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

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

Renée Hlozek11affiliation: Department of Astrophysics, Oxford University, Oxford, UK OX1 3RH , Joanna Dunkley11affiliation: Department of Astrophysics, Oxford University, Oxford, UK OX1 3RH 22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 33affiliation: Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ USA 08544 , Graeme Addison11affiliation: Department of Astrophysics, Oxford University, Oxford, UK OX1 3RH , John William Appel22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 , J. Richard Bond44affiliation: Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON, Canada M5S 3H8 , C. Sofia Carvalho55affiliation: IPFN, IST, Av. RoviscoPais, 1049-001Lisboa, Portugal & RCAAM, Academy of Athens, Soranou Efessiou 4, 11-527 Athens, Greece , Sudeep Das66affiliation: Berkeley Center for Cosmological Physics, LBL and Department of Physics, University of California, Berkeley, CA, USA 94720 22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 33affiliation: Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ USA 08544 , Mark J. Devlin77affiliation: Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA, USA 19104 , Rolando Dünner88affiliation: Departamento de Astronomía y Astrofísica, Facultad de Física, Pontificía Universidad Católica de Chile, Casilla 306, Santiago 22, Chile , Thomas Essinger-Hileman22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 , Joseph W. Fowler22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 99affiliation: NIST Quantum Devices Group, 325 Broadway Mailcode 817.03, Boulder, CO, USA 80305 , Patricio Gallardo88affiliation: Departamento de Astronomía y Astrofísica, Facultad de Física, Pontificía Universidad Católica de Chile, Casilla 306, Santiago 22, Chile , Amir Hajian44affiliation: Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON, Canada M5S 3H8 33affiliation: Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ USA 08544 22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 , Mark Halpern1010affiliation: Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z4 , Matthew Hasselfield1010affiliation: Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z4 , Matt Hilton1111affiliation: School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD , Adam D. Hincks22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 , John P. Hughes1212affiliation: Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ USA 08854-8019 , Kent D. Irwin99affiliation: NIST Quantum Devices Group, 325 Broadway Mailcode 817.03, Boulder, CO, USA 80305 , Jeff Klein77affiliation: Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA, USA 19104 , Arthur Kosowsky1313affiliation: Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, USA 15260 , Tobias A. Marriage1414affiliation: Dept. of Physics and Astronomy, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218-2686 33affiliation: Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ USA 08544 , Danica Marsden77affiliation: Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA, USA 19104 , Felipe Menanteau1212affiliation: Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ USA 08854-8019 , Kavilan Moodley1515affiliation: Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of KwaZulu-Natal, Durban, 4041, South Africa , Michael D. Niemack99affiliation: NIST Quantum Devices Group, 325 Broadway Mailcode 817.03, Boulder, CO, USA 80305 22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 , Michael R. Nolta44affiliation: Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON, Canada M5S 3H8 , Lyman A. Page22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 , Lucas Parker22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 , Bruce Partridge1616affiliation: Department of Physics and Astronomy, Haverford College, Haverford, PA, USA 19041 , Felipe Rojas88affiliation: Departamento de Astronomía y Astrofísica, Facultad de Física, Pontificía Universidad Católica de Chile, Casilla 306, Santiago 22, Chile , Neelima Sehgal1717affiliation: Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA, USA 94305-4085 , Blake Sherwin22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 , Jon Sievers44affiliation: Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON, Canada M5S 3H8 , David N. Spergel33affiliation: Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ USA 08544 , Suzanne T. Staggs22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 , Daniel S. Swetz77affiliation: Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA, USA 19104 99affiliation: NIST Quantum Devices Group, 325 Broadway Mailcode 817.03, Boulder, CO, USA 80305 , Eric R. Switzer1818affiliation: Kavli Institute for Cosmological Physics, Laboratory for Astrophysics and Space Research, 5620 South Ellis Ave., Chicago, IL, USA 60637 22affiliation: Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ, USA 08544 , Robert Thornton77affiliation: Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA, USA 19104 1919affiliation: Department of Physics , West Chester University of Pennsylvania, West Chester, PA, USA 19383 , Ed Wollack2020affiliation: Code 553/665, NASA/Goddard Space Flight Center, Greenbelt, MD, USA 20771
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).

Figure 1.— Stepping up in power: we show schematically the angular power spectrum (lower panel) resulting from building up the primordial power spectrum in bins (top panel), from  Mpc (left-most curve in the top panel) to  Mpc (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 spectrum (plotted as  mK in the bottom panel) also tends to a spectrum characterized by , also shown as the grey dashed curve.

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 CAMB111http://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 CosmoMC222http://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 aaFor one-tailed distributions, the upper 95% confidence limit is given, whereas the 68% limits are shown for two-tailed distributions.bbThe 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
Table 1Estimated power spectrum bands in units of

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).

