The First Galaxies

Abraham Loeb and Steven R. Furlanetto

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

To our families

Contents Preface

This book captures the latest exciting developments concerning one of the unsolved mysteries about our origins: how did the first stars and galaxies form? Most research on this question has been theoretical so far. But the next few years will bring about a new generation of large telescopes with unprecedented sensitivity that promise to supply a flood of data about the infant Universe during its first billion years after the Big Bang. Among the new observatories are the James Webb Space Telescope (JWST) – the successor to the Hubble Space Telescope, and three extremely large telescopes on the ground (ranging from 24 to 42 meters in diameter), as well as several new arrays of dipole antennae operating at low radio frequencies. The fresh data on the first galaxies and the diffuse gas in between them will test existing theoretical ideas about the formation and radiative effects of the first galaxies, and might even reveal new physics that has not yet been anticipated. This emerging interface between theory and observation will constitute an ideal opportunity for students considering a research career in astrophysics or cosmology. With this in mind, the book is intended to provide a self-contained introduction to research on the first galaxies at a technical level appropriate for a graduate student.

Various introductory sections of this book are based on an undergraduate-level book, entitled “How Did the First Stars and Galaxies Form?” by one of us (A.L.), which followed a cosmology class that he had taught over the past decade in the Astronomy and Physics departments at Harvard University. Other parts relate to overviews that both of us wrote over the past decade in the form of review articles. Where necessary, selected references are given to advanced papers and other review articles in the scientific literature.

The writing of this book was made possible thanks to the help we received from a large number of individuals. First and foremost, ... Special thanks go to ... for their careful reading of the book and detailed comments. We also thank Joey Munoz and Ramesh Narayan for their help with two plots. Finally, we are particularly grateful to our families for their support and patience during our lengthy pregnancy period with the book.

–A. L. & S. F.

Chapter One

Introduction

1.1 PRELIMINARY REMARKS

As the Universe expands, galaxies get separated from one another, and the average density of matter over a large volume of space is reduced. If we imagine playing the cosmic movie in reverse and tracing this evolution backwards in time, we would infer that there must have been an instant when the density of matter was infinite. This moment in time is the “ Big Bang”, before which we cannot reliably extrapolate our history. But even before we get all the way back to the Big Bang, there must have been a time when stars like our Sun and galaxies like our Milky Way^{i} did not exist, because the Universe was denser than they are. If so, how and when did the first stars and galaxies form?

Primitive versions of this question were considered by humans for thousands of years, long before it was realized that the Universe expands. Religious and philosophical texts attempted to provide a sketch of the big picture from which people could derive the answer. In retrospect, these attempts appear heroic in view of the scarcity of scientific data about the Universe prior to the twentieth century. To appreciate the progress made over the past century, consider, for example, the biblical story of Genesis. The opening chapter of the Bible asserts the following sequence of events: first, the Universe was created, then light was separated from darkness, water was separated from the sky, continents were separated from water, vegetation appeared spontaneously, stars formed, life emerged, and finally humans appeared on the scene.^{ii} Instead, the modern scientific order of events begins with the Big Bang, followed by an early period in which light (radiation) dominated and then a longer period dominated by matter, leading to the appearance of stars, planets, life on Earth, and eventually humans. Interestingly, the starting and end points of both versions are the same.

Cosmology is by now a mature empirical science. We are privileged to live in a time when the story of genesis (how the Universe started and developed) can be critically explored by direct observations. Because of the finite time it takes light to travel to us from distant sources, we can see images of the Universe when it was younger by looking deep into space through powerful telescopes.

Existing data sets include an image of the Universe when it was 400 thousand

iA star is a dense, hot ball of gas held together by gravity and powered by nuclear fusion reactions. A galaxy consists of a luminous core made of stars or cold gas surrounded by an extended halo of dark matter.

iiOf course, it is possible to interpret the biblical text in many possible ways. Here I focus on a plain reading of the original Hebrew text.

Figure 1.1 Image of the Universe when it first became transparent, 400 thousand years after the Big Bang, taken over five years by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite (http://map.gsfc.nasa.gov/). Slight density inhomogeneities at the level of one part in ∼ 10^{5} in the otherwise uniform early Universe imprinted hot and cold spots in the temperature map of the cosmic microwave background on the sky. The fluctuations are shown in units of μK, with the unperturbed temperature being 2.73 K. The same primordial inhomogeneities seeded the large-scale structure in the present-day Universe. The existence of background anisotropies was predicted in a number of theoretical papers three decades before the technology for taking this image became available.

years old (in the form of the cosmic microwave background in Figure ??), as well as images of individual galaxies when the Universe was older than a billion years. But there is a serious challenge: in between these two epochs was a period when the Universe was dark, stars had not yet formed, and the cosmic microwave background no longer traced the distribution of matter. And this is precisely the most interesting period, when the primordial soup evolved into the rich zoo of objects we now see. How can astronomers see this dark yet crucial time?

