Part I: The Expanding Universe | Chapter 1

The FLRW Metric & Friedmann Equations

A rigorous derivation of the Friedmann-Lemaître-Robertson-Walker metric from symmetry principles, and the Friedmann equations from Einstein's field equations

Historical Context

In 1922, Alexander Friedmann discovered the first non-static cosmological solutions to Einstein's field equations, showing that the universe could expand or contract. In 1927, Georges Lemaître independently derived the same solutions and connected them to Hubble's observations of galaxy recession. Howard P. Robertson (1935) and Arthur G. Walker (1936) independently proved that the FLRW metric is the most general metric compatible with spatial homogeneity and isotropy.

Einstein himself initially introduced the cosmological constant Λ to enforce a static universe, calling it his “greatest blunder” after Hubble's discovery of expansion. Ironically, Λ returned in 1998 when Type Ia supernovae revealed the universe's accelerating expansion.

1. The Cosmological Principle

The foundation of modern cosmology rests on a single assumption: on sufficiently large scales (≳100 Mpc), the universe is spatially homogeneous and isotropic. This is the cosmological principle.

Homogeneity

The universe looks the same at every spatial point. There is no preferred location. Mathematically, the spatial geometry admits a transitive group of isometries: for any two points P and Q on a spatial hypersurface, there exists a symmetry transformation mapping P to Q.

A space is homogeneous if and only if it possesses at least 3 independent Killing vector fields that generate translations (i.e., at least a 3-dimensional group of isometries acting transitively).

Isotropy

The universe looks the same in every direction from any given point. The CMB confirms this to $\Delta T/T \sim 10^{-5}$. At each point, the spatial geometry is invariant under the rotation group SO(3).

Isotropy at every point implies homogeneity (but not vice versa). If the universe is isotropic about every point, it must also be homogeneous.

Maximally Symmetric Spaces and Killing Vectors

A d-dimensional Riemannian manifold is maximally symmetric if it possesses the maximum number of independent Killing vectors:

$$N_{\text{max}} = \frac{d(d+1)}{2}$$

For our 3-dimensional spatial slices, this gives $N_{\text{max}} = 6$: three translations (homogeneity) plus three rotations (isotropy). A maximally symmetric 3-space has constant sectional curvature K everywhere. By rescaling coordinates, K can be normalized to $k = +1, 0, -1$.

The Riemann curvature tensor of a maximally symmetric space with curvature K takes the unique form:

$$R_{ijkl} = K\left(\gamma_{ik}\gamma_{jl} - \gamma_{il}\gamma_{jk}\right)$$

where $\gamma_{ij}$ is the spatial metric. This is the most constrained form of the Riemann tensor, leaving only one free parameter K.

2. Constructing the FLRW Metric from Symmetry

2.1 Foliation of Spacetime

We begin by assuming that spacetime can be foliated into a family of non-intersecting spacelike hypersurfaces $\Sigma_t$, each labeled by cosmic time t. This is possible whenever a timelike Killing vector or a preferred family of observers (the “comoving” observers) exists.

The most general line element respecting this foliation is:

$$ds^2 = -c^2 N^2(t,\mathbf{x})\,dt^2 + 2 N_i(t,\mathbf{x})\,dt\,dx^i + h_{ij}(t,\mathbf{x})\,dx^i dx^j$$

ADM decomposition: N is the lapse function, $N_i$ is the shift vector, $h_{ij}$ is the induced 3-metric

The cosmological principle forces drastic simplifications:

Isotropy eliminates the shift vector: A nonzero shift vector $N_i$ would define a preferred spatial direction at each point, violating isotropy. Thus $N_i = 0$.
Homogeneity fixes the lapse: The lapse N cannot depend on spatial coordinates $\mathbf{x}$. We can absorb any remaining function N(t) into a redefinition of the time coordinate, giving $N = 1$ in cosmic time.
The spatial metric factorizes: Homogeneity and isotropy require $h_{ij}(t,\mathbf{x}) = a^2(t)\,\gamma_{ij}(\mathbf{x})$, where$\gamma_{ij}$ is a time-independent metric of constant curvature.

The metric therefore reduces to:

$$ds^2 = -c^2\,dt^2 + a^2(t)\,d\sigma^2$$

where $d\sigma^2 = \gamma_{ij}\,dx^i dx^j$ is the line element of a 3-space of constant curvature

2.2 Metrics of Constant Curvature in 3D

The metric $d\sigma^2$ on a 3-dimensional space of constant curvature K can be constructed by embedding in 4-dimensional flat space. Consider the constraint surface in$\mathbb{R}^4$:

$$x_1^2 + x_2^2 + x_3^2 + \epsilon\, x_4^2 = R_0^2$$

where $\epsilon = +1$ for k = +1 (sphere), $\epsilon = -1$ for k = -1 (hyperboloid), and $K = k/R_0^2$

Introducing spherical-type coordinates on this surface and normalizing$r \to r/R_0$, the three maximally symmetric 3-geometries are:

k = +1 (Closed)

$$d\sigma^2 = \frac{dr^2}{1-r^2} + r^2\,d\Omega^2$$

3-sphere $S^3$. Finite volume $V = 2\pi^2 a^3$. Circumnavigable.

k = 0 (Flat)

$$d\sigma^2 = dr^2 + r^2\,d\Omega^2$$

Euclidean $\mathbb{R}^3$. Infinite volume. Planck 2018: $|\Omega_k| < 0.002$.

k = -1 (Open)

$$d\sigma^2 = \frac{dr^2}{1+r^2} + r^2\,d\Omega^2$$

Hyperbolic space $H^3$. Infinite volume. Divergent parallel geodesics.

Combining all three cases into a single expression with$d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2$:

The FLRW Line Element

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

The most general metric compatible with spatial homogeneity and isotropy. The entire dynamics of the universe is encoded in the single function a(t).

