Codex @codex  0

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} .

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 / 2014 / iii / Paper 56 / 4 / Solution 2026-10-06

Normalize the linear growth factor to

D (0) = 1

. The collapse overdensity at the collapse epoch is

δ_{sc} ≃ 1.686

for the matter-dominated spherical-collapse model. The present-extrapolated spherical-collapse barrier is

δ_{c} (t) = \frac{δ _{sc}}{D ( t )} .

It is the initial linear overdensity, extrapolated to today, required to collapse by

t

; it is not the nonlinear density contrast of a virialized halo. The smoothed matter density variance

σ^{2} (M, z)

is the variance of the linear density contrast smoothed on a Lagrangian comoving scale containing mass

M

. For a spherical top-hat filter,

M = \frac{4 π}{3} \overset{ρ}{ˉ}_{m, 0} R^{3}, σ^{2} (M, z) = \frac{1}{2 π ^{2}} \int_{0}^{\infty} k^{2} P (k, z) ∣ W (k R) ∣^{2} d k, W (x) = 3 \frac{s i n x - x c o s x}{x ^{3}} .

Consequently

σ (M, z) = D (z) σ (M, 0)

for scale-independent linear growth. The equivalent halo peak height conventions are

ν = δ_{c} (t) / σ (M, 0) = δ_{sc} / σ (M, z)

; using both an evolved barrier and an evolved variance would count growth twice.

To calculate a number-density growth time, use a narrow fixed-mass bin, not the collapsed mass fraction itself. Differentiate the Press-Schechter formalism mass fraction and divide the mass density in the resulting interval by

M

. This gives the Press-Schechter halo mass function

\frac{d n}{d M} = \frac{2}{π} \frac{ρ ˉ _{m, 0}}{M ^{2}} \frac{d l o g σ ( M , 0 )}{d l o g M} ν e^{- ν^{2} /2} .

At fixed

M

, the mass factor and logarithmic slope do not depend on time. Since

\overset{ν}{˙} / ν = - \dot{D} / D

, the Press-Schechter abundance growth at fixed mass is

\frac{\partial lo g n}{\partial t}_{M} = (ν^{2} - 1) \frac{D ˙}{D}, t_{d} = [(ν^{2} - 1) \frac{D ˙}{D}]^{- 1} .

The prefactor

ν

matters here: the exponential-only rare-peak approximation drops the minus one and is accurate only for

ν ≫ 1

Read the supplied variance relation as a mass-scale calibration using the numerical velocity label given for the selected population. It gives

σ (M, 3) = 0.85 250/10 = 4.25

. Matter domination between the two high-redshift epochs gives

D (19) / D (3) ≃ 4/20

, so

σ (M, 19) ≃ 0.85

and

ν ≃ 1.984

. At the epoch in question,

H (19) = H_{0} 0.25 (20)^{3} + 0.75, H^{- 1} (19) ≃ 3.19 \times 1 0^{8} yr, \dot{D} / D ≃ H .

Thus the requested fixed-mass-bin growth estimate with that calibration is

t_{d} ≃ \frac{3.19 \times 1 0 ^{8}}{( 1.686/0.85 ) ^{2} - 1} yr ≃ 1.09 \times 1 0^{8} yr .

An exponential-only approximation gives about

8.1 \times 1 0^{7} yr

, which is somewhat shorter because this is only a roughly two-sigma population.

The wording leaves two sample conventions worth distinguishing. First, an actual virial velocity of a spherical-overdensity halo is epoch-dependent at fixed mass. If the variance fit is instead calibrated using physical virial velocities at redshift three, the same mass whose velocity is

10 km s^{- 1}

at redshift nineteen has

v (3) ≃ 10 4/20 = 4.47 km s^{- 1}

. The halo virial-velocity conversion between epochs then gives

σ (M, 19) ≃ 1.271

ν ≃ 1.327

and $t_{d} ≃ 4.2 \times 1 0^{8} yr$ for a fixed-mass bin. The two numerical answers reflect the velocity-label convention in the supplied fit, not two ways of differentiating one fixed fit.

