## How did some early black holes get so big so fast?

The supermassive black holes (SMBHs) found in the centers of large galaxies can be astonishingly large. The closest example to us is in the giant elliptical galaxy M87, and it’s estimated to be 6.6 billion solar masses (M). More distant examples can be even larger, more than 10 billion M (at distances ~300 million light-years).

Those are extremes. 1 or 2 billion M SMBHs are a little more common in our neighborhood, though still rare. Rather surprisingly, however, SMBHs that large can be found even in the very early universe. The largest yet discovered is about 2 billion M, and it’s 12.9 billion light-years away, at a redshift z=7.085. That SMBH reached its observed size only 765 million years after the big bang, i. e. perhaps 500 million years after the very first stars formed. It’s been a difficult problem to understand how SMBHs that large could have formed so quickly. A recently announced computer simulation of a large part of the very early universe may have come up with a good answer.

It is only barely possible to detect very bright objects (such as quasars or large galaxies) at redshifts z~7 with the best telescope technology today, and impossible to detect less bright objects (even the brightest stars) or objects at higher redshifts. So direct observation of the earliest stars – which may have begun to form as early as z~30, 100 million years after the big bang – is currently impossible, and computer simulations must be used to understand their properties and the process in which they formed.

Primordial black holes may have formed in the big bang itself. Otherwise there must be mechanisms for black holes to form later, so that by the time SMBHs can be detected powering the brightest quasars at z~7 they can have masses at high as 109 M. Black holes, of course, cannot be “seen” at all by themselves. They are only detectable through their effects on other things, such as stars and clouds of gas and dust in galaxies – in particular, when they form the central engine that powers quasars.

Quasars form inside galaxies when large quantities of galactic gas and dust are sucked by gravity into the accretion disks around SMBHs. This happens long after the first generation of stars has formed and spread elements heavier than hydrogen and helium into their surroundings, in which later generations of stars, and eventually galaxies and quasars can form.

Although we still don’t have very secure knowledge of how big the first generation of stars were, even the largest would leave behind black holes of mass at most about 100 M after exploding as supernovae. It is also possible that black holes as large as 105 M could have formed directly from streams of matter in the very early universe (z~20). If there actually were large primordial black holes formed in the big bang itself, they could have had almost any amount of mass.

The recently published computer simulation shows that black hole seeds having mass in the range of 100 to 105 M at z~20 could have grown to as large as 109 M or more by z~7, when they can actually be detected powering quasars. Starting from 100 to 105 M it takes factors of growth between 104 and 107 to reach 109 M. That’s between about 17 and 27 doublings in size. The simulation needed to determine whether that’s possible, and if so, how.

The simulation, named Massive Black, looked at a cube about 2.5 billion light-years on a side. It had a high resolution, with more than 65 billion particles initially. That’s still very diffuse – only about 4 particles per cubic light-year. However, the particles are assumed to be very massive, involving equal numbers of particles of ordinary and dark matter, with masses 5.7×107 M and 2.8×108 M, respectively. (That reflects the mass ratio between ordinary and dark matter in the universe as a whole.) The initial conditions correspond to z=159, and the simulation was run up to z=4.75, corresponding to times from 8 million to 1.275 billion years after the big bang.

The masses of matter particles, of course, do not correspond to actual physical particles of matter. The idea is to keep the number of particles small enough for practical computation. From the computed behavior of these particles, the average density of matter in small regions can be calculated. This density is then used to calculate the maximum rate of growth of the black holes seeds (which are initially much less massive than each particle). The calculation is based on this formula for the maximum possible growth rate:

$\frac{d}{dt}M_{BH} = \frac{4\pi G^{2}M_{BH}^{2}\rho}{(c_{s}^{2}+v^{2})^{3/2}}$

Here MBH is the black hole mass, G is the gravitational constant, ρ is the gas density, cs is the speed of sound in the gas (which depends on ρ), and v is the velocity of the black hole with respect to the surrounding gas.

The formula has been known for a long time. It’s derived by balancing the inward force of gravity with the outward pressure of gas heated by the conversion of potential energy to internal kinetic energy. Since the gas is atomic (or molecular) hydrogen and helium, collisions between atoms lead to intense photon emissions, which add radiation pressure. The formula expresses the upper limit on growth rate – known as the Eddington limit. The gas can become extremely hot, because the efficiency of conversion of mass into energy is about 10%, which is far above what occurs in thermonuclear fusion. (Since dark matter cannot radiate photons, it cannot lose kinetic energy and hence is accreted much more slowly.)

The maximum growth rate is very high, since it’s proportional to the square of the mass. So it’s even faster than exponential. Obviously, that can’t go on forever, and will eventually slow only when the gas in the whole region around the growing black hole is hot enough to disrupt the strong infalling flow and lower its density. Since the rate is proportional to density, growth slows when the density begins to fall.

Where did the necessary matter come from? The simulation itself provides the answer. In the early universe, the dominant dark matter had broken apart into numerous filamentary streams, which could flow like fast rivers. Where two or more streams intersected, the streams slowed somewhat, so density was especially high.

Stream intersections like that are where black hole seeds could initially form. The first seeds may have been remnants of supernova explosions of the first generation of stars, or they could have formed directly as black holes. (The simulation showed that growth by merger between black holes was unlikely.) As with exponential growth in general, the rate is relatively moderate for a long period of time before it begins to accelerate.

However, there’s plenty of time available. The first generation of stars and the first black holes could have formed at z~30, 100 million years after the big bang. Black holes that were initially supernova remnants would be around 100 M, while black holes that formed directly could be as much as 105 M to begin with.

The simulation introduced black hole seeds of 105 M at varying times up to z~15, 275 million years after the big bang, depending on when and where it found the densest concentrations of matter. That leaves almost half a billion years to grow into quasar-powering black holes of 109 M that are detectable at z~7. The most massive SMBH found in the simulation was actually 3×109 M. It’s represented in the right panel of the image above, along with filaments that fed its growth. There were 10 SMBHs around 109 M, and many smaller ones by z~6.

In the simulation, rapid growth, at the Eddington limit, starts around z~10, 480 million years after the big bang. It only stops when the entire halo of gas around the growing SMBH becomes heated sufficiently to disrupt the streams of inflowing cold gas. The streams are so strong that this doesn’t happen until about z~6, when the growth rate falls well below the Eddington limit and levels off around or below 10 M per year.

While the SMBHs are growing, galaxies are also forming around them, as stars form. However, star formation rates are slower, so that by the time SMBHs have stopped growing rapidly, the galaxies around them are small compared to galaxies today. Consequently, the ratio of SMBH mass to galaxy mass was typically higher at z~6 or 7 than it is now.

Impressive graphical representations of the simulation have been produced, and are described in a separate paper.

 Di Matteo, T., Khandai, N., DeGraf, C., Feng, Y., Croft, R., Lopez, J., & Springel, V. (2012). COLD FLOWS AND THE FIRST QUASARS The Astrophysical Journal, 745 (2) DOI: 10.1088/2041-8205/745/2/L29