First results from the Very Small Array – IV. Cosmological parameter estimation
Abstract
We investigate the constraints on basic cosmological parameters set by the first compactconfiguration observations of the Very Small Array (VSA), and other cosmological data sets, in the standard inflationary CDM model. Using the weak priors km s Mpc and , we find that the VSA and COBEDMR data alone produce the constraints , , and at the 68 per cent confidence level. Adding in the type Ia supernovae constraints, we additionally find and . These constraints are consistent with those found by the BOOMERanG, DASI and MAXIMA experiments. We also find that, by combining all these CMB experiments and assuming the HST key project limits for (for which the Xray plus Sunyaev–Zel’dovich route gives a similar result), we obtain the tight constraints and , which are consistent with, but independent of, those obtained using the supernovae data.
keywords:
cosmology: observations – cosmic microwave background1 Introduction
One of the central aims of cosmology is to determine the values of the fundamental cosmological parameters that describe our Universe. A unique opportunity to achieve this goal is provided by the observation of anisotropies in the cosmic microwave background (CMB) radiation. By comparing such observations with the predictions of our current theories of structure formation and the evolution of the Universe, we may place constraints on the cosmological parameters that appear in these models.
The currently most favoured theoretical model for describing our Universe is based on the idea of inflation (guth82), which provides a natural mechanism for producing initial density fluctuations described by a powerlaw spectrum with a slope close to unity. The simplest versions of inflation also predict the Universe to be spatially flat. The initial spectrum of adiabatic density fluctuations is modulated through acoustic oscillations in the plasma phase prior to recombination and the resulting inhomogeneities are then imprinted as anisotropies in the CMB. In the basic inflationary scenario, the CMB temperature anisotropies are predicted to follow a multivariate Gaussian distribution, and so may be completely described in terms of their angular power spectrum. Moreover, the acoustic oscillations in the plasma phase lead to a characteristic series of harmonic peaks in the power spectrum, which are a robust indicator of the existence of fluctuations on superhorizon scales.
Although the presence of acoustic peaks in the CMB power spectrum is a generic prediction of inflationary models, detailed features of the power spectrum, such as the relative positions and heights of the peaks, depend strongly on a wide range of cosmological parameters, see e.g. hu97. Indeed, this sensitivity to the parameters is the reason why observations of the CMB provide such a powerful means of constraining theoretical models.
Thus measurement of the CMB power spectrum is a major goal of observational cosmology and numerous experiments have provided estimates of the spectrum on a range of angular scales. It is only recently, however, that observations by the BOOMERanG (netterfield01), DASI (halverson02) and MAXIMA (lee01) experiments have provided measurements of the CMB power spectrum with sufficient accuracy over a wide range of scales to allow tight constraints to be placed on a wide range of cosmological parameters (see, for example, dupis). This parameter estimation process is performed by comparing the observed bandpowers with a wide range of theoretical power spectra corresponding to different sets of values of the cosmological parameters, which can be accurately calculated using the Cmbfast (zaldarriaga00) or Camb (alcamb) software packages. The comparison of the observed and predicted power spectrum is usually carried out in a Bayesian context by evaluating the likelihood function of the data as a function of the cosmological parameters.
In this paper we perform this parameter estimation process using, as the main CMB datasets, the flat bandpower estimates of the CMB power spectrum measured by the Very Small Array (VSA) in its compact configuration, which has been described in the sequence of earlier papers VSApaperI, VSApaperII and VSApaperIII (Papers I, II and III), and the COBEDMR bandpowers for low normalisation. We also combine the VSA data with other recent CMB experiments, and constraints from the HST Key Project on and observations of type Ia supernovae, to tighten further the constraints on the cosmological parameters. Two different methods are used to perform the parameter estimation procedure. First, we employ the traditional technique of evaluating the likelihood function on a large grid in parameter space. Second, we consider a more flexible approach in which the likelihood function is explored by MarkovChain Monte Carlo (MCMC) sampling. The latter method has a great potential in terms of expanding the dimension of the parameter set which can be investigated. Here we use it to demonstrate the robustness of the results from the standard grid approach and also to provide a novel visualisation of the range of uncertainty in our parameter estimates.