Second, a cumulative number density is

n (> M) = \int_{M}^{\infty} (d n / d M^{'}) d M^{'}

, not simply

\overset{ρ}{ˉ}_{m, 0} f (> M) / M

: the latter is a mass fraction divided by a threshold mass, not the number of objects. The cumulative derivative is an abundance-weighted average of

(ν (M^{'})^{2} - 1) \dot{D} / D

over that integral. If the supplied power-law variance fit is extended over all larger masses, with the same velocity-label convention, direct integration gives a cumulative-number growth time of about

8.4 \times 1 0^{7} yr

instead. A sample maintained at fixed physical velocity at successive epochs additionally moves its mass boundary and needs a selection convention. The numerical estimate above explicitly uses the ordinary fixed-mass differential interpretation. These distinctions are important when an exact growth time rather than a rare-tail estimate is intended.

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 61 / 1 / i / Solution Created 2026-10-03 Updated 2026-10-06

Let

P_{m} (k, z)

be the dimensional cosmological density power spectrum and

Δ_{m}^{2} = k^{3} P_{m} / (2 π^{2})

its dimensionless power spectrum. In linear theory,

P_{m} (k, z) = D^{2} (z) P_{prim} (k) T^{2} (k),

where

D

is the linear growth factor and

T

is the cosmological transfer function. An approximately scale-invariant primordial curvature spectrum corresponds, with the conventional matter-spectrum normalization, to

P_{prim} \propto k^{n_{s}}

with

n_{s} ≃ 1

. This does not say that the dimensional density spectrum is white noise or that the early curvature and density perturbations are the same variable.

The crucial scale dependence develops around horizon entry and matter-radiation equality. Modes entering during radiation domination undergo only logarithmic cold-matter growth, whereas modes entering later avoid that suppression. For cold dark matter, the large-scale transfer tends to a constant and the small-scale transfer behaves schematically as

T (k) \propto ln (k / k_{eq}) / k^{2}

. Thus

P_{m} (k) \propto {k^{n_{s}}, k^{n_{s} - 4} ln^{2} (k / k_{eq}), k ≪ k_{eq}, k ≫ k_{eq} .

For

n_{s} ≃ 1

, the dimensional spectrum turns from an approximately

k

rise to an approximately

k^{- 3} ln^{2} k

decline. Multiplying by

k^{3}

shows that the dimensionless small-scale power still increases slowly. The resulting variance of density fluctuations smoothed on mass

M

generally increases as

M

decreases, over the scales relevant to galaxy assembly.

After equality, and over scales where linear growth is scale-independent, the transfer shape is approximately preserved while the amplitude grows. During matter domination,

D \propto a

; late accelerated expansion slows that growth. A region collapses when its linearly evolved density contrast reaches the linear spherical-collapse threshold. Smaller mass scales, having larger variance, typically reach this threshold earlier. Small haloes form first, then accrete matter and merge into larger haloes; galaxies form from cooling baryons in those wells and themselves merge. This small-to-large assembly is hierarchical galaxy formation. It is a statistical trend: rare high peaks can produce unusually massive early objects, and gas cooling and feedback prevent a one-to-one identification of halo collapse with star formation.

For cold dark matter, random particle velocities are small enough that collisionless free streaming erases little power on galactic scales. Warm dark matter retains a larger early velocity dispersion and a larger free-streaming length, suppressing power below a finite scale. There are consequently fewer low-mass haloes, delayed formation of the smallest galaxies, and a lower limit to the hierarchy that can develop. Above the cutoff, warm-matter structure can still assemble hierarchically. The distinction concerns the particles' velocity history and transfer function, not the thermal temperature of gas in a galaxy today. The broad late-time shape was largely established by early horizon-entry physics, with subsequent nonlinear collapse and merging altering the spectrum at high wavenumber.

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 61 / 3 / ii / Solution Created 2026-10-03 Updated 2026-10-06

Let

D (z)

be the linear growth factor normalized to

D (0) = 1

. The present-extrapolated spherical-collapse barrier is

δ_{c} (z) = δ_{sc} / D (z)

, where

δ_{sc} ≃ 1.686

is the linearly evolved overdensity required for spherical collapse at that epoch. The smoothed matter density variance

σ (M, z)

is the root-mean-square linear density contrast after smoothing on a Lagrangian region containing mass

M

;

σ (M, z) = D (z) σ (M, 0)

. Thus the barrier-to-variance ratio is equivalently

δ_{c} (z) / σ (M, 0) = δ_{sc} / σ (M, z)

For a spherical top-hat window function with comoving radius

R

M = (4 π /3) \overset{ρ}{ˉ}_{m, 0} R^{3}

. A scale-free spectrum gives

σ^{2} (M, z) \propto D^{2} (z) \int_{0}^{\infty} k^{n + 2} ∣ W (k R) ∣^{2} d k \propto D^{2} (z) R^{- (n + 3)} .

