Horizon Problem

The cosmic microwave background is the most uniform large-scale radiation field we know. Strip away the dipole our own motion imprints on it, and its temperature $T_0=2.725\,\mathrm{K}$ is the same in every direction to a few parts in $10^5$ — hotter and colder patches differ by microkelvins on a two-and-a-half-kelvin sea. In the standard hot Big Bang, built on the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, that uniformity is not reassuring but alarming. The photons arriving from opposite sides of the sky left the surface of last scattering at redshift $z_{\rm ls}\approx1090$ , and when you trace their past light cones back to the singularity you find they never touch. Two such regions had no common causal past: no signal of any kind could have passed between them in the whole prior age of the universe. Nothing was available to negotiate a common temperature — and yet they agree to five decimal places. The engine of the puzzle is a single monotonic quantity: the comoving Hubble radius $(aH)^{-1}$ , the comoving distance over which causal physics can act in one expansion time. Ordinary matter and radiation both satisfy the strong energy condition $\rho+3p>0$ , so gravity decelerates the expansion, $\ddot a<0$ , and $(aH)^{-1}$ grows without pause. Wind the film backwards and the causal horizon shrinks toward zero; every scale we survey today was, at early times, larger than the region any signal could have crossed. Project the horizon at last scattering onto the sky and it subtends barely $1$ - $2^\circ$ , so the full sky holds roughly $10^4$ such patches, each an island that never traded a single quantum with its neighbours, all somehow tuned to one temperature. That is not physics doing its work; it is a fine-tuned initial condition — exactly the coincidence a good cosmology should dissolve. The resolution, cosmic inflation, does so by making $(aH)^{-1}$ shrink for a while, and in the same stroke seeds the fluctuations that grew into galaxies.

Key takeaways

In the standard hot Big Bang the comoving particle horizon at last scattering subtends only about $1$ - $2^\circ$ , yet the CMB is isotropic to $\sim10^{-5}$ across the whole $4\pi$ steradians.
Roughly $10^4$ causally disconnected patches at recombination share one blackbody temperature without ever exchanging a signal — a tuned initial condition, not a dynamical result.
The root cause is deceleration: matter and radiation obey $\rho+3p>0$ , so the comoving Hubble radius $(aH)^{-1}$ grows with time and the largest scales we see today lay outside the horizon early on.
Quantitatively $\eta_{\rm ls}\approx284\,\mathrm{Mpc}$ against a distance to last scattering $\eta_0\approx14\,300\,\mathrm{Mpc}$ , giving $\theta_H\approx1.2^\circ$ and $N_{\rm patches}\sim10^4$ .
Cosmic inflation cures it with an early epoch of accelerated expansion ( $\ddot a>0$ , $w<-\tfrac13$ ) during which $(aH)^{-1}$ shrinks, so today's observable universe descends from one thermalized patch after $N_e\gtrsim60$ e-folds — it is not a flaw in general relativity but a demand that smooth initial data be explained, not assumed.
Key idea: in a decelerating universe the comoving Hubble radius $(aH)^{-1}$ grows, so the particle horizon at last scattering is far smaller than the distance to it; CMB regions more than $\sim1$ - $2^\circ$ apart were never in causal contact yet share one temperature — a fine-tuning that inflation removes by shrinking $(aH)^{-1}$ .

The core insight

In a decelerating (radiation- or matter-dominated) universe the comoving Hubble radius $(aH)^{-1}$ grows with time, so the particle horizon at last scattering is far smaller than the comoving distance to the last-scattering surface. Regions of the CMB separated by more than $\sim1$ - $2^\circ$ were never in causal contact, yet share the same temperature — a fine-tuning that standard cosmology cannot explain but inflation can.

Where this lives today

The horizon problem is no longer just a motivation slide — it is a live constraint on precision cosmology. Its clearest fingerprint is the low-multipole plateau of the CMB power spectrum: the Sachs-Wolfe region at $\ell\lesssim40$ encodes fluctuations correlated on scales larger than the $1$ - $2^\circ$ causal horizon, which no post-Big-Bang process could make but which arise naturally when quantum modes freeze outside the Hubble radius during inflation. WMAP and Planck measured this plateau and the acoustic peaks beyond it, and their consistency with a nearly scale-invariant, adiabatic, Gaussian spectrum ( $n_s\approx0.965$ ) is the strongest evidence that inflation, not fine-tuned initial data, smoothed the sky. The requirement $N_e\gtrsim50$ - $60$ also feeds model building: with the tensor-to-scalar ratio $r$ it maps viable inflaton potentials in the $n_s$ - $r$ plane that CMB-S4, LiteBIRD, and the Simons Observatory now target through B-mode polarization — and since ekpyrotic and bouncing models also restore causal contact by shrinking $(aH)^{-1}$ under $w\gg1$ , measuring B-modes and $f_{\rm NL}$ is in part a test of how the horizon problem was solved.

