arXiv:astro-ph/0502330v2 4 Nov 2005 o.Nt .Ato.Soc. Astron. R. Not. Mon.
ua Rajguru Nutan methods estimation comparison parameter A and VSA: maps and coincident CBI of the from observations CMB nstoishv rvdivlal netbihn h cu (CMB) the establishing background in invaluable microwave proved have cosmic anisotropies of Measurements INTRODUCTION 1 irnCleary Kieran nhn Lasenby Anthony lv Dickinson Clive oahn .Sievers L. Jonathan. cetd— eevd— noiia om1 uut2018 August 15 form original in —; received —; Accepted ihr .E Saunders E. D. Richard nhn .S Readhead S. C. Anthony ihe .Hobson P. Michael lzbt Waldram Elizabeth c + † × ⋄ ⋆ 7 6 5 ‡ 4 ◦ 3 2 1 rsn drs:AtooyCnr,Uiest fSse,B Sussex, of University Centre, Astronomy address: Present rsn drs:Dprmn fPyis h ey Wilkinso Denys The Physics, of Department address: Present ainlRdoAtooyOsraoy OBx2 re Bank, Green 2, Box PO Observatory, Astronomy Radio National nttt eAstrof de Instituto aionaIsiueo ehooy alCd 0-4 Pasa 105-24, Code Mail Technology, of Institute California aainIsiuefrTertclAtohsc,Univers Astrophysics, Theoretical for Institute Canadian orl akOsraoy acefil,Cehr K19DL, SK11 Cheshire Macclesfield, Observatory, Bank Jodrell ainlRdoAtooyOsraoy oor,N 70,U 87801, NM Socorro, Observatory, Astronomy Radio National srpyisGop aeds aoaoy aige Road Madingley Laboratory, Cavendish Group, Astrophysics rsn drs:JtPouso aoaoy 80OkGrove Oak 4800 Laboratory, Propulsion Jet address: Present -al [email protected] E-mail: rsn drs:Dprmn fPyis nvriyo Cali of University Physics, of Department address: Present 05RAS 2005 rsn drs:H il hsc aoaoy ydl Aven Tyndall Laboratory, Physics Wills HH address: Present rsn drs:Fclyo ahmtc n hsc,Unive Physics, and Mathematics of Faculty address: Present ´ sc eCnra,320L aua eeie Spain. Tenerife, Laguna, La 38200 Canarias, de isica 3 , 1 ⋄ 3 ,⋆ al .Contaldi R. Carlo , 000 , 5 tvnT Myers T. Steven , 1 iad Genova-Santos Ricardo , ra .Mason S. Brian , 1 –0(05 rne 5Ags 08(NL (MN 2018 August 15 Printed (2005) 1–10 , 1 ihe .Jones E. Michael , oetA Watson A. Robert , 4 Anˇze Slosar , idw,sosta h w ehd il ossetresults. consistent words: yield Key estimates methods parameter two appro VSA the the an that of function, re-evaluation shows window A windows, bandpower group. the the VSA is the of by difference incre place key with in The also function group. consistent We each window observations. be of summer method to during estimation shown data rameter VSA is on This errors simulations. calibration Carlo Monte from evdb ahgop w r on ob necletagreement. excellent in be the 2 to Of a F bandpowers. found is the are flat and there two into plane group, data each map CMB by t the of served in (CBI) compression tested Imager usual is the Background datasets to Cosmic full and the of (VSA) consistency Array Small Backg Microwave Very Cosmic the the of observations coincident present We ABSTRACT 1 ihr .Savage S. Richard , 5 aalRebolo Rafael , σ iceac ewe h orlto ftedt n h ee exp level the and data the of correlation the between discrepancy omlg:osrain omcmcoaebackground microwave cosmic – observations cosmology: 1 2 , 7 4 × ihr .Battye A. Richard , t fTrno 0S.Gog tet oot,Otro 5 3 M5S Ontario, Toronto, Street, George St. 60 Toronto, of ity ioh .Pearson J. Timothy , o .Davies D. Rod , neaC Taylor C. Angela , ona ekly A97070,USA. 94720-7300, CA Berkeley, fornia, ea A915 USA. 91125, CA dena, r- 1 abig B H,UK. 0HE, CB3 Cambridge , iho,B19H UK. 9QH, BN1 righton, , st fLulaa 00Lulaa Slovenia. Ljubljana, 1000 Ljubljana, of rsity ulig el od xod X R,UK. 3RH, OX1 Oxford, Road, Keble Building, n rv,Psdn,C 10,USA. 91109, CA Pasadena, Drive, e rso S T,UK. 1TL, BS8 Bristol ue, UK. ‡ R¨udiger Kneissl , 3 SA. lhaWilkinson Althea , 6 V294 USA. 24944, WV rc Rocha Graca , 6 03 eooe l 04 edede l 04.A im- An 2004). al. al. et et Readhead 2004; Goldstein al. 2003; et al. pa- Rebolo et cosmological in 2003; (Spergel used widely estimation are rameter and model ΛCDM rent et Grainge Keith , 1 , † naScaife Anna , A T E 3 tl l v2.2) file style X ihr .Davis J. Richard , 1 3 , 1 .RcadBond Richard J. , ‡ 1 o´ let Rubi˜no-Martin Jos´e Alberto , 5 ai Titterington David , , u .Pooley G. Guy , ◦ 1 ayLancaster Katy , 1 ae .Hafez A. Yaser , 3 alF Scott F. Paul , . ntetidmosaic, third the In on CB from (CMB) round s ftevariance the of use uirpae prior plane, ourier sn bandpower using , he oac ob- mosaics three osdrtepa- the consider 3 lsoe.The elescopes. iainused ximation , sdphase ased 8 Canada. H8, 4 1 1 , , , 3 1 , , + ected 1 , , 6 , 2 Nutan Rajguru et al.