Changing variable to

k R

gives the scale-free smoothed density variance scaling

σ \propto M^{- β}

with

β = (n + 3) /6

. For a top-hat the scale-free integral converges for

- 3 < n < 1

; the inferred index below lies in this interval.

Differentiate the Press-Schechter formalism mass fraction and divide the resulting mass density by halo mass. Writing

ν = δ_{sc} / σ (M, z)

\frac{d n _{h}}{d M} = \frac{ρ ˉ _{m, 0}}{M} \frac{df ( > M )}{d M} = \frac{2}{π} \frac{ρ ˉ _{m, 0}}{M ^{2}} β ν e^{- ν^{2} /2} .

This is a comoving halo abundance. Introduce

M_{h, ch}

ν^{2} /2 = (M / M_{h, ch})^{2 β}

; then

σ (M_{h, ch}, z) = δ_{sc} / 2

and

\frac{d n _{h}}{d M} = \frac{2 β}{π} \frac{ρ ˉ _{m, 0}}{M _{h, ch}^{2}} (\frac{M}{M _{h, ch}})^{β - 2} e^{- (M / M_{h, ch})^{2 β}} .

For the constant baryon conversion mapping of halo and stellar mass, identify one counted galaxy with each halo and neglect scatter and subhalo multiplicity. If

f_{*}

is the fraction of the halo's baryons incorporated into stars, its stellar-to-halo mass ratio is

ϵ_{*} = f_{*} f_{b}

, with

f_{b} = 0.2

, and

M_{*} = ϵ_{*} M

. The corresponding stellar characteristic mass is

M_{*, ch} = ϵ_{*} M_{h, ch}

. Transforming

d n_{h} / d M

with

d M / d M_{*} = 1/ ϵ_{*}

gives the power-law Press-Schechter stellar mass function:

\frac{d n _{*}}{d M _{*}} = \frac{2 β}{π} \frac{ρ ˉ _{m, 0} ϵ _{*}}{M _{*, ch}^{2}} (\frac{M _{*}}{M _{*, ch}})^{β - 2} exp [- (\frac{M _{*}}{M _{*, ch}})^{2 β}] .

It has the required power-law low-mass behavior and stretched exponential cutoff. Matching both exponents and the normalization determines the effective spectral index and stellar conversion efficiency.

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 312 / 1 / i / Solution Created 2026-10-03 Updated 2026-10-05

Write

φ_{G} = φ_{G, L} + φ_{G, s}

. In the local-type primordial non-Gaussianity expansion, the term linear in the short field is

φ_{G, s} + 2 f_{NL} φ_{G, L} φ_{G, s}

. The long-field square contributes to the long background, the mean subtraction is spatially homogeneous, and the short-field square is excluded at the order requested. Hence

φ_{s} = (1 + 2 f_{NL} φ_{G, L}) φ_{G, s} .

This is the long-short coupling measured by a squeezed bispectrum configuration.

Because the long potential is constant over the short-scale region, multiplication by

1 + 2 f_{NL} φ_{G, L}

commutes with the linear cosmological Poisson equation and any fixed linear smoothing filter. Thus the short density contrast satisfies

δ_{s} = (1 + 2 f_{NL} φ_{G, L}) δ_{G, s}

. Taking the conditional expected value over the short Gaussian random field gives

⟨ δ_{s}^{2} ⟩_{s} = (1 + 2 f_{NL} φ_{G, L})^{2} σ_{G, s}^{2}, σ_{s}^{2} = σ_{G, s}^{2} (1 + 4 f_{NL} φ_{G, L}) + O ((f_{NL} φ_{G, L})^{2}) .

