Paper 346 (Galaxy Formation)

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 .

(1)

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.

(2)

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.

(3)

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.

b

 0  0

Solution

 0  0

The shell equation is conservative because its radial acceleration is the negative derivative of the specific potential energy

U (r) = - \frac{GM}{r} - \frac{Λ r ^{2}}{6} .

(1)

Multiplying the equation of motion by

\overset{r}{˙}

gives

\frac{d}{d t} (\frac{1}{2} \overset{r}{˙}^{2} - \frac{GM}{r} - \frac{Λ r ^{2}}{6}) = \overset{r}{˙} (\overset{r}{¨} + \frac{GM}{r ^{2}} - \frac{Λ r}{3}) = 0.

(2)

Hence

E = \frac{1}{2} \overset{r}{˙}^{2} - \frac{GM}{r} - \frac{Λ r ^{2}}{6} = constant .

(3)

This is energy per unit shell mass. The total energy of a uniform sphere requires integrating the energies of its shells.

For a sphere of physical radius

R

, the shell theorem gives

M (r) = M (r / R)^{3}

and

d M = 3 M r^{2} d r / R^{3}

. Assemble the sphere shell by shell to obtain its Newtonian gravitational potential energy:

W_{G} (R) = - \int_{0}^{R} \frac{GM ( r )}{r} d M = - \frac{3 G M ^{2}}{R ^{6}} \int_{0}^{R} r^{4} d r = - \frac{3}{5} \frac{G M ^{2}}{R} .

(4)

The cosmological constant contributes a prescribed one-body potential

- Λ r^{2} /6

, rather than a pair interaction, so there is no additional factor

1/2

W_{Λ} (R) = - \frac{Λ}{6} \int_{0}^{R} r^{2} d M = - \frac{Λ M}{2 R ^{3}} \int_{0}^{R} r^{4} d r = - \frac{1}{10} Λ M R^{2} .

(5)

Evaluating at the turnaround radius gives

W_{G, ta} = - \frac{3}{5} \frac{G M ^{2}}{r _{ta}}, W_{Λ, ta} = - \frac{1}{10} Λ M r_{ta}^{2} .

(6)

As in the question,

Λ

here includes the factor

c^{2}

that would multiply the geometrical cosmological constant in SI units.

For the final equilibrium apply the scalar virial theorem. The gravitational virial is

W_{G}

, whereas the cosmological constant force contributes

\sum_{α} r_{α} \cdot F_{Λ, α} = \frac{Λ}{3} \int r^{2} d M = - 2 W_{Λ} .

(7)

Thus, neglecting a surface-pressure term,

2 T_{f} + W_{G, f} - 2 W_{Λ, f} = 0.

(8)

At fixed

R

, a positive cosmological constant reduces the kinetic energy needed for virial equilibrium because its outward acceleration partially offsets self-gravity.

For the uniform-sphere collapse with a cosmological constant, assume a uniform final sphere of the same mass, with no energy loss or escaping matter. Put

x = \frac{r _{f}}{r _{ta}}, A = \frac{3 G M ^{2}}{5 r _{ta}}, B = \frac{Λ M r _{ta}^{2}}{10} .

(9)

The sphere has no bulk kinetic energy at turnaround, so

E_{ta} = - A - B

. The final virial theorem gives

T_{f} = \frac{A}{2 x} - B x^{2}, E_{f} = T_{f} - \frac{A}{x} - B x^{2} = - \frac{A}{2 x} - 2 B x^{2} .

(10)

Since

ρ_{ta} = 3 M / (4 π r_{ta}^{3})

\frac{B}{A} = \frac{Λ r _{ta}^{3}}{6 GM} = \frac{η}{2}, η = \frac{Λ}{4 π G ρ _{ta}} .

(11)

Conservation of energy therefore gives

1 + \frac{η}{2} = \frac{1}{2 x} + η x^{2}, 2 η x^{3} - (2 + η) x + 1 = 0.

(12)

Select the collapsing branch continuous from

x = 1/2

η = 0

, rather than an unrelated root of the cubic equation.

For

∣ η ∣ ≪ 1

, the root lies close to

1/2

. Linearize

x^{3}

there:

x^{3} = \frac{1}{8} + \frac{3}{4} (x - \frac{1}{2}) + O ((x - \frac{1}{2})^{2}) = \frac{3}{4} x - \frac{1}{4} + O ((x - \frac{1}{2})^{2}) .