The situation is similar to having a photo album of a person that begins with the first ultra-sound image of him or her as an unborn baby and then skips to some additional photos of his or her years as teenager and adult. The late photos do not simply show a scaled up version of the first image. We are currently searching for the missing pages of the cosmic photo album that will tell us how the Universe evolved during its infancy to eventually make galaxies like our own Milky Way.

The observers are moving ahead along several fronts. The first involves the construction of large infrared telescopes on the ground and in space that will provide us with new (although rather expensive!) photos of galaxies in the Universe at intermediate ages. Current plans include ground-based telescopes which are 24-42 meter in diameter, and NASA’s successor to the Hubble Space Telescope, the James Webb Space Telescope. In addition, several observational groups around the globe are constructing radio arrays that will be capable of mapping the three-dimensional distribution of cosmic hydrogen left over from the Big Bang in the infant Universe. These arrays are aiming to detect the long-wavelength (redshifted 21-cm) radio emission from hydrogen atoms. Coincidentally, this long wavelength (or low frequency) overlaps with the band used for radio and television broadcasting, and so these telescopes include arrays of regular radio antennas that one can find in electronics stores. These antennas will reveal how the clumpy distribution of neutral hydrogen evolved with cosmic time. By the time the Universe was a few hundreds of millions of years old, the hydrogen distribution had been punched with holes like swiss cheese. These holes were created by the ultraviolet radiation from the first galaxies and black holes, which ionized the cosmic hydrogen in their vicinity.

Theoretical research has focused in recent years on predicting the signals expected from the above instruments and on providing motivation for these ambitious observational projects.

1.2 STANDARD COSMOLOGICAL MODEL

1.2.1 Cosmic Perspective

In 1915 Einstein came up with the general theory of relativity. He was inspired by the fact that all objects follow the same trajectories under the influence of gravity (the so-called “equivalence principle,” which by now has been tested to better than one part in a trillion), and realized that this would be a natural result if space-time is curved under the influence of matter. He wrote down an equation describing how the distribution of matter (on one side of his equation) determines the curvature of space-time (on the other side of his equation). He then applied his equation to describe the global dynamics of the Universe.

Back in 1915 there were no computers available, and Einstein’s equations for the Universe were particularly difficult to solve in the most general case. It was therefore necessary for Einstein to alleviate this difficulty by considering the simplest possible Universe, one that is homogeneous and isotropic. Homogeneity means uniform conditions everywhere (at any given time), and isotropy means the same conditions in all directions when looking out from one vantage point. The combination of these two simplifying assumptions is known as the cosmological principle.

The universe can be homogeneous but not isotropic: for example, the expansion rate could vary with direction. It can also be isotropic and not homogeneous: for example, we could be at the center of a spherically-symmetric mass distribution. But if it is isotropic around every point, then it must also be homogeneous. Isotropy is well established for the distribution of faint radio sources, optical galaxies, the Xray background, and most importantly the CMB. The constraints on homogeneity are less strict, but a cosmological model in which the Universe is isotropic and significantly inhomogeneous in spherical shells around our special location, is also excluded based on surveys of galaxies and quasars.

Under the simplifying assumptions associated with the cosmological principle, Einstein and his contemporaries were able to solve the equations. They were looking for their “lost keys” (solutions) under the “lamppost” (simplifying assumptions), but the real Universe is not bound by any contract to be the simplest that we can imagine. In fact, it is truly remarkable in the first place that we dare describe the conditions across vast regions of space based on the blueprint of the laws of physics that describe the conditions here on Earth. Our daily life teaches us too often that we fail to appreciate complexity, and that an elegant model for reality is often too idealized for describing the truth (along the lines of approximating a cow as a spherical object).