The physical picture

Picture two observers on the last-scattering surface, one on each side of our sky. The oldest news either could possibly have received travelled at most one particle-horizon distance — projected onto our sky, only a couple of degrees wide. So the two are like villages that never traded, never sent a runner, never saw each other's smoke, yet keep their clocks synchronized to five decimal places. Someone must have set every clock before the villages were founded, and in FLRW cosmology there is no such someone: the smoothness has to be written into the initial data by hand. The deep reason is that matter and radiation gravitate attractively and slow the expansion. In a decelerating universe the region a signal can sweep out in one Hubble time — the comoving Hubble radius $(aH)^{-1}=c/\dot a$ — grows as the universe ages, and the particle horizon tracks it, so both were minuscule early on. The first figure is the classic conformal diagram making the disconnection literal: our past light cone, the last-scattering surface, and the tiny non-overlapping causal patches of two points on it.

Figure

Conformal diagram of the horizon problem: our past light cone (blue) meets the last-scattering surface at two points whose own past light cones (amber) reach back to the Big Bang without ever overlapping — so the two patches, separated by more than the horizon angle θ_H ≈ 1.2°, were never in causal contact yet radiate at the same T = 2.725 K

The same message reads off scale space. Every comoving scale we observe today was, at early enough times, larger than the comoving Hubble radius — "outside the horizon". Inflation flips the acceleration's sign: for a stretch of early history $\ddot a>0$ and $(aH)^{-1}$ shrinks, so a thermalized patch is blown up until it swallows everything we see. The second figure plots $(aH)^{-1}$ against the scale factor: our observable universe exits the shrinking horizon during inflation and re-enters the growing horizon billions of years later.

Figure

The comoving Hubble radius (aH)⁻¹ versus scale factor a: during inflation it shrinks (slope set by ä greater than 0), then grows all through the radiation era as (aH)⁻¹ ∝ a and the matter era as ∝ a^(1/2). A fixed comoving scale (green, our observable universe) exits the horizon during inflation and re-enters today, requiring N_e ≳ 60 e-folds of shrinking to work

One image ties the two together: conformal time $\eta$ is a causal clock, since the comoving distance light travels is exactly $\Delta\eta$ . In standard cosmology $\eta$ starts at zero at the singularity and accumulates slowly through the decelerating eras, so by last scattering horizons are tiny; inflation effectively pushes the Big Bang to large negative conformal time, giving light a head start that stretches the causal horizon past the whole sky before the hot phase begins.

The core equations

The horizon problem is a competition between two comoving lengths — how far light could travel before last scattering, and how far that surface is now. The first is the comoving particle horizon, identical to the conformal time elapsed since the Big Bang.

\eta(t)=\int_0^{t}\frac{c\,\mathrm{d}t'}{a(t')}=\int_0^{a}\frac{c\,\mathrm{d}a'}{a'^2\,H(a')} \qquad (1)

where: $\eta$ — comoving particle horizon, equal to the conformal time and the comoving distance light has travelled since $t=0$ ; $a$ — scale factor, $a_0=1$ today; $H=\dot a/a$ — Hubble parameter; $c$ — speed of light. Reading: the integral is dominated by late times, so in a decelerating universe the horizon was tiny early and grows monotonically. The comoving Hubble radius is the horizon's local, one-expansion-time cousin; in a decelerating universe the two agree up to an order-unity factor, which is why they are so often conflated.

(aH)^{-1}=\frac{c}{\dot a} \qquad (2)

where: $(aH)^{-1}$ — comoving Hubble radius, the length over which causal physics acts in one Hubble time; $\dot a$ — time derivative of the scale factor. Reading: as $c/\dot a$ , it shrinks exactly when $\dot a$ grows, i.e. when the expansion accelerates. Differentiating (2) turns the problem into a statement about the sign of the acceleration, and hence — via the second Friedmann equation — the equation of state.