2 Models, parameter space and methods
In the analyses presented in this paper we restrict our attention to the case in which the initial fluctuations are adiabatic with a simple powerlaw spectrum; such perturbations are naturally produced in the standard singlefield inflationary model. Moreover, as is now common practice, we consider models in which the contents of the Universe are divided into three components: ordinary baryonic matter; cold dark matter (CDM), which interacts with baryonic matter solely through its gravitational effect; and an intrinsic vacuum energy. The presentday contributions of these components, measured as a fraction of the critical density required to make the Universe spatially flat, are denoted by , and respectively. It is possible that some of the dark matter may, in fact, be in the form of relativistic neutrinos (hot dark matter), but the presence of a hot component has a negligible effect on the power spectrum, given the sensitivity and angular resolution of current CMB experiments (dodelson96). We therefore assume that all dark matter is cold and set .
Following the current theoretical expectation (lyth97), we also assume that the contribution of tensor mode perturbations is very small compared with the scalar fluctuations, and so we ignore their effects. This assumption is consistent with current observations. Since tensor modes contribute primarily to low multipoles, , the only existing measurement that would be particularly sensitive to their presence is the level of the CMB power spectrum at low observed by the COBEDMR experiment (Smoot92). If the tensor component made up a large fraction of this observed power, the value of the spectral index for scalar perturbations would need to exceed unity by a considerable margin in order to provide the level of power at higher measured by numerous other CMB experiments. Such a large value of is, however, ruled out by largescale structure studies (lssns). Nevertheless, it must be remembered that this argument only holds if the initial perturbation spectrum is indeed a simple powerlaw.
Given the assumptions outlined above, there remain seven degrees of freedom in the description of the standard inflationary CDM model. The parameterisation of this sevendimensional model space can be performed in numerous ways, but we shall adopt the most common choice, which is defined by the following parameters: the physical density of baryonic matter (); the physical density of CDM (); the vacuum energy density due to a cosmological constant (); the total density (); the spectral index of the initial powerlaw spectrum of scalar perturbations (); the optical depth to the lastscattering surface due to reionisation (); and the overall normalisation of the power spectrum as measured by and is quoted relative to , as determined from the 4year COBEDMR data by bunnwhite. This choice of parameters is similar to that made in the analysis of the CMB bandpower measurements obtained by the BOOMERanG, MAXIMA and DASI experiments. We note that, in this parameterisation, the reduced Hubble parameter ( km s Mpc/100) is auxiliary and is given by .
In comparing the observed CMB flat bandpowers measured by the VSA with the above multidimensional model, we adopt a Bayesian approach based on the evaluation of the likelihood function for the data as a function of the cosmological parameters, which for brevity we denote by the vector . To set proper constraints on the values of these parameters, it is necessary to explore the sevendimensional model space over a region large enough to encompass those models with significant likelihood. To that end, we employ two different techniques to explore the likelihood function: a traditional approach in which the likelihood function is evaluated on a grid of points in parameter space, and a MarkovChain Monte Carlo (MCMC) technique in which a set of samples are drawn from the likelihood function. These techniques are described below. A common requirement of both methods, however, is the accurate evaluation of the likelihood function at any given point in the parameter space, and we begin by describing how this computation is performed.
2.1 Evaluation of the likelihood function
As described in Paper III, the constraints imposed by the VSA compact array observations on the CMB power spectrum have been obtained using the Madcow analysis package (klaus). This provides a onedimensional likelihood distribution for each flat bandpower , conditioned on the values of the other bandpowers at the joint maximum of the likelihood function. In addition the Hessian matrix at the joint maximum is also calculated, which may be inverted to obtain the elements of the covariance matrix of the flat bandpower estimates under the assumption that the likelihood function is well approximated by a multivariate Gaussian near its peak.
Since the VSA measures power only on small angular scales (), it does not constrain the amplitude and tilt of the power spectrum at low, which leads to pronounced degeneracies in the parameter space. Therefore, in Section 3, we also include in our analysis the 28 COBEDMR bandpowers provided in the Radpack distribution (RADPACK). Moreover, in Section 4, we further include CMB bandpower measurements obtained by the BOOMERanG, MAXIMA and DASI experiments.