2.3 Alternative Coordinate Form

Using the substitution $r = S_k(\chi)$ where:

$$S_k(\chi) = \begin{cases} \sin\chi & k=+1 \\ \chi & k=0 \\ \sinh\chi & k=-1 \end{cases}$$

The metric becomes:

$$ds^2 = -c^2\,dt^2 + a^2(t)\left[d\chi^2 + S_k^2(\chi)\,d\Omega^2\right]$$

Here $\chi$ is the comoving radial distance (dimensionless). For k = +1, $\chi \in [0,\pi]$; for k = 0 and k = -1, $\chi \in [0,\infty)$.

3. Metric Tensor Components

Working in coordinates $x^\mu = (ct, r, \theta, \phi)$, the metric tensor$g_{\mu\nu}$ for the FLRW spacetime has components:

$$g_{\mu\nu} = \text{diag}\!\left(-1,\;\frac{a^2}{1-kr^2},\;a^2 r^2,\;a^2 r^2 \sin^2\theta\right)$$

$$g_{00} = -1, \qquad g_{11} = \frac{a^2(t)}{1-kr^2}$$

$$g_{22} = a^2(t)\,r^2, \qquad g_{33} = a^2(t)\,r^2\sin^2\theta$$

The inverse metric is:

$$g^{00} = -1, \qquad g^{11} = \frac{1-kr^2}{a^2(t)}$$

$$g^{22} = \frac{1}{a^2(t)\,r^2}, \qquad g^{33} = \frac{1}{a^2(t)\,r^2\sin^2\theta}$$

4. Christoffel Symbols — Full Computation

The Christoffel symbols (Levi-Civita connection coefficients) are defined by:

$$\Gamma^\lambda{}_{\mu\nu} = \frac{1}{2}\,g^{\lambda\sigma}\!\left(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu}\right)$$

Since the FLRW metric is diagonal, many components vanish. We now compute all non-zero Christoffel symbols. We use $x^0 = ct$ and denote $\dot{a} = da/dt$.

4.1 Mixed temporal-spatial components

The time derivative of the spatial metric generates the “Hubble” connection coefficients. From $\partial_0 g_{ij} = 2a\dot{a}\,\gamma_{ij}/c$:

$$\Gamma^0{}_{11} = \frac{a\dot{a}}{c^2(1-kr^2)}, \qquad \Gamma^0{}_{22} = \frac{a\dot{a}\,r^2}{c^2}, \qquad \Gamma^0{}_{33} = \frac{a\dot{a}\,r^2\sin^2\theta}{c^2}$$

Compact form: $\Gamma^0{}_{ij} = \frac{a\dot{a}}{c^2}\,\gamma_{ij}$

$$\Gamma^1{}_{01} = \Gamma^1{}_{10} = \frac{\dot{a}}{ca}, \qquad \Gamma^2{}_{02} = \Gamma^2{}_{20} = \frac{\dot{a}}{ca}, \qquad \Gamma^3{}_{03} = \Gamma^3{}_{30} = \frac{\dot{a}}{ca}$$

Compact form: $\Gamma^i{}_{0j} = \frac{\dot{a}}{ca}\,\delta^i{}_j = \frac{H}{c}\,\delta^i{}_j$

4.2 Purely spatial components

These arise from the curvature of the spatial slices and are identical to those of the time-independent metric $\gamma_{ij}$:

$$\Gamma^1{}_{11} = \frac{kr}{1-kr^2}, \qquad \Gamma^1{}_{22} = -r(1-kr^2), \qquad \Gamma^1{}_{33} = -r(1-kr^2)\sin^2\theta$$

$$\Gamma^2{}_{12} = \Gamma^2{}_{21} = \frac{1}{r}, \qquad \Gamma^2{}_{33} = -\sin\theta\cos\theta$$

$$\Gamma^3{}_{13} = \Gamma^3{}_{31} = \frac{1}{r}, \qquad \Gamma^3{}_{23} = \Gamma^3{}_{32} = \cot\theta$$

Key observation: All $\Gamma^i{}_{0j}$ components are proportional to the Hubble parameter $H = \dot{a}/a$. When $H > 0$ (expanding universe), freely falling particles experience an effective “Hubble friction” that redshifts their momenta as $p \propto 1/a$.

5. The Ricci Tensor

The Ricci tensor is computed from the Christoffel symbols via:

$$R_{\mu\nu} = \partial_\lambda \Gamma^\lambda{}_{\mu\nu} - \partial_\nu \Gamma^\lambda{}_{\mu\lambda} + \Gamma^\lambda{}_{\lambda\sigma}\Gamma^\sigma{}_{\mu\nu} - \Gamma^\lambda{}_{\nu\sigma}\Gamma^\sigma{}_{\mu\lambda}$$

5.1 The $R_{00}$ component

The temporal-temporal component measures how geodesics converge or diverge in time. Computing term by term:

$$R_{00} = \partial_\lambda \Gamma^\lambda{}_{00} - \partial_0 \Gamma^\lambda{}_{0\lambda} + \Gamma^\lambda{}_{\lambda\sigma}\Gamma^\sigma{}_{00} - \Gamma^\lambda{}_{0\sigma}\Gamma^\sigma{}_{0\lambda}$$

Since $\Gamma^\lambda{}_{00} = 0$ (the metric component $g_{00} = -1$ is constant), the first and third terms vanish. We are left with:

$$R_{00} = -\partial_0 \Gamma^i{}_{0i} - \Gamma^i{}_{0j}\Gamma^j{}_{0i}$$

With $\Gamma^i{}_{0j} = \frac{\dot{a}}{ca}\delta^i{}_j$, the trace is $\Gamma^i{}_{0i} = \frac{3\dot{a}}{ca}$, and:

$$\partial_0 \Gamma^i{}_{0i} = \frac{1}{c}\frac{d}{dt}\!\left(\frac{3\dot{a}}{ca}\right) = \frac{3}{c^2}\frac{d}{dt}\!\left(\frac{\dot{a}}{a}\right) = \frac{3}{c^2}\left(\frac{\ddot{a}}{a} - \frac{\dot{a}^2}{a^2}\right)$$

And $\Gamma^i{}_{0j}\Gamma^j{}_{0i} = 3\left(\frac{\dot{a}}{ca}\right)^2 = \frac{3\dot{a}^2}{c^2 a^2}$. Combining:

$$\boxed{R_{00} = -\frac{3\ddot{a}}{c^2\,a}}$$

5.2 The $R_{ij}$ components

The spatial components of the Ricci tensor carry information about both the time evolution (through $\dot{a}$ and $\ddot{a}$) and the spatial curvature (through k). The full calculation yields:

$$\boxed{R_{ij} = \frac{1}{c^2}\left(a\ddot{a} + 2\dot{a}^2 + 2kc^2\right)\gamma_{ij}}$$

where $\gamma_{ij}$ is the spatial metric of constant curvature. Explicitly: $R_{11} = \frac{a\ddot{a} + 2\dot{a}^2 + 2kc^2}{c^2(1-kr^2)}$, etc.