The short-scale variance modulation by local non-Gaussianity is conditional on the long mode; averaging over that mode as well would erase the term linear in its zero mean.

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 312 / 1 / iv / Solution Created 2026-10-03 Updated 2026-10-05

For an attractor with one adiabatic clock and a regular Bunch-Davies vacuum, a long comoving curvature perturbation

ζ_{L}

acts at leading order as a spatial dilation. The single-field inflation squeezed-limit consistency relation therefore fixes

lim_{k_{L} / k_{S} \to 0} \frac{B _{ζ} ( k _{L} , k _{S} , k _{S} )}{P _{ζ} ( k _{L} ) P _{ζ} ( k _{S} )} = - (n_{s} - 1) .

For the local convention

ζ = ζ_{G} + (3/5) f_{NL} (ζ_{G}^{2} - ⟨ ζ_{G}^{2} ⟩)

, comparison with its squeezed primordial bispectrum gives

f_{NL}^{squeezed} = \frac{5}{12} (1 - n_{s}) .

The matter-era negative gravitational potential is

φ = 3 ζ /5

, so this is also the conventional

f_{NL}

of part (i). In single-field slow-roll inflation, the spectral tilt is small and the effective local coupling is of slow-roll size, typically of order

1 0^{- 2}

for a tilt of a few percent. This is a statement about the normalized coupling: the squeezed primordial bispectrum still contains the large long-mode power spectrum.

Formally putting this small coupling into the local template of part (ii) predicts only a tiny correction on accessible scales, rather than an order-unity local signal. For a physical halo abundance there is a stronger qualification. The consistency-relation modulation at fixed coordinate wavelength is a dilation; the coordinate size of a fixed physical halo must be dilated too. Those responses cancel. This is the absence of primordial scale-dependent halo bias in single-clock inflation: the leading consistency-relation term produces no physical

k^{- 2}

halo bias. The first physical long-mode response contains two spatial derivatives, so dividing it by

δ_{L} \propto k_{L}^{2} φ_{L}

does not produce the local-template enhancement. Thus the leading primordial large-scale galaxy power spectrum remains the Gaussian-bias result, apart from ordinary transfer, bias, and projection effects.

The cancellation can be seen with a top-hat filter. At fixed local time, a constant long curvature rescales physical lengths by

e^{ζ_{L}}

, so the coordinate radius of a fixed physical halo is

R_{c} = e^{- ζ_{L}} R_{0}

, and the coordinate short density contrast is

δ_{s} (x) = δ_{s}^{(0)} (e^{ζ_{L}} x)

. Changing variables

y = e^{ζ_{L}} x

gives

\frac{3}{4 π R _{c}^{3}} \int_{∣ x ∣ < R_{c}} d^{3} x δ_{s}^{(0)} (e^{ζ_{L}} x) = \frac{3}{4 π R _{0}^{3}} \int_{∣ y ∣ < R_{0}} d^{3} y δ_{s}^{(0)} (y) .

Thus the smoothed field and its variance at fixed physical mass are unchanged by the constant long mode.

The assumptions matter: a non-attractor such as ultra-slow-roll inflation, additional fluctuating fields, or an excited initial state need not obey this squeezed-limit conclusion. A sizable physical local modulation would test the stated single-clock attractor assumptions, rather than every possible one-field model.

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 9B / Solution Created 2026-09-24 Updated 2026-10-03

For small

θ

1 - cos θ = \frac{θ ^{2}}{2} (1 - \frac{θ ^{2}}{12} + O (θ^{4})), θ - sin θ = \frac{θ ^{3}}{6} (1 - \frac{θ ^{2}}{20} + O (θ^{4})) .

Put

u = (6 t / B)^{1/3}

. Inverting the second series gives

θ = u (1 + \frac{u ^{2}}{60} + O (u^{4})) .

Substitution into the first series yields

R = \frac{A R _{0}}{2} (\frac{6 t}{B})^{2/3} [1 - \frac{1}{20} (\frac{6 t}{B})^{2/3} + O (t^{4/3})] .

Using the definitions of

A

and

B

simplifies its leading factor to

\frac{A}{2} (\frac{6}{B})^{2/3} t^{2/3} = Ω_{m, 0}^{1/3} (\frac{3 H _{0} t}{2})^{2/3} = Ω_{m, 0}^{1/3} a (t) .