In order to compare a set of observed flat bandpowers with our theoretical model, it is necessary to compute the corresponding predicted values of these bandpowers, given a particular set of parameter values . If is the corresponding theoretical power spectrum for this set of parameter values, the predicted value of the th flat bandpower is given by
where and are the window functions for the experiment under consideration. The VSA window functions are presented in Paper III.
In the comparison of the observed and predicted bandpowers it is necessary to take proper account of the uncertainties in our estimates of the . In fact, the uncertainties in bandpower estimates are, in general, nonGaussian and this precludes us from calculating a simple statistic. It is possible, however, to make a transformation from the flat bandpowers to a related set of ‘offset lognormal’ variables for which the uncertainties are Gaussian to a very good approximation (bond00). This requires the calculation of an additional set of quantities from the data, which represent the uncertainty due to instrumental noise. These are straightforwardly calculated for the VSA bandpowers, and will be available on our website^{1}^{1}1http://www.mrao.cam.ac.uk/telescopes/vsa. The corresponding quantities for the DMR and DASI bandpowers are included in the Radpack package, and those for MAXIMA are available on their website. For BOOMERanG, however, the values are not yet publicly available, and so we assume simply that the likelihood is a multivariate Gaussian.
The observed bandpowers are then transformed as
and similarly for the predicted bandpowers. It is straightforward to show that the elements of the covariance matrix for the new variables are related to the covariance matrix of the original variables by
The likelihood function is then taken to be a multivariate Gaussian in the transformed variables, so that
where the misfit statistic is given by
We find that this assumed form for the likelihood function provides an excellent fit to the true likelihood distribution for the VSA bandpowers. This is illustrated in Fig. 1, in which we plot the true likelihood and the corresponding offset lognormal approximation for the flat bandpower in the first VSA spectral bin for the combined data from the VSA1, VSA2 and VSA3 fields.
We note that the use of this transformation not only allows us to find the bestfit point in the parameter space , but also allows us to use the value of at this point as a measure of goodness of fit.
2.2 Exploration of the parameter space: gridbased approach
Once one has the facility to calculate accurately the likelihood function for the cosmological parameters at any given point in the parameter space, one must devise a strategy to explore the likelihood distribution throughout this space.
From the above discussion, it is clear that the evaluation of the likelihood function at each point in parameter space requires one to calculate the theoretical power spectrum corresponding to that point. The calculation of this spectrum is performed using version 4.0 of the Cmbfast code (zaldarriaga00), which uses a space splitting technique that accelerates the calculation of the spectrum by using a flat model to compute the high multipoles for models with . Nevertheless, the calculation of a typical spectrum for a model still requires around 30 sec of CPU time on one of the processors of the Cosmos SGI Origin 2000 computer. Even with the 16 processors available to us, this computational cost severely limits the total number of points in parameter space at which the likelihood function can reasonably be evaluated. We note that, in fact, two complete grids of models (and two independent pipelines for the cosmological parameters estimation) were set up in Tenerife and Cambridge.
The traditional approach is thus to calculate the spectra for a grid of models that is as fine as CPUtime limitations allows, while being sufficiently large to encompass the entire probability distribution. For our model space, we calculated spectra on a sixdimensional grid corresponding to the values for each parameter given in Table 1.
:  0.010  0.015  0.020  0.025  0.030  0.035  0.040  0.045  0.050  

:  0.02  0.06  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  1.1 
:  0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  
:  0.7  0.75  0.80  0.85  0.90  1.00  1.05  1.10  1.15  1.20  1.30  
:  0.7  0.8  0.9  1.0  1.1  1.2  1.3  1.4  
:  0.0  0.025  0.05  0.075  0.1  0.2  0.3  0.5 
The sixdimensional grid contains spectra. Since the different values of the scalar spectral index can all be obtained on a single call to Cmbfast, this required runs in total and took days of CPU time on Cosmos. The overall normalisation parameter need not be precomputed on a grid since it produces a simple linear scaling of the spectra. In this parameter, the likelihood function was calculated at 10 points in the range 0.45 to 1.44, with a step size of 0.11.
