended to be taken as the qualitative musings of a typical graduate student in astrophysics on a difficult and unsolved problem, not as a complete, coherent, quantitative, scientifically accurate solution to the problem.

3We have neglected the role of binary stars in the discussion, but binaries tend to have a stabilizing effect on clusters. The orbits of binary stars are elastic enough to be able to absorb energy, so that clusters which might otherwise be marginally unbound can stay bound (Ford 1998). Sufficiently little is known about the relative abundance of binary population III stars that any predictions of their number would be wild speculation, but in the spirit of the paper we suggest that the delicate nature of the clouds and the rapid dispersal of gas that we predict may be less conducive to binary star systems than in z = 0 clouds.

References

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