Back in 1915 Einstein had the wrong notion of the Universe; at the time people associated the Universe with the Milky Way galaxy and regarded all the “nebulae,” which we now know are distant galaxies, as constituents within our own Milky Way galaxy. Because the Milky Way is not expanding, Einstein attempted to reproduce a static universe with his equations. This turned out to be possible after adding a cosmological constant, whose negative gravity would exactly counteract that of matter. However, later Einstein realized that this solution is unstable: a slight enhancement in density would make the density grow even further. As it turns out, there are no stable static solution to Einstein’s equations for a homogeneous and isotropic Universe. The Universe must either be expanding or contracting. Less than a decade later, Edwin Hubble discovered that the nebulae previously considered to be constituents of the Milky Way galaxy are receding away from us at a speed v that is proportional to their distance r, namely v = H_{0}r with H_{0} a spatial constant (which could evolve with time), commonly termed the Hubble constant.^{iii} Hubble’s data indicated that the Universe is expanding.

iiiThe redshift data examined by Hubble was mostly collected by Vesto Slipher a decade earlier and only partly by Hubble’s assistant, Milton L. Humason. The linear local relation between redshift and distance was first formulated by Georges Lemaître in 1927, two years prior to the observational paper written by Hubble and Humason.

Einstein was remarkably successful in asserting the cosmological principle. As it turns out, our latest data indicates that the real Universe is homogeneous and isotropic on the largest observable scales to within one part in a hundred thousand. Fortuitously, Einstein’s simplifying assumptions turned out to be extremely accurate in describing reality: the keys were indeed lying next to the lamppost. Our Universe happens to be the simplest we could have imagined, for which Einstein’s equations can be easily solved.

Why was the Universe prepared to be in this special state? Cosmologists were able to go one step further and demonstrate that an early phase transition, called cosmic inflation – during which the expansion of the Universe accelerated exponentially, could have naturally produced the conditions postulated by the cosmological principle. One is left to wonder whether the existence of inflation is just a fortunate consequence of the fundamental laws of nature, or whether perhaps the special conditions of the specific region of space-time we inhabit were selected out of many random possibilities elsewhere by the prerequisite that they allow our existence. The opinions of cosmologists on this question are split.

1.2.2 Origin of Structure

Hubble’s discovery of the expansion of the Universe has immediate implications with respect to the past and future of the Universe. If we reverse in our mind the expansion history back in time, we realize that the Universe must have been denser in its past. In fact, there must have been a point in time where the matter density was infinite, at the moment of the so-called Big Bang. Indeed we do detect relics from a hotter denser phase of the Universe in the form of light elements (such as deuterium, helium and lithium) as well as the Cosmic Microwave Background (CMB). At early times, this radiation coupled extremely well to the cosmic gas and obtained a spectrum known as blackbody, that was predicted a century ago to characterize matter and radiation in equilibrium. The CMB provides the best example of a blackbody spectrum we have.

To get a rough estimate of when the Big Bang occurred, we may simply divide the distance of all galaxies by their recession velocity. This gives a unique answer, ∼ r/v ∼ 1/H_{0}, which is independent of distance.^{iv} The latest measurements of the Hubble constant give a value of H_{0} ≈ 70 kilometers per second per Megaparsec,^{v} implying a current age for the Universe 1/H_{0} of 14 billion years (or 5 × 10^{17} seconds).

The second implication concerns our future. A fortunate feature of a sphericallysymmetric Universe is that when considering a sphere of matter in it, we are allowed to ignore the gravitational influence of everything outside this sphere. If we empty the sphere and consider a test particle on the boundary of an empty void

ivAlthough this is an approximate estimate, it turns out to be within a few percent of the true age of our Universe owing to a coincidence. The cosmic expansion at first decelerated and then accelerated with the two almost canceling each other out at the present-time, giving the same age as if the expansion were at a constant speed (as would be strictly true only in an empty Universe).

vA megaparsec (abbreviated as ‘Mpc’) is equivalent to 3.086 × 10^{24} centimeter, or roughly the distance traveled by light in three million years.

embedded in a uniform Universe, the particle will experience no net gravitational acceleration. This result, known as Birkhoff’s theorem, is reminiscent of Newton’s “iron sphere theorem.” It allows us to solve the equations of motion for matter on the boundary of the sphere through a local analysis without worrying about the rest of the Universe. Therefore, if the sphere has exactly the same conditions as the rest of the Universe, we may deduce the global expansion history of the Universe by examining its behavior. If the sphere is slightly denser than the mean, we will infer how its density contrast will evolve relative to the background Universe.

