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Testing the CMB Data for Systematic Effects
Louise M. Griffiths and Charles H. Lineweaver Department of Astrophysics and Optics, School of Physics, University of New South Wales, Sydney, NSW 2052, Australia ABSTRACT

arXiv:astro-ph/0301490 v1 24 Jan 2003

Under the assumption that the concordance cold dark matter (CDM) mo del is the correct mo del, we test the cosmic microwave background (CMB) anisotropy data for systematic effects by examining the band pass temperature residuals with respect to this mo del. Residuals are plotted as a function of , galactic latitude, frequency, calibration source, instrument type and several other variables that may be asso ciated with potential systematic effects. Linear fitting to the residuals indicates no significant identifiable systematic errors asso ciated with these variables, except for the case of galactic latitude. We find evidence for a trend asso ciated with the absolute galactic latitude range at more than the 2- level. This may be indicative of galactic contamination and may require a 2% reduction in the normalisation of low galactic latitude observations. Subject headings: cosmic microwave background - cosmology: observations

1.

INTRODUCTION

The cosmic microwave background (CMB) power spectrum is a particularly potent probe of cosmology. As long as the systematic errors asso ciated with these observations are small, the detected signal has direct cosmological importance. The ever-tightening network of constraints from CMB and non-CMB observations favours a concordant cold dark matter (CDM) mo del that is commonly accepted as the standard cosmological mo del (Table 1). Since the anisotropy power spectrum is playing an increasingly large role in establishing and refining this mo del, it is crucial to check the CMB data for possible systematic errors in as many ways as possible. Systematic errors and selection effects are notoriously difficult to identify and quantify. Calibration and/or beam uncertainties dominate current CMB measurements and there may be lower level systematic errors of which we are not aware (Page 2001). Individual experimental groups have developed various ways to check their CMB observations for systematic


­2­ effects (e.g. Kogut et al. 1996; Miller et al. 2002), including the use of multiple calibration sources, multiple frequency channels and extensive beam calibrating observations. Internal consistency is the primary concern of these checks. Testing for consistency with other CMB observations is another important way to identify possible systematic errors. When the areas of the sky observed overlap, this can be done by comparing CMB temperature maps (e.g. Ganga et al. 1994a; Lineweaver et al. 1995; Xu et al. 2001). When similar angular scales are being observed one can compare power spectra (e.g. Sievers et al. 2002, Figure 11). A prerequisite for the extraction of useful estimates for cosmological parameters from the combined CMB data set is the mutual consistency of the observational data points (Wang et al. 2002a); the best-fit must also be a go o d fit. Wang et al. (2002a) and Sievers et al. (2002) have recently explored the consistency of various CMB observations with respect to power spectrum mo dels and concluded that the CMB fluctuation data is consistent with several minor exceptions. Although individual observational groups vigorously test their data sets for systematic errors, the entire CMB observational data set has not yet been collectively tested. Here we check for consistency of the concordance mo del (Table 1) with respect to possible sources of systematic error. Under the assumption that the concordance mo del is the correct mo del (i.e. more correct than the best-fit to the CMB data alone), we explore residuals of the observational data with respect to this mo del to see if any patterns emerge. We attempt to identify systematic errors in the data that may have been ignored or only partially corrected f o r. With only a few independent band power measurements the usefulness of such a strategy is compromised by low number statistics. However, we now have 203 measurements of band power on scales of 2 < < 2000 from over two dozen autonomous and semi-autonomous groups. There are enough CMB fluctuation detections from independent observations that subtle systematic effects could appear above the noise in regression plots of the data residuals. This is particularly the case when one has a better idea of the underlying mo del than provided by the CMB data alone. The history of the estimates of the position of the CMB dipole illustrates the idea. Once a relatively precise direction of the dipole was established, the positional scatter elongated in the direction of the galactic centre could be distinguished unambiguously from statistical scatter and more reliable corrections for galactic contamination could be made (Lineweaver 1997, Figure 2). We aim to ascertain whether the use of the concordance mo del as a prior can help to separate statistical and systematic errors in the CMB anisotropy data. In §2 we discuss constraints on cosmological parameters, the current concordance mo del and how simultaneously analysing combinations of independent observational data sets can


­3­ tighten cosmological constraints. Our analytical metho dology is detailed in §3. In §4 possible sources of systematic uncertainty are discussed. In §5 and §6 our results are discussed and summarised.

2.

