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4.3 The physics of Inflation
A key characteristic of inflation is that all physical quantities are slowly varying, despite fact that the space is expanding rapidly. Let us write the time derivative of the comoving Hubble radius as
(4.33) 𝑑/𝑑𝑡 (𝑎𝐻)^{1} = (𝑎̇𝐻 + 𝑎Ḣ)/(𝑎𝐻)^{2} = 1/𝑎(1  𝜀),
where we have introduced the slowroll parameter
(4.34) 𝜀 ¡Õ Ḣ/𝐻^{2} = 𝑑 ln 𝐻/𝑑𝑁,
with 𝑑𝑁 ¡Õ 𝑑 ln 𝑎 = 𝐻𝑑𝑡. This shows that a shrinking Hubble radius, ¡Ó_{𝑡}(𝑎𝐻) < 0, is associated with 𝜀 < 1. In other words, inflation occurs if the fractional change of the Hubble parameter, ∆ln 𝐻 = ∆𝐻/𝐻, per 𝑒folding of expansion, ∆𝑁, is small. Moreover, as we will see in Chapter 8, the near scaleinvariance of the observed fluctuations in fact requires that 𝜀 ¡ì 1. In the limit 𝜀 ¡æ 0, the dynamics becomes timetranslation invariant, 𝐻 = const, and the spacetime is de Sitter space
(4.35) 𝑑𝑠^{2} = 𝑑𝑡^{2} + 𝑒^{2𝐻𝑡}𝑑𝐱^{2}.
Because inflation has to end, so that timetranslation symmetry must be broken and the spacetime must ferive from a perfect de Sitter space. However, for small , but finite 𝜀 ¡Á 0, the line element (4.35) is still a good approximation to the inflationary background. This is why inflation is also often referred to as a quaside Sitter period.
Finally, we want inflation to last for a sufficient long time (usually at least 40 to 60 𝑒folds), which requires that 𝜀 remains small for a sufficiently large number of Hubble times. This condition is measured by a second slowroll parameter^{7}
(4.36) 𝜅 ¡Õ 𝑑(ln 𝜀)/𝑑𝑁 = 𝜀̇/𝐻𝜀.
For ∣𝜅∣ < 1, the fractional change of 𝜀 per 𝑒fold is small and inflation persists. In the next section, we will duscuss what microscopic physics can lead to the conditions 𝜀 < 1 and ∣𝜅∣ < 1,
Exercise 4.2 Consider a perfect fluid with density 𝜌 and pressure 𝑃. Using the Friedmann and continuity equations, show that
(4.378) 𝜀 = 3/2 (1 + 𝑃/𝜌), 𝜀 = 1/2 𝑑(ln 𝜌)/𝑑(ln 𝑎).
This shows that inflation occurs if the pressure is negative and the density is nearby constant. Conventional matter sources all dilute with expansion, so we need to look for something more unusual.
[Solution] From the definition of the slowroll parameter, we have
(a) 𝜀 ¡Õ Ḣ/𝐻^{2} = 1/𝐻^{2} 𝑑/𝑑𝑡(𝑎̇/𝑎) = 1/𝐻^{2} (𝑎̇^{2}/𝑎^{2} + (𝑑𝑎̇/𝑑𝑡)/𝑎 = 1  (𝑑𝑎̇/𝑑𝑡)/𝑎 1/𝐻^{2}.
Using the Friedmann equations for a flat universe
(b) 𝐻^{2} = 8¥ð𝐺/3 𝜌, (𝑑𝑎̇/𝑑𝑡)/𝑎 = 4¥ð𝐺/3 (1 + 𝑃/𝜌), ¢¡ (𝑑𝑎̇/𝑑𝑡)/𝑎 = 𝐻^{2}/2 (1 + 3𝑃/𝜌),
(c) 𝜀 = 1  [𝐻^{2}/2 (1 + 3𝑃/𝜌)] 1/𝐻^{2} = 3/2 (1 + 𝑃/𝜌).
Using the continuity equation (2.106), 𝜌̇ + 3𝐻(𝜌 + 𝑃) = 0 with 𝑐 ¡Õ 1, we can write
(d) 𝜀 = 3/2 [1 + (𝜌̇/3𝐻  𝜌)/𝜌] = 3/2 (1/3𝐻 𝜌̇/𝜌) = 1/2𝐻 𝜌̇/𝜌 = 1/2𝐻 𝑑(ln 𝜌)/𝑑𝑡 = 1/2 𝑑(ln 𝜌)/𝑑(ln 𝑎). ▮
4.3.1 Scalar Field Dynamics
The simplest models of inflation implement the timedependent dynamics during inflation in terms of the evolution of a scalar field, 𝜙(𝑡, 𝐱), called the inflaton. [𝑡: time, 𝐱: position] Associated with each value is a potential energy density 𝑉(𝜙) (see Fig. 4.8) When the field changes with time then it it also carries a kinetic energy density 1/2 ᾡ^{2}. If the energy associated with the scalar field dominates the universe, then it sources the evolution of the FRW background. We want to determine under which conditions this can derive an inflationary expansion. [RE Wikipedia inflaton]
Let us begin with a scalar field in Minkowski space. Its action is [RE Wikipedia Scalar field theory]
(4.39) 𝑆 = ¡ò 𝑑𝑡 𝑑^{3}𝑥 [1/2 ᾡ^{2}  1/2 (¡Ô𝜙)^{2}  𝑉(𝜙)],
where we have also included the gradient energy, 1/2 (¡Ô𝜙)^{2}, associated with a spatially varying field. To determine the equation of the scalar field, we consider the variation 𝜙 ¡æ 𝜙 + 𝛿𝜙. Under this variation, the action changes as
(4.40) 𝛿𝑆 = ¡ò 𝑑𝑡 𝑑^{3}𝑥 [𝜙̇𝛿𝜙̇  ¡Ô𝜙 ⋅ 𝛿¡Ô𝜙  𝑑𝑉/𝑑𝜙 𝛿𝜙]
{= ¡ò 𝑑𝑡 𝑑^{3}𝑥 [𝑑/𝑑𝑡 𝜙̇ 𝛿𝜙 (¡Ô ⋅ ¡Ô𝜙 𝛿𝜙)  𝑑𝑉/𝑑𝜙 𝛿𝜙]} [Notice that ¡Ô is used instead of 𝑑/𝑑𝑡 in the second term for applying Hamilton's principle.]