The equation describing the motion of a spherical shell of matter is identical to the equation of motion of a rocket launched from the surface of the Earth. The rocket will escape to infinity if its kinetic energy exceeds its gravitational binding energy, making its total energy positive. However, if its total energy is negative, the rocket will reach a maximum height and then fall back. In order to figure out the future evolution of the Universe, we need to examine the energy of a spherical shell of matter relative to the origin. With a uniform density ρ, a spherical shell of radius r would have a total mass M = ρ × (\frac{4π}{3} r^{3}) enclosed within it. Its energy per unit mass is the sum of the kinetic energy due to its expansion speed v = Hr, \frac{1}{2}v^{2}, and its potential gravitational energy, −GM/r (where G is Newton’s constant), namely E = \frac{1}{2}v^{2} − \frac{GM}{r}. By substituting the above relations for v and M, it can be easily shown that E = \frac{1}{2}v^{2}(1 − Ω), where Ω = ρ/ρ_{c} and ρ_{c} = 3H^{2}/8πG is defined as the critical density. We therefore find that there are three possible scenarios for the cosmic expansion. The Universe has either: (i) Ω > 1, making it gravitationally bound with E < 0 – such a “closed Universe” will turn-around and end up collapsing towards a “big crunch”; (ii) Ω < 1, making it gravitationally unbound with E > 0 – such an “open Universe” will expand forever; or the borderline case (iii) Ω = 1, making the Universe marginally bound or “flat” with E = 0.

Einstein’s equations relate the geometry of space to its matter content through the value of Ω: an open Universe has a geometry of a saddle with a negative spatial curvature, a closed Universe has the geometry of a spherical globe with a positive curvature, and a flat Universe has a flat geometry with no curvature. Our observable section of the Universe appears to be flat.

Now we are at a position to understand how objects, like the Milky Way galaxy, have formed out of small density inhomogeneities that get amplified by gravity.

Let us consider for simplicity the background of a marginally bound (flat) Universe which is dominated by matter. In such a background, only a slight enhancement in density is required for exceeding the critical density ρ_{c}. Because of Birkhoff’s theorem, a spherical region that is denser than the mean will behave as if it is part of a closed Universe and increase its density contrast with time, while an underdense spherical region will behave as if it is part of an open Universe and appear more vacant with time relative to the background, as illustrated in Figure ??. Starting with slight density enhancements that bring them above the critical value ρ_{c}, the overdense regions will initially expand, reach a maximum radius, and then collapse upon themselves (like the trajectory of a rocket launched straight up, away from the center of the Earth). An initially slightly inhomogeneous Universe − − − − − − − − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − − − − − − − −

Figure 1.2 Top: Schematic illustration of the growth of perturbations to collapsed halos through gravitational instability. Once the overdense regions exceed a threshold density contrast above unity, they turn around and collapse to form halos. The material that makes the halos originated in the voids that separate them. Middle: A simple model for the collapse of a spherical region. The dynamical fate of a rocket which is launched from the surface of the Earth depends on the sign of its energy per unit mass, E = \frac{1}{2} v^{2} − GM_{⊕}/r. The behavior of a spherical shell of matter on the boundary of an overdense region (embedded in a homogeneous and isotropic Universe) can be analyzed in a similar fashion. Bottom: A collapsing region may end up as a galaxy, like NGC 4414, shown here (image credit: NASA and ESA). The halo gas cools and condenses to a compact disk surrounded by an extended dark matter halo.

will end up clumpy, with collapsed objects forming out of overdense regions. The material to make the objects is drained out of the intervening underdense regions, which end up as voids.

The Universe we live in started with primordial density perturbations of a fractional amplitude ∼ 10^{−5}. The overdensities were amplified at late times (once matter dominated the cosmic mass budget) up to values close to unity and collapsed to make objects, first on small scales. We have not yet seen the first small galaxies that started the process that eventually led to the formation of big galaxies like the Milky Way. The search for the first galaxies is a search for our origins.

Life as we know it on planet Earth requires water. The water molecule includes oxygen, an element that was not made in the Big Bang and did not exist until the first stars had formed. Therefore our form of life could not have existed in the first hundred millions of years after the Big Bang, before the first stars had formed. There is also no guarantee that life will persist in the distant future.

