doi: /j x Accepted 2005 August 8. Received 2005 July 27; in original form 2005 February 16

Transcription

1 Mon. Not. R. Astron. Soc. 363, (2005) doi: /j x Cosmic microwave background observations from the Cosmic Background Imager and Very Sma Array: a comparison of coincident maps and parameter estimation methods Nutan Rajguru, 1 Steven T. Myers, 2 Richard A. Battye, 3 J. Richard Bond, 4 Kieran Ceary, 3 Caro R. Contadi, 4 Rod D. Davies, 3 Richard J. Davis, 3 Cive Dickinson, 3,5 Ricardo Genova-Santos, 6 Keith Grainge, 1 Yaser A. Hafez, 3 Michae P. Hobson, 1 Michae E. Jones, 1 Rüdiger Kneiss, 1 Katy Lancaster, 1 Anthony Lasenby, 1 Brian S. Mason, 7 Timothy J. Pearson, 5 Guy G. Pooey, 1 Anthony C. S. Readhead, 5 Rafae Reboo, 6 Graca Rocha, 1 José Aberto Rubiño-Martin, 6 Richard D. E. Saunders, 1 Richard S. Savage, 1 Anna Scaife, 1 Pau F. Scott, 1 Jonathan L. Sievers, 4 Anže Sosar, 1 Angea C. Tayor, 1 David Titterington, 1 Eizabeth Wadram, 1 Robert A. Watson 3 and Athea Wikinson 3 1 Astrophysics Group, Cavendish Laboratory, Madingey Road, Cambridge CB3 0HE 2 Nationa Radio Astronomy Observatory (NRAO), Socorro, NM 87801, USA 3 Jodre Bank Observatory, Maccesfied, Cheshire SK11 9DL 4 Canadian Institute for Theoretica Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario, M5S 3H8, Canada 5 Caifornia Institute of Technoogy, Mai Code , Pasadena, CA 91125, USA 6 Instituto de Astrofísica de Canarias (IAC), La Laguna, Tenerife, Spain 7 NRAO, PO Box 2, Green Bank, WV 24944, USA Accepted 2005 August 8. Received 2005 Juy 27; in origina form 2005 February 16 ABSTRACT We present coincident observations of the cosmic microwave background (CMB) from the Very Sma Array (VSA) and Cosmic Background Imager (CBI) teescopes. The consistency of the fu data sets is tested in the map pane and the Fourier pane, prior to the usua compression of CMB data into fat bandpowers. Of the three mosaics observed by each group, two are found to be in exceent agreement. In the third mosaic, there is a 2σ discrepancy between the correation of the data and the eve expected from Monte Caro simuations. This is shown to be consistent with increased phase caibration errors on VSA data during summer observations. We aso consider the parameter estimation method of each group. The key difference is the use of the variance window function in pace of the bandpower window function, an approximation used by the VSA group. A re-evauation of the VSA parameter estimates, using bandpower windows, shows that the two methods yied consistent resuts. Key words: cosmic microwave background cosmoogy: observations. E-mai: nr243@mrao.cam.ac.uk Present address: Jet Propusion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA. Present address: Department of Physics, The Denys Wikinson Buiding, Kebe Road, Oxford OX1 3RH. Present address: Department of Physics, University of Caifornia, Berkeey, CA , USA. Present address: HH Wis Physics Laboratory, Tynda Avenue, Bristo BS8 1TL. 1 I N T RO D U C T I O N Measurements of cosmic microwave background (CMB) anisotropies have proved invauabe in estabishing the current cod dark matter (CDM) mode and are widey used in cosmoogica Present address: Astronomy Centre, University of Sussex, Brighton BN1 9QH. Present address: Facuty of Mathematics and Physics, University of Ljubjana, 1000 Ljubjana, Sovenia. C 2005 RAS

2 1126 N. Rajguru et a. parameter estimation (Godstein et a. 2003; Sperge et a. 2003; Reboo et a. 2004; Readhead et a. 2004). An important check on the accuracy of CMB measurements is to compare the data obtained from instruments which are subject to different systematic effects. The Cosmic Background Imager (CBI) and Very Sma Array (VSA) instruments have significant design differences and the comparison of our data is a check that known systematic effects have been accuratey corrected for, and that neither data set is seriousy contaminated by unrecognized systematic errors. In the CMB community, the focus has been on the comparison of power spectra. However, maps of CMB anisotropies aso contain important information. In particuar, they are used in tests of Gaussianity (see, for exampe, Aiaga et a. 2003; Komatsu et a. 2003; Savage et a. 2004), a key assumption of CMB anaysis. It is therefore an important exercise to check the correation between maps of CMB anisotropies. To this end, the VSA has undertaken a programme of observing fieds previousy imaged by the CBI. In this paper we present the resuts of these observations and assess their consistency with the CBI data. Key scientific resuts from the measurement of CMB anisotropies are the cosmoogica parameters. An essentia ingredient in converting the fat bandpower estimates into cosmoogica parameters is the window functions, which aow a theoretica power spectrum to predict a fat bandpower. Both the CBI