(4.41) = ¡ò 𝑑𝑡 𝑑^{3}𝑥 [𝑑𝜙̇/𝑑𝑡 + ¡Ô^{2}𝜙  𝑑𝑉/𝑑𝜙]𝛿𝜙
where we have integrated by parts and dropped a boundary term. If the variation around the classical field configuration, then the principle of leastaction states that 𝛿𝑆 = 0. For this to be valid for an arbitrary field variation 𝑑𝜙, [RE Wikipedia Hamilton's principle: 𝛿𝑆 = ¡ò_{𝑡1 }^{𝑡2} 𝜀 ⋅ (¡Ó𝐿/¡Ó𝑝  𝑑/𝑑𝑡 ¡Ó𝐿/¡Ó𝑝̇) 𝑑𝑡, (using 𝑝 instead of 𝑞). When 𝑆 is at minimum or maximum or saddle point, 𝛿𝑆 = 0 or (¡Ó𝐿/¡Ó𝑝  𝑑/𝑑𝑡 ¡Ó𝐿/¡Ó𝑝̇) = 0.] the expression in the square brackets in (4.41) must vanish:
(4.42) 𝑑𝜙̇/𝑑𝑡  ¡Ô^{2}𝜙 = 𝑑𝑉/𝑑𝜙
This is the KleinGordon equation.
It is straightforward to extend this analysis to the evolution in an expanding FRW spacetime. For simplicity we will ignore spatial curvature.Physical coordinates are related to comoving coordinates by the scale factor, 𝑎(𝑡)𝐱.Taking this into account, it is easy to guess that the generalization of the action (4.49) to an FRW background is
(4.43) 𝑆 = ¡ò 𝑑𝑡 𝑑^{3}𝑥 𝑎^{3}(𝑡) [1/2 ᾡ^{2}  1/2𝑎^{2}(𝑡) (¡Ô𝜙)^{2}  𝑉(𝜙)],
We are interested in the evolution of a homogeneous field configurations, 𝜙 = 𝜙(𝑡), in which case the action reduces to
(4.44) 𝑆 = ¡ò 𝑑𝑡 𝑑^{3}𝑥 𝑎^{3}(𝑡) [1/2 ᾡ^{2}  𝑉(𝜙)],
Performing the same variation of the action as before, we find
(4.45) 𝛿𝑆 = ¡ò 𝑑𝑡 𝑑^{3}𝑥 𝑎^{3}(𝑡) [𝜙̇𝛿𝜙̇  𝑑𝑉/𝑑𝜙 𝛿𝜙].
(4.46) = ¡ò 𝑑𝑡 𝑑^{3}𝑥 [𝑑/𝑑𝑡 (𝑎^{3}𝜙̇)  𝑎^{3} 𝑑𝑉/𝑑𝜙]𝛿𝜙
and the principle of least action leads to the following KleinGordon equation
(4.47) 𝑑𝜙̇/𝑑𝑡 + 3𝐻𝜙̇ = 𝑑𝑉/𝑑𝜙.
The expansion has introduced one new feature, the socalled Hubble frictionassociated with the term 3𝐻ᾡ. This friction will play a crucial role in the inflationary dynamics.
Let us now assume that this scalar field dominates the universe and determine its effect on the expansion.Given the form of the action in (4.44), it is natural to guess that the energy density is the sum of kinetic and potential energy densities
(4.48) 𝜌_{𝜙} = 1/2 ᾡ^{2} + 𝑉(𝜙).
Taking the time derivative of the energy density, we find
(4.49) 𝜌̇_{𝜙} = (𝑑𝜙̇/𝑑𝑡 + 𝑑𝑉/𝑑𝜙)𝜙̇ = 3𝐻𝜙̇^{2},
where the KleinGordon equation (4.47) was used. Comparing this to the continuity equation, ῤ_{𝜙} = 3𝐻(𝜌_{𝜙} + 𝑃_{𝜙}) we infer the pressure induced by the field is
(4.50) 𝑃_{𝜙} = 1/2 ᾡ^{2}  𝑉(𝜙).
This pressure determines the acceleration of the expansion 𝑑𝑎̇/𝑑𝑡 ¡ð (𝜌_{𝜙} + 3𝑃_{𝜙}). We see that the density and pressure are in general not related by a constant equation of state as for a perfect fluid. Notice that if the kinetic energy of the inflation is much smaller than its potential energy, then 𝑃_{𝜙} ≈ 𝜌_{𝜙}. The inflationary potential then acts like a temporary cosmological constant, sourcing a period of exponential expansion.
Exercise 4.3 The action of scalar field in a general curved spacetime is
(4.51) 𝑆 = ¡ò 𝑑^{4}𝑥¡î(𝑔) [1/2 𝑔^{𝜇𝜈}¡Ó_{𝜇}𝜙¡Ó_{𝜈}𝜙  𝑉(𝜙)],
where 𝑔 ¡Õ det 𝑔_{𝜇𝜈}, It is easy to check that this reduces to (4.43) for a flat FRW metric. Under a variation of the (inverse) metric, 𝑔^{𝜇𝜈} ¡æ 𝑔^{𝜇𝜈} + 𝛿𝑔^{𝜇𝜈}, the action changes
(4.52) 𝛿𝑆 = 1/2 ¡ò 𝑑^{4}𝑥¡î(𝑔) 𝛵_{𝜇𝜈}𝛿𝑔^{𝜇𝜈},
where 𝛵_{𝜇𝜈} is the energymomentum tensor.
Show that the energymomentum tensor for a scalar field is
(4.53) 𝛵_{𝜇𝜈} = ¡Ó_{𝜇}𝜙¡Ó_{𝜈}𝜙  𝑔^{𝜇𝜈}(1/2 𝑔^{𝛼𝛽}¡Ó_{𝛼}𝜙¡Ó_{𝛽}𝜙 + 𝑉(𝜙)).
and confirm that this leads to the expressions for 𝜌_{𝜙} and 𝑃_{𝜙} found above. Hints: We have to use that 𝛿¡î(𝑔) = 1/2 ¡î(𝑔) (𝑔_{𝜇𝜈}𝛿𝑔^{𝜇𝜈}).
Show that the conservation of the energymomentum tensor, ¡Ô_{𝜇}𝛵^{𝜇𝜈} = 0, implies the KleinGordon equation (4.47).
