Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation
The combination of 7-year data from \map and improved astrophysical data rigorously tests the standard cosmological model and places new constraints on its basic parameters and extensions. By combining the \map data with the latest distance measurements from the Baryon Acoustic Oscillations (BAO) in the distribution of galaxies (Percival et al., 2009) and the Hubble constant () measurement (Riess et al., 2009), we determine the parameters of the simplest 6-parameter CDM model. The power-law index of the primordial power spectrum is (68% CL) for this data combination, a measurement that excludes the Harrison-Zel’dovich-Peebles spectrum by 99.5% CL. The other parameters, including those beyond the minimal set, are also consistent with, and improved from, the 5-year results. We find no convincing deviations from the minimal model. The 7-year temperature power spectrum gives a better determination of the third acoustic peak, which results in a better determination of the redshift of the matter-radiation equality epoch. Notable examples of improved parameters are the total mass of neutrinos, , and the effective number of neutrino species, (68% CL), which benefit from better determinations of the third peak and . The limit on a constant dark energy equation of state parameter from \map+BAO+, without high-redshift Type Ia supernovae, is (68% CL). We detect the effect of primordial helium on the temperature power spectrum and provide a new test of big bang nucleosynthesis by measuring (68% CL). We detect, and show on the map for the first time, the tangential and radial polarization patterns around hot and cold spots of temperature fluctuations, an important test of physical processes at and the dominance of adiabatic scalar fluctuations. The 7-year polarization data have significantly improved: we now detect the temperature--mode polarization cross power spectrum at 21, compared to 13 from the 5-year data. With the 7-year temperature--mode cross power spectrum, the limit on a rotation of the polarization plane due to potential parity-violating effects has improved by 38% to (68% CL). We report significant detections of the Sunyaev-Zel’dovich (SZ) effect at the locations of known clusters of galaxies. The measured SZ signal agrees well with the expected signal from the X-ray data on a cluster-by-cluster basis. However, it is a factor of 0.5 to 0.7 times the predictions from “universal profile” of Arnaud et al., analytical models, and hydrodynamical simulations. We find, for the first time in the SZ effect, a significant difference between the cooling-flow and non-cooling-flow clusters (or relaxed and non-relaxed clusters), which can explain some of the discrepancy. This lower amplitude is consistent with the lower-than-theoretically-expected SZ power spectrum recently measured by the South Pole Telescope collaboration.
Subject headings:cosmic microwave background, cosmology: observations, early universe, dark matter, space vehicles, space vehicles: instruments, instrumentation: detectors, telescopes
A simple cosmological model, a flat universe with nearly scale-invariant adiabatic Gaussian fluctuations, has proven to be a remarkably good fit to ever improving cosmic microwave background (CMB) data (Hinshaw et al., 2009; Reichardt et al., 2009; Brown et al., 2009), large-scale structure data (Reid et al., 2010a; Percival et al., 2009), supernova data (Hicken et al., 2009b; Kessler et al., 2009), cluster measurements (Vikhlinin et al., 2009b; Mantz et al., 2010c), distance measurements (Riess et al., 2009), measurements of strong (Suyu et al., 2010; Fadely et al., 2009) and weak (Massey et al., 2007; Fu et al., 2008; Schrabback et al., 2009) gravitational lensing effects.
Observations of CMB have been playing an essential role in testing the model and constraining its basic parameters. The \map satellite (Bennett et al., 2003a, b) has been measuring temperature and polarization anisotropies of the CMB over the full sky since 2001. With 7 years of integration, the errors in the temperature spectrum at each multipole are dominated by cosmic variance (rather than by noise) up to , and the signal-to-noise at each multipole exceeds unity up to (Larson et al., 2010). The power spectrum of primary CMB on smaller angular scales has been measured by other experiments up to (Reichardt et al., 2009; Brown et al., 2009; Lueker et al., 2010; Fowler et al., 2010).
The polarization data show the most dramatic improvements over our earlier \map results: the temperature-polarization cross power spectra measured by \map at are still dominated by noise, and the errors in the 7-year cross power spectra have improved by nearly 40% compared to the 5-year cross power spectra. While the error in the power spectrum of the cosmological -mode polarization (Seljak & Zaldarriaga, 1997; Kamionkowski et al., 1997) averaged over is cosmic-variance limited, individual multipoles are not yet cosmic-variance limited. Moreover, the cosmological -mode polarization has not been detected (Nolta et al., 2009; Komatsu et al., 2009a; Brown et al., 2009; Chiang et al., 2010).
The temperature-polarization (TE and TB) power spectra offer unique tests of the standard model. The TE spectrum can be predicted given the cosmological constraints from the temperature power spectrum, and the TB spectrum is predicted to vanish in a parity-conserving universe. They also provide a clear physical picture of how the CMB polarization is created from quadrupole temperature anisotropy. We show the success of the standard model in an even more striking way by measuring this correlation in map space, rather than in harmonic space.