and VSA groups use a maximum-ikeihood method of estimating the bandpowers, with some differences in the impementation, but there are important differences in the type of window functions used for parameter estimation. The CBI group computes the bandpower window function, which fuy takes into account the anticorreations between neighbouring bins (Myers et a. 2003). The VSA group use variance windows as an approximation to the bandpower windows when computing parameter estimates (Rubiño-Martin et a. 2003; Sosar et a. 2003; Reboo et a. 2004), athough anticorreations are not accounted for. We assess the bias resuting from this approximation by re-evauating VSA parameter estimates using bandpower window functions. In Section 2 we summarize the key differences between the VSA and CBI instruments and the impications for observing strategies. In Section 3 we describe how the differenced maps are produced and present the resuts of the comparison. Section 4 contains our assessment of the impact of window functions on the VSA parameter estimation. Finay, in Section 5 we present our concusions. 2 T H E T E L E S C O P E S The CBI is an interferometric teescope ocated at an atitude of 5000 m in the Atacama Desert in northern Chie. The instrument operates in 10 1-GHz frequency bands over GHz. The antennas have ow-noise high eectron mobiity transistor (HEMT) ampifiers and typica system temperatures incuding the CMB, ground and atmosphere are 30 K. The 13 Cassegrain antennas, each 0.9 m in diameter are co-mounted on a 6-m tracking patform (Padin et a. 2002). The VSA is sited at the Teide Observatory, in Tenerife, at an atitude of 2400 m. The VSA operates in a singe 1.5-GHz channe at a centra frequency of 33 GHz. The 14 antennas aso have HEMT ampifiers and the typica system temperature is 35 K. In the extended array configuration, the VSA uses mirrors of diameter m. The VSA horn-refector antennas are mounted on a tit tabe hinged aong east west, but each antenna individuay tracks the observed fied by rotating its horn axis perpendicuary to the tabe hinge, and wavefront coherence is maintained with an eectronic path compensator system (Watson et a. 2003). This is a key design difference to the CBI. The individua tracking of the VSA antennas aows for the fitering of contaminating signas. These may be ceestia sources such as the Sun and Moon, or ground-spi and other ground-based spurious signas (Watson et a. 2003). The fringe rates of contaminating signas differ sufficienty from those of the target to aow effective fitering whist retaining most of the data. For exampe, fitering the Sun at a distance of 20 from the target removes approximatey 25 per cent of the data. The VSA aso uses a ground-shied to minimize ground-spi. The consequences of these design differences are twofod. First, the VSA is abe to observe 24 hours a day and its extended array (as used here) can fiter out emission from the Sun and Moon when they are as cose as 9. The CBI is imited to observing at night and to fieds which are more than 60 away from the Moon (Padin et a. 2002). Secondy, the VSA is unaffected by ground-spi contamination for fieds within 35 of the zenith and so is abe to make direct images of the sky. The raw CBI data are contaminated by ground-spi and this is eiminated by means of a differencing scheme. The method of differencing the CBI data used in this anaysis works as foows. A ead fied is observed foowed by a trai fied at the same decination but separated by 8 min in RA. The trai fied visibiities are then subtracted from those of the ead fied. This has the effect of removing the contaminating signa which is constant on an 8-min time-scae, whist preserving the statistica distribution of the sky Fourier modes (Padin et a. 2002). Both sky maps and the power spectrum are estimated from the differenced data. The CBI mounting patform aows the orientation of the baseines to be changed by rotating the patform about the optica axis. The rotation of the tracking tabe together with the broad bandwidth ensures a we-fied aperture pane and a circuary symmetric synthesized beam. Fig. 1 shows the typica UV coverage and synthesized beams of each teescope for the observations presented in this paper, in the range of anguar scaes common to both experiments. The UV coverage of the VSA is ess compete and the beam is ess circuary symmetric than that of the CBI, for these fieds. This is party due to the ow decination of the observations ( 3. 5), which is cose to the owest eevation that the VSA is abe to observe. The teescope is ocated at a atitude of 28 and at the higher decinations of the main primordia fieds, UV coverage is significanty better (Tayor et a. 2003; Dickinson et a. 2004). Unike the CBI, the VSA mounting tabe does not aow for rotation about the optica axis. Together with the ower bandwidth, this imits the coverage of the aperture pane, athough it minimizes the number of redundant baseines at a given frequency. A summary of the specification of the two teescopes is shown in Tabe 1. 