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The relevant comparison is continuing hierarchical galaxy formation in the Lambda-CDM model against an idealized monolithic galaxy formation model in which both formation and assembly were essentially complete early. Each observation below concerns a population or a fossil record, rather than a currently merging pair. None individually proves that every galaxy formed through the same route.

Fossil substructure in stellar halos. Old stars in the Milky Way's stellar halo include coherent groups in orbital-integral space, and external halos contain extended debris and shells. Phase mixing can erase spatial concentration while retaining correlations of orbital energy or angular momentum. Such stellar streams, when linked to a disrupted dwarf progenitor, record accretion of an already formed stellar population. A smooth single collapse followed by quiet passive evolution has no equally direct mechanism for creating several dynamically and chemically distinct accreted populations. This supports a composite assembly history, without requiring that every halo star was accreted or counting a current merging pair as evidence.

Growth of the quiescent population. The galaxy red sequence contains old stellar populations, yet its inferred stellar mass density grows between approximately redshift one and the present. Pure fading of an already fixed early population changes luminosities and colours, but cannot increase the mass in that population. Quenching adds newly passive systems, while subsequent assembly combines older stellar populations into larger galaxies. The observed evolution therefore requires demographic evolution after the putative completed collapse epoch. It is consistent with hierarchical galaxy formation plus quenching, although red-sequence growth alone does not distinguish assembly from all possible histories of in-situ star formation; stellar-mass estimates and survey selection must be treated consistently.

Evolution of galaxy morphology. Distant galaxy samples contain a larger prevalence of irregular and peculiar structures than a simple non-evolving extrapolation of the local population. Settled discs and smooth spheroids become common outcomes as time proceeds. Continued gas supply, dynamical interactions, and internal rearrangement in hierarchical galaxy formation naturally allow this galaxy morphological evolution. A model in which structure became fixed during one early collapse and then only stellar light faded has difficulty producing the population change. Rest-frame wavelength, resolution, and surface-brightness selection can affect the observed classifications, and clumpiness is not unique evidence for a galaxy merger: unstable gas-rich discs can also be irregular.

Evolution of sizes at fixed stellar mass. Massive high-redshift galaxies, including concentrated systems, are often more compact than comparable populations at low redshift. Galaxy size evolution implies structural growth beyond passive luminosity fading. In particular, deposition of accreted stars at large radii in minor galaxy mergers can enlarge the outer stellar body while preserving an old central stellar population; gas accretion can grow discs from the inside outward. A completed early collapse with a fixed stellar structure does not naturally supply those new outskirts. Changes in sample membership and expansion associated with stellar mass loss also contribute, so this is evidence requiring continued structural evolution rather than a unique quantitative measurement of a merger rate.

Diffuse intracluster stars. Intracluster light and extended outer envelopes show that some stars no longer belong to individual cluster galaxies. Their irregular distribution and substructure retain evidence of redistribution within an assembling cluster. Tidal stripping and disruption of member systems supply stars to the common environment, as expected when smaller bound systems are incorporated into a growing hierarchy. Passive evolution of independent galaxies formed in isolated early collapses cannot itself move their stars into the space between galaxies. Early stellar formation followed by later stripping remains possible, but admitting that later dynamical assembly already departs from the strict completed-collapse picture. This evidence concerns the cumulative stellar distribution, not merely the existence of an occasional present-day merging pair.

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 60 / 4 / ii / Solution Created 2026-10-03 Updated 2026-10-14

Low-mass main sequence stars live for much of the Milky Way's age, retaining representatives of many generations, while phase mixing disperses their birth associations around their orbits. In an approximately axisymmetric galactic disk, a sufficiently complete local census can therefore sample the mixed annular stellar population at the Sun's radius despite its small volume. This is a representative local-radius sample, not automatically an unbiased whole-disc census: different vertical scale heights, selection effects and radial migration require corrections when inferring integrated population fractions.

Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 56 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

In the Lambda-CDM model, nearly Gaussian primordial density contrasts grow under gravity within an expanding universe containing cold dark matter, ordinary baryons and a cosmological constant. The cold dark matter is effectively collisionless and has negligible pressure on galactic scales. Before recombination, baryons are coupled to the photon fluid: radiation pressure and acoustic oscillations prevent their perturbations from behaving like pressureless matter. After recombination they can fall into the gravitational potentials already established by dark matter, subject to gas pressure and the Jeans mass.

For small density contrasts, evolution is linear. On pressure-free scales the growing mode is multiplied by the linear growth factor

D (t)

, with

D \propto a

during matter domination. The cosmological density power spectrum can be written

P (k, t) = D^{2} (t) P_{prim} (k) T^{2} (k) .

Here

T (k)

is the cosmological transfer function. For nearly scale-invariant initial conditions, the large-scale matter spectrum behaves approximately as

P \propto k

, whereas well inside the matter-radiation equality scale it falls approximately as

k^{- 3} lo g^{2} k

, until the microscopic dark-matter cutoff matters. This fall of the dimensional

P (k)

does not imply less fluctuation power on every smaller mass scale: the power per logarithmic wavenumber is

Δ^{2} = k^{3} P / (2 π^{2})

, and the smoothed matter density variance is obtained by integrating it against a mass-dependent window. Over the relevant cold-dark-matter hierarchy, smaller mass windows generally have larger variance.

The hierarchical galaxy formation picture follows: fluctuations on small mass scales typically reach the nonlinear collapse threshold first, while larger objects assemble later through accretion and dark-matter halo mergers. It is a statistical ordering, not a claim that every small object precedes every large rare peak. When

∣ δ ∣

becomes order unity, the linear growth factor is no longer a solution for the local density. Collisionless dark matter develops multistream regions and bound dark matter halos; phase mixing and violent relaxation redistribute orbital energies, and virialized structures approximately obey the virial theorem.

Baryons have an additional nonlinear route. Infall and shocks convert bulk kinetic energy into thermal energy, with characteristic virial temperature

k_{B} T_{vir} \sim \frac{μ m _{p} G M _{h}}{2 R _{h}} .

Unlike collisionless dark matter, the gas can lose this energy through radiative cooling. The optically thin gas cooling time is the thermal-energy density divided by the radiative loss rate, for example

t_{cool} = \frac{3 n _{tot} k _{B} T /2}{n _{e} n _{i} Λ ( T )}, t_{dyn} \sim (G ρ_{h})^{- 1/2} .

The density convention in

Λ

must agree with the denominator; the astrophysical cooling function can also be defined using

n_{H}^{2}

instead. The cooling criterion for galaxy formation compares this time with the collapse or supply time. Rapidly cooling gas loses pressure support, contracts and can form stars. Slowly cooling gas remains in a hot atmosphere. Stable virial shocks are not obligatory in every low-mass system: gas can also arrive in cold streams and cool while being accreted.

Angular momentum prevents indefinite radial contraction. Tidal torque theory supplies an initial halo spin, and later mergers change it. If gas radiates energy while retaining much of its specific angular momentum, it settles into a rotationally supported galactic disk rather than reaching the centre. The relation

j \sim R v_{c}

explains why modest halo spin can set a disk radius much smaller than its virial radius of a dark-matter halo. Torques, bars and gravitational encounters can transport angular momentum outwards and feed central concentrations; radiative cooling alone does not remove it.

Galaxy mergers alter stellar structure as well as assembling mass. A major galaxy merger can strongly disturb or destroy an existing galactic disk, randomizing stellar orbits and creating a spheroid through violent relaxation. A gas-rich galaxy merger also permits dissipation, inflow and a burst of star formation; gas left over or accreted afterwards can rebuild a galactic disk. Minor galaxy mergers add stars to outer components, thicken disks and grow bulges. Dry galaxy mergers add stellar mass and can increase size without much new star formation. Halo merging therefore does not imply instantaneous merging of its galaxies: satellite orbital decay requires Chandrasekhar dynamical friction and can take a substantial time.

