There are several cosmological observations which suggests the existence of a non zero cosmological constant with different levels of reliability. Most of these determine either the value of NR or some combination of NR and . When combined with the strong evidence from the CMBR observations that the tot = 1 (see section 6) or some other independent estimate of NR, one is led to a non zero value for . The most reliable ones seem to be those based on high redshift supernova [72, 73, 74] and structure formation models [75, 76, 77]. We shall now discuss some of these evidence.
3.1. Observational evidence for accelerating universe
Figure 2 shows that the evolution of a universe with 0 changes from a decelerating phase to an accelerating phase at late times. If H(a) can be observationally determined, then one can check whether the real universe had undergone an accelerating phase in the past. This, in turn, can be done if dL(z), say, can be observationally determined for a class of sources. Such a study of several high redshift supernova has led to the data which is shown in figures 5, 9.
Bright supernova explosions are brief events (~ 1 month) and occurs in our galaxy typically once in 300 years. These are broadly classified as two types. Type-Ia supernova occurs when a degenerate dwarf star containing CNO enters a stage of rapid nuclear burning cooking iron group elements (see eg., chapter 7 of ). These are the brightest and most homogeneous class of supernova with hydrogen poor spectra. An empirical correlation has been observed between the sharply rising light curve in the initial phase of the supernova and its peak luminosity so that they can serve as standard candles. (Type II supernova, which occur at the end of stellar evolution, are not useful for this purpose.)
For any supernova, one can directly observe the apparent magnitude m [which is essentially the logarithm of the flux F observed] and its redshift. The absolute magnitude M of the supernova is again related to the actual luminosity L of the supernova in a logarithmic fashion. Hence the relation F = (L / 4 dL2) can be written as
The numerical factors arise from the astronomical conventions used in the definition of m and M. Very often, one will use the dimensionless combination (H0 dL(z) / c) rather than dL(z) and the above equation will change to m(z) = + 5 log10 Q(z). The results below will be often stated in terms of the quantity , related to M by
If the absolute magnitude of a class of Type I supernova can be determined from the study of its light curve, then one can obtain the dL for these supernova at different redshifts. Knowing dL, one can determine the geometry of the universe.
To understand this effect in a simple context, let us compare the luminosity distance for a matter dominated model (NR = 1, = 0)
with that for a model driven purely by a cosmological constant (NR = 0, = 1):
It is clear that at a given z, the dL is larger for the cosmological constant model. Hence, a given object, located at a fixed redshift, will appear brighter in the matter dominated model compared to the cosmological constant dominated model. Though this sounds easy in principle, the statistical analysis turns out to be quite complicated.
The high-z supernova search team (HSST) discovered about 16 supernova in the redshift range (0.16 - 0.62) and another 34 nearby supernova  and used two separate methods for data fitting. The supernova cosmology project (SCP) has discovered  42 supernova in the range (0.18 - 0.83). Assuming NR + = 1, the analysis of this data gives NR = 0.28 ± 0.085 (stat) ± 0.05 (syst).
Figure 5 shows the dL(z) obtained from the supernova data and three theoretical curves all of which are for k = 0 models containing non relativistic matter and cosmological constant. The data used here is based on the redshift magnitude relation of 54 supernova (excluding 6 outliers from a full sample of 60) and that of SN 1997ff at z = 1.755; the magnitude used for SN 1997ff has been corrected for lensing effects. The best fit curve has NR 0.32, 0.68. In this analysis, one had treated NR and the absolute magnitude M as free parameters (with NR + = 1) and has done a simple best fit for both. The corresponding best fit value for is = 23.92 ± 0.05. Frame (a) of figure 6 shows the confidence interval (for 68 %, 90 % and 99 %) in the NR - for the flat models. It is obvious that most of the probability is concentrated around the best fit value. We shall discuss frame (b) and frame (c) later on. (The discussion here is based on .)