1.2.3 Geometry of Space

How can we tell the difference between the flat surface of a book and the curved surface of a balloon? A simple way would be to draw a triangle of straight lines between three points on those surfaces and measure the sum of the three angles of the triangle. The Greek mathematician Euclid demonstrated that the sum of these angles must be 180 degrees (or π radians) on a flat surface. Twenty-one centuries later, the German mathematician Bernhard Riemann extended the field of geometry to curved spaces, which played an important role in the development of Einstein’s general theory of relativity. For a triangle drawn on a positively curved surface, like that of a balloon, the sum of the angles is larger than 180 degrees. (This can be easily figured out by examining a globe and noticing that any line connecting one of the poles to the equator opens an angle of 90 degrees relative to the equator. Adding the third angle in any triangle stretched between the pole and the equator would surely result in a total of more than 180 degrees.) According to Einstein’s equations, the geometry of the Universe is dictated by its matter content; in particular, the Universe is flat only if the total Ω equals unity. Is it possible to draw a triangle across the entire Universe and measure its geometry?

Remarkably, the answer is yes. At the end of the twentieth century cosmologists were able to perform this experiment^{1} by adopting a simple yardstick provided by the early Universe. The familiar experience of dropping a stone in the middle of a pond results in a circular wave crest that propagates outwards. Similarly, perturbing the smooth Universe at a single point at the Big Bang would have resulted in a spherical sound wave propagating out from that point. The wave would have traveled at the speed of sound, which was of order the speed of light c (or more precisely, \frac{1}{\sqrt{3}}c) early on when radiation dominated the cosmic mass budget. At any given time, all the points extending to the distance traveled by the wave are affected by the original pointlike perturbation. The conditions outside this “sound horizon” will remain uncorrelated with the central point, because acoustic information has not been able to reach them at that time. The temperature fluctuations of the CMB trace the simple sum of many such pointlike perturbations that were generated in the Big Bang. The patterns they delineate would therefore show a characteristic correlation scale, corresponding to the sound horizon at the time when the CMB was produced, 400 thousand years after the Big Bang. By measuring the apparent angular scale of this “standard ruler” on the sky, known as the acoustic peak in the CMB, and comparing it to theory, experimental cosmologists inferred from the simple geometry of triangles that the Universe is flat.

The inferred flatness is a natural consequence of the early period of vast expansion, known as cosmic inflation, during which any initial curvature was flattened. Indeed a small patch of a fixed size (representing our current observable region in the cosmological context) on the surface of a vastly inflated balloon would appear nearly flat. The sum of the angles on a non-expanding triangle placed on this patch would get arbitrarily close to 180 degrees as the balloon inflates.

1.2.4 Observing our Past: Cosmic Archaeology

Our Universe is the simplest possible on two counts: it satisfies the cosmological principle, and it has a flat geometry. The mathematical description of an expanding, homogeneous, and isotropic Universe with a flat geometry is straightforward. We can imagine filling up space with clocks that are all synchronized. At any given snapshot in time the physical conditions (density, temperature) are the same everywhere. But as time goes on, the spatial separation between the clocks will increase. The stretching of space can be described by a time-dependent scale factor, a(t). A separation measured at time t_{1} as r(t_{1}) will appear at time t_{2} to have a length r(t_{2}) = r(t_{1})[a(t_{2})/a(t_{1})].

A natural question to ask is whether our human bodies or even the solar system, are also expanding as the Universe expands. The answer is no, because these systems are held together by forces whose strength far exceeds the cosmic force. The mean density of the Universe today, ρ̄, is 29 orders of magnitude smaller than the density of our body. Not only are the electromagnetic forces that keep the atoms in our body together far greater than gravity, but even the gravitational self-force of our body on itself overwhelms the cosmic influence. Only on very large scales does the cosmic gravitational force dominate the scene. This also implies that we cannot observe the cosmic expansion with a local laboratory experiment; in order to notice the expansion we need to observe sources which are spread over the vast scales of millions of light years.

Einstein’s equations relate the geometry of space to its matter content. Recent data indicates that our observable section of the Universe is flat (meaning that the sum of the angles in a triangle is 180^{∘}). The inferred flatness is a natural consequence of the early period of vast expansion, known as cosmic inflation, during which any initial curvature was flattened. Indeed a small patch of a fixed size (representing our current observable region in the cosmological context) on the surface of a vastly inflated balloon would appear nearly flat. The sum of the angles on a non-expanding triangle placed on this patch would get arbitrarily close to 180 degrees as the balloon inflates.

