Part I: The Expanding Universe | Chapter 5

Cosmological Horizons

Particle horizons, event horizons, and the causal structure of an expanding universe — from Rindler's classification to conformal diagrams

1. Introduction: Causal Structure of the Expanding Universe

In Minkowski spacetime, the causal structure is straightforward: the past light cone of any event encompasses an ever-growing region, and in principle every event in the past can eventually influence every event in the future. In an expanding universe governed by general relativity, the situation is fundamentally different. The expansion of space introduces cosmological horizons — boundaries that limit the region of spacetime accessible to any given observer.

There are three distinct notions of “horizon” in cosmology, each capturing a different aspect of causal accessibility:

Particle Horizon

The maximum proper distance from which light could have reached the observer since the Big Bang. It defines the observable universe — the boundary of our past light cone at $t = 0$.

Event Horizon

The maximum proper distance from which light emitted now will ever be able to reach the observer in the infinite future. It exists only in universes with accelerating expansion.

Hubble Horizon

The distance at which the recession velocity equals $c$. Unlike the other horizons, this is not a true causal boundary — light can and does cross it.

These concepts were first systematically classified by Wolfgang Rindler in 1956. Understanding them is essential for appreciating the horizon problem (a key motivation for inflation), the thermodynamics of de Sitter space, and the causal structure revealed by conformal (Penrose-Carter) diagrams.

Notation and Conventions

We work with the FLRW metric in the form $ds^2 = -c^2\,dt^2 + a(t)^2\left[\frac{dr^2}{1-kr^2} + r^2 d\Omega^2\right]$, where $a(t)$ is the scale factor normalized so that $a(t_0) = 1$ today. Comoving coordinates are denoted by $\chi$ (radial comoving distance) and proper distances by $d$. The Hubble parameter is $H(t) = \dot{a}/a$.

2. Derivation 1: The Particle Horizon

2.1 General Derivation

Consider a photon emitted at the Big Bang ($t = 0$) travelling radially toward a comoving observer at the origin. Light travels on null geodesics with $ds^2 = 0$, giving:

$$0 = -c^2\,dt^2 + a(t)^2\,d\chi^2 \quad \Rightarrow \quad d\chi = \frac{c\,dt}{a(t)}$$

Integrating from the Big Bang to time $t$, the comoving distance traversed by the photon is:

$$\chi_{\text{PH}}(t) = \int_0^{t} \frac{c\,dt'}{a(t')}$$

The proper distance to the particle horizon at time $t$ is obtained by multiplying the comoving distance by the scale factor:

Particle Horizon (Proper Distance)

$$\boxed{d_{\text{PH}}(t) = a(t) \int_0^{t} \frac{c\,dt'}{a(t')}}$$

The particle horizon exists whenever this integral converges — i.e., whenever $a(t') \to 0$ slowly enough as $t' \to 0$ that the integrand remains integrable.

Note the crucial distinction: the comoving particle horizon $\chi_{\text{PH}}$ depends only on the expansion history from $t' = 0$ to $t' = t$, while the properparticle horizon $d_{\text{PH}}$ also factors in the current scale factor.

2.2 Power-Law Cosmologies

For a universe dominated by a single component with equation of state $w$, the scale factor evolves as $a(t) \propto t^{2/[3(1+w)]}$. Let us compute the particle horizon for the two most important cases.

Radiation Domination ($w = 1/3$)

With $a(t) = (t/t_0)^{1/2}$, the integral becomes:

$$\chi_{\text{PH}} = \int_0^{t} \frac{c\,dt'}{(t'/t_0)^{1/2}} = \frac{c\,t_0^{1/2}}{1/2}\,t^{1/2} = 2ct_0^{1/2}\,t^{1/2}/t_0^{1/2} = 2ct$$

More carefully: writing $a = (t/t_0)^{1/2}$,

$$d_{\text{PH}} = a(t) \cdot \chi_{\text{PH}} = \left(\frac{t}{t_0}\right)^{1/2} \cdot 2c\,t_0^{1/2}\,t^{1/2} = 2ct$$

$$d_{\text{PH}}^{\text{rad}} = 2ct$$

Matter Domination ($w = 0$)