Derivation sketch for $R_{11}$:

Starting from $R_{11} = \partial_\lambda \Gamma^\lambda{}_{11} - \partial_1 \Gamma^\lambda{}_{1\lambda} + \Gamma^\lambda{}_{\lambda\sigma}\Gamma^\sigma{}_{11} - \Gamma^\lambda{}_{1\sigma}\Gamma^\sigma{}_{1\lambda}$:

1. $\partial_0 \Gamma^0{}_{11} = \frac{1}{c}\frac{d}{dt}\!\left(\frac{a\dot{a}}{c^2(1-kr^2)}\right) = \frac{\dot{a}^2 + a\ddot{a}}{c^3(1-kr^2)}\times\frac{1}{c}$

2. Spatial Christoffel derivatives: $\partial_1 \Gamma^1{}_{11} = \frac{k(1+kr^2)}{(1-kr^2)^2}$

3. Cross terms: $\Gamma^0{}_{00}\Gamma^0{}_{11} = 0$, $\Gamma^i{}_{0i}\Gamma^0{}_{11} = \frac{3\dot{a}}{ca}\cdot\frac{a\dot{a}}{c^2(1-kr^2)}$

4. After collecting all 16+ terms and using $g_{11} = a^2/(1-kr^2)$:

$R_{11} = \frac{a\ddot{a} + 2\dot{a}^2 + 2kc^2}{c^2(1-kr^2)}$

5.3 The Ricci Scalar and Einstein Tensor

The Ricci scalar is $R = g^{\mu\nu}R_{\mu\nu}$:

$$R = g^{00}R_{00} + g^{ij}R_{ij} = \frac{3\ddot{a}}{c^2 a} + \frac{3}{a^2}\cdot\frac{1}{c^2}\left(a\ddot{a} + 2\dot{a}^2 + 2kc^2\right)$$

$$\boxed{R = \frac{6}{c^2}\left(\frac{\ddot{a}}{a} + \frac{\dot{a}^2}{a^2} + \frac{kc^2}{a^2}\right)}$$

The Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$ has components:

Time-time component:

$$G_{00} = \frac{3}{c^2}\left(\frac{\dot{a}^2}{a^2} + \frac{kc^2}{a^2}\right) = \frac{3}{c^2}\left(H^2 + \frac{kc^2}{a^2}\right)$$

Space-space components:

$$G_{ij} = -\frac{1}{c^2}\left(2a\ddot{a} + \dot{a}^2 + kc^2\right)\gamma_{ij}$$

6. Einstein's Field Equations

Einstein's field equations with cosmological constant relate the geometry of spacetime to its matter-energy content:

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}$$

Left side: geometry (Einstein tensor + cosmological constant). Right side: matter (stress-energy tensor).

The Perfect Fluid Stress-Energy Tensor

The cosmological principle requires that the matter content, like the geometry, be spatially homogeneous and isotropic. The most general such stress-energy tensor is that of a perfect fluid:

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

where $\rho$ is the energy density, p is the isotropic pressure, and $u^\mu = (c, 0, 0, 0)$ for comoving observers

In the comoving frame, the components are:

$$T_{00} = \rho c^2, \qquad T_{0i} = 0$$

$$T_{ij} = p\,g_{ij} = p\,a^2\,\gamma_{ij}$$

Or equivalently: $T^\mu{}_\nu = \text{diag}(-\rho c^2, p, p, p)$

7. Deriving the Friedmann Equations

7.1 The First Friedmann Equation

The (0,0) component of Einstein's equations gives:

$$G_{00} + \Lambda g_{00} = \frac{8\pi G}{c^4}\,T_{00}$$

$$\frac{3}{c^2}\!\left(\frac{\dot{a}^2}{a^2} + \frac{kc^2}{a^2}\right) - \Lambda = \frac{8\pi G}{c^4}\cdot\rho c^2$$

$$\frac{3\dot{a}^2}{c^2 a^2} + \frac{3k}{a^2} - \Lambda = \frac{8\pi G\rho}{c^2}$$

Dividing through by $3/c^2$:

First Friedmann Equation

$$\boxed{H^2 \equiv \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\,\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}}$$

The expansion rate $H = \dot{a}/a$ is determined by the energy density $\rho$, spatial curvature k, and the cosmological constant $\Lambda$.

7.2 The Second Friedmann Equation (Acceleration Equation)

The spatial (i,j) components of Einstein's equations, combined with the first Friedmann equation, yield the acceleration equation. Taking the trace of the spatial Einstein equation:

$$G_{ij} + \Lambda g_{ij} = \frac{8\pi G}{c^4}\,T_{ij}$$

$$-\frac{1}{c^2}\!\left(2a\ddot{a} + \dot{a}^2 + kc^2\right)\gamma_{ij} + \Lambda a^2 \gamma_{ij} = \frac{8\pi G}{c^4}\,p\,a^2\gamma_{ij}$$

Dividing by $a^2\gamma_{ij}$ and using the first Friedmann equation to eliminate $\dot{a}^2/a^2$:

Second Friedmann Equation (Acceleration Equation)

$$\boxed{\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\!\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}}$$

The deceleration/acceleration of expansion depends on both energy density and pressure. Matter with $\rho + 3p/c^2 > 0$ decelerates; $\Lambda > 0$ accelerates.

Physical Insight: The Strong Energy Condition

The quantity $\rho + 3p/c^2$ is the active gravitational mass density. In GR, pressure gravitates — this is a purely relativistic effect with no Newtonian analog.