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Figure 1. Typical uv-coverage. Top left: VSA. Each point represents the full 1.5 GHz bandwidth. Top right: CBI. Each point represents a single 1 GHz channel. Bottom: Synthesised beams for typical uv-coverage. portant check on the accuracy of CMB measurements is to power. Both the CBI and VSA groups use a maximum like- compare the data obtained from instruments which are sub- lihood method of estimating the bandpowers, with some dif- ject to different systematic effects. The CBI and VSA in- ferences in the implementation, but there are important dif- struments have significant design differences and the com- ferences in the type of window functions used for param- parison of our data is a check that known systematic effects eter estimation. The CBI group computes the bandpower have been accurately corrected for, and that neither dataset window function which fully takes into account the anti- is seriously contaminated by unrecognised systematic errors. correlations between neighbouring bins (Myers et al. 2003). In the CMB community, the focus has been on the compar- The VSA group use variance windows as an approximation ison of power spectra. However, maps of CMB anisotropies to the bandpower windows when computing parameter esti- also contain important information. In particular, they are mates (Rubi˜no-Martin et al. 2003; Slosar et al. 2003; Rebolo used in tests of Gaussianity (see for example, Komatsu et al. et al. 2004), although anti-correlations are not accounted for. 2003; Aliaga et al. 2003; Savage et al. 2004), a key assump- We assess the bias resulting from this approximation by re- tion of CMB analysis. It is therefore an important exercise evaluating VSA parameter estimates using bandpower win- to check the correlation between maps of CMB anisotropies. dow functions. To this end the VSA has undertaken a programme of ob- serving fields previously imaged by the CBI. In this paper we present the results of these observations and assess their consistency with the CBI data. In section 2 we summarise the key differences between the VSA and CBI instruments and the implications for ob- Key scientific results from the measurement of CMB serving strategies. In section 3 we describe how the differ- anisotropies are the cosmological parameters. An essential enced maps are produced and present the results of the com- ingredient in converting the flat bandpower estimates into parison. Section 4 contains our assessment of the impact of cosmological parameters are the window functions which al- window functions on the VSA parameter estimation. Finally, low a theoretical power spectrum to predict a flat band- in section 5 we present our conclusions.
c 2005 RAS, MNRAS 000, 1–10 CMB observations from the VSA & CBI 3
Table 1. A summary of the specifications of the CBI and the VSA extended array.