With $a(t) = (t/t_0)^{2/3}$:

$$\chi_{\text{PH}} = \int_0^{t} \frac{c\,dt'}{(t'/t_0)^{2/3}} = c\,t_0^{2/3} \cdot \frac{t'^{1/3}}{1/3}\bigg|_0^t = 3c\,t_0^{2/3}\,t^{1/3}$$

The proper distance:

$$d_{\text{PH}} = \left(\frac{t}{t_0}\right)^{2/3} \cdot 3c\,t_0^{2/3}\,t^{1/3} = 3ct$$

$$d_{\text{PH}}^{\text{mat}} = 3ct$$

In general, for $a \propto t^n$ with $n < 1$ (decelerating expansion), we get$d_{\text{PH}} = \frac{ct}{1-n}$. The particle horizon is always finite in decelerating cosmologies — the integral converges because $a \to 0$ sufficiently fast.

2.3 de Sitter Universe

For a de Sitter universe with $a(t) = e^{Ht}$ (pure cosmological constant, $H = \text{const}$):

$$\chi_{\text{PH}} = \int_0^{t} \frac{c\,dt'}{e^{Ht'}} = \frac{c}{H}\left(1 - e^{-Ht}\right)$$

As $t \to \infty$: $\chi_{\text{PH}} \to c/H$. The proper distance is:

$$d_{\text{PH}} = e^{Ht} \cdot \frac{c}{H}\left(1 - e^{-Ht}\right) = \frac{c}{H}\left(e^{Ht} - 1\right)$$

For $Ht \gg 1$: $d_{\text{PH}} \approx \frac{c}{H}e^{Ht}$, which grows exponentially. But the comoving particle horizon saturates at $c/H$ — only a finite comoving region is ever accessible.

2.4 Particle Horizon at Recombination

A key application is computing the particle horizon at the time of recombination ($z_{\text{rec}} \approx 1100$). The comoving particle horizon at recombination determines the maximum angular scale over which regions on the CMB could have been in causal contact.

Converting the time integral to an integral over scale factor $a$, using $dt = da/(aH)$:

$$\chi_{\text{PH}}(a_{\text{rec}}) = \int_0^{a_{\text{rec}}} \frac{c\,da}{a^2 H(a)}$$

In the radiation+matter era, $H^2 = H_0^2\left(\Omega_r\,a^{-4} + \Omega_m\,a^{-3}\right)$, and with$a_{\text{rec}} = 1/(1+z_{\text{rec}}) \approx 9.1 \times 10^{-4}$:

$$\chi_{\text{PH}}(a_{\text{rec}}) \approx \frac{c}{H_0} \int_0^{a_{\text{rec}}} \frac{da}{\sqrt{\Omega_r + \Omega_m\,a}} \approx 280\;\text{Mpc (comoving)}$$

The angular diameter distance to recombination is $d_A \approx 12.8\;\text{Gpc}$. The angular size of the particle horizon on the CMB sky is therefore:

$$\theta_{\text{PH}} \approx \frac{\chi_{\text{PH}}(a_{\text{rec}})}{d_A(z_{\text{rec}})} \approx \frac{280\;\text{Mpc}}{12800\;\text{Mpc}} \approx 0.022\;\text{rad} \approx 1.3°$$

(A more precise calculation including the sound horizon gives the first acoustic peak at $\ell \approx 200$, corresponding to $\sim 1°$). The key point: the particle horizon at recombination subtends only about 2° on the sky (accounting for the full causal horizon rather than just the sound horizon). Yet the CMB is uniform to $10^{-5}$ across the entire sky — this is the horizon problem.

3. Derivation 2: The Cosmological Event Horizon

3.1 General Derivation

While the particle horizon looks backward in time, the event horizon looks forward. It answers: what is the maximum (proper) distance of an event occurring now that the observer can ever receive a signal from?

