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A
Lagrangian Density
The starting point in field theory is the Lagrangian density L(ϕi , ∂ µ ϕi ). The simplest case is
the scalar field ϕ, for which
1
(1)
L = − ∂µ ϕ ∂ µ ϕ − V (ϕ) .
2
Here V (ϕ) is the potential of the field. If
1
V (ϕ) = m2 ϕ2 ,
2
(2)
the particle corresponding to the field ϕ will have mass m. In general, the mass of the particle
is given by m2 = V ′′ (ϕ). We write
V ′ (ϕ) ≡
dV
dϕ
and
V ′′ (ϕ) ≡
d2 V
.
dϕ2
(3)
The particles corresponding to scalar fields are spin-0 bosons. Spin- 12 particles correspond
to spinor fields and spin-1 particles to vector fields.
The field equation is obtained from the Lagrangian by minimizing (or extremizing) the action
Z
Ld4 x ,
(4)
which leads to the Euler–Lagrange equation
∂L
∂L
− ∂µ
= 0.
∂ϕi (x)
∂[∂µ ϕi (x)]
(5)
For the above scalar field we get the field equation
∂µ ∂ µ ϕ − V ′ (ϕ) = 0 .
(6)
For a massless noninteracting field, V (ϕ) = 0, and the field equation is just the wave equation
∂µ ∂ µ ϕ = −ϕ̈ + ∇2 ϕ = 0 .
(7)
The Lagrangian also gives us the energy tensor
T µν = −
∂L
∂ ν ϕ + gµν L .
∂(∂µ ϕ)
(8)
For the scalar field
1
∂ρ ϕ∂ ρ ϕ + V (ϕ) .
2
In particular, the energy density and pressure of a scalar field are
T µν = ∂ µ ϕ∂ ν ϕ − gµν
1 2 1
ϕ̇ + ∇ϕ2 + V (ϕ)
2
2
1 11
1
1
22
p =
T + T + T 33 = ϕ̇2 − ∇ϕ2 − V (ϕ) .
3
2
6
ρ = T 00 =
(We are in Minkowski space, so that g µν = diag(−1, 1, 1, 1)).
1
(9)
(10)
(11)
B GENERATION OF PRIMORDIAL PERTURBATIONS FROM INFLATION
B
2
Generation of Primordial Perturbations from Inflation
Subhorizon scales during inflation are microscopic and therefore quantum effects are important.
Thus we should study the behavior of the inflaton field using quantum field theory. To warm
up we first consider quantum field theory of ascalar field in Minkowski space.
B.1
Vacuum fluctuations in Minkowski space
The field equation for a massive free (i.e. V (ϕ) = 12 m2 ϕ2 ) real scalar field in Minkowski space is
ϕ̈ − ∇2 ϕ + m2 ϕ = 0 ,
(12)
ϕ̈k + Ek2 ϕk = 0 ,
(13)
or
where Ek2 = k2 + m2 , for Fourier components. We recognize Eq. (13) as the equation for
a harmonic oscillator. Thus each Fourier component of the field behaves as an independent
harmonic oscillator.
In the quantum mechanical treatment of the harmonic oscillator one introduces the creation
and annihilation operators, which raise and lower the energy state of the system. We can do
the same here.
Now we have a different pair of creation and annihilation operators â†k , âk for each Fourier
mode k. We denote the ground state of the system by |0i, and call it the vacuum. As discussed
earlier, particles are quanta of the oscillations of the field. The vacuum is a state with no
particles. Operating on the vacuum with the creation operator â†k , we add one quantum with
momentum k and energy Ek to the system, i.e., we create one particle. We denote this state
with one particle, whose momentum is k by |1k i. Thus
â†k |0i = |1k i .
(14)
This particle has a well-defined momentum k, and therefore it is completely unlocalized (Heisenberg’s uncertainty principle). The annihilation operator acting on the vacuum gives zero, i.e.,
not the vacuum state but the zero element of Hilbert space (the space of all quantum states),
âk |0i = 0 .
(15)
We denote the hermitian conjugate of the vacuum state by h0|. Thus
h0|âk = h1k |
and
h0|â†k = 0 .
(16)
The commutation relations of the creation and annihilation operators are
[â†k , â†k′ ] = [âk , âk′ ] = 0,
[âk , â†k′ ] = δkk′ .
(17)
When going from classical physics to quantum physics, classical observables are replaced by
operators. One can then calculate expectation values for these observables using the operators.
Here the classical observable
X
ϕ(t, x) =
ϕk (t)eik·x
(18)
is replaced by the field operator
ϕ̂(t, x) =
X
ϕ̂k (t)eik·x
(19)
B GENERATION OF PRIMORDIAL PERTURBATIONS FROM INFLATION
where1
3
ϕ̂k (t) = wk (t)âk + wk∗ (t)â†−k
(20)
1
e−iEk t
2Ek
(21)
and
wk (t) = V −1/2 √
is the mode function, a normalized solution of the field equation (13). We are using the Heisenberg picture, i.e. we have time-dependent operators; the quantum states are time-independent.
Classically the ground state would be one where ϕ = const. = 0, but we know from the
quantum mechanics of a harmonic oscillator, that there are oscillations even in the ground
state. Likewise, there are fluctuations of the scalar field, vacuum fluctuations, even in the
vacuum state.