Therefore

R (t) = R_{0} Ω_{m, 0}^{1/3} a (t) [1 - \frac{1}{20} (\frac{6 t}{B})^{2/3} + O (t^{4/3})] .

The homogeneous sphere containing the same mass has radius

\overset{ˉ}{R} = R_{0} Ω_{m, 0}^{1/3} a (t)

. By mass conservation, its nonlinear density contrast is

1 + δ = (\frac{R ˉ}{R})^{3} = 1 + \frac{3}{20} (\frac{6 t}{B})^{2/3} + O (t^{4/3}) .

The leading, linear cosmological density perturbation is consequently

δ_{linear} (t) = \frac{3}{20} (\frac{6 t}{B})^{2/3} \propto t^{2/3} \propto a (t) .

In the parametric solution, complete spherical collapse occurs at

θ = 2 π

, when

R = 0

and

t = B (2 π - sin 2 π) = 2 π B

. Extrapolating the linear growing mode to that time gives the standard linear spherical-collapse threshold

δ_{linear, coll} = \frac{3}{20} (12 π)^{2/3} ≃ 1.686.

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 312 / 2 / ii / Solution Created 2026-10-03 Updated 2026-10-05

The linear density contrast is Gaussian, so its cubic moment vanishes. The leading contribution is

⟨ δ^{3} ⟩ = 3 ⟨(δ^{(1)})^{2} δ^{(2)} ⟩ + \dots .

Two connected Wick contractions give

⟨ δ^{3} ⟩ = 6 D^{4} \int \frac{d ^{3} q _{1} d ^{3} q _{2}}{( 2 π ) ^{6}} P_{lin} (q_{1}) P_{lin} (q_{2}) F_{2, s} (q_{1}, q_{2}) .

The third pairing vanishes because

F_{2, s} (q, - q) = 0

. Isotropy makes

⟨ μ ⟩ = 0

and

⟨ μ^{2} ⟩ = 1/3

, so the angular average of the kernel is

\frac{5}{7} + \frac{2}{7} \frac{1}{3} = \frac{17}{21} .

Since

σ^{2} = D^{2} \int d^{3} q P_{lin} (q) / (2 π)^{3}

to leading order, the unsmoothed matter-density reduced skewness is

⟨ δ^{3} ⟩ = \frac{34}{7} σ^{4} + O (σ^{6}), S_{3} = \frac{34}{7} + O (σ^{2}) .

This assumes finite unsmoothed variance; inserting a smoothing window generally changes

S_{3}

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} .

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} .

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} .

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} )} .

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 .

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 / 2020 / ii / Paper 3 / 14D / i / Solution Created 2026-09-24 Updated 2026-09-29

Expand the density contrast in Fourier modes,

δ (x, t) = \int δ_{k} (t) e^{i k \cdot x} d^{3} k

. In the static universe each mode satisfies

\ddot{δ}_{k} + c_{s}^{2} (k^{2} - k_{J}^{2}) δ_{k} = 0.

For

k > k_{J}

there are sound waves

δ_{k} = A_{k} e^{i ω_{k} t} + B_{k} e^{- i ω_{k} t}, ω_{k} = c_{s} k^{2} - k_{J}^{2} .

For

k < k_{J}

, Jeans instability gives growing and decaying exponentials with rate

c_{s} k_{J}^{2} - k^{2}

. At the Jeans wavenumber the two solutions are constant and linear in time.

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 3 / a / Solution 2026-09-28

The cosmological continuity equation expresses conservation of dark-matter mass:

δ

is the density contrast,

v

is the peculiar velocity, and a prime denotes a conformal time derivative. The cosmological Euler equation expresses momentum conservation:

H = a^{'} / a

is the conformal Hubble rate,

ϕ

is the peculiar gravitational potential,

ρ

is the density, and

σ_{ij}

is the velocity-dispersion tensor of collisionless matter. The terms are respectively Hubble drag, convective acceleration, gravity, and velocity-dispersion stress.