3 O B S E RVAT I O N S The CBI data used in this comparison are the mosaics 02H, 14H and 20H and first reported in Pearson et a. (2003). The CBI data aso incude two deep fied observations, which fa within the 14H and 20H mosaics and are described in Mason et a. (2003). Both CBI mosaicked and deep fied data were coected during the period 2000 January December. Each mosaic consists of 42 differenced fieds. The VSA observations were carried out at a ater epoch. The data were made between 2002 May and 2004 May with the teescope in the extended array configuration. The arger primary beam size of the extended array mirrors aows each mosaic to be covered in three pointings. Tabe 2 shows the coordinates and effective

3 CMB observations from the VSA and CBI 1127 Figure 1. Typica UV coverage. Top eft: VSA. Each point represents the fu 1.5-GHz bandwidth. Top right: CBI. Each point represents a singe 1-GHz channe. Bottom: synthesized beams for typica UV coverage. Tabe 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 channes 1 10 Number of antennas Number of baseines Range of baseine engths m m range Primary beam (FWHM) arcmin 31 GHz/ν System temperature 35 K 30 K Mirror diameters 0.32 m 0.90 m Synthesized beam (FWHM) 11 arcmin 5 arcmin Fux sensitivity 50 Jy s 1/2 1.5 Jy s 1/2 integration times for the VSA observations. The VSA data reduction and caibration procedure are described in Dickinson et a. (2004) and references therein. 3.1 Foreground contamination At frequencies of GHz, the dominant cosmoogica contamination to CMB observations comes from gaactic foregrounds and Tabe 2. Ceestia coordinates for the VSA observations of the CBI 02H, 14H and 20H mosaics. The effective integration time is cacuated after fagging and fitering of the data. Fied RA (J2000) Dec. (J2000) t int (h) VSA-02H-a VSA-02H-b VSA-02H-c VSA-14H-a VSA-14H-b VSA-14H-c VSA-20H-a VSA-20H-b VSA-20H-c extragaactic radio sources. The diffuse gaactic foregrounds incude both bremsstrahung and synchrotron emission. However, this emission is concentrated in the gaactic pane and contamination may be avoided by observing at high gaactic atitudes. The observations presented here are at gaactic atitudes >20. In addition, the data are insensitive to the arge anguar scaes where gaactic contamination is significant. Therefore, the main contaminant is ikey to be extragaactic point sources.

4 1128 N. Rajguru et a. The standard VSA source-subtraction strategy invoves an initia survey with the Rye Teescope (RT) at 15 GHz (Wadram et a. 2003). Sources identified by the RT are then monitored with the source subtractor at 33 GHz. The observations are carried out simutaneousy with the CMB fied observations to take account of the variabiity of the sources. A statistica correction is aso appied to the power spectrum to remove the sma effect of the remaining, fainter sources. As the RT is ocated in Cambridge at a atitude of +52, we were unabe to survey fieds at the ow decinations of the CBI observations. For this reason, a more imited eve of source subtraction was impemented. This invoved the subtraction from the data of both groups, the fuxes obtained by the CBI group from observations at 31 GHz with the 40-m teescope at the Owens Vaey Radio Observatory (OVRO). These observations were carried out, simutaneousy, in so far as was possibe, with the CBI observations. As noted above, the VSA observations were carried out a ater epoch and no further source observations were carried out to account for the variabiity of sources. The typica sensitivity achieved in the OVRO data was 2 mjy (rms) which aows for a competeness estimate of 90 per cent at S 31 > 16 mjy (Mason et a. 2003). To achieve high measurements of the power spectrum, a deeper eve of discrete source subtraction is required. To achieve this, the usua approach of the CBI group is to empoy the strategy of constraint matrices to project out sources at known positions but with unknown fuxes (Bond, Jaffe & Knox 1998). A sources >3.4 mjy in the 1.4-GHz NRAO Very Large Array (VLA) Sky Survey (NVSS) cataogue are projected out of the CBI data. A statistica correction based on the CBI source count is then appied for sources <3.4 mjy at 1.4 GHz (Mason et a. 2003). For this investigation, the OVRO fuxes have been subtracted from the CBI data but the constraint matrix strategy has not been empoyed. This is due to the imited resoution of the data incuded in the comparison and to avoid removing a significant fraction of the data, which is unnecessary for this anaysis. The different methods usuay empoyed to remove the effects of contaminating sources are a key