For radiation ($p = \rho c^2/3$), the effective gravitational source is$\rho + 3p/c^2 = 2\rho$ — twice the energy density. For a cosmological constant ($p = -\rho c^2$), we get$\rho + 3p/c^2 = -2\rho < 0$, producing repulsive gravity.

7.3 Alternative Derivation via the Trace

A quicker route: take the trace of Einstein's equations $G^\mu{}_\mu + 4\Lambda = \frac{8\pi G}{c^4}T^\mu{}_\mu$:

$$-R + 4\Lambda = \frac{8\pi G}{c^4}\left(-\rho c^2 + 3p\right)$$

Substituting $R = \frac{6}{c^2}\!\left(\frac{\ddot{a}}{a} + \frac{\dot{a}^2}{a^2} + \frac{kc^2}{a^2}\right)$ and using the first Friedmann equation for $\dot{a}^2/a^2 + kc^2/a^2$:

$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\!\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3} \quad \checkmark$$

8. The Continuity Equation

Energy-momentum conservation $\nabla_\mu T^{\mu\nu} = 0$ provides a third equation. The $\nu = 0$ component gives:

$$\nabla_\mu T^{\mu 0} = \partial_\mu T^{\mu 0} + \Gamma^\mu{}_{\mu\lambda}T^{\lambda 0} - \Gamma^\lambda{}_{\mu 0}T^{\mu\lambda} = 0$$

$$\partial_0 T^{00} + \Gamma^i{}_{i0}\,T^{00} + \Gamma^0{}_{00}\,T^{00} - \Gamma^0{}_{i0}\,T^{i0} - \Gamma^i{}_{00}\,T^{0i} - \Gamma^0{}_{00}\,T^{00} + \Gamma^i{}_{i0}\,T^{00} - \cdots = 0$$

After careful bookkeeping of all nonzero terms:

$$\frac{1}{c}\dot{\rho} + 3\frac{\dot{a}}{ca}\left(\rho + \frac{p}{c^2}\right) = 0$$

Continuity Equation (Fluid Equation)

$$\boxed{\dot{\rho} + 3H\!\left(\rho + \frac{p}{c^2}\right) = 0}$$

This is the first law of thermodynamics in an expanding universe: $d(\rho c^2 a^3) = -p\,d(a^3)$

These three equations (first Friedmann, second Friedmann, continuity) are not independent: any one can be derived from the other two. A common strategy is to work with the first Friedmann equation and the continuity equation, specifying an equation of state to close the system.

9. Equation of State and Solutions for Each Component

Specifying the equation of state $p = w\rho c^2$ with constant w, the continuity equation can be integrated analytically:

$$\dot{\rho} + 3\frac{\dot{a}}{a}\rho(1+w) = 0 \quad\Rightarrow\quad \frac{d\rho}{\rho} = -3(1+w)\frac{da}{a}$$

$$\boxed{\rho(a) = \rho_0\,a^{-3(1+w)}}$$

Component	w	$\rho(a)$	$a(t)$ (flat, single component)	Physical origin
Non-relativistic matter	$0$	$\rho_0\,a^{-3}$	$a \propto t^{2/3}$	Number density dilution
Radiation	$1/3$	$\rho_0\,a^{-4}$	$a \propto t^{1/2}$	Dilution + redshift
Cosmological constant	$-1$	$\rho_0 = \text{const}$	$a \propto e^{Ht}$	Vacuum energy
Curvature (effective)	$-1/3$	$\rho_0\,a^{-2}$	$a \propto t$	Spatial curvature term
Stiff matter	$+1$	$\rho_0\,a^{-6}$	$a \propto t^{1/3}$	Kinetic-dominated scalar field

10. Density Parameters and the Friedmann Constraint

Define the critical density as the density for which the universe is spatially flat (k = 0, $\Lambda = 0$):

$$\rho_c \equiv \frac{3H^2}{8\pi G} \approx 1.88 \times 10^{-29}\,h^2\;\text{g/cm}^3$$

where $h \equiv H_0/(100\;\text{km/s/Mpc}) \approx 0.674$

The density parameters are the fractional contributions of each component:

$$\Omega_m = \frac{\rho_m}{\rho_c}, \qquad \Omega_r = \frac{\rho_r}{\rho_c}, \qquad \Omega_\Lambda = \frac{\Lambda c^2}{3H^2}, \qquad \Omega_k = -\frac{kc^2}{a^2 H^2}$$

The first Friedmann equation then takes the elegant form:

$$\boxed{\Omega_m + \Omega_r + \Omega_\Lambda + \Omega_k = 1}$$

The Friedmann constraint: the density parameters always sum to unity.

Using the scaling relations, the Friedmann equation becomes:

$$H^2(a) = H_0^2\!\left[\Omega_{r,0}\,a^{-4} + \Omega_{m,0}\,a^{-3} + \Omega_{k,0}\,a^{-2} + \Omega_{\Lambda,0}\right]$$

The entire expansion history is encoded in four numbers: $\Omega_{r,0}$, $\Omega_{m,0}$, $\Omega_{k,0}$, $\Omega_{\Lambda,0}$

Planck 2018 Best-Fit Values ($\Lambda$CDM)

$\Omega_{m,0}$

0.315

$\Omega_{\Lambda,0}$

0.685

$\Omega_{r,0}$

$9.1 \times 10^{-5}$

$H_0$

67.4 km/s/Mpc

11. Exact Analytical Solutions

11.1 Matter-Dominated (Einstein–de Sitter) Universe

For a flat (k = 0), matter-only ($\Lambda = 0$, $\Omega_m = 1$) universe:

$$H^2 = \frac{8\pi G}{3}\rho_0 a^{-3} \quad\Rightarrow\quad \dot{a}^2 = H_0^2 a^{-1}$$

$$a(t) = \left(\frac{3H_0}{2}\,t\right)^{2/3}, \qquad t_0 = \frac{2}{3H_0} \approx 9.3\;\text{Gyr}$$

The universe decelerates: $\ddot{a} < 0$. Age = $2/(3H_0)$, shorter than observed.

