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 Incoming links: P(X, phi) scalar field theory

K-essence 2026-10-14

K-essence is a cosmological scalar-field mechanism in which noncanonical kinetic dynamics supply negative pressure or accelerated expansion. Its action is a particular use of a P(X, phi) scalar field theory; that general action also includes theories without kinetic dark energy.

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 311 / 2 / d / iii / Solution Created 2026-10-03 Updated 2026-10-06

For the P(X, phi) scalar field theory specialized to

F (X)

, vary the inverse metric tensor while holding

ϕ

fixed. With

q_{a} = \partial_{a} ϕ

,

δ X = - \frac{1}{2} q_{a} q_{b} δ g^{ab}, δ - g = - \frac{1}{2} - g g_{ab} δ g^{ab} .

Using

T_{ab} = - 2 (- g)^{- 1/2} δ S / δ g^{ab}

yields

T_{ab} = F^{'} (X) q_{a} q_{b} + F (X) g_{ab} .

In any local Lorentz frame, set

A = F^{'} (X) \geq 0

,

Y = X F^{'} (X) - F (X) \geq 0

,

s = \sum_{i} q_{i}^{2}

, and

X = (q_{0}^{2} - s) /2

. Then

ρ = T_{00} = A q_{0}^{2} - F = \frac{A}{2} (q_{0}^{2} + s) + Y \geq 0, T_{0 i} = A q_{0} q_{i} .

A direct calculation gives

ρ^{2} - i \sum T_{0 i}^{2} = A^{2} X^{2} + A (q_{0}^{2} + s) Y + Y^{2} \geq 0.

Hence

ρ \geq T_{0 i} T_{0 i}

in every local Lorentz frame, proving the dominant energy condition. This proof covers timelike, null, spacelike, and zero

\nabla ϕ

alike; it does not assume

X > 0

or use a perfect fluid in general relativity rest frame. The hypotheses need only hold at every attained value of

X

; the stated all-

X

assumptions are sufficient. For example,

F = X - V_{0}

,

V_{0} \geq 0

, satisfies them.

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 312 / 1 / ii / Solution Created 2026-10-03 Updated 2026-10-05

For the P(X, phi) scalar field theory, the background pressure is

p = P

and the energy density is

ρ = 2 X P_{X} - P

. Differentiating at fixed

ϕ

gives

p_{X} = P_{X}

and

ρ_{X} = P_{X} + 2 X P_{XX}

, so the Sound speed of a P(X, phi) scalar perturbation is

c_{s}^{2} = \frac{p _{X}}{ρ _{X}} = \frac{P _{X}}{P _{X} + 2 X P _{XX}} .

This is the propagation speed of the scalar fluctuation; the derivative is taken at fixed field, rather than along an arbitrary background trajectory.

On the fixed expanding metric,

X = \frac{1}{2} \dot{\overset{ˉ}{ϕ}}^{2} + \dot{\overset{ˉ}{ϕ}} δ \dot{ϕ} + \frac{1}{2} (δ \dot{ϕ})^{2} - \frac{1}{2 a ^{2}} (\partial_{i} δ ϕ)^{2} .

Write the comoving curvature perturbation in the paper's convention as

ζ = - Hδ ϕ / \dot{\overset{ˉ}{ϕ}}

. Neglecting slow-roll derivatives of

H

and

\dot{\overset{ˉ}{ϕ}}

yields

δ X = \frac{X ˉ}{H ^{2}} [- 2 H \dot{ζ} + \dot{ζ}^{2} - a^{- 2} (\partial_{i} ζ)^{2}] .

With

δ X_{1} = - 2 \overset{ˉ}{X} \dot{ζ} / H

and

δ X_{2} = \overset{ˉ}{X} [\dot{ζ}^{2} - a^{- 2} (\partial_{i} ζ)^{2}] / H^{2}

, the only cubic terms are

P_{XX} δ X_{1} δ X_{2} + \frac{1}{6} P_{XXX} δ X_{1}^{3} = \frac{2 X ˉ ^{2} P _{XX}}{H ^{3}} \dot{ζ} a^{- 2} (\partial_{i} ζ)^{2} - (\frac{2 X ˉ ^{2} P _{XX}}{H ^{3}} + \frac{4 X ˉ ^{3} P _{XXX}}{3 H ^{3}}) \dot{ζ}^{3} .

The identities

ϵ = \overset{ˉ}{X} P_{X} / H^{2}

and

(1 - c_{s}^{2}) / c_{s}^{2} = 2 \overset{ˉ}{X} P_{XX} / P_{X}

give the cubic curvature action for a P(X, phi) scalar field

S_{3} = \int d t d^{3} x \frac{a ^{3} ϵ ( 1 - c _{s}^{2} )}{H c _{s}^{2}} [a^{- 2} \dot{ζ} (\partial_{i} ζ)^{2} + A \dot{ζ}^{3}], A = - 1 - \frac{2 X ˉ P _{XXX}}{3 P _{XX}} .

If

P_{XX} = 0

, the unfactored expression above supplies the regular limiting result.

For the requested primordial bispectrum, define

C = ϵ (1 - c_{s}^{2}) / (H c_{s}^{2})

and

\int_{p} = \int d^{3} p / (2 π)^{3}

. The cubic Hamiltonian and

d t = a d τ

combine to give

a H_{int} = C a \int_{p_{1}, p_{2}, p_{3}} (2 π)^{3} δ^{(3)} (p_{1} + p_{2} + p_{3}) (p_{2} \cdot p_{3}) ζ_{p_{1}}^{'} ζ_{p_{2}} ζ_{p_{3}} .