The constraints on the basic 6 parameters of a flat CDM model (see Table 1), as well as those on the parameters beyond the minimal set (see Table 2), continue to improve with the 7-year \map temperature and polarization data, combined with improved external astrophysical data sets. In this paper, we shall give an update on the cosmological parameters, as determined from the latest cosmological data set. Our best estimates of the cosmological parameters are presented in the last columns of Table 1 and 2 under the name “\map+BAO+.” While this is the minimal combination of robust data sets such that adding other data sets does not significantly improve most parameters, the other data combinations provide better limits than \map+BAO+ in some cases. For example, adding the small-scale CMB data improves the limit on the primordial helium abundance, (see Table 3 and Section 4.8), the supernova data are needed to improve limits on properties of dark energy (see Table 4 and Section 5), and the power spectrum of Luminous Red Galaxies (LRGs; see Section 3.2.3) improves limits on properties of neutrinos (see footnotes g, h, and i in Table 2 and Sections 4.6 and 4.7).
The CMB can also be used to probe the abundance as well as the physics of clusters of galaxies, via the SZ effect (Zel’dovich & Sunyaev, 1969; Sunyaev & Zel’dovich, 1972). In this paper, we shall present the \map measurement of the averaged profile of SZ effect measured towards the directions of known clusters of galaxies, and discuss implications of the \map measurement for the very small-scale () power spectrum recently measured by the South Pole Telescope (SPT; Lueker et al., 2010) and Atacama Cosmology Telescope (ACT; Fowler et al., 2010) collaborations.
This paper is one of six papers on the analysis of the \map 7-year data: Jarosik et al. (2010) report on the data processing, map-making, and systematic error limits; Gold et al. (2010) on the modeling, understanding, and subtraction of the temperature and polarized foreground emission; Larson et al. (2010) on the measurements of the temperature and polarization power spectra, extensive testing of the parameter estimation methodology by Monte Carlo simulations, and the cosmological parameters inferred from the \map data alone; Bennett et al. (2010) on the assessments of statistical significance of various “anomalies” in the \map temperature map reported in the literature; and Weiland et al. (2010) on \map’s measurements of the brightnesses of planets and various celestial calibrators.
This paper is organized as follows. In Section 2, we present results from the new method of analyzing the polarization patterns around temperature hot and cold spots. In Section 3, we briefly summarize new aspects of our analysis of the \map 7-year temperature and polarization data, as well as improvements from the 5-year data. In Section 4, we present updates on various cosmological parameters, except for dark energy. We explore the nature of dark energy in Section 5. In Section 6, we present limits on primordial non-Gaussianity parameters . In Section 7, we report detection, characterization, and interpretation of the SZ effect toward locations of known clusters of galaxies. We conclude in Section 8.
\map 7-year ML
\map 7-year Mean
||13.79 Gyr||13.76 Gyr||Gyr||Gyr|
||No Running Ind.||
|Section 4.2||Running Index||No Grav. Wave||
|Section 4.5||Parity Violation||
|Section 4.7||Relativistic Species||
2. CMB Polarization on the Map
Electron-photon scattering converts quadrupole temperature anisotropy in the CMB at the decoupling epoch, , into linear polarization (Rees, 1968; Basko & Polnarev, 1980; Kaiser, 1983; Bond & Efstathiou, 1984; Polnarev, 1985; Bond & Efstathiou, 1987; Harari & Zaldarriaga, 1993). This produces a correlation between the temperature pattern and the polarization pattern (Coulson et al., 1994; Crittenden et al., 1995). Different mechanisms for generating fluctuations produce distinctive correlated patterns in temperature and polarization:
Adiabatic scalar fluctuations predict a radial polarization pattern around temperature cold spots and a tangential pattern around temperature hot spots on angular scales greater than the horizon size at the decoupling epoch, . On angular scales smaller than the sound horizon size at the decoupling epoch, both radial and tangential patterns are formed around both hot and cold spots, as the acoustic oscillation of the CMB modulates the polarization pattern (Coulson et al., 1994). As we have not seen any evidence for non-adiabatic fluctuations (Komatsu et al., 2009a, see Section 4.4 for the 7-year limits), in this section we shall assume that fluctuations are purely adiabatic.
Tensor fluctuations predict the opposite pattern: the temperature cold spots are surrounded by a tangential polarization pattern, while the hot spots are surrounded by a radial pattern (Crittenden et al., 1995). Since there is no acoustic oscillation for tensor modes, there is no modulation of polarization patterns around temperature spots on small angular scales. We do not expect this contribution to be visible in the \map data, given the noise level.
Defect models predict that there should be minimal correlations between temperature and polarization on (Seljak et al., 1997). The detection of large-scale temperature polarization fluctuations rules out any causal models as the primary mechanism for generating the CMB fluctuations (Spergel & Zaldarriaga, 1997). This implies that the fluctuations were either generated during an accelerating phase in the early universe or were present at the time of the initial singularity.