The likelihood function for a given dataset can be evaluated at each of these grid points as described in section 2.1, and the location of the maximum determined. For each cosmological parameter, the onedimensional marginalised distribution is then obtained by numerically integrating over the other parameters. These marginal distributions are then interpolated with a cubic spline, and determine the constraints placed on the cosmological parameters.
Wherever possible, we also include the calibration uncertainties of the CMB experiments under consideration as extra parameters in our analysis. The prior on such parameters is taken as a Gaussian centred on unity, with a standard deviation of the appropriate width. An analytic marginalisation is performed over this parameter, using the method proposed by bridlemarg. This analytical procedure assumes, in addition to a Gaussian prior on the calibration parameter, that the likelihood function is Gaussian. Unfortunately, in neither the original bandpowers, nor the offset lognormal variables, are both these functions precisely Gaussian, and so some (small) approximation is introduced. In this paper, the analytical marginalisation is performed before transforming to offset lognormal variables.
2.3 Exploration of the parameter space: MCMC approach
In a parameter space of large dimensionality, a natural alternative to the gridbased approach is instead to sample from the likelihood distribution. An efficient procedure for obtaining such samples is to construct a Markov chain whose equilibrium distribution corresponds to the likelihood function in parameter space (see e.g. knoxage). Thus after propagating the Markov chain for a given ‘burnin’ period, one obtains samples from the likelihood distribution, provided the chain has converged.
The MCMC sampling procedure may be implemented most straightforwardly by using a simplified version of the Metropolis–Hastings algorithm. At each step in the chain, the next state is chosen by first sampling a candidate point from some proposal distribution . The candidate point is then accepted with probability
where is the posterior distribution for the parameters and is simply the product of the likelihood and and some prior ; we take the latter to be uniform unless otherwise stated. If the candidate point is accepted, the next state in the chain becomes , but if the candidate point is rejected, the chain does not move, and . In theory, the convergence of the chain to the limiting distribution is independent of the choice of the proposal distribution but this choice is crucial in determining both the rate of convergence to the limiting distribution and the efficiency of the subsequent sampling. An effective approach is to set the proposal distribution to a multivariate Gaussian, centred on the current point in parameter space.
As mentioned above, the states of the chain can be regarded as samples from the limiting (i.e. likelihood) distribution only after some initial burnin period for the chain to reach equilibrium. The topic of convergence is still a matter of statistical research and no definitive formula exists for calculating the required length of the burnin period. Nevertheless, several convergence diagnostics have been proposed (burnin), which may be used as a guide.
After burnin, the sample density is directly proportional to the likelihood distribution, and so the samples may be used straightforwardly to obtain estimates of the parameter values and confidence limits. In particular, one may easily obtain onedimensional marginalised distributions for each parameter separately, obviating the need for computerintensive numerical integrations. Moreover, the computational requirements of MCMC procedures are almost independent of dimensionality of parameter space and thus allow a large number of parameters to be constrained simultaneously by the data.
The strength of MCMC methods lies in the fact that useful parameter estimations can be achieved with considerably fewer likelihood evaluations as compared to the traditional gridbased methods. Nevertheless, the density of samples must be high enough so that the estimation of the underlying posterior probability distribution is not plagued by Poisson noise and is independent of the kernel density estimation method. Usually one requires on the order of several tens of thousands of (accepted) samples. The efficiency with which these samples may be obtained depends strongly on the shape of the posterior likelihood function and in practice the basic Metropolis–Hastings algorithm can be augmented by the use of various speedups. Most notably, the sampling efficiency is improved by considering several simultaneous correlated Markov chains and by specific random point generators that attempt to follow the shape of the likelihood distribution posterior. The main MCMC implementation used here is that provided by the Bayesys software (Skilling, private communication), which employs several enhancements of the basic Metropolis–Hastings algorithm and has the ability to sample using multiple chains.
