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IV. Cosmological Parameter Estimation 3 Nificant Likelihood Mon. Not. R. Astron. Soc. 000, 1–10 (2002) Printed 26 October 2018 (MN LATEX style file v2.2) First results from the Very Small Array – IV. Cosmological parameter estimation Jose Alberto Rubi˜no-Martin1, Rafael Rebolo1,2, Pedro Carreira3, †, Kieran Cleary3, Rod D. Davies3, Richard J. Davis3, Clive Dickinson3, Keith Grainge4, Carlos M. Guti´errez1, Michael P. Hobson4, Michael E. Jones4, R¨udiger Kneissl4, Anthony Lasenby4, Klaus Maisinger4, Carolina Odman¨ 4, Guy G. Pooley4, Pedro J. Sosa Molina1, Ben Rusholme4,⋆, Richard D.E. Saunders4, Richard Savage4, Paul F. Scott4, Anˇze Slosar4, Angela C. Taylor4, David Titterington4, Elizabeth Waldram4, Robert A. Watson3,† and Althea Wilkinson3 1Instituto de Astrof´ısica de Canarias, 38200 La Laguna, Tenerife, Canary Islands 2Consejo Superior de Investigaciones Cient´ıficas, Spain 3Jodrell Bank Observatory, University of Manchester, UK 4Astrophysics Group, Cavendish Laboratory, University of Cambridge, UK †Present address: Instituto de Astrof´ısica de Canarias ⋆Present address: Stanford University, Palo Alto, CA, USA Accepted —; received —; in original form 26 October 2018 ABSTRACT We investigate the constraints on basic cosmological parameters set by the first compact-configuration observations of the Very Small Array (VSA), and other cos- mological data sets, in the standard inflationary ΛCDM model. Using the weak priors −1 −1 40 <H0 < 90 km s Mpc and 0 <τ < 0.5, we find that the VSA and COBE-DMR +0.12 2 +0.009 2 data alone produce the constraints Ωtot = 1.03−0.12, Ωbh = 0.029−0.009, Ωcdmh = +0.08 +0.11 0.13−0.05 and ns = 1.04−0.08 at the 68 per cent confidence level. Adding in the type . arXiv:astro-ph/0205367v2 9 May 2003 +0 09 +0 07 Ia supernovae constraints, we additionally find Ωm = 0.32−0.06 and ΩΛ = 0.71−0.07. 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 H0 (for which the X-ray plus Sunyaev– +0.14 Zel’dovich route gives a similar result), we obtain the tight constraintsΩm =0.28−0.07 +0.07 and ΩΛ = 0.72−0.13, which are consistent with, but independent of, those obtained using the supernovae data. Key words: cosmology: observations – cosmic microwave background 1 INTRODUCTION scribing our Universe is based on the idea of inflation (?), which provides a natural mechanism for producing initial One of the central aims of cosmology is to determine the density fluctuations described by a power-law spectrum with values of the fundamental cosmological parameters that de- a slope close to unity. The simplest versions of inflation also scribe our Universe. A unique opportunity to achieve this predict the Universe to be spatially flat. The initial spec- goal is provided by the observation of anisotropies in the trum of adiabatic density fluctuations is modulated through cosmic microwave background (CMB) radiation. By com- acoustic oscillations in the plasma phase prior to recombina- paring such observations with the predictions of our current tion and the resulting inhomogeneities are then imprinted as theories of structure formation and the evolution of the Uni- anisotropies in the CMB. In the basic inflationary scenario, verse, we may place constraints on the cosmological param- the CMB temperature anisotropies are predicted to follow eters that appear in these models. a multivariate Gaussian distribution, and so may be com- The currently most favoured theoretical model for de- c 2002 RAS 2 Rubi˜no-Martin et al. pletely described in terms of their angular power spectrum. model. Moreover, as is now common practice, we consider Moreover, the acoustic oscillations in the plasma phase lead models in which the contents of the Universe are divided to a characteristic series of harmonic peaks in the power into three components: ordinary baryonic matter; cold dark spectrum, which are a robust indicator of the existence of matter (CDM), which interacts with baryonic matter solely fluctuations on super-horizon scales. through its gravitational effect; and an intrinsic vacuum en- Although the presence of acoustic peaks in the CMB ergy. The present-day contributions of these components, power spectrum is a generic prediction of inflationary mod- measured as a fraction of the critical density required to els, detailed features of the power spectrum, such as the make the Universe spatially flat, are denoted by Ωb, Ωcdm relative positions and heights of the peaks, depend strongly and ΩΛ respectively. It is possible that some of the dark on a wide range of cosmological parameters, see e.g. ?. In- matter may, in fact, be in the form of relativistic neutrinos deed, this sensitivity to the parameters is the reason why (hot dark matter), but the presence of a hot component has observations of the CMB provide such a powerful means of a negligible effect on the power spectrum, given the sensitiv- constraining theoretical models. ity and angular