VSA CBI
Observing Frequency 33 GHz 31.5 GHz Bandwidth 1.5 GHz 10 GHz Number of Channels 1 10 Number of Antennas 14 13 Number of Baselines 91 78 Range of baseline lengths 0.6m−2.5 m 1.0m−5.5m ℓ range ≈ 300 − 1500 ≈ 300 − 3500 Primary Beam (FWHM) 2.11◦ 45.2 arcmin × 31 GHz/ν System temperature ≈ 35 K ≈ 30 K Mirror diameters 0.32 m 0.90 m Synthesised beam (FWHM) ≈ 11 arcmin ≈ 5 arcmin Flux sensitivity 50 Jy s−1/2 1.5 Jy s−1/2
2 THE TELESCOPES separated by 8 minutes in RA. The trail field visibilities are then subtracted from those of the lead field. This has the ef- The CBI is an interferometric telescope located at an al- fect of removing the contaminating signal which is constant titude of 5000 m in the Atacama Desert in northern Chile. on an 8 minute timescale, whilst preserving the statistical The instrument operates in ten 1-GHz frequency bands over distribution of the sky Fourier modes (Padin et al. 2002). 26−36 GHz. The antennas have low-noise high electron mo- Both sky maps and the power spectrum are estimated from bility transistor (HEMT) amplifiers and typical system tem- the differenced data. peratures including the CMB, ground and atmosphere are The CBI mounting platform allows the orientation of ≈30 K. The 13 Cassegrain antennas, each 0.9 m in diame- the baselines to be changed by rotating the platform about ter are co-mounted on a 6 m tracking platform (Padin et al. the optical axis. The rotation of the tracking table together 2002). with the broad bandwidth ensures a well-filled aperture The VSA is sited at the Teide Observatory, in Tener- plane and a circularly symmetric synthesised beam. Figure ife, at an altitude of 2400 m. The VSA operates in a single 1 shows the typical uv-coverage and synthesised beams of 1.5-GHz channel at a central frequency of 33 GHz. The 14 each telescope for the observations presented in this paper, antennas also have HEMT amplifiers and the typical system in the range of angular scales common to both experiments. temperature is ≈35 K. In the extended array configuration, The uv-coverage of the VSA is less complete and the beam the VSA uses mirrors of diameter 0.322 m. The VSA horn- is less circularly symmetric than that of the CBI, for these reflector antennas are mounted on a tilt table hinged along fields. This is partly due to the low declination of the obser- east-west, but each antenna individually tracks the observed vations (−3.5◦) which is close to the lowest elevation that field by rotating its horn axis perpendicularly to the table the VSA is able to observe. The telescope is located at a hinge, and wavefront coherence is maintained with an elec- latitude of 28◦ and at the higher declinations of the main tronic path compensator system (Watson et al. 2003). This primordial fields, uv-coverage is significantly better (Taylor is a key design difference to the CBI. et al. 2003; Dickinson et al. 2004). Unlike the CBI, the VSA The individual tracking of the VSA antennas allows for mounting table does not allow for rotation about the optical the filtering of contaminating signals. These may be celes- axis. Together with the lower bandwidth this limits the cov- tial sources such as the Sun and Moon, or ground-spill and erage of the aperture plane, although minimises the number other ground based spurious signals (Watson et al. 2003). of redundant baselines at a given frequency. A summary of The fringe rates of contaminating signals differ sufficiently the specification of the two telescopes is shown in Table 1. from those of the target to allow effective filtering whilst re- taining most of the data. For example, filtering the Sun at a ◦ distance of 20 from the target removes approximately 25% 3 OBSERVATIONS of the data. The VSA also uses a ground-shield to minimise ground-spill. The consequences of these design differences The CBI data used in this comparison are the mosaics 02H, are two-fold. Firstly, the VSA is able to observe 24 hours 14H and 20H and first reported in Pearson et al. (2003). The a day and its extended array (as used here) can filter out CBI data also include two deep field observations which fall emission from the Sun and Moon when they are as close as within the 14H and 20H mosaics and are described in Mason 9◦. The CBI is limited to observing at night and to fields et al. (2003). Both CBI mosaiced and deep field data were which are more than 60◦ away from the Moon (Padin et al. collected during the period January - December 2000. Each 2002). Secondly, the VSA is unaffected by ground-spill con- mosaic consists of 42 differenced fields. tamination for fields within 35◦ of the zenith and so is able The VSA observations were carried out at a later epoch. to make direct images of the sky. The raw CBI data are con- The data were made between May 2002 and May 2004 with taminated by ground-spill and this is eliminated by means of the telescope in the extended array configuration. The larger a differencing scheme. The method of differencing the CBI primary beam size of the extended array mirrors allows each data used in this analysis works as follows. A lead field is mosaic to be covered in three pointings. Table 2 shows the observed followed by a trail field at the same declination but coordinates and effective integration times for the VSA ob-
c 2005 RAS, MNRAS 000, 1–10 4 Nutan Rajguru et al.