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 Riess et al. (2009) 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  Mpc. The horizontal solid line shows a scale-invariant spectrum, while the dashed black line shows the best-fit CDM power-law with from Dunkley et al. (2010), with the spectra corresponding to the variation in spectral index indicated by solid band. The constraints are summarized in Table 1.

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).

Figure 3.— Mapping primordial power to the angular power spectrum: the constraints on the primordial power spectrum from Figure  2 translate into the angular power spectrum of the temperature CMB fluctuations, shown as  mK (left panel) to highlight the higher order peaks. The dashed vertical lines show the multipoles corresponding to the wavenumbers under consideration, using ; these wavenumbers as shown for the high bands. The dark (light) band shows the region for the 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 et al. (2010). The right panel shows the corresponding power spectrum, plotted here as together with WMAP data and data from the QUaD experiment (Brown 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 aaFor one-tailed distributions, the upper 95% confidence limit is given, whereas the 68% limits are shown for two-tailed distributions. ACT+WMAP Power-lawbbThe power-law model for the primordial spectrum is ACT + WMAP binned with prior
Primary
Secondary
Table 2Estimated model parameters and confidence limits for the ACT 2008 Southern Survey data combined with WMAP 

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 , the dark energy density , and the matter density . The light solid curves show the constraints on the parameters from ACT in combination with WMAP data for the CDM case — the vertical lines in the power spectrum panels show the values the power spectrum would take assuming the best-fit power-law from Dunkley et al. (2010). The parameter constraints for this power-law 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 () is . The dashed curves indicate the degeneracy between low values of and primordial power in modes around the position of the first peak.

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.

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 CDM model with spectral index of computed using CAMB and HALOFIT (Smith et al., 2003). The data points between  Mpc 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 et al. (2010) value of 1.3, as we are explicitly using the Hubble constant measurement from Riess et al. (2011) to make a change of units from Mpc to Mpc. The constraints from CMB lensing (Das et al., 2011), from cluster measurements from ACT (Sehgal et al., 2011), CCCP (Vikhlinin et al., 2009) and BCG halos (Tinker et al., 2011), and the power spectrum constraints from measurements of the Lyman– forest (McDonald et al., 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, from Riess et al. (2011). The bottom panel shows the same data plotted on axes where we relate the power spectrum to a mass variance, and illustrates how the range in wavenumber (measured in Mpc) 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 . This highlights the consistency of power spectrum measurements by an array of cosmological probes over a large range of scales.

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.

This work was supported by the U.S. National Science Foundation through awards AST-0408698 for the ACT project, and PHY-0355328, AST-0707731 and PIRE-0507768. Funding was also provided by Princeton University and the University of Pennsylvania, Rhodes Trust (RH), RCUK Fellowship (JD), ERC grant 259505 (JD), NASA grant NNX08AH30G (SD, AH and TM), NSERC PGSD scholarship (ADH), NSF AST-0546035 and AST-060697 (AK), NSF Physics Frontier Center grant PHY-0114422 (ES), SLAC no. DE-AC3-76SF0051 (NS), and the Berkeley Center for Cosmological Physics (SD) Computations were performed on the GPC supercomputer at the SciNet HPC Consortium. We thank Reed Plimpton, David Jacobson, Ye Zhou, Mike Cozza, Ryan Fisher, Paula Aguirre, Omelan Stryzak and the Astro-Norte group for assistance with the ACT observations. We also thank Jacques Lassalle and the ALMA team for assistance with observations. RH thanks Seshadri Nadathur for providing the best-fit power spectrum void models and Chris Gordon, David Marsh and Joe Zuntz for useful discussions. ACT operates in the Chajnantor Science Preserve in northern Chile under the auspices of the Commission Nacional de Investigaci—n Cientifica y Tecnol—gica (CONICYT). Data acquisition electronics were developed with assistance from the Canada Foundation for Innovation.