\frac{\mathrm{d}}{\mathrm{d}t}(aH)^{-1}=-\frac{c\,\ddot a}{\dot a^{2}},\qquad \ddot a=-\frac{4\pi G}{3}\bigl(\rho+3p\bigr)\,a \qquad (3)

where: $\ddot a$ — cosmic acceleration; $\rho,p$ — energy density and pressure; $G$ — Newton's constant. Reading: $(aH)^{-1}$ decreases iff $\ddot a>0$ , i.e. $w<-\tfrac13$ . Matter ( $w=0$ ) and radiation ( $w=\tfrac13$ ) both give $\ddot a<0$ , so the horizon grows — the horizon problem in one line. Before recombination the universe is a matter-plus-radiation mixture, so the horizon integral (1) has a closed form once the matter term is factored out.

\eta_{\rm ls}=\frac{2c}{H_0\sqrt{\Omega_{m,0}}}\left(\sqrt{a_{\rm ls}+a_{\rm eq}}-\sqrt{a_{\rm eq}}\right) \qquad (4)

where: $\eta_{\rm ls}$ — comoving horizon at last scattering; $a_{\rm ls}=1/(1+z_{\rm ls})$ , $z_{\rm ls}\approx1090$ ; $a_{\rm eq}=\Omega_{r,0}/\Omega_{m,0}$ , equality ( $z_{\rm eq}\approx3400$ ); $\Omega_{m,0}\approx0.31$ ; $H_0\approx67\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Reading: with these numbers $\eta_{\rm ls}\approx284\,\mathrm{Mpc}$ comoving. The observable consequence is the angle this horizon subtends across the comoving distance to the last-scattering surface.

\theta_H\simeq\frac{\eta_{\rm ls}}{\eta_0-\eta_{\rm ls}}\approx\frac{\eta_{\rm ls}}{\eta_0} \qquad (5)

where: $\theta_H$ — angle subtended by the causal horizon at last scattering; $\eta_0$ — comoving distance to the last-scattering surface (the present conformal time), $\approx14.3\,\mathrm{Gpc}$ . Reading: a $\sim0.28\,\mathrm{Gpc}$ horizon over a $\sim14\,\mathrm{Gpc}$ distance is a few hundredths of a radian, roughly $1^\circ$ . A matter-domination shortcut ( $\eta\propto a^{1/2}$ ) gives the same order of magnitude from the redshift.

\theta_H\;\sim\;\sqrt{\frac{a_{\rm ls}}{a_0}}\;=\;(1+z_{\rm ls})^{-1/2}\;\approx\;1.7^\circ \qquad (6)

where: $z_{\rm ls}\approx1090$ . Reading: $(1091)^{-1/2}\approx0.030\,\mathrm{rad}\approx1.7^\circ$ ; exact geometry and radiation pull this toward $1.2^\circ$ , so the answer is "one to two degrees". Tiling the sphere with patches of this radius counts the causally independent regions.

N_{\rm patches}\sim\frac{4\pi}{\pi\,\theta_H^{2}}=\frac{4}{\theta_H^{2}}\gtrsim10^{4} \qquad (7)

where: $N_{\rm patches}$ — number of never-in-contact regions tiling the CMB; $\theta_H\approx0.02$ - $0.03\,\mathrm{rad}$ . Reading: $4/\theta_H^2$ lands near ten thousand — the puzzle's headline figure. Finally, the cure is quantified by the e-folds — the logarithmic expansion so the observable universe fits inside one pre-inflationary Hubble patch.

N_e=\ln\frac{a_{\rm end}}{a_{\rm start}}\gtrsim 60 \qquad (8)

where: $N_e$ — e-folds during inflation; $a_{\rm end},a_{\rm start}$ — scale factors at end and start. Reading: for GUT-scale inflation the stretch is $e^{60}\approx10^{26}$ in linear size.

Derivation AThe comoving horizon is conformal time, and it stays small while gravity attracts

The goal is to establish the two facts on which the whole puzzle rests: that the greatest comoving distance any causal signal can have travelled since the Big Bang is exactly the conformal time $\eta$ , and that in a decelerating universe this $\eta$ is dominated by late times, so it was arbitrarily small early on. We integrate a radial null ray in the FLRW metric, then read off how $\eta$ scales with the equation of state.