The atomic and molecular cooling thresholds for galaxy formation supply a lower characteristic scale. Primordial atomic gas cools inefficiently below roughly

1 0^{4} K

because electronic excitation is suppressed. The corresponding halo mass is of order

1 0^{8} M_{⊙}

at a redshift of order ten, with approximate dependence

M_{atomic} \propto (1 + z)^{- 3/2}

at fixed threshold temperature. Molecular hydrogen can cool gas at hundreds of kelvin and permit smaller early objects, provided it forms and survives dissociating radiation. Metal-line cooling changes these thresholds after enrichment. Thus the atomic threshold is not an absolute minimum mass for all stellar systems.

At the other end, sufficiently massive dark matter halos have high virial temperatures and low-density hot gas. Above the strong atomic-line-cooling interval, thermal bremsstrahlung has

Λ \propto T^{1/2}

. At comparable halo gas density,

t_{cool} \propto T^{1/2} \propto M_{h}^{1/3}

, while

t_{dyn}

depends mainly on formation density. Cooling therefore becomes less able to condense all the gas within the available time. This upper galaxy mass from gas cooling argument selects galaxy-sized condensations, broadly halo masses around

1 0^{12}

–

1 0^{13} M_{⊙}

in simple low-redshift estimates, rather than single luminous galaxies containing every baryon in a group or cluster. Its numerical scale depends on epoch, metallicity and gas profile. Subsequent galaxy mergers can build larger stellar systems; cooling is not an absolute upper bound on their final mass.

Finally, a Press-Schechter halo mass function has many low-mass objects and a steep high-mass cutoff, while the luminosity function of galaxies also has a faint component and a bright cutoff, often described by a Schechter function. The two shapes are related through the halo-to-galaxy luminosity mapping, not by identifying luminosity with total halo mass. In the idealized one-central-galaxy, no-scatter limit,

Φ (L) = \frac{d n _{h}}{d M _{h}} \frac{d M _{h}}{d L} .

(4)

d n_{h} / d M_{h} \propto M_{h}^{- a}

and

L \propto M_{h}^{p}

, then

Φ (L) \propto L^{- 1 - (a - 1) / p}

. A constant conversion gives similar shapes. Actual stellar feedback, inefficient low-temperature cooling and reionization suppress faint galaxies relative to small haloes, while long cooling times and active-galactic-nucleus feedback suppress luminosity at large halo masses. Satellites, scatter, stellar populations and dust further affect the correspondence. The halo hierarchy supplies the gravitational framework; cooling, angular momentum and feedback determine which parts become luminous galaxies.

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 63 / 1 / Solution Created 2026-10-03 Updated 2026-10-14

Take the sky frame and disk frame to be right-handed orthonormal bases, with positive angular displacement measured from

X

toward

Y

. In particular, positive

Z

is toward the observer, so material crossing the sky plane with positive

\dot{Z}

is at the ascending node. Write

c_{d} = cos Ω_{d}

s_{d} = sin Ω_{d}

C_{d} = cos I_{d}

, and

S_{d} = sin I_{d}

. The disk frame's first unit vector lies along its ascending node, and its second has positive

Z

component. The orbital-frame rotation from inclination and node therefore gives

T = R_{Z} (Ω_{d}) R_{X} (I_{d}) = c_{d} s_{d} 0 - s_{d} C_{d} c_{d} C_{d} S_{d} s_{d} S_{d} - c_{d} S_{d} C_{d} .

Each column is a disk-frame unit vector in sky coordinates, so

X = T x

and the inverse is

x = T^{T} X

. At the disk's ascending node, the

Z

component of its tangential velocity is positive, confirming the sign of the tilt.