Ideally, one would like to calculate a theoretical power spectrum using one of the popular packages, such as Camb or Cmbfast for each sample. However, this is computationally extremely expensive. As a practical alternative, one can use a precomputed grid of theoretical spectra, such as that discussed in section 2.2, and calculate the required spectrum by suitable interpolation between neighbouring grid points. The density of grid points in the parameter space must be small enough that the dominant error in the estimated cosmological parameters comes from the errors in the measured CMB power spectrum and not from the interpolation between the grid points. We tested the accuracy of the grid discussed in section 2.2 by calculating the exact CMB power spectrum using Cmbfast for 1000 randomly chosen sets of parameters and comparing them with grid interpolation. We find that the rms error on the predicted bandpowers resulting from the interpolation is around 4 per cent, which is very small as compared with the uncertainties in the bandpowers from the current CMB experiments.
The Bayesys sampler provides a powerful general purpose MCMC engine, which allows one to explore complicated likelihood functions of large dimensionality. We find, however, that the present accuracy of the CMB experiments results in likelihood surfaces that are relatively smooth and highly convex, which can be adequately explored using a simple MCMC sampler based solely on the straightforward Metropolis–Hastings algorithm. This allows the possibility of tailoring an MCMC algorithm to the specific problem of cosmological parameter estimation from CMB bandpower measurements, and taking advantage of our prior knowledge concerning which parameter combinations can be calculated quickly using Cmbfast or Camb and which directions in parameter space to avoid. Such an approach has been implemented in the software package Cosmomc (lewisbrid), and allows one to perform an MCMC exploration of the parameter space in which the use of a grid is bypassed altogether, and the exact theoretical spectrum is calculated at each sample point. We have made use of this approach as an additional crosscheck of our results.
In addition to using the MCMC approach to provide useful checks of the parameter constraints obtained from the standard gridbased method, we have exploited the insensitivity of the MCMC method to the dimensionality of the problem by including calibration uncertainty in our numerical analysis, rather than performing an approximate analytical marginalisation over it, as performed in the gridbased approach. A new parameter is introduced which is the ratio of the real telescope calibration to the experimental best estimate. A Gaussian prior is assumed for , with its centre at unity and with a standard deviation corresponding to the calibration uncertainty of the experiment under consideration. Whenever the MCMC algorithm requires a sample for a given value of , the data are rescaled accordingly.
3 Cosmological parameter constraints from the VSA
We first consider the constraints placed on the values of the cosmological parameters using only the VSA bandpowers, and the low normalisation provided by the 28 COBEDMR bandpowers provided in the RADPACK package. The precise VSA data used were the bandpower estimates and associated covariance matrices for each of the three separate 3field mosaics VSA1, VSA2 and VSA3.
3.1 Gridbased approach
Prior  

{}      
{} + {}      
{}      
{} + {}      
{} + {}  
{SNIa} + {} 
Using the approach outlined in section 2.2, we calculate the corresponding likelihood function over the sixdimensional grid summarised in Table 2 and the normalisation parameter . In addition, we include the calibration uncertainty of the VSA bandpowers as an extra parameter in our analysis. The prior on this parameter is taken as a Gaussian centred on unity, with a standard deviation corresponding to the known calibration error for the VSA of 7 per cent in .
After analytically marginalising over calibration uncertainty, the bestfit model is characterised by the parameter values , but no particular significance should be attached to this model. It is the marginalised constraints on the individual parameters that are most important. Nevertheless, it is of interest to determine the goodnessoffit for this model. At the peak, the value of was found to be for the VSA plus 28 COBEDMR bandpowers. Assuming that a full 7 degrees of freedom are lost to the fit (which is unlikely given the form of the theoretical power spectra), we thus have 51 remaining degrees of freedom. The value lies at the 38 per cent point on the cumulative distribution function, which is entirely acceptable and shows that model to be a good fit to the data.
