The Atacama Cosmology Telescope: a measurement of the primordial power spectrum
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 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 Mpc. 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.
1. Introduction
The cosmic microwave background (CMB) is light from the nascent universe, which probes early-universe physics. Measurements of the small-scale anisotropies of this radiation provide us with powerful constraints on many cosmological parameters, e.g., Reichardt et al. (2009); Sievers et al. (2009); Komatsu et al. (2010); Lueker et al. (2010); Dunkley et al. (2010).
In particular, the CMB constrains the power spectra of scalar and tensor perturbations, the relic observables associated with a period of inflation in the early universe (Wang et al., 1999; Tegmark & Zaldarriaga, 2002; Bridle et al., 2003; Mukherjee & Wang, 2003; Easther & Peiris, 2006; Kinney et al., 2006; Bridges et al., 2007; Shafieloo & Souradeep, 2007; Spergel et al., 2007; Verde & Peiris, 2008; Reichardt et al., 2009; Chantavat et al., 2010; Bridges et al., 2009; Peiris & Verde, 2010; Vazquez et al., 2011). The standard models of inflation predict a power spectrum of adiabatic scalar perturbations close to scale-invariant. Such models are often described in terms of a spectral index and an amplitude of perturbations as where is a pivot scale. A wide variety of models, however, predict features in the primordial spectrum of perturbations, which alter the fluctuations in the CMB (Amendola et al., 1995; Kates et al., 1995; Atrio-Barandela et al., 1997; Wang et al., 1999; Einasto et al., 1999; Kinney, 2001; Adams et al., 2001; Matsumiya et al., 2002; Blanchard et al., 2003; Lasenby & Doran, 2003; Hunt & Sarkar, 2007; Barnaby & Huang, 2009; Achúcarro et al., 2011; Nadathur & Sarkar, 2010; Chantavat et al., 2010), which can be constrained using reconstruction of the primordial power.
Primordial fluctuations evolve over cosmic time to form the large scale structures that we see today. Therefore, a precision measurement of the power spectrum of these fluctuations, imprinted on the CMB, impacts all aspects of cosmology. Recent measurements of the CMB temperature and polarization spectra have put limits on the deviation from scale invariance including a variation in power-law with scale (e.g., a running of the spectral index, Kosowsky & Turner (1995)); in particular data from the Atacama Cosmology Telescope (ACT) (Das et al., 2010; Dunkley et al., 2010) combined with WMAP satellite data (Larson et al., 2010) find no evidence for running of the spectral index with scale and disfavor a scale-invariant spectrum with at .
In this work we probe a possible deviation from power-law fluctuations by considering the general case where the power spectrum is parameterized as bandpowers within bins in wavenumber (or ) space. This ‘agnostic’ approach allows for a general form of the primordial power spectrum without imposing any specific model of inflation on the power spectrum, and facilitates direct comparison with a wide range of models. Such tests of the primordial power have been considered by various groups (Wang et al., 1999; Tegmark & Zaldarriaga, 2002; Bridle et al., 2003; Hannestad, 2003; Sealfon et al., 2005; Spergel et al., 2007; Verde & Peiris, 2008; Peiris & Verde, 2010; Vazquez et al., 2011). We revisit the calculation because of ACT high sensitivity over a broad range in angular scale.
This paper is based on data from 296 square degrees of the ACT 2008 survey in the southern sky, at a central frequency of 148 GHz. The resulting maps have an angular resolution of 1.4’ and a noise level of between and K per arcmin. A series of recent papers has described the analysis of the data and scientific results. The ACT experiment is described in Swetz et al. (2010), the beams and window functions are described in Hincks et al. (2009), while the calibration of the ACT data to WMAP is discussed in Hajian et al. (2010). The power spectra measured at 148 GHz and 218 GHz are presented in Das et al. (2010), and the constraints on cosmological parameters are given in Dunkley et al. (2010). A high-significance catalog of clusters detected through their Sunyaev-Zel’dovich (SZ) signature is presented in Marriage et al. (2010); the clusters are followed up with multi-wavelength observations described in Menanteau et al. (2010); the cosmological interpretation of these clusters is presented in Sehgal et al. (2011).