resolution of current CMB experiments (?). Thus measurement of the CMB power spectrum is a We therefore assume that all dark matter is cold and set major goal of observational cosmology and numerous exper- Ων = 0. iments have provided estimates of the spectrum on a range Following the current theoretical expectation (?), we of angular scales. It is only recently, however, that obser- also assume that the contribution of tensor mode perturba- vations by the BOOMERanG (?), DASI (?) and MAXIMA tions is very small compared with the scalar fluctuations, (?) experiments have provided measurements of the CMB and so we ignore their effects. This assumption is consistent power spectrum with sufficient accuracy over a wide range with current observations. Since tensor modes contribute of scales to allow tight constraints to be placed on a wide primarily to low multipoles, ℓ, the only existing measure- range of cosmological parameters (see, for example, ?). This ment that would be particularly sensitive to their presence parameter estimation process is performed by comparing the is the level of the CMB power spectrum at low-ℓ observed observed band-powers with a wide range of theoretical power by the COBE-DMR experiment (?). If the tensor compo- spectra corresponding to different sets of values of the cos- nent made up a large fraction of this observed power, the mological parameters, which can be accurately calculated value of the spectral index ns for scalar perturbations would using the Cmbfast (?) or Camb (?) software packages. The need to exceed unity by a considerable margin in order to comparison of the observed and predicted power spectrum provide the level of power at higher ℓ measured by numer- is usually carried out in a Bayesian context by evaluating ous other CMB experiments. Such a large value of ns is, the likelihood function of the data as a function of the cos- however, ruled out by large-scale structure studies (?). Nev- mological parameters. ertheless, it must be remembered that this argument only In this paper we perform this parameter estimation pro- holds if the initial perturbation spectrum is indeed a simple cess using, as the main CMB datasets, the flat band-power power-law. estimates of the CMB power spectrum measured by the Very Given the assumptions outlined above, there remain Small Array (VSA) in its compact configuration, which has seven degrees of freedom in the description of the stan- been described in the sequence of earlier papers ?, ? and ? dard inflationary CDM model. The parameterisation of this (Papers I, II and III), and the COBE-DMR band-powers for seven-dimensional model space can be performed in numer- low-ℓ normalisation. We also combine the VSA data with ous ways, but we shall adopt the most common choice, which other recent CMB experiments, and constraints from the is defined by the following parameters: the physical density 2 HST Key Project on H0 and observations of type Ia super- of baryonic matter (Ωbh ≡ ωb); the physical density of 2 novae, to tighten further the constraints on the cosmological CDM (Ωcdmh ≡ ωcdm); the vacuum energy density due to parameters. Two different methods are used to perform the a cosmological constant (ΩΛ); the total density (Ωtot); the parameter estimation procedure. First, we employ the tra- spectral index of the initial power-law spectrum of scalar ditional technique of evaluating the likelihood function on a perturbations (ns); the optical depth to the last-scattering large grid in parameter space. Second, we consider a more surface due to reionisation (τ); and the overall normalisation flexible approach in which the likelihood function is explored of the power spectrum as measured by Q ≡ 5C2/(4π) and by Markov-Chain Monte Carlo (MCMC) sampling. The lat- is quoted relative to QCOBE, as determinedp from the 4-year ter method has a great potential in terms of expanding the COBE-DMR data by ?. This choice of parameters is similar dimension of the parameter set which can be investigated. to that made in the analysis of the CMB band-power mea- Here we use it to demonstrate the robustness of the results surements obtained by the BOOMERanG, MAXIMA and from the standard grid approach and also to provide a novel DASI experiments. We note that, in this parameterisation, −1 −1 visualisation of the range of uncertainty in our parameter the reduced Hubble parameter h (≡ H0 km s Mpc /100) estimates. is auxiliary and is given by h = (ωb + ωcdm)/(Ωtot − ΩΛ). In comparing the observed CMBp flat band-powers mea- sured by the VSA with the above multidimensional model, 2 MODELS, PARAMETER SPACE AND we adopt a Bayesian approach based on the evaluation of METHODS the likelihood function for the data as a function of the cosmological parameters, which for brevity we denote by In the analyses presented in this paper we restrict our atten- the vector θ = (ωb,ωcdm, ΩΛ, Ωtot, ns,τ,Q/QCOBE).
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