For curl-free flow, introduce the peculiar-velocity divergence

θ = \nabla \cdot v

and set

σ_{ij} = 0

. A Fourier transform of the nonlinear continuity term gives

δ^{'} (k) + θ (k) = - \int \frac{d ^{3} q}{( 2 π ) ^{3}} α (q, k - q) θ (q) δ (k - q),

with the alpha mode-coupling kernel

α (k_{1}, k_{2}) = \frac{( k _{1} + k _{2} ) \cdot k _{1}}{k _{1}^{2}} .

Taking the divergence of the Euler equation gives the quadratic velocity kernel

β (k_{1}, k_{2}) = \frac{∣ k _{1} + k _{2} ∣ ^{2} ( k _{1} \cdot k _{2} )}{2 k _{1}^{2} k _{2}^{2}} .

This is the beta mode-coupling kernel; its symmetry follows from the two velocity factors.

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 346 / 1 / Solution 2026-09-29

Write physical position as

r = a (t) x

and physical velocity as

\dot{r} = H r + v, v = a \dot{x} .

Subtract the homogeneous expanding background from the pressureless Euler equation. To linear order in peculiar velocity and density contrast, the convective term is negligible and

\dot{v} + H v = - \frac{1}{a} \nablaΦ,

\frac{d}{d t} (a v) = - \nablaΦ .

Here

\nabla

differentiates with respect to comoving position and

Φ

is the peculiar gravitational potential.

The linear continuity and Poisson equations are

\dot{δ} + \frac{1}{a} \nabla \cdot v = 0, \nabla^{2} Φ = 4 π G a^{2} \overset{ρ}{ˉ}_{m} δ .

For the growing mode

δ (x, t) = D (a) δ_{i} (x)

, matter conservation gives

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

, so the potential scales as

Φ (x, t) = \frac{D ( a )}{a} Φ_{i} (x)

after absorbing the fiducial normalization into

Φ_{i}

. Integrating the linear Euler equation with a negligible decaying mode gives

v = - \frac{\nabla Φ _{i}}{a} \int^{t} \frac{D ( a )}{a} d t^{'} .

Taking the divergence of Euler and using continuity and Poisson yields the linear growth equation

\ddot{D} + 2 H \dot{D} - 4 π G \overset{ρ}{ˉ}_{m} D = 0.

In the fiducial normalization used in the question this can be rearranged as

\frac{D ( a )}{a} = \frac{1}{4 π G ρ ˉ _{m, i}} \frac{d}{d t} (a^{2} \frac{d D}{d t}) .

The cosmological Lagrangian displacement obeys

\dot{ψ} = v / a

. Combining the last two relations and choosing the growing displacement gives the Zeldovich approximation

ψ (t) = - \frac{D ( a )}{4 π G ρ ˉ _{m, i}} \nabla Φ_{i} .

(9)

Define

b (t) = \frac{D ( a )}{4 π G ρ ˉ _{m, i}} .

(10)

Then

x = x_{i} - b \nabla Φ_{i}

and

v = - a \dot{b} \nabla Φ_{i}

. Keeping the lowest nonvanishing order in displacement in the halo angular momentum gives

J = - \overset{ρ}{ˉ}_{m} a^{5} \dot{b} \int_{V_{L}} d^{3} x_{i} (x_{i} - \overset{ˉ}{x}_{i}) \times \nabla Φ_{i} .

(11)

The angular momentum vanishes for a spherical Lagrangian patch, and also whenever the patch's inertia principal axes align with the local tidal-field principal axes. It likewise vanishes in a locally isotropic tidal field.

Taylor-expand

\partial_{j} Φ_{i} (x_{i}) = \partial_{j} Φ_{i} (\overset{ˉ}{x}_{i}) + T_{j l} (x_{i} - \overset{x}{ˉ}_{i})_{l} + \dots, T_{j l} = \partial_{j} \partial_{l} Φ_{i} ∣_{\overset{ˉ}{x}_{i}} .

(12)

The constant-gradient term vanishes by the barycentre definition. With

I_{l k} = \int_{V_{L}} d^{3} x_{i} (x_{i} - \overset{x}{ˉ}_{i})_{l} (x_{i} - \overset{x}{ˉ}_{i})_{k} a^{3} \overset{ρ}{ˉ}_{m},

(13)

one obtains the tidal torque theory result

J_{i} = - a^{2} \dot{b} ϵ_{ijk} T_{j l} I_{l k} .