Step 1. Take the flat FLRW line element $\mathrm{d}s^2=-c^2\mathrm{d}t^2+a^2(t)\,\mathrm{d}\chi^2$ for a radial ray and impose $\mathrm{d}s^2=0$ , giving $c\,\mathrm{d}t=a\,\mathrm{d}\chi$ . Why: a null geodesic is the fastest any influence can propagate, so its comoving reach bounds every causal process.

Step 2. Integrate from the Big Bang to time $t$ : $\chi_{\rm hor}=\int_0^{t}c\,\mathrm{d}t'/a(t')\equiv\eta$ — the maximum comoving distance a signal could have travelled is exactly the conformal time. Why: this is the precise meaning of "particle horizon" — beyond $\eta$ lies matter we could never have influenced.

Step 3. Change variables with $\mathrm{d}t=\mathrm{d}a/(aH)$ to get $\eta=\int_0^a c\,\mathrm{d}a'/(a'^2H)$ . For $a\propto t^{n}$ , $H\propto a^{-1/n}$ , so the integrand goes as $a'^{(1/n)-2}$ and converges at the lower limit precisely when $n<1$ . Why: $n<1$ is deceleration; the integral is then finite and dominated by its upper limit, which is why the horizon was tiny early and grows with $a$ .

Step 4. The two decelerating cases give $\eta\propto a$ (radiation) and $\eta\propto a^{1/2}$ (matter), both increasing; $(aH)^{-1}\propto a^{1-1/n}$ tracks them up to an order-unity factor. Why: this lets us use the Hubble radius as a proxy for the causal horizon in the standard eras — and warns that the two diverge the moment expansion accelerates. *Verified: $\eta=\int_0^a c\,\mathrm{d}a'/(a'^2H)$ ; $\eta\propto a$ (radiation), $\eta\propto a^{1/2}$ (matter); integral finite iff $n<1$ . Matches Weinberg Cosmology ch. 1 and Dodelson ch. 2.*

Go deeper · Derivation B · 1 minThe two-degree causal patch on the microwave sky

Derivation BThe two-degree causal patch on the microwave sky

The goal is to turn the abstract horizon into the number an observer measures: the angular size of the largest causally connected region on the CMB and the count of independent patches tiling the sky. We compute the comoving horizon at last scattering from the matter-plus-radiation Friedmann equation, the distance to that surface from the full $\Lambda$ CDM expansion, take their ratio for the angle, and divide the sphere. Use $a_0=1$ and Planck-era parameters.

Step 1. Write the pre-recombination Hubble rate by factoring out matter, $H(a)\approx H_0\sqrt{\Omega_{m,0}}\,a^{-3/2}\sqrt{1+a_{\rm eq}/a}$ , with $a_{\rm eq}=\Omega_{r,0}/\Omega_{m,0}\approx2.9\times10^{-4}$ , dropping $\Omega_\Lambda$ since $a_{\rm ls}\ll1$ . Why: at last scattering dark energy is $\sim10^{-9}$ of the density, so only matter and radiation set the expansion.

Step 2. Insert this into $\eta_{\rm ls}=\int_0^{a_{\rm ls}}c\,\mathrm{d}a/(a^2H)$ to get $\eta_{\rm ls}=\dfrac{2c}{H_0\sqrt{\Omega_{m,0}}}\left(\sqrt{a_{\rm ls}+a_{\rm eq}}-\sqrt{a_{\rm eq}}\right)\approx284\,\mathrm{Mpc}$ . Why: this is the entire causal reach at recombination — the radius of the largest region that could have thermalized itself.

Step 3. Compute the distance to last scattering from the full expansion, $\eta_0-\eta_{\rm ls}=\int_{a_{\rm ls}}^{1}c\,\mathrm{d}a/(a^2H)$ with $H^2=H_0^2[\Omega_{r,0}a^{-4}+\Omega_{m,0}a^{-3}+\Omega_\Lambda]$ : $\approx14\,300\,\mathrm{Mpc}$ . Why: this is the lever arm onto which we project the tiny horizon, fifty times larger.