[Solution] Under a variation of the inverse metric, the given action changes as follow
(a) 𝛿𝑆 = ¡ò 𝑑^{4}𝑥 [𝛿¡î(𝑔) (1/2 𝑔^{𝛼𝛽}¡Ó_{𝛼}𝜙¡Ó_{𝛽}𝜙  𝑉(𝜙))  ¡î(𝑔) 1/2𝛿𝑔^{𝜇𝜈}¡Ó_{𝜇}𝜙¡Ó_{𝜈}𝜙] = 1/2 ¡ò 𝑑^{4}𝑥 ¡î(𝑔) [𝑔_{𝜇𝜈}(1/2 𝑔^{𝛼𝛽}¡Ó_{𝛼}𝜙¡Ó_{𝛽}𝜙 + 𝑉(𝜙))  ¡Ó_{𝜇}𝜙¡Ó_{𝜈}𝜙]𝛿𝑔^{𝜇𝜈},
where we used 𝛿¡î(𝑔) = 1/2 ¡î(𝑔) (𝑔_{𝜇𝜈}𝛿𝑔^{𝜇𝜈}). Comparing this (4.52) we get
(b) 𝛵_{𝜇𝜈} = ¡Ó_{𝜇}𝜙¡Ó_{𝜈}𝜙  𝑔_{𝜇𝜈}(1/2 𝑔^{𝛼𝛽}¡Ó_{𝛼}𝜙¡Ó_{𝛽}𝜙 + 𝑉(𝜙).
Raising one index, this yield
(c) 𝛵_{𝜇}_{𝜈} = 𝑔^{𝜇𝛼}¡Ó_{𝛼}𝜙¡Ó_{𝛽}𝜙  𝛿_{𝜇}_{𝜈}(1/2 𝑔^{𝛼𝛽}¡Ó_{𝛼}𝜙¡Ó_{𝛽}𝜙 + 𝑉(𝜙).
For a homogeneous field configuration 𝜙(𝑡), this takes the form of a perfect fluid
(d) 𝛵_{𝜇}_{𝜈} = diag (𝜌_{𝜙}, 𝑃_{𝜙}, 𝑃_{𝜙}, 𝑃_{𝜙}).
(e) 𝜌_{𝜙} = 𝛵_{0}_{0} = 𝑔^{00}¡Ó_{0}𝜙¡Ó_{0}𝜙 + 𝛿_{0}_{0}(1/2 𝑔^{00}¡Ó_{0}𝜙¡Ó_{0}𝜙 + 𝑉(𝜙) = 1/2 ᾡ^{2} + 𝑉(𝜙). [with 𝑔^{00} = 1, ¡Ó_{0} ¡Õ ¡Ó/¡Ó(𝑡)]
(f) 𝑃_{𝜙} = 1/3 𝛵_{𝑖}_{𝑖} = 1/3 𝛿_{𝑖}_{𝑖} (1/2 𝑔^{00}¡Ó_{0}𝜙¡Ó_{0}𝜙 + 𝑉(𝜙)) = 1/2 ᾡ^{2}  𝑉(𝜙).
This confirms the expressions derived in the text. We can now compute the covariant derivative of (4.53 or c)
(g) ¡Ô_{𝜇}𝛵^{𝜇𝜈} = 𝑔^{𝜇𝛼}𝑔^{𝜈𝛽}¡Ô_{𝜇}(¡Ó_{𝛼}𝜙¡Ó_{𝛽}𝜙)  𝑔^{𝜇𝜈}¡Ô_{𝜇}(1/2 𝑔^{𝛼𝛽}¡Ó_{𝛼}𝜙¡Ó_{𝛽}𝜙 + 𝑉(𝜙) = 𝑔^{𝜇𝛼}𝑔^{𝜈𝛽}[¡Ó_{𝜇}(¡Ó_{𝛼}𝜙¡Ó_{𝛽}𝜙)  (𝛤^{𝜌}_{𝜇𝛼}¡Ó_{𝛽}𝜙 + 𝛤^{𝜌}_{𝜇𝛽}¡Ó_{𝛼}𝜙)¡Ó_{𝜌}𝜙]  𝑔^{𝜇𝜈}(1/2 𝑔^{𝛼𝛽}¡Ó_{𝛼}𝜙¡Ó_{𝛽}𝜙 + 𝑉(𝜙)).
Using that 𝑔^{0𝑖} = 0 and ¡Ó_{𝑖}𝜙 = 0, this expression simplifies to
(h) ¡Ô_{𝜇}𝛵^{𝜇𝜈} = 𝛿^{𝜈0}¡Ó_{0}ᾡ^{2} + 𝛤^{0}_{𝜇𝛼}𝑔^{𝜇𝛼}𝛿^{𝜈0}ᾡ^{2} + 𝛤^{0}_{0𝛽}𝑔^{𝜈𝛽}ᾡ^{2} + 𝛿^{0𝜈}¡Ó_{0}(1/2 ᾡ^{2} + 𝑉(𝜙)).
Recall that the Christoffel symbols with two times indies vanish. This implies that 𝛤^{0}_{0𝛽} = 0 and 𝛤^{0}_{𝜇𝛼} = 𝛤^{0}_{𝑖𝑗}𝑔^{𝑖𝑗} = 𝐻𝑔_{𝑖𝑗}𝑔^{𝑖𝑗} = 3𝐻, so that
(i) ¡Ô_{𝜇}𝛵^{𝜇𝜈} = 𝛿^{𝜈0}[2𝜙̇ 𝑑𝜙̇/𝑑𝑡 + 3𝐻𝜙̇^{2} + (𝜙̇𝑑𝜙̇/𝑑𝑡 + 𝜙̇ 𝑑𝑉/𝑑𝜙)] = 𝛿^{𝜈0}𝜙̇(𝑑𝜙̇/𝑑𝑡 + 3𝐻ᾡ + 𝑑𝑉/𝑑𝜙).
The conservation of the energymomentum tensor, ¡Ô_{𝜇}𝛵^{𝜇𝜈} = 0, then implies the KleinGordon equation
(j) 𝑑𝜙̇/𝑑𝑡 + 3𝐻𝜙̇ + 𝑑𝑉/𝑑𝜙 = 0. ▮
4.3.2 SlowRoll Inflation
The dynamics during inflation is then determined by a combination of the Friedmann and Klein Gordon equations
(4.54) 𝐻^{2} = 1/3𝑀_{𝑃𝑙}^{2} [1/2 ᾡ^{2} + 𝑉], [RE (2.136) and (4.48)]
(4.55) 𝑑𝜙̇/𝑑𝑡 + 3𝐻𝜙̇ = 𝑑𝑉/𝑑𝜙
These equations are coupled. The energy stored in the field determines the Hubble rate, [𝐻], which in turn induceds friction and hence affects the evolution of the field. We would like to determine the precise conditions under which this feedback leads to inflation.