(13)

Substitution into the cubic equation gives

(η /2 - 2) x + 1 - η /2 ≃ 0

, whence

\frac{r _{f}}{r _{ta}} ≃ \frac{1 - η /2}{2 - η /2} = \frac{1}{2} - \frac{η}{8} + O (η^{2}) .

(14)

The rational expression is an approximation, not an exact solution of the cubic equation. In fact both it and the exact collapsing root agree through order

η^{2}

; their first difference is of order

η^{3}

With zero cosmological constant, spherical collapse gives

r_{f} = r_{ta} /2

. For small positive

Λ

, the same sphere with the same turnaround radius reaches a smaller final radius. This does not mean that repulsion promotes collapse: as the sphere contracts,

W_{Λ}

becomes less negative, so some released gravitational binding energy pays the energetic cost of moving inward against the repulsive force. Combined with the modified virial theorem, energy conservation requires a more compact endpoint. Positive

Λ

can prevent turnaround altogether; an inward acceleration at

r_{ta}

requires

- \frac{GM}{r _{ta}^{2}} + \frac{Λ}{3} r_{ta} < 0, η < 1.

(15)

The perturbative radius formula applies well inside this limit, for spheres that actually turn around. A small negative cosmological constant instead gives

r_{f} / r_{ta} > 1/2

at fixed turnaround data.

2

 0  0

Solution

 0  0

Ignoring gravitational focusing, a galaxy sweeps a cylinder of volume

π R^{2} v_{rel} T

. Multiplying this by the number density

N

of target galaxies and averaging the relative speed gives the mean encounter count

μ = π R^{2} ⟨ v_{rel} ⟩ NT .

(1)

For independent encounters modeled by a Poisson process, the probability of at least one galaxy merger is

1 - e^{- μ}

. The printed expression

P = μ

is its rare-encounter approximation

μ ≪ 1

; it is not an exact probability for arbitrary

T

For an illustrative present-day field population take an effective merger impact parameter

R = 20 kpc = 0.02 Mpc

N = 1 0^{- 2} Mpc^{- 3}

, and

⟨ v_{rel} ⟩ = 200 km s^{- 1}

. These are assumed order-of-magnitude inputs, not precise observational measurements. Taking a Hubble time of

T ≃ 1.4 \times 1 0^{10} yr ≃ 4.4 \times 1 0^{17} s

, the given distance conversion yields

⟨ v_{rel} ⟩ T ≃ \frac{200 ( 4.4 \times 1 0 ^{17} )}{3 \times 1 0 ^{19}} Mpc ≃ 2.9 Mpc,

(2)

and hence

P ≃ μ ≃ π (0.02)^{2} (1 0^{- 2}) (2.9) ≃ 3.7 \times 1 0^{- 5} .

(3)

Thus the geometric field estimate is of order

1 0^{- 5}

1 0^{- 4}

per Hubble time. It scales as

R^{2} N ⟨ v_{rel} ⟩ T

and is very sensitive to environment and the adopted effective merger radius. Enhanced density in groups, gravitational focusing, and the evolution of the galaxy population are all omitted; this number is not a prediction of the full cosmological merger fraction.

Now use the stipulated rapid, distant encounter. The consistent rectilinear trajectory is

R (t) = r_{p} (t) = (p, 0, v t), R (t) = p^{2} + v^{2} t^{2} .

(4)

With the stated impact parameter along

x

and velocity along

z

, this trajectory lies in the

x z

-plane. The original PDF's reference to the

x y

-plane is a typo; the local TeX also corrupts the impact-parameter direction. The original PDF fixes that direction as

x

Use the positive potential

G m_{p} /∣ R - r ∣

, whose gradient is the attractive acceleration under the question's sign convention. Its multipole expansion is

\frac{1}{∣ R - r ∣} = \frac{1}{R} + \frac{r \cdot R}{R ^{3}} + \frac{3 ( r \cdot R ) ^{2} - R ^{2} r ^{2}}{2 R ^{5}} + O (\frac{r ^{3}}{R ^{4}}) .

(5)

The first term has no force; the linear term accelerates the entire galaxy and disappears in the frame following its centre of mass. The leading internal tidal potential is therefore

ψ (r, t) = \frac{G m _{p}}{R ^{3}} [- \frac{r ^{2}}{2} + \frac{3 ( r \cdot R ) ^{2}}{2 R ^{2}}] .