Step 4. Form the angle $\theta_H\approx\eta_{\rm ls}/\eta_0\approx284/14\,300\approx0.020\,\mathrm{rad}\approx1.2^\circ$ ; the shortcut $(1+z_{\rm ls})^{-1/2}\approx1.7^\circ$ brackets it from above, so the patch is $1$ - $2^\circ$ across, and $N_{\rm patches}\sim4/\theta_H^2\approx10^{4}$ . Why: a full Moon is half a degree, so the largest causally connected patch is a few Moon-widths — yet $\sim10^4$ such islands, never in contact, all radiate at $2.725\,\mathrm{K}$ to a part in $10^5$ . *Verified: $\eta_{\rm ls}=284\,\mathrm{Mpc}$ , $\eta_0=14\,300\,\mathrm{Mpc}$ , $\theta_H=1.2^\circ$ (ratio) to $1.7^\circ$ (scaling), $N_{\rm patches}\sim10^4$ , using $\Omega_m=0.31$ , $H_0=67$ , $z_{\rm ls}=1090$ , $z_{\rm eq}=3400$ . Reproduces Ryden ch. 10 and Dodelson ch. 6.*

Go deeper · Derivation C · 1 minHow much inflation cures the horizon: sixty e-folds

Derivation CHow much inflation cures the horizon: sixty e-folds

The goal is to derive the minimum inflationary expansion needed so that the entire observable universe descends from a single causally connected patch. The condition is that the comoving Hubble radius at the start of inflation was at least as large as today's, so everything now inside our horizon once sat inside one thermalized Hubble patch. We impose that inequality, use the near-constancy of $H$ during inflation to turn it into a ratio of scale factors, and connect the end of inflation to the present through reheating.

Step 1. Require $(a_iH_i)^{-1}\ge(a_0H_0)^{-1}$ : the largest scale we observe must have been inside the horizon before inflation began. Why: only then could ordinary causal physics have smoothed the region that later becomes our whole sky — the horizon problem is solved by construction.

Step 2. During inflation $H\approx$ const (near-de Sitter), so the comoving Hubble radius shrinks purely through the growth of $a$ : $\dfrac{a_iH_i}{a_{\rm end}H_{\rm end}}\approx e^{-N_e}$ , with $N_e=\ln(a_{\rm end}/a_i)$ . Why: this defines the e-fold number as the logarithmic measure of how much the accelerating phase contracts every scale relative to the horizon.

Step 3. Connect the end of inflation to today: after reheating, radiation then matter domination regrow $(aH)^{-1}$ by $a_{\rm end}H_{\rm end}/(a_0H_0)\sim T_0/T_{\rm reh}$ (up to $g_*$ factors), since $aH\propto T$ in radiation domination. Why: the horizon we must overshoot is today's, not the tiny one at reheating, so we undo all the post-inflationary growth of $(aH)^{-1}$ .

Step 4. Collecting factors, $N_e\gtrsim\ln(T_{\rm reh}/T_0)$ . For GUT-scale inflation, $T_{\rm reh}\sim10^{16}\,\mathrm{GeV}$ and $T_0\approx2.35\times10^{-13}\,\mathrm{GeV}$ , so $N_e\gtrsim67$ , conventionally $N_e\gtrsim50$ - $60$ . Why: a few dozen e-folds inflate one thermalized sub-horizon patch into everything we will ever see; the exact number slides with the energy scale. *Verified: $N_e\gtrsim\ln(T_{\rm reh}/T_0)\approx60$ - $67$ for GUT-scale reheating; lower-scale inflation needs fewer. Matches Guth 1981 and Baumann's lectures.*

Limiting cases

Everything hinges on the equation of state $w=p/\rho$ through the sign of $\rho+3p\propto(1+3w)$ , which fixes whether the comoving Hubble radius grows or shrinks.

Case	Result	Meaning
Radiation domination ( $w=\tfrac13$ )	$a\propto t^{1/2},\ (aH)^{-1}\propto t^{1/2}\propto a$	$\rho+3p>0$ : decelerating. Horizon grows; the problem is present through most of the pre-recombination era.
Matter domination ( $w=0$ )	$a\propto t^{2/3},\ (aH)^{-1}\propto t^{1/3}\propto a^{1/2}$	$\rho+3p>0$ : still decelerating. Horizon keeps growing; recombination itself happens here, so the problem persists to last scattering.
de Sitter / inflation ( $w=-1$ )	$a\propto e^{Ht},\ (aH)^{-1}=\dfrac{c\,e^{-Ht}}{H}\;\downarrow$	$\rho+3p<0$ : accelerating. Comoving Hubble radius shrinks exponentially; one causal patch grows to cover the sky — problem solved.
General power law ( $a\propto t^{n}$ )	$(aH)^{-1}\propto t^{1-n}$ , shrinks iff $n>1\Leftrightarrow w<-\tfrac13$	Accelerated expansion ( $\ddot a>0$ ) is the exact and only criterion for the comoving Hubble radius to decrease and cure the horizon problem.