THE CONCORDANCE COSMOLOGY 2.1. Observational concordance

The CMB has the potential to simultaneously constrain a number of cosmological parameters that are the ingredients of the hot big bang mo del. Unfortunately, particular parameter combinations can pro duce indistinguishable C spectra (Efstathiou & Bond 1999). For example, cosmological mo dels with different matter content but the same geometry can have nearly identical anisotropies. Such mo del degeneracies limit parameter extraction from the CMB alone. A number of recent analyses combine information from a range of independent observational data sets (e.g. Efstathiou et al. 2002; Lewis & Bridle 2002; Sievers et al. 2002; Wang et al. 2002a,b), enabling certain degeneracies of the individual data sets to be resolved. As the observational data become more precise and diverse they form an increasingly tight network of parameter constraints. Analyses of a variety of astrophysical observations are beginning to refine an observationally concordant cosmological mo del. Efstathiou et al. (2002) perform a combined likeliho o d analysis of the power spectra of the 2-degree Field Galaxy Redshift Survey (2dFGRS) and the CMB anisotropies under the assumptions that the galaxy power spectrum on large scales is directly proportional to the linear matter power spectrum and that the initial fluctuations were adiabatic, Gaussian and well described by power-laws with scalar and tensor indeces of ns and nt . 11 cosmological parameter combinations are simultaneously considered; the curvature parameter , the contribution to the overall energy density from the cosmological constant , b = b h2 (where b is the baryon energy density and h the Hubble parameter), c = c h2 (where c is the energy density of CDM), the amplitude of scalar perturbations As , the ratio of tensor to scalar perturbations At /As , the spectral index of scalar perturbations ns , that of tensor perturbations nt , the reionisation optical depth , the shape parameter that defines the turn-over of the matter power spectrum m h (where m is the energy density of non-relativistic matter) and the bias parameter b. The big bang nucleosynthesis (BBN) constraint on the baryon content (Burles et al. 2001) is added to the analysis as a prior. The results for the CMB alone and combined analyses are given in Table 1. Wang et al. (2002a) perform a similar analysis using the CMB data, the decorrelated


­4­ linear power spectrum extracted from the PSCz survey (Hamilton & Tegmark 2002) and the Hubble Key Pro ject (HKP) prior for h (Freedman et al. 2001). Instead of assuming that the dark matter contribution to the energy density is entirely composed of cold dark matter, they explore the parameter combination d = d h2 and intro duce a new parameter, f = /d , that is the fraction of dark matter that is hot. Their constraints from the CMB alone and combined analyses are also given in Table 1. Another recent analysis of the CMB data set is that of Sievers et al. (2002). They perform a likeliho o d analysis over 7 cosmological parameters (total = 1 - , , b , c , As , ns , ) applying a sequence of increasingly strong prior probabilities successively to the likeliho o d functions. These are a flat prior = 0 in accordance with the predictions of the simplest inflationary scenarios, a large scale structure prior that involves a constraint 2 on the amplitude 8 and shape of the matter power spectrum, the HKP prior for h and the m - priors from supernova type Ia (SNIa) observations (Riess et al. 1998; Perlmutter et al. 1999). Their results for the CMB alone and the CMB+priors likeliho o d analyses are also given in Table 1. Lewis & Bridle (2002) implement a Markov chain Monte Carlo metho d to parameter flat mo dels using a subsample of the CMB data set together with the and SNIa priors. They also perform a combined analysis with the 2dFGRS. The joint analysis of the CMB data set and the 2dFGRS is that of Wang et al. (2 results of both these analysis are also given in Table 1. constrain 9 BBN, HKP most recent 002b). The

Due to the degeneracies in the anisotropy power spectrum, the CMB alone is only able to provide weak constraints on particular cosmological parameters. Combining the CMB constraints with the results of independent observational data sets can tighten these constraints. The results of the joint likeliho o d analyses discussed in this section suggest the observationally concordant cosmology; 0, 0.7 (m = b + c 0.3), b h2 0.02, ns 1 and h 0.68 with At , and taken to be zero. With more precise and diverse cosmological observations, the ability of the standard CDM cosmology to describe the observational universe will be extended and tested for inconsistencies.

2.2.

