The Baryon Fraction Test

The most obvious difference between CDM and MOND in the context of the microwave background anisotropies is the baryon fraction. CDM is thought to outweigh ordinary baryonic matter by a factor of $\sim 10$ (Evrard, Metzler, & Navarro 1996EMN). The precise value depends on the Hubble constant and on the type of system examined (McGaugh & de Blok 1998aMBa). If instead MOND is the cause of the observed mass discrepancies, there is no CDM. The difference between a baryon fraction $f_b \approx 0.1$ and unity should leave a distinctive imprint on the microwave background.

The main impact of varying the baryon fraction is on the relative amplitude of the peaks in the angular power spectrum of the microwave background (as expanded in spherical harmonics). In general, increasing f_b increases the baryon drag, which enhances the amplitude of compressional (odd numbered) peaks while suppressing rarefaction (even numbered) peaks (Hu, Sugiyama, & Silk 1997HSS). The precise shape of the power spectrum is thus very sensitive to f_b.

In order to investigate this aspect of the problem, I have used CMBFAST (Seljak & Zaldarriagacmbfast 1996) to compute the expected microwave background power spectrum in several representative cases. These have reasonable baryon fractions for each case and baryon-to-photon ratios consistent with primordial nucleosynthesis. Several specific cases are illustrated in Figure 1. These have $\Omega_b = 0.01$, 0.02, and 0.03 with $\Omega_{\rm CDM} = 0.2$or 0 (so f_b = 0.05, 0.1, 0.15 or 1). Other model parameters are held fixed (h = 0.75, T_CMB = 2.726 K, Y_p = 0.24, $N_{\nu}$ = 3), and adiabatic initial conditions are assumed. As a check, models with $\Omega_{\rm CDM} = 0.3$ and 0.4 were also run with the same baryon fractions and H₀ scaled to maintain the same baryon-to-photon ratio. As expected, these resulted in power spectra which are indistinguishable in shape.

 

Figure 1: The power spectrum of temperature anisotropies in the microwave background with (a) and without (b) CDM. Three choices for the baryon density are illustrated in each case. The highest (lowest) baryon content corresponds to the highest (lowest) curve. CDM models with $\Omega_{\rm CDM} = 0.2$, 0.3, and 0.4 all gave indistinguishable result s provided the baryon fraction was the same and H₀ was scaled to maintain the same baryon-to-photon ratio. CDM models have several distinct peaks before $\ell = 1000$ while in the pure baryon cases representing MOND the even numbered peaks have disappeared. Also shown are current measurements with errors $\Delta T < 40 \mu$K from the compilation of Tegmark (http://www.sns.ias.edu/~max/main.html#CMB) as of March 1999.  

I am interested in the shape of the power spectrum, not the absolute positions of the peaks. The latter depends mostly on the scale and geometry of the universe. For purposes of computation, I assume the universe is flat, with $\Omega_{\Lambda} = 1 - \Omega_{\rm CDM} - \Omega_b$. This results in a CDM universe close to the current ``concordant'' model (e.g., Ostriker & SteinhardtOS 1995). In the case of MOND, the resulting model is very close to the de Sitter case. This is a plausible case for a MOND universe (indeed, the relation of inertial mass to a finite vacuum energy density has been suggested as a possible physical basis for MOND: MilgromM99 1999), but is by no means the only possibility. A model with no cosmological constant and $\Omega_m = \Omega_b \approx 0.02$is plausible, but would be very open if the geometry were Robertson-Walker. The position of the first peak in the power spectrum moves to smaller angular scales in open universes because of the dependence of the angular diameter distance on $\Omega_m$. For such low $\Omega_m$ with $\Omega_{\Lambda} = 0$,the position of the first peak occurs at $\ell_1 \gt 1000$. This is inconsistent with recent observations which constrain $\ell_1$ to be near 200 (Miller et al.MAT 1999). However, the geometry in MOND might not be Robertson-Walker, so the position of the first peak is not uniquely specified. It is important to realize that while the position of the first peak provides an empirical constraint on the geometry traversed by the microwave background photons, in the context of MOND this does not necessarily translate into a measure of $\Omega_m$.

The test is therefore not in the absolute positions of the peaks, but in the shape of the spectrum. As the baryon fraction becomes very high ($f_b \rightarrow 1$), the even numbered peaks are suppressed to the point of disappearing. One is left with a spectrum that looks rather like a stretched version of the standard CDM case.