A photon emitted at time $t$ from comoving coordinate $\chi$ reaches the observer at the origin at time $t_{\text{obs}}$ determined by:

$$\chi = \int_t^{t_{\text{obs}}} \frac{c\,dt'}{a(t')}$$

The event horizon corresponds to the maximum $\chi$ for which $t_{\text{obs}} \to \infty$:

$$\chi_{\text{EH}}(t) = \int_t^{\infty} \frac{c\,dt'}{a(t')}$$

Event Horizon (Proper Distance)

$$\boxed{d_{\text{EH}}(t) = a(t) \int_t^{\infty} \frac{c\,dt'}{a(t')}}$$

The event horizon exists if and only if this integral converges — requiring sufficiently rapid expansion at late times.

3.2 Convergence Criterion

For $a(t) \propto t^n$, the integral $\int_t^{\infty} dt'/t'^n$ converges only if $n > 1$, i.e., accelerating expansion($\ddot{a} > 0$).

No Event Horizon

Radiation: $a \propto t^{1/2}$ ⇒ $\int_t^\infty t'^{-1/2}\,dt'$ diverges.

Matter: $a \propto t^{2/3}$ ⇒ $\int_t^\infty t'^{-2/3}\,dt'$ diverges.

In these universes, given enough time, every event is eventually observable.

Event Horizon Exists

de Sitter: $a = e^{Ht}$ ⇒ $\int_t^\infty e^{-Ht'}\,dt' = \frac{1}{H}e^{-Ht}$ converges.

ΛCDM (late time): dominated by $\Lambda$, so event horizon emerges.

Events beyond the event horizon are forever inaccessible.

3.3 de Sitter Event Horizon

For de Sitter space with $a = e^{Ht}$:

$$d_{\text{EH}} = e^{Ht} \int_t^{\infty} \frac{c\,dt'}{e^{Ht'}} = e^{Ht} \cdot \frac{c}{H}\,e^{-Ht} = \frac{c}{H}$$

$$d_{\text{EH}}^{\text{dS}} = \frac{c}{H}$$

The de Sitter event horizon is constant in time and equals the Hubble radius. This is a special coincidence specific to de Sitter space — in general, the event horizon and Hubble horizon are distinct.

3.4 Event Horizon in ΛCDM

In our ΛCDM universe, the event horizon at the present epoch can be computed numerically. Converting to an integral over scale factor:

$$d_{\text{EH}}(t_0) = a_0 \int_{a_0}^{\infty} \frac{c\,da}{a^2\,H(a)} = \int_1^{\infty} \frac{c\,da}{a^2\,H_0\,E(a)}$$

where $E(a) = \sqrt{\Omega_m\,a^{-3} + \Omega_\Lambda}$ at late times. Evaluating this integral gives:

$$d_{\text{EH}}(t_0) \approx 16.7\;\text{Gly} \approx 5.1\;\text{Gpc}$$

Events currently happening at proper distances beyond $\sim 16\;\text{Gly}$ will never be observable to us. Galaxies we see today (which are currently beyond 16 Gly due to expansion) are already sending their last observable photons.

4. Derivation 3: The Hubble Horizon

4.1 Definition and Physical Meaning

The Hubble horizon (or Hubble radius) is defined as:

Hubble Horizon

$$\boxed{d_H(t) = \frac{c}{H(t)}}$$

The physical interpretation follows from Hubble's law. At proper distance $d$, the recession velocity is $v_{\text{rec}} = H(t)\,d$. Setting $v_{\text{rec}} = c$:

$$v_{\text{rec}} = H\,d = c \quad \Rightarrow \quad d = \frac{c}{H} = d_H$$

The Hubble horizon is the distance at which the recession velocity equals the speed of light. However, this does not mean that objects beyond $d_H$ are unobservable! Photons emitted from beyond the Hubble radius can still reach us if the Hubble sphere subsequently expands to encompass them. Superluminal recession is a property of the expanding coordinate grid, not a violation of special relativity.