2. Methodology
2.1. Angular Power spectrum
Following the work of Wang et al. (1999); Tegmark & Zaldarriaga (2002); Bridle et al. (2003); Mukherjee & Wang (2003) and Spergel et al. (2007), we parameterize the primordial power spectrum using bandpowers in 20 bins, logarithmically spaced in mode from to Mpc , with . To ensure the power spectrum is smooth within bins, we perform a cubic spline such that:
where the are the power spectrum amplitudes within bin normalised so that corresponds to scale invariance. The is a normalized amplitude of scalar density fluctuations, which we take to be (Larson et al., 2010), and is the amplitude for a power-law spectrum around a pivot scale of Mpc. We do not vary the amplitude in our analysis as the power in the individual bands is degenerate with the overall amplitude; if a higher value was used, the estimated bandpowers would be lower by the corresponding amount, as we are measuring the total primordial power within a bin. The coefficients are the second derivatives of the input binned power spectrum data (Press et al., 1992), is the width of the step and and We do not impose a ‘smoothness penalty’ as discussed in Verde & Peiris (2008) and in Peiris & Verde (2010). Adding more parameters to the parameter set makes it easier for the model to fit bumps and wiggles in the spectrum, hence as the number of bins increases, this parameterization will fit the noise in the data, particularly on large scales (small values of ). This in turn is expected to increase the goodness-of-fit of the model by approximately one per additional parameter. Hence, a model that fits the data significantly better than the standard CDM power-law would yield an increase in the likelihood of more than one per additional parameter in the model. The logarithmic spacing in means that this is less of a problem at high multipoles, as many measurements are used to estimate the power in each band.
The primordial power spectrum is related to the CMB power spectra through the radiation transfer functions , and (defined as in Komatsu & Spergel (2001)) as:
(2) |
where and index or , corresponding to temperature or the two modes of polarization. The correspondence between multipole and mode (in Mpc) is roughly , where Mpc is the comoving distance to the last scattering surface. Figure 1 shows schematically how the primordial power spectrum translates to the temperature angular power spectrum. In each case, a single step function is used for the primordial power spectrum in Eq. (2).
Previous analyses have only constrained the primordial power out to Mpc (Bridle et al., 2003; Spergel et al., 2007; Peiris & Verde, 2010). The arcminute resolution of ACT means that one can constrain the primordial power out to larger values of Mpc). The primary CMB power spectrum decreases exponentially due to Silk damping (Silk, 1968) at multipoles greater than , while the power spectrum from diffuse emission of secondary sources begins to rise from . The ACT measurement window between provides a new window with which to constrain any deviation from a standard power-law spectrum, as this is in the range of scales before the power from secondary sources dominates. We use the 148 GHz measurements from the 2008 ACT Southern Survey, and include polarization and temperature measurements from the WMAP satellite with a relative normalization determined by Hajian et al. (2010). We use the ACT likelihood described in Dunkley et al. (2010) and the WMAP likelihood found in Larson et al. (2010).
2.2. Parameter estimation
Our cosmological models are parameterized using:
(3) |
where is the cold dark matter density; is the baryon density; is the dimensionless Hubble parameter such that kmsMpc; is the ratio of the sound horizon to the angular diameter distance at last scattering, and is a measure of the angular scale of the first acoustic peak in the CMB temperature fluctuations; is the optical depth at reionization, which we consider to be ‘instantaneous’ (equivalent to assuming a redshift range of for CMB fluctuations) and is the vector of bandpowers where describes a scale-invariant power spectrum. We assume a flat universe in this analysis. In addition, we add three parameters, to model the secondary emission from the Sunyaev-Zel’dovich effect from clusters, Poisson-distributed and clustered point sources respectively, marginalizing over templates as described in Dunkley et al. (2010) and Fowler et al. (2010). We impose positivity priors on the amplitudes of these secondary parameters. We modify the standard Boltzmann code CAMB^{1}^{1}1http://cosmologist.info/camb (Lewis & Challinor, 2002) to include a general form for the primordial power spectrum, and generate lensed theoretical CMB spectra to , above which we set the spectra to zero for computational efficiency, as the signal is less than of the total power.
The likelihood space is sampled using Markov chain Monte Carlo methods. The probability distribution is smooth, single-moded and close to Gaussian in most of the parameters. These properties make the 27-dimensional likelihood space less demanding to explore than an arbitrary space of this size: the number of models in the Markov chain required for convergence scales approximately linearly with the number of dimensions. Sampling of the parameter space is performed using CosmoMC^{2}^{2}2http://cosmologist.info/cosmomc (Lewis & Bridle, 2002). The analysis is performed on chains of length We sample the chains and test for convergence following the prescription in Dunkley et al. (2005), using an optimal covariance matrix determined from initial runs.
Wavenumber (Mpc) | Power spectrum band ^{a}^{a}For one-tailed distributions, the upper 95% confidence limit is given, whereas the 68% limits are shown for two-tailed distributions.^{b}^{b}The primordial power spectrum is normalized by a fixed overall amplitude (Larson et al., 2010). | WMAP only binned | ACT+WMAP binned |
---|---|---|---|
0.0010 | |||
0.0014 | |||
0.0019 | |||
0.0025 | |||
0.0034 | |||
0.0047 | |||
0.0064 | |||
0.0087 | |||
0.0118 | |||
0.0160 | |||
0.0218 | |||
0.0297 | |||
0.0404 | |||
0.0550 | |||
0.0749 | |||
0.1020 | |||
0.1388 | |||
0.1889 | |||
0.2571 | |||
0.3500 |
We impose limiting values on the power spectrum bands for all . To avoid exploring regions of parameter space inconsistent with current astronomical measurements, we impose a Gaussian prior on the Hubble parameter today of from Riess et al. (2009).