A slowly rolling field
First, (4.54) and (4.55) can be combined into an expression for the evolution of the Hubble parameter
(4.56) Ḣ = 1/2 𝜙̇^{2}/𝑀_{𝑃𝑙}^{2}. [from (4.54), 2𝐻Ḣ = 1/3𝑀_{𝑃𝑙}^{2} [𝜙̇ 𝑑𝜙̇/𝑑𝑡 + 𝑑𝑉/𝑑𝜙 𝜙̇] ¡æ Ḣ = 1/6𝐻Ḣ𝑀_{𝑃𝑙}^{2} [𝜙̇ 𝑑𝜙̇/𝑑𝑡  (𝑑𝜙̇/𝑑𝑡 + 3𝐻𝜙̇)𝜙̇] = 1/2 𝜙̇^{2}/𝑀_{𝑃𝑙}^{2}.]
Talking the ratio of (4.56) and (4.54), we then find
(4.57) 𝜀 ¡Õ Ḣ/𝐻^{2} = (1/2 ᾡ^{2})/𝑀_{𝑃𝑙}^{2}𝐻^{2} = (3/2 ᾡ^{2})/(1/2 𝜙̇^{2} + 𝑉).
Inflation (𝜀 ¡ì 1) therefore occurs if the kinetic energy density, 1/2 𝜙̇^{2}, only makes a small contribution to the total energy density, 𝜌_{𝜙} = 1/2 ᾡ^{2} + 𝑉. For obvious reasons, this situation is called slowroll inflation.
In order for slowroll behavior to persist, the acceleration of the scalar field also has to be small. To assess this, it is useful to define the dimensionless acceleration per Hubble time
(4.58) 𝛿 ¡Õ (𝑑𝜙̇/𝑑𝑡)/𝐻𝜙̇.
When 𝛿 is small, the friction term in (4.55) dominates and the inflation speed is determined by the slope of the potential. As long as the inflation kinetic energy stays subdominant and the inflationary expansion continues. To see this more explicitly, we take the time derivative of (4.57),
(4.59) 𝜀̇ = 𝜙̇ 𝑑𝜙̇/𝑑𝑡/𝑀_{𝑃𝑙}^{2}𝐻^{2}  ᾡ^{2}Ḣ/𝑀_{𝑃𝑙}^{2}𝐻^{3},
and substitute it into (4.36):
(4.60) 𝜅 = 𝜀̇/𝐻𝜀 = 2 (𝑑𝜙̇/𝑑𝑡)/𝐻𝜙̇  2 Ḣ/𝐻^{2} = 2(𝜀  𝛿).
This shows that {𝜀,∣𝛿∣} ¡ì 1 implies {𝜀,∣𝜅∣} ¡ì 1. If both the speed and the acceleration of the inflation field are small, then the inflationary expansion will last for a long time.
Slowroll approximation
Now, we will use these condition to simplify the equations of motion. This is called the slowroll approximation.
First, we note that 𝜀 ¡ì 1 implies ᾡ^{2} ¡ì 𝑉, which leads to the following simplification of Friedmann equation (4.54)
(4.61) 𝐻^{2} ≈ 𝑉/3𝑀_{𝑃𝑙}^{2}.
Next, we see that the condition ∣𝛿∣ ¡ì 𝑉 simplifies the KleinGordon equation (4.55) to
(4.62) 3𝐻𝜙̇ ≈ 𝑉_{,𝜙},
where 𝑉_{,𝜙} ¡Õ 𝑑𝑉/𝑑𝜙. This provides a simple relationship between the slope of the potential and the speed of the inflation. Finally, substituting (4.61) and (4.62) into (4.57) gives
(4.63) 𝜀 = (1/2 ᾡ^{2})/3𝑀_{𝑃𝑙}^{2}𝐻^{2} ≈ 𝑀_{𝑃𝑙}^{2}/2 (𝑉_{,𝜙}/𝑉),
which express the parameter 𝜀 purely in terms of the potential.
To evaluate the parameter 𝛿 in (4.58), in the slowroll approximation, we take the time derivative of (4.62), 3Ḣᾡ + 3𝐻ᾥ = 𝑉_{,𝜙𝜙}ᾡ. This leads to
(4.64) 𝜂 ¡Õ 𝛿 + 𝜀 = ᾥ/𝐻ᾡ  Ḣ/𝐻^{2} ≈ 𝑀_{𝑃𝑙}^{2} 𝑉_{,𝜙𝜙}/𝑉.
Hence, a convenient way to judge whether a given potential 𝑉(𝜙) can leads to slowroll inflation is to compute the potential slowroll parameters
(4.65) 𝜀_{𝑉} ¡Õ 𝑀_{𝑃𝑙}^{2}/2 (𝑉_{,𝜙}/𝑉)^{2}, 𝜂_{𝑉} ¡Õ 𝑀_{𝑃𝑙}^{2} 𝑉_{,𝜙𝜙}/𝑉.
Successful inflation occurs when these parameters are much smaller than unity.
Exercise 4.4 Show that in the slowroll regime, the potential slowroll parameters and the Hubble slowroll parameters are related as follows:
(4.66) 𝜀_{𝑉} ≈ 𝜀 and 𝜂_{𝑉} ≈ 2𝜀  1/𝜅.
[Solution] In the slowroll approximation, the Friedmann and KleinGordon equations are
(a) 𝐻^{2} ≈ 𝑉/3𝑀_{𝑃𝑙}^{2}, 3𝐻𝜙̇ ≈ 𝑉_{,𝜙}
(b) 𝜀 = (1/2 𝜙̇^{2})/3𝑀_{𝑃𝑙}^{2}𝐻^{2} ≈ 𝑀_{𝑃𝑙}^{2}/2 (𝑉_{,𝜙}/𝑉), 𝜅 = 𝜀̇/𝐻𝜀 = 2 (𝑑𝜙̇/𝑑𝑡)/𝐻𝜙̇  2 Ḣ/𝐻^{2} = 2(𝜀  𝛿), 𝛿 ¡Õ (𝑑𝜙̇/𝑑𝑡)/𝐻ᾡ.