4.2 Distinguishing the Three Horizons

Horizon	Definition	Question Answered	Causal Boundary?
Particle	$a \int_0^t c\,dt'/a$	What can we see from the past?	Yes (past)
Event	$a \int_t^\infty c\,dt'/a$	What will we ever see?	Yes (future)
Hubble	$c/H$	Where is recession superluminal?	No

4.3 The Comoving Hubble Radius and Inflation

A quantity of particular importance for inflation is the comoving Hubble radius:

$$R_H^{\text{com}}(t) = \frac{c}{a(t)\,H(t)} = (aH)^{-1}$$

For power-law expansion $a \propto t^n$, we have $H = n/t$, so:

$$R_H^{\text{com}} = \frac{t}{n\,t^n} \propto t^{1-n}$$

The behavior depends critically on whether $n < 1$ or $n > 1$:

Standard Deceleration ($n < 1$)

Radiation ($n=1/2$): $R_H^{\text{com}} \propto t^{1/2}$ — grows.

Matter ($n=2/3$): $R_H^{\text{com}} \propto t^{1/3}$ — grows.

More and more comoving scales “enter the horizon” over time.

Inflation ($n > 1$ or exponential)

de Sitter: $R_H^{\text{com}} = (e^{Ht} \cdot H)^{-1} \propto e^{-Ht}$ — shrinks.

The comoving Hubble radius decreases during inflation.

Comoving scales “exit the horizon” during inflation, then re-enter after inflation ends. This is the mechanism by which inflation solves the horizon problem.

Key Insight: Horizon Exit and Re-entry

During inflation, the comoving Hubble radius $(aH)^{-1}$ shrinks dramatically. A perturbation with comoving wavelength $\lambda$ is “inside the horizon” when $\lambda < (aH)^{-1}$. During inflation, $(aH)^{-1}$ decreases until $\lambda > (aH)^{-1}$ — the mode “exits the horizon.” After inflation, $(aH)^{-1}$ grows again, and the mode eventually “re-enters.” This mechanism explains why the CMB has correlations on scales larger than the naïve particle horizon at recombination.

5. Derivation 4: Conformal Diagrams and Causal Structure

5.1 Conformal Time

The key to visualizing causal structure is the conformal time:

Conformal Time

$$\boxed{\eta(t) = \int_0^{t} \frac{c\,dt'}{a(t')}}$$

Note: $\eta$ is precisely the comoving particle horizon distance! This is no coincidence.

In terms of conformal time, the FLRW metric becomes:

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

This is conformally flat: the metric is just the Minkowski metric $\eta_{\mu\nu}$ multiplied by $a^2$. Since null geodesics ($ds^2 = 0$) are unaffected by a conformal factor, light rays travel at 45° in the $(\eta, \chi)$ plane, just as in Minkowski space.

5.2 Conformal Time for Different Cosmologies

Radiation Domination

$a \propto t^{1/2}$, so $\eta = \int_0^t c\,dt'/t'^{1/2} \propto t^{1/2}$, giving $a \propto \eta$. The conformal time range is $\eta \in [0, \infty)$.

Matter Domination

$a \propto t^{2/3}$, so $\eta \propto t^{1/3}$, giving $a \propto \eta^2$. Again $\eta \in [0, \infty)$.

de Sitter ($a = e^{Ht}$)

$\eta = \int_{-\infty}^t c\,dt'/e^{Ht'} = -\frac{c}{H}\,e^{-Ht}$. As $t \to -\infty$: $\eta \to -\infty$; as$t \to +\infty$: $\eta \to 0^-$. Range: $\eta \in (-\infty, 0)$.

The future boundary ($\eta = 0$) is spacelike — this is the signature of an event horizon.

5.3 Penrose-Carter Diagrams

A Penrose-Carter diagram (or conformal diagram) maps the entire spacetime onto a finite region by further compactifying the conformal coordinates. The key properties:

Light rays travel at 45°
Timelike worldlines are steeper than 45°
Spatial infinity, temporal infinity, and null infinity are mapped to finite boundaries
The particle horizon appears as the past light cone of the observer at the present time, traced back to $t = 0$
The event horizon appears as the past light cone of the observer at $t = \infty$

Radiation/Matter Universe

The diagram is a triangle: the Big Bang ($\eta = 0$) is a horizontal spacelike line at the bottom, and $\eta \to \infty$ is the apex at the top. There is a particle horizon but no event horizon. The past light cone of the observer at late times eventually encompasses the entire spatial slice at$\eta = 0$.