3. Results
3.1. Primordial Power
Figure 2 shows the constraints on the primordial power spectrum from measurements of the cosmic microwave background. The shaded bands are the constraints on the power spectrum from WMAP measurements alone. Over this range of scales there is no indication of deviation from power-law fluctuations. As was shown in Spergel et al. (2007), the lack of data at multipole moments larger than restricts any constraints on the primordial power spectrum at Mpc. In contrast, the combined ACT/WMAP constraints are significantly improved, particularly for the power at scales Mpc. The resulting power spectrum is still consistent with a power-law shape, with (the best-fit value from Dunkley et al. (2010)). Despite the fact that we have added 18 extra degrees of freedom to the fit, a scale-invariant spectrum (, shown by the horizontal line on Fig. 2) is disfavored at . In addition, we find no evidence for a significant feature in the small-scale power. The bands at Mpc in the ACT+WMAP case are largely unconstrained by the data. Including 218 GHz ACT data will improve the measurements of the primordial power, since it will relieve the degeneracies between the binned primordial power, the clustered IR source power, and the Poisson source power, all of which provide power at .
The estimated primordial power spectrum values are summarized in Table 1. The CMB angular power spectra corresponding to the allowed range in the primordial power spectrum (at ) are shown in Figure 3 for WMAP-alone compared to WMAP and ACT combined. The temperature-polarization cross spectra corresponding to these allowed models are also shown in Figure 3, indicating how little freedom remains in the small-scale spectrum. This allowed range in at multipoles will be probed by future CMB polarization experiments such as Planck (Planck Collaboration et al., 2011), ACTPol (Niemack et al., 2010) and SPTPol (Carlstrom et al., 2009).
In this analysis, we use a prior on the Hubble constant. Removing this prior reveals a degeneracy between the primordial power on scales Mpc(bands in Figure 4), and the set of parameters describing the contents and expansion rate of the universe. Both affect the first acoustic peak. This degeneracy was previously noted in, e.g., Blanchard et al. (2003); Hunt & Sarkar (2007); Nadathur & Sarkar (2010), where a power spectrum model “bump” at Mpc was found to be consistent with observations in the context of a low , and without any dark energy. Along this degeneracy, the primordial spectrum can be modified to move the position of the first peak to larger scales (relative to power-law), also increasing its relative amplitude. Since the first peak position is well measured by WMAP, this increase in angular scale is compensated by decreasing . In a flat universe, this corresponds to a decrease in the Hubble constant and the cosmological constant. The matter density increases to maintain the first peak amplitude. The baryon density then decreases to maintain the relative peak heights. Imposing a prior on the Hubble constant has the effect of breaking this degeneracy. Alternatively, one could impose a prior on the baryon density from Big Bang Nucleosynthesis () which would disfavor the low- models, as indicated in the top left panel of the bottom rows in Figure 4.
It is worth noting that the increase in the matter density along this degeneracy also increases the gravitational lensing deflection power, as a universe with a larger matter content exhibits stronger clustering at a given redshift. ACT maps have sufficient angular resolution to measure this deflection of the CMB (Das et al. 2011). Even without a strong prior on the Hubble constant, models with a bump in the primordial spectrum and (with Dark Energy density) are disfavored at from the lensing measurement alone, a result similar to that discussed in Sherwin et al. (2011). Although parameterized differently, the same argument applies to the primordial spectrum considered by Hunt & Sarkar (2007), motivated by phase transitions during inflation, that eliminates dark energy but which is also disfavored at by the lensing for a standard cold dark matter model.
Parameter ^{a}^{a}For one-tailed distributions, the upper 95% confidence limit is given, whereas the 68% limits are shown for two-tailed distributions. | ACT+WMAP Power-law^{b}^{b}The power-law model for the primordial spectrum is | ACT + WMAP binned with prior | |
---|---|---|---|
Primary | |||
Secondary | |||
The estimated cosmological parameters are given in Table 2 and the marginalized one-dimensional likelihoods are shown in Figure 4. While the binned model adds 18 additional parameters to the parameter set, only 13 of those parameters are well constrained. All cosmological parameters in the binned power spectrum model are consistent with those derived using the concordance 6-parameter model with a power-law primordial spectrum. The addition of 13 new parameters which are substantially constrained by the data increases the likelihood of the model such that Using a simple model comparison criterion like the Akaike Information Criterion (e.g., Liddle, 2004; Takeuchi, 2000), the binned power spectrum model is disfavored over the standard concordance model. Further, we find a power-law slope fit to the 13 constrained bands in power spectrum space (bands labeled from to in Figure 4) of , which is, as expected, consistent with the constraints on the spectral index from Dunkley et al. (2010).
