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Determining Fundamental Cosmological Parameters

At multipoles $\ell \lower.5ex\hbox{$\; \buildrel \gt \over \sim \;$}100$, the CMB anisotropies become sensitive to fundamental cosmological parameters, such as the spatial curvature of the Universe, the baryon density and the Hubble constant. Some specific examples have been shown in Figure 1.9a. With the high sensitivity, angular resolution and large sky coverage afforded by PLANCK it is possible, with certain general assumptions, to determine these fundamental parameters to a precision of a few percent (see eg, Bond, Efstathiou & Tegmark 1997, Zaldarriaga, Spergel & Seljak 1997, Efstathiou & Bond 1999).

The unique ability of Planck to distinguish between theoretical models with very similar cosmological parameters is illustrated in figure 1.9a. In this figure, we compare CMB power spectra for spatially flat adiabatic CDM models with different parameters and compare the ability of MAP and Planck to differentiate between them.

Figure 1.9a: Simulations of the CMB power spectrum of a cold dark matter model illustrating how Planck can determine cosmological parameters to high precision. The solid curves in the upper panel shows the CMB power spectrum for an adiabatic CDM model with baryon density b = b h2= 0.0125$, CDM density c= c h2=0.2375, zero cosmological constant, Hubble constant H0 = 50 km s-1 Mpc-1, scale-invariant spectra, ns = 1, nt=0, and a ratio of r=0.2 for the tensor to scalar amplitudes. The dashed lines (barely distinguishable from the solid lines) show spatially flat models with the parameters listed above each figure. The differences in these power spectra are plotted on an expanded scale in the lower panels. The points show simulated observations and 1 errors for the original specifications of MAP, the current improved MAP design (designated MAP+) and for the current design specifications of Planck HFI. These models are marginally distinguishable by MAP but are easily distinguishable by Planck high significance levels.

The Table below displays the 1 errors in estimates of cosmological parameters (assuming a spatially flat universe) expected from MAP and Planck. It is important to emphasise that the examples shown in the Table ignore any systematic errors from Galactic foregrounds, extragalactic point sources, sidelobe leakage, etc. As already demonstrated in in the Phase A report, Planck has been carefully designed to minimise these sources of systematic error. In particular, the wide frequency coverage provided by the Planck LFI and HFI instruments allows accurate subtraction of the Galaxy and extragalactic foregrounds from the primordial cosmological signal. Furthermore, LFI and HFI the have different beam profiles, noise characteristics and both sample the 100 GHz frequency range at comparable sensitivities. The Planck instruments have been designed to allow a large number of cross-checks of the data. Such consistency checks are essential for a comprehensive and convincing analysis of cosmological parameters.
ParameterMAPMAP+PLANCK LFIPLANCK HFI
b/ b 0.110.05 0.016 0.0068
c/ c0.210.11 0.04 0.0063
0.15 0.0810.035 0.0049
Q 0.0140.0046 0.00139
r 0.81 0.67 0.130.49
ns 0.0660.032 0.01 0.005
nt 0.74 0.72 0.57
0.220.190.18

1 errors in estimates of cosmological parameters (spatially flat universe). We use the notation i = i h2 to denote physical densities ( b, c and are the baryon, CDM and cosmological constant densities, respectively). Q denotes the quadrupole amplitude, ns (nt) stands for the scalar (tensor) spectral index, while is the Thomson optical depth back to the redshift at which the universe reionized.

Determining fundamental cosmological parameters with high accuracy would lead to a profound change in our understanding of cosmology. For example:

These and many other questions will, for the first time, be open to quantitative analysis. PLANCK would truly revolutionize cosmology, turning it from a qualitative science fraught with systematic errors and uncertainties, into a quantitative science in which most of the key parameters are constrained to high precision.

Figure 1.9b shows examples of the correlations between some of these parameters. In Figure 1.9b we have assumed a universe with $\Omega_0=1$,h = 0.5, $\Omega_b = 0.05$, spectral index ns=1, r=1, $\tau=0$,and a normalization of $Q_{rms}=20 \mu$K to match the amplitude of the second year COBE measurements at large angular scales. The figure shows likelihood contours for representative pairs of parameters marginalized over all other parameters for CMB experiments at two angular resolutions, $\theta_{FWHM}=10^\prime$ and $\theta_{FWHM} = 1^\circ$.In both cases we have assumed that 1/3 of the sky is mapped at a sensitivity of $\Delta T/T = 2 \times 10^{-6}$per resolution element.


Figure 1.9b: The contours show 50, 5, 2 and 0.1 percentile likelihood contours for pairs of parameters determined from fits to the CMB power spectrum. The figures to the left show results for an experiment with resolution FWHM = 1°. Those to the right for a higher resolution resolution experiment with FWHM = 10' plotted with the same scales (central column) and with expanded scales (rightmost column). Notice how the accuracy of parameter estimation increases dramatically at the higher angular resolution. In these examples, we have assumed that 1/3 of the sky is observed at a sensitivity of T/T = 2x10-6 per resolution element. (See Figure 1.5 for simulations of the CMB power spectra for these experimental configurations.)

This figure shows that: (1) the parameters $\Omega_0$, h and $\Omega_b$ are relatively weakly coupled with each other and that the accuracy with which these parameters can be measured is strongly dependent on angular resolution; (2) some parameters are strongly coupled, for example the amplitude Qrms and spectral index ns, but again with the angular resolution of PLANCK the uncertainty on each parameter can be reduced to a few percent; (3) the uncertainty on the amplitude of the gravitational wave component of the fluctuations, r, depends only weakly on angular resolution because it is determined by low order multipoles (cf. e.g. Knox 1995).

In summary, the analysis of this section shows that observations of the CMB anisotropies with PLANCK are capable of determining fundamental cosmological parameters to high precision. The only assumptions involved are that the primordial fluctuations are adiabatic and characterized by an approximately power-law spectral index, assumptions which themselves can be verified from the PLANCK maps of the CMB anisotropies. The physics underlying these predictions is extremely well understood, involving only linear perturbation theory, and hence the theoretical predictions presented here should be realistic. The analysis provided in From Observations to Scientific Information shows that the frequency coverage and high sensitivity of PLANCK will allow subtraction of foregrounds and discrete sources, so that the CMB anisotropies should be retrieved over at least 1/3 of the sky with a sensitivity of $\Delta T/T
\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}2 \times 10^{-6}$, as assumed in this Section. No other technique is capable of yielding results of such precision for parameters that have yet to be measured reliably, despite many decades of research.


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[last update: 1 August 1999 by P. Fosalba]