ΛCDM Universe

The diagram is a rectangle (or square): the Big Bang is the bottom edge and the future de Sitter boundary ($\eta \to \eta_{\max}$) is the top edge. Both are spacelike. The finite maximum conformal time reflects the existence of an event horizon. The past light cone from the future boundary separates events that are eventually observable from those that are not.

Horizons on the Conformal Diagram

On a conformal diagram with the observer at the center of the bottom edge:

Particle horizon at time $\eta_0$: the intersection of the observer's past light cone (45° lines from $\eta_0$) with the Big Bang surface $\eta = 0$.
Event horizon: the 45° line from the observer at the top edge ($\eta_{\max}$), projected down. Events below this line (closer to the observer) are eventually observable; events above are not.
Hubble horizon: the curve where $v_{\text{rec}} = c$, which is nota straight line on the conformal diagram and does not separate causally connected from disconnected regions.

6. Applications

6.1 The Horizon Problem and Inflation

The particle horizon at recombination subtends only $\sim 2°$ on the sky. Yet the CMB is isotropic to$10^{-5}$ across the full $360°$ sky. There are roughly$(360/2)^2 \sim 30{,}000$ causally disconnected patches on the last scattering surface. How did they “know” to have the same temperature?

Inflation solves this by postulating a period of exponential expansion before the radiation era. During inflation, the comoving Hubble radius $(aH)^{-1}$ shrinks, so that scales much larger than the horizon at recombination were actually inside the Hubble radius at earlier times. All observed patches shared a common causal past before inflation stretched them apart.

Quantitative Requirement

The number of e-folds of inflation needed to solve the horizon problem is:

$$N = \ln\frac{a_{\text{end}}}{a_{\text{start}}} \gtrsim 60$$

This ensures that the comoving Hubble radius at the start of inflation was larger than our entire observable universe today, so all points on the CMB sky shared a common causal past.

6.2 Information Loss at Cosmological Horizons

The de Sitter event horizon shares deep similarities with a black hole event horizon. Gibbons and Hawking (1977) showed that the de Sitter horizon radiates thermally at temperature:

$$T_{\text{dS}} = \frac{\hbar\,H}{2\pi\,k_B\,c}$$

For the observed value $H_0 \approx 67.4\;\text{km/s/Mpc}$, this gives$T_{\text{dS}} \sim 10^{-30}\;\text{K}$ — utterly negligible compared to the CMB temperature. However, as the universe approaches pure de Sitter space in the far future, the CMB redshifts away and the de Sitter temperature becomes the dominant thermal background.

The associated entropy is:

$$S_{\text{dS}} = \frac{k_B\,c^3\,A}{4\,G\,\hbar} = \frac{\pi\,k_B\,c^5}{G\,\hbar\,H^2} \sim 10^{122}\;k_B$$

This is the maximum entropy of the observable universe according to the holographic principle, and it coincides with the Bekenstein-Hawking entropy formula applied to the cosmological horizon area$A = 4\pi(c/H)^2$.

6.3 Holographic Cosmology

The holographic principle, inspired by black hole thermodynamics, suggests that the maximum number of degrees of freedom in a region of space is proportional to the surface area of its boundary (in Planck units), not its volume. Applied to cosmology:

The cosmological event horizon area $A = 4\pi d_{\text{EH}}^2$ bounds the entropy of the observable universe.
The Bousso bound (covariant entropy bound) generalizes this to arbitrary cosmological spacetimes using light sheets.
In the far future de Sitter phase, the universe asymptotes to a finite-dimensional Hilbert space of dimension $\sim e^{S_{\text{dS}}}$ — a profoundly finite number despite the infinite spatial extent.