3.2. Reconstructed
The primordial power spectrum translates to the angular power spectrum of the CMB, but can in addition be mapped to the late-time matter power spectrum through the growth of perturbations:
(4) |
where gives the growth of matter perturbations, is the matter transfer function, and the are the fitted values as in Eq. (1). This mapping enables the constraints on the power spectrum from the CMB to be related to power spectrum constraints from other probes at (Tegmark & Zaldarriaga, 2002; Bird et al., 2010). We illustrate the power spectrum constraints from the ACT and WMAP data in Figure 5. We take and from a CDM model, but neither varies significantly as the cosmological parameters are varied within their errors in the flat cosmology we consider in this work. The constraints from the CMB alone overlap well with the power spectrum measurements from the SDSS DR7 LRG sample (Reid et al., 2010), which have been deconvolved from their window functions. The ACT data allow us to probe the power spectrum today at scales Mpc using only the CMB, improving on previous constraints using microwave data. In addition, the lensing deflection power spectrum also provides a constraint on the amplitude of matter fluctuations at a comoving wavenumber of Mpc at a redshift . The recent measurement of CMB lensing by ACT (Das et al., 2011) is shown as Mpc Mpc on Figure 5. These two measurements are consistent with each other and come from two independent approaches: the lensing deflection power is a direct probe of the matter content at this scale (with only a minor projection from to ), while the primordial power is projected from the scales at the surface of last scattering at to the power spectrum today.
Finally, cluster measurements provide an additional measurement of the matter power spectrum on a characteristic scale , corresponding to the mass of the cluster, where is the matter density of the universe today. We compute the amplitude of the power spectrum at the scale from reported values as
(5) |
where is the concordance CDM value (Larson et al., 2010). We use the measurement of from clusters detected by ACT, at a characteristic mass of (Sehgal et al., 2011), as well as measurements from the Chandra Cosmology Cluster Project (CCCP) (Vikhlinin et al., 2009), measured from the 400 square degree ROSAT cluster survey (Burenin et al., 2007). The quoted value of is given at a characteristic mass of In addition, we illustrate constraints from galaxy clustering calibrated with weak lensing mass estimates of brightest cluster galaxies (BCG) (Tinker et al., 2011), quoted as In this case, we compute the characteristic mass (and hence ) from the inverse variance weighted average mass of the halos (from Table 2 in Tinker et al. (2011)) as . To remove the dependence on cosmology, the CCCP and BCG mass measurements are multiplied by a factor of (where is taken from the recent Riess et al. (2011) result). The ACT cluster measurement, however, is already expressed in solar mass units, and hence this operation is not required.
Power spectrum constraints from measurements of the Lyman– forest are shown at the smallest scales probed. The slanted errorbars for the SDSS and Lyman– data reflect the uncertainty in the power spectrum measurement from the Hubble constant uncertainty alone. Again, these data are normally plotted as a function of Mpc, hence we propagate the error on the Hubble parameter from Riess et al. (2011) through to the plotted error region in both wavenumber and power spectrum.
Transforming from units of power spectrum to mass variance (indicated in the bottom panel of Figure 5), allows one to visualize directly the relationship between mass scale and variance. While for galaxies, the variance decreases as the mass increases and we probe the largest scales, covering ten orders of magnitude in the range of masses of the corresponding probes.
4. Conclusions
We constrained the primordial power spectrum as a function of scale in 20 bands using a combination of data from the 2008 Southern Survey of the Atacama Cosmology Telescope and WMAP data. We make no assumptions about the smoothness of the power spectrum, beyond a spline interpolation between power spectrum bands. The arcminute resolution of ACT constrains the power spectrum at scales Mpc which had not yet been well constrained by microwave background experiments. This allows us to test for deviations from scale invariance in a model-independent framework. We find no significant evidence for deviation from a power-law slope. When a power-law spectrum is fit to our well-constrained bands, our best-fit slope of is consistent with that determined directly from a standard parameter space of CDM models with a power-law spectrum, using the same data. Mapping the primordial power to the late-time power spectrum using the fluctuations in the matter density, we obtain measurements of the power spectrum today from the cosmic microwave background which are consistent with results from galaxy redshift surveys, but which also probe the power spectrum to much larger scales, Mpc, over mass ranges . Finally, the allowed range in the primordial power from the high- ACT temperature power spectrum measurements constrains the allowed range in the polarization-temperature cross spectrum, which will be probed with future polarization experiments.
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