The Atacama Cosmology Telescope: a measurement of the primordial power spectrum

TL;DR

The paper addresses whether the primordial power spectrum of adiabatic fluctuations departs from a simple power-law form. It introduces a model-independent, 20-bin reconstruction of $\mathcal{P}(k)$ connected to CMB observables via transfer functions, using cubic-spline interpolation and Markov-chain Monte Carlo parameter estimation with ACT 2008 data plus WMAP. The main finding is no compelling evidence for deviations from a power-law up to $k\sim 0.19$ Mpc$^{-1}$, with a best-fit tilt $n_s\approx 0.963$–$0.965$ and scale invariance disfavored at about $2\sigma$. Mapping to the late-time matter power spectrum, the results are consistent with galaxy clustering and lensing measurements, reinforcing the concordance $\Lambda$CDM framework and illustrating ACT's enhanced sensitivity to small-scale primordial power. The work also highlights the potential of future polarization data to further constrain the primordial spectrum, particularly the $TE$ cross-spectrum at high multipoles.

Abstract

We present constraints on the primordial power spectrum of adiabatic fluctuations using data from the 2008 Southern Survey of the Atacama Cosmology Telescope (ACT). The angular resolution of ACT provides sensitivity to scales beyond \ell = 1000 for resolution of multiple peaks in the primordial temperature power spectrum, which enables us to probe the primordial power spectrum of adiabatic scalar perturbations with wavenumbers up to k \simeq 0.2 Mpc^{-1}. We find no evidence for deviation from power-law fluctuations over two decades in scale. Matter fluctuations inferred from the primordial temperature power spectrum evolve over cosmic time and can be used to predict the matter power spectrum at late times; we illustrate the overlap of the matter power inferred from CMB measurements (which probe the power spectrum in the linear regime) with existing probes of galaxy clustering, cluster abundances and weak lensing constraints on the primordial power. This highlights the range of scales probed by current measurements of the matter power spectrum.

Figures (5)

• Figure 1: Stepping up in power: we show schematically the angular power spectrum (lower panel) resulting from building up the primordial power spectrum $\mathcal{P}(k)$ in bins (top panel), from $k = 0.007$ Mpc$^{-1}$ (left-most curve in the top panel) to $k=0.22$ Mpc$^{-1}$ (right-most curve). The power in each case is normalized to a single amplitude before the step function, and set to zero afterwards, so that as more bins are added to the primordial spectrum, it tends towards a scale-invariant spectrum (shown as the dashed line). Correspondingly, the $C_\ell$ spectrum (plotted as $\ell^3(\ell+1)C^{TT}_\ell/2\pi$ mK$^2$ in the bottom panel) also tends to a spectrum characterized by $n_s=1$, also shown as the grey dashed curve.
• Figure 2: Primordial power constraints: the constraints on the primordial power spectrum from the ACT data in addition to WMAP data compared to the WMAP constraints alone. In both cases, a prior on the Hubble parameter from riessHubble was included. Where the marginalised distributions are one-tailed, the upper errorbars show the 95$\%$ confidence upper limits. On large scales the power spectrum is constrained by the WMAP data, while at smaller scales the ACT data yield tight constraints up to $k=0.19$ Mpc$^{-1}$. The horizontal solid line shows a scale-invariant spectrum, while the dashed black line shows the best-fit $\Lambda$CDM power-law with $n_s = 0.963$ from dunkley/etal:prep, with the spectra corresponding to the $2\sigma$ variation in spectral index indicated by solid band. The constraints are summarized in Table \ref{['pktable']}.
• Figure 3: Mapping primordial power to the angular power spectrum: the constraints on the primordial power spectrum from Figure \ref{['fig:primkplot']} translate into the angular power spectrum of the temperature CMB fluctuations, shown as $\ell^3(\ell+1)C^{TT}_\ell/2\pi$ mK${^2}$ (left panel) to highlight the higher order peaks. The dashed vertical lines show the multipoles corresponding to the wavenumbers under consideration, using $\ell=kd$; these wavenumbers as shown for the high$-k$ bands. The dark (light) band shows the $1\sigma$ region for the $C^{TT}_\ell$ spectra for the ACT+WMAP (WMAP only) data. The best-fit curve using the combination of ACT and WMAP data is shown as the dark solid curve and the dashed black curve shows the best-fit power-law spectrum from dunkley/etal:prep. The right panel shows the corresponding $C^{TE}_\ell$ power spectrum, plotted here as $\ell(\ell+1)C^{TE}_\ell/2\pi~\mu\mathrm{K}^{2},$ together with WMAP data and data from the QUaD experiment brown/etal:2010.
• Figure 4: Parameter constraints: marginalized one dimensional distributions for the parameters determined from the ACT and WMAP data. The top 20 panels in the figure show the likelihoods for the power spectrum parameters directly determined using MCMC methods, while the lower 10 panels show the primary and secondary cosmological parameters and 3 derived quantities: the Hubble parameter $H_0$, the dark energy density $\Omega_\Lambda$, and the matter density $\Omega_m$. The light solid curves show the constraints on the parameters from ACT in combination with WMAP data for the $\Lambda$CDM case --- the vertical lines in the power spectrum panels show the values the power spectrum would take assuming the best-fit $n_s=0.963$ power-law from dunkley/etal:prep. The parameter constraints for this power-law $\Lambda$CDM model is shown as the light curves. The solid dark lines show the distributions from ACT and WMAP data, assuming a prior on the Hubble constant. The best-fit value of the power-law spectral index obtained from fitting the well-constrained bands ($P_5 - P_{17}$) is $n_s = 0.965$. The dashed curves indicate the degeneracy between low values of $\theta_A$ and primordial power in modes around the position of the first peak.
• Figure 5: The reconstructed matter power spectrum: the stars show the power spectrum from combining ACT and WMAP data (top panel). The solid and dashed lines show the nonlinear and linear power spectra respectively from the best-fit ACT $\Lambda$CDM model with spectral index of $n_s = 0.96$ computed using CAMB and HALOFIT smith/etal:2002. The data points between $0.02< k < 0.19$ Mpc$^{-1}$ show the SDSS DR7 LRG sample, and have been deconvolved from their window functions, with a bias factor of 1.18 applied to the data. This has been rescaled from the reid/etal:2010 value of 1.3, as we are explicitly using the Hubble constant measurement from riess/etal:2011 to make a change of units from $h^{-1}$Mpc to Mpc. The constraints from CMB lensing dasetal:2011, from cluster measurements from ACT sehgal/etal:2010b, CCCP vikhlinin/etal:2009 and BCG halos tinker/etal:2011, and the power spectrum constraints from measurements of the Lyman--$\alpha$ forest mcdonald/etal:2006 are indicated. The CCCP and BCG masses are converted to solar mass units by multiplying them by the best-fit value of the Hubble constant, $h=0.738$ from riess/etal:2011. The bottom panel shows the same data plotted on axes where we relate the power spectrum to a mass variance, $\Delta_M/M,$ and illustrates how the range in wavenumber $k$ (measured in Mpc$^{-1}$) corresponds to range in mass scale of over 10 orders of magnitude. Note that large masses correspond to large scales and hence small values of $k$. This highlights the consistency of power spectrum measurements by an array of cosmological probes over a large range of scales.