Please Note: the e-mail address(es) and any external links in this paper were correct when it was written in 1995, but may no longer be valid.
Mullard Radio Astronomy Observatory, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UNITED KINGDOM
Most models of structure formation in the universe predict anisotropies in the CMBR. Since the COBE detection of large angular scale anisotropies, a great deal of attention has been devoted to constraining these models. Meanwhile other detections have been claimed from smaller angular scales. Thus the next step would be comparing experimental results with theoretical predictions and constraining the model parameters (e.g. the primordial power spectrum of fluctuations, the matter density in our universe, the baryonic content, the nature of non-luminous matter, the biasing, the process of structure formation, the statistics of the temperature fluctuations etc.). In this comparision three different aspects must be considered: (1) what is the dominant anisotropy effect, (2) theoretical errors, (3) observational strategy. With respect to (1), for the experiments probing large angular scales (larger than the horizon size at recombination) the anisotropy is mainly due to the so called Sachs-Wolfe effect (due to potential fluctuations in the last scattering surface) whereas on smaller angular scales these are due to the Doppler effect resulting from peculiar velocities of the electrons scattering off the CMBR photons during recombination. For (2), the theory can only predict the statistics of the temperature fluctuations and ensemble average values but not its value in our universe. Therefore one must take into account not only the theoretically expected value of the temperature fluctuation but also the so-called cosmic variance, i.e. the theoretical error due to observing only one universe. In smaller scale experiments it is usually assumed that the cosmic variance is small (i.e. only higher multipoles contribute in the temperature fluctuations expansion) (Scaramela & Vittorio (1993)) but this only applies to measurements over the whole sky, otherwise the result is affected by the so called sample variance, that is the error due to the fact that one is sampling part of the sky not using all available information to compute, e.g. the . For (3), the different observational strategies may consist of single difference, double differences (beam switching) or interferometry. To proceed with the comparision one may use two methods: (1) Monte Carlo simulations (simulation of the observed sky for a given experiment, computation of , cosmic variance, sample variance and the theoretically expected value of ) or (2) an analytical approach (using semi-analytical expression to compute the cosmic variance for each angular scale and each experiment (Cayón et al. (1991)). The sample variance is then computed using an approximate relation with cosmic variance, or directly (Scott et al. (1994)). The models considered are those of: Bond & Efstathiou (1987) (hereafter BE), Holtzmann (1989), Peebles (1987) and topological defect models as opposed to the gravitational instabilities.
We have consider the following experiments (with the corresponding FWHM): OVRO, OVRO/RING (); MSAM ( ); MAX3Peg, MAX3GUM, MAX4GUM, MAX4ID, MAX4SH (); SP(ACME/HEMT); MIT (); Tenerife (); COBE (). In this paper we give the predicted expected values of ( for BE models in a plot comparing the predictions of one of the BE models normalised to the COBE detection and the experimental results or upper limits for some experiments. This gives an overall idea of what is happening with ( versus angular scale (more precisely FWHM). Work has been done with the other models and it will be published shortly. With respect to theoretical uncertainty, we give a brief account of the analytical techniques already existent (Cayón et al. (1991),Scott et al. (1994)) and the Monte Carlo simulations in progress (of which the results will be published in the near future).