This section presents the first direct measurement of the predicted pattern of adiabatic scalar fluctuations in CMB polarization maps. We stack maps of Stokes and around temperature hot and cold spots to show the expected polarization pattern at the statistical significance level of 8. While we have detected the TE correlations in the first year data (Kogut et al., 2003), we present here the direct real space pattern around hot and cold spots. In Section 2.5, we discuss the relationship between the two measurements.
2.2. Measuring Peak-Polarization Correlation
We first identify temperature hot (or cold) spots, and then stack the polarization data (i.e., Stokes and ) on the locations of the spots. As we shall show below, the resulting polarization data is equivalent to the temperature peak-polarization correlation function which is similar to, but different in an important way from, the temperature-polarization correlation function.
and : Transformed Stokes Parameters
Our definitions of Stokes and follow that of Kogut et al. (2003): the polarization that is parallel to the Galactic meridian is and . Starting from this, the polarization that is rotated by from east to west (clockwise, as seen by an observer on Earth looking up at the sky) has and , that perpendicular to the Galactic meridian has and , and that rotated further by from east to west has and . With one more rotation we go back to and . We show this in Figure 1.
As the predicted polarization pattern around temperature spots is either radial or tangential, we find it most convenient to work with and first introduced by Kamionkowski et al. (1997):
These transformed Stokes parameters are defined with respect to the new coordinate system that is rotated by , and thus they are defined with respect to the line connecting the temperature spot at the center of the coordinate and the polarization at an angular distance from the center (also see Figure 1). Note that we have used the small-angle (flat-sky) approximation for simplicity of the algebra. This approximation is justified as we are interested in relatively small angular scales, .
The above definition of is equivalent to the so-called “tangential shear” statistic used by the weak gravitational lensing community. By following what has been already done for the tangential shear, we can find the necessary formulae for and . Specifically, we shall follow the derivations given in Jeong et al. (2009).
where is the angle between and the line of Galactic latitude, . Note that we have included the negative signs on the left hand side because our sign convention for the Stokes parameters is opposite of that used in equation (38) of Zaldarriaga & Seljak (1997). The transformed Stokes parameters are given by
The stacking of and at the locations of temperature peaks can be written as
where the angle bracket, , denotes the average over the locations of peaks, is the surface number density of peaks (of the temperature fluctuation) at the location , is the total number of temperature peaks used in the stacking analysis, and is equal to 0 at the masked pixels and 1 otherwise. Defining the density contrast of peaks, , we find
where is the fraction of sky outside of the mask, and we have used .
In Appendix B, we use the statistics of peaks of Gaussian random fields to relate to the temperature--mode polarization cross power spectrum , to the temperature--mode polarization cross power spectrum , and the stacked temperature profile, , to the temperature power spectrum . We find
where and are the harmonic transform of window functions, which are a combination of the experimental beam, pixel window, and any other additional smoothing applied to the temperature and polarization data, respectively, and is the “scale-dependent bias” of peaks found by Desjacques (2008) averaged over peaks. See Appendix B for details.
Prediction and Physical Interpretation
What do and look like? The map is expected to be non-zero for a cosmological signal, while the map is expected to vanish in a parity-conserving universe unless some systematic error rotates the polarization plane uniformly.
To understand the shape of as well as its physical implications, let us begin by showing the smoothed spectra and the corresponding temperature- correlation functions, , in Figure 2. (Note that and can be computed from equations (11) and (12), respectively, with and .) This shows three distinct effects causing polarization of CMB (see Hu & White, 1997, for a pedagogical review):
, where is the angular size of the radius of the horizon size at the decoupling epoch. Using the comoving horizon size of Gpc and the comoving angular diameter distance to the decoupling epoch of Gpc as derived from the \map data, we find . As this scale is so much greater than the sound horizon size (see below), only gravity affects the physics. Suppose that there is a Newtonian gravitational potential, , at the center of a perturbation, . If it is overdense at the center, , and thus it is a cold spot according to the Sachs–Wolfe formula (Sachs & Wolfe, 1967), . The photon fluid in this region will flow into the gravitational potential well, creating a converging flow. Such a flow creates the quadrupole temperature anisotropy around an electron at , producing polarization that is radial, i.e., . Since the temperature is negative, we obtain , i.e., anti-correlation (Coulson et al., 1994). On the other hand, if it is overdense at the center, then the photon fluid moves outward, producing polarization that is tangential, i.e., . Since the temperature is positive, we obtain , i.e., anti-correlation. The anti-correlation at is a smoking-gun for the presence of super-horizon fluctuations at the decoupling epoch (Spergel & Zaldarriaga, 1997), which has been confirmed by the \map data (Peiris et al., 2003).
, where is the angular size of the radius of the sound horizon size at the decoupling epoch. Using the comoving sound horizon size of Gpc and Gpc as derived from the \map data, we find . Again, consider a potential well with at the center. As the photon fluid flows into the well, it compresses, increasing the temperature of the photons. Whether or not this increase can reverse the sign of the temperature fluctuation (from negative to positive) depends on whether the initial perturbation was adiabatic. If it was adiabatic, then the temperature would reverse sign at . Note that the photon fluid is still flowing in, and thus the polarization direction is radial, . However, now that the temperature is positive, the correlation reverses sign: . A similar argument (with the opposite sign) can be used to show the same result, , for at the center. As an aside, the temperature reverses sign on smaller angular scales for isocurvature fluctuations.