Common trapThe horizon problem is about the particle horizon, not the Hubble radius at one instant, and not the event horizon.

Three distinct lengths get muddled here. The particle horizon is how far a signal could have travelled since $t=0$ — an integral over all past history. The Hubble radius $(aH)^{-1}$ is a one-instant length. The event horizon is how far a signal can ever reach in the future. In matter or radiation domination the particle horizon and Hubble radius coincide up to an order-unity factor, so people use them interchangeably — but in an accelerating universe they diverge: the particle horizon can be enormous while the Hubble radius is small. The horizon problem is about the particle horizon at last scattering being small. Two further traps: the CMB dipole ( $\sim10^{-3}$ ) is kinematic and must be removed before quoting the $\sim10^{-5}$ intrinsic anisotropy; and the " $1$ - $2^\circ$ " figure is the particle horizon, a few times larger than the sound horizon ( $\sim1^\circ$ , the acoustic-peak scale) because sound travels slower than light.

Worked examples

1 · The two-degree patch, worked end to end (fully numeric)

Using $H_0=67\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ , $\Omega_{m,0}=0.31$ , $\Omega_{r,0}=9.1\times10^{-5}$ , $z_{\rm ls}=1090$ , compute the comoving horizon at last scattering, the distance to it, the angle $\theta_H$ , and the number of causal patches.

Step 1. Fix the anchors: $c/H_0=299\,792/67=\mathbf{4474\ Mpc}$ ; equality at $a_{\rm eq}=\Omega_{r,0}/\Omega_{m,0}=\mathbf{2.94\times10^{-4}}$ ( $z_{\rm eq}\approx3400$ ); last scattering at $a_{\rm ls}=1/1091=\mathbf{9.17\times10^{-4}}$ .

Step 2. Horizon at last scattering from equation (4): prefactor $2c/(H_0\sqrt{\Omega_{m,0}})=2\times4474/\sqrt{0.31}=16\,073\,\mathrm{Mpc}$ ; bracket $\sqrt{1.211\times10^{-3}}-\sqrt{2.94\times10^{-4}}=0.03480-0.01715=0.01765$ . So $\eta_{\rm ls}=16\,073\times0.01765=\mathbf{284\ Mpc}$ .

Step 3. Integrating the full $\Lambda$ CDM expansion from $a_{\rm ls}$ to $1$ gives the distance to last scattering $\eta_0-\eta_{\rm ls}\approx\mathbf{14\,300\ Mpc}$ (dominated by the recent matter and dark-energy eras).

Step 4. The angle: $\theta_H\approx\eta_{\rm ls}/\eta_0=284/14\,300=0.0199\,\mathrm{rad}=\mathbf{1.14^\circ}$ . The shortcut $(1091)^{-1/2}=0.0303\,\mathrm{rad}=1.73^\circ$ brackets it from above, so $\theta_H\approx1$ - $2^\circ$ .

Step 5. Count patches: $N_{\rm patches}\sim4/\theta_H^2=4/(0.0199)^2=\mathbf{1.0\times10^{4}}$ . The CMB is tiled from about ten thousand patches of angular radius ≈ 1.2°, none of which had ever exchanged a signal, yet all share T = 2.725 K to one part in 10⁵.

2 · How many e-folds for GUT-scale inflation (fully numeric)

Inflation reheats near the grand-unification scale, $T_{\rm reh}\sim10^{16}\,\mathrm{GeV}$ . How many e-folds are needed so the observable universe descends from one causal patch, and what linear stretch is that?

Step 1. The requirement (equation 8 and Derivation 3) is $N_e\gtrsim\ln(T_{\rm reh}/T_0)$ . Convert today's temperature: $T_0=2.725\,\mathrm{K}\times(8.617\times10^{-5}\,\mathrm{eV/K})=\mathbf{2.35\times10^{-13}\ GeV}$ .

Step 2. Take the logarithm: $N_e\gtrsim\ln\!\left(\dfrac{10^{16}}{2.35\times10^{-13}}\right)=\ln(4.3\times10^{28})=\mathbf{66}$ , i.e. the familiar $N_e\gtrsim60$ .