Goodness of fit of the concordance cosmology to the CMB

We perform a simple 2 calculation (see Appendix A) to determine the go o dness-of-fit of this new standard CDM cosmology to the CMB, employing the band power temperature measurements in Table 2 and their asso ciated window functions. We limit our analysis to 2 < < 2000 because secondary anisotropy contributions, such as the Sunyaev-Zel'dovich


­5­ (Sunyaev & Zel'dovich 1970) effect, may dominate at > 2000 (e.g. Bond et al. 2002). The mo del radiation angular power spectrum is calculated using cmbfast (Seljak & Zaldarriaga 1996). However, rather than adopting the cmbfast COBE-DMR normalisation, we implement the numerical approximation to marginalisation (see Appendix A) to find the optimal normalisation of the theoretical mo del to the full observational data set. We also similarly treat the beam uncertainties of BOOMERanG98 and MAXIMA1 as given by Lesgourgues & Liddle (2001) and the calibration uncertainties asso ciated with the observations, treating them as free parameters with Gaussian distributions about their nominal values (see Eq. A7). The minimised 2 for the concordance mo del is 174.2. In order to determine how go o d a fit this mo del is to the observational data we need to know the number of degrees of freedom of the analysis. Although 203 degrees of freedom are provided by the number of observational data points (assuming they are uncorrelated), these are reduced by the number of concordance parameters that are constrained using the CMB data alone. The flatness of the concordance mo del ( 0) and the scale invariance of the primordial power spectrum of scalar perturbations (ns 1) are extracted almost entirely from the CMB data. The remaining concordant parameters are more strongly constrained by non-CMB observations. We therefore estimate that 2 degrees of freedom should be subtracted from the original 203. Within our analysis we marginalise over a number of nuisance parameters. We fit for 23 individual calibration constants, 2 beam uncertainties (those of BOOMERanG-98 and MAXIMA-1) and an overall normalisation. Thus a further 26 degrees of freedom must be subtracted leaving 175 degrees of freedom. The 2 per degree of freedom is then 1.0, indicating that the concordance cosmology provides a go o d fit to the CMB data alone. Data correlations other than the correlated beam and calibration uncertainties of individual experiments, that we take to have no inter-experiment dependence, are not considered in our analysis. Including such correlations would further reduce the number of degrees of freedom, increasing the 2 per degree of freedom. However, our result is in agreement with the joint likeliho o d analyses that find that the cosmological mo del that best fits the CMB data is a better fit at the 1 or 2 level than fit to the concordance mo del (Wang et al. 2 0 0 2 a ). Q10 We find the normalisation of the concordance mo del to the full CMB data set to be = 16.3 ± 0.1 µK, where Q10 is defined through the relation (Linewaver & Barbosa 1998), 10(10 + 1)C
10

=

24 Q20 1 . 2 5 TC M B

(1 )

The normalised concordance mo del is plotted with the calibrated and beam corrected observational data in Figure 1. It is difficult to distinguish the most important measurements


­6­ because there are so many CMB data points the largest error bars. Therefore, for clarity, The binned observations are plotted with the x-axis emphasises the detail at small angular peaks. on the plot and it is dominated by those with we bin the data as described in Appendix B. concordance cosmology in Figure 2. The linear scales, clarifying measurements of the acoustic

3.

EXAMINING THE RESIDUALS

Our analysis is based on the assumption that the combined cosmological observations used to determine the concordance mo del are giving us a more accurate estimate of cosmological parameters, and therefore of the true C spectrum, than is given by the CMB data alone. Under this assumption, the residuals of the individual observed CMB band powers and the concordance CDM mo del become to ols to identify a variety of systematic errors. To this end, we create residuals, Ri , of the observed band power temperature anisotropies o Tiobs ± i bs with respect to the concordant band powers Tith such that, Ri = Tiobs - Tith obs ± i th . Tith Ti (2 )

Systematic errors are part of the CMB band power estimates at some level. We examine our data residuals as functions of the instrument type, receivers, scan strategy and attitude control. Possible sources of systematic uncertainty are discussed in the following section and the instrumental and observational details that may be associated with systematic errors are listed in Table 3. We lo ok for any linear trends that may identify systematic effects that are correlated with these details of the experimental design. We quote the 2 per degree of freedom of the best fitting line and the significance of the fit for each regression in Table 4. If the analysis determines that a linear trend can pro duce a significantly improved fit in comparison to that of a zero gradient line (zero-line) through the data, it may be indicative of an unidentified systematic source of uncertainty. The zero-line through all the residual data gives a 2 of 174.2. The analysis that determines the go o dness-of-fit of the concordance mo del to the CMB data has 175 degrees of freedom. If the gradient and intercept were independent of the parameters varied to pro duce the concordance fit, the degrees of freedom would be further reduced by 2. However, the intercept of any line that fits the residual data will depend on the normalisation of the concordance mo del. We therefore subtract only one further degree of freedom, giving 174 degrees of freedom. The best fitting zero-line fit to all the residual data has a 2 per degree of freedom


