Inflaton

A canonical homogeneous inflaton

ϕ (t)

has

ρ = \frac{1}{2} \dot{ϕ}^{2} + V (ϕ), P = \frac{1}{2} \dot{ϕ}^{2} - V (ϕ),

and obeys

\ddot{ϕ} + 3 H \dot{ϕ} + V^{'} (ϕ) = 0

.

Table of contents

Inflaton equation of motion

For a homogeneous canonical inflaton in an expanding FLRW metric,

\ddot{ϕ} + 3 H \dot{ϕ} + V^{'} (ϕ) = 0.

The expansion contributes the damping term called Hubble friction.

Frozen scalar fields and the Friedmann constraint

For a homogeneous canonical scalar field, the cosmological perfect-fluid continuity equation reduces to

\dot{ϕ} (\ddot{ϕ} + 3 H \dot{ϕ} + V^{'}) = 0

. It implies the inflaton equation of motion wherever

\dot{ϕ} \neq = 0

, and by continuity at isolated turning points. It cannot by itself exclude a frozen field on a time interval. Indeed, choose

ϕ = ϕ_{0}

with

V (ϕ_{0}) > 0

but

V^{'} (ϕ_{0}) \neq = 0

; a de Sitter scale factor in flat slicing with

H^{2} = V (ϕ_{0}) / (3 M_{Pl}^{2})

satisfies the homogeneous Friedmann equation and Friedmann acceleration equation, but not the scalar equation. Varying the inflaton action in an expanding universe supplies the missing equation also for a frozen field, requiring

V^{'} (ϕ_{0}) = 0

.

Exact hyperbolic-sine scalar-field cosmology

In a flat universe containing a homogeneous canonical scalar field, the ansatz

ϕ (t) = ϕ_{0} lo g tanh (λ t)

has the exact solution

H = λ \frac{ϕ _{0}^{2}}{M _{Pl}^{2}} coth (2 λ t), a = a_{0} [sinh (2 λ t)]^{ϕ_{0}^{2} / (2 M_{Pl}^{2})} .

It interpolates between decelerating power-law expansion at early times and exponential accelerated expansion at late times when

2/3 < ϕ_{0}^{2} / M_{Pl}^{2} < 2

.

Inflaton action in an expanding universe

For a canonical scalar field in a spatially flat expanding universe,

S [ϕ] = \int d^{3} x d t a^{3} [\frac{1}{2} \dot{ϕ}^{2} - \frac{c ^{2}}{2 a ^{2}} ∣\nabla ϕ ∣^{2} - V (ϕ)] .

The field Euler-Lagrange equation is

\ddot{ϕ} + 3 H \dot{ϕ} - \frac{c ^{2}}{a ^{2}} \nabla^{2} ϕ + V^{'} (ϕ) = 0.

Inflationary scalar Fourier mode

For nearly constant inflaton potential and constant

H_{inf}

, the spatial Fourier transform obeys

\ddot{ϕ}_{k} + 3 H_{inf} \dot{ϕ}_{k} + \frac{c ^{2} k ^{2}}{a ^{2}} ϕ_{k} = 0.

Canonically rescaled de Sitter scalar mode

For

a = - 1/ (H_{inf} τ)

, the rescaled mode

ϕ_{k} = a ϕ_{k}

satisfies

ϕ_{k}^{''} + (c^{2} k^{2} - \frac{2}{τ ^{2}}) ϕ_{k} = 0.

Conformal-time quadratic action for an inflaton perturbation ( $S^{(2)} = \frac{1}{2} \int d τ d^{3} x {(f^{'})^{2} - (\nabla f)^{2} + (a^{''} / a - a^{2} V^{''}) f^{2}}$ )

Neglecting metric perturbations, the rescaling

f = a δ ϕ

removes the expansion friction from the scalar perturbation action after a boundary term is discarded. Its canonical momentum is

f^{'}

, and each real Fourier oscillator has time-dependent squared frequency

k^{2} + a^{2} V^{''} - a^{''} / a

. The oscillator can become inverted on superhorizon scales; the canonical commutation relations still apply.