For a multidimensional likelihood function calculated on a grid, one has already implicitly assumed tophat priors on each of the parameters, corresponding to the edges of the grid. It should also be bourne in mind that, in principle, if the grid does not encompass all of the likelihood in any parameter, then that tophat prior becomes relevant for all of the parameters. In addition to the implicit prior arising from the grid, we may also impose explicit priors on the cosmological parameters values, according to our existing knowledge (or prejudices) concerning their values. In particular, we consider combinations of the following five priors: (i) a weak tophat prior on (); (ii) a strong prior Gaussian on (, hstfinal); (iii) a weak tophat prior on age (); (iv) a strong prior on optical depth (); (v) a prior in the plane from highredshift Type IA supernovae (where ) (perlmutter99). After adopting (combinations of) these priors, we then obtain the onedimensional marginalised distribution for each cosmological parameter by direct numerical integration; the successive integrals over the parameter directions are evaluated in turn by first performing a cubic spline interpolation onto a fine regularlyspace grid of points.
An illustration of our results is shown in Fig. 2 for the two cases in which we assume prior (i) above on , with and without the additional prior (iv) on . The solid lines represent results treating as a free parameter, whereas the dotdashed lines correspond to setting . It is clear from the figure that the effect of this latter prior is minimal, leading only to minor changes in the constraints on and , which is to be expected from the wellknown degeneracy between these two parameters (nsobh).
The 68 per cent confidence limits derived from these marginal distributions on the cosmological parameters , , and are given in the first two rows of Table 2. The upper and lower limit in each case is defined such that the corresponding interval contains 68 per cent of the total probability, and the likelihood function evaluated at each limit has the same value; the quoted bestfit value is the mode of the corresponding marginalised distribution. Also listed in the table are the 68 per cent confidence limits on these parameters resulting from (combinations of) the different priors listed above. We note that the constraints on these four parameters are relatively insensitive to the inclusion of increasingly stringent priors. In particular, we see that the constraints on are all consistent with the constraint (burles01) resulting from the observed primordial hydrogendeuterium ratio and the theory of nucleosynthesis. Depending on the prior assumed, the preferred value of is typically around 0.12, with an uncertainty of 0.05, and lends very strong support to existing evidence for the existence of nonbaryonic dark matter. We also note that, for all the priors considered, the constraints on the total density are consistent with the Universe being spatially flat. Finally, and again independently of the particular prior assumed, the constraints on the scalar spectral index are consistent with the scaleinvariant (Harrison–Zel’dovich) initial power spectrum, which is preferred by standard inflationary models.
In the first four rows of Table 2, the priors are unable to break the wellknown degeneracy of CMB data in the plane, and so no constraints are given. Nevertheless, with the assumption of our strong prior on and a weak prior on the age of the Universe, we see that one begins to break this degeneracy. Indeed, one finds that the preferred values of and correspond to a roughly equal partitioning of the critical density between matter (baryonic and dark) and vacuum energy, although the lower limit for and the upper limit for extend some way from the bestfit values. Finally, the assumption of the highredshift SNIa constraint (perlmutter99) and our weak prior on , succeed in cleanly breaking the CMB degeneracy and we find the partitioning of the critical density between matter and vacuum energy is well constrained in the approximate proportions onethird to twothirds.
3.2 MCMC approach
Using precisely the same data as analysed in the previous section, we also explored the sevendimensional parameter space using the MCMC techniques outline in section 2.3. First, we used the Bayesys algorithm and, for each sample, calculated the theoretical spectrum by interpolating from the precomputed grid. Since the shape of the likelihood function is very simple, it was enough to run just eight simultaneous Markov chains, with each walk requiring only a very short burnin period. After burnin, we collected 3000 samples from each chain, thus obtaining 24000 samples in total from the likelihood distribution. As one would hope, we find that all parameter constraints calculated from these samples are fully consistent with those obtained above for each set of imposed priors. We therefore do not reproduce them here, although they do provide a useful check on our earlier results.
As mentioned in section 2.3, to illustrate the flexibility of the MCMC approach, we included the calibration uncertainty of the VSA bandpower measurements as an additional parameter in our numerical analysis. The prior on this parameter was taken to be a Gaussian centred at unity, with standard deviation . In Fig. 3, we plot this prior distribution, together with the marginalised distribution on the parameter after analysing the data. The mean of this distribution lies at and has a standard deviation of 0.071. Thus, we see that, within the class of models considered, the measured CMB bandpowers are consistent with our estimated calibration uncertainty.