In the slowroll approximation we have
(c) 𝛿 + 𝜀 = (𝑑𝜙̇/𝑑𝑡)/𝐻𝜙̇  Ḣ/𝐻^{2} ≈ 𝑀_{𝑃𝑙}^{2} 𝑉_{,𝜙𝜙}/𝑉 ¡Õ 𝜂_{𝑉}
and hence
(d) 𝜀_{𝑉} ≈ 𝜀, 𝜂_{𝑉} ≈ 𝛿 + 𝜀 = (𝜅/2  𝜀) + 𝜀 = 2𝜀  1/2 𝜅. ▮
The total number of 𝑒foldings of accelerated expansion is
(4.67) 𝑁_{tot} ¡Õ ¡ò_{𝑎𝑖}^{𝑎𝑒} 𝑑 ln 𝑎 = ¡ò_{𝑡𝑖}^{𝑡𝑒} 𝐻(𝑡) 𝑑𝑡 = ¡ò_{𝜙𝑖}^{𝜙𝑒} 𝐻/𝜙̇ 𝑑𝜙,
where 𝑡_{𝑖} and 𝑡_{𝑒} are defined as the time when 𝜀(𝑡_{𝑖}) = 𝜀(𝑡_{𝑒}) ¡Õ 1. In the slowroll regime, we can use (4.63) to write the integral over the field space as
(4.68) 𝑁_{tot} ≈ ¡ò_{𝜙𝑖}^{𝜙𝑒} 1/¡î(2𝜀_{𝑉}) ∣𝑑𝜙∣/𝑀_{𝑃𝑙},
where 𝜙_{𝑖} and 𝜙_{𝑒} are the field values at the boundaries of the interval where 𝜀_{𝑉} < 1. As we have seen above, a solution to the horizon problem requires 𝑁_{tot} ≳ 60, which provide an important constraint on successful inflation models.
Case study: quadratic inflation
As an example, let us give the slowroll analysis of arguably the simplest model of inflation: singlefield inflation driven by a mass term
(4.69) 𝑉(𝜙) = 1/2 𝑚^{2}𝜙^{2}.
As we will see in Chapter 8, this models is ruled out by CMB observation, but it still provides a useful example to illustrate the mechanism of slowroll inflation. The slowroll parameter for the potential are
(4.70) 𝜀_{𝑉}(𝜙) = 𝜂_{𝑉}(𝜙) = 2 (𝑀_{𝑃𝑙}/𝜙)^{2}.
To satisfy the slowroll conditions {𝜀_{𝑉}, ∣𝜂_{𝑉}∣} < 1, we need to consider superPlanckian value for the inflation
(4.71) 𝜙 > ¡î2𝑀_{𝑃𝑙} ¡Õ 𝜙_{𝑒}.
Let the initial field value be 𝜙_{𝑖}, The 𝑒foldings of inflationary expansion are
(4.72) 𝑁_{tot} = ¡ò_{𝜙𝑖}^{𝜙𝑒} 𝑑𝜙/𝑀_{𝑃𝑙} 1/¡î(2𝜀_{𝑉}) = 𝜙^{2}/4𝑀_{𝑃𝑙}^{2}∣_{𝜙𝑒}^{𝜙𝑖} = 𝜙_{𝑖}^{2}/4𝑀_{𝑃𝑙}^{2}  1/2.
To obtain 𝑁_{tot} > 60, the initial field value must satisfy
(4.73) 𝜙_{𝑖} > 2¡î60 𝑀_{𝑃𝑙} ~ 15𝑀_{𝑃𝑙}.
We note that the total field excursion is superPlanckian, ∆𝜙 = 𝜙_{𝑖}  𝜙_{𝑒} ¡í 𝑀_{𝑃𝑙}. In Section 4.4.1, we will discuss whether this large field variation should be a cause for concern.
4.3.3 Creating the Hot Universe
Most of the energy density during inflation is the form of the inflaton potential 𝑉(𝜙). Inflation ends when the potential sleeps and the field picks up kinetic energy. The energy in the inflation sector then has to transferred to the particles of the Standard Model. This process is called reheating and starts the hot BB.
Once the inflation field reaches the bottom of the potential, it begins to oscillate. Near the minimum, the potential can be approximated as 𝑉(𝜙) ≈ 1/2 𝑚^{2}𝜙^{2} and the equation of motion of the inflation is
(4.74) 𝑑𝜙̇/𝑑𝑡 + 3𝐻𝜙̇ = 𝑚^{2}𝜙. [RE (4.47) ]
The energy density evolves according to the continuity equation [RE (4.50) (4.69)]
(4.75) 𝜌̇_{𝜙} + 3𝐻𝜌_{𝜙} = 3𝐻𝑃_{𝜙} = 3/2 𝐻 (𝑚^{2}𝜙^{2}  ᾡ^{2}),
where the righthand side averages to zero over one oscillation period. This averaging has ignored the Hubble friction in (4.74), which is justified on timescales shorter than the expansion time. The oscillating field behaves like pressureless matter, with 𝜌_{𝜙} ¡ð 𝑎^{3} (See problem 4.1 for the derivation). As the energy density drops, the amplitude of the oscillation decreases.
To avoid that the universe ends up completely empty, the inflaton has to couple to Standard Model fields. The energy stored in the inflaton filed will then be transferred to ordinary particles. If the decay is slow, then the inflaton's energy density follows the equation
(4.76) 𝜌̇_{𝜙} + 3𝐻𝜌_{𝜙} = 𝛤_{𝜙}𝜌_{𝜙},
where 𝛤_{𝜙} parameterizes the inflaton decay rate. A slow decay of the inflaton typically occurs if the coupling is only to fermions. If the inflaton can also decay into bosons then the decay rate may be enhanced by Bose condensation and parametric resonance effects. This kind of rapid decay is called preheating, since the bosons are created far from thermal equilibrium.
the new particles will interact with each other and eventually reach the thermal state that characterizes the hot Big Bang. The energy density at the end of reheating epoch is 𝜌_{𝑅} < 𝜌_{𝜙,𝑒} is the energy density at the end of inflation, and reheating temperature 𝛵_{𝑅} is determined by
(4.77) 𝜌_{𝑅} = ¥ð^{2}/30 𝑔_{*}(𝛵_{𝑅}) 𝛵_{𝑅}^{4}.
If reheating takes a long time, then 𝜌_{𝑅} ¡ì 𝜌_{𝜙,𝑒} and the reheating temperature gets smaller. At a minimum, the reheating temperature has to be larger than 1 MeV to allow for successful BBN, and most likely it is much larger than this to also allow for baryongenesis after inflation.
