302
19 Inflation and Some Questions
19.2 Use the scalar field Lagrangian in (19.9) to obtain the dynamical equations
for the inflaton field in (19.10). This is also done in Appendix 2.
19.3 Work out the Lagrangian in (19.9) in ordinary units, that is in which and c
are not taken to be 1. Is it clear why natural units are preferred by theorists?
19.4 Consider one idealized over-simplified case for the inflaton field equation (19.11). Take the Hubble function to be zero and the potential V to
be linearly decreasing, and solve for ϕ(t). Does this clarify the physical role
of V ?
19.5 Consider another idealized over-simplified case for (19.11). Take the Hubble
function to be constant and the potential V to be zero, and solve for ϕ(t).
Does this clarify the physical role of H?
19.6 The integral of the Lagrangian has the dimensions of an action in (19.9). Use
this fact to work out the dimensions of the scalar field and the potential. Do
this for both ordinary and natural units. Are they consistent with the energy
density and pressure of the inflaton field in (19.12)?
19.7 Sketch the Inflaton Potentials in (19.13).
19.8 Take the inflaton self-interaction potential V to be that in (19.13a); show that
the equation of motion is the Klein Gordon equation that describes a free
particle of mass m in quantum field theory. You might want to assume the
flat space of special relativity for simplicity.
19.9 Use dimensional analysis to show that the amplitude of fluctuations of the
inflaton field during a hubble time period is of order |δϕ| ∼ H (Linde 2007).
19.10 Consider a universe with a scale factor that is a power t
m as in (19.2) and
Appendix 1. How would such a universe fit into the picture in Fig. 19.5 if m
is large?
19.11 According to classical theoretical physics the hydrogen atom would not be
stable since the electron would radiate energy and fall onto the proton. Use
the UP to show how it is stable according to quantum theory, and obtain an
estimate of the ground state energy.
19.12 What if the uncertainties in (19.27) add as squares? That is
x
2
tot
p
2
+
L
2
P
p
2
.
Which version do you think is a more reasonable guess? Show that the conclusions of Sect. 19.6 concerning a minimal length do not change significantly
if this version is used.
19 Inflation and Some Questions
19.2 Use the scalar field Lagrangian in (19.9) to obtain the dynamical equations
for the inflaton field in (19.10). This is also done in Appendix 2.
19.3 Work out the Lagrangian in (19.9) in ordinary units, that is in which and c
are not taken to be 1. Is it clear why natural units are preferred by theorists?
19.4 Consider one idealized over-simplified case for the inflaton field equation (19.11). Take the Hubble function to be zero and the potential V to
be linearly decreasing, and solve for ϕ(t). Does this clarify the physical role
of V ?
19.5 Consider another idealized over-simplified case for (19.11). Take the Hubble
function to be constant and the potential V to be zero, and solve for ϕ(t).
Does this clarify the physical role of H?
19.6 The integral of the Lagrangian has the dimensions of an action in (19.9). Use
this fact to work out the dimensions of the scalar field and the potential. Do
this for both ordinary and natural units. Are they consistent with the energy
density and pressure of the inflaton field in (19.12)?
19.7 Sketch the Inflaton Potentials in (19.13).
19.8 Take the inflaton self-interaction potential V to be that in (19.13a); show that
the equation of motion is the Klein Gordon equation that describes a free
particle of mass m in quantum field theory. You might want to assume the
flat space of special relativity for simplicity.
19.9 Use dimensional analysis to show that the amplitude of fluctuations of the
inflaton field during a hubble time period is of order |δϕ| ∼ H (Linde 2007).
19.10 Consider a universe with a scale factor that is a power t
m as in (19.2) and
Appendix 1. How would such a universe fit into the picture in Fig. 19.5 if m
is large?
19.11 According to classical theoretical physics the hydrogen atom would not be
stable since the electron would radiate energy and fall onto the proton. Use
the UP to show how it is stable according to quantum theory, and obtain an
estimate of the ground state energy.
19.12 What if the uncertainties in (19.27) add as squares? That is
x
2
tot
p
2
+
L
2
P
p
2
.
Which version do you think is a more reasonable guess? Show that the conclusions of Sect. 19.6 concerning a minimal length do not change significantly
if this version is used.