Step 3. The linear stretch is $e^{N_e}=e^{60}=\mathbf{1.1\times10^{26}}$ . A proton-sized patch ( $\sim10^{-15}\,\mathrm{m}$ ) before inflation is blown up to $\sim10^{11}\,\mathrm{m}$ — the Earth-Sun distance — during inflation alone, then expands by the usual hot-Big-Bang factor to today's Hubble volume. About 60 e-folds, a linear blow-up of ~10²⁶, converts one thermalized sub-atomic patch into the entire observable universe.

Physics of the portrait

The horizon problem is a puzzle three generations of relativists circled before anyone had a cure — named and answered in the same 1981 paper that launched inflationary cosmology.

Origins · 1 minAlan Guth and how this was discovered

Alan Guth

born 1947 · MIT · inflationary cosmology and the naturalness puzzles of the hot Big Bang

The causal structure of expanding universes was first laid out by Wolfgang Rindler in 1956, whose paper "Visual horizons in world-models" cleanly separated the particle horizon (how far we can see) from the event horizon (how far we can ever influence) — the very distinction the horizon problem turns on. Through the 1960s the smoothness on super-horizon scales was a known embarrassment; Charles Misner tried to tame it dynamically in 1969 with his "Mixmaster" universe, hoping chaotic early oscillations would mix distant regions, but the particle horizon in a decelerating universe is simply too small. The decisive turn came from particle physics. Alan Guth, then a postdoc working on grand unified theories and the overproduction of magnetic monopoles, realized in late 1979 that a supercooled false-vacuum phase would drive exponential, accelerated expansion — and that the same de Sitter stretch which diluted the monopoles would solve the flatness and horizon problems at a stroke. His 1981 paper named the horizon problem in its title and gave it a mechanism: a region far smaller than the pre-inflationary Hubble radius, already thermalized, blown up to encompass the whole observable universe. Guth's "old inflation" could not gracefully end, but within a year Andrei Linde, and independently Andreas Albrecht with Paul Steinhardt, replaced it with slow-roll "new inflation", in which a scalar field rolls smoothly down a flat potential. That framework, tested against the CMB by COBE, WMAP, and Planck, remains the standard resolution: the smoothness Rindler could only describe, and Misner could not engineer, is now the fossil of an accelerating epoch before the hot Big Bang.

History

1956

— Rindler defines the particle and event horizons of expanding world-models

1969

— Misner proposes the Mixmaster universe, an attempt to erase the horizon problem dynamically

1981

— Guth introduces inflation and names the horizon and flatness problems together

1982

— Linde, and Albrecht and Steinhardt, devise slow-roll "new" inflation

2018

— Planck's final maps confirm the near-scale-invariant, super-horizon fluctuations inflation predicts

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., ch. 10 — horizon setup and horizon-distance calculation · Dodelson & Schmidt, Modern Cosmology, 2nd ed., ch. 6 — comoving horizon and the case for inflation · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 29-30 — the cosmological puzzles · Mo, van den Bosch & White, Galaxy Formation and Evolution, ch. 3 — horizons in FLRW · Kolb & Turner, The Early Universe, ch. 8 — the quantitative horizon problem · Weinberg, Cosmology, ch. 1 and 4 · Baumann, Cosmology (2022), ch. 4 — modern inflationary treatment.

Provenance. Horizon-distance and angular estimates follow Ryden ch. 10 and Dodelson ch. 6. A verifier should recompute: (1) $\eta_{\rm ls}=\frac{2c}{H_0\sqrt{\Omega_m}}(\sqrt{a_{\rm ls}+a_{\rm eq}}-\sqrt{a_{\rm eq}})=284\,\mathrm{Mpc}$ with $\Omega_m=0.31$ , $H_0=67$ , $z_{\rm ls}=1090$ , $z_{\rm eq}=3400$ ; (2) $\eta_0\approx14\,300\,\mathrm{Mpc}$ from the full $\Lambda$ CDM integral; (3) $\theta_H\approx1.1^\circ$ - $1.7^\circ$ , $N_{\rm patches}\sim10^4$ ; (4) the growth criterion $\mathrm{d}(aH)^{-1}/\mathrm{d}t=-c\,\ddot a/\dot a^2$ ; (5) $N_e\gtrsim\ln(T_{\rm reh}/T_0)\approx60$ - $67$ for GUT-scale reheating. The scalings ( $a\propto t^{1/2},t^{2/3},e^{Ht}$ ) are standard FLRW results (Weinberg ch. 1). All figures use Planck-era parameters; none are fabricated.