Einstein’s general relativity (GR) equations do not admit a stable steady-state (non-expanding or contracting) solution. A decade after Einstein’s invention of GR, Hubble demonstrated that our Universe is indeed expanding. The space-time of an expanding, homogeneous and isotropic, flat Universe can be described very simply. Because the cosmological principle, we can establish a unique time coordinate throughout space by distributing clocks which are all synchronized throughout the Universe, so that each clock would measure the same time t since the Big Bang. The space-time (4–dimensional) line element ds, commonly defined to vanish for a photon, is described by the Friedmann-Robertson-Walker (FRW) metric,

ds^{2} = c^{2}dt^{2} − dℓ^{2},  (1.1)

where c is the speed of light and dℓ is the spatial line-element. The cosmic expansion can be incorporated through a scale factor a(t) which multiples the fixed (x, y, z) coordinates tagging the clocks which are themselves “comoving” with the cosmic expansion. For a flat space,

dℓ^{2} = a(t)^{2}(dx^{2} + dy^{2} + dz^{2}) = a^{2}(t)(dR^{2} + R^{2}dΩ),  (1.2)

where dΩ = dθ^{2} + sin^{2} θdϕ^{2} with (R, θ, ϕ) being the spherical coordinates centered on the observer, and (x, y, z) = R(cos θ, sin θ cos ϕ, sin θ sin ϕ)

A source located at a separation r = a(t)R from us would move at a velocity v = dr/dt = ȧR = (ȧ/a)r, where ȧ = da/dt. Here r is a time-independent tag, denoting the present-day distance of the source. Defining H = ȧ/a which is constant in space, we recover the Hubble expansion law v = Hr.

Edwin Hubble measured the expansion of the Universe using the Doppler effect. We are all familiar with the same effect for sound waves: when a moving car sounds its horn, the pitch (frequency) we hear is different if the car is approaching us or receding away. Similarly, the wavelength of light depends on the velocity of the source relative to us. As the Universe expands, a light source will move away from us and its Doppler effect will change with time. The Doppler formula for a nearby source of light (with a recession speed much smaller than the speed of light) gives

\frac{Δν}{ν} ≈ −\frac{Δv}{c} = − (\frac{ȧ}{a}) (\frac{r}{c} ) = −\frac{(ȧΔt)}{a} = −\frac{Δa}{a},  (1.3)

with the solution, ν ∝ a^{−1}. Correspondingly, the wavelength scales as λ = (c/ν) ∝ a. We could have anticipated this outcome since a wavelength can be used as a measure of distance and should therefore be stretched as the Universe expands. The redshift z is defined through the factor (1 + z) by which the photon wavelength was stretched (or its frequency reduced) between its emission and observation times. If we define a = 1 today, then a = 1/(1 + z) at earlier times. Higher redshifts correspond to a higher recession speed of the source relative to us (ultimately approaching the speed of light when the redshift goes to infinity), which in turn implies a larger distance (ultimately approaching our horizon, which is the distance traveled by light since the Big Bang) and an earlier emission time of the source in order for the photons to reach us today.

We see high-redshift sources as they looked at early cosmic times. Observational cosmology is like archaeology – the deeper we look into space the more ancient the clues about our history are (see Figure ??). But there is a limit to how far back we can see. In principle, we can image the Universe only as long as it was transparent,

Figure 1.3 Cosmic archaeology of the observable volume of the Universe, in comoving coordinates (which factor out the cosmic expansion). The outermost observable boundary (z = ∞) marks the comoving distance that light has traveled since the Big Bang. Future observatories aim to map most of the observable volume of our Universe, and improve dramatically the statistical information we have about the density fluctuations within it. Existing data on the CMB probes mainly a very thin shell at the hydrogen recombination epoch (z ∼ 10^{3}, beyond which the Universe is opaque), and current large-scale galaxy surveys map only a small region near us at the center of the diagram. The formation epoch of the first galaxies that culminated with hydrogen reionization at a redshift z ∼ 10 is shaded grey. Note that the comoving volume out to any of these redshifts scales as the distance cubed.

corresponding to redshifts z < 10^{3} for photons. The first galaxies are believed to have formed long after that.

