19.3 Origin of Structure
287
Our goal in this section is quite modest since the theory of structure formation
via inflation is large in scope (Kinney 2002; Linde 2007). Our aim is only to obtain
a rough qualitative understanding of the behavior of the fluctuations as the universe
expands. To do this we consider two length scales: one scale is the Hubble length
L H = c/H , which determines a fundamental causal region, and the other is the
physical spatial size of the fluctuations.
A basic property of the FLRW geometry is that the physical separation L of
co-moving objects increases proportional to the scale factor, and is given by
L = aσ, σ = constant coordinate separation.
(19.14)
In this section we will take the scale factor to be 1 at some arbitrary initial time,
rather than at the present time t 0 , and we will not use units with c = 1. Thus the
velocity of separation between co-moving points and a Hubble type law are
v = L
= a
σ =
a
a
(aσ ) = H L =
c
L H
L , prime denotes
d
dt
.
(19.15)
From this it is clear that the Hubble length defines a causal region: within this region
co-moving objects move apart at less than c and outside of it they move apart faster
than c and can have no causal influence on each other. Thus the Hubble length defines
a cosmological horizon. (Be aware that the word horizon is used for a number of
different things in physics and cosmology.) During the inflationary era the Hubble
length is constant at L H = c/H I so the causal region does not change. During later
times such as the radiation and matter dominated eras it grows linearly with time,
specifically
L H = 2ct, radiation era, L H =
3
2
ct, matter era.
(19.16)
Loosely speaking nothing larger than the Hubble distance can be considered coherent.
The second length to consider is the spatial size of the inflaton field fluctuations.
We may think of a fluctuation as composed of modes with wavelengths λ. It is clear
from the above comments that only modes with λ less than about the Hubble length
can have a direct physical effect in producing structure; those with larger λ act like
constants and have negligible gradients, so do not produce physical effects such
as concentrations of energy density. This is indicated in Fig. 19.3. As the universe
expands the physical wavelength λ of a mode will stretch like a wavelength of light
for the same reasons we discussed in Sect. 13.3; that is, the mode wavelength will
increase proportional to the scale factor, which increases very rapidly during inflation.
Thus an initial mode wavelength λ i will grow according to
λ(t) = a(t)λ i , λ i initial wavelength.
(19.17)
287
Our goal in this section is quite modest since the theory of structure formation
via inflation is large in scope (Kinney 2002; Linde 2007). Our aim is only to obtain
a rough qualitative understanding of the behavior of the fluctuations as the universe
expands. To do this we consider two length scales: one scale is the Hubble length
L H = c/H , which determines a fundamental causal region, and the other is the
physical spatial size of the fluctuations.
A basic property of the FLRW geometry is that the physical separation L of
co-moving objects increases proportional to the scale factor, and is given by
L = aσ, σ = constant coordinate separation.
(19.14)
In this section we will take the scale factor to be 1 at some arbitrary initial time,
rather than at the present time t 0 , and we will not use units with c = 1. Thus the
velocity of separation between co-moving points and a Hubble type law are
v = L
= a
σ =
a
a
(aσ ) = H L =
c
L H
L , prime denotes
d
dt
.
(19.15)
From this it is clear that the Hubble length defines a causal region: within this region
co-moving objects move apart at less than c and outside of it they move apart faster
than c and can have no causal influence on each other. Thus the Hubble length defines
a cosmological horizon. (Be aware that the word horizon is used for a number of
different things in physics and cosmology.) During the inflationary era the Hubble
length is constant at L H = c/H I so the causal region does not change. During later
times such as the radiation and matter dominated eras it grows linearly with time,
specifically
L H = 2ct, radiation era, L H =
3
2
ct, matter era.
(19.16)
Loosely speaking nothing larger than the Hubble distance can be considered coherent.
The second length to consider is the spatial size of the inflaton field fluctuations.
We may think of a fluctuation as composed of modes with wavelengths λ. It is clear
from the above comments that only modes with λ less than about the Hubble length
can have a direct physical effect in producing structure; those with larger λ act like
constants and have negligible gradients, so do not produce physical effects such
as concentrations of energy density. This is indicated in Fig. 19.3. As the universe
expands the physical wavelength λ of a mode will stretch like a wavelength of light
for the same reasons we discussed in Sect. 13.3; that is, the mode wavelength will
increase proportional to the scale factor, which increases very rapidly during inflation.
Thus an initial mode wavelength λ i will grow according to
λ(t) = a(t)λ i , λ i initial wavelength.
(19.17)