In order to compute the expected temperature fluctuation for a given experiment and model (i.e. () one uses the two-point angular correlation function C() which is defined as being the expected average of the product of the temperature fluctuations in two directions in the sky separated by an angle : , where . The expression of depends on the experimental configuration (e.g. beam size, switching pattern etc.). With a two-beam experiment one measures the temperature difference between beams separated by an angle on the sky usually called single difference: and its expected variance is written in terms of the temperature autocorrelation function: ; while a three-beam experiment measures the difference between a field point, , and the mean value of the temperatures in two directions which are separated from the field point by an angle , usually called double difference: ) and its expected variance is given by: . All these expressions are true for an ideal experiment with infinitely narrow beam, for a real one it is necessary to consider the beam smearing due to the finite resolution of the antenna. This is usually taken into account by convolving the radiation intensity with a beam approximated by a Gaussian : where is the dispersion of the Gaussian. The correlation function of the convolved radiation field is given by BE:
where is the modified Bessel function. So for a single difference one has:
and for a double difference:
where is the beamthrow. One is able to compute using equations (2.1) and (2.2) or (2.3) according to the experimental configuration. If, for example, for a given model an analytical expression (e.g. fitting parameter expression) for the angular correlation function is given or e.g. a parameterized expression for the transfer function of , , is given to be used in where and are spherical Bessel functions, , is the horizon distance. An alternative method consists of using the angular power spectrum, , which are the coefficients of the Legendre polynomials expansion of . Commonly one expresses the temperature fluctuations on the celestial sphere using an expansion in spherical harmonics (completeness property): Inflationary models predict Gaussian initial perturbations, in this case the coefficients are stochastic variables with random phase, zero mean and variance given by: . The expectation value of the temperature correlation function (the ensemble average) is given by:
For a more realistic scenario one must take into account the smearing due to the finite resolution of the antenna: and
where (Scaramela & Vittorio (1993)). Once given the 's one can compute the correlation using (2.4) or (2.5) and according to (2.2) or (2.3) for the switching experiments. These 's may be obtained directly using the solutions to the equations describing temperature fluctuations evolution by where are the coefficients of the Legendre polynomials expansion of, , the radiation intensity fluctuations and is the conformal time (Efstathiou (1987)). In particular in the case of large angular scales for and an initial power-law form of the density fluctuations power spectrum P(k) an expression for the may be used (Efstathiou (1987),Scaramela & Vittorio (1988)):
with , , ; this is important mainly because one is able to predict and its error bands as function of providing a direct test of the Harrison-Zeldovich spectrum (scale-invariant, ) although this is only adequate for experiments probing angular scales larger than the horizon size at recombination () where only Sachs-Wolfe or isocurvature effects are dominant. (As an example it was applied to put constraints on by comparing a simulated Tenerife experiment normalised to the COBE detection with the actual experiment (Hancock et al. (1994)). An alternative method consists of using the expression of the expected temperature autocorrelation function and the orthogonality property of the Legendre polynomials:
These methods were applied to BE models using the fitting parameter expression;
where and A, , , , , are tabulated for each model (BE). Inserting (2.8) in (2.7) and computing the integral one obtains the . These are in good agreement with those plotted in BE. The calculations of using both (2.1) and the methods are in good agreement, the results are in Table 1. Figure 1 shows both one BE model prediction (normalised to COBE) and observational results (detection and instrumental error or upper limits) via versus FWHM. The same method is to be applied to other models and experiments. As most of the experimental results are given as the result of a likelihood analysis, they are a measure of intrinsic anisotropies i.e. values before beam convolution. In order to compare properly with theoretical predictions the observed values (convolved with the beam and single or double difference as appropriate) must be computed and used in Figure 1. The actual figure uses the values as published by each experiment.
Table 1: ( for BE models ( is beamthrow)
As previously mentioned this comparision is not complete until one compares the predicted mean values with the theoretical uncertainties. Once this is accomplished we may look for intersections of the experimental and theoretical bands for each model and experiment and decide upon consistency. The methods to compute theoretical errors are to be described in next section.
Figure 1: T/T versus FWHM: predictions and experimental values
The temperature correlation function, , is a random variable with ensemble average given by (2.5). It can be expressed as , where , with a distribution and ; , variance . To compute the cosmic variance we use two approaches: (1) analytical and (2) Monte Carlo simulations. The first method uses the probability density function of the temperature autocorrelation given in a semi-analytical form (Cayón et al. (1991)):
where , with , P(k) = Ak. This expression applies to a single beam experiment, for beam-switching experiments new variables , are defined. The distribution function is defined using the same expression with the new defined according to the switching pattern. The sample variance () is obtained using an approximate relation with the cosmic variance (Scott et al. (1994)):
where A is