Prerequisites. Thermal History of the Universe · Recombination and the Saha Equation · Equilibrium Thermodynamics of the Early Universe. Leads to. Inflation · CMB Anisotropies and the Power Spectrum · The Flatness Problem.

Ask the tutor: Ask me to draw your past light cone on the CMB and show why two patches a few degrees apart could never have swapped a photon — and how sixty e-folds of inflation rewrite that diagram.

Is the horizon problem a failure of general relativity or of the Big Bang?

Neither theory is "wrong". General relativity and the hot Big Bang describe the observed expansion, the light-element abundances, and the CMB blackbody spectrum superbly. The horizon problem is a fine-tuning of initial conditions, not a contradiction: standard cosmology must assume the primordial data were smooth on scales never in causal contact, with no dynamical reason. It is the absence of an explanation that is unsatisfying, not any error — which is why the fix is extra early-universe physics (inflation, or a contracting alternative) rather than a change to gravity or the Big Bang.

How can CMB fluctuations be correlated on scales larger than the horizon?

This is the sharpest form of the puzzle. The CMB is not just uniform; its tiny fluctuations are correlated across separations larger than the $1$ - $2^\circ$ causal horizon, visible as the Sachs-Wolfe plateau at low multipoles. No causal process after $t=0$ in a decelerating universe can imprint super-horizon correlations. Inflation supplies them naturally: quantum fluctuations of the inflaton are generated on sub-horizon scales, stretched outside the shrinking Hubble radius where their amplitude freezes, and re-enter much later as coherent classical perturbations. Observing them is a direct vindication of the inflationary resolution.

Does dark energy solve the horizon problem, since it also accelerates the expansion?

No — it accelerates far too late and by far too little. Dark energy makes $(aH)^{-1}$ begin to shrink again only in the last few billion years, long after last scattering, by which time the horizon problem was already baked into the CMB. To cure the puzzle the accelerated phase must occur before the hot Big Bang and provide $\sim60$ e-folds of shrinking Hubble radius, whereas the dark-energy era has so far delivered less than one. Dark energy and inflation share $w\approx-1$ but act at opposite ends of cosmic history — only the early one can homogenize the CMB.

What is the difference between the particle horizon and the sound horizon here?

Both are causal scales at last scattering, but travel at different speeds. The particle horizon is set by light and gives the $\sim1$ - $2^\circ$ figure for causal contact. The sound horizon is how far a pressure wave in the photon-baryon plasma travels, at $c_s\approx c/\sqrt3$ , so it is a few times smaller — about $1^\circ$ — and sets the acoustic-peak scale. "The horizon at last scattering is a degree" sometimes means the sound horizon (the acoustic ruler) and sometimes the particle horizon (causal contact); the horizon problem is about the latter, the larger of the two.

Do inflation's alternatives — bouncing or ekpyrotic models — really also solve it?

Yes, at the level of the horizon problem alone. Any mechanism that shrinks the comoving Hubble radius, or gives light a large conformal-time head start, restores causal contact. Ekpyrotic and cyclic models do it with a slow contraction under an ultra-stiff equation of state ( $w\gg1$ ), so the horizon problem does not single out inflation. What distinguishes them is the fluctuation spectrum: inflation predicts a near-scale-invariant tilt and an observable tensor background, contracting models different non-Gaussianity and weaker tensors. With the measured $n_s$ , adiabaticity, and Gaussianity the data favour inflation — but the horizon problem is the shared entry ticket, not the deciding vote.

Why does the comoving Hubble radius, not the physical one, tell the story?

Because comoving coordinates factor out the overall expansion and let you compare scales at different epochs on one ruler. The physical Hubble radius $c/H$ almost always grows, hiding the competition. The comoving Hubble radius $(aH)^{-1}$ instead tracks whether a fixed comoving scale is inside or outside causal reach: when it grows, scales that were outside come inside (the standard eras, creating the puzzle); when it shrinks, inside scales go outside (inflation, curing it). Plotting $(aH)^{-1}$ against $a$ , as in the second figure, is the cleanest way to watch a scale exit the horizon during inflation and re-enter billions of years later.