distinction in the anaysis of the two groups. Athough we are unabe to make a direct comparison of these strategies, due to the positions of the fieds, the 33-GHz source counts estimated from the main VSA source monitoring programme have been used to evauate the CBI source subtraction strategy. Based on VSA source counts Ceary et a. (2004) find the residua correction due to sources beow the detection threshod of 3.4 mjy at 1.4 GHz to be 0.03 Jy 2 sr 1, which is consistent with the CBI group estimate of 0.08 ± 0.04 Jy 2 sr Maps As a consequence of the need to difference the CBI visibiities, maps from the two teescopes may ony be compared if the differencing scheme is aso impemented on the VSA data. Further to the usua data reduction, the VSA data were processed in severa ways to enabe this. The VSA has a much arger primary beam than the CBI and covers the same area of sky in fewer pointings. To produce data equivaent to each CBI ead and trai fied, the foowing procedure was impemented. The first step was to shift the VSA fied centres to each of the CBI fied centres. The compex visibiity measured by an interferometer is defined as V (u) = d 2 x A(x) I (x) e 2πiu x + N(u) (1) where A(x) is the primary beam, I(x) is sky intensity, u = (u, v) is the baseine ength, measured in units of the waveength and N(u) is the instrumenta noise (Thompson, Moran & Swenson 2001). The fied shift was achieved by rotating the phase of the VSA visibiities so that the direction cosines, x = ( x, y), were defined with respect to a new phase centre x = cos δ sin(α α 0 ) y = sin δ cos δ 0 cos δ sin δ 0 cos(α α 0 ) where α, δ and α 0, δ 0 are the right ascension and decination of the VSA and CBI fied centres, respectivey. At this stage, before differencing can be carried out, the arger primary beam of the VSA must be taken into account. The effect of observing a imited area of sky is to convove the sky Fourier modes with the aperture function (the Fourier transform of the primary beam). To mimic observations with the CBI beam, which has a FWHM of 45.2 arcmin (31 GHz/ν), the VSA visibiities were convoved with the CBI aperture function. This was modeed as a Gaussian with a cut-off at 0.45 m (the outer radius of the antenna) and a centra region with no iumination at r < m (corresponding to bockage by the secondary mirror). The 84 ead and trai fieds for each mosaic were produced by convoving the phase rotated data with the CBI aperture function. Corresponding ead and trai fieds were then differenced. It is important to note that matching the UV coverage of the two interferometers is critica in making maps of the CMB. This is a consequence of samping a random Gaussian fied. Not ony must the same range of anguar scaes be used, but they must sampe the same region of UV space. Fig. 1 iustrates the mismatch in UV coverage between the VSA and CBI in the range of common UV scaes. To overcome this, the data were binned and reweighted. As the data were obtained from a convoution of the sky Fourier modes with the CBI aperture function, which has a FWHM of 67λ, a ce size of 17λ was chosen to ensure that the aperture function was more than adequatey samped according to the Nyquist samping theorem. Data ces with a match in both sets were assigned a weight of the geometric mean of their individua weights. Data ces adjacent to, but without a direct match, were downweighted by a factor of 2. Ces without a direct or adjacent match were assigned a weight of zero. Maps were then produced from the reweighted visibiities using the AIPS package and are shown in Fig. 2. The CBI data have been standardized to 33 GHz with the assumption of a backbody spectrum. Both data sets 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 effective beam size, σ eff, is given by 1 σ 2 eff = σc 2 σv 2 where σ C and σ V are the CBI and VSA primary beam sizes, respectivey. The effective beam size is smaer than the CBI beam size by 5 per cent. This is because the sky signa measured by the VSA data has been mutipied by the CBI and VSA primary beams. Fig. 3 shows the sensitivity maps for each mosaic. The minimum noise in the VSA 02H, 14H and 20H mosaics is 7.29, 7.44 and 6.03 mjy beam 1, respectivey. The corresponding noise eves for the CBI mosaics are 3.88, 1.82 and 1.08 mjy beam 1. The noise eves in the CBI maps are approximatey a factor of 2 ower than those of the VSA maps, with a wider gap in the region of the CBI deep fieds, which are ceary visibe in the sensitivity maps. The banking threshod of 15 mjy beam 1 is for iustrative purposes

5 CMB observations from the VSA and CBI 1129 Figure 2. The centra region of the 42-fied mosaics of differenced maps. Each map covers an area of The RA scae refers to the position of the ead fied. Left: VSA data. Right: CBI data. Top: 02H mosaic. Centre: 14H mosaic. Bottom: 20H mosaic.