. Again, consider a potential well with at the center. At , the pressure of the photon fluid is so great that it can slow down the flow of the fluid. Eventually, at , the pressure becomes large enough to reverse the direction of the flow (i.e., the photon fluid expands). As a result the polarization direction becomes tangential, ; however, as the temperature is still positive, the correlation reverses sign again: .
On even smaller scales, the correlation reverses sign again (see Figure 2 of Coulson et al., 1994) because the temperature gets too cold due to expansion. We do not see this effect in Figure 2 because of the smoothing. Lastly, there is no correlation between and at because of symmetry.
These features are essentially preserved in the peak-polarization correlation as measured by the stacked polarization profiles. We show them in Figure 3 for various values of the threshold peak heights. The important difference is that, thanks to the scale-dependent bias , the small-scale trough at is enhanced, making it easier to observe. On the other hand, the large-scale anti-correlation is suppressed. We can therefore conclude that, with the \map data, we should be able to measure the compression phase at , as well as the reversal phase at . We also show the profiles calculated from numerical simulations (gray solid lines). The agreement with equation (11) is excellent. We also show the predicted profiles of the stacked temperature data in Figure 4.
2.3. Analysis Method
We use the foreground-reduced VW temperature map at the HEALPix resolution of to find temperature peaks. First, we smooth the foreground-reduced temperature maps in 6 differencing assemblies (DAs) (V1, V2, W1, W2, W3, W4) to a common resolution of (FWHM) using
where is the appropriate beam transfer function for each DA (Jarosik et al., 2010), and is the pixel window function for , , times the spherical harmonic transform of a Gaussian with . We then coadd the foreground-reduced V- and W-band maps with the inverse noise variance weighting, and remove the monopole from the region outside of the mask (which is already negligibly small, K). For the mask, we combine the new 7-year KQ85 mask, KQ85y7 (defined in Gold et al., 2010, also see Section 3.1) and P06 masks, leaving 68.7% of the sky available for the analysis.
We find the locations of minima and maxima using the software “hotspot” in the HEALPix package (Gorski et al., 2005). Over the full sky (without the mask), we find 20953 maxima and 20974 minima. As the maxima and minima found by hotspot still contain negative and positive peaks, respectively, we further select the “hot spots” by removing all negative peaks from maxima, and the “cold spots” by removing all positive peaks from minima. This procedure corresponds to setting the threshold peak height to ; thus, our prediction for is the top-left panel of Figure 3.
Outside of the mask, we find 12387 hot spots and 12628 cold spots. The
r.m.s. temperature fluctuation is .
What does the theory predict? Using equation (B15) with the
power spectrum where is the
noise bias of the VW map before Gaussian smoothing and is the 5-year best-fitting power-law CDM temperature
power spectrum, we find for
and ; thus, the number of observed hot and
cold spots is consistent with the predicted number.
As for the polarization data, we use the raw (i.e., without foreground cleaning) polarization maps in V and W bands. We have checked that the cleaned maps give similar results with slightly larger error bars, which is consistent with the excess noise introduced by the template foreground cleaning procedure (Page et al., 2007; Gold et al., 2009, 2010). As we are focused on relatively small angular scales, , in this analysis, the results presented in this section would not be affected by a potential systematic effect causing an excess power in the W-band polarization data on large angular scales, . However, note that this excess power could just be a statistical fluctuation (Jarosik et al., 2010). We form two sets of the data: (i) V, W, and VW band maps smoothed to a common resolution of , and (ii) V, W, and VW band maps without any additional smoothing. The first set is used only for visualization, whereas the second set is used for the analysis.
We extract a square region of around each temperature hot or cold spot. We then coadd the extracted images with uniform weighting, and and images with the inverse noise variance weighting. We have eliminated the pixels masked by KQ85y7 and P06 from each region when we coadd images, and thus the resulting stacked image has the smallest noise at the center (because the masked pixels usually appear near the edge of each image). We also accumulate the inverse noise variance per pixel as we coadd and maps. The coadded inverse noise variance maps of and will be used to estimate the errors of the stacked images of and per pixel, which will then be used for the analysis.
