General relativity - Cosmology and Dark Energy

Cosmology Introduction Cosmology is the study of the universe as a whole—its origin, evolution, and large-scale structure. Modern cosmology is built on Einstein's theory of general relativity and supported by precise astronomical observations. Over the past century, we've discovered that the universe is expanding, began with an extremely hot and dense state called the Big Bang, and is dominated by mysterious components called dark matter and dark energy. This section explores how we describe the universe mathematically, what we observe, and how observations confirm our models. Einstein's Field Equations and the Cosmological Constant Einstein's field equations form the foundation of modern cosmology. These equations relate the curvature of spacetime to the distribution of matter and energy: $$G{\mu\nu} + \Lambda g{\mu\nu} = 8\pi G T{\mu\nu}$$ The key term here is $\Lambda$, the cosmological constant. Einstein originally introduced this term to prevent his equations from predicting a collapsing universe. The cosmological constant represents a uniform energy density throughout space that affects the universe's expansion rate. While Einstein later abandoned it, modern observations have revealed that the cosmological constant (or something like it) is indeed necessary to explain our universe. This equation tells us that spacetime geometry is directly influenced by matter and energy. For cosmology, this means we can use observations of how the universe expands to infer what matter and energy it contains—and vice versa. The FLRW Models: Describing an Expanding Universe To apply Einstein's equations to the universe as a whole, cosmologists make two crucial assumptions: the universe is homogeneous (looks the same everywhere on large scales) and isotropic (looks the same in all directions). These assumptions are supported by observations of galaxy distributions and the cosmic microwave background. Under these assumptions, Einstein's equations yield a family of solutions called the Friedmann–Lemaître–Robertson–Walker (FLRW) metrics. These metrics describe how spacetime expands or contracts with time. The key result is the Friedmann equations, which describe the expansion rate of the universe: $$H^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2} + \frac{\Lambda}{3}$$ Here, $H$ is the Hubble parameter (the expansion rate), $\rho$ is the density of matter and energy, $a$ is the scale factor (which grows with time as the universe expands), and $k$ determines the geometry of space (whether it's flat, closed, or open). The FLRW models predict that the universe evolved from an extremely hot, dense state (the Big Bang) roughly 14 billion years ago. As the universe expanded, it cooled, allowing complex structures like atoms, stars, and galaxies to form. This framework has become the standard model of cosmology because it successfully explains nearly all observations of the large-scale universe. Testing Cosmological Models: Observational Parameters FLRW models contain several parameters—such as the matter density, the expansion rate today (the Hubble constant), and the amount of dark energy—that cannot be derived from theory alone. Instead, we determine these parameters from astronomical observations. A small set of measurements constrains all the others: The Hubble constant $H0$ describes the universe's current expansion rate The matter density determines how much gravitational pull exists The dark energy density controls whether the expansion accelerates or decelerates By measuring these parameters from observations, we can test whether FLRW models correctly describe our universe. When predictions match observations, we gain confidence that our model is correct. When predictions disagree with data, we must revise our understanding. Key Predictions Confirmed by Observation Primordial Nucleosynthesis In the first few minutes after the Big Bang, the universe was so hot that protons and neutrons fused together to form light elements. Big Bang nucleosynthesis (BBN) theory predicts the abundance of hydrogen, helium, lithium, and other light elements produced in this early phase. When we measure the abundances of these elements in the oldest stars and gas clouds, we find that they match BBN predictions remarkably well. This agreement provides strong evidence that the universe really did begin in a hot, dense state. If the early universe were cold or had different properties, the predicted abundances would differ significantly from what we observe. The Cosmic Microwave Background (CMB) The early universe was so hot that it was opaque—filled with a glowing plasma of electrons and photons, much like the interior of a star. As the universe expanded and cooled, electrons combined with protons to form neutral hydrogen atoms. This event, called recombination, happened roughly 380,000 years after the Big Bang. After recombination, the universe became transparent. The photons from that era have been traveling toward us ever since, but the expansion of the universe has stretched their wavelengths. Today, we observe these ancient photons as the cosmic microwave background (CMB), a faint glow of radiation in all directions with a temperature of about $2.73\,\mathrm{K}$. The CMB is direct evidence that the early universe was indeed hot and dense. Its properties—its near-perfect thermal spectrum and tiny temperature variations from place to place—match predictions from FLRW models with extraordinary precision. These small variations in temperature also contain clues about the composition and geometry of the universe, which modern observations decode to determine cosmological parameters. Large-Scale Structure On the largest scales, the universe is a web of galaxy clusters connected by filaments, with vast voids in between. This structure emerged from tiny density variations in the early universe. Where regions were slightly denser than average, gravity pulled more matter in, creating overdensities. Conversely, underdense regions became even emptier. FLRW models predict that this structure grows from seed perturbations present at the time of recombination. These seed perturbations are observable in the CMB's temperature variations. The pattern of galaxy clustering we observe today matches what theory predicts we should see if these models are correct. This agreement represents another confirmation that FLRW cosmology describes reality. Dark Matter: The Universe's Missing Mass When astronomers measure the speeds of stars orbiting in galaxies, they find something puzzling: stars at large distances from a galaxy's center move faster than they should based on the observable matter (stars and gas) alone. According to Newton's law of gravity, distant stars should move slowly because less mass lies within their orbits. Yet they don't. This discrepancy, seen in galactic rotation curves, indicates that galaxies contain far more matter than we can see. This invisible matter is called dark matter. Dark matter's properties are remarkable: It exerts gravitational force like ordinary matter It does not emit, absorb, or reflect light (hence "dark") It does not interact electromagnetically with ordinary matter It comprises roughly 85% of all matter in the universe (or about 27% of the universe's total energy density) Several lines