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4.3. Type 1a Supernovae and the value of Lambda

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

Equation 35 (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 OmegaLambda and Omegatotal 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 appeq - 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 (ltapprox 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 chi2 statistic

Equation 36 (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 OmegaLambda is strongly preferred. For instance, treating type 1a supernovae as standard candles and then using distance estimates to 42 moderately high redshift supernovae with zltapprox 0.83, Perlmutter et al. (1998b) find that the joint probability distribution of the parameters OmegaLambda & Omegam is well approximated by the relationship (valid for Omegam leq 1.5)


The best-fit confidence region in the Omegam - OmegaLambda 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 Omegam, OmegaLambda peaks near Omegam + OmegaLambda appeq 1.

Figure 6

Figure 6. Best-fit confidence regions in the Omegam - OmegaLambda plane obtained from the analysis of Type 1a high redshift supernovae of Perlmutter et al. (1998b). The upper-left shaded region corresponds to the singularity free `bouncing universe' models discussed in section 3.1.

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 Lambda dominated epoch, also passed through an inflexion point at which the expansion of the universe changed from deceleration (addot < 0) to acceleration (addot > 0). From (3) & (4) it can be shown that this occurred at a redshift when Lambda 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

Equation 37 (37)

On the other hand the epoch of equality between rhom and Lambda occurred at the redshift

Equation 38 (38)

where OmegaLambda = Lambda c2 / 3H02. Substituting the `best-fit' values obtained by Perlmutter et al. (1998b) for a flat universe Omegam appeq 0.28, OmegaLambda appeq 0.72 we get z* appeq 0.726 and z* appeq 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 appeq - 0.5).

Supernovae data can also be used to constrain time dependent Lambda models of the kind discussed in section 8. In the case of scalar field models with potentials V(phi) ~ phi-p and V(phi) ~ (e1 / phi - 1) the scalar field density in a spatially flat universe is constrained to lie in the range [159, 203] Omegaphi in (0.6, 0.7) and the effective equation of state is wphi < - 0.6 (at the 95% confidence level).

A combined analysis of gravitational lensing and Type 1a supernovae gives the best-fit value Omegam appeq 0.33 for a spatially flat universe [201]. Attempts to constrain the decay rate of a time-dependent cosmological term Lambda = Lambda0(1 + z)m result in 0.24 ltapprox Omegam ltapprox 0.38 and m ltapprox 0.85 (at the 68% confidence level) which in turn places constrains on the cosmic equation of state w = m/3 - 1 ltapprox - 0.72. The combined supernovae & lensing data therefore convincingly rule out a network of tangled cosmic strings (w appeq - 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 Lambda is firmly established. (7) As we shall show in the next section, much stronger constraints on Omegam and OmegaLambda 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 appeq 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.

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