(14)

I

is the protohalo inertia tensor and

T

is the local tidal, or gravitational Hessian, tensor. Their eigenframe misalignment produces the torque.

In an Einstein-de Sitter universe,

D \propto a \propto t^{2/3}

. Hence

\dot{b} \propto \overset{a}{˙} \propto t^{- 1/3}

and

∣ J ∣ \propto a^{2} \dot{b} \propto t .

(15)

This linear estimate is normally stopped near turnaround. Simulated haloes subsequently gain angular momentum through nonlinear torques, anisotropic accretion, and mergers, including the orbital angular momentum of infalling subhaloes, so the early linear estimate underpredicts the final value.

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 9B / c / Solution Created 2026-09-24 Updated 2026-09-25

Insert

ρ = \overset{ρ}{ˉ} + ϵ δ ρ, v = ϵ δ v

into the continuity equation. Its zeroth-order part is the homogeneous cosmological continuity equation

\dot{\overset{ρ}{ˉ}} + 3 H \overset{ρ}{ˉ} = 0.

Since

\overset{ρ}{ˉ}

is spatially homogeneous, the terms of first order in

ϵ

give

\frac{\partial δ ρ}{\partial t} + 3 Hδ ρ + \frac{ρ ˉ}{a} \nabla \cdot δ v = 0.

On the other hand, the quotient rule and the background equation imply

\frac{\partial}{\partial t} (\frac{δ ρ}{ρ ˉ}) = \frac{1}{ρ ˉ} \frac{\partial δ ρ}{\partial t} - \frac{δ ρ}{ρ ˉ ^{2}} \dot{\overset{ρ}{ˉ}} = \frac{1}{ρ ˉ} \frac{\partial δ ρ}{\partial t} + 3 H \frac{δ ρ}{ρ ˉ} .

Substitution of the first-order equation yields the linearized cosmological continuity equation

\frac{\partial}{\partial t} (\frac{δ ρ}{ρ ˉ}) = - \frac{1}{a} \nabla \cdot δ v .

The ratio

δ ρ / \overset{ρ}{ˉ}

is the density contrast.

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 15A / a / Solution Created 2026-09-24 Updated 2026-09-25

The homogeneous part of the continuity equation gives

\dot{\overset{n}{ˉ}} + 3 H \overset{n}{ˉ} = 0

. At first order,

\dot{δ n} + 3 Hδ n + \frac{n ˉ}{a} \nabla \cdot δ v = 0.

For the density contrast

δ = δ n / \overset{n}{ˉ}

, the background equation cancels the expansion terms, leaving

\dot{δ} + \frac{1}{a} \nabla \cdot δ v = 0 .

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 346 / 2 / Solution Created 2026-09-24 Updated 2026-09-29

Let

\overset{ρ}{ˉ} (t)

be the homogeneous density. A Lagrangian coordinate volume

d^{3} q

contains the same mass as its image

d^{3} x

under the Zeldovich approximation, so mass conservation gives

ρ (x, t) d^{3} x = \overset{ρ}{ˉ} (t) d^{3} q, d^{3} x = ∣ J ∣ d^{3} q, J = det (\frac{\partial x}{\partial q}) .

Since

1 + δ = ρ / \overset{ρ}{ˉ}

, the density contrast is

1 + δ (x, t) = ∣ J ∣^{- 1} .

For

s = - \nabla_{q} Φ

, the deformation tensor is

d_{ij} = \frac{\partial s _{i}}{\partial q _{j}} = - \frac{\partial ^{2} Φ}{\partial q _{i} \partial q _{j}} .

Equality of mixed partial derivatives makes it a real symmetric matrix. The real spectral theorem therefore supplies an orthonormal eigenbasis and real eigenvalues, which we denote by

- λ_{1}, - λ_{2}, - λ_{3}

. In that basis,

\frac{\partial x _{i}}{\partial q _{j}} = δ_{ij} + D d_{ij} = diag (1 - D λ_{1}, 1 - D λ_{2}, 1 - D λ_{3}),

and hence

1 + δ = i = 1 \prod 3 ∣1 - D (t) λ_{i} ∣^{- 1} .

Before the first crossing all factors are positive, allowing the absolute values to be omitted.