Here the positive sign comes from the two spatial derivatives and

H_{int} = - L_{int}

at cubic order. Wick theorem gives two contractions for each choice of which external leg meets

ζ^{'}

. At the external time zero,

⟨ ζ_{k} (0) ζ_{p} (τ)⟩ = (2 π)^{3} δ^{(3)} (k + p) u_{k} (0) u_{k}^{*} (τ) .

Thus, after removing the momentum-conserving delta function, the Hamiltonian contraction is

2 C a \prod_{j = 1}^{3} u_{k_{j}} (0) \sum_{cyc} (k_{2} \cdot k_{3}) u_{k_{1}}^{*'} (τ) u_{k_{2}}^{*} (τ) u_{k_{3}}^{*} (τ) .

Put

A_{k} = H / 4 ϵ c_{s} k^{3}

. Then

u_{k} (0) = A_{k}

and

u_{k}^{*'} = A_{k} c_{s}^{2} k^{2} τ e^{ik c_{s} τ}

. The

τ

cancels the

1/ τ

in

a = - 1/ (H τ)

; moreover,

k_{2} \cdot k_{3} = \frac{k _{1}^{2} - k _{2}^{2} - k _{3}^{2}}{2} .

Inserting these expressions into the in-in formalism and conjugating the expression inside the real part gives

⟨ ζ_{k_{1}} (0) ζ_{k_{2}} (0) ζ_{k_{3}} (0)⟩ = (2 π)^{3} δ^{(3)} (k_{1} + k_{2} + k_{3}) \frac{( 1 - c _{s}^{2} ) H ^{4}}{32 ϵ ^{2} c _{s}^{3} ( k _{1} k _{2} k _{3} ) ^{3}} \times Re [- i \int_{- \infty (1 + i 0^{+})}^{0} d τ e^{- i K c_{s} τ} cyc \sum k_{1}^{2} (k_{1}^{2} - k_{2}^{2} - k_{3}^{2}) (1 + i k_{2} c_{s} τ) (1 + i k_{3} c_{s} τ)],

where

K = k_{1} + k_{2} + k_{3}

. Conjugating the integrand also conjugates the contour: the original positive-frequency-product integral has lower limit

- \infty (1 - i 0^{+})

, whereas the negative exponential displayed here requires

- \infty (1 + i 0^{+})

. This supplies the convergent interpretation of the paper's integral.

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 1 / i / Solution Created 2026-09-24 Updated 2026-09-25

Write

ϕ = \overset{ˉ}{ϕ} + φ

and evaluate derivatives of

P

on the homogeneous background

\overset{ˉ}{X} = \frac{ϕ ˉ ^{'2}}{2 a ^{2}} .

The first- and second-order changes in

X

are

δ_{1} X = \frac{ϕ ˉ ^{'} φ ^{'}}{a ^{2}}, δ_{2} X = \frac{φ ^{'2} - ( \nabla φ ) ^{2}}{2 a ^{2}} .

After using the background equation to remove the linear action, the quadratic action of the P(X, phi) scalar field theory is

S_{2} = \frac{1}{2} \int d τ d^{3} x {a^{2} A φ^{'2} - a^{2} P_{X} (\nabla φ)^{2} + 2 a^{2} P_{Xϕ} \overset{ˉ}{ϕ}^{'} φ φ^{'} + a^{4} P_{ϕϕ} φ^{2}},

where

A = P_{X} + 2 \overset{ˉ}{X} P_{XX} .

Varying gives

(a^{2} A φ^{'})^{'} - a^{2} P_{X} \nabla^{2} φ + [(a^{2} P_{Xϕ} \overset{ˉ}{ϕ}^{'})^{'} - a^{4} P_{ϕϕ}] φ = 0.

The Sound speed of a P(X, phi) scalar perturbation is

c_{s}^{2} = \frac{P _{X}}{P _{X} + 2 X ˉ P _{XX}} = \frac{P _{X}}{A} .

At leading slow variation, take

H

,

c_{s}

, and the kinetic coefficients as nearly constant and neglect the effective mass and their logarithmic derivatives. The Fourier equation then becomes

φ_{k}^{''} + 2 \frac{a ^{'}}{a} φ_{k}^{'} + c_{s}^{2} k^{2} φ_{k} = 0.

For

u_{k} = a φ_{k}

and de Sitter spacetime

a^{''} / a = 2/ τ^{2}

, this is

u_{k}^{''} + (c_{s}^{2} k^{2} - \frac{2}{τ ^{2}}) u_{k} = 0.

Two independent solutions are

u_{k}^{\pm} = (1 \pm \frac{i}{c _{s} k τ}) e^{\pm i c_{s} k τ} .

The Bunch-Davies vacuum selects the positive-frequency behavior

e^{- i c_{s} k τ}

- c_{s} k τ \to \infty

. Canonical normalization of

v = a A φ

gives the general amplitude; with the field normalization

P_{X} ≃ 1

, so

A ≃ c_{s}^{- 2}

, it reduces, up to an overall phase, to

φ_{k} (τ) = \frac{H}{2 c _{s} k ^{3}} (1 + i c_{s} k τ) e^{- i c_{s} k τ} .

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P(X, phi) scalar field theory Created 2026-09-24 Updated 2026-10-14

P (X, ϕ)

scalar theory has Lagrangian density

P

depending on the field and

X = - g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ /2

. It includes canonical scalar theory and models with noncanonical kinetic interactions.

A k-essence model uses noncanonical kinetic terms for cosmological acceleration; a general P(X, phi) scalar field theory need not have that application.

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