Table 2. Celestial coordinates for the VSA observations of the sources > 3.4 mJy in the 1.4 GHz NVSS catalogue are pro- CBI 02H, 14H and 20H mosaics. The effective integration time is jected out of the CBI data. A statistical correction based on calculated after flagging and filtering of the data. the CBI source count is then applied for sources < 3.4 mJy at 1.4 GHz (Mason et al. 2003). For this investigation, the
Field RA (J2000) DEC (J2000) tint (hrs) OVRO fluxes have been subtracted from the CBI data but the constraint matrix strategy has not been employed. This VSA-02H-a 02 44 24.00 -03 30 00.0 169.9 is due to the limited resolution of the data included in the VSA-02H-b 02 50 00.00 -03 30 00.0 86.2 comparison and to avoid removing a significant fraction of VSA-02H-c 02 55 36.00 -03 30 00.0 49.7 the data, which is unnecessary for this analysis. VSA-14H-a 14 44 24.00 -03 30 00.0 37.1 VSA-14H-b 14 50 00.00 -03 30 00.0 73.3 The different methods usually employed to remove the VSA-14H-c 14 55 36.00 -03 30 00.0 44.9 effects of contaminating sources are a key distinction in the VSA-20H-a 20 44 24.00 -03 30 00.0 154.2 analysis of the two groups. Although we are unable to make VSA-20H-b 20 50 00.00 -03 30 00.0 94.2 a direct comparison of these strategies, due to the positions VSA-20H-c 20 55 36.00 -03 30 00.0 74.4 of the fields, the 33 GHz source counts estimated from the main VSA source monitoring programme have been used to evaluate the CBI source subtraction strategy. Based on VSA source counts Cleary et al. (2004) find the residual correction servations. The VSA data reduction and calibration proce- due to sources below the detection threshold of 3.4 mJy at dure are described in Dickinson et al. (2004) and references 1.4 GHz to be 0.03 Jy2 sr−1 which is consistent with the CBI therein. group estimate of 0.08 ± 0.04 Jy2 sr−1.
3.1 Foreground contamination 3.2 Maps At frequencies of 26-36 GHz, the dominant cosmological con- As a consequence of the need to difference the CBI visibil- tamination to CMB observations comes from galactic fore- ities, maps from the two telescopes may only be compared grounds and extragalactic radio sources. The diffuse galactic if the differencing scheme is also implemented on the VSA foregrounds include both bremsstrahlung and synchrotron data. Further to the usual data reduction, the VSA data emission. However, this emission is concentrated in the were processed is several ways to enable this. galactic plane and contamination may be avoided by observ- The VSA has a much larger primary beam than the ing at high galactic latitudes. The observations presented CBI and covers the same area of sky in fewer pointings. To ◦ here are at galactic latitudes > 20 . In addition, the data produce data equivalent to each CBI lead and trail field, the are insensitive to the large angular scales where galactic con- following procedure was implemented. The first step was to tamination is significant. Therefore, the main contaminant shift the VSA field centres to each of the CBI field centres. is likely to be extragalactic point sources. The complex visibility measured by an interferometer is de- The standard VSA source-subtraction strategy involves fined as: an initial survey with the Ryle Telescope (RT) at 15 GHz (Waldram et al. 2003). Sources identified by the RT are then 2 −2πiu·x monitored with the source subtractor at 33 GHz. The obser- V (u)= d x A(x) I(x) e + N(u) (1) Z vations are carried out simultaneously with the CMB field observations to take account of the variability of the sources. where A(x) is the primary beam, I(x) is sky intensity, A statistical correction is also applied to the power spectrum u = (u, v) is the baseline length, measured in units of the to remove the small effect of the remaining, fainter sources. wavelength and N(u) is the instrumental noise (Thompson As the RT is located in Cambridge at a latitude of +52◦, we et al. 2001). The field shift was achieved by rotating the were unable to survey fields at the low declinations of the phase of the VSA visibilities so that the direction cosines, CBI observations. For this reason, a more limited level of x = (∆x, ∆y), were defined with respect to a new phase source subtraction was implemented. This involved the sub- centre: traction from the data of both groups, the fluxes obtained ∆y = sin δ cos δ0 − cos δ sin δ0 cos(α − α0) by the CBI group from observations at 31 GHz with the ∆x = cos δ sin(α − α0) 40-metre telescope at the Owens Valley Radio Observatory (OVRO). These observations were carried out, simultane- where α, δ and α0, δ0 are the right ascension and declination ously, in so far as was possible, with the CBI observations. of the VSA and CBI field centres respectively. As noted above, the VSA observations were carried out a At this stage, before differencing can be carried out, later epoch and no further source observations were carried the larger primary beam of the VSA must be taken into ac- out to account for the variability of sources. count. The effect of observing a limited area of sky is to con- The typical sensitivity achieved in the OVRO data was volve the sky Fourier modes with the aperture function (the 2 mJy (rms) which allows for a completeness estimate of Fourier transform of the primary beam). To mimic observa- 90% at S31 > 16 mJy (Mason et al. 2003). To achieve high tions with the CBI beam which has a FWHM of 45.2 arcmin- ℓ measurements of the power spectrum, a deeper level of utes × (31 GHz/ν), the VSA visibilities were convolved with discrete source subtraction is required. To achieve this the the CBI aperture function. This was modelled as a Gaussian usual approach of the CBI group is to employ the strat- with a cut-off at 0.45 m (the outer radius of the antenna) and egy of constraint matrices to ‘project out’ sources at known a central region with no illumination at r < 0.0774 m (corre- positions but with unknown fluxes (Bond et al. 1998). All sponding to blockage by the secondary mirror). The 84 lead
c 2005 RAS, MNRAS 000, 1–10 CMB observations from the VSA & CBI 5
Figure 2. The central region of the 42-field mosaics of differenced maps. Each map covers an area of 2.4◦ x 2.4◦. The RA scale refers to the position of the lead field. Left: VSA data Right: CBI data. Top: 02H mosaic Centre: 14H mosaic Bottom: 20H mosaic
c 2005 RAS, MNRAS 000, 1–10 6 Nutan Rajguru et al.