References

  • Achúcarro et al. (2011) Achúcarro, A., Gong, J., Hardeman, S., Palma, G. A., & Patil, S. P. 2011, JCAP, 1, 30, 1010.3693
  • Adams et al. (2001) Adams, J., Cresswell, B., & Easther, R. 2001, Phys. Rev. D, 64, 123514, arXiv:astro-ph/0102236
  • Amendola et al. (1995) Amendola, L., Gottloeber, S., Muecket, J. P., & Mueller, V. 1995, ApJ, 451, 444, arXiv:astro-ph/9408104
  • Atrio-Barandela et al. (1997) Atrio-Barandela, F., Einasto, J., Gottlöber, S., Müller, V., & Starobinsky, A. 1997, Soviet Journal of Experimental and Theoretical Physics Letters, 66, 397, arXiv:astro-ph/9708128
  • Barnaby & Huang (2009) Barnaby, N., & Huang, Z. 2009, Phys. Rev. D, 80, 126018, 0909.0751
  • Bird et al. (2010) Bird, S., Peiris, H. V., Viel, M., & Verde, L. 2010, ArXiv e-prints, 1010.1519
  • Blanchard et al. (2003) Blanchard, A., Douspis, M., Rowan-Robinson, M., & Sarkar, S. 2003, Astron. Astrophys., 412, 35, astro-ph/0304237
  • Bridges et al. (2009) Bridges, M., Feroz, F., Hobson, M. P., & Lasenby, A. N. 2009, MNRAS, 400, 1075, ADS, 0812.3541
  • Bridges et al. (2007) Bridges, M., Lasenby, A. N., & Hobson, M. P. 2007, MNRAS, 381, 68, ADS, arXiv:astro-ph/0607404
  • Bridle et al. (2003) Bridle, S. L., Lewis, A. M., Weller, J., & Efstathiou, G. 2003, MNRAS, 342, L72, ADS
  • Brown et al. (2009) Brown, M. L. et al. 2009, ApJ, 705, 978, ADS, 0906.1003
  • Burenin et al. (2007) Burenin, R. A., Vikhlinin, A., Hornstrup, A., Ebeling, H., Quintana, H., & Mescheryakov, A. 2007, ApJS, 172, 561, ADS, arXiv:astro-ph/0610739
  • Carlstrom et al. (2009) Carlstrom, J. E. et al. 2009, ArXiv e-prints, ADS, 0907.4445
  • Chantavat et al. (2010) Chantavat, T., Gordon, C., & Silk, J. 2010, ArXiv e-prints, 1009.5858
  • Das et al. (2010) Das, S. et al. 2010, arXiv:1009.0847, ADS, 1009.0847
  • Das et al. (2011) —. 2011, ArXiv e-prints, 1103.2124
  • Dunkley et al. (2005) Dunkley, J., Bucher, M., Ferreira, P. G., Moodley, K., & Skordis, C. 2005, MNRAS, 356, 925, ADS, arXiv:astro-ph/0405462
  • Dunkley et al. (2010) Dunkley, J. et al. 2010, arXiv:1009.0866, ADS, 1009.0866
  • Easther & Peiris (2006) Easther, R., & Peiris, H. 2006, JCAP, 0609, 010, astro-ph/0604214
  • Einasto et al. (1999) Einasto, J., Einasto, M., Tago, E., Starobinsky, A. A., Atrio-Barandela, F., Müller, V., Knebe, A., & Cen, R. 1999, ApJ, 519, 469, arXiv:astro-ph/9812249
  • Fowler et al. (2010) Fowler, J. W. et al. 2010, ApJ, 722, 1148, ADS, 1001.2934
  • Hajian et al. (2010) Hajian, A. et al. 2010, arXiv:1009.0777, ADS, 1009.0777
  • Hannestad (2003) Hannestad, S. 2003, JCAP, 0305, 004, astro-ph/0303076
  • Hincks et al. (2009) Hincks, A. D. et al. 2009, arXiv:0907:0461, ADS, 0907.0461
  • Hunt & Sarkar (2007) Hunt, P., & Sarkar, S. 2007, Phys. Rev. D, 76, 123504, 0706.2443
  • Kates et al. (1995) Kates, R., Muller, V., Gottlober, S., Mucket, J. P., & Retzlaff, J. 1995, MNRAS, 277, 1254, arXiv:astro-ph/9507036
  • Kinney (2001) Kinney, W. H. 2001, Phys. Rev. D, 63, 043001, arXiv:astro-ph/0005410
  • Kinney et al. (2006) Kinney, W. H., Kolb, E. W., Melchiorri, A., & Riotto, A. 2006, Phys. Rev., D74, 023502, astro-ph/0605338
  • Komatsu et al. (2010) Komatsu, E. et al. 2010, arXiv:1001.4538, ADS, 1001.4538
  • Komatsu & Spergel (2001) Komatsu, E., & Spergel, D. N. 2001, Phys. Rev. D, 63, 063002, ADS, arXiv:astro-ph/0005036
  • Kosowsky & Turner (1995) Kosowsky, A., & Turner, M. S. 1995, Phys. Rev. D, 52, 1739