Figure 5. The luminosity distance of a set of type Ia supernova at different redshifts and three theoretical models with NR + = 1. The best fit curve has NR = 0.32, = 0.68.
Figure 6. Confidence contours corresponding to 68 %, 90 % and 99 % based on SN data in the NR - plane for the flat models with NR + = 1. Frame (a) on the left uses all data while frame (b) in the middle uses low redshift data and the frame (c) in the right uses high redshift data. While neither the low-z or high-z data alone can exclude the NR = 1, = 0 model, the full data excludes it to a high level of significance.
The confidence intervals in the - NR plane are shown in figure 7 for the full data. The confidence regions in the top left frame are obtained after marginalizing over . (The best fit value with 1 error is indicated in each panel and the confidence contours correspond to 68 %, 90 % and 99 %.) The other three frames show the corresponding result with a constant value for rather than by marginalizing over this parameter. The three frames correspond to the mean value and two values in the wings of 1 from the mean. The dashed line connecting the points (0,1) and (1,0) correspond to a universe with NR + = 1. From the figure we can conclude that: (i) The results do not change significantly whether we marginalize over or whether we use the best fit value. This is a direct consequence of the result in frame (a) of figure (6) which shows that the probability is sharply peaked. (ii) The results exclude the NR = 1, = 0 model at a high level of significance in spite of the uncertainty in .
Figure 7. Confidence contours corresponding to 68 %, 90 % and 99 % based on SN data in the NR - plane. The top left frame is obtained after marginalizing over while the other three frames uses fixed values for . The values are chosen to be the best-fit value for and two others in the wings of 1 limit. The dashed line corresponds to the flat model. The unbroken slanted line corresponds to H0 dL(z = 0.63) = constant. It is clear that: (i) The data excludes the NR = 1, = 0 model at a high significance level irrespective of whether we marginalize over or use an accepted 1 range of values for . (ii) The shape of the confidence contours are essentially determined by the value of the luminosity distance at z 0.6.
The slanted shape of the probability ellipses shows that a particular linear combination of NR and is selected out by these observations . This feature, of course, has nothing to do with supernova and arises purely because the luminosity distance dL depends strongly on a particular linear combination of and NR, as illustrated in figure 8. In this figure, NR, are treated as free parameters [without the k = 0 constraint] but a particular linear combination q (0.8NR - 0.6) is held fixed. The dL is not very sensitive to individual values of NR, at low redshifts when (0.8NR - 0.6) is in the range of (- 0.3, - 0.1). Though some of the models have unacceptable parameter values (for other reasons), supernova measurements alone cannot rule them out. Essentially the data at z < 1 is degenerate on the linear combination of parameters used to construct the variable q. The supernova data shows that most likely region is bounded by -0.3 (0.8NR - 0.6) - 0.1. In figure 7 we have also over plotted the line corresponding to H0 dL(z = 0.63) = constant. The coincidence of this line (which roughly corresponds to dL at a redshift in the middle of the data) with the probability ellipses indicates that it is this quantity which essentially determines the nature of the result.
Figure 8. The luminosity distance for a class of models with widely varying NR, but with a constant value for q (0.8NR - 0.6) are shown in the figure. It is clear that when q is fixed, low redshift observations cannot distinguish between the different models even if NR and vary significantly.
We saw earlier that the presence of cosmological constant leads to an accelerating phase in the universe which - however - is not obvious from the above figures. To see this explicitly one needs to display the data in the vs a plane, which is done in figure 9. Direct observations of supernova is converted into dL(z) keeping M a free parameter. The dL is converted into dH(z) assuming k = 0 and using (17). A best fit analysis, keeping (M, NR) as free parameters now lead to the results shown in figure 9, which confirms the accelerating phase in the evolution of the universe. The curves which are over-plotted correspond to a cosmological model with NR + = 1. The best fit curve has NR = 0.32, = 0.68.