Let

T_{p} = R_{Z} (Ω_{p}) R_{X} (I_{p})

define a provisional planet frame whose first axis is the planet's sky-plane ascending node. Its normal is

z_{p} = (sin Ω_{p} sin I_{p}, - cos Ω_{p} sin I_{p}, cos I_{p})

. Expressing that normal in the disk orthonormal basis, with

ΔΩ = Ω_{p} - Ω_{d}

, gives

T^{T} z_{p} = sin I_{p} sin ΔΩ - cos I_{d} sin I_{p} cos ΔΩ + sin I_{d} cos I_{p} cos I_{d} cos I_{p} + sin I_{d} sin I_{p} cos ΔΩ .

The last component is the dot product of the two orbital normals, hence the cosine of the mutual inclination. The mutual ascending node points along

z_{d} \times z_{p}

: motion there has positive disk-normal component. Set

A = cos I_{d} sin I_{p} cos ΔΩ - sin I_{d} cos I_{p}, B = sin I_{p} sin ΔΩ.

Then

x' = (A x + B y) / A^{2} + B^{2}

, and the quadrant-correct longitude of ascending node is

Ω_{m} = atan2 (B, A)

. Thus

cos I_{m} = cos I_{d} cos I_{p} + sin I_{d} sin I_{p} cos ΔΩ, tan Ω_{m} = \frac{sin I _{p} sin ΔΩ}{cos I _{d} sin I _{p} cos ΔΩ - sin I _{d} cos I _{p}} .

(4)

The tangent alone cannot fix the quadrant, and

Ω_{m}

is undefined for exactly coincident orbital planes.

For the two rotation routes, define

ψ

as the oriented angle, within the planet's orbital plane, from its sky-plane ascending node to its disk-plane ascending node. If

x_{p}, y_{p}

are the first two columns of

T_{p}

, take

ψ = atan2 (x' \cdot y_{p}, x' \cdot x_{p})

. Both routes describe the same planet orthonormal basis:

T R_{z} (Ω_{m}) R_{x} (I_{m}) = T_{p} R_{z} (ψ) .

(5)

The left route goes through the disk frame; the right goes through the sky-node planet frame. Comparing their third columns gives the two boxed relations, while comparison of first columns fixes the extra in-plane angle.

Figure 1.
Sky-node directions and the mutual ascending node of the disk and planet
. **Details**.

Put

I_{d} = π /2 + δ_{d}

and

I_{p} = π /2 + δ_{p}

. A second-order Taylor expansion gives

cos I_{m} = 1 - [(δ_{p} - δ_{d})^{2} + Δ Ω^{2}] /2 + O (δ^{4})

, and therefore the mutual inclination of nearly edge-on orbits satisfies

I_{m}^{2} = (I_{p} - I_{d})^{2} + (Ω_{p} - Ω_{d})^{2} + O (δ^{4}) .

(6)

All expansion angles are in radians. For plotting in degrees this is

I_{m} ≃ (I_{p} - 8 7^{\circ})^{2} + (4^{\circ})^{2}

. The given planet has exact

I_{m} ≃ 4.11 9^{\circ}

and

Ω_{m} ≃ 76.04 4^{\circ}

. The exact minimum occurs at

I_{p} = atan2 (sin I_{d} cos ΔΩ, cos I_{d}) ≃ 86.99 3^{\circ}

, with

I_{m, m i n} ≃ 3.99 5^{\circ}

. At

I_{p} = 8 5^{\circ}

and

9 5^{\circ}

, the second-order values are

20^{\circ} ≃ 4.47 2^{\circ}

and

80^{\circ} ≃ 8.94 4^{\circ}

Figure 2.
Mutual inclination versus the planet's observed inclination
. **Details**.

For the disk evolution, neglect disk self-gravity, collisions that align the planes, and back-reaction on the planet. In the initial disk plane, use the complex inclination

w = I e^{i Ω}

for a ring and

w_{p} = I_{m} e^{i Ω_{m}}

for the planet. Linear Laplace-Lagrange secular theory has forced inclination

w_{p}

and solution

w (a, t) = w_{p} [1 - e^{- i ν (a) t}]

for initially flat rings. For an inner circular perturber, the Laplace coefficient formula is

ν = n (M_{p} / M_{⋆}) α b_{3/2}^{(1)} (α) /4

, with

α = a_{p} / a

. Its quadrupole approximation uses

b_{3/2}^{(1)} ≃ 3 α

, yielding

ν (a) ≃ \frac{3}{4} n \frac{M _{p}}{M _{⋆}} (\frac{a _{p}}{a})^{2} \propto a^{- 7/2} .