6 1130 N. Rajguru et a. Figure 3. Sensitivity maps for each mosaic shown in Fig. 2. Each map covers an area of and is banked at a threshod of 15 mjy beam 1. The RA scae refers to the position of the ead fied. Left: VSA data. Right: CBI data. Top: 02H mosaic. Centre: 14H mosaic. Bottom: 20H mosaic. ony. In cacuating the map correations, a cut was appied when the power in the primary beam reached 1/e of the maximum eve. There are a number of statistics which may be used to quantify the consistency between data sets. The measure used here for both the map pane and UV pane tests is the product-moment correation coefficient (e.g. Barow 1989): r = xy x y σ x σ y.

7 CMB observations from the VSA and CBI 1131 In both panes a weighting scheme was used. In correating the maps, each pixe was weighted by the power of the primary beam at the centre of the pixe. In the UV pane, each ce was weighted by the inverse of the noise squared. As the sky is rea, the underying Fourier modes a(u) = a ( u). Consequenty, correations exist between visibiities that ie on opposite sides of the UV pane. To account for this, the conjugate symmetry of the visibiities was used to refect the data into one-haf of the pane and the correation was carried out ony in this region. The rea and imaginary parts of the visibiities were treated independenty. One of the disadvantages usuay cited with this measure of correation is the difficuty of interpretation. This has been overcome by using Monte Caro simuations of the data to predict the expected correation. The input mode for these simuations was a standard CDM mode with noise eves and UV coverage appropriate to the actua observations. The method of producing the VSA differenced data from the simuations was carried out in the same manner as for the actua observations. The correation of the 02H and 14H mosaics is shown in Tabe 3. There is exceent agreement between the observed and expected correations in the 02H and 14H mosaics in both the map pane and the UV pane. The expected correation of the 20H mosaics is higher than for the 02H and 14H mosaics, which refects the greater signa-to-noise in these observations. The actua correation, however, ies 2σ away from the expected eve (see Tabe 4). In contrast to the first two mosaics, the VSA data in the 20H mosaic were argey coected during the summer period and between the hours of 8 am and 6 pm. In this period, the accuracy of the phase caibration is known to deteriorate and typica phase errors are some 20. The source of these errors is thought to be due to a sight warping of the tit tabe in the heat. A of the VSA data in the 02H and 14H mosaics were made outside the summer daytime period, when typica phase errors are around 3 4. The VSA phase errors are estimated from the change in phase caibration factors over the course of a day. Severa caibration observations are carried out each day but there may be significant change in the phase caibration between observations. The caibra- Tabe 3. Correation of the 02H and 14H mosaics. The expected correation is in exceent agreement with the actua correation of the 02H and 14H mosaics. Expected 02H Map pane 0.23 ± ± 0.03 UV pane 0.12 ± ± 0.02 Expected 14H Map pane 0.23 ± ± 0.03 UV pane 0.12 ± ± 0.02 tors are either point sources or resoved sources for which a mode is known. The phase errors have a weak dependence on baseine ength but show a high degree of repeatabiity. The expected correation is, of course, revised downwards by modeing the summer phase errors in the simuations. The phase errors fuy account for the reduced correation seen in the 20H mosaics. In principe, the power spectrum is insensitive to the phases of the visibiities, athough this depends upon the eve of correation of the phase errors. The phase errors on the VSA data are found to be highy correated. Athough typica summer phase errors are some 20, the rms about the mean phase error for each baseine is ony 2 3. A further consideration is the number of baseines which contribute to each UV ce. If a arge number of ces have contributions from severa baseines, then the phase errors wi be ess correated and it is possibe that this coud affect the power spectrum estimate. However, simuations of VSA shaow fied observations, which incude the effect of the crossing UV tracks and summer phase errors, show that this has a negigibe effect on the power spectrum. However, as a further safeguard the VSA group discard the worst affected portion of summer data. Fig. 4 shows the joint power spectrum for the 02H, 14H and 20H mosaics in the range of scaes common to both instruments. The VSA power spectrum was estimated using the direct VSA observations, prior to the impementation of the differencing and reweighting scheme. There is exceent agreement in the first six bins. As the contribution of point sources increases as 2, the discrepancy in the fina bin may be due to the variabiity of sources, as the data were coected at different epochs. Tabe 5 shows the binning and bandpowers for the power spectra. (+1)C /2π (µk 2 ) Figure 4. The power spectrum of the 02H, 14H and 20H mosaics from the undifferenced VSA data (open circes) and the differenced CBI data (soid circes). The concordance mode which best fits the Wikinson Microwave Anisotropy Probe data is shown for comparison. Tabe 4. Correation of the 20H mosaic. The expected correation in the 20H mosaic given the greater weight of data in these fieds is shown in the first coumn, and in the second coumn this is modified by the incusion of the VSA summer phase errors. The actua correation, in coumn 3, is consistent with the eve expected given the phase caibration errors. Expected Expected 20H (inc. phase errors) Map pane 0.27 ± ± ± 0.03 UV pane 0.14 ± ± ± 0.02 Tabe 5. The CMB bandpowers [T 2 0 ( + 1)C /2π] µk 2 for the joint mosaic power spectra shown in Fig. 4. eff is the centroid of the bandpower window function. Bin -range VSA VSA eff CBI CBI eff ± ± ± ± ± ± ± ± ± ± ± ± ± ±

