Constraining Large Scale Structure Theories with the Cosmic Background Radiation

TL;DR

This paper demonstrates how joint CMB and LSS observations, analyzed through Bayesian inference across inflationary model sequences (tilted $\Lambda$CDM, $\Lambda$hCDM, and $o$CDM) and constrained by COBE and current bandpowers, can tightly bound the primordial fluctuation spectra and the matter content of the universe. It finds a near scale-invariant initial spectrum ($n_s \approx 1$) and strong evidence for a nonzero cosmological constant ($\Omega_\Lambda \approx 0.7$) in a 13 Gyr framework, while open models are disfavored; these conclusions are strengthened by incorporating LSS priors. The work also details the role of radiative transport, secondary anisotropies, and topology constraints, and it provides forecasts showing that future MAP/Planck-era data could dramatically improve parameter precision under idealized conditions. Overall, the study highlights the power of CMB+LSS synergy for discriminating inflationary scenarios and guiding the design of next-generation cosmological surveys.

Abstract

We review the relevant 10+ parameters associated with inflation and matter content; the relation between LSS and primary and secondary CMB anisotropy probes; COBE constraints on energy injection; current anisotropy band-powers which strongly support the gravitational instability theory and suggest the universe could not have reionized too early. We use Bayesian analysis methods to determine what current CMB and CMB+LSS data imply for inflation-based Gaussian fluctuations in tilted $Λ$CDM, $Λ$hCDM and oCDM model sequences with age 11-15 Gyr, consisting of mixtures of baryons, cold (and possibly hot) dark matter, vacuum energy, and curvature energy in open cosmologies. For example, we find the slope of the initial spectrum is within about 5% of the (preferred) scale invariant form when just the CMB data is used, and for $Λ$CDM when LSS data is combined with CMB; with both, a nonzero value of $Ω_Λ$ is strongly preferred ($\approx 2/3$ for a 13 Gyr sequence, similar to the value from SNIa). The $o$CDM sequence prefers $Ω_{tot}<1 $, but is overall much less likely than the flat $Ω_Λ\ne 0$ sequence with CMB+LSS. We also review the rosy forecasts of angular power spectra and parameter estimates from future balloon and satellite experiments when foreground and systematic effects are ignored.

Figures (5)