(7)

In the

(I cos Ω, I sin Ω)

plot referenced to the initial disk, every ring runs clockwise on the same circle centered at

w_{p}

, starting at the origin. Inner rings advance more quickly. If the axes are instead referenced to the planet's plane, the relative tilt is

w - w_{p} = - w_{p} e^{- i ν t}

: these circles are centered at the origin, and the mutual inclination is constant. Both conventions describe the same nodal precession; they should not be mixed.

Figure 3.
Differential nodal precession in initial-disk and planet-plane inclination coordinates
. **Details**.

At an intermediate epoch, rings with

ν t ≳ 1

have tilted substantially while outer rings with

ν t ≪ 1

remain near the original disk plane. This radial variation is a planet-induced debris-disk warp. The characteristic affected radius grows like

t^{2/7}

. Conservative differential nodal precession preserves each ring's tilt relative to the planet, so it does not by itself align every orbit; an unresolved inner region can acquire a mean plane near the planetary plane through phase mixing. The secular interpretation is also described in Wyatt's planetary dynamics lectures.

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 316 / 1 / iv / Solution Created 2026-10-03 Updated 2026-10-06

The printed line-density statement is incorrect for a phase-mixed orbit. A steady line density on a Kepler orbit is inversely proportional to speed: for a cross-sectional-area current

J_{σ}

, the area per unit arc length is

d σ / d s = J_{σ} / v

. Equivalently, phase mixing gives

d σ = (σ_{tot} / P) d t

, uniform in mean anomaly, where

P = 2 π a^{3} / μ

is the orbital period.

For an optically thin population of blackbodies in radiative equilibrium, a fragment absorbs

σ L_{⋆} / (4 π r^{2})

and reradiates the same luminosity. Thus the fractional luminosity of a phase-mixed eccentric wire is

f = \frac{σ _{tot}}{4 π P} \int_{0}^{P} \frac{d t}{r ^{2}} .

The specific angular momentum gives

d t / r^{2} = d θ / h

, so the integral is

2 π / h

. Since

P h = 2 π a^{2} 1 - e^{2}

f = \frac{σ _{tot}}{4 π a ^{2} 1 - e ^{2}} .

This derives the intended result after explicitly correcting the density to

1/ v

The error is consequential. If one instead imposes the literal density

d σ / d s \propto v

, its time weighting is

v^{2} d t

. Using

⟨ v^{2} ⟩ = μ / a

⟨ r^{- 2} ⟩ = [a^{2} 1 - e^{2}]^{- 1}

and

⟨ r^{- 3} ⟩ = [a^{3} (1 - e^{2})^{3/2}]^{- 1}

gives

f_{λ_{ℓ} \propto v} = \frac{σ _{tot}}{4 π a ^{2}} \frac{1 + e ^{2}}{( 1 - e ^{2} ) ^{3/2}} .

e = 1/2

this is

5/3

times the printed result. The inverse-radius averages can also be obtained directly with eccentric anomaly

E

r = a (1 - e cos E)

and

d t = (1 - e cos E) d E / n

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 316 / 1 / ii / Solution Created 2026-10-03 Updated 2026-10-05

For an unperturbed Kepler orbit, successive revolutions advance the planet-relative orientation by

- n_{p} T_{c}

modulo

2 π

, where

T_{c}

is the comet's orbital period. An irrational period ratio makes these orientations dense; sufficiently distant mean-motion resonance and adequate observation time justify the corresponding phase mixing approximation.

At each radius between its pericentre distance and apocentre distance, the comet eventually visits every azimuth in the rotating reference frame. The spatial projection is therefore

q \leq r \leq Q, 0 \leq θ < 2 π .