Bunch-Davies vacuum

The Bunch-Davies vacuum selects the mode that approaches the positive-frequency ground-state mode of angular frequency

c k

in the far past

τ \to - \infty

.

Cosmological horizon exit

A Fourier mode exits the Hubble horizon during inflation when

k = a H

. On superhorizon scales

k ≪ a H

, the growing mode of a sufficiently light scalar perturbation approaches a constant.

Superhorizon scale

A cosmological mode is on a superhorizon scale when its physical wavenumber is smaller than the Hubble expansion rate,

k / a ≪ H

, equivalently

k ≪ a H

.

Scale-invariant inflationary power spectrum

A Fourier-space variance

P_{k}

in three dimensions is scale invariant when the dimensionless power per logarithmic wavenumber interval,

Δ^{2} (k) = \frac{k ^{3} P _{k}}{2 π ^{2}},

is independent of

k

. For a massless inflaton mode at late times,

P_{k} ⟶ \frac{ℏ H _{inf}^{2}}{2 c ^{3} k ^{3}},

so

Δ^{2} (k)

is constant.

Infrared qualification of a scale-invariant covariance ( $C (r) - C (r_{0}) = - \frac{A}{2 π ^{2}} lo g (r / r_{0})$ )

An exact

P (k) = A / k^{3}

has divergent unsubtracted point variance. Its finite isotropic covariance difference is the displayed logarithm, unchanged by simultaneous dilation of

r, r_{0}

. Thus the nonzero-mode scaling statement has a natural subtraction interpretation. Exact dilation invariance of a continuous finite-variance isotropic covariance itself would instead force a constant covariance, corresponding only to a zero-mode spectral measure. Cutoffs or a zero-mode subtraction must accompany the nontrivial spectrum.

Equal-time two-point function of a de Sitter scalar ( $P_{δ ϕ} (k, τ) = H^{2} (1 + k^{2} τ^{2}) / (2 k^{3})$ )

For a light scalar in the Bunch-Davies vacuum with metric perturbations neglected, the mode

e^{- ik τ} (1 - i / (k τ)) / 2 k

gives the displayed equal-time spectrum. The dimensionless power tends to

H^{2} / (4 π^{2})

on superhorizon scales. The exact massless infinite-volume correlation has an infrared divergence; a finite-volume or finite-duration infrared regulator, and smearing for coincident products, makes the correlators well defined.

Primordial scalar amplitude ( $A_{s}$ )

At leading slow-roll order, the dimensionless primordial curvature-perturbation amplitude at horizon exit is

A_{s} = \frac{H ^{2}}{8 π ^{2} ϵ _{V} M _{Pl}^{2}} = \frac{V}{24 π ^{2} M _{Pl}^{4} ϵ _{V}} .

Scalar spectral index ( $n_{s}$ )

The scalar spectral index describes the departure of the primordial scalar spectrum from scale invariance. To first order in potential slow-roll parameters,

n_{s} - 1 ≃ - 6 ϵ_{V} + 2 η_{V} .

Scalar spectral index in potential slow-roll parameters ( $n_{s} ≃ 1 - 6 ϵ_{V} + 2 η_{V}$ )

The leading scalar spectrum is proportional to

V^{3} / V^{'2}

. Along a slow-roll trajectory,

d ϕ / d ln a ≃ - M_{pl}^{2} V^{'} / V

and

d ln k / d ln a = 1 - ϵ_{H}

. Differentiating at horizon exit and keeping first order gives

n_{s} - 1 = - 6 ϵ_{V} + 2 η_{V}

. The derivative-factor correction contributes at second order.

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The term "inflaton" refers to a hypothetical field or particle that is proposed to be responsible for the rapid expansion of the universe during a phase of cosmic inflation. This inflationary period is thought to have occurred in the very early universe, approximately 10^-36 to 10^-32 seconds after the Big Bang. According to inflationary theory, the universe underwent an exponential expansion that smoothed out any initial irregularities and set the stage for the large-scale structure we observe today.

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