Figure 3. Sensitivity maps for each mosaic shown in 2. Each map covers an area of 2.4◦ x 2.4◦ and is blanked at a threshold of 15 mJy beam−1. The RA scale refers to the position of the lead field. Left: VSA data Right: CBI data. Top: 02H mosaic Centre: 14H mosaic Bottom: 20H mosaic
c 2005 RAS, MNRAS 000, 1–10 CMB observations from the VSA & CBI 7 and trail fields for each mosaic were produced by convolv- Table 3. Correlation of the 02H and 14H mosaics. The expected ing the phase rotated data with the CBI aperture function. correlation is in excellent agreement with the actual correlation Corresponding lead and trail fields were then differenced. of the 02H and 14H mosaics. It is important to note that matching the uv-coverage of the two interferometers is critical in making maps of the Expected 02H CMB. This is a consequence of sampling a random Gaus- Map plane 0.23 ± 0.02 0.25 ± 0.03 sian field. Not only must the same range of angular scales uv-plane 0.12 ± 0.02 0.14 ± 0.02 be used, but they must sample the same region of uv-space. Expected 14H Figure 1 illustrates the mismatch in uv-coverage between Map plane 0.23 ± 0.02 0.24 ± 0.03 the VSA and CBI in the range of common uv-scales. To uv-plane 0.12 ± 0.02 0.13 ± 0.02 overcome this, the data were binned and re-weighted. As the data were obtained from a convolution of the sky fourier modes with the CBI aperture function, which has a FWHM of 67λ, a cellsize of 17λ was chosen to ensure that the aper- Table 4. Correlation of the 20H mosaic. The expected correlation ture function was more than adequately sampled according in the 20H mosaic given the greater weight of data in these fields to the Nyquist sampling theorem. Data cells with a match is shown in the first column, and in the second column this is in both sets were assigned a weight of the geometric mean of modified by the inclusion of the VSA summer phase errors. The their individual weights. Data cells adjacent to, but without actual correlation, in column three, is consistent with the level a direct match, were downweighted by a factor of two. Cells expected given the phase calibration errors. without a direct or adjacent match were assigned a weight of zero. Expected Expected 20H Maps were then produced from the re-weighted visibili- (incl. phase errors) ties using the AIPS package and are shown in Figure 2. The Map plane 0.27 ± 0.02 0.23 ± 0.02 0.22 ± 0.03 CBI data have been standardised to 33 GHz with the as- uv-plane 0.14 ± 0.02 0.11 ± 0.02 0.10 ± 0.02 sumption of a blackbody spectrum. Both datasets have been corrected for the primary beam. In the case of the VSA data, the data were corrected for an effective primary beam where the sky is real, the underlying fourier modes a(u)= a∗(−u). the effective beamsize, σeff , is given by: Consequently, correlations exist between visibilities that lie on opposite sides of the uv-plane. To account for this, the 1 1 1 conjugate symmetry of the visibilities was used to reflect the 2 = 2 + 2 data into one half of the plane and the correlation was car- σeff σC σV ried out only in this region. The real and imaginary parts of where σC and σV are the CBI and VSA primary beam sizes the visibilities were treated independently. respectively. The effective beam size is smaller than the CBI One of the disadvantages usually cited with this mea- beam size by 5%. This is because the sky signal measured sure of correlation is the difficulty of interpretation. This by the VSA data has been multiplied by the CBI and VSA has been overcome by using Monte-Carlo simulations of the primary beams. data to predict the expected correlation. The input model Figure 3 shows the sensitivity maps for each mosaic. for these simulations was a standard ΛCDM model with The minimum noise in the VSA 02H, 14H and 20H mosaics noise levels and uv-coverage appropriate to the actual ob- is 7.29 mJy beam−1, 7.44 mJy beam−1 and 6.03 mJy beam−1 servations. The method of producing the VSA differenced respectively. The corresponding noise levels for the data from the simulations was carried out in the same man- CBI mosaics are 3.88 mJy beam−1, 1.82 mJy beam−1 and ner as for the actual observations. 