We find that the stacked images of and have constant offsets, which is not surprising. Since these affect our determination of polarization directions, we remove monopoles from the stacked images of and . The size of each pixel in the stacked image is , and the number of pixels is .
|Hot, , VW||661.9|
|Hot, , VW||661.1|
|Hot, , VW||694.2|
|Hot, , VW||629.2|
|Cold, , VW||668.3|
|Cold, , VW||682.7|
|Cold, , VW||682.2|
|Cold, , VW||657.8|
|Hot, , VW||559.8|
|Hot, , VW||629.8|
|Hot, , VW||662.2|
|Hot, , VW||567.0|
|Cold, , VW||584.0|
|Cold, , VW||668.2|
|Cold, , VW||616.0|
|Cold, , VW||636.9|
In Figure 5 and 6, we show the stacked images of , , , , and around temperature cold spots and hot spots, respectively. The peak values of the stacked temperature profiles agree with the predictions (see the dashed line in the top-left panel of Figure 4). A dip in temperature (for hot spots; a bump for cold spots) at is clearly visible in the data. While the Stokes and measured from the data exhibit the expected features, they are still fairly noisy. The most striking images are the stacked (and ). The predicted features are clearly visible, particularly the compression phase at and the reversal phase at in : the polarization directions around temperature cold spots are radial at and tangential at , and those around temperature hot spots show the opposite patterns, as predicted.
How significant are these features? Before performing the quantitative analysis, we first compare and using both the (VW)/2 sum map (here, VW refers to the inverse noise variance weighted average) as well as the (VW)/2 difference map (bottom panels of Figure 5 and 6). The map (which is expected to be non-zero for a cosmological signal) shows clear differences between the sum and difference maps, while the map (which is expected to vanish in a parity-conserving universe unless some systematic error rotates the polarization plane uniformly) is consistent with zero in both the sum and difference maps.
Next, we perform the standard analysis. We summarize the results in Table 5. We report the values of measured with respect to zero signal in the second column, where the number of degrees of freedom (DOF) is 625. For each sum map combination, we fit the data to the predicted signal to find the best-fitting amplitude.
The largest improvement in is observed for , as expected from the visual inspection of Figure 5 and 6: we find and for the stacking of around hot and cold spots, respectively. The improvement in is and , respectively; thus, we detect the expected polarization patterns around hot and cold spots at the level of 5.4 and 6, respectively. The combined significance exceeds 8.
On the other hand, we do not find any evidence for . The values with respect to zero signal per DOF are (hot spots) and (cold spots), and the probabilities of finding larger values of are 44.5% and 18%, respectively. But, can we learn anything about cosmology from this result? While the standard model predicts and hence , models in which the global parity symmetry is violated can create (Lue et al., 1999; Carroll, 1998; Feng et al., 2005). Therefore, we fit the measured to the predicted , finding a null result: and (68% CL), or equivalently and (68% CL) for hot and cold spots, respectively. Averaging these numbers, we obtain (68% CL), which is consistent with (although not as stringent as) the limit we find from the full analysis presented in Section 4.5. Finally, all the values measured from the difference maps are consistent with a null signal.
How do these results compare to the full analysis of the TE power spectrum? By fitting the 7-year data to the same power spectrum used above (5-year best-fitting power-law CDM model from to , i.e., DOF777), we find the best-fitting amplitude of and , i.e., a 21 detection of the TE signal. This is reasonable, as we used only the V- and W-band data for the stacking analysis, while we used also the Q-band data for measuring the TE power spectrum; is insensitive to information on (see top left panel of Figure 3); and the smoothing suppresses the power at (see left panel of Figure 2). Nevertheless, there is probably a way to extract more information from by, for example, combining data at different threshold peak heights and smoothing scales.
If the temperature fluctuations of the CMB obey Gaussian statistics and global parity symmetry is respected on cosmological scales, the temperature--mode polarization cross power spectrum, , contains all the information about the temperature-polarization correlation. In this sense, the stacked polarization images do not add any new information.
The detection and measurement of the temperature- mode polarization cross-correlation power spectrum, (Kovac et al., 2002; Kogut et al., 2003; Spergel et al., 2003), can be regarded as equivalent to finding the predicted polarization patterns around hot and cold spots. While we have shown that one can write the stacked polarization profile around temperature spots in terms of an integral of , the formal equivalence between this new method and is valid only when temperature fluctuations obey Gaussian statistics, as the stacked and maps measure correlations between temperature peaks and polarization. So far there is no convincing evidence for non-Gaussianity in the temperature fluctuations observed by \map (Komatsu et al., 2003, see Section 6 for the 7-year limits on primordial non-Gaussianity, and Bennett et al. 2010 for discussion on other non-Gaussian features).
Nevertheless, they provide striking confirmation of our understanding of the physics at the decoupling epoch in the form of radial and tangential polarization patterns at two characteristic angular scales that are important for the physics of acoustic oscillation: the compression phase at and the reversal phase at .
Also, this analysis does not require any analysis in harmonic space, nor decomposition to and modes. The analysis is so straightforward and intuitive that the method presented here would also be useful for null tests and systematic error checks. The stacked image of should be particularly useful for systematic error checks.
