4.3. Type 1a Supernovae and the value of
The luminosity distance also plays a crucial role in determining cosmological parameters once the absolute brightness of a class of objects is known. Of considerable importance in this context is the magnitude-redshift relation which relates the apparent magnitude m of an object to its absolute magnitude M
![]() | (35) |
where µ is known as the distance modulus.
Since dL depends upon the geometry of space
and its material content,
the magnitude-redshift relation (35) can, in principle be used to
determine
and
total if
both m and M are known
within reasonable limits.
(6)
The recent discovery that type Ia supernovae may be used as calibrated
standard candles for obtaining estimates of the luminosity distance
dL through (35) has aroused great interest.
Type Ia supernovae are explosions which arise as a white dwarf star crosses
the Chandrasekhar stability limit while accreting matter from a companion
star [94,
4,
40].
The high absolute luminosity of SNe Ia
(MB -
19.5 mag) suggests that they can be seen out to large distances making
them ideal candidates for measuring and constraining cosmological
parameters
[39,
21].
Of crucial import to using type Ia supernovae for estimating the luminosity
distance dL
has been the observation that: (i) the dispersion in their luminosity at
maximum light is extremely small
(
0.3 mag);
(ii) the width of the supernova light
curve is strongly correlated with its intrinsic luminosity:
a brighter supernova will have a broader light curve indicative of a more
gradual decline in its brightness
[161].
Both (i) and (ii) reduce the scatter in the absolute luminosity of type 1a
supernovae to ~ 10% making them excellent standard candles
[21].
Nearby type 1a supernovae have been used to determine the value of
H0 whereas those further away are used to
obtain reasonable estimates of cosmological parameters by minimizing
the 2 statistic
![]() | (36) |
where µp are model dependent `predicted' values
of the distance modulus obtained from (28) and (35), and
µ0(zi) are the observed values.
At least two groups - the Supernova Cosmology Project
[157] and
the High-Z Supernova Search Team
[165]
have been engaged in both finding and
calibrating supernovae at low and high redshifts. At the time of
writing, both groups have analyzed data for several dozen type Ia supernovae
and a consensus seems to be emerging that a positive value of
is
strongly preferred.
For instance, treating type 1a supernovae as standard candles and then
using distance estimates to 42 moderately high redshift supernovae with
z
0.83,
Perlmutter et al. (1998b)
find that the joint probability distribution of the parameters
&
m is well
approximated by the relationship (valid for
m
1.5)
![]() |
The best-fit confidence region in the
m -
plane shown in
(6) appears to favour a
closed universe. However, as we shall see in the next section, when
combined with the results of cosmic microwave experiments, the combined
likelihood of
m,
peaks near
m +
1.
![]() |
Figure 6. Best-fit confidence regions in
the |
These results provide an interesting insight into the expansion dynamics
of the universe during its recent past. For instance,
a cosmological model which passed through an epoch of matter domination
before the present
dominated epoch, also passed through an inflexion point
at which the expansion of the universe changed from deceleration
(
< 0)
to acceleration
(
> 0). From (3)
& (4) it can be shown that this occurred at a redshift when
was still not
dominating the expansion dynamics of the universe.
For instance from (4) we find that
deceleration is succeeded by acceleration at the epoch
![]() | (37) |
On the other hand the epoch of equality between
m
and
occurred at the
redshift
![]() | (38) |
where =
c2 / 3H02.
Substituting the `best-fit' values obtained by Perlmutter et al. (1998b)
for a flat universe
m
0.28,
0.72
we get z*
0.726 and
z*
0.37 so that
z* < z * . From
(14)) we also get
q0 = - 0.58 for the deceleration parameter, indicating
an accelerating universe (the combined Sn+CMB data give a slightly
larger value
q0
-
0.5).
Supernovae data can also be used to constrain time dependent
models
of the kind discussed in section 8. In the case of
scalar field models with potentials
V(
) ~
-p and
V(
) ~
(e1 /
- 1)
the scalar field density in a spatially flat universe
is constrained to lie in the range
[159,
203]
(0.6, 0.7) and the effective
equation of state is
w
< -
0.6 (at the 95% confidence level).
A combined analysis of gravitational lensing and Type 1a supernovae gives
the best-fit value
m
0.33 for a spatially
flat universe
[201].
Attempts to constrain the decay rate of a time-dependent
cosmological term
=
0(1 +
z)m result in
0.24
m
0.38
and m
0.85 (at
the 68% confidence level) which in turn places constrains on the cosmic
equation of state w = m/3 - 1
- 0.72.
The combined supernovae & lensing data therefore convincingly rule
out a network of tangled cosmic strings (w
- 1/3) and strongly
favour a cosmological constant (w = - 1).
The results obtained by both the Supernova Cosmology Project and
the High-Z Supernova Search Team team present the strongest `direct'
evidence for a non-zero cosmological constant. However much work
needs to be done both in understanding systematic uncertainties
as well as Sn Ia properties before the case for a positive
is firmly
established.
(7)
As we shall show in the next section, much stronger constraints on
m and
emerge if we combine the supernovae results with observations of the
cosmic microwave background.
6 In practice
(35) must be corrected for effects associated with the
redshifting of light as it travels to us, commonly called the K-correction.
For instance photons being detected using a red filter would originally
have had a `blue spectrum' if the source was located at
z 1.
Other possible sources of systematic errors include luminosity evolution,
intergalactic extinction,
Malmquist bias, the aperture correction, weak lensing etc.
A more complete discussion of these issues can be found in
[142,
154].
Back.
7 An accelerating universe can also be accommodated within the framework of the Quasi-Steady State Cosmology of Hoyle, Burbidge and Narlikar (1993). Back.