­7­ of 1.0 (= 174.2/174). In order to determine the significance of a better fit provided by a linear trend, an understanding of the statistical effects of intro ducing the 2 parameters to the line-fitting analysis is required. For a 2-dimensional Gaussian distribution, the difference between the 2 of the best-fit mo del and a mo del within the 68% confidence region of the best-fit mo del is less than 2.3 and for a mo del that is within the 95% confidence region of the best-fit mo del, this difference is less than 6.17 (Press et al. 1992). Our 68% and 95% contours in Figures 3 to 20 are so defined. The further the horizontal concordance zero-line is from the best fitting slope, the stronger the indication of a possible systematic error.

4.

POSSIBLE SOURCES OF SYSTEMATIC UNCERTAINTY 4.1. Foregrounds

If foreground emission is present, it will raise the observed power. Galactic and extragalactic signals from synchrotron, bremsstrahlung and dust emission have frequency dependencies that are different from that of the CMB (e.g. Tegmark & Efstathiou 1996). If such contamination is present in the data, it may be revealed by a frequency dependence of the residuals (Figure 6). Multiple frequency observations provide various frequency lever-arms that allow individual groups to identify and correct for frequency dependent contamination. Experiments with broad frequency coverage may be better able to remove this contamination than those with narrow frequency coverage. We therefore examine the residuals as a function of the frequency lever-arm (max - min )/main (Figure 7). Observations taken at lower absolute galactic latitudes, |b|, will be more prone to galactic contamination. In Figures 4, 16 and 5 we check for this effect by examining the residuals as a function of |b| (Figures 4 and 5) and galactic longitude (Figure 16). Less likely would be a signal asso ciated with the narrowness of the band pass of the main frequency channel main /main (Figure 17).

4.2.

Angular Scale-dependent effects by plotting the residuals as a function of is most critical when the curvature of the acoustic oscillations). We therefore explore the filter functions in space / (Figure

We examine scale-dependent uncertainties (Figure 3). The shape of the window function power spectrum is large (at the extrema of the the residuals as a function of the narrowness of 1 5 ).


­8­ The area of the sky observed determines the lowest probed while the beam size beam determines the highest . The resolution of the instrument and the pointing uncertainty become increasingly important as fluctuations are measured at smaller angular scales. Small beams may be sub ject to unidentified smearing effects that may show up as a trend in the residual data with respect to beam / ef f . Thus we examine the residuals as a function of the area of sky probed (Figure 12), beam / ef f (Figure 10) and pointing uncertainty (Figure 11) to lo ok for hints of systematic errors asso ciated with these factors.

4.3.

Calibration

To analyse various experiments, knowledge of the calibration uncertainty of the measurements is necessary. Independent observations that calibrate off the same source will have calibration uncertainties that are correlated at some level and therefore a fraction of their freedom to shift upwards or downwards will be shared. For example, ACME-MAX, BOOMERanG97, CBI, MSAM, OVRO, TOCO and CBI all calibrate off Jupiter, so part of the quoted calibration uncertainties from these experiments will come from the brightness uncertainty of this source. The remainder will be due to detector noise and sample variance and should not have any such inter-experiment correlations. Wang et al. (2002a) perform a joint analysis of the CMB data making the approximation that the entire contribution to the calibration uncertainty from Jupiter's brightness uncertainty is shared by the experiments that use this calibration source. The true correlation will be lower since the independent experiments observed Jupiter at different frequencies. Inter-experiment correlations are not considered in our analysis, since we are unable to separate out the fraction of uncertainty that is shared by experiments. Instead we test for any calibration dependent systematics by examining the data residuals with respect to the calibration source (Figure 8). We note that including correlations between data points would reduce the number of degrees of freedom of our 2 analysis.

4.4.

Instrument type, platform and altitude

The experiments use combinations of 3 types of detector that operate over different frequency ranges. We classify the data with respect to their instrument type; HEMT interferometers (HEMT/Int), HEMT amplifier based non-interferometric instruments (HEMT), HEMT based amplifier and SIS based mixer combination instruments (HEMT/SIS), bolometric instruments and bolometric interferometers (Bol/Int). We check for receiver specific


­9­ systematic effects by plotting the residuals as a function of instrument type (Figure 9). Water vapour in the atmosphere is a large source of contamination for ground based instruments. There may also be systematic errors asso ciated with the temperature and stability of the thermal environment. We therefore explore instrument altitude (Figure 14) and platform (Figure 13) dependencies of the data residuals.

