19.3 Origin of Structure
289
Inflation: a = Ae
H I t
, L H =
c
H I
, lnL H = ln
c
H I
(19.18a)
Radiation: a = Bt
1/2
, L H =
2c
B 2 a
2
, lnL H = 2 lna + ln
2c
B 2
(19.18b)
Matter: a = Ct
2/3
, L H =
3c
2C 3/2 a
3/2
, lnL H =
3
2
lna + ln
3c
2C
3/2
(19.18c)
De Sitter: a = De
H d t
, L H =
c
H d
, lnL H = ln
c
H d
.
( 19.18d)
Notice that the logarithmic relations in (19.18) are all conveniently linear.
Figure 19.5 is similar to Fig. 19.4 and shows a plot of the Hubble length from
(19.18) as four straight lines. It is clear from the figure how the wavelength of a
mode leaves the Hubble region during inflation and reenters during the radiation or
matter era. The part of the plot where the wavelength is below the Hubble curve is
where the fluctuations interact with the material of the universe and have an effect
on density variations. Note that the longer wavelengths reenter and become effective
at later times than the shorter wavelengths.
We have only considered in this brief discussion some aspects of the fluctuations
leading to structure. Calculation of the nature and magnitude of the fluctuations
and their effect on the CMB and later structure is beyond our present scope. For a
comprehensive discussion the reader may consult Linde (2007) and Kinney (2002).
Finally we note that some theorists speculate that the same fluctuation mechanism
that produces structure should also give rise to many entire universes that are not
connected in any evident way to our own (Linde 2007). This idea faces the serious
objection that such other universes may not be subject to observational test and thus
not subject to scientific study. In the next section we will mention one aspect of the
multiverse idea, that it is related to the calculability of the values of some of the
fundamental constants of nature.
Fig. 19.5 This shows Fig. 19.4 in logarithmic form and includes the matter era. The Hubble length
and the mode wavelength are plotted versus the scale factor. The causal region is where λ < L H .
The figure is a qualitative sketch and very much not to scale
289
Inflation: a = Ae
H I t
, L H =
c
H I
, lnL H = ln
c
H I
(19.18a)
Radiation: a = Bt
1/2
, L H =
2c
B 2 a
2
, lnL H = 2 lna + ln
2c
B 2
(19.18b)
Matter: a = Ct
2/3
, L H =
3c
2C 3/2 a
3/2
, lnL H =
3
2
lna + ln
3c
2C
3/2
(19.18c)
De Sitter: a = De
H d t
, L H =
c
H d
, lnL H = ln
c
H d
.
( 19.18d)
Notice that the logarithmic relations in (19.18) are all conveniently linear.
Figure 19.5 is similar to Fig. 19.4 and shows a plot of the Hubble length from
(19.18) as four straight lines. It is clear from the figure how the wavelength of a
mode leaves the Hubble region during inflation and reenters during the radiation or
matter era. The part of the plot where the wavelength is below the Hubble curve is
where the fluctuations interact with the material of the universe and have an effect
on density variations. Note that the longer wavelengths reenter and become effective
at later times than the shorter wavelengths.
We have only considered in this brief discussion some aspects of the fluctuations
leading to structure. Calculation of the nature and magnitude of the fluctuations
and their effect on the CMB and later structure is beyond our present scope. For a
comprehensive discussion the reader may consult Linde (2007) and Kinney (2002).
Finally we note that some theorists speculate that the same fluctuation mechanism
that produces structure should also give rise to many entire universes that are not
connected in any evident way to our own (Linde 2007). This idea faces the serious
objection that such other universes may not be subject to observational test and thus
not subject to scientific study. In the next section we will mention one aspect of the
multiverse idea, that it is related to the calculability of the values of some of the
fundamental constants of nature.
Fig. 19.5 This shows Fig. 19.4 in logarithmic form and includes the matter era. The Hubble length
and the mode wavelength are plotted versus the scale factor. The causal region is where λ < L H .
The figure is a qualitative sketch and very much not to scale