7. Historical Context

Timeline of Key Developments

1917 — de Sitter: Willem de Sitter found the first vacuum solution to Einstein's equations with a cosmological constant, later recognized as an exponentially expanding spacetime with both particle and event horizons.
1956 — Rindler: Wolfgang Rindler published the definitive classification of cosmological horizons in his paper “Visual Horizons in World-Models,” introducing the terms “particle horizon” and “event horizon” in a cosmological context. This paper remains the standard reference.
1963–1967 — Penrose & Carter: Roger Penrose and Brandon Carter developed conformal compactification techniques for representing infinite spacetimes as finite diagrams, revolutionizing the study of causal structure. Penrose-Carter diagrams became the standard tool for visualizing horizons.
1969 — Misner: Charles Misner formulated the horizon problem explicitly, noting that the observed isotropy of the CMB could not be explained by causal processes in a standard Big Bang cosmology.
1977 — Gibbons & Hawking: Extended Hawking's black hole radiation to cosmological horizons, showing that de Sitter space has a temperature$T = \hbar H / (2\pi k_B c)$ and entropy proportional to the horizon area.
1981 — Guth: Alan Guth proposed cosmic inflation as a solution to the horizon and flatness problems, showing that a period of exponential expansion naturally produces a causally connected observable universe.
1998 — Riess/Perlmutter: The discovery of the accelerating expansion of the universe confirmed that our universe does possess an event horizon, making all of the above considerations physically relevant rather than merely theoretical.

8. Python Simulation: Cosmological Horizons in ΛCDM

The following simulation computes and plots the particle horizon, event horizon, and Hubble horizon as functions of cosmic time and redshift for a ΛCDM cosmology. It also generates a conformal spacetime diagram showing the past light cone of an observer today. We use only numpy for numerical integration (simple trapezoidal rule) and no external plotting libraries — the output is printed as a table for the built-in chart renderer.

Python

script.py146 lines

#!/usr/bin/env python3
"""
cosmological_horizons.py
Compute particle horizon, event horizon, and Hubble horizon vs cosmic time
for a flat LCDM cosmology. Also compute conformal diagram coordinates.
Uses only numpy (no scipy).
"""
import numpy as np

# =============================================
# Cosmological Parameters (Planck 2018)
# =============================================
H0_km = 67.4              # km/s/Mpc
Mpc_m = 3.0857e22         # meters per Mpc
H0_si = H0_km * 1e3 / Mpc_m  # H0 in 1/s
c_si = 2.998e8             # m/s
Gyr_s = 3.1557e16          # seconds per Gyr
Gly_m = c_si * Gyr_s       # meters per Gly

Omega_r = 9.1e-5
Omega_m = 0.315
Omega_L = 0.685
Omega_k = 0.0

# =============================================
# Dimensionless Hubble parameter E(a) = H(a)/H0
# =============================================
def E(a):
    return np.sqrt(Omega_r / a**4 + Omega_m / a**3 + Omega_k / a**2 + Omega_L)

# =============================================
# Cosmic time t(a) via trapezoidal integration
# dt/da = 1/(a H(a))
# =============================================
def compute_cosmic_time(a_arr):
    """Return t(a) in Gyr for an array of scale factors."""
    t = np.zeros_like(a_arr)
    for i in range(1, len(a_arr)):
        da = a_arr[i] - a_arr[i-1]
        f0 = 1.0 / (a_arr[i-1] * E(a_arr[i-1]) * H0_si)
        f1 = 1.0 / (a_arr[i] * E(a_arr[i]) * H0_si)
        t[i] = t[i-1] + 0.5 * (f0 + f1) * da
    return t / Gyr_s  # convert to Gyr

# =============================================
# Comoving particle horizon chi_PH(a) = int_0^a c da'/(a'^2 H(a'))
# =============================================
def compute_particle_horizon_comoving(a_arr):
    """Return comoving particle horizon in Gly."""
    chi = np.zeros_like(a_arr)
    for i in range(1, len(a_arr)):
        da = a_arr[i] - a_arr[i-1]
        f0 = c_si / (a_arr[i-1]**2 * E(a_arr[i-1]) * H0_si)
        f1 = c_si / (a_arr[i]**2 * E(a_arr[i]) * H0_si)
        chi[i] = chi[i-1] + 0.5 * (f0 + f1) * da
    return chi / Gly_m  # convert to Gly