4.5.

Random controls

We use a number of control regressions to check that our analysis is working as expected. To this end, the residuals are examined with respect to the publication date of the band power data (Figure 20), the number of letters in the first author's surname (Figure 18) and the affiliation of the last author (Figure 19). We expect the line fitted to these control regressions to be consistent with a zero-line through the residual data. Any significant improvement provided by a linear fit to these residuals may be indicative of a problem in the software or metho dology.

5.

RESULTS

For the regressions plotted, the residual data is binned as described in Appendix B so that any trends can be more effectively visualised. Since the data binning pro cess may wash out any discrepancies between experiments, the linear fit analyses are performed on the unbinned data residuals. In Figures 3 to 20, the line that best-fits the data is plotted (solid white) and the 68% (dark grey) and 95% (light grey) confidence regions of the best-fit line are shaded. For each plot, we report the 2 and the 2 per degree of freedom for the best-fit line, the probability of finding a mo del that better fits the data and comment on the significance of the deviation of the zero-line (dashed black). Our results are listed in Table 4 and imply that the most significant linear trend observable in the residuals is with respect to the absolute galactic latitude |b| of the observations (see Figure 4). This trend is not eliminated by the removal of any one experiment and may be indicative of a source of galactic emission that has not been appropriately treated. The weighted average of points |b| > 40 is -1%, while it is +1% for |b| < 40 . If this is due to galactic contamination, then the normalisation Q10 may have to be reduced to 16.1 µK. For this regression, the errors in both the y and x direction are used in the fit. We have defined |b| to be that of the centre of the observations and the uncertainties to extend to edges of the range. This allows the observations some freedom of the x-co ordinate in the


­ 10 ­ line-fitting analysis and may over-weight those detections that span small ranges in absolute galactic latitude. It is therefore also interesting to examine the residuals with respect to the central |b| to determine the significance of the trend with the x-co ordinate freedom removed (see Figure 5). The most plausible galactic latitude regression will be somewhere between the two plots. Removing the x-co ordinate freedom removes the significance of the trend. This result implies that experiments that observe over small ranges in galactic latitude are dominating the trend and we therefore can not simply correct for the systematic that is implied in (Figure 4). The comparison of rms levels in galactic dust (Finkbeiner et al. 1999) and synchrotron1 maps over the areas of CMB observations may help to clarify the interpretation of the trend. Such a technique has recently been applied to the MAXIMA1 data (Jaffe et al. 2003) but has yet to be performed on the full CMB data set. Other plots also show some evidence for systematic errors. Figures 2 and 3 indicate that the 6 bins between 1000 2000 prefer a lower normalisation. This could be due to underestimates of beam sizes or pointing uncertainties or unidentified beam smearing effects at high for small beams. However, Figures 10 and 11 show no evidence for any trends, although limiting the pointing uncertainty analysis to the 5 points with the largest uncertainties would indicate a trend, suggesting that the largest pointing uncertainties may have been underestimated.

6.

SUMMARY

Although individual observational groups vigorously test their data sets for systematic errors, the entire CMB observational data set has not yet been collectively tested. Under the assumption that the concordance mo del is the correct mo del, we have explored residuals of the observational data with respect to this mo del to see if any patterns emerge that may indicate a source of systematic error. We have performed linear fits on the residual data with respect to many aspects of the observational techniques and, for the ma jority, we have found little or no evidence for any trends. However, there is significant evidence for an effect asso ciated with |b| that is not eliminated with the exclusion of any one data set. The data prefers a linear trend that is inconsistent at more than 95% confidence with a zero gradient line through the residuals. The best-fit line to the data suggests that CMB observations made closer to the galactic
1

http://astro.berkeley.edu/dust


­ 11 ­ plane may be over-estimated by approximately 2%. A more detailed analysis of galactic dust and synchrotron maps may clarify the source of the indicated systematic uncertainty. LMG thanks Martin Kunz for useful discussions and is grateful to the University of Sussex where part of the work was carried out. LMG acknowledges support from the Royal So ciety and PPARC. CHL acknowledges a research fellowship from the Australian Research Council.

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