Supplemental Material

Interpreting BOOMerang:
the Quintessential Guide

P. J. Steinhardt

Princeton University, April 29, 2000

Overview of Conclusions

As discussed below, BOOMerang results:

• are consistent with flatness (a prediction of inflation)
• are consistent with adiabatic fluctuations (a prediction of inflation)
• combined with measured density of galaxy clusters, MORE strongly rule out standard CDM (Omega_matter=1) than before (despite one's first impression, given the low first acoustic peak in BOOMerang) -- click here
• favor Quintessence over Cosmological Constant (based on the angular scale of the first acoustic peak) -- click here
• fit Quintessence reasonably without requiring non-standard choices for the Hubble constant, baryon density, etc. -- click here
• combined with the measured density of galaxy clusters, favor intermediate mass density (Omega_matter ~ 1/3) over low matter density (Omega_matter < 0)

The MAXIMA results:

• strengthen the case for a first peak and are consistent with flatness
• provide new evidence for adiabaticity by suggesting a second and even possibly a third peak (a prediction of inflation)
• make the case against standard CDM (Omega_matter=1) stronger by suggesting an even higher first peak (more inconsistent with high matter density) than BOOMerang;
• do NOT favor Quintessence over Cosmological Constant (based on the angular scale of the first acoustic peak) although the data appears to have significantly less resolution over the first peak
• fit Quintessence (or Cosmological constant) reasonably without requiring non-standard choices for the Hubble constant, baryon density, etc. -- click here
• weaken the case against low mass density (relative to BOOMerang) by allowing a higher first peak, although intermediate mass density appears to fit the combined data sets best (Omega_matter ~ 1/3) over low matter density (Omega_matter < 0.2)

Figure 1: The small panel above and the first below show MAXIMA and and BOOMerang data (and, in the second panel below, COBE) compared to Quintessence and Cosmological Constant models that fits all current constraints on the Hubble parameter (h=0.7), baryon density (Omega_bh^2=0.021), and COBE sigma8 matches the cluster value. The models have a small tilt (n=0.97) and gravitational waves (T/S=0.21) consistent with inflation. The value of the equation-of-state parameter, w, is -2/3. (This is not a "formal" best-fit model, but a good example to illustrate what the current data suggests.)

(N.B. Quintessence is a dynamical, time-evolving, spatially inhomogeneous form of dark energy, such as the energy associated with a time-evolving scalar field, in contrast to a Cosmological Constant, which is a static, uniform vacuum density. Both have negative pressure and can cause the expansion of the universe to accelerate. -- click here for review )

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Forewarned is Forearmed:

The BOOMerang group has presented in Nature (de Bernardis et al., Nature, 27 April 2000) the results from <10% of their Antarctic flight. In their Nature paper, they only discuss directly the first acoustic peak and its implication that the universe is flat, in accordance with inflation. The result represents a verification and significant refinement of the earlier evidence for a first acoustic peak obtained by the MAT experiment (Miller et al, astro-ph/9906421).

The BOOMerang group also showed a figure of the temperature anisotropy power spectrum which, in addition to showing evidence of a first acoustic peak, also displayed some curious anomalies. The following discussion focuses on those anomalies and what they might mean.

However, the reader is forewarned that this discussion is, at this stage, purely academic since the BOOMerang group has not made clear whether the power spectrum shown is representative of the entire data set. Hence, the following is only intended to explain what the data might mean if the true power spectrum matched the small fraction of data analyzed thus far. Also note that the discussion below is based on considering only BOOMerang, and not the other measures on large and small angular scales. By combining them statistically, effects discussed below are reduced in strength to some degree.

What are the anomalies?

The BOOMerang power spectrum in Fig. 2 of their Nature paper has three anomalies when compared to the favored "concordance" model,

a universe with 5% baryons, 30% dark matter, and 65% vacuum density (or cosmological constant)

1st acoustic peak is too low

1st acoustic peak is too far to the left

2nd peak appears to be significantly smaller than anticipated

It is critical to take account of all three anomalies. While the small 2nd peak is what initially captured the attention of most cosmologists, the anomalies with the first peak are important in their own right and have a major impact on any attempt to solve the 2nd peak problem, as will be discussed below.

It is probably also the case the data related to the peak is less prone to systematic errors (esp. scanning and beam errors) that affect the smaller angular scale data associated with the 2nd peak.

1st Peak Anomalies?

Compared to the "concordance model" the low 1st peak could mean:

• tilt of the primordial spectrum
• larger Hubble constant (shorter age of the universe)
• high matter density (could this be a revival of Einstein-de Sitter?? --I think not. **)
• the dark energy is "quintessence" rather than cosmological constant

In the three panels below, each of these degrees of freedom have been explored. The upper panel shows examples of quintessence models, which resolve the 1st peak problems without requiring any change in the Hubble parameter, baryon density or tilt from their previously favored values. Quintessence has the advantage that it both lowers the first peak and shifts it to the left, in agreement with what has been reported.

The second panel of Figure 2 shows the next best fit models -- models with cosmological constant. The problem with these models is that the 1st peak is shifted too far right if the dark energy is cosmological constant, as compared to the BOOMerang reported analysis. (This is not apparent from looking at the figure.)

