The No-CDM model

I had proposed that, in the abscence of a known cosmological solution
in a satisfactory relativistic theory of MOND, MOND might be
approximated by a conventional model with no CDM. While this
approximation should fail at some level, I do expect that
it provides a reasonable expectation value for the amplitude ratios
of the peaks. With no non-baryonic dark matter to provide a net driving
term, there is little left to play with (presuming any MOND GR does not
deviate too greatly from regular GR).

So, how do the predictions of the no-CDM model stack
up?

Basically, one expects that, in the absence of the driving term contributed
by CDM, the second peak will be smaller relative to the first than it would
be with CDM. Qualitatively, this is what is seen (and had been seen by
previous experiments). WMAP now
provides data accurate enough for a quantitative test:

Model | L_{2:1} | A_{1:2} | R_{1:2} |

1999 LCDM | 2.4 | 1.8 | 3.5 * |

No CDM | 2.6 | 2.4 | 5.4 * |

WMAP Data | 2.48 | 2.34 | 5.56 |

- L
_{2:1}is the ratio of 2nd to 1st peak position.

- A
_{1:2}is the ratio of 1st to 2nd peak absolute amplitude.

- R
_{1:2}is the ratio of 1st to 2nd peak amplitude relative to the intervening trough.

- 1999 LCDM: most favorable ratios expected for LCDM when No CDM prediction
was made.

- No CDM: best guess baryon density (in 1999).

- * The more general expectation
was that LCDM would have R
_{1:2}< 4 and No CDM would have R_{1:2}> 5.

- WMAP data from Page et al. (2003).
- The complete table of the expected peak ratios published in McGaugh (1999) is here.

Within the uncertainties, both theoretical and experimental, the peak amplitude ratio is bang on what I predicted it would be (McGaugh 1999) for the case of no CDM. When BOOMERanG first reported observations of the first and second peaks, the smallness of the second peak came as a great surprise. No CDM was the only model to correctly predict this in advance. Moreover, the follow-on prediction that precisely this ratio would be observed when the second peak became well resolved has now been realized.

What about the third peak?

This remains the billion Eruo question (or however much it will cost to
fly the Planck mission).
Page et al. (2003) do not offer a
WMAP measurement of the third peak.
They do quote a compilation of other experiments by
Wang et al. (2003).
Taking this number at face value, the second to third peak amplitude
ratio is A_{2:3} = 1.03 +/- 0.20. The LCDM expectation value for
this quantity was 1.1, while the No-CDM expectation was 1.9. By this
measure, LCDM is clearly preferable, in contradiction to the better measured
first-to-second peak ratio.

It is far from obvious that we can trust the measurements of the third peak yet. Various experiments give various answers, and the uncertainty emerging from any compilation depends on the accuracy of the calibrations of the experiments which contribute to it. Anyone who has tried to measure things on the sky knows how hard this is to do. And those who haven't had this experience only need look at the calibration correction made by the BOOMERanG experiment between their 2000 and 2001 data release. Simply looking at the various data around L ~ 800, it does look like there could well be systematic calibration discrepancies between different experiments. This is why, for a proper test, you want all three peaks to be measured accurately by the same experiment. This has not happened.

LCDM fits the WMAP data very well.
Why do the 1:2 and 2:3 peak ratios give different answers?

WMAP power spectrum with LCDM fit |

In LCDM models, the amplitude of the second peak is suppressed relative to the
first by increasing the baryon density. So, in addition to predicting
the correct amplitude ratio, I further
predicted that LCDM fits would find high baryon densities.
This has come to pass, with O_{b}h^{2}=0.024 fequently
being quoted. That is a lot of baryons, and a long way from the best
value for the best determined quantity in cosmology as of only a few years
ago. The CMB measurements are still not competetive with abundance
measurements for constraining the baryon density, as there is a further
degeneracy in the "tilt" (*n*) of the power spectrum. Some fits are
now finding *n* < 1, which also has the effect of reducing the second
peak amplitude. The best fit
WMAP model
(Spergel et al. 2003)
does a bit of both: O_{b}h^{2}=0.022
and *n* = 0.93. So long as we're willing to play this game,
there is no test for the existence of CDM in
the CMB data. A negative test only occurs if CMB and BBN results
clearly conflict, or if the CMB data themselves can not be fit
with a CDM component. Marginal agreement (the situation currently)
certainly does not falsify CDM, but neither does it confirm it.

That said, it is not obvious to me that a simple
No-CDM model can match the *absolute* amplitudes of the peaks.
It may well be that the WMAP
are now good enough to demand a proper treatment with a relativistic
version of MOND. If so, these data provide strong empirical constraints
on the form any such theory must take. The lack of such a theory is
certainly unfortunate, but in its absence the CMB data can neither
confirm nor falsify MOND.

It does appear that whatever the underlying [MOND?] cosmology is, it has
to share many properties of LCDM. A Friedmann model with the correct
LCDM parameters would appear to be the correct *effective*
cosmology. It would seem rather unlikely that this would just fall out
of the "right" relativistic MOND cosmology.
By the same coin, if LCDM is really right,
it must explain the observed MOND phenomena in indidual galaxies.
Though few people seem to appreciate it, this is extraordinarilly
difficult to do. Hence, both alternatives seem impossibly difficult.

Many puzzles

Is the fact that No-CDM predicted the right 1st:2nd peak ratio just a fluke? I might think so, except that it can not be a fluke every time MOND fits a rotation curve. There has to be a physical reason for this regularity in the data, for which no plausible explanation has been offered. Indeed, the problem is so difficult that no one who has considered the full problem has found a solution (Sanders & Begeman 1994; McGaugh & de Blok 1998), while those who claim to offer solutions don't even address the root problem (van den Bosch & Dalcanton 2000; Kaplinghat & Turner 2002).

There remain some substantial puzzles for cosmology as well.
Not all is well for LCDM in the CMB data. The model which best fits the
C_{L} power spectrum fails to fit the real space power
spectrum C(theta) at high significance (>99% c.l.)
(Spergel et al. 2003). The data have power consistent with zero
on scales >60 degrees, a very peculiar result in the context of
any inflationary model. (This same result had previously been obtained by
COBE.)

To me, the greater question is whether the paradigm is right. There have been plenty of occassions on which humans have been quite sure the had the right cosmological model. Though we don't burn people at the stake any more, I am not so sure we are all that much smarter than our forebearers who made this mistake of hubris. I would be more comfortable if the best fit LCDM parameters made any sense, but they do not (having been called "preposterous" by even the most vocal advocates). Yes, a Friedmann model fits the cosmic scale data, but we have had to invoke two tooth fairies in order to accomplish this: dark matter and dark energy. The existence of both these components remains supposition based on the presumption of the correctness of the underlying model. Perhaps the silliness of the parameters we find is an indication that something is broken. Certainly, we have no business claiming that cosmology is a solved problem at least until we confirm the existence of non-baryonic dark matter by laboratory experiments. The LCDM model which works so well to describe the CMB data only does so if this stuff really exists, and in the correct amount.

Finally, cosmological data are approaching the accuracy of rotation curves.

© *Stacy McGaugh, February 17, 2003;
last edited February 21, 2003*

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