1.08 mJy beam−1. The noise levels in the CBI maps are The correlation of the 02H and 14H mosaics is shown in approximately a factor of 2 lower than those of the VSA Table 3. There is excellent agreement between the observed maps, with a wider gap in the region of the CBI deep fields and expected correlations in the 02H and 14H mosaics in which are clearly visible in the sensitivity maps. The blank- both the map plane and the uv-plane. ing threshold of 15 mJy beam−1 is for illustrative purposes The expected correlation of the 20H mosaics is higher only. In calculating the map correlations, a cut was applied than for the 02H and 14H mosaics, which reflects the greater when the power in the primary beam reached 1/e of the signal-to-noise in these observations. The actual correlation maximum level. however, lies 2σ away from the expected level - see Table There are a number of statistics which may be used 4. In contrast to the first two mosaics, the VSA data in the to quantify the consistency between datasets. The measure 20H mosaic were largely collected during the summer period used here for both the map plane and uv-plane tests is the and between the hours of 8am and 6pm. In this period, the product-moment correlation coefficient (e.g. Barlow 1989). accuracy of the phase calibration is known to deteriorate and typical phase errors are some 20◦. The source of these xy − x y errors is thought to be due to a slight warping of the tilt r = table in the heat. All of the VSA data in the 02H and 14H σxσy mosaics were made outside of the summer daytime period, In both planes a weighting scheme was used. In cor- when typical phase errors are around 3-4◦. relating the maps, each pixel was weighted by the power of The VSA phase errors are estimated from the change the primary beam at the centre of the pixel. In the uv-plane, in phase calibration factors over the course of a day. Sev- each cell was weighted by the inverse of the noise squared. As eral calibration observations are carried out each day but
c 2005 RAS, MNRAS 000, 1–10 8 Nutan Rajguru et al.
6000
0.02 5000
4000 ) 2 K µ ( π
/2 3000 l +1)C l ( l
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0 0 200 400 600 800 1000 1200 1400 1600 l Figure 4. The power spectrum of the 02H, 14H and 20H mo- saics from the undifferenced VSA data (open circles) and the dif- ferenced CBI data (solid circles). The concordance model which 0 best fits WMAP data is shown for comparison. 0 500 1000 1500
2 2 Table 5. The CMB band-powers (T0 ℓ(ℓ+1)Cℓ/2π [µK ]) for the joint mosaic power spectra shown in figure 4. ℓeff is the centroid 0.02 of the bandpower window function.
VSA CBI Bin ℓ-range VSA ℓeff CBI ℓeff
1 0 − 400 3730 ± 669 306 3455 ± 973 311 0.01 2 400 − 586 1798 ± 372 487 1993 ± 432 490 3 586 − 772 1858 ± 386 679 1955 ± 367 673 4 772 − 957 2363 ± 490 861 2396 ± 405 858 5 957 − 1143 1382 ± 580 1044 1099 ± 244 1045 6 1143 − 1329 1709 ± 907 1232 1873 ± 309 1226 0 7 1329 − 1515 15 ± 817 1416 1055 ± 251 1434
there may be significant change in the phase calibration be- 0 500 1000 1500 tween observations. The calibrators are either point sources or resolved sources for which a model is known. The phase Figure 5. top: VSA variance window functions. bottom: VSA errors have a weak dependence on baseline length but show bandpower window functions. Alternate curves are solid and dot- a high degree of repeatability. The expected correlation is, of ted lines for clarity. course, revised downwards by modelling the summer phase errors in the simulations. The phase errors fully account for the reduced correlation seen in the 20H mosaics. agreement in the first