• Figure 1: The bands in comoving wavenumber $k$ probed by CMB primary and secondary anisotropy experiments, in particular by the satellites COBE, MAP and Planck, and by large scale structure (LSS) observations are contrasted. The width of the CMB photon decoupling region and the sound crossing radius ($\Delta \tau_{\gamma dec}, c_s\tau_{\gamma dec}$) define the effective acoustic peak range for primary anistropies (those involving linear fluctuations). Secondary anisotropies arise only once matter has gone nonlinear. Sample (linear) gravitational potential power spectra (actually ${\cal P}_\Phi^{1/2}(k)$) are also plotted, and the $y$-axis values refer to ${\cal P}_\Phi^{1/2}/10^{-5}$ (which is dimensionless). The horizontal dotted line is the post-inflation scale invariant power spectrum, which is bent down as the universe evolves by an amount dependent upon the matter content. The hatched region at low $k$ gives the 4 year DMR error bar on the $\Phi$ amplitude in the COBE regime. The solid data point in the cluster-band denotes the $\Phi$ constraint from the abundance of clusters (for $\Omega_{tot}$=1,$\Omega_{\Lambda}$=0). The open circles are estimates of the linear $\Phi$ power from current galaxy clustering data by Peacock (1997). A bias is "allowed" to (uniformly) raise the shapes to match the observations. The corresponding linear density power spectra, ${\cal P}_\rho^{1/2}(k)$, are also shown rising to high $k$. Models shown in Fig. \ref{['fig:probes']} are the "standard" $n_s=1$ CDM model (labelled $\Gamma=0.5$ with $\Omega_{nr}=1$, ${\rm h}=0.5$), a tilted ($n_s=0.6$, $\Gamma=0.5$) CDM model and a model with the shape modified ($\Gamma=0.25$) by changing the matter content of the Universe, e.g., $\Omega_{nr}=0.36$, ${\rm h=0.7}$. The bands at high $k$ associated with object formation (cls, gal, etc. ) and the filters showing the bands various CMB experiments probe are discussed in the text.
• Figure 2: The ${\cal C}_\ell$ anisotropy bandpower data for experiments up to summer 1998 are shown in the upper left panel. The data are optimally combined into 9 bandpower estimates (with one-sigma errors) shown in the upper right panel. To guide the eye an untilted COBE-normalized sCDM model is repeated in all panels. The rest of the panels show forecasts of how accurate ${\cal C}_\ell$ will be determined for this model from balloon and satellite experiments, with parameters given in Table \ref{['tab:exptparams']}.
• Figure 3: The 9 bandpower estimates from current anisotropy data are compared with various 13 Gyr model sequences: (1) $H_0$ from 50 to 90, $\Omega_{\Lambda}$, 0 to 0.87, for an untilted $\Lambda$CDM sequence; (2) $n_s$ from 0.85 to 1.25 for the $H_0=70$$\Lambda$CDM model ($\Omega_{\Lambda}=.66$); (3) $\Omega_B{\rm h}^2$ from 0.003 to 0.05 for the $H_0=70$$\Lambda$CDM model; (4) $H_0$ from 50 to 65, $\Omega_{k}$ from 0 to 0.84 for the untilted oCDM sequence; (5) the same for a $n_s=0.9$ oCDM sequence, clearly at odds with the data; (6) $H_0$=50 sequence with neutrino fractions varying from 0.1 to 0.95; (7) shows an isocurvature CDM sequence with positive isocurvature tilts ranging from 0 to 0.8; (8) shows that sample defect ${\cal C}_\ell$'s from Pen, Seljak and Turok (1997) do not fare well compared with the current data; ${\cal C}_\ell$'s from (Allen et al. 1997) are similar. The bottom right panel is extended to low values to show the magnitude of secondary fluctuations from the thermal SZ effect for the $\Lambda$CDM model. The kinematic SZ ${\cal C}_\ell$ is significantly lower. Dusty emission from early galaxies may lead to high signals, but the power is concentrated at higher $\ell$, with possibly a weak tail because galaxies are correlated extending into the $\ell \mathrel{\hbox{$\mathchar"218$} \hbox{$\mathchar"13C$}} 2000$ regime.
• Figure 4: The first column shows unfiltered $140^\circ$ diameter dmr$A$+$B$ maps centered on the North Galactic Pole, the second shows them after Wiener-filtering (with monopole, dipole and quadrupole removed), the third the South Pole version, with the $nth$ contour as noted and negative contours heavier than positive ones. The Wiener maps use a model which fits the correlation function and amplitude of the DMR data (specifically, the $n_s$=1 sCDM model was used, but insensitive to even rather significant variations). The maps have been smoothed by a $1.7^\circ$ Gaussian filter. all is 53+90+31$A$+$B$. Although higher noise results in filtering on greater angular scales, the large scale features of all maps are the same. This is also borne out by detailed statistical comparisons map to map. The last column shows some theoretical realizations, after optimal filtering. The first two rows are the NGP and SGP for a $n_s$=1 CDM model. The lower two rows are for a 3-torus topology, with repetition length $d_T=9000 {\, {\rm h}^{-1}~\rm Mpc}$, 1.5 times the horizon radius, in all three directions, a model strongly ruled out because of the high degree of positive correlation between the North and Southern hemispheres that the periodicity induces (Bond, Pogosyan & Souradeep 1998). Highly correlated patterns also exist for small compact hyperbolic models and lead to constraints on manifold size.
• Figure 5: Likelihood curves for fixed-age $\Lambda$CDM and oCDM sequences, marginalized over $\sigma_8$ and $n_s$. A Gaussian approximation to the likelihood places 1,2,3$\sigma$ at the horizontal dashed lines. The 1$\sigma$ ranges are explicitly given in the text. The dotted curves are for CMB only, solid for CMB+LSS. The right panels are equivalent to the left, but translated to $\Omega_{nr}$ ($1-\Omega_\Lambda$ for $\Lambda$CDM, $\Omega_{tot}$ for oCDM). The curves shown are for no GW, but there is little difference if GW are included. The sequence labelled $m_\nu$ is for the 13 Gyr $\Lambda$hCDM sequence with a fixed $\Omega_{m\nu}/\Omega_{nr}$ ratio of 0.2, and 2 degenerate neutrino species. Even this case favours slightly a nonzero $\Omega_\Lambda$. In the upper left panels, the 68% confidence limits for the values of $\Omega_m$ estimated using the luminosity-distance relation for Type I supernovae from Perlmutter et al. and Craig et al. are also shown. Note that the CMB data alone slightly favours a value of $\Omega_{tot}<1$. The absolute likelihood for the CMB+LSS data strongly favours the $\Lambda$CDM over the oCDM sequences. When just the fully analyzed DMR+SK95+SP94 data are used with LSS, the marginalized results for $\Lambda$CDM are remarkably similar, with the long baseline between DMR and SK95 fixing the freedom in $n_s$ (although values of $n_s$ about 1.1 are now preferred). When only DMR is used along with LSS, $n_s$ is not nailed down, and the resulting freedom implies $\Omega_{nr}=1$ models are not disfavoured. The horizontal error bars in the upper right panel show the $1\sigma$ range of $\Omega_{nr}$ for $\Omega_{nr}+\Omega_\Lambda=1$ models inferred from the supernova Ia observations of Perlmutter et al. 1998 (upper) and Reiss et al. 1998 (lower).