We shall now calculate the power spectrum of these vacuum fluctuations. The power spectrum
is defined as the expectation value
Pϕ (k) = V
k3
h|ϕk |2 i
2π 2
(22)
and it gives the variance of ϕ(x) as
2
hϕ(x) i =
Z
∞
0
dk
Pϕ (k) .
k
(23)
For the vacuum state |0i the expectation value of |ϕk |2 is
h0|ϕ̂k ϕ̂†k |0i =
|wk |2 h0|âk â†k |0i + wk2 h0|âk â−k |0i + (wk∗ )2 h0|â†−k â†k |0i + |wk |2 h0|â†−k â−k |0i
= |wk |2 h1k |1k i = |wk |2
(24)
since all but the first term give 0, and our states are normalized so that h1k |1k′ i = δkk′ . From
Eq. (21) we have that |wk |2 = 1/(2V Ek ). Our main result is that
Pϕ (k) = V
k3
|wk |2
2π 2
(25)
for vacuum fluctuations, which we shall now apply to inflation, where the mode functions wk (t)
are different.
B.2
Vacuum fluctuations during inflation
During inflation the field equation (for inflaton perturbations) is, from Eq. (??),
"
#
a0 k 2
′′
δϕ̈k + 3Hδϕ̇k +
+ V (ϕ̄) δϕk = 0 .
a
(26)
There are oscillations only in the perturbation δϕ, the background ϕ̄ is homogeneous and evolving slowly in time. For the particle point of view, the background solution represents the
vacuum,2 i.e., particles are quanta of oscillations around that value.
1
We skip the detailed derivation of the field operator, which belongs to a course of quantum field theory. See
e.g. Peskin & Schroeder, section 2.3 (note different normalizations of operators and states, related to doing Fourier
integrals rather than sums and considerations of Lorentz invariance.)
2
This is not the vacuum state in the sense of being the ground state of the system. The true ground state has
ϕ̄ at the minimum of the potential. However there are no particles related to the background evolution ϕ̄(t).
B GENERATION OF PRIMORDIAL PERTURBATIONS FROM INFLATION
4
After making the approximations H = const. and
V ′′
= 3η ≈ 0
H2
we found that the two independent solutions for δϕk (t) are
ia0 k
a0 k
−1/2 H
√
wk (t) = V
i+
exp
aH
aH
2k3
(27)
(28)
and its complex
conjugate wk∗ (t), where the time dependence is in a = a(t) ∝ eHt . The factor
√
V −1/2 H/ 2k3 is here for normalization purposes (V = L3 being the reference volume, not the
inflaton potential).
When the scale k is well inside the horizon, a0 k ≫ aH, δϕk (t) oscillates rapidly compared
to the Hubble time H −1 . If we consider distance and time scales much smaller than the Hubble
scale, spacetime curvature does not matter and things should behave like in Minkowski space.
Considering Eq. (28) in this limit, one finds (exercise) that wk (t) indeed becomes equal to the
Minkowski space mode function, Eq. (21). (We cleverly chose the normalization in Eq. (28) so
that the normalizations would agree.) Therefore the wk (t) of Eq. (28) is our mode function. We
can use it to follow the evolution of the mode functions as the scale approaches and exits the
horizon.
The field operator for the inflaton perturbations is
δϕ̂k (t) = wk (t)âk + wk∗ (t)â†−k ,
(29)
and the power spectrum of inflaton fluctuations is
Pϕ (k) = V
k3
|wk |2 .
2π 2
(30)
Well before horizon exit, a0 k ≫ aH, observed during timescales ≪ H −1 , the field operator
δϕ̂k (t) becomes the Minkowski space field operator and we have standard vacuum fluctuations
in δϕ.
Well after horizon exit, a0 k ≪ aH, the mode function becomes a constant
iH
,
wk (t) → V −1/2 √
2k3
the vacuum fluctuations “freeze”, and the power spectrum acquires the constant value
2
H
k3
2
.
Pϕ (k) = V 2 |wk | =
2π
2π
(31)
(32)
After this we can, for our purposes, forget further quantum effects and treat the later evolution of the inflaton field, both the background and the perturbation, classically. The effect of the
vacuum fluctuations was to produce “out of nothing” the perturbations δϕk . We can’t predict
their individual values; their production from quantum fluctuations is a random process. We
can only calculate their statistical properties. Closer investigation reveals that this is a Gaussian
random process. All δϕk acquire their values as independent random variables (except for the
reality condition δϕ−k = δϕ∗k ) with a Gaussian probability distribution. Thus all statistical
information is contained in the power spectrum Pϕ (k).
The result (32) was obtained treating H as a constant. However, over long time scales,
H does change. The main purpose of our discussion was to follow the inflaton perturbations
through the horizon exit. After the perturbation is well outside the horizon, we switch to other
B GENERATION OF PRIMORDIAL PERTURBATIONS FROM INFLATION
5
variables, namely the curvature perturbation Rk , which, unlike δϕk , remains constant outside
the horizon, even though H changes. Therefore we have to use for each scale k a value of H
which is representative for the evolution of that particular scale through the horizon. That is,
we choose the value of H at horizon exit3 , so that aH = a0 k. Thus we write our power spectrum
result as
2
H
k3
2
,
(33)
Pϕ (k) = V 2 |wk | =
2π
2π aH=a0 k
to signify that the value of H for each k is to be taken at horizon exit of that particular scale.
Equation (33) is our main result from inflaton fluctuations.
3
One can do a more precise calculation, where one takes into account the evolution of H(t). The result is that
one gets a correction to the amplitude of PR (k), which is first order in slow-roll parameters, and a correction to
its spectral index n which is second order in the slow-roll parameters.