As a second illustration of the usefulness of the MCMC approach, we plot in Fig. 4 a novel representation of the sets of cosmological models consistent with the VSA plus COBEDMR data, which is produced as follows. Each of the MCMC samples corresponds to a theoretical spectrum. Thus from the samples it is straightforward to construct a onedimensional distribution of the power at each multipole . In Fig. 4, the position of the maximum of the distribution at each is shown by the solid line, while the dashed and dotdashed lines indicate the 68 and 95 per cent confidence limits of the distribution respectively, determined in the same manner as in section 3.1. This plot assumes our earlier weak prior on the age of the Universe. We note that the first peak is very well defined, and that there is also good evidence for the second peak.
As mentioned in section 2.3, we also check our cosmological parameter constraints by using the straightforward Cosmomc algorithm, which is optimised for the problem at hand and bypasses the need for a grid. Once again our results are fully consistent with those found above, and provide another useful check on our analysis.
4 Combining VSA with other CMB experiments
So far we have only combined the VSA data with the COBEDMR experiment in order to place limits on cosmological parameters from the CMB. It is clear, however, that tighter constraints may be obtained if we additionally include information from other CMB experiments, in particular BOOMERanG (bernardispeaks), DASI (cosmoparamsfromdasi), and MAXIMA (stompor01).
In Fig. 5, we begin by simply plotting the maximumlikelihood estimates for , , and obtained by the VSA and these other recent CMB experiments, together with the reported 68 per cent confidence intervals (except for MAXIMA, for which we plot as error bars of the reported 95 per cent confidence limits). We note that the confidence limits for each experiment include the effects of calibration and beam uncertainty, where appropriate, and each experiment also assumes the COBEDMR bandpowers and similar weak priors to those adopted in section 3.1.
In general, the constraints on each individual parameter agree within error bars. Moreover, since each of these experiments uses different observing techniques and has observed different regions of the sky, we may consider each measurement as an independent estimate of the corresponding parameter. Assuming further that the individual likelihoods are Gaussian, they may be immediately combined to produce a joint constraint on each parameter, which is also shown in the figure.
In the top panel, corresponding to the parameter , we also plot the 68 per cent confidence limits arising from the nucleosynthesis constraint (burles01). The combined measurement from all CMB experiments is consistent with the BBN constraint and fully supports the case for a low value of the primordial deuterium abundance. It also favours a primordial helium mass fraction of . In the second panel, we see that all the experiments agree with the prediction of standard inflationary models. Remarkably, the combined CMB measurement has an error bar of only 3 per cent. The combined value for also agrees at the 1 level with the scaleinvariant HarrisonZel’dovich initial power spectrum, i.e. with . Finally, we see that the combined value is tightly constrained to be around 5 times larger than the value for , which is a strong indication for the existence of nonbaryonic matter.
In Fig. 6 we compare the VSA+SNIa constraints on and , with those published by the other recent CMB experiments. It is important to note, however, that the published DASI value does not make use of any SNIa data, so in order to enable a proper comparison to be made, we have repeated the complete gridbased parameter estimation procedure for DASI performed by cosmoparamsfromdasi, using their published covariance matrices and window functions, but including the SNIa prior from perlmutter99 and a weak prior on . Incidentally, we found that the parameter constraints we derived for DASI before including the SNIa prior were in complete agreement with those obtained by cosmoparamsfromdasi. In Fig. 6, we plot the 68 per cent confidence limits from each experiment, and we see that they are all in good agreement. The combined constraint is and .
As noted previously by cosmoparamsfromdasi, by combining all the available CMB datasets together with a strong prior on , it is possible to break cleanly the CMB degeneracy in the plane without using the SNIa prior. As our CMB datasets, we use the VSA and COBEDMR bandpowers, the RADPACK compilation for DASI, plus the published results from BOOMERanG and MAXIMA. By performing a simple gridbased parameter estimation procedure, as outlined in section 2.2, we find that one can break the plane CMB degeneracy by assuming only our weak prior on the age of the Universe (10 Gyr age 20 Gyr) and our strong Gaussian prior on from the HST key project. Our resulting constraints are and , which are similar to those derived above, but are independent of the SNIa data. Hence, this result is not subject to the usual uncertainties associated with using highredshift type IA supernovae as standard candles. We stress that a very similar result would be obtained by instead combining the CMB data with the prior on from Sunyaev–Zel’dovich and Xray observations of clusters (jones02).