The difference between the CDM and MOND cases is obvious by inspection (Figure 1). However, from an observer's perspective, it is not so easy to distinguish them. The second peak has disappeared in the MOND case, so what would have been the third peak we would now count as the second peak. The absolute positions of the peaks are not specified a priori by either theory. The absolute amplitude in the CDM case is constrained by the need to match large scale structure at z=0. The mechanics to do a similar exercise with MOND do not currently exist, so the absolute amplitude is also not specified a priori. We must therefore rely on the relative amplitudes and positions of the peaks to measure the difference. Since the third peak becomes the second peak in MOND, the observable difference is rather more difficult to perceive than one might have expected, at least for the assumptions made here.

The ratios of the positions and amplitudes of the peaks are given in Table 1. The peak position ratios depend on the sound horizon at recombination, which should not depend on MOND (for constant a₀) because this is well before the universe approaches the low acceleration regime. Other parameters do matter a bit, which can complicate matters.

One difference we could hope to distinguish is in the ratio of the positions of the first and second peaks. In the CDM models, $\ell_2/\ell_1 \approx 2.35$, while in the case of MOND $\ell_2/\ell_1 \approx 2.66$. This requires a positional accuracy determination of $\sim 5\%$beyond $\ell \gt 500$, no small feat.

If we can recognize that second peak is actually missing, so that what we called the second peak in MOND actually corresponds to the third peak in CDM, then the distinction is greater: for CDM, $\ell_3/\ell_1 \approx 3.6$, which should be compared to MOND's 2.66. It is not clear how to do this observationally. Once the position of the first peak is tied down, the given ratio predicts the expected position of the second observable peak (under the assumptions made here). This is not very different in the two cases.

The ratios of the positions of the next observable peaks help not at all. For CDM, $\ell_3/\ell_2 = 1.54$. For MOND, $\ell_3/\ell_2 = 1.57$.

The ratio of the absolute amplitudes of the peaks can also distinguish the two cases, but require comparable accuracy. In CDM, $(C_{{\ell},1}/C_{{\ell},2})_{abs} \approx 1.7$,while in MOND $(C_{{\ell},1}/C_{{\ell},2})_{abs} \approx 2.4$.This may appear to be a substantial difference, but recall that what is measured is the temperature anisotropy. Since $\Delta T \propto \sqrt{C_{\ell}}$,one requires $\sim 7\%$ accuracy to distinguish the two cases at the $2 \sigma$ level. The amplitude ratios of the second and third peaks have a bit more power to distinguish between CDM and MOND, but are more difficult to measure. The precise value of this ratio is very sensitive to f_b in the CDM case. In CDM, $(C_{{\ell},2}/C_{{\ell},3})_{abs} < 1.6$ for f_b > 0.05, while in MOND $(C_{{\ell},2}/C_{{\ell},3})_{abs} \approx 1.9$.

Using the absolute amplitude of the peak heights does not untilize all the information available. In the purely baryonic MOND cases, there is a longer drop from the first peak to the first trough, and a shorter rise to the second peak than in the CDM cases. Therefore, measuring the peak heights relative to the bottom of the intervening trough may be a better approach. To do this, we define $(C_{\ell,n}/C_{\ell,n+1})_{rel} = (C_{\ell,n} - C_{\ell,min})/(C_{\ell,n+1} - C_{\ell,min})$to be the ratio of the amplitudes at maxima n and n+1 less the amplitude of the intervening minimum. This does indeed appear more promising. The purely baryonic MOND cases all have $(C_{\ell,1}/C_{\ell,2})_{rel} \gt 5$, while the CDM cases have $(C_{\ell,1}/C_{\ell,2})_{rel} < 4$(Table 1). This is a nice test, for in most cases this ratio falls well on one side or the other (for $\Omega_b = 0.02$,$(C_{\ell,1}/C_{\ell,2})_{rel}^{MOND}/(C_{\ell,1}/C_{\ell,2})_{rel}^{CDM} = 2$).

By inspection of Figure 1, one might also think that the width of the first peak could be a discriminant, as measured at the amplitude of the first minimum. This is a bit more sensitive to how other parameters shift or stretch the power spectrum. It is also very sensitive to the neutrino mass. Baryonic models with zero neutrino mass have perceptibly broader peaks than the equivalent CDM model, but zero CDM models with finite neutrino mass have peaks which are similar in width to those in the CDM models.