11.2 Radiation-Dominated Universe

$$H^2 = \frac{8\pi G}{3}\rho_0 a^{-4} \quad\Rightarrow\quad a(t) = \left(2H_0 t\right)^{1/2}$$

Dominant in the first ~47,000 years. The Hubble time is $t_0 = 1/(2H_0)$. Temperature scales as $T \propto a^{-1} \propto t^{-1/2}$.

11.3 de Sitter Universe (Dark Energy Dominated)

$$H^2 = \frac{\Lambda c^2}{3} = \text{const} \quad\Rightarrow\quad a(t) = a_0\,e^{H_\Lambda t}$$

$$\text{where}\quad H_\Lambda = c\sqrt{\Lambda/3}$$

Exponential expansion. This is the future attractor of our universe as matter dilutes. Also describes the inflationary epoch at $t \sim 10^{-36}$ s.

12. Numerical Integration: Python Implementation

The following Python code numerically integrates the Friedmann equation for the full$\Lambda$CDM model with radiation, matter, and dark energy. It computes the scale factor evolution, Hubble parameter, deceleration parameter, and cosmological distances.

Python

friedmann_solver.py127 lines

#!/usr/bin/env python3
"""
friedmann_solver.py
Full numerical integration of the Friedmann equations for ΛCDM cosmology.
Computes scale factor, Hubble parameter, deceleration parameter, and age of universe.
"""
import numpy as np
from scipy.integrate import solve_ivp, quad

# ============================
# Cosmological Parameters (Planck 2018)
# ============================
H0_km = 67.4          # Hubble constant [km/s/Mpc]
H0 = H0_km * 3.2408e-20  # Convert to [1/s]
Omega_r = 9.1e-5       # Radiation density parameter
Omega_m = 0.315        # Matter density parameter
Omega_L = 0.685        # Dark energy (cosmological constant)
Omega_k = 1.0 - Omega_r - Omega_m - Omega_L  # Curvature

c = 2.998e8            # Speed of light [m/s]
Mpc = 3.0857e22        # 1 Mpc in meters
Gyr = 3.1557e16        # 1 Gyr in seconds

def E(a):
    """
    Dimensionless Hubble parameter E(a) = H(a)/H0.
    From the first Friedmann equation:
      H²(a) = H0² [Ω_r a⁻⁴ + Ω_m a⁻³ + Ω_k a⁻² + Ω_Λ]
    """
    return np.sqrt(Omega_r / a**4
                 + Omega_m / a**3
                 + Omega_k / a**2
                 + Omega_L)

def dadt(t_H0, a):
    """
    da/d(H0*t) = a * E(a)
    ODE for scale factor evolution in units of 1/H0.
    """
    return a * E(a)

def deceleration_q(a):
    """
    Deceleration parameter q = -a*ä/(ȧ²) = -1 - (ȧ/a)'/H²
    From the Friedmann equations:
      q = (Ω_r/a⁴ + 0.5*Ω_m/a³ - Ω_Λ) / E²(a)
    """
    E2 = E(a)**2
    return (Omega_r / a**4 + 0.5 * Omega_m / a**3 - Omega_L) / E2

# ============================
# 1. Compute the age of the universe
# ============================
def age_integrand(a):
    """dt = da / (a * H(a)) = da / (a * H0 * E(a))"""
    return 1.0 / (a * E(a))

age_H0, _ = quad(age_integrand, 0, 1.0)  # Age in units of 1/H0
age_Gyr = age_H0 / (H0 * Gyr)

# ============================
# 2. Solve ODE for scale factor a(t)
# ============================
a_init = 1e-6
t_span_H0 = [0, 3.0]  # In units of 1/H0
sol = solve_ivp(dadt, t_span_H0, [a_init],
                rtol=1e-10, atol=1e-12,
                dense_output=True, max_step=0.001)

t_arr = np.linspace(0, 3.0, 5000)
a_arr = sol.sol(t_arr)[0]

# Find t corresponding to a = 1 (today)
t_today_idx = np.argmin(np.abs(a_arr - 1.0))
t_today = t_arr[t_today_idx]
t_phys = (t_arr - t_today) / (H0 * Gyr)  # Convert to Gyr from today

# ============================
# 3. Compute derived quantities
# ============================
H_arr = H0 * np.array([E(a) for a in a_arr])  # H(t) in 1/s
q_arr = np.array([deceleration_q(a) for a in a_arr])

# ============================
# 4. Cosmological distances
# ============================
z_arr = np.linspace(0, 10, 500)

def comoving_dist(z):
    """d_C = (c/H0) ∫₀ᶻ dz'/E(1/(1+z'))"""
    integrand = lambda zp: 1.0 / E(1.0 / (1 + zp))
    result, _ = quad(integrand, 0, z)
    return c / (H0 * Mpc) * result  # in Mpc

d_C = np.array([comoving_dist(z) for z in z_arr])
d_L = d_C * (1 + z_arr)              # Luminosity distance
d_A = d_C / (1 + z_arr)              # Angular diameter distance

# ============================
# Print results
# ============================
print("=" * 60)
print("  FRIEDMANN EQUATION SOLVER — ΛCDM COSMOLOGY")
print("=" * 60)
print(f"\n  Cosmological Parameters (Planck 2018):")
print(f"    H₀  = {H0_km} km/s/Mpc")
print(f"    Ωᵣ  = {Omega_r}")
print(f"    Ωₘ  = {Omega_m}")
print(f"    ΩΛ  = {Omega_L}")
print(f"    Ωₖ  = {Omega_k:.6f}")
print(f"\n  Derived Quantities:")
print(f"    Age of universe: {age_Gyr:.3f} Gyr")
print(f"    Hubble time 1/H₀: {1.0/(H0*Gyr):.3f} Gyr")
print(f"    Hubble radius c/H₀: {c/(H0*Mpc):.1f} Mpc")
print(f"\n  Current deceleration parameter q₀ = {deceleration_q(1.0):.4f}")
print(f"    (q < 0 → accelerating expansion)")
print(f"\n  Scale factor at matter-radiation equality:")
a_eq = Omega_r / Omega_m
z_eq = 1.0/a_eq - 1
print(f"    a_eq = {a_eq:.6f},  z_eq = {z_eq:.0f}")
print(f"\n  Distances to z = 1:")
print(f"    Comoving distance:  {comoving_dist(1.0):.1f} Mpc")
print(f"    Luminosity distance: {comoving_dist(1.0)*2:.1f} Mpc")
print(f"    Angular diameter distance: {comoving_dist(1.0)/2:.1f} Mpc")
print(f"\n  Distances to z = 1100 (CMB):")
print(f"    Comoving distance: {comoving_dist(1100):.1f} Mpc")
print("\n" + "=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