  • Larson et al. (2010) Larson, D. et al. 2010, arXiv:1001.4635, ADS, 1001.4635
  • Lasenby & Doran (2003) Lasenby, A., & Doran, C. 2003, astro-ph/0307311
  • Lewis & Bridle (2002) Lewis, A., & Bridle, S. 2002, Phys. Rev. D, 66, 103511, arXiv:astro-ph/0205436
  • Lewis & Challinor (2002) Lewis, A., & Challinor, A. 2002, Phys. Rev. D, 66, 023531, arXiv:astro-ph/0203507
  • Liddle (2004) Liddle, A. R. 2004, Monthly Notices of the Royal Astronomical Society, 351, L49
  • Lueker et al. (2010) Lueker, M. et al. 2010, ApJ, 719, 1045, ADS, 0912.4317
  • Marriage et al. (2010) Marriage, T. A. et al. 2010, ArXiv e-prints, ADS, 1010.1065
  • Matsumiya et al. (2002) Matsumiya, M., Sasaki, M., & Yokoyama, J. 2002, Phys. Rev. D, 65, 083007, arXiv:astro-ph/0111549
  • McDonald et al. (2006) McDonald, P. et al. 2006, ApJS, 163, 80, ADS, arXiv:astro-ph/0405013
  • Menanteau et al. (2010) Menanteau, F. et al. 2010, ApJ, 723, 1523, ADS, 1006.5126
  • Mukherjee & Wang (2003) Mukherjee, P., & Wang, Y. 2003, ApJ, 599, 1, ADS
  • Nadathur & Sarkar (2010) Nadathur, S., & Sarkar, S. 2010, ArXiv e-prints, 1012.3460
  • Niemack et al. (2010) Niemack, M. D. et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7741, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 1006.5049, ADS
  • Peiris & Verde (2010) Peiris, H. V., & Verde, L. 2010, Phys. Rev. D, 81, 021302, 0912.0268
  • Planck Collaboration et al. (2011) Planck Collaboration et al. 2011, ArXiv e-prints, ADS, 1101.2022
  • Press et al. (1992) Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. (Cambridge University Press)
  • Reichardt et al. (2009) Reichardt, C. L. et al. 2009, ApJ, 694, 1200, ADS, 0801.1491
  • Reid et al. (2010) Reid, B. A. et al. 2010, MNRAS, 404, 60, ADS, 0907.1659
  • Riess et al. (2011) Riess, A. G. et al. 2011, ApJ, 730, 119, ADS, 1103.2976
  • Riess et al. (2009) —. 2009, ApJ, 699, 539, ADS, 0905.0695
  • Sealfon et al. (2005) Sealfon, C., Verde, L., & Jimenez, R. 2005, Phys. Rev. D, 72, 103520, ADS, arXiv:astro-ph/0506707
  • Sehgal et al. (2011) Sehgal, N. et al. 2011, ApJ, 732, 44, 1010.1025
  • Shafieloo & Souradeep (2007) Shafieloo, A., & Souradeep, T. 2007, ArXiv e-prints, 709, ADS, 0709.1944
  • Sherwin et al. (2011) Sherwin, B. D. et al. 2011, ArXiv e-prints, ADS, 1105.0419
  • Sievers et al. (2009) Sievers, J. L. et al. 2009, arXiv:0901.4540, ADS, 0901.4540
  • Silk (1968) Silk, J. 1968, ApJ, 151, 459
  • Smith et al. (2003) Smith, R. E. et al. 2003, MNRAS, 341, 1311, ADS, arXiv:astro-ph/0207664
  • Spergel et al. (2007) Spergel, D. N. et al. 2007, ApJS, 170, 377, arXiv:astro-ph/0603449
  • Swetz et al. (2010) Swetz, D. S. et al. 2010, arXiv:1007.0290, ADS, 1007.0290
  • Takeuchi (2000) Takeuchi, T. 2000, Astrophysics and Space Science, 271, 213
  • Tegmark & Zaldarriaga (2002) Tegmark, M., & Zaldarriaga, M. 2002, Phys. Rev. D, 66, 103508, arXiv:astro-ph/0207047
  • Tinker et al. (2011) Tinker, J. L. et al. 2011, ArXiv e-prints, ADS, 1104.1635
  • Vazquez et al. (2011) Vazquez, J. A., Lasenby, A. N., Bridges, M., & Hobson, M. P. 2011, ArXiv e-prints, ADS, 1103.4619
  • Verde & Peiris (2008) Verde, L., & Peiris, H. 2008, JCAP, 7, 9, ADS, 0802.1219
  • Vikhlinin et al. (2009) Vikhlinin, A. et al. 2009, ApJ, 692, 1060, ADS, 0812.2720
  • Wang et al. (1999) Wang, Y., Spergel, D. N., & Strauss, M. A. 1999, ApJ, 510, 20, ADS, arXiv:astro-ph/9802231
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
53278
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description