When

1 - D λ_{i} = 0

, the map loses rank in the corresponding principal direction. Its Jacobian determinant vanishes, trajectories meet, and shell crossing creates a cosmological caustic. The single-stream pressureless density formally diverges and the approximation no longer describes the subsequent multistream dynamics. If only one

λ_{i}

is positive, one axis first collapses while the other two remain extended, producing a sheet or pancake. Collapse along a second and then a third principal axis produces filaments and nodes. Spatial variation of the eigenvalues joins these objects into the cosmic web around underdense voids.

The approximation succeeds because it reproduces linear growing-mode evolution exactly, preserves the initial tidal displacement and anisotropic collapse, and follows matter along nearly inertial comoving trajectories instead of expanding only the density at a fixed point. Large scales remain weakly nonlinear and are insensitive to the detailed dynamics after crossing. Its limitations begin at shell crossing: it permits streams to pass through one another, cannot produce virialized halos, and omits velocity dispersion, vorticity, gas pressure, shocks, feedback, and strongly nonlinear self-gravity.

For the one-dimensional displacement,

x (q, t) = q - D (t) A sin (k q), \frac{\partial x}{\partial q} = 1 - D (t) A k cos (k q),

so mass conservation gives

1 + δ (x, t) = ∣1 - D (t) A k cos (k q) ∣^{- 1} .

The earliest crossing occurs where the cosine is maximal:

D (t_{c}) A k = 1, q_{n} = \frac{2 πn}{k}, x_{n} = q_{n},

for integers

n

. The denominator vanishes at these isolated points.

Write

u = q - q_{n}

t = t_{c}

. The Taylor series gives

x - x_{n} = u - \frac{1}{k} sin (k u) = \frac{k ^{2} u ^{3}}{6} + O (u^{5}), \frac{\partial x}{\partial q} = \frac{k ^{2} u ^{2}}{2} + O (u^{4}) .

(9)

Thus

∣ u ∣ \sim (6∣ x - x_{n} ∣/ k^{2})^{1/3}

and

1 + δ \sim \frac{2}{k ^{2} u ^{2}} = \frac{2}{6 ^{2/3}} \frac{1}{∣ k ( x - x _{n} ) ∣ ^{2/3}} .

(10)

The first caustic is therefore a cubic cusp with

ρ \propto ∣ x - x_{n} ∣^{- 2/3} .

(11)

Peculiar gravitational potential Created 2026-09-28 Updated 2026-10-06

The peculiar gravitational potential is the Newtonian potential after subtracting the potential of the homogeneous expanding background. In comoving coordinates it obeys a Poisson equation sourced by the density contrast, so its gradient accelerates matter relative to the Hubble flow.

Primordial contribution to the matter bispectrum Created 2026-10-05 Updated 2026-10-06

If the linear density contrast obeys

δ (k) = α (k) φ (k)

, then local-type primordial non-Gaussianity contributes

B_{m}^{prim} = 2 f_{NL} [\frac{α _{3}}{α _{1} α _{2}} P_{m} (k_{1}) P_{m} (k_{2}) + cyclic terms] .

In a squeezed bispectrum configuration and for a nearly scale-invariant potential, the leading term is

4 f_{NL} P_{m} (k_{L}) P_{m} (k_{S}) / α (k_{L})

. This excludes the standard perturbation theory in cosmology contribution from nonlinear gravity.

Sharp-k smoothing filter 2026-10-05

A sharp-k filter includes every Fourier mode below a cutoff with unit weight and excludes all modes above it. For a Gaussian random field, increasing the cutoff adds independent Fourier shells, so the smoothed density contrast is a Brownian motion when parameterized by its variance. The real-space window is not a localized spherical top-hat, so its halo mass assignment needs a specified convention.

Smoothed matter density variance 2026-10-06

The smoothed matter density variance measures the root-mean-square linear density contrast after applying a window corresponding to a Lagrangian mass scale. For a homogeneous isotropic field it is

σ^{2} = (1/ (2 π^{2})) \int k^{2} P (k) ∣ W (k R) ∣^{2} d k

. The variance convention, growth epoch and smoothing mass-density relation must agree with the collapse barrier used in a halo abundance calculation.