3. Summary of 7-year Parameter Estimation
||TE||Detected at 21||Detected at 13|
|BB||(95% CL)||(95% CL)|
|EE/BB||(95% CL)||(95% CL)|
|All||TE/EE/BB||(95% CL)||(95% CL)|
3.1. Improvements from the 5-year Analysis
Foreground Mask. The 7-year temperature analysis masks, KQ85y7 and KQ75y7, have been slightly enlarged to mask the regions that have excess foreground emission, particularly in the HII regions Gum and Ophiuchus, identified in the difference between foreground-reduced maps at different frequencies. As a result, the new KQ85y7 and KQ75y7 masks eliminate an additional 3.4% and 1.0% of the sky, leaving 78.27% and 70.61% of the sky for the cosmological analyses, respectively. See Section 2.1 of Gold et al. (2010) for details. There is no change in the polarization P06 mask (see Section 4.2 of Page et al., 2007, for definition of this mask), which leaves 73.28% of the sky.
Point Sources and the SZ Effect. We continue to marginalize over a contribution from unresolved point sources, assuming that the antenna temperature of point sources declines with frequency as (see equation (5) of Nolta et al., 2009). The 5-year estimate of the power spectrum from unresolved point sources in Q band in units of antenna temperature, , was (Nolta et al., 2009), and we used this value and the error bar to marginalize over the power spectrum of residual point sources in the 7-year parameter estimation. The subsequent analysis showed that the 7-year estimate of the power spectrum is (Larson et al., 2010), which is somewhat lower than the 5-year value because more sources are resolved by \map and included in the source mask. The difference in the diffuse mask (between KQ85y5 and KQ85y7) does not affect the value of very much: we find instead of if we use the 5-year diffuse mask and the 7-year source mask. The source power spectrum is sub-dominant in the total power. We have checked that the parameter results are insensitive to the difference between the 5-year and 7-year residual source estimates.
We continue to marginalize over a contribution from the SZ effect using the same template as for the 3- and 5-year analyses (Komatsu & Seljak, 2002). We assume a uniform prior on the amplitude of this template as , which is now justified by the latest limits from the SPT collaboration, (68% CL; Lueker et al., 2010), and the ACT collaboration, (95% CL; Fowler et al., 2010).
High- Temperature and Polarization. We increase the multipole range of the power spectra used for the cosmological parameter estimation from to for the TT power spectrum, and from to for the TE power spectrum. We use the 7-year V- and W-band maps (Jarosik et al., 2010) to measure the high- TT power spectrum in . While we used only Q- and V-band maps to measure the high- TE and TB power spectra for the 5-year analysis (Nolta et al., 2009), we also include W-band maps in the 7-year high- polarization analysis.
With these data, we now detect the high- TE power spectrum at 21, compared to 13 for the 5-year high- TE data. This is a consequence of adding two more years of data and the W-band data. The TB data can be used to probe a rotation angle of the polarization plane, , due to potential parity-violating effects or systematic effects. With the 7-year high- TB data we find a limit (68% CL). For comparison, the limit from the 5-year high- TB power spectrum was (68% CL; Komatsu et al., 2009a). See Section 4.5 for the 7-year limit on from the full analysis.
Low- Temperature and Polarization. Except for using the 7-year maps and the new temperature KQ85y7 mask, there is no change in the analysis of the low- temperature and polarization data: we use the Internal Linear Combination (ILC) map (Gold et al., 2010) to measure the low- TT power spectrum in , and calculate the likelihood using the Gibbs sampling and Blackwell-Rao (BR) estimator (Jewell et al., 2004; Wandelt, 2003; Wandelt et al., 2004; O’Dwyer et al., 2004; Eriksen et al., 2004, 2007a, 2007b; Chu et al., 2005; Larson et al., 2007). For the implementation of the BR estimator in the 5-year analysis, see Section 2.1 of Dunkley et al. (2009). We use Ka-, Q-, and V-band maps for the low- polarization analysis in , and evaluate the likelihood directly in pixel space as described in Appendix D of Page et al. (2007).
To get a feel for improvements in the low- polarization data with two additional years of integration, we note that the 7-year limits on the optical depth, and the tensor-to-scalar ratio and rotation angle from the low- polarization data alone, are (68% CL; see Larson et al., 2010), (95% CL; see Section 4.1), and (68% CL; see Section 4.5), respectively. The corresponding 5-year limits were (Dunkley et al., 2009), (see Section 4.1), and (Komatsu et al., 2009a), respectively.
In Table 6, we summarize the improvements from the 5-year data mentioned above.
3.2. External data sets
The \map data are statistically powerful enough to constrain 6 parameters of a flat CDM model with a tilted spectrum. However, to constrain deviations from this minimal model, other CMB data probing smaller angular scales and astrophysical data probing the expansion rates, distances, and growth of structure are useful.
Small-scale CMB Data
The best limits on the primordial helium abundance, , are obtained when the \map data are combined with the power spectrum data from other CMB experiments probing smaller angular scales, .
We use the temperature power spectra from the Arcminute Cosmology Bolometer Array Receiver (ACBAR; Reichardt et al., 2009) and QUEST at DASI (QUaD) (Brown et al., 2009) experiments. For the former, we use the temperature power spectrum binned in 16 band powers in the multipole range . For the latter, we use the temperature power spectrum binned in 13 band powers in .