The expansion history of the Universe is captured by the scale factor a(t). We can write a simple equation for the evolution of a(t) based on the behavior of a small region of space. For that purpose we need to incorporate the fact that in Einstein’s theory of gravity, not only does mass density ρ gravitate but pressure p does too. In a homogeneous and isotropic Universe, the quantity ρ_{grav} = (ρ + 3p/c^{2}) plays the role of the gravitating mass density ρ of Newtonian gravity.^{2} There are several examples to consider. For a radiation fluid,^{vi} p_{rad}/c^{2} = \frac{1}{3}ρ_{rad}, implying that ρ_{grav} = 2ρ_{rad}. On the other hand, for a constant vacuum density (the so-called “cosmological constant”), the pressure is negative because by opening up a new volume increment ΔV one gains an energy ρc^{2}ΔV instead of losing energy, as is the case for normal fluids that expand into more space. In thermodynamics, pressure is derived from the deficit in energy per unit of new volume, which in this case gives p_{vac}/c^{2} = −ρ_{vac}. This in turn leads to another reversal of signs, ρ_{grav} = (ρ_{vac} + 3p_{vac}/c^{2}) = −2ρ_{vac}, which may be interpreted as repulsive gravity! This surprising result gives rise to the phenomenon of accelerated cosmic expansion, which characterized the early period of cosmic inflation as well as the latest six billions years of cosmic history.

As the Universe expands and the scale factor increases, the matter mass density declines inversely with volume, ρ_{matter} ∝ a^{−3}, whereas the radiation energy density (which includes the CMB and three species of relativistic neutrinos) decreases as ρ_{rad}c^{2} ∝ a^{−4}, because not only is the density of photons diluted as a^{−3}, but the energy per photon hν = hc/λ (where h is Planck’s constant) declines as a^{−1}. Today ρ_{matter} is larger than ρ_{rad} (assuming massless neutrinos) by a factor of ∼ 3, 300, but at (1 + z) ∼ 3, 300 the two were equal, and at even higher redshifts the radiation dominated. Since a stable vacuum does not get diluted with cosmic expansion, the present-day ρ_{vac} remained a constant and dominated over ρ_{matter} and ρ_{rad} only at late times (whereas the unstable “false vacuum” that dominated during inflation has decayed when inflation ended).

1.3 MILESTONES IN COSMIC EVOLUTION

The gravitating mass, M_{grav} = ρ_{grav}V, enclosed by a spherical shell of radius a(t) and volume V = \frac{4π}{3} a^{3}, induces an acceleration

\frac{d^{2}a}{dt^{2}} = −\frac{GM_{grav}}{a^{2}} .  (1.4)

Since ρ_{grav} = ρ+3p/c^{2}, we need to know how pressure evolves with the expansion factor a(t). This is obtained from the thermodynamic relation mentioned above between the change in the internal energy d(ρc^{2}V) and the pdV work done by the pressure, d(ρc^{2}V) = −pdV. This relation implies −3paȧ/c^{2} = a^{2}ρ̇ + 3ρaȧ,

viThe momentum of each photon is \frac{1}{c} of its energy. The pressure is defined as the momentum flux along one dimension out of three, and is therefore given by \frac{1}{3} ρ_{rad}c^{2}, where ρ_{rad} is the mass density of the radiation.

where a dot denotes a time derivative. Multiplying equation (??) by ȧ and making use of this relation yields our familiar result

E = \frac{1}{2}ȧ^{2} − \frac{GM}{a} ,  (1.5)

where E is a constant of integration and M ≡ ρV. As discussed before, the spherical shell will expand forever (being gravitationally unbound) if E ≥ 0, but will eventually collapse (being gravitationally bound) if E < 0. Making use of the Hubble parameter, H = ȧ/a, equation (??) can be re-written as

\frac{E}{\frac{1}{2}ȧ^{2}} = 1 − Ω,  (1.6)

where Ω = ρ/ρ_{c}, with

ρ_{c} = \frac{3H^{2}}{8πG} = 9.2 × 10^{−30} \frac{g}{cm^{3}} (\frac{H}{70 km s^{−1}Mpc^{−1}} )^{2} .  (1.7)

With Ω_{m}, Ω_{Λ}, and Ω_{r} denoting the present contributions to Ω from matter (including cold dark matter as well as a contribution Ω_{b} from ordinary matter of protons and neutrons, or “baryons”), vacuum density (cosmological constant), and radiation, respectively, a flat universe satisfies

\frac{H(t)}{H_{0}} = [\frac{Ω_{m}}{a^{3}} + Ω_{Λ} + \frac{Ω_{r}}{a^{4}} ]^{1/2} ,  (1.8)

where we define H_{0} and Ω_{0} = (Ω_{m} + Ω_{Λ} + Ω_{r}) = 1 to be the present-day values of H and Ω, respectively.