This completes our highly oversimplified sketch of the reheating phenomenon. In reality, the dynamics during reheating can be very rich, often involving nonperturbative effects captured by numerical simulation (for review see [25]). Describing this phase in the history is essential for understanding how the hot Big Bang began.With accelerated expansion having ended, there is no "cosmic amplifier" to make the microscopic physics of reheating easily accessible on cosmological length scales. Moreover, the high energy scale associated with the end of inflation, as well as the subsequent thermalization, can further hide details of this era from our lowenergy probes. Nevertheless, in some cases, the dynamics during this period can generate relics such as isocurvature perturbations, stochastic gravitational waves, nonGaussianities, dark matter/radiation, primordial black holes, topological and nontopological solitons, matter/antimatter asymmetry, and primordial magnetic fields. Detecting any of these relics would give us an interesting window into the reheating era.
4.1 Open Problems^{*}
Despite its phonomenological success, inflation is not yet a complete theory. To achieve inflation we had to postulate new physics at energies far above those probed by particle colliders and its success is sensitive to assumption about the physics at even higher energies.
4.4.1 Ultraviolet Sensitivity
The most conservative way to describe the inflationary era is in terms of an effective field theory (EFT). In the EFT approach, we admit that we don't know the details of the highenergy theory and instead parameterize our ignorance. We begin by defining the degrees of freedom and symmetries of the inflationary theory. This theory is valid below a cutoff scale 𝛬, and we should ask how the unknown physics above the scale 𝛬 can affect the lowenergy dynamics during inflation. This means writing down all possible corrections to the lowenergy theory that are consistent with the assume symmetries. The effective Lagrangian of the theory can then be written as
(4.78) 𝓛_{eff}[𝜙] = 𝓛_{0}[𝜙] + ¢²_{𝑛} 𝑐_{𝑛} 𝑂_{𝑛}[𝜙]/𝛬^{𝛿𝑛  4},
where 𝓛_{0} is the Lagrangian of the inflationary model and 𝑂_{𝑛} are a set of "perator" that parameterize the corrections coming from the couplings to additional highenergy degrees of freedom. The dimensionless parameters 𝑐_{𝑛} are usually taken to be of order one.^{8} If the cutoff scale 𝛬 is much larger than the typical energy 𝐸 at which the lowenergy theory is being probed, then the corrections are small, suppressed by powers of 𝐸/𝛬. This is why quantum gravity doesn't affect our everyday lives, or even those of our friends working at particle colliders.^{9} A remarkable feature of inflation, however, is that it is extremely sensitive even to effects suppressed by the Planck scale.
We have seen that slowroll inflation requires a flat potential (in Planck scale), but his is hard to control and sensitive to very small corrections. In the absence of any special symmetries, the EFT of slowroll inflation takes the following form
(4.79) 𝓛_{eff}[𝜙] = 1/2 (𝜕_{𝜇}𝜙)^{2}  𝑉(𝜙)  ¢²_{𝑛} 𝑐_{𝑛} 𝑉(𝜙) 𝜙^{2𝑛}/𝛬^{4𝑛} + ∙ ∙ ∙ ,
where a representative set of higherdimension operators is written. If the inflation field value is smaller than the cutoff scale we can truncate the 𝐸𝐹𝛵 expansion in (4.79) and the leading effect comes from the dimensionsix operator
(4.80) ∆𝑉 = 𝑐_{1}𝑉(𝜙) 𝜙^{2}/𝛬^{2}.
Since 𝜙 ¡ì 𝛬, this is a small correction to the inflation potential, ∆𝑉 ¡ì 𝑉. Nevertheless, it affects the delicate flatness of the potential and hence is relevant for the dynamics during inflation. In particular, the slowroll parameter 𝜂_{𝑉} receives the correction
(4.81) ∆𝜂_{𝑉} = 𝑀_{𝑃𝑙}^{2}/𝑉 (∆𝑉)_{,𝜙𝜙} ≈ 2𝑐_{1} (𝑀_{𝑃𝑙}/𝛬)^{2} > 1,
where the final inequality comes from 𝑐_{1} = 𝑂(1) and 𝛬 < 𝑀_{𝑃𝑙}. Notice that this problem is independent of the energy scale of inflation. All inflationary models have to address this socalled eta problem.
One promising way to solve the eta problem is to postulate a shift symmetry for he inflation, 𝜙 ¡æ 𝜙 + 𝑐, where 𝑐 is constant. If this symmetry is respected by the couplings to massive fields, then having 𝑐_{𝑛} ¡ì 1 is technically natural [6]. Another way to ameliorate the eta problem is supersymmetry. Above the scale 𝐻, the theory is supersymmetric and the contribution from bosons and fermions precisely cancel as a minor miracle. However, supersymmetry is spontaneously broken during inflation, leading yo an inflaton mass of order the Hubble scale, 𝑚_{𝜙} ~ 𝐻, and an eta parameter of order one, ∆𝜂_{𝑉} ~ 1. Although the eta problem is still there, it is much less severe and requires just a percentlevel finetuning of inflaton mass parameter.
We have seen that in quadratic inflation the inflaton moves over a superPlanckian distance in field space, ∆𝜙 > 𝑀_{𝑃𝑙}. This is a general feature of all inflationary models with observable gravitational wave signals. The sperPlanckian excursion of the field should make us nervous. When the field value isn't smaller than the cutoff, we cannot trust the EFT expansion in (4.79). Models of largefield inflation therefore usually invoke symmetriesshuch as an approximate shift symmetry 𝜙 ¡æ 𝜙 + 𝑐to suppress the couplings of the inflaton to massive degree off freedom and reduce the size of the corrections in (4.79). An important question is whether the assumed symmetries are respected by the UVcompletion.
The UV sensitivity of inflation is a challenge, because we either need to work in a UVcomplte theory of quantum gravity or make assumptions about the form that such a UVcompletion might take. It is also an opportunity,, because it suggsts the exciting possibility of using cosmological observations to learn about fundamental aspects of quantum gravity [7].
4.4.2 Initial Conditions
First, Imagine that the inflation field initially takes different values in different regions of space, some at top of the potential, some at the bottom. The regions of space with large vacuum energy will inflate and grow. Weighted by volume, these regions will dominate, explaining why most f space experience inflation. Alternatively, it is sometimes assumed that slowroll inflation was proceeded by an earlier phase of false vacuum domination (see Fig. 4.9. In quantum mechanics, there is a small probability that the field will tunnel through the potential energy barrier. so the question becomes why the quantum mechanical tunneling is more likely to put the inflaton field at the top of the potential than at its minimum. We should still ask why the universe started in the highenergy false vacuum. Most ambitiously, the noboundary proposal by Hartle and Hawking gives a prescription for evaluating the provability that a universe is spontaneousely created from nothing [8].