It is an annulus, with a nonuniform radial residence probability, rather than a uniformly filled area. Strictly, the full position-velocity phase space is not this annulus: at each radius the energy and angular momentum constrain the velocity, with inward and outward branches. The annulus is its position-space projection. This description assumes the orbital elements have not yet been substantially changed by planetary scattering.

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 316 / 1 / v / Solution Created 2026-10-03 Updated 2026-10-05

A coplanar target sweeps a strip of width

2 R_{p}

through the comet's local position probability. Hence the relevant geometric collision cross-section is a length, and the rare collision rate per comet is

Γ_{col} = 2 R_{p} Σ_{c} (a_{p}) v_{rel}

Using the corrected phase-mixed comet surface density and the notation

s = q / a_{p}

γ = 3 - 2 2 s

, the coplanar comet collision time is

t_{col} ≃ \frac{π ^{2} Q ^{3/2} a _{p} 1 - s}{2 R _{p} γ G M _{⋆}} .

For

q ≪ a_{p}

, this becomes

π^{2} Q^{3/2} a_{p} / (23 R_{p} G M_{⋆})

, so

t_{col} \propto Q^{3/2} M_{p}^{- 1/3} a_{p}

at fixed stellar mass and planetary mass density. This is the mean waiting time for one orbiting comet; the mean interval between impacts from

N_{c}

such independent comets is

t_{col} / N_{c}

. It assumes rare encounters and orbital phase mixing, not a Poisson process derived from a single deterministic trajectory.

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 346 / 1 / a / Solution Created 2026-10-03 Updated 2026-10-05

Write

x

for a comoving coordinate,

u

for the peculiar velocity,

H = \overset{a}{˙} / a

for the Hubble parameter, and

δ = δ ρ / \overset{ρ}{ˉ}

for the density contrast. For a homogeneous self-gravitating fluid, the linearized continuity equation, Euler equations for an inviscid fluid, and cosmological Poisson equation are

\dot{δ} = - \frac{1}{a} \nabla \cdot u, \dot{u} + H u = - \frac{c _{s}^{2}}{a} \nabla δ - \frac{1}{a} \nabla ϕ, \nabla^{2} ϕ = 4 π G a^{2} \overset{ρ}{ˉ} δ .

Here

c_{s}

is the physical adiabatic sound speed, gradients refer to comoving coordinates, and the homogeneous background force has already been subtracted. Differentiating the first equation and eliminating

u

and

ϕ

gives

\ddot{δ} + 2 H \dot{δ} - \frac{c _{s}^{2}}{a ^{2}} \nabla^{2} δ - 4 π G \overset{ρ}{ˉ} δ = 0.

A Fourier mode with comoving wavenumber

k

therefore obeys

\ddot{δ}_{k} + 2 H \dot{δ}_{k} + (\frac{c _{s}^{2} k ^{2}}{a ^{2}} - 4 π G \overset{ρ}{ˉ}) δ_{k} = 0.

The Jeans wavenumber occurs where the coefficient in parentheses vanishes. Since a comoving wavelength is

2 π / k

, the result is

k_{J} = \frac{a 4 π G ρ ˉ}{c _{s}}, λ_{J}^{com} = \frac{c _{s}}{a} \frac{π}{G ρ ˉ} .

(4)

Below this comoving Jeans length, pressure restores a displaced fluid element: there are acoustic waves rather than growing Jeans instability. In a static background they oscillate; expansion changes their amplitude and frequency, and dissipation can damp them. Pressure support alone does not erase them. In the pre-recombination photon-baryon fluid, Silk damping supplies a separate erasure mechanism.

For collisionless matter, replace the sound speed by a characteristic velocity dispersion. For an isotropic Maxwell-Boltzmann velocity distribution, defining

σ^{2} = ⟨ u_{x}^{2} ⟩ = ⟨ ∣ u ∣^{2} ⟩ /3

, the static collisionless marginal-stability calculation gives

λ_{J, collisionless}^{com} = \frac{σ}{a} \frac{π}{G ρ ˉ} = \frac{v _{rms}}{a} \frac{π}{3 G ρ ˉ} .