13. High-Precision Solver: Fortran Implementation

For production-grade cosmological calculations requiring $10^6$+ evaluations (e.g., MCMC parameter estimation, N-body initial conditions), Fortran's speed is advantageous. The following code implements a 4th-order Runge-Kutta integrator for the Friedmann equation:

Fortran + Python: Friedmann Equations with Visualization

Python

Compiles and runs a Fortran RK4 solver for LCDM cosmology, then plots scale factor, Hubble parameter, deceleration parameter, and density evolution

friedmann_fortran_plot.py211 lines

import subprocess, sys, os, csv, io
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import numpy as np

# --- Embedded Fortran source ---
fortran_source = r"""
program friedmann_rk4
  implicit none
  integer, parameter :: dp = selected_real_kind(15, 307)

real(dp), parameter :: c_light = 2.99792458d8
  real(dp), parameter :: Mpc     = 3.08567758d22
  real(dp), parameter :: Gyr     = 3.15576d16

! Cosmological parameters (Planck 2018)
  real(dp), parameter :: H0_km   = 67.4d0
  real(dp), parameter :: H0      = H0_km * 1.0d3 / Mpc
  real(dp), parameter :: Omega_r = 9.1d-5
  real(dp), parameter :: Omega_m = 0.315d0
  real(dp), parameter :: Omega_L = 0.685d0
  real(dp), parameter :: Omega_k = 1.0d0 - Omega_r - Omega_m - Omega_L

integer, parameter  :: nsteps = 100000
  integer, parameter  :: out_every = 100
  real(dp) :: a, t, dt, a_new
  real(dp) :: k1, k2, k3, k4
  real(dp) :: t_today, age_Gyr_val, H_now, q_now
  integer  :: i
  logical  :: found_today

a = 1.0d-8
  t = 0.0d0
  dt = 3.0d0 / dble(nsteps)
  found_today = .false.

write(*,'(A)') '=================================================='
  write(*,'(A)') '  FRIEDMANN RK4 SOLVER - LCDM COSMOLOGY (Fortran)'
  write(*,'(A)') '=================================================='
  write(*,'(A,F8.2,A)') '  H0       = ', H0_km, ' km/s/Mpc'
  write(*,'(A,F8.4)')   '  Omega_m  = ', Omega_m
  write(*,'(A,F8.4)')   '  Omega_L  = ', Omega_L
  write(*,'(A,ES12.4)') '  Omega_r  = ', Omega_r

write(*,'(A)') 'DATA_START'
  write(*,'(A)') 't_H0,a,H_over_H0,q,Omega_m_eff,Omega_L_eff,Omega_r_eff'

do i = 1, nsteps
    if (.not. found_today .and. a >= 1.0d0) then
      t_today = t
      found_today = .true.
    end if

if (mod(i, out_every) == 0) then
      write(*,'(ES16.8,A,ES16.8,A,ES16.8,A,ES16.8,A,ES16.8,A,ES16.8,A,ES16.8)') &
        t, ',', a, ',', E_func(a), ',', q_func(a), ',', &
        Omega_m/(a**3 * E_func(a)**2), ',', &
        Omega_L/E_func(a)**2, ',', &
        Omega_r/(a**4 * E_func(a)**2)
    end if

k1 = dt * rhs(a)
    k2 = dt * rhs(a + 0.5d0*k1)
    k3 = dt * rhs(a + 0.5d0*k2)
    k4 = dt * rhs(a + k3)
    a_new = a + (k1 + 2.0d0*k2 + 2.0d0*k3 + k4) / 6.0d0
    a = a_new
    t = t + dt
  end do

write(*,'(A)') 'DATA_END'

age_Gyr_val = t_today / (H0 * Gyr)
  q_now = q_func(1.0d0)

write(*,'(A)') ''
  write(*,'(A,F10.4,A)') '  Age of universe  = ', age_Gyr_val, ' Gyr'
  write(*,'(A,F10.4,A)') '  Hubble time 1/H0 = ', 1.0d0/(H0*Gyr), ' Gyr'
  write(*,'(A,F10.4)')   '  q0 (deceleration)= ', q_now
  write(*,'(A)') '=================================================='

contains
  function E_func(a_val) result(E)
    real(dp), intent(in) :: a_val
    real(dp) :: E
    E = sqrt(Omega_r/a_val**4 + Omega_m/a_val**3 + Omega_k/a_val**2 + Omega_L)
  end function E_func

function rhs(a_val) result(f)
    real(dp), intent(in) :: a_val
    real(dp) :: f
    f = a_val * E_func(a_val)
  end function rhs

function q_func(a_val) result(q)
    real(dp), intent(in) :: a_val
    real(dp) :: q, E2
    E2 = E_func(a_val)**2
    q = (Omega_r/a_val**4 + 0.5d0*Omega_m/a_val**3 - Omega_L) / E2
  end function q_func
end program friedmann_rk4
"""

# Write, compile, and run the Fortran code
with open('friedmann_rk4.f90', 'w') as f:
    f.write(fortran_source)

comp = subprocess.run(['gfortran', '-O2', '-o', './friedmann_rk4', 'friedmann_rk4.f90'],
                      capture_output=True, text=True)
if comp.returncode != 0:
    print("Compilation error:")
    print(comp.stderr)
    sys.exit(1)

result = subprocess.run(['./friedmann_rk4'], capture_output=True, text=True, timeout=30)
stdout = result.stdout