We marginalize over the beam and calibration errors of each experiment: for ACBAR, the beam error is 2.6% on a 5 arcmin (FWHM) Gaussian beam and the calibration error is 2.05% in temperature. For QUaD, the beam error combines a 2.5% error on 5.2 and 3.8 arcmin (FWHM) Gaussian beams at 100 GHz and 150 GHz, respectively, with an additional term accounting for the sidelobe uncertainty (see Appendix A of Brown et al., 2009, for details). The calibration error is 3.4% in temperature.
The ACBAR data are calibrated to the \map 5-year temperature data, and the QUaD data are calibrated to the BOOMERanG data (Masi et al., 2006) which are, in turn, calibrated to the \map 1-year temperature data. (The QUaD team takes into account the change in the calibration from the 1-year to the 5-year \map data.) The calibration errors quoted above are much greater than the calibration uncertainty of the \map 5-year data (0.2% Hinshaw et al., 2007). This is due to the noise of the ACBAR, QUaD, and BOOMERanG data. In other words, the above calibration errors are dominated by the statistical errors that are uncorrelated with the \map data. We thus treat the \map, ACBAR, and QUaD data as independent.
We do not use the other, previous small-scale CMB data, as their statistical errors are much larger than those of ACBAR and QUaD, and thus adding them would not improve the constraints on the cosmological parameters significantly. The new power spectrum data from the SPT (Lueker et al., 2010) and ACT (Fowler et al., 2010) collaborations were not yet available at the time of our analysis.
Hubble Constant and Angular Diameter Distances
There are two main astrophysical priors that we shall use in this paper: the Hubble constant and the angular diameter distances out to and .
A Gaussian prior on the present-day Hubble constant, (68% CL; Riess et al., 2009). The quoted error includes both statistical and systematic errors. This measurement of is obtained from the magnitude-redshift relation of 240 low- Type Ia supernovae at . The absolute magnitudes of supernovae are calibrated using new observations from HST of 240 Cepheid variables in six local Type Ia supernovae host galaxies and the maser galaxy NGC 4258. The systematic error is minimized by calibrating supernova luminosities directly using the geometric maser distance measurements. This is a significant improvement over the prior that we adopted for the 5-year analysis, , which is from the Hubble Key Project final results (Freedman et al., 2001).
Gaussian priors on the distance ratios, and , measured from the Two-Degree Field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Survey Data Release 7 (SDSS DR7) (Percival et al., 2009). The inverse covariance matrix is given by equation (5) of Percival et al. (2009). These priors are improvements from those we adopted for the 5-year analysis, and (Percival et al., 2007).
The above measurements can be translated into a measurement of at a single, “pivot” redshift: (Percival et al., 2009). Kazin et al. (2010) used the two-point correlation function of SDSS-DR7 LRGs to measure at . They found , which is an excellent agreement with the above measurement by Percival et al. (2009) at a similar redshift. The excellent agreement between these two independent studies, which are based on very different methods, indicates that the systematic error in the derived values of may be much smaller than the statistical error.
Here, is the comoving sound horizon size at the baryon drag epoch ,
where is the proper (not comoving) angular diameter distance:
where , , and for (; positively curved), (; flat), and (; negatively curved), respectively. The Hubble expansion rate, which has contributions from baryons, cold dark matter, photons, massless and massive neutrinos, curvature, and dark energy, is given by equation (27) in Section 3.3.
The cosmological parameters determined by combining the \map data, BAO, and will be called “\map+BAO+,” and they constitute our best estimates of the cosmological parameters, unless noted otherwise.
Note that, when redshift is much less than unity, the effective distance approaches . Therefore, the effect of different cosmological models on do not appear until one goes to higher redshifts. If redshift is very low, is simply measuring the Hubble constant.
Power Spectrum of Luminous Red Galaxies
A combination of the \map data and the power spectrum of Luminous Red Galaxies (LRGs) measured from the SDSS DR7 is a powerful probe of the total mass of neutrinos, , and the effective number of neutrino species, (Reid et al., 2010a, b). We thus combine the LRG power spectrum (Reid et al., 2010a) with the \map 7-year data and the Hubble constant (Riess et al., 2009) to update the constraints on and reported in Reid et al. (2010a). Note that BAO and the LRG power spectrum cannot be treated as independent data sets because a part of the measurement of BAO used LRGs as well.
The luminosity distances out to high- Type Ia supernovae have been the most powerful data for first discovering the existence of dark energy (Riess et al., 1998; Perlmutter et al., 1999) and then constraining the properties of dark energy, such as the equation of state parameter, (see Frieman et al., 2008, for a recent review). With more than 400 Type Ia supernovae discovered, the constraints on the properties of dark energy inferred from Type Ia supernovae are now limited by systematic errors rather than by statistical errors.