(5)

The coefficient depends on the distribution and the convention for velocity dispersion. Small-scale suppression is now collisionless free streaming and phase mixing, rather than collisional acoustic waves. An instantaneous collisionless Jeans length must be distinguished from the accumulated comoving distance

\int v d t / a

traveled since decoupling.

For the requested thermal histories, an Einstein-de Sitter universe describes the matter-era limit: it cannot literally also have a radiation-dominated epoch. Interpret the question as a spatially flat universe with negligible cosmological constant, passing from radiation domination,

a \propto t^{1/2}

, to matter domination,

a \propto t^{2/3}

. For the usual schematic dynamical Jeans scale use

\overset{ρ}{ˉ}_{tot}

in the estimate, so

λ_{dyn}^{com} \propto {c_{s} a, c_{s} a^{1/2}, t ≪ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 4}, t ≫ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 3} .

(6)

Near the particle horizon a relativistic perturbation calculation replaces the Newtonian derivation; the scaling estimate still identifies the relevant sound-crossing or streaming scale.

Before cosmological recombination, adiabatic initial conditions keep the entropy per baryon fixed in the tightly coupled photon-baryon fluid. With

R_{b} = 3 \overset{ρ}{ˉ}_{b} / (4 \overset{ρ}{ˉ}_{γ}) \propto a

, its photon-baryon sound speed is

c_{s}^{2} = \frac{c ^{2}}{3 ( 1 + R _{b} )} .

(7)

For example, this follows from

δ ρ_{γ} / \overset{ρ}{ˉ}_{γ} = (4/3) δ ρ_{b} / \overset{ρ}{ˉ}_{b}

and

δ P = c^{2} δ ρ_{γ} /3

. In the radiation-dominated limit

R_{b} ≪ 1

c_{s} ≃ c / 3

; in the strongly baryon-loaded matter-era limit

R_{b} ≫ 1

c_{s} \propto a^{- 1/2}

. Thus the baryon Jeans length across recombination has the asymptotic history

λ_{J, b}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, approximately constant, a^{- 1/2} \propto t^{- 1/3}, t ≪ t_{eq}, t_{eq} ≪ t < t_{rec}, t > t_{rec} with adiabatic gas cooling .

(8)

The transition to the plateau is smooth: retaining

1 + R_{b}

gives

λ_{J, b}^{com} \propto a^{1/2} / 1 + R_{b}

in matter domination. If baryon loading is still small, this intermediate segment instead rises as

a^{1/2}

until loading becomes important.

At photon decoupling, radiation ceases to provide pressure support to the baryons, so the sound speed drops from the coupled-fluid value to that of a nonrelativistic ideal gas,

c_{s, b}^{2} = γ k_{B} T_{b} / (μ m_{p})

, with

γ = 5/3

. The drop in comoving Jeans length at fixed density is the same ratio of sound speeds, typically several orders of magnitude. Subsequently an isentropic process gives

T_{b} \propto \overset{ρ}{ˉ}_{b}^{γ - 1} \propto a^{- 2}

, so

c_{s, b} \propto a^{- 1}

and

λ_{J, b}^{com} \propto a^{- 1/2}

. Residual energy exchange through Compton scattering can initially keep

T_{b} \propto a^{- 1}

, producing a short approximately flat gas segment before the decreasing segment. The idealized sketch assumes immediate adiabatic cooling:

log(comoving baryon Jeans scale)
  ^
  |                   __________________
  |                 /                   |
  |               /                     |  loss of photon pressure
  |             /                       |
  |           /                         +---\
  |         /                                \
  +-------------------|-----------------|----------> log(time)
                    t_eq              t_rec
        slope 1/2          slope ~0        slope -1/3