Second, if the initial inflaton velocity is nonnegligible, then it is possible that the field will overshoot the region of the potential where inflation is supposed to occur, without actually sourcing accelerated expansion. In smallfield models the Hubble friction is often not efficient enough to slow down the field before it reaches the region of interest. But in largefield models, on the other hand, Hubble friction is usually very efficient and the slowroll solution becomes an attractor.
Finally, since inflation is supposed to explain the homogeneous initial conditions of the hot BB, we must ask what happens when allow for large inhomogeneities in the inflaton field, These inhomogeneities carry gradient energy, which might hinder the accelerated expansion. The sensitivity of an inflationary model to initial perturbations of the inflaton field is best addressed by numerical simulations. In 1989 the earliest such simulations were performed in 1 + 1 dimensions [9, 10]. Recently the problem was revisited using simulation in 3 + 1 dimensions [11, 12]. It was again found that largefield models are more robust than smallfield models.
4.4.3 Eternal Inflation
A rather dramatic consequence of the quantum dynamics of inflation may be that globally that it never ends, but is eternal. There are two types of eternal inflation, both are illustrated in Fig. 4.9. First, inflation can occur while the field is stuck in false vacuum. Any region of space has a finite probability to decay and reach the true vacuum via quantum tunneling. If the decay rate is 𝛤, then the probability that the region will survive in the inflationary state is 𝑒^{𝛤𝑡}. The volume of inflating region increases as 𝑒^{3𝑡}. If the rate of exponential expansion is larger than the rate of decay, then inflation does not end globally. There are many pockets of space where inflation does endone of these pockets is socalled our "universe"but they are embedded in a much larger "multiverse." The space between the pocket universes is rapidly expanding, creating new space, where new pocket universes can form. This never ends.
We will study the effects of quantum mechanics on the dynamics of inflation and see that the field experiences random fluctuations up and down the potential. Eternal inflation occurs if these quantum fluctuation dominate over the classical rolling. Consider the evolution of the field in one Hubble volume 𝐻^{3} during one Hubble time ∆𝑡 = 𝐻^{1}. Classically, the field will change by ∆𝜙_{𝑐𝑙} ~ ᾡ𝐻^{1}. At the same time, quantum mechanics induces random jitters ∆𝜙_{𝑄}, so that the total change of the field is
(4.82) ∆𝜙 = ∆𝜙_{𝑐𝑙} + ∆𝜙_{𝑄}.
How big does this probability have to be in order for inflation to be eternal? During a Hubble time, the volume increases by a factor of 𝑒^{3} ≈ 20, meaning that the inflationary region breaks up into 20 Hubblesized regions. As we will see in Chapter 8, averaged over one of these 20 regions, inflationary quantum fluctuations have a Gaussian probability distribution with width 𝐻/2¥ð. If the probability to have ∆𝜙 < 0 is greater than 1 in 20, then the volume of space that is inflating will increase. When
(4.83) ∆𝜙_{𝑄} = 𝐻/2¥ð > 0.6 ∆𝜙_{𝑐𝑙} = 0.6 ∣ᾡ∣𝐻^{1}.
If the condition is satisfied, inflation will continue forever, This arguments for slowroll eternal inflation was admittedly somewhat heuristic and it is sill being debated the reliability.
Exercise 4.5 Show that, in quadratic inflation, slowroll eternal inflation occurs for
(4.84) 𝜙^{2}/𝑀_{𝑃𝑙}^{2} ≳ 13 𝑀_{𝑃𝑙}/𝑚. ^{[verification needed]}
In Chapter 8, we will find that in order for the predicted CMB fluctuations to have the right amplitude, we must have 𝑚 ~ 10^{6} 𝑀_{𝑃𝑙}. This implies 𝜙 > 3600 𝑀_{𝑃𝑙}. In that case the regime of eternal inflation would not exist. ^{[verification needed]}
[Solution] The condition for eternal inflation is and assuming a quadratic potential 𝑉(𝜙) = 𝑚^{2}𝜙^{2}/2
(a) ∆𝜙_{𝑄} = 𝐻/2¥ð > 0.6 ∆𝜙_{𝑐𝑙} = 0.6 ∣𝜙̇∣𝐻^{1} ¢¡ 𝐻_{2} > 0.6 ¡¿ 2¥ð∣𝜙̇∣ ≈ 3.8∣𝜙̇∣
(b) 𝐻^{2} ≈ 𝑉/3𝑀_{𝑃𝑙}^{2} = 𝑚^{2}𝜙^{2}/3𝑀_{𝑃𝑙}^{2}
(c) ∣ᾡ∣ ≈ ∣𝑉_{,𝜙}∣/3𝐻 ≈ ¡î3𝑀_{𝑃𝑙}∣𝑉_{,𝜙}∣/3¡î𝑉 = 𝑀_{𝑃𝑙} 𝑑(𝑚^{2}𝜙^{2}/2)/𝑑𝜙 ¡¿ 1/[¡î(3/2)𝑚𝜙] = ¡î(2/3) 𝑚𝑀_{𝑃𝑙}.
The condition 𝐻_{2} > 3.8∣ᾡ∣ implies
(d) 𝑚^{2}𝜙^{2}/3𝑀_{𝑃𝑙}^{2} > 3.8¡î(2/3)𝑚𝑀_{𝑃𝑙} ¢¡ 𝜙^{2}/𝑀_{𝑃𝑙}^{2} > 6/𝑚^{2} ¡¿ 3.8¡î(2/3)𝑚𝑀_{𝑃𝑙} = 3.8¡î24 𝑀_{𝑃𝑙}/𝑚 ≈ 18.6 𝑀_{𝑃𝑙}/𝑚. ^{[verification needed]}
Using 𝑚 ~ 10^{6}𝑀_{𝑃𝑙}, this implies 𝜙 > 4300 𝑀_{𝑃𝑙}. ▮ ^{[verification needed]}
The concept of eternal inflation is almost as old as inflation itself, it recently revived in the context of string theory. the string theory seems to give rise to a large number of metastable solutions.The space vacua is called the string landscape. The physics may be different in each of these vacua. Eternal inflation provides the mechanism by which this landscape populated. The fact that eternal inflation combined with the string landscape allows for a solution the the cosmological constant problem (see Section 2.3.3) is the single most important reason to take either of these ideas seriously.
Let us finally address the elephant in the room: the measure problem of the inflationary multiverse. We have seen that eternal inflation produces an infinite number of universe. The fraction of universes with a certain observable property is equal to infinity divided by infinity. Unfortunately, the answer tends to depend sensitively on type of regulator that is being used.