of evidence confirm dark matter's existence beyond rotation curves: Gravitational lensing: The gravity of galaxy clusters bends light from distant objects, and the amount of bending requires more mass than we see CMB anisotropies: Temperature variations in the CMB are sensitive to the total matter density, which observations confirm is dominated by dark matter Large-scale structure: The growth of galaxies and clusters matches predictions only if dark matter provides most of the gravitational force The nature of dark matter remains unknown. Leading candidates include weakly interacting massive particles (WIMPs) and axions, but these have not yet been directly detected. Nevertheless, the evidence that dark matter exists is overwhelming. Dark Energy and Cosmic Acceleration For most of the twentieth century, astronomers expected that gravity would slow the universe's expansion. The question was whether it would eventually stop and reverse (in a closed universe) or expand forever at a decreasing rate (in an open universe). In 1998, observations of distant Type Ia supernovae delivered a stunning surprise: the universe's expansion is accelerating. Distant supernovae appeared dimmer than expected, indicating they were farther away than their redshifts suggested. The only explanation is that the universe has been speeding up its expansion. What could cause this acceleration? Ordinary matter and radiation only provide attractive gravity. The observed acceleration implies a component of the universe with negative pressure—it pushes spacetime apart rather than pulling it together. The simplest candidate is the cosmological constant $\Lambda$ we mentioned earlier. If the cosmological constant explains the acceleration, then observations tell us that dark energy (the vacuum energy associated with $\Lambda$) comprises about 68% of the universe's total energy density. This makes dark energy the dominant component of the universe by far. The nature of dark energy is profound mystery. Why does the universe have a cosmological constant? Why is its value so small (so the universe hasn't expanded away all structure) yet nonzero (so we observe acceleration)? These questions remain open, though several alternative theories propose modified gravity or exotic forms of matter instead of $\Lambda$. Cosmic Inflation: Solving the Universe's Puzzles FLRW models raise a puzzle: why does the universe appear so homogeneous? Why do the temperature and density appear the same everywhere, even in regions that never had time to exchange information with each other (since light can only travel so far in the universe's age)? This is the horizon problem. Additionally, observations show the universe is remarkably flat—meaning space follows Euclidean geometry rather than being noticeably curved. Yet FLRW models predict that unless the density is fine-tuned to extraordinary precision, the universe should be noticeably curved. Why is the universe so flat? In the early 1980s, physicist Alan Guth proposed that these puzzles dissolve if the universe underwent a brief period of exponential expansion very early on, around $10^{-36}$ seconds after the Big Bang. This is cosmic inflation. During inflation, a quantum field called the inflaton dominates the universe's energy density and drives exponential expansion. A tiny region expands to become vast, smoothing out any initial irregularities. This solves the horizon problem: regions we see today were once adjacent, so they had time to come into thermal equilibrium before inflation stretched them apart. Inflation also solves the flatness problem: exponential expansion flattens out any curvature, like stretching a balloon's surface smooth. This naturally explains why the universe is so flat. Most importantly, quantum fluctuations in the inflaton field during inflation leave imprints—tiny density variations that seed the large-scale structure we observe today. These variations appear as temperature variations in the CMB. Recent precise measurements of the CMB, particularly from the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck satellite, have provided evidence supporting inflation. The pattern of temperature variations matches predictions from inflation theory, giving us confidence that inflation really occurred. <extrainfo> A note on inflation's complications: While inflation successfully solves several puzzles, it also raises new questions. What field is the inflaton? Why did inflation start and stop? These remain active areas of research. Several variations of inflation theory exist, and astronomers continue to test them against observations. </extrainfo> Large-Scale Structure Formation The universe today is lumpy—it contains galaxies, clusters, and superclusters. Yet observations of the CMB show that the early universe was remarkably uniform. How did the large-scale structure form? The answer lies in gravitational instability. The early universe wasn't perfectly uniform; tiny density variations existed, seeded by quantum fluctuations during inflation. Where density was slightly higher than average, gravity pulled in more matter, enhancing the overdensity. Where density was lower, matter flowed away, creating deeper voids. FLRW models predict how these density perturbations grow over cosmic time. Starting from the tiny variations imprinted by inflation, gravity amplifies them into galaxies and clusters. The growth rate depends on what the universe contains: ordinary matter, dark matter, and dark energy all influence how structures form. The observed distribution of galaxies matches predictions from FLRW models remarkably well when we include dark matter and dark energy with their measured amounts. This provides strong evidence that our understanding of cosmology is fundamentally correct, even though dark energy's nature remains mysterious. Computer simulations using N-body methods (which follow the gravitational evolution of billions of particles) can reproduce the observed cosmic web of galaxies, clusters, and voids. These simulations further confirm that FLRW cosmology with dark matter and dark energy successfully describes how the large-scale universe evolved from an early, nearly uniform state to the lumpy universe we see today. Summary Modern cosmology rests on Einstein's general relativity, described by the FLRW models under the assumptions that the universe is homogeneous and isotropic on large scales. These models make testable predictions about the universe's composition, expansion history, and structure formation. Observations of the cosmic microwave background, the abundances of light elements, distant supernovae, and galaxy clustering all confirm the FLRW framework. These observations reveal that the universe is composed primarily of dark matter (about 27% of total energy density) and dark energy (about 68%), with ordinary matter making up only about 5%. The universe began about 14 billion years ago in a hot, dense state called the Big Bang, underwent rapid exponential expansion called inflation, cooled enough for atoms and stars to form, and recently began accelerating its expansion due to dark energy. Understanding these elements requires accepting that most of the universe consists of matter and energy we do not yet fully understand—a humbling reminder that cosmology remains an active frontier of science.