Figure 9. Observations of SN are converted into the `velocity' of the universe using a fitting function. The curves which are over-plotted corresponds to a cosmological model with NR + = 1. The best fit curve has NR = 0.32, = 0.68.
In the presence of the cosmological constant, the universe accelerates at low redshifts while decelerating at high redshift. Hence, in principle, low redshift supernova data should indicate the evidence for acceleration. In practice, however, it is impossible to rule out any of the cosmological models using low redshift (z 0.2) data as is evident from figure 9. On the other hand, high redshift supernova data alone cannot be used to establish the existence of a cosmological constant. The data for (z 0.2) in figure 9 can be moved vertically up and made consistent with the decelerating = 1 universe by choosing the absolute magnitude M suitably. It is the interplay between both the high redshift and low redshift supernova which leads to the result quoted above.
This important result can be brought out more quantitatively along the following lines. The data displayed in figure 9 divides the supernova into two natural classes: low redshift ones in the range 0 < z 0.25 (corresponding to the accelerating phase of the universe) and the high redshift ones in the range 0.25 z 2 (in the decelerating phase of the universe). One can repeat all the statistical analysis for the full set as well as for the two individual low redshift and high redshift data sets. Frame (b) and (c) of figure 6 shows the confidence interval based on low redshift data and high redshift data separately. It is obvious that the NR = 1 model cannot be ruled out with either of the two data sets! But, when the data sets are combined - because of the angular slant of the ellipses - they isolate a best fit region around NR 0.3. This is also seen in figure 10 which plots the confidence intervals using just the high-z and low-z data separately. The right most frame in the bottom row is based on the low-z data alone (with marginalisation over ) and this data cannot be used to discriminate between cosmological models effectively. This is because the dL at low-z is only very weakly dependent on the cosmological parameters. So, even though the acceleration of the universe is a low-z phenomenon, we cannot reliably determine it using low-z data alone. The top left frame has the corresponding result with high-z data. As we stressed before, the NR = 1 model cannot be excluded on the basis of the high-z data alone either. This is essentially because of the nature of probability contours seen earlier in frame (c) of figure 6. The remaining 3 frames (top right, bottom left and bottom middle) show the corresponding results in which fixed values of - rather than by marginalising over . Comparing these three figures with the corresponding three frames in 7 in which all data was used, one can draw the following conclusions: (i) The best fit value for is now = 24.05 ± 0.38; the 1 error has now gone up by nearly eight times compared to the result (0.05) obtained using all data. Because of this spread, the results are sensitive to the value of that one uses, unlike the situation in which all data was used. (ii) Our conclusions will now depend on . For the mean value and lower end of , the data can exclude the NR = 1, = 0 model [see the two middle frames of figure 10]. But, for the high-end of allowed 1 range of , we cannot exclude the NR = 1, = 0 model [see the bottom left frame of figure 10].
Figure 10. Confidence contours corresponding to 68 %, 90 % and 99 % based on SN data in the NR - plane using either the low-z data (bottom right frame) or high-z data (the remaining four frames). The bottom right and the top left frames are obtained by marginalizing over while the remaining three uses fixed values for . The values are chosen to be the best-fit value for and two others in the wings of 1 limit. The dashed line corresponds to the flat model. The unbroken slanted line corresponds to H0 dL(z = 0.63) = constant. It is clear that: (i) The 1 error in top left frame (0.38) has gone up by nearly eight times compared to the value (0.05) obtained using all data (see figure 7) and the results are sensitive to the value of . (ii) The data can exclude the NR = 1, = 0 model if the mean or low-end value of is used [see the two middle frames]. But, for the high-end of allowed 1 range of , we cannot exclude the NR = 1, = 0 model [see the bottom left frame]. (iii) The low-z data [bottom right] cannot exclude any of the models.
While these observations have enjoyed significant popularity, certain key points which underly these analysis need to be stressed. (For a sample of views which goes against the main stream, see [81, 82]).