Although we have remarkable observational evidence that something like inflation occurred in the early universe (see Chapter 8), inflation cannot yet be considered a part of the standard model of cosmology, Inflation is still a work in progress.
4.5 Summary
In this chapter, we have studied inflation. We stared with a discussion of the causality problems of the hot BB, highlighting the crucial role played by the evolution of the comoving Hubble radius, (𝑎𝐻)^{1}. We showed that this evolution determines the particle horizonthe maximal distance that light can propagate between an initial time 𝑡_{𝑖} and a later time 𝑡:
(4.84) 𝑑_{𝘩}(𝜂) = ¡ò_{𝑡𝑖}^{𝑡} 𝑑𝑡/𝑎(𝑡) = ¡ò_{𝑁𝑖}^{𝑁} (𝑎𝐻)^{1} 𝑑𝑁,
where 𝑁 = ln 𝑎.Since the comoving Hubble radius is monotonically increasing in the conventional BB theory, the particle horizon is dominated by the contributions from late times. Most parts of the CMB would then never have been causal contact, yet they have nearly the same temperature. This is the horizon problem.
Inflation is a period of accelerated expansion during which the comoving Hubble radius is shrinking. The particle horizon receives most of its contribution from early times and can be much larger than naively assumed. Causal contact is established in the time before the hot BB. The shrinking of the comoving Hubble radius also solves the closely related flatness problem and allows a causal mechanism to generate superhorizon correlations.
Inflation occurs if all physical quantities vary slowly with time. For example, the fractional change of the Hubble rate during inflation must be small,
(4.86) 𝜀 ¡Õ Ḣ/𝐻^{2} ¡ì 1.
To solve the problems of the ht BB, this condition must be maintained for about 60 𝑒foldings of expansion. A popular way to achieve this is through a slowly rolling scalar field (the inflaton). The evolution of the inflaton is governed by the KleinGordon equation
(4.87) 𝑑𝜙̇/𝑑𝑡 + 3𝐻ᾡ = 𝑉_{,𝜙}.
The field rolls slowly if the dynamics is dominated by the friction term, ∣𝑑𝜙̇/𝑑𝑡∣ ¡ì ∣3𝐻𝜙̇∣ ≈ ∣𝑉_{,𝜙}∣. The size of the friction is determined by the energy density associated with the inflation itself, which by the Friedmann equation determines the expansion rate
(4.88) 𝐻^{2} = 1/3𝑀_{𝑃𝑙}^{2} [1/2 ᾡ^{2} + 𝑉].
Inflation requires that the kinetic energy is much smaller than the potential energy, 1/2 ᾡ^{2} ¡ì 𝑉. The conditions for successful slowroll inflation can be characterized by the slowroll parameters
(4.89) 𝜀_{𝑉} ¡Õ 𝑀_{𝑃𝑙}^{2}/2 (𝑉_{,𝜙}/𝑉)^{2}, 𝜂_{𝑉} ¡Õ 𝑀_{𝑃𝑙}^{2} 𝑉_{,𝜙𝜙}/𝑉.
Inflation will occur in regions of potential where {𝜀_{𝑉}, ∣𝜂_{𝑉}∣} ¡ì 1. Achieving such flat potentials in a theory of quantum gravity is challenging.
^{7} This parameter is often denoted 𝜂. the parameter 𝜀 and 𝜅 are often called Hubble slowroll parameters to distinguish potential slowroll parameters defined in Section 4.3.1.
^{8} Some new degrees of freedom must appear below the Planck scale, 𝛬 ≲ 𝑀_{𝑃𝑙}, because gravity is not renormalizable and needs a highenergy (or "ultraviolet") completion. If 𝜙 has orderone couplings to any heavy fields, with masses of order 𝛬, then integrating out these fields yields the effective action (4.78), with orderone couplings 𝑐_{𝑛}.
^{9} Note that the largest energies proved at the Large Hadron Collier (LHC) are still tiny relative to the Planck scale, 𝐸/𝑀_{𝑃𝑙} < 10^{16}.
Problem 4.1 Oscillating scalar field
The action for a scalar field in a curved space time is
(a) 𝑆 = ¡ò 𝑑^{4} 𝑥¡î(𝑔) [1/2 𝑔^{𝜇𝜈}¡Ó_{𝜇}𝜙¡Ó_{𝜈}𝜙  𝑉(𝜙)],
where 𝑔 ¡Õ det 𝑔_{𝜇𝜈} is the determinant of the metric tensor.
1. Evaluate the action for the homogeneous field 𝜙 = 𝜙(𝑡) in a flat FRW spacetime and determine the equation of the motion for the field.
[Solution] For the FRW metric,
(b) 𝑑𝑠^{2} = 𝑑𝑡^{2} + 𝑎^{2}(𝑡) 𝑑𝐱^{2}.
the determinant of the matrix, 𝑔_{𝜇𝜈} = diag (1, 𝑎^{2}, 𝑎^{2}, 𝑎^{2}), is 𝑔 = 𝑎^{6}(𝑡). The scalar field is constrained by the symmetries of the FRW spacetime to evolve over only in time 𝜙(𝑡, 𝐱) = 𝜙(𝑡). The Lagrangian in the example is then
(c) 𝐿 = 𝑎^{3}[1/2 ᾡ^{2}  𝑉(𝜙)]
We use the EulerLagrange equation
(d) 𝑑/𝑑𝑡 (¡Ó𝐿/¡Ó𝜙̇)  ¡Ó𝐿/¡Ó𝜙 = 0, ¡Ó𝐿/¡Ó𝜙̇ = 𝑎^{3}ᾡ, ¡Ó𝐿/¡Ó𝜙 = 𝑎^{3}¡Ó𝑉/¡Ó𝜙.
(e) 𝑑/𝑑𝑡 (𝑎^{3}𝜙̇) = 3𝑎^{2} 𝑑𝑎/𝑑𝑡 𝜙̇ + 𝑎^{3}𝑑𝜙̇/𝑑𝑡
(f) 𝑎^{3}(3𝑎̇/𝑎 𝜙̇ + 𝑑𝜙̇/𝑑𝑡 + ¡Ó𝑉/¡Ó𝜙) = 𝑎^{3}(3𝐻𝜙̇ + 𝑑𝜙̇/𝑑𝑡 + ¡Ó𝑉/¡Ó𝜙) = 0 ¢¡ 𝑑𝜙̇/𝑑𝑡 + 3𝐻𝜙̇ = ¡Ó𝑉/¡Ó𝜙. [(4.47) KleinGordon equation] ▮