8 1132 N. Rajguru et a. 4 W I N D OW F U N C T I O N S It is standard practice for experiments observing a imited sky area to assume a fat bandpower when estimating the CMB power spectrum (Hobson & Maisinger 2002; Kuo et a. 2002; Myers et a. 2003). However, the theoretica power spectra are not fat and in estimating the cosmoogica parameters it is necessary to define a window function that aows a theoretica prediction of the fat bandpower q B, which may be compared to the experimenta vaues. The theoretica prediction is the expectation vaue of q B for a given input mode: ( ) W B q B = C (a p ). (2) To estimate the bandpowers, the CBI group use the secondorder Tayor expansion of the og-ikeihood function around the maximum-ikeihood bandpowers. A quadratic approximation for the change in the bandpower, δq B, is used iterativey, to move towards the maximum, with the second derivative of the og-ikeihood function repaced by its expectation vaue, the Fisher matrix. The expectation vaue of the bandpowers is then given by q B = 1 2 B [F 1 ] B B Tr [( C 1 C S B C 1 ) C S], (3) where C is the covariance matrix with independent contributions from a signa sources: the CMB signa C S, the instrumenta noise C N and the foreground signas C src and C res. C S B is the CMB signa from each band (Myers et a. 2003). Because C S C S B = C S C, (4) C B the bandpower window functions can be cacuated, once the maximum-ikeihood bandpowers have been obtained, from W B / = 1 [ ] (C [F 1 ] B B Tr 1 C S B 2 C 1) C S, (5) C B where F is the Fisher matrix of the fine-binned C estimates (Knox 1999). The VSA group aso use a maximum-ikeihood method to estimate the bandpowers but do not use the approximations described above. Instead, the exact ikeihood function is cacuated as a function of each bandpower through the maximum-ikeihood point, with some speed-up achieved by the use of the signa-to-noise eigenbasis (Hobson & Maisinger 2002). For parameter estimates, the VSA group do not compute the bandpower window functions but use as an approximation the variance window function, which is rapidy constructed from the overap integras of the aperture functions for pairs of visibiities and is defined as W V = 2π 0 S(, φ) 2 dφ. (6) Here, = 2π u and S(u) = w k à eff (u v k ), (7) k where w k is the noise weighting of the visibiities and à eff is the effective aperture function (Scott et a. 2003) Knox (1999) has noted that the bandpower window function is distinct from the variance window function where the signa or noise is correated. Here we investigate the effect of the window function by estimating the cosmoogica parameters from the VSA data set with each set of window functions. We have used the fu compact Tabe 6. The CMB bandpowers (in µk 2 ) for the compete VSA data set combining both compact and extended array data. Bin -range eff T 2 0 ( + 1)C /2π(µK 2 ) and extended array data sets. The binning scheme and maximumikeihood bandpowers are shown in Tabe 6. The bandpower window functions have been computed using the CBI software (Myers et a. 2003), which has been adapted for use with the VSA specifications, incuding the observing frequency, bandwidth and primary beam size. Fig. 5 shows the variance and window functions for the VSA. Each variance window function is normaized to unit area. The bandpower window functions are, by definition, normaized to unit area within the band imits and to zero outside the band. The bandpower windows show negative sideobe features, which indicate the anticorreations of C vaues in adjacent bins. 4.1 Parameter estimation We consider the set of cosmoogica modes described by the foowing six free parameters: the physica baryon density ω b = b h 2 ; the physica dark matter density ω dm = dm h 2 ; the ratio of the sound horizon to the anguar diameter distance θ; the optica depth to the surface of ast scattering τ; the ampitude of scaar modes A s ; and the spectra index of scaar modes n s. We assume spatia fatness, setting the curvature density k = 1 tot = 0. We aso set w = 1 describing the equation of state of dark energy (p = wρ) and do not consider tensor modes or massive neutrinos in this anaysis. The parameter estimation was performed using the 2004 October version of the COSMOMC software package (Lewis & Bride 2002). This uses a new parametrization with θ instead of H 0 as a basic parameter. As θ is ess correated with other parameters than H 0, this has the advantage of aowing the Markov chains to converge more quicky.