# =============================================
# Comoving event horizon chi_EH(a) = int_a^inf c da'/(a'^2 H(a'))
# =============================================
def compute_event_horizon_comoving(a_arr, a_future):
    """Compute comoving event horizon by integrating from a to a_max."""
    # First compute the full integral from each a to a_max
    # We integrate backwards from the future
    chi_future = np.zeros(len(a_future))
    # Integrate from a_max down to each a
    total = 0.0
    for i in range(len(a_future)-1, 0, -1):
        da = a_future[i] - a_future[i-1]
        f0 = c_si / (a_future[i-1]**2 * E(a_future[i-1]) * H0_si)
        f1 = c_si / (a_future[i]**2 * E(a_future[i]) * H0_si)
        total += 0.5 * (f0 + f1) * da
        chi_future[i-1] = total
    # Interpolate onto a_arr
    chi_eh = np.interp(a_arr, a_future, chi_future)
    return chi_eh / Gly_m  # Gly

# =============================================
# Hubble horizon d_H = c/H(a) in proper distance
# =============================================
def compute_hubble_horizon(a_arr):
    """Return Hubble horizon c/H in Gly (proper distance)."""
    return c_si / (E(a_arr) * H0_si) / Gly_m

# =============================================
# Main computation
# =============================================
N = 5000
a_min = 1e-6
a_max_future = 100.0  # far future for event horizon integral

# Scale factor array from a_min to today (a=1) and beyond
a_past = np.linspace(a_min, 1.0, N)
a_future_full = np.linspace(a_min, a_max_future, N * 10)

# Cosmic time
t_past = compute_cosmic_time(a_past)

# Comoving particle horizon
chi_ph = compute_particle_horizon_comoving(a_past)
# Proper particle horizon
d_ph = a_past * chi_ph

# Comoving event horizon
chi_eh = compute_event_horizon_comoving(a_past, a_future_full)
# Proper event horizon
d_eh = a_past * chi_eh

# Hubble horizon (proper)
d_hub = compute_hubble_horizon(a_past)

# =============================================
# Print table: t(Gyr), d_PH(Gly), d_EH(Gly), d_H(Gly)
# =============================================
print("t_Gyr d_PH_Gly d_EH_Gly d_H_Gly")
step = max(1, len(a_past) // 150)
for i in range(0, len(a_past), step):
    if t_past[i] > 0.001:  # skip very early times
        print(f"{t_past[i]:.4f} {d_ph[i]:.4f} {d_eh[i]:.4f} {d_hub[i]:.4f}")

# Print current values
print()
print("=== Present-day values (a=1, t=t0) ===")
print(f"Age of universe: {t_past[-1]:.2f} Gyr")
print(f"Particle horizon (proper): {d_ph[-1]:.2f} Gly")
print(f"Event horizon (proper): {d_eh[-1]:.2f} Gly")
print(f"Hubble horizon: {d_hub[-1]:.2f} Gly")
print(f"Comoving particle horizon: {chi_ph[-1]:.2f} Gly")
print(f"Comoving event horizon: {chi_eh[-1]:.2f} Gly")

# =============================================
# Part 2: Conformal diagram data
# Past light cone: chi(a) = eta_0 - eta(a)
# where eta(a) = comoving particle horizon = chi_ph(a)
# =============================================
print()
print("=== Conformal Diagram: Past Light Cone ===")
print("eta_conf chi_comoving")
eta_0 = chi_ph[-1]  # conformal time today = comoving particle horizon
# Past light cone traced backward
for i in range(0, len(a_past), step):
    eta_i = chi_ph[i]
    chi_lc = eta_0 - eta_i  # comoving distance of photon at conformal time eta_i
    if chi_lc >= 0:
        print(f"{eta_i:.4f} {chi_lc:.4f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

The simulation above produces three key outputs:

Particle horizon ($d_{\text{PH}}$): grows monotonically, reaching $\sim 46$ Gly today (the radius of the observable universe).
Event horizon ($d_{\text{EH}}$): starts large and decreases, converging to $c/H_\infty \approx 16$ Gly as the universe approaches de Sitter.
Hubble horizon ($d_H = c/H$): not monotonic; initially grows (deceleration phase), then turns over and asymptotes to $c/H_\infty$.