(6)

This is a quadrupole approximation valid for

r ≪ p

, rather than an exact equality for every

r < p

. Its acceleration tidal tensor acts on

r

\nabla ψ = G m_{p} [- \frac{r}{R ^{3}} + \frac{3 ( r \cdot R ) R}{R ^{5}}] .

(7)

In the impulse approximation, each stellar position is held fixed during the flyby and the velocity kick is the integral of this acceleration. Put

s = v t

; terms odd in

s

integrate to zero, while

\int_{- \infty}^{\infty} \frac{d s}{( p ^{2} + s ^{2} ) ^{3/2}} = \frac{2}{p ^{2}}, \int_{- \infty}^{\infty} \frac{d s}{( p ^{2} + s ^{2} ) ^{5/2}} = \frac{4}{3 p ^{4}}, \int_{- \infty}^{\infty} \frac{s ^{2} d s}{( p ^{2} + s ^{2} ) ^{5/2}} = \frac{2}{3 p ^{2}} .

(8)

The integrated diagonal coefficients of the tidal tensor are consequently

(2, - 2, 0) / (v p^{2})

, giving

Δ v = \frac{2 G m _{p}}{v p ^{2}} (x, - y, 0) .

(9)

The encounter stretches the galaxy along the impact parameter, compresses it along

y

, and gives no net leading kick along the flyby direction.

Denote the pre-encounter stellar velocity by

u

, to distinguish it from the relative flyby speed

v

. The instantaneous change of specific kinetic energy is

Δ e = u \cdot Δ v + \frac{1}{2} ∣Δ v ∣^{2} .

(10)

Uncorrelated kicks with zero mean cross term imply

⟨ u \cdot Δ v ⟩ = 0

, so the phase-averaged heating at a specified position is

⟨ Δ e ⟩ = \frac{2 G ^{2} m _{p}^{2}}{v ^{2} p ^{4}} (x^{2} + y^{2}) .

(11)

For an individual star the cross term need not vanish: the formula is an ensemble or orbital-phase average. Integrating over a spherical galaxy of mass

m_{g}

, symmetry gives

⟨ x^{2} ⟩ = ⟨ y^{2} ⟩ = ⟨ z^{2} ⟩ = ⟨ r^{2} ⟩ /3

. Therefore the total tidal heating is

Δ E_{g} = \frac{4 G ^{2} m _{p}^{2} m _{g}}{3 v ^{2} p ^{4}} ⟨ r^{2} ⟩ .

(12)

For two identical galaxies, each receives this heating with

m_{p} = m_{g}

. Adding both contributions gives

Δ E_{int} = \frac{8 G ^{2} m _{g}^{3}}{3 v ^{2} p ^{4}} ⟨ r^{2} ⟩ .

(13)

In the centre of mass frame, equal masses approach with speeds

v /2

v

is their relative speed at infinity. The initial orbital energy is

E_{orb} = 2 [\frac{1}{2} m_{g} (\frac{v}{2})^{2}] = \frac{1}{2} (\frac{m _{g}}{2}) v^{2} = \frac{1}{4} m_{g} v^{2},

(14)

where

m_{g} /2

is the reduced mass. The orbital potential energy vanishes at infinite separation. Within the weak-deflection approximation this asymptotic speed is also the nearly constant speed used in the flyby calculation.

Conservation of energy transfers the positive internal heating out of the relative orbit. Tidal capture of galaxies occurs in this model if

E_{orb} - Δ E_{int} < 0

, which gives

\frac{8 G ^{2} m _{g}^{3} ⟨ r ^{2} ⟩}{3 v ^{2} p ^{4}} > \frac{1}{4} m_{g} v^{2}, p v < [\frac{32}{3} G^{2} m_{g}^{2} ⟨ r^{2} ⟩]^{1/4} .

(15)

The equality is marginal capture. A bound pair still needs subsequent evolution to coalesce, so the result is an approximate capture criterion used here as a merger criterion. Its right-hand side has dimensions of length times speed, as required.