In the particularly simple case of a flat Universe, we find that if matter dominates then a ∝ t^{2/3}, if radiation dominates then a ∝ t^{1/2}, and if the vacuum density dominates then a ∝ exp{H_{vac}t} with H_{vac} = (8πGρ_{vac}/3)^{1/2} being a constant. In the beginning, after inflation ended, the mass density of our Universe ρ was at first dominated by radiation at redshifts z > 3, 300, then it became dominated by matter at 0.3 < z < 3, 300, and finally was dominated by the vacuum at z < 0.3. The vacuum started to dominate ρ_{grav} already at z < 0.7 or six billion years ago. Figure ?? illustrates the mass budget in the present-day Universe and during the epoch when the first galaxies had formed.

The above results for a(t) have two interesting implications. First, we can figure out the relationship between the time since the Big Bang and redshift since a = (1 + z)^{−1}. For example, during the matter-dominated era (1 < z < 10^{3}),

t ≈ \frac{2}{3H_{0}Ω_{m}^{1/2}(1 + z)^{3/2}} = \frac{0.95 × 10^{9} years}{[(1 + z)/7]^{3/2}}.  (1.9)

Second, we note the remarkable exponential expansion for a vacuum dominated phase. This accelerated expansion serves an important purpose in explaining a few puzzling features of our Universe. We already noticed that our Universe was prepared in a very special initial state: nearly isotropic and homogeneous, with Ω close to unity and a flat geometry. In fact, it took the CMB photons nearly the entire age of the Universe to travel towards us. Therefore, it should take them twice as long to bridge across their points of origin on opposite sides of the sky. How is it possible then that the conditions of the Universe (as reflected in the nearly uniform CMB temperature) were prepared to be the same in regions that were never in causal contact before? Such a degree of organization is highly unlikely to occur at random. If we receive our clothes ironed out and folded neatly, we know that there must have a been a process that caused it. Cosmologists have identified an analogous “ironing process” in the form of cosmic inflation. This process is associated with an early period during which the Universe was dominated temporarily by the mass density of an elevated vacuum state, and experienced exponential expansion by at least ∼ 60 e-folds. This vast expansion “ironed out” any initial curvature of our environment, and generated a flat geometry and nearly uniform conditions across a region far greater than our current horizon. After the elevated vacuum state decayed, the Universe became dominated by radiation.

The early epoch of inflation is important not just in producing the global properties of the Universe but also in generating the inhomogeneities that seeded the formation of galaxies within it.^{3} The vacuum energy density that had driven inflation encountered quantum mechanical fluctuations. After the perturbations were stretched beyond the horizon of the infant Universe (which today would have occupied the size no bigger than a human hand), they materialized as perturbations in the mass density of radiation and matter. The last perturbations to leave the horizon during inflation eventually entered back after inflation ended (when the scale factor grew more slowly than ct). It is tantalizing to contemplate the notion that galaxies, which represent massive classical objects with ∼ 10^{67} atoms in today’s Universe, might have originated from sub-atomic quantum-mechanical fluctuations at early times.

After inflation, an unknown process, called “baryo-genesis” or “lepto-genesis”, generated an excess of particles (baryons and leptons) over anti-particles.^{vii} As the Universe cooled to a temperature of hundreds of MeV (with 1MeV/k_{B} = 1.1604 × 10^{10}K), protons and neutrons condensed out of the primordial quarkgluon plasma through the so-called QCD phase transition. At about one second after the Big Bang, the temperature declined to ∼ 1 MeV, and the weakly interacting neutrinos decoupled. Shortly afterwards the abundance of neutrons relative to protons froze and electrons and positrons annihilated. In the next few minutes, nuclear fusion reactions produced light elements more massive than hydrogen, such as deuterium, helium, and lithium, in abundances that match those observed today in regions where gas has not been processed subsequently through stellar interiors. Although the transition to matter domination occurred at a redshift z ∼ 3, 300 the Universe remained hot enough for the gas to be ionized, and electron-photon scattering effectively coupled ordinary matter and radiation. At z ∼ 1, 100 the temperature dipped below ∼ 3, 000K, and free electrons recombined with protons to form neutral hydrogen atoms. As soon as the dense fog of free electrons was depleted, the Universe became transparent to the relic radiation, which is observed at present as the CMB. These milestones of the thermal history are depicted in Figure

viiAnti-particles are identical to particles but with opposite electric charge. Today, the ordinary matter in the Universe is observed to consist almost entirely of particles. The origin of the asymmetry in the cosmic abundance of matter over anti-matter is stil an unresolved puzzle.

INTRODUCTION 15

Figure 1.4 Following inflation, the Universe went through sev