six bins. As the contribution of point In principle, the power spectrum is insensitive to the sources increases as ℓ2, the discrepancy in the final bin may phases of the visibilities, although this depends upon the be due to the variability of sources, as the data were col- level of correlation of the phase errors. The phase errors on lected at different epochs. Table 5 shows the binning and the VSA data are found to be highly correlated. Although bandpowers for the power spectra. typical summer phase errors are some 20◦, the rms about the mean phase error for each baseline is only 2-3◦. A further consideration is the number of baselines which contribute to each uv-cell. If a large number of cells have contributions 4 WINDOW FUNCTIONS from several baselines then the phase errors will be less cor- It is standard practice for experiments observing a limited related and it is possible that this could affect the power sky area to assume a flat bandpower when estimating the spectrum estimate. However, simulations of VSA shallow CMB power spectrum (Kuo et al. 2002; Hobson & Maisinger field observations which include the effect of the crossing 2002; Myers et al. 2003). However, the theoretical power uv-tracks and summer phase errors show that this has a spectra are not flat and in estimating the cosmological pa- negligible effect on the power spectrum. However, as a fur- rameters it is necessary to define a window function which al- ther safeguard the VSA group discard the worst affected lows a theoretical prediction of the flat bandpower qB which portion of summer data. may be compared to the experimental values. The theoreti- Figure 4 shows the joint power spectrum for the 02H, cal prediction is the expectation value of qB for a given input 14H and 20H mosaics in the range of scales common to both model: instruments. The VSA power spectrum was estimated using B the direct VSA observations, prior to the implementation of Wℓ hqBi = C (ap) , (2) ℓ ℓ the differencing and reweighting scheme. There is excellent Xℓ
c 2005 RAS, MNRAS 000, 1–10 CMB observations from the VSA & CBI 9
To estimate the bandpowers, the CBI group use the second Table 6. The CMB band-powers (in µK2) for the complete VSA order Taylor expansion of the log likelihood function around data set combining both compact and extended array data. the maximum likelihood bandpowers. A quadratic approx- imation for the change in the bandpower, δqB , is used it- 2 2 Bin ℓ-range ℓeff T ℓ(ℓ + 1)C /2π [µK ] eratively, to move towards the maximum, with the second 0 ℓ +1616 derivative of the log likelihood function replaced by its ex- 1 100 − 190 156 3626−1150 +1561 pectation value, the Fisher matrix. The expectation value of 2 190 − 250 220 5561−1232 +1123 the bandpowers is then given by 3 250 − 310 281 5131−959 +438 4 310 − 370 333 2531−411 5 370 − 450 410 1570+246 1 −1 −1 S −1 S −219 +383 hqB i = F ′ Tr (C CB′ C ) C , (3) − 2 BB 6 450 500 475 1811−356 XB′ h i +356 7 500 − 580 537 2212−274 +356 where C is the covariance matrix with independent con- 8 580 − 640 611 1736−301 S +329 tributions from all signal sources: the CMB signal C , the 9 640 − 700 670 1614−301 N src +411 instrumental noise C and the foreground signals C and 10 700 − 750 721 1628−356 res S +301 C . CB′ is the CMB signal from each band (Myers et al. 11 750 − 850 794 2486−246 +274 2003). Since 12 850 − 950 902 1553−274 +274 13 950 − 1050 987 1135−246 +274 S 14 1050 − 1200 1123 677−246 S S ∂C +356 C ≡ CB = Cℓ, (4) 15 1200 − 1350 1267 937−329 ∂Cℓ +657 XB Xℓ 16 1350 − 1700 1440 758−603 the bandpower window functions can be calculated, once the maximum likelihood bandpowers have been obtained, from Table 7. The priors assumed for the basic parameters. The no- S tation (a, b) for parameter x denotes a top-hat prior in the range B 1 −1 −1 S −1 ∂C C C ′ C Wℓ /ℓ = F BB′ Tr ( B ) , (5) a
c 2005 RAS, MNRAS 000, 1–10 10 Nutan Rajguru et al.