The impulse approximation requires the flyby duration

τ \sim p / v

to be short compared with the stellar dynamical time

Ω^{- 1}

, namely

v / p ≫ Ω

. If

v / p < Ω

, a star moves substantially during the encounter. Its orbital adiabatic invariance of an orbital action suppresses net heating by a slowly varying tidal field: adiabatic shielding of tidal encounters replaces the frozen-position calculation. The divergent

v^{- 2}

heating predicted by extrapolating the impulsive formula to slow encounters is therefore spurious. Close passages with

p

comparable to the galaxy size or strong gravitational focusing also lie outside the derivation.

3

 0  0

Solution

 0  0

Assume nonrelativistic pressureless matter, an initial irrotational vector field of growing-mode linear cosmological density perturbations, a homogeneous expanding background, and no shell crossing. All gradients below refer to initial comoving coordinates. Let

ρ_{0} = \overset{ρ}{ˉ}_{m} (t) a^{3} (t) = \overset{ρ}{ˉ}_{m} (t_{i})

, with

a (t_{i}) = 1

, and normalize the linear growth factor by

D (t_{i}) = 1

. The paper calls

D

a growth rate, but it is the dimensionless growth factor;

\dot{D}

is its time derivative.

Writing

x_{i}

for the actual position at

t_{i}

, the Zeldovich approximation is, to first order,

x (t) = x_{i} + [D (t) - 1] s (x_{i}), \dot{x} = \dot{D} s (x_{i}) .

(1)

This explicitly gives

x (t_{i}) = x_{i}

. The usual notation

x = q + D s (q)

instead uses the uniform reference Lagrangian coordinate

q

, with

x_{i} = q + s (q)

. The two labels can be interchanged inside a first-order displacement, but not in the unperturbed position without the initial displacement correction.

To determine

s

, mass conservation between the initial and current positions gives

1 + δ (t) = \frac{1 + δ _{i}}{d e t [ 1 + ( D - 1 ) \nabla s ]} = 1 + δ_{i} - (D - 1) \nabla \cdot s + O (δ_{i}^{2}) .

(2)

Demanding

δ (t) = D (t) δ_{i}

gives

\nabla \cdot s = - δ_{i}

. The irrotational vector field assumption makes

s

a gradient; the initial cosmological Poisson equation is

\nabla^{2} Φ_{i} = 4 π G ρ_{0} δ_{i}

. With the same boundary conditions for these potential equations, and after removing an irrelevant uniform translation,

s = - \frac{\nabla Φ _{i}}{4 π G ρ _{0}}, \dot{x} = - \frac{D ˙}{4 π G ρ ˉ _{m} a ^{3}} \nabla Φ_{i} .

(3)

The evolution of

D

follows from the linearized peculiar-motion equation

\ddot{x} + 2 H \dot{x} = - \nablaΦ/ a^{2}

. Since cosmological Poisson equation gives

Φ (t) = (D / a) Φ_{i}

to this order, substitution yields the linear growth equation

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

(4)

Thus the approximation extrapolates the linear growing displacement along a fixed initial direction, even as its density mapping becomes nonlinear.

The early spin of a dark matter halo comes from an external gravitational torque on its nonspherical initial mass region. A uniform external acceleration moves its centre of mass without spinning it; the spatially varying tidal tensor exerts different forces on different parts. The tidal torque theory requires misalignment of the region's shape and the surrounding tidal field. A local irrotational vector field of velocity does not imply zero integrated angular momentum for a nonspherical region.

Put

y = x_{i} - \hat{x}_{i}

and use the physical peculiar velocity

v = a \dot{x}

. The physical lever arm is

a (x - \hat{x})

, while a leading-order mass element is

ρ_{0} d^{3} x_{i}

. Substituting the velocity formula into the supplied definition of angular momentum about the centre of mass gives

J = - \frac{a ^{2} D ˙}{4 π G} \int_{V_{L}} y \times \nabla Φ_{i} d^{3} x_{i} .

(5)

Terms from the perturbed mass measure are higher order. Subtracting the barycentre's velocity also changes nothing because

\int_{V_{L}} y d^{3} x_{i} = 0

at this order.

Apply the divergence theorem componentwise:

ϵ_{jk l} \int_{V_{L}} y_{k} \partial_{l} Φ_{i} d^{3} x_{i} = ϵ_{jk l} \int_{Σ_{L}} Φ_{i} y_{k} d S_{l} - ϵ_{jk l} \int_{V_{L}} Φ_{i} δ_{k l} d^{3} x_{i} = ϵ_{jk l} \int_{Σ_{L}} Φ_{i} y_{k} d S_{l} .

