Foundations of Modern Cosmology and the 'Dark' Problem

Foundations of Modern Cosmology and the 'Dark' Problem

1.1 The 95% Problem: A Universe We Cannot See

The most humbling fact in all of modern physics is also the simplest to state. When we tally the energy content of the universe using the standard cosmological model — calibrated against the cosmic microwave background (CMB), Type Ia supernovae, baryon acoustic oscillations, and large-scale galaxy surveys — we find that roughly 95% of the energy budget is attributed to components that have never been directly observed in any laboratory. Approximately 68% is "dark energy," a component that drives the accelerated expansion of space; approximately 27% is "dark matter," a non-luminous, non-absorbing form of matter that shapes the growth of cosmic structure; and only the remaining 5% is the ordinary baryonic matter that constitutes stars, planets, and people. This is not a peripheral uncertainty or a "to be resolved later" footnote. It is the central, irreducible output of the most data-rich scientific model humanity has ever built.

The figure is not in dispute among practitioners. The tension lies elsewhere: in the interpretation. The fact that the Friedmann–Lemaître–Robertson–Walker (FLRW) framework, when confronted with high-precision observations, demands a "dark" sector of unknown physical origin is either (a) a profound discovery about the actual composition of the cosmos, or (b) a signal that the framework itself is incomplete or incorrect. Resolving this tension — and, more importantly, understanding the empirical and theoretical chain that forces the dark-sector conclusion upon us — is the task of the chapters that follow.

The goal of this opening chapter is to install the conceptual and mathematical scaffolding required for the rest of the book. We will derive the FLRW metric and the Friedmann equations from the Einstein field equations, define the density parameters that quantify the cosmic inventory, walk through the historical evolution that led to the current "concordance" model, and reconstruct numerically the cosmic energy budget from the 2018 Planck satellite data release. By the end, you should be able to derive the Friedmann equations yourself, compute the cosmic energy inventory from observational parameters, and articulate the historical and logical chain that led to the acceptance of dark-sector physics.

> 关键术语 / Key Terms > - FLRW metric / FLRW 度规: The geometric description of a homogeneous, isotropic universe governed by General Relativity. > - Friedmann equations / 弗里德曼方程: The two differential equations governing the scale factor $a(t)$ in an FLRW universe. > - Concordance cosmology / 共识宇宙学 (also called $\Lambda$CDM): The standard cosmological model with a cosmological constant $\Lambda$ and cold dark matter. > - Density parameter $\Omega_i$ / 密度参数: The ratio of a component's energy density to the critical density.

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1.2 From Einstein to Friedmann: Deriving the Master Equations

1.2.1 The Einstein Field Equations as a Starting Point

Modern cosmology is, at its foundation, an exercise in applied General Relativity (GR). The governing relation is Einstein's field equation,

$$G_{\mu\nu} + \Lambda g_{\mu\nu} \equiv R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu},$$

where $G_{\mu\nu}$ is the Einstein tensor encoding spacetime curvature, $R_{\mu\nu}$ is the Ricci curvature tensor, $R$ is the Ricci scalar, $g_{\mu\nu}$ is the metric tensor, $\Lambda$ is the cosmological constant, $T_{\mu\nu}$ is the energy–momentum tensor, and the right-hand side couples spacetime geometry to the matter-energy content through Newton's gravitational constant $G$ and the speed of light $c$. This single tensor equation — ten components, six independent after symmetry — is the most rigorously tested non-trivial equation in physics, and every cosmological model discussed in this book is a solution of it (or a generalization of it).

The cosmological constant $\Lambda$ deserves a brief comment. Introduced by Einstein in 1917 to permit a static universe, $\Lambda$ is mathematically a free parameter of the field equation itself, not a dynamical variable. It can be interpreted physically as the energy density of the vacuum, with energy density $\rho_{\Lambda} = \Lambda c^2 / (8\pi G)$ and equation of state $p_{\Lambda} = -\rho_{\Lambda} c^2$. For most of the 20th century, $\Lambda$ was set to zero by Occamist preference; the post-1998 revival of $\Lambda$ as a non-zero quantity is one of the great reversals in modern physics.

1.2.2 The Cosmological Principle: Why the Metric Must Be FLRW

The Einstein field equation is a set of ten coupled, non-linear partial differential equations. To extract a tractable cosmological model, we impose two symmetry assumptions that drastically reduce its complexity:

1. Homogeneity: At any given cosmic time, the universe looks statistically the same from every spatial location. 2. Isotropy: At any given spatial location, the universe looks statistically the same in every direction.

These two statements together constitute the Cosmological Principle (sometimes called the Copernican Principle in its strong form). Note that homogeneity does not strictly imply isotropy in all possible spacetimes, but together they uniquely select a metric of Robertson–Walker form, up to a single function of time $a(t)$ and a single discrete curvature parameter $k \in \{-1, 0, +1\}$.

In comoving spherical coordinates $(t, r, \theta, \phi)$, the FLRW metric takes the form

$$ds^2 = -c^2 dt^2 + a^2(t)\left[\frac{dr^2}{1-kr^2} + r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right)\right].$$

The function $a(t)$ is the scale factor, which encodes the relative size of the universe as a function of time. By convention, $a(t_0) \equiv 1$ at the present cosmic time $t_0$, so that all lengths scale with $a$ relative to the present. The parameter $k$ is the spatial curvature: $k = +1$ corresponds to a closed (spherical) universe of finite volume, $k = 0$ to a flat (Euclidean) universe of infinite volume, and $k = -1$ to an open (hyperbolic) universe. The factor $a^2(t)$ in front of the spatial part means that proper distances between comoving observers grow as $a$ evolves, even though the comoving coordinates themselves remain fixed.

> 物理直觉 / Physical Intuition: The scale factor $a(t)$ is not a "stretching of space" in the rubber-sheet sense often invoked in popular accounts. It is a coordinate-invariant measure of the rate at which proper distances between comoving observers change. The metric itself — including the $a^2(t)$ factor — is invariant under redefinitions of $a$; only dimensionless ratios like $a(t)/a(t_0)$ are physically meaningful.

1.2.3 Energy Content: The Perfect Fluid Assumption

To close the field equations, we must specify $T_{\mu\nu}$. The Cosmological Principle, together with the assumption of thermodynamic equilibrium on large scales, demands that $T_{\mu\nu}$ take the perfect-fluid form

$$T_{\mu\nu} = \left(\rho + \frac{p}{c^2}\right) u_\mu u_\nu + p\, g_{\mu\nu},$$

where $\rho(t)$ is the total energy density, $p(t)$ is the pressure, and $u^\mu$ is the four-velocity of the cosmic fluid. In the comoving frame where the fluid is at rest, $u^\mu = (c, 0, 0, 0)$ and the off-diagonal components vanish.

Different cosmic components have different equations of state $p = w \rho c^2$:

• **Pressureless matter** ("dust," including both baryons and cold dark matter): $w = 0$.
• **Radiation** (photons and massless neutrinos in the early universe): $w = 1/3$.
• **Cosmological constant / vacuum energy**: $w = -1$.
• **Generic dark energy**: $w \neq -1$ in principle; observationally very close to $-1$.

The fluid conservation equation $T^{\mu\nu}{}_{;\nu} = 0$, applied in the FLRW background, gives the continuity equation

$$\dot{\rho} + 3\frac{\dot a}{a}\left(\rho + \frac{p}{c^2}\right) = 0,$$

which implies, for a constant equation-of-s