For the cold-dark-matter kinetic Jeans scale, take the stated toy history to mean that collisions maintain a particle temperature equal to the photon temperature until kinetic decoupling. This is an assumption about thermal coupling; a negligible collision rate after decoupling makes the subsequent treatment collisionless. Assume the ordering

t_{NR} < t_{dec} < t_{eq} < t_{rec}

. While relativistic, the characteristic particle speed is constant; while nonrelativistic but thermally coupled,

σ \propto T \propto a^{- 1/2}

; after kinetic decoupling, particle momentum redshifts as

a^{- 1}

and

σ \propto a^{- 1}

. Combining these with the background dynamical density gives

λ_{dyn, CDM}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, a^{1/2} \propto t^{1/4}, constant, a^{- 1/2} \propto t^{- 1/3}, t < t_{NR}, t_{NR} < t < t_{dec}, t_{dec} < t < t_{eq}, t > t_{eq} .

(9)

There is no sudden change at cosmological recombination: cold dark matter has already decoupled. Its small initial velocity dispersion makes its suppression scale much smaller than the pre-recombination baryon scale.

log(comoving CDM dynamical Jeans scale)
  ^
  |                        _______________
  |                    ___/               \
  |                ___/                    \       no jump
  |              /                          \      at t_rec
  |            /                             \
  +-----------|------------|--------------|----|------> log(time)
            t_NR         t_dec          t_eq t_rec
   slope 1/2    slope 1/4       slope 0       slope -1/3

The density convention matters. If one instead defines a species' self-gravitating collisionless Jeans length using only its own nonrelativistic density

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

, its thermally coupled segment is constant and its decoupled segment decreases as

a^{- 1/2}

even during radiation domination. The constant radiation-era segment in the second sketch is the usual instantaneous background dynamical streaming estimate, not an exact single-species self-gravity threshold. A complete multicomponent calculation uses the separate perturbation equations and their combined cosmological Poisson equation.

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 2 / vi / Solution 2026-09-25

The impact gives fragments a spread of orbital energy and specific angular momentum. Their resulting spread of mean motion lets Keplerian shear stretch a compact dust clump into an arc and then a ring, while the spread of orbital frequencies causes phase mixing. Size-dependent radiation-pressure coefficients immediately place small grains on different eccentric or even radiation-pressure blowout orbits; Poynting–Robertson drag and stellar-wind drag then alter their orbits on longer timescales. Further collisions grind or disperse the clump, and planetary perturbations can accelerate mixing.

These processes depend strongly on

D_{f}

. Small grains have larger radiation-force-to-gravity ratios and generally shorter collisional or drag lifetimes, while larger fragments remain closer to the parent orbit but can preserve a velocity-dispersion-driven clump for longer. The lifetime also depends on collision location, ejection velocities, optical depth, and orbital radius. A universal fixed

Δ t

is therefore a useful population-model approximation, not a literal property of every collision; a size- and event-dependent lifetime distribution is more realistic.

Planet-induced debris-disk warp 2026-10-06

An inclined inner planet causes exterior debris disk rings to undergo nodal precession at different rates. In the quadrupole approximation,

ν ≃ (3/4) n (M_{p} / M_{⋆}) (a_{p} / a)^{2} \propto a^{- 7/2}

. Rings that have precessed appreciably tilt away from their original orbital plane, whereas distant rings remain near it, making an evolving warp. Conservative secular perturbations alone do not damp every ring into the planet's plane; unresolved phase mixing can instead produce a thick inner disk whose mean normal is near the forced plane.

Stellar stream 2026-10-14

A stellar stream is a correlated collection of stars commonly originating in the disruption of a bound progenitor. Phase mixing can erase spatial contrast while leaving coherence in orbital integrals. Such fossil debris constrains a galaxy's past accretion, although not every stream is from a galaxy: star clusters also produce streams.

Violent relaxation Created 2026-10-06 Updated 2026-10-14

Violent relaxation is rapid redistribution of the orbital energies of collisionless matter in a time-dependent collective gravitational potential. It can occur during collapse or a major galaxy merger. It differs from collisional two-body relaxation and from phase mixing in a stationary potential; it does not imply perfect final thermal equilibrium.