There is an indication that the constraints on dark energy parameters are different when different methods are used to fit the light curves of Type Ia supernovae (Hicken et al., 2009b; Kessler et al., 2009). We also found that the parameters of the minimal 6-parameter CDM model derived from two compilations of Kessler et al. (2009) are different: one compilation uses the light curve fitter called SALT-II (Guy et al., 2007) while the other uses the light curve fitter called MLCS2K2 (Jha et al., 2007). For example, derived from \map+BAO+SALT-II and \map+BAO+MLCS2K2 are different by nearly , despite being derived from the same data sets (but processed with two different light curve fitters). If we allow the dark energy equation of state parameter, , to vary, we find that derived from \map+BAO+SALT-II and \map+BAO+MLCS2K2 are different by .
At the moment it is not obvious how to estimate systematic errors and properly incorporate them in the likelihood analysis, in order to reconcile different methods and data sets.
In this paper, we shall use one compilation of the supernova data called the “Constitution” samples (Hicken et al., 2009b). The reason for this choice over the others, such as the compilation by Kessler et al. (2009) that includes the latest data from the SDSS-II supernova survey, is that the Constitution samples are an extension of the “Union” samples (Kowalski et al., 2008) that we used for the 5-year analysis (see Section 2.3 of Komatsu et al., 2009a). More specifically, the Constitution samples are the Union samples plus the latest samples of nearby Type Ia supernovae optical photometry from the Center for Astrophysics (CfA) supernova group (CfA3 sample; Hicken et al., 2009a). Therefore, the parameter constraints from a combination of the \map 7-year data, the latest BAO data described above (Percival et al., 2009), and the Constitution supernova data may be directly compared to the “\map+BAO+SN” parameters given in Table 1 and 2 of Komatsu et al. (2009a). This is a useful comparison, as it shows how much the limits on parameters have improved by adding two more years of data.
However, given the scatter of results among different compilations of the supernova data, we have decided to choose the “\map+BAO+” (see Section 3.2.2) as our best data combination to constrain the cosmological parameters, except for dark energy parameters. For dark energy parameters, we compare the results from \map+BAO+ and \map+BAO+SN in Section 5. Note that we always marginalize over the absolute magnitudes of Type Ia supernovae with a uniform prior.
Can we measure angular diameter distances out to higher redshifts? Measurements of gravitational lensing time delays offer a way to determine absolute distance scales (Refsdal, 1964). When a foreground galaxy lenses a background variable source (e.g., quasars) and produces multiple images of the source, changes of the source luminosity due to variability appear on multiple images at different times.
The time delay at a given image position for a given source position , , depends on the angular diameter distances as (see, e.g., Schneider et al., 2006, for a review)
where , , and are the angular diameter distances out to a lens galaxy, to a source galaxy, and between them, respectively, and is the so-called Fermat potential, which depends on the path length of light rays and gravitational potential of the lens galaxy.
The biggest challenge for this method is to control systematic errors in our knowledge of , which requires a detailed modeling of mass distribution of the lens. One can, in principle, minimize this systematic error by finding a lens system where the mass distribution of lens is relatively simple.
The lens system B1608+656 is not a simple system, with two lens galaxies and dust extinction; however, it has one of the most precise time-delay measurements of quadruple lenses. The lens redshift of this system is relatively large, (Myers et al., 1995). The source redshift is (Fassnacht et al., 1996). This system has been used to determine to 10% accuracy (Koopmans et al., 2003).
Suyu et al. (2009) have obtained more data from the deep HST Advanced Camera for Surveys (ACS) observations of the asymmetric and spatially extended lensed images, and constrained the slope of mass distribution of the lens galaxies. They also obtained ancillary data (for stellar dynamics and lens environment studies) to control the systematics, particularly the the so-called “mass-sheet degeneracy,” which the strong lensing data alone cannot break. By doing so, they were able to reduce the error in (including the systematic error) by a factor of two (Suyu et al., 2010). They find a constraint on the “time-delay distance,” , as
where the number is found from a Gaussian fit to the likelihood of
Likelihood of out to the lens system B1608+656 given by Suyu et al. (2010),
where , , , and . This likelihood includes systematic errors due to the mass-sheet degeneracy, which dominates the total error budget (see Section 6 of Suyu et al., 2010, for more details). Note that this is the only lens system for which (rather than ) has been constrained.
3.3. Treating Massive Neutrinos in Exactly
When we evaluate the likelihood of external astrophysical data sets, we often need to compute the Hubble expansion rate, . While we treated the effect of massive neutrinos on approximately for the 5-year analysis of the external data sets presented in Komatsu et al. (2009a), we treat it exactly for the 7-year analysis, as described below.
The total energy density of massive neutrino species, , is given by (in natural units)
where is the mass of each neutrino species. Using the comoving momentum, , and the present-day neutrino temperature, K, we write
Throughout this paper, we shall assume that
all massive neutrino species have the equal mass , i.e.,
for all .
When neutrinos are relativistic, one may relate to the photon energy density, , as
where is the effective number of neutrino species.
Note that for the standard neutrino
The limits of this function are