# Print text output (outside DATA markers)
in_data = False
data_lines = []
for line in stdout.splitlines():
    if line.strip() == 'DATA_START':
        in_data = True
        continue
    elif line.strip() == 'DATA_END':
        in_data = False
        continue
    if in_data:
        data_lines.append(line)
    else:
        print(line)

# Parse CSV data
reader = csv.reader(data_lines)
header = next(reader)
rows = []
for row in reader:
    try:
        rows.append([float(x) for x in row])
    except ValueError:
        continue

data = np.array(rows)
t_H0    = data[:, 0]
a_t     = data[:, 1]
H_H0    = data[:, 2]
q_t     = data[:, 3]
Om_m    = data[:, 4]
Om_L    = data[:, 5]
Om_r    = data[:, 6]

# Convert dimensionless time to Gyr
H0_val = 67.4e3 / 3.08567758e22
Gyr_s  = 3.15576e16
t_Gyr  = t_H0 / (H0_val * Gyr_s)

# --- Plot with dark theme ---
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0f172a')
fig.suptitle('Friedmann Equations: LCDM Cosmology (Fortran RK4 Solver)',
             color='#e2e8f0', fontsize=15, fontweight='bold', y=0.98)

for ax in axes.flat:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

# Panel 1: Scale factor a(t)
ax = axes[0, 0]
ax.plot(t_Gyr, a_t, color='#a78bfa', linewidth=2)
ax.axhline(y=1.0, color='#f97316', linestyle='--', alpha=0.6, label='a=1 (today)')
ax.set_xlabel('Time (Gyr)', color='#cbd5e1')
ax.set_ylabel('Scale factor a(t)', color='#cbd5e1')
ax.set_title('Scale Factor Evolution', color='#e2e8f0', fontweight='bold')
ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
ax.set_xlim(0, max(t_Gyr))

# Panel 2: Hubble parameter H(t)/H0
ax = axes[0, 1]
ax.semilogy(t_Gyr, H_H0, color='#22d3ee', linewidth=2)
ax.set_xlabel('Time (Gyr)', color='#cbd5e1')
ax.set_ylabel('H(t) / H0', color='#cbd5e1')
ax.set_title('Hubble Parameter', color='#e2e8f0', fontweight='bold')

# Panel 3: Deceleration parameter q(t)
ax = axes[1, 0]
ax.plot(t_Gyr, q_t, color='#fb923c', linewidth=2)
ax.axhline(y=0, color='#ef4444', linestyle='--', alpha=0.6, label='q=0 (accel/decel boundary)')
ax.set_xlabel('Time (Gyr)', color='#cbd5e1')
ax.set_ylabel('Deceleration q(t)', color='#cbd5e1')
ax.set_title('Deceleration Parameter', color='#e2e8f0', fontweight='bold')
ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)

# Panel 4: Density parameter evolution
ax = axes[1, 1]
ax.plot(t_Gyr, Om_r, color='#f472b6', linewidth=2, label='Omega_r (radiation)')
ax.plot(t_Gyr, Om_m, color='#34d399', linewidth=2, label='Omega_m (matter)')
ax.plot(t_Gyr, Om_L, color='#818cf8', linewidth=2, label='Omega_L (dark energy)')
ax.set_xlabel('Time (Gyr)', color='#cbd5e1')
ax.set_ylabel('Density parameter', color='#cbd5e1')
ax.set_title('Density Evolution', color='#e2e8f0', fontweight='bold')
ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)
ax.set_ylim(-0.05, 1.1)

plt.tight_layout(rect=[0, 0, 1, 0.95])
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
print("\nPlot saved to output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Compilation & Output

$ gfortran -O3 -o friedmann friedmann_rk4.f90

$ ./friedmann

# Produces friedmann_output.dat with columns:

# t/t_H0 a(t) H(t)/H0 q(t)

The Fortran code runs ~50× faster than the Python equivalent for the same number of steps, making it suitable for integration into N-body codes and MCMC samplers like CosmoMC or MontePython.

14. Conformal Time and the FLRW Metric

A particularly useful time coordinate is the conformal time$\eta$, defined by $c\,d\eta = c\,dt/a(t)$:

$$ds^2 = a^2(\eta)\!\left[-c^2 d\eta^2 + d\chi^2 + S_k^2(\chi)\,d\Omega^2\right]$$

In conformal time, the FLRW metric is conformally flat (for k = 0): the metric equals a time-dependent scale factor times the Minkowski metric.

Conformal time has deep physical significance:

Light cones are 45° lines in a conformal spacetime diagram (just as in special relativity), since null geodesics satisfy $d\chi = \pm c\,d\eta$.
The particle horizon at cosmic time t is$\chi_{\text{ph}} = c\eta(t) = c\int_0^t dt'/a(t')$ — the maximum comoving distance from which light could have reached us.
Cosmological perturbation theory is most naturally formulated in conformal time, where the perturbed equations take their simplest form.

The Friedmann equation in conformal time (denoting $\mathcal{H} = a'/a = aH$, where primes are $d/d\eta$):

$$\mathcal{H}^2 = \frac{8\pi G}{3}\rho\,a^2 - kc^2 + \frac{\Lambda c^2}{3}\,a^2$$

$$\mathcal{H}' = -\frac{4\pi G}{3}\!\left(\rho + \frac{3p}{c^2}\right)a^2 + \frac{\Lambda c^2}{3}\,a^2$$

15. Summary of Key Results

The FLRW Metric

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

First Friedmann Equation

$$H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$$

Second Friedmann Equation

$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\!\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$$

Continuity Equation

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

Friedmann Constraint

$$\Omega_r + \Omega_m + \Omega_\Lambda + \Omega_k = 1$$

The Logical Chain of Derivation

1. Cosmological Principle (homogeneity + isotropy) → maximally symmetric spatial slices

2. Maximally symmetric 3-spaces have constant curvature K = k/R² with k ∈ {-1, 0, +1}

3. Foliation + symmetry → the FLRW metric $ds^2 = -c^2\,dt^2 + a^2(t)\,d\sigma_k^2$

4. Compute Christoffel symbols $\Gamma^\lambda{}_{\mu\nu}$ for the FLRW metric

5. Compute Ricci tensor $R_{\mu\nu}$, Ricci scalar R, Einstein tensor $G_{\mu\nu}$

6. Model matter as perfect fluid: $T_{\mu\nu} = (\rho + p/c^2)u_\mu u_\nu + p\,g_{\mu\nu}$

7. Insert into Einstein's equations $G_{\mu\nu} + \Lambda g_{\mu\nu} = (8\pi G/c^4)\,T_{\mu\nu}$

8. (0,0) component → First Friedmann equation; (i,j) components → Second Friedmann equation