Table 8. Parameter estimates and 68% confidence intervals from dictions to a maximum of 28%. Differences of this size would the marginalised distributions. When the marginalised posterior be likely to bias the parameter estimates. However, the level for a parameter does not contain a peak, 95% confidence limits of agreement between the bandpower predictions is not a are given. simple function of bin size and will fluctuate depending on the exact binning used. Variance windows Bandpower windows The VSA is currently undergoing an upgrade to increase the sensitivity of the instrument and to enable a measure- 0.006 0.006 ωb 0.029±0.006 0.027±0.006 0.04 0.04 ment of the power spectrum up to ℓ = 2500. The upgrade ωdm 0.14±0.04 0.13±0.04 0.09 0.07 will involve the fitting of larger mirrors with a diameter of Ωm 0.42±0.17 0.39±0.17 0.17 0.17 0.55 m. In this case, the fraction of non-zero elements in the ΩΛ 0.58±0.09 0.61±0.07 0.02 0.02 covariance matrix will increase slightly to 7.5%. The suit- θ 1.04±0.02 1.05±0.02 0.10 0.09 ability of the approximation also depends upon the signal- h 0.69±0.10 0.71±0.10 0.07 0.07 to-noise achieved. The upgraded VSA will have increased ns 0.90±0.07 0.93±0.08 10 0.2 0.2 ln10 As 3.4±0.2 3.4±0.2 sensitivity, but taken in conjunction with the reduced signal, τ (0.04, 0.47) (0.04, 0.48) then the overall signal-to-noise level will remain about the 10 11 zre 21±10 22±11 same. Given the significant difference between the expected 0.9 0.9 Age 13.1±0.9 13.2±0.9 bandpower values which may arise and to ensure that this does not bias future VSA parameter estimates, analysis of the upgraded VSA data will be carried out using bandpower window functions. In addition to the priors on the basic parameters, listed in Table 7, we also impose priors on some of the derived parameters. Specifically, we use top-hat priors on the age of the Universe lying between 10 and 20 Gyr, and of H0 lying 5 CONCLUSIONS between 40 km s−1 Mpc−1 and 100 km s−1 Mpc−1. In this paper we have presented three sets of coincident CMB The CosmoMC software was run on a 24-node linux clus- observations observed at different epochs by the VSA and ter. The chains were run until the largest eigenvalue returned CBI telescopes. We have chosen to analyse the full datasets, by the Gelman-Rubin convergence test reached 0.08. After rather than focusing on the power spectra, in order to inves- burn-in a total of more than 100,000 samples were collected. tigate any possible systematic effects which may not other- As successive samples in a Markov chain are, by nature, cor- wise be revealed. The correlation of the datasets from each related, the samples were thinned by a factor of 25 resulting group was found to be as expected for the 02H and 14H in approximately 4000 independent samples. These samples mosaics. In the third mosaic the data disagreed with the were then used to calculate the marginalised distributions Monte Carlo simulations at a level of 2σ. However, this is and parameter estimates (see table 8). consistent with the phase calibration errors expected from Figure 6 shows the marginalised distributions for each VSA data during the summer months. It has been estab- parameter. It is clear that no significant bias has been intro- lished that this does not affect the power spectrum estima- duced as a result of using variance windows to approximate tion due to the correlated nature of the phase errors. The bandpower windows. The largest discrepancy is seen in the results of this analysis reaffirm that both groups have cor- parameter estimate for ω where the bandpower windows b rectly characterised the noise properties and systematics of reduce the estimate by one-third of the 1-sigma error. For the telescopes as well as other, potential data contaminants. all other parameters the estimates are consistent to a much We have investigated the use of variance windows as an smaller fraction of the error. The width of each distribution approximation to bandpower windows, for the VSA, and is also shown to be unchanged by any significant amount. found that for the data obtained so far, this is a valid Furthermore, the correlations of the parameter estimates are approximation. We note that alternative binning schemes also consistent using both methods. may reduce the suitability of this method and plan to use These results are perhaps to be expected given the bandpower window functions for parameter estimation from sparse nature of the covariance matrix. For the VSA ex- super-extended VSA data. tended array data, approximately 5% of the elements of the matrix are non-zero and in the limit of a diagonal co- variance matrix the bandpower and variance windows are equivalent. However, the validity of the approximation also ACKNOWLEDGEMENTS depends on the signal-to-noise level and on the binning used. The level of correlations between the bins and the gradient We thank Sarah Smith for useful discussions. We thank of the power spectrum across the bin also has an impact the staff of the Mullard Radio Astronomy Observatory, the on the bandpower predicted. For example, using a standard Jodrell Bank Observatory and the Teide Observatory for ΛCDM power spectrum and the binning used above, we can invaluable assistance in the commissioning and operation compare the bandpowers predicted from each window func- of the VSA. The VSA is supported by PPARC and the tion. In the majority of bins, the agreement is within 5% IAC. N.Rajguru, A.Scaife, K.Lancaster and R. S. Savage but there is a 13% difference in bin 3 where the gradient acknowledge the support of PPARC studentships. A. Slozar of the power spectrum is steep and there is a strong anti- acknowledges the support of St. Johns College, Cambridge. correlation with bin 4. The effect of doubling the size of the G. Rocha acknowledges a Leverhulme Fellowship at the Uni- bins increases the discrepancy between the bandpower pre- versity of Cambridge.
c 2005 RAS, MNRAS 000, 1–10 CMB observations from the VSA & CBI 11
Figure 6. The one-dimensional marginalised probability distributions for cosmological parameters estimated using variance windows (solid lines) and bandpower windows (dotted lines).
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