9 CMB observations from the VSA and CBI Tabe 8. Parameter estimates and 68 per cent confidence intervas from the marginaized distributions. When the marginaized posterior for a parameter does not contain a peak, 95 per cent confidence imits are given. Variance windows Bandpower windows ω b 0.029± ± ω dm 0.14± ± m 0.42± ± ± ± θ 1.04± ± h 0.69± ± n s 0.90± ± n10 10 A s 3.4± ± τ (0.04, 0.47) (0.04, 0.48) z re 21± ± Age 13.1± ± Figure 5. Top: VSA variance window functions. Bottom: VSA bandpower window functions. Aternate curves are soid and dotted ines for carity. Tabe 7. The priors assumed for the basic parameters. The notation (a, b) for parameter x denotes a top-hat prior in the range a < x < b. Basic parameter Prior ω b (0.005, 0.100) ω dm (0.01, 0.99) θ (0.5, 10.0) τ (0.01, 0.50) n s (0.5, 1.5) n(10 10 A s ) (2.7, 4.6) In addition to the priors on the basic parameters, isted in Tabe 7, we aso impose priors on some of the derived parameters. Specificay, we use top-hat priors on the age of the Universe ying between 10 and 20 Gyr, and of H 0 ying between 40 and 100 km s 1 Mpc 1. The COSMOMC software was run on a 24-node Linux custer. The chains were run unti the argest eigenvaue returned by the Geman Rubin convergence test reached After burn-in, a tota of more than sampes were coected. As successive sampes in a Markov chain are, by nature, correated, the sampes were thinned by a factor of 25 resuting in approximatey 4000 independent sampes. These sampes were then used to cacuate the marginaized distributions and parameter estimates (see Tabe 8). Fig. 6 shows the marginaized distributions for each parameter. It is cear that no significant bias has been introduced as a resut of using variance windows to approximate bandpower windows. The argest discrepancy is seen in the parameter estimate for ω b where the bandpower windows reduce the estimate by one-third of the 1σ error. For a other parameters, the estimates are consistent to a much smaer fraction of the error. The width of each distribution is aso shown to be unchanged by any significant amount. Furthermore, the correations of the parameter estimates are aso consistent using both methods. These resuts are perhaps to be expected given the sparse nature of the covariance matrix. For the VSA extended array data, approximatey 5 per cent of the eements of the matrix are non-zero and in the imit of a diagona covariance matrix the bandpower and variance windows are equivaent. However, the vaidity of the approximation aso depends on the signa-to-noise eve and on the binning used. The eve of correations between the bins and the gradient of the power spectrum across the bin aso has an impact on the bandpower predicted. For exampe, using a standard CDM power spectrum and the binning used above, we can compare the bandpowers predicted from each window function. In the majority of bins, the agreement is within 5 per cent but there is a 13 per cent difference in bin 3 where the gradient of the power spectrum is steep and there is a strong anticorreation with bin 4. The effect of doubing the size of the bins increases the discrepancy between the bandpower predictions to a maximum of 28 per cent. Differences of this size woud be ikey to bias the parameter estimates. However, the eve of agreement between the bandpower predictions is not a simpe function of bin size and wi fuctuate depending on the exact binning used. The VSA is currenty undergoing an upgrade to increase the sensitivity of the instrument and to enabe a measurement of the power spectrum up to = The upgrade wi invove the fitting of arger mirrors with a diameter of 0.55 m. In this case, the fraction of non-zero eements in the covariance matrix wi increase sighty to