9. $\nabla_\mu T^{\mu\nu} = 0$ → Continuity equation (not independent)

10. Equation of state $p = w\rho c^2$ closes the system → analytical or numerical solutions

Bibliography

Textbooks & Monographs

Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley. — Classic derivation of the FLRW metric and Friedmann equations from first principles.
Weinberg, S. (2008). Cosmology. Oxford University Press. — Comprehensive modern treatment including perturbation theory and observational constraints.
Dodelson, S. & Schmidt, F. (2020). Modern Cosmology, 2nd ed. Academic Press. — Standard graduate textbook with detailed derivations of the Friedmann equations and cosmological distances.
Ryden, B. (2017). Introduction to Cosmology, 2nd ed. Cambridge University Press. — Accessible undergraduate treatment of the expanding universe and Friedmann models.
Carroll, S.M. (2019). Spacetime and Geometry: An Introduction to General Relativity, 2nd ed. Cambridge University Press. — Excellent GR background for the Einstein field equations and cosmological solutions.
Mukhanov, V. (2005). Physical Foundations of Cosmology. Cambridge University Press. — Rigorous treatment of FRW dynamics, thermodynamics, and the deceleration parameter.
Kolb, E.W. & Turner, M.S. (1990). The Early Universe. Addison-Wesley. — Foundational reference for Friedmann cosmology and the matter-radiation transition.
Peebles, P.J.E. (1993). Principles of Physical Cosmology. Princeton University Press. — Comprehensive treatment by a pioneer of modern cosmology.

Key Papers

Friedmann, A. (1922). “Über die Krümmung des Raumes,” Zeitschrift für Physik 10, 377–386. — The original paper deriving the expansion equations from GR.
Lemaître, G. (1927). “Un Univers homogène de masse constante et de rayon croissant,” Annales de la Société Scientifique de Bruxelles A47, 49–59. — Independent derivation of the expansion law, predating Hubble’s observational confirmation.
Hubble, E. (1929). “A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae,” Proceedings of the National Academy of Sciences 15, 168–173. — The observational discovery of cosmic expansion.
Robertson, H.P. (1935). “Kinematics and World-Structure,” Astrophysical Journal 82, 284–301; Walker, A.G. (1937). “On Milne’s Theory of World-Structure,” Proceedings of the London Mathematical Society 42, 90–127. — Formal proof of the uniqueness of the FLRW metric.
Planck Collaboration (2020). “Planck 2018 results. VI. Cosmological parameters,” Astronomy & Astrophysics 641, A6. arXiv:1807.06209. — Definitive measurement of cosmological parameters used throughout this chapter.

← Course Overview Back to Cosmology →

Component	w	\(\rho(a)\)	\(a(t)\) (flat, single component)	Physical origin
Non-relativistic matter	\(0\)	\(\rho_0\,a^{-3}\)	\(a \propto t^{2/3}\)	Number density dilution
Radiation	\(1/3\)	\(\rho_0\,a^{-4}\)	\(a \propto t^{1/2}\)	Dilution + redshift
Cosmological constant	\(-1\)	\(\rho_0 = \text{const}\)	\(a \propto e^{Ht}\)	Vacuum energy
Curvature (effective)	\(-1/3\)	\(\rho_0\,a^{-2}\)	\(a \propto t\)	Spatial curvature term
Stiff matter	\(+1\)	\(\rho_0\,a^{-6}\)	\(a \propto t^{1/3}\)	Kinetic-dominated scalar field

Cosmology

The FLRW Metric & Friedmann Equations

Historical Context

1. The Cosmological Principle

Homogeneity

Isotropy

Maximally Symmetric Spaces and Killing Vectors

2. Constructing the FLRW Metric from Symmetry

2.1 Foliation of Spacetime

2.2 Metrics of Constant Curvature in 3D

k = +1 (Closed)

k = 0 (Flat)

k = -1 (Open)

The FLRW Line Element

2.3 Alternative Coordinate Form

3. Metric Tensor Components

4. Christoffel Symbols — Full Computation

4.1 Mixed temporal-spatial components

4.2 Purely spatial components

5. The Ricci Tensor

5.1 The \(R_{00}\) component

5.2 The \(R_{ij}\) components

Derivation sketch for \(R_{11}\):

5.3 The Ricci Scalar and Einstein Tensor

6. Einstein's Field Equations

The Perfect Fluid Stress-Energy Tensor

7. Deriving the Friedmann Equations

7.1 The First Friedmann Equation

First Friedmann Equation

7.2 The Second Friedmann Equation (Acceleration Equation)

Second Friedmann Equation (Acceleration Equation)

Physical Insight: The Strong Energy Condition

7.3 Alternative Derivation via the Trace

8. The Continuity Equation

Continuity Equation (Fluid Equation)

9. Equation of State and Solutions for Each Component

10. Density Parameters and the Friedmann Constraint

Planck 2018 Best-Fit Values (\(\Lambda\)CDM)

11. Exact Analytical Solutions

11.1 Matter-Dominated (Einstein–de Sitter) Universe

11.2 Radiation-Dominated Universe

11.3 de Sitter Universe (Dark Energy Dominated)

12. Numerical Integration: Python Implementation

13. High-Precision Solver: Fortran Implementation

Fortran + Python: Friedmann Equations with Visualization

Compilation & Output

14. Conformal Time and the FLRW Metric

15. Summary of Key Results

The FLRW Metric

First Friedmann Equation

Second Friedmann Equation

Continuity Equation

Friedmann Constraint

The Logical Chain of Derivation

Bibliography