Model independent early expansion history and dark energy

We examine model independent constraints on the high redshift and prerecombination expansion history from upcoming cosmic microwave background observations, using a combination of principal component analysis and other techniques. This can be translated to model independent limits on early dark energy and the number of relativistic species $N_{\mathrm{eff}}$. Models such as scaling (Doran-Robbers), dark radiation ($\Delta N_{\mathrm{eff}}$), and barotropic aether fall into distinct regions of eigenspace and can be easily distinguished from each other. Incoming CMB data will map the expansion history from $z=0–{10}^5$, achieving subpercent precision around recombination, and enable determination of the amount of early dark energy and valuable guidance to its nature.

Figure 1

(Top) The bin basis for expansion variations $\delta (a_i)$ is plotted vs $\log a$. (Bottom) The scale factors $a_i$ of the center of each bin in the top panel (with matching colors) are approximately mapped into multipoles by $\ell \approx {\eta }_0/\eta (a_i)$, with the CMB temperature power spectrum overplotted. (Amplitudes of the colored lines are arbitrary.)

Figure 2

The fractional Fisher sensitivities, $\sqrt{F_{ii,\ell }}$, to the expansion variations $\delta (a_i)$ in each bin, color coded as in Fig. 1, are plotted vs multipole. The top panels use log scale in multipole; the bottom panels a linear scale to highlight different regions. The left panels are for the temperature power spectrum only, while the right panels use the variance weighted sum of the T, E, and TE power spectra entering the Fisher information matrix. Polarization information fills in the sensitivity gaps due to the acoustic troughs.

Figure 3

The Fisher information submatrix (left) corresponding to the expansion variation parameters $\delta (a_i)$ is plotted with redder shades representing larger absolute values (more information). The color bar gives the log of the absolute value of elements. The covariance matrix (middle), marginalized over other cosmological parameters, follows the same color scheme, so the best determined parameters (smallest errors) are bluest. Diagonal elements of the Fisher matrix (right), as a fraction of the largest diagonal element, quantify for which redshifts the CMB data are most sensitive to the expansion history. The bump at $\log a\approx -1.1$ reflects reionization.

Figure 4

The fractional precision with which the expansion history can be determined by projected Planck CMB data is plotted vs scale factor, for two different bandwidths. The top (bottom) curve is for 10 (2) bins per decade in scale factor. Subpercent precision can be achieved around decoupling but large swaths of the cosmic history will remain unknown.

Figure 5

Principal components of the CMB observational sensitivity to expansion history are plotted vs $\log a$, for the first 10 and last two modes. The bottom plot shows the ${\sigma }_i$ for all modes.

Figure 6

Three models of early dark energy are analyzed—dark radiation (top row), aether (middle row), and Doran-Robbers (bottom row)—with different recombination era behaviors. The leftmost column shows $\delta (a)$ built up out of PCs, with the thick red line giving the exact model. The second column gives the PC amplitudes $m_i$ and the third column shows the cumulative $S/N$, summing Eq. (17) over the first $i$ PCs in quadrature. The rightmost column shows the bias squared (falling curve), variance (rising curve), and risk squared (top curve) when including the first $i$ PCs. One might choose to keep either those PCs that give the largest jumps in $S/N$ or all those up to the minimum in the risk curve.

Figure 7

(Top) Differences between each PC amplitude $m_i$ are shown for pairs $(X,Y)$ of early dark energy models, with solid curves comparing Doran-Robbers to dark radiation, dashed Doran-Robbers to aether, and dash-dotted dark radiation to aether. The highest peaks indicate the modes with strong discriminating power. (Bottom) Amplitudes of modes 1 and 7 are plotted for the three models, with the $+$’s indicating the values $m_i$ and the ellipses showing the uncertainties ${\sigma }_i$. These two modes can clearly distinguish between each of the three models.

Figure 8

PCA of three models of early dark energy—dark radiation (top row), aether (middle row), and Doran-Robbers (bottom row)—is compared to the exact models in terms of the observable CMB power spectrum. The left column shows the deviation in the temperature power spectrum for the sum of the first 4, 8, 12, 15 modes using $S/N$ ordering of the modes (see Fig. 6). The ${\chi }^2$ of the deviations summing over multipoles is in the middle panels, with the dashed curve using ordering by ${\sigma }_i$ instead. Reconstruction of the theoretical $\delta (a)$ appears in the right panels; note how only certain epochs need be fit well to reproduce the observable CMB. For the aether model (middle row) dashed curves show also the ${\sigma }_i$ ordering results for the $N=8$, 15 cases, which do much worse than $S/N$ ordering.