10 1134 N. Rajguru et a. Figure 6. The one-dimensiona marginaized probabiity distributions for cosmoogica parameters estimated using variance windows (soid ines) and bandpower windows (dotted ines). 7.5 per cent. The suitabiity of the approximation aso depends upon the signa-to-noise achieved. The upgraded VSA wi have increased sensitivity, but taken in conjunction with the reduced signa, then the overa signa-to-noise eve wi remain about the same. Given the significant difference between the expected bandpower vaues which may arise and to ensure that this does not bias future VSA parameter estimates, anaysis of the upgraded VSA data wi be carried out using bandpower window functions. 5 C O N C L U S I O N S In this paper we have presented three sets of coincident CMB observations observed at different epochs by the VSA and CBI teescopes. We have chosen to anayse the fu data sets, rather than focusing on the power spectra, in order to investigate any possibe systematic effects which may not otherwise be reveaed. The correation of the data sets from each group was found to be as expected for the 02H and 14H mosaics. In the third mosaic, the data disagreed with the Monte Caro simuations at a eve of 2σ. However, this is consistent with the phase caibration errors expected from VSA data during the summer months. It has been estabished that this does not affect the power spectrum estimation due to the correated nature of the phase errors. The resuts of this anaysis reaffirm that both groups have correcty characterized the noise properties and systematics of the teescopes as we as other, potentia data contaminants. We have investigated the use of variance windows as an approximation to bandpower windows, for the VSA, and found that, for the data obtained so far, this is a vaid approximation. We note that aternative binning schemes may reduce the suitabiity of this method and pan to use bandpower window functions for parameter estimation from superextended VSA data. AC K N OW L E D G M E N T S We thank Sarah Smith for usefu discussions. We thank the staff of the Muard Radio Astronomy Observatory, the Jodre Bank Observatory and the Teide Observatory for invauabe assistance in the commissioning and operation of the VSA. The VSA is supported by the UK Partice Physics and Astronomy Research Counci (PPARC) and the IAC. NR, AS, KL and RSS acknowedge the support of PPARC studentships. AS acknowedges the support of St Johns Coege, Cambridge. GR acknowedges a Leverhume Feowship at the University of Cambridge. R E F E R E N C E S Aiaga A. M., Martínez-Gonzáez E., Cayón L., Argüeso F., Sanz J. L., Barreiro R. B., 2003, NewAR, 47, 821 Barow R. J., 1989, Statistics. A Guide to the Use of Statistica Methods in the Physica Sciences. Wiey, New York Bond J. R., Jaffe A. H., Knox L., 1998, Phys. Rev. D, 57, 2117 Ceary K. et a., 2004, MNRAS, 360, 340 Dickinson C. et a., 2004, MNRAS, 353, 732 Godstein J. H. et a., 2003, ApJ, 599, 773 Hobson M. P., Maisinger K., 2002, MNRAS, 334, 569 Knox L., 1999, Phys. Rev. D, 60, Komatsu E. et a., 2003, ApJS, 148, 119

11 CMB observations from the VSA and CBI 1135 Kuo C.-L. et a., 2002, Bu. Am. Astron. Soc., 34, 1324 Lewis A., Bride S., 2002, Phys. Rev. D, 66, Mason B. S. et a., 2003, ApJ, 591, 540 Myers S. T. et a., 2003, ApJ, 591, 575 Padin S. et a., 2002, PASP, 114, 83 Pearson T. J. et a., 2003, ApJ, 591, 556 Readhead A. C. S. et a., 2004, ApJ, 609, 498 Reboo R. et a., 2004, MNRAS, 353, 747 Rubiño-Martin J. A. et a., 2003, MNRAS, 341, 1084 Savage R. et a., 2004, MNRAS, 349, 973 Scott P. F. et a., 2003, MNRAS, 341, 1076 Sosar A. et a., 2003, MNRAS, 341, L29 Sperge D. N. et a., 2003, ApJS, 148, 175 Tayor A. C. et a., 2003, MNRAS, 341, 1066 Thompson A. R., Moran J. M., Swenson G. W., Jr, 2001, Interferometry and Synthesis in Radio Astronomy. Wiey, New York Wadram E. M., Pooey G. G., Grainge K. J. B., Jones M. E., Saunders R. D. E., Scott P. F., Tayor A. C., 2003, MNRAS, 342, 915 Watson R. A. et a., 2003, MNRAS, 341, 1057 This paper has been typeset from a TEX/LATEX fie prepared by the author.