(6)

The antisymmetry of the Levi-Civita symbol kills the last term. Consequently, for the ordinary oriented comoving surface element

d S

J = - \frac{a ^{2} D ˙}{4 π G} \int_{Σ_{L}} Φ_{i} y \times d S .

(7)

The original PDF prints a different coefficient,

- \dot{D} / (4 π G \overset{ρ}{ˉ}_{m} a^{2})

. That coefficient does not follow from its own mass measure and velocity equation and has the wrong dimensions for total angular momentum. The corrected coefficient above also gives the time dependence consistent with the paper's later tensor expression; this is a source typo, not a TeX transcription issue.

For a spherical

V_{L}

centered on its barycentre, the outward normal is parallel to

y

, so

y \times d S = 0

pointwise. Hence

J = 0 for a spherical initial region .

(8)

For an equipotential surface,

Φ_{i} = Φ_{*}

Σ_{L}

, and

ϵ_{jk l} \int_{Σ_{L}} y_{k} d S_{l} = ϵ_{jk l} \int_{V_{L}} \partial_{l} y_{k} d^{3} x_{i} = ϵ_{jk l} δ_{k l} Vol (V_{L}) = 0.

(9)

Therefore

J = 0 if the initial boundary is an equipotential .

(10)

The second conclusion holds for any boundary shape at this order, not just a sphere.

To extract the leading tidal torque, expand the initial peculiar gravitational potential about the barycentre using its Taylor series:

Φ_{i} (\hat{x}_{i} + y) = Φ_{0} + g_{k} y_{k} + \frac{1}{2} H_{km} y_{k} y_{m} + \dots, H_{km} = \partial_{k} \partial_{m} Φ_{i} (\hat{x}_{i}) .

(11)

The constant has no force;

g

has zero integrated torque because the first mass moment vanishes. With the mass second-moment tensor

I_{km} = ρ_{0} \int_{V_{L}} y_{k} y_{m} d^{3} x_{i},

(12)

we obtain

J_{j} = - \frac{a ^{2} D ˙}{4 π G ρ _{0}} ϵ_{jk l} H_{l m} I_{km} = \frac{D ˙}{4 π G ρ ˉ _{m} a} ϵ_{jk l} H_{km} I_{m l} .

(13)

The second equality swaps the indices

k, l

and uses the symmetry of both tensors. Define the initial acceleration tidal tensor by

T_{km} = - \partial_{k} \partial_{m} Φ_{i} (\hat{x}_{i}) = - H_{km} .

(14)

Then the formula in the question has exactly the stated sign:

J_{j} = - \frac{D ˙}{4 π G ρ ˉ _{m} a} ϵ_{jk l} T_{km} I_{m l} .

(15)

If one defines

T

as the positive Hessian matrix of

Φ_{i}

instead, the displayed contraction has a plus sign. Defining the convention is essential. Also, the paper's

I

is a mass second-moment tensor; the mechanical inertia tensor is

tr (I) 1 - I

Only the anisotropic parts contribute: a multiple of the identity produces zero contraction with the Levi-Civita symbol. In axes where

I = diag (I_{1}, I_{2}, I_{3})

, for example,

J_{1} = \frac{D ˙}{4 π G ρ ˉ _{m} a} T_{23} (I_{2} - I_{3}) .

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Thus a spherical region or tensors sharing principal axes have zero leading torque; unequal principal moments together with off-diagonal tidal components create spin. The Taylor series truncation assumes that higher spatial derivatives of the tidal field are sufficiently small across the region.

In an Einstein-de Sitter universe,

a \propto t^{2/3}

D \propto a

, and

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

. The initial

I

and

T

are time independent, so

J \propto a^{2} \dot{D} \propto t^{4/3} t^{- 1/3} \propto t .

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This is early-time growth for fixed initial matter, before collapse invalidates the extrapolation. It does not predict indefinite linear spin growth for a virialized halo.

Two reasons for only approximate agreement with simulations are that nonlinear collapse and shell crossing change the trajectories and the tidal field, invalidating the first-order displacement and fixed leading tidal tensor; and that real halos undergo dark-matter halo mergers, anisotropic accretion, and exchange of material and angular momentum, so the halo identified at a later time need not be the same isolated collection of initial particles. These processes can change both the magnitude and the direction of the spin.

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 346 (Galaxy Formation)

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 346 (Galaxy Formation)

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