5 Discussion
In Figs 5 and 6 we have presented results of cosmological parameter extraction from various CMB experiments. These experiments use a variety of observational techniques and operate at a range of frequencies and have therefore widely different systematic effects. Nevertheless, the figures show remarkable agreement between different experiments. This may indicate the importance of the assumed weak priors (which are often common) and possibly the fitting of too many parameters given the constraining power of individual experiments. Indeed, the reduced is below for most experiments (see individual papers). Nevertheless, when one combines experiments (VSA, DASI, BOOMERanG, MAXIMA and DMR data) most cosmological parameters become constrained at the level ranging between 5–20 per cent, if one neglects the possibility of tensor modes. This accuracy rivals the discriminatory power of the upcoming CMB satellite experiments, such as MAP (map), with the added bonus that the residual systematic effects of the various experiments are diluted.
We also verify that a constraint on the vacuum energy component of the Universe may be obtained independently of the Type Ia supernovae data, by combining the results from CMB experiments alone, together with the a prior on from the HST key project (hstfinal). Moreover, the resulting values for and are consistent with those obtained when supernovae data are included. It is often assumed that CMB experiments cannot constrain the vacuum energy, as a result of the well known CMB degeneracy in the plane, and the SNIa data are usually employed to break this degeneracy. The supernovae data are, however, still somewhat plagued by uncertain systematic effects (such as extinction along the line of sight and the evolutionary effects due to metallicity), although the issues are gradually being resolved (snresolv). Therefore, obtaining independent consistent results on the value of the cosmological constant increases our confidence in both the supernovae and CMB results.
6 Conclusions
In our analysis of the newly available data from the VSA compact array, and other recent cosmological results, we have found the following.

Traditional gridbased methods and MarkovChain Monte Carlo (MCMC) sampling techniques have been applied to the cosmological parameter estimation problem and found to yield consistent results.

The VSA observations, when combined with the COBEDMR data and a weak tophat prior on () give , , and at the 68 per cent confidence level.

Adding in observations of type Ia supernovae, the CMB degeneracy in the plane may be broken to yield the constraints and .

The BOOMERanG, DASI and MAXIMA experiments, which have different approaches and systematics, yield consistent constraints on cosmological parameters, which is gratifying. Combining the results of these recent CMB experiments with the VSA data, and assuming weaks priors on and the age of the Universe, gives , , and .

Adding in the type Ia supernovae constraint to the combined CMB result, the plane degeneracy is cleanly broken to give and .

One can equally well break the plane degeneracy without the SNIa data, by assuming the strong prior on from either the HST Key Project or Sunyaev–Zel’dovich and Xray observations of clusters. This yields the constraints and .
Acknowledgements
We thank Sarah Bridle for providing her gridbased likelihood programs, Antony Lewis for making available his Cosmomc code and John Skilling for allowing us to use his Bayesys MCMC sampler. This research was conducted in cooperation with SGI utilising the HEFCEsupported COSMOS supercomputer. We also acknowledge the IAC Computer Centre for helping with the setup of a 10processor Beowulf system dedicated to this research. We thank the staff of the Mullard Radio Astronomy Observatory, Jodrell Bank Observatory and the Teide Observatory for invaluable assistance in the commissioning and operation of the VSA. The VSA is supported by PPARC and the IAC. Partial financial support was provided by the Spanish Ministry of Science and Technology project AYA20011657. A. Taylor, R. Savage, B. Rusholme and C. Dickinson acknowledge support by PPARC studentships. K. Cleary and J.A. RubiñoMartin acknowledge Marie Curie Fellowships of the European Community programme EARASTARGAL, ’The Evolution of Stars and Galaxies’, under contract HPMTCT200000132. K. Maisinger acknowledges a Marie Curie Fellowship of the European Community. A. Slosar acknowledges the support of St. Johns College, Cambridge. We thank Professor Jasper Wall for assistance and advice throughout the project.