Interpreting the Conformal Diagram

The conformal diagram output plots $\eta$ (conformal time) vs $\chi$ (comoving distance). The past light cone appears as a curve starting at $(\eta_0, 0)$ (us, today) and reaching back to $(\eta = 0, \chi_{\max})$ (the Big Bang surface). In conformal coordinates, the light cone is a straight 45° line, confirming the conformal flatness of the FLRW metric. The maximum comoving distance reached is $\eta_0 \approx 46$ Gly, which is the comoving radius of the observable universe.

References

Textbooks

Weinberg, S. (2008). Cosmology. Oxford University Press. — Rigorous treatment of horizons and causal structure in Ch. 1.
Mukhanov, V. (2005). Physical Foundations of Cosmology. Cambridge University Press. — Excellent discussion of conformal diagrams and inflation.
Dodelson, S. & Schmidt, F. (2020). Modern Cosmology, 2nd ed. Academic Press. — Clear treatment of horizons and the horizon problem.
Ryden, B. (2017). Introduction to Cosmology, 2nd ed. Cambridge University Press. — Accessible introduction with good horizon discussions.

Key Papers

Rindler, W. (1956). “Visual Horizons in World-Models,”Monthly Notices of the Royal Astronomical Society 116, 662–677. — The definitive classification of cosmological horizons.
Gibbons, G.W. & Hawking, S.W. (1977). “Cosmological event horizons, thermodynamics, and particle creation,” Physical Review D 15, 2738–2751. — Thermodynamics of de Sitter horizons.
Davis, T.M. & Lineweaver, C.H. (2004). “Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe,”Publications of the Astronomical Society of Australia 21, 97–109. — Essential reading on the distinction between different horizons.
Guth, A.H. (1981). “Inflationary universe: A possible solution to the horizon and flatness problems,” Physical Review D 23, 347–356. — The original inflation paper.
Planck Collaboration (2020). “Planck 2018 results. VI. Cosmological parameters,” Astronomy & Astrophysics 641, A6. arXiv:1807.06209. — Cosmological parameters used in this chapter.

← Distance Measures Thermal History →

Horizon	Definition	Question Answered	Causal Boundary?
Particle	\(a \int_0^t c\,dt'/a\)	What can we see from the past?	Yes (past)
Event	\(a \int_t^\infty c\,dt'/a\)	What will we ever see?	Yes (future)
Hubble	\(c/H\)	Where is recession superluminal?	No

Cosmological Horizons

1. Introduction: Causal Structure of the Expanding Universe

Particle Horizon

Event Horizon

Hubble Horizon

Notation and Conventions

2. Derivation 1: The Particle Horizon

2.1 General Derivation

2.2 Power-Law Cosmologies

Radiation Domination (\(w = 1/3\))

Matter Domination (\(w = 0\))

2.3 de Sitter Universe

2.4 Particle Horizon at Recombination

3. Derivation 2: The Cosmological Event Horizon

3.1 General Derivation

3.2 Convergence Criterion

No Event Horizon

Event Horizon Exists

3.3 de Sitter Event Horizon

3.4 Event Horizon in ΛCDM

4. Derivation 3: The Hubble Horizon

4.1 Definition and Physical Meaning

4.2 Distinguishing the Three Horizons

4.3 The Comoving Hubble Radius and Inflation

Standard Deceleration (\(n < 1\))

Inflation (\(n > 1\) or exponential)

Key Insight: Horizon Exit and Re-entry

5. Derivation 4: Conformal Diagrams and Causal Structure

5.1 Conformal Time

5.2 Conformal Time for Different Cosmologies

Radiation Domination

Matter Domination

de Sitter (\(a = e^{Ht}\))

5.3 Penrose-Carter Diagrams

Radiation/Matter Universe

ΛCDM Universe

Horizons on the Conformal Diagram

6. Applications

6.1 The Horizon Problem and Inflation

Quantitative Requirement

6.2 Information Loss at Cosmological Horizons

6.3 Holographic Cosmology

7. Historical Context

Timeline of Key Developments

8. Python Simulation: Cosmological Horizons in ΛCDM

Interpreting the Conformal Diagram

References