The third panel shows the worst fit -- the standard cold dark matter (standard CDM model). It may seem peculiar, at first, that standard CDM is the worst fit since it predicts a smaller first acoustic peak than models with either quintessence or cosmological constant. But, closer inspection reveals that the peak shape is really all wrong (see third panel, Figure 2).

Figure 2: comparing fits to three classes of models -- In each case, we first find best fit to BOOMerang. Don't be fooled if the fits below all look reasonable. Parameters were chosen to force them to be that way. But, then we consider whether the best-fit models also satisfy the constraints on the first peak position and on sigma8, and we find that only the quintessence models (first panel) satisfy both.

Combining our two best measurements on large scales, CMB/BOOMerang and constraints on sigma8 from cluster abundance and cluster evolution:

Quintessence fits better than cosmological constant, and both fit much better than the standard CDM model.

2nd Peak too small? and the baryon density...

Whereas the BOOMerang group directs our attention to the first peak, one cannot help but notice the small second peak in their data. The panel below shows that this can be solved by increasing the baryon density by 50% above the nucleosynthesis value, and, in some cases, changing tilt and the Hubble parameter. This is true for ALL of the models.

The models with cosmological constant or quintessence give reasonable values of sigma8 and fit other cosmological obervations. BUT, so far as BOOMerang is concerned, the standard CDM model has gone from very bad to worse: even after adjusting parameters to fit BOOMerang, the conflict between the CMB value of sigma8 and the cluster value remains (within statistical uncertainty) the same or worse. That is, one overpredicts the number of clusters today and in the past by a factor of 100 compared to observations. This is not new -- standard CDM had this problem before BOOMerang. However, after BOOMerang, we are further constrained from adjusting parameters to solve the sigma8 problem. (It could have been otherwise -- it could have been forced us to adjust parameters so that, at the same time we match BOOMerang, we resolve the sigma8 problem. Instead, it adds more weight against standard CDM.)

Of course, the standard CDM is in conflict with many other measures as well, including the Supernovae measurements and the evidence that the universe is undergoing cosmic acceleration.

In my view, the conservative approach is to place more weight on the first peak data (which is less subject to problems with beam and scanning errors); hence, a wait and see attitude about the 2nd peak is probably the most sensible approach.
Addendum (5/9/2000) This conservative attitude seems especially wise in lieu of the MAXIMA results which suggest somewhat more power in the 2nd peak

Figure 3: An ilustration showing that each type of model can be made to match the 2nd peak PROVIDED that the baryon density is increased by 50% over the nucleosynthesis bound. The models with cosmological constant and quintessence also match sigma8 well. When adjusted to match to BOOMerang, the standard CDM model overshoots sigma8 even worse than before.



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What does this tell us about the matter density?

One can go further with the Boomerang data: in debating the energy density of the universe, cosmologists have recently broken up into three camps: those that think the density is very low (Omega_matter < 0.25, say), those that think that Omega_matter = 1 (a rapidly diminishing corps), and those that think Omega_matter is around 1/3, an intermediate value.

Figure 3 above shows that the Omega_matter =1 possibility can be made to match BOOMerang, but only at the expense of violating cluster limits on sigma8 by more than a factor of two (as well as a large number of other observational constraints, including SNe, x-ray limits on baryon fraction, Lyman-alpha absorption, etc.).

The panel below illustrates the problem for the low density case. The model shown in the first panel below has a good fit to BOOMerang and to sigma8, but the Hubble parameter is h=1. In particular, sigma8 as extrapolated from the CMB matches sigma8 obtained from cluster abundance adn evolution.

But, if we try to move a reasonable value of h, say h=0.8, the first problem is that the COBE extrapolated sigma8 is .815 which does not match up with the cluster value, which is 1.145 +/- .2 (2 sigma).

The second problem is that the fit to BOOM, which was ok for h=1.0,is now poor for h=0.8. In Figure 4b below, you can see the red curve is for h=1.0 and the curve for h=0.8, and that the latter is a significantly worse fit --- it is too high.

But, now I claim you are trapped. It is necessary to raise the CMB extrapolated sigma8 value but move the CMB curve down. These push in diametrically opposite directions. The changes needed to recover a good fit with h=0.8 for BOOMerang -- more tilt, increasing w -- all make the CMB sigma8 smaller, an even worse fit. Decreasing h further makes both BOOMerang and sigma8 problems worse.

So, that is essentially a rigorous proof that we cannot reconcile low density models with BOOMerang + clusters (sigma8) constraints.

Hence, combining all three figures, BOOMerang + clusters eliminates the low density and high density regime, forcing us towards the intermediate (Omega_m ~ 1/3) regime.

Figure 4a and 4b: Shows the power spectrum for a low density (Omega_matter =0.2) model that fits BOOMerang and sigma8 well, but with h=1.0. If we change to a more reasonable h=0.8, the BOOMerang fit is much worse (see Fig. 4b), and sigma8 does not fit. Any effort to improve the BOOMerang fit in 3b causes the sigma8 problem to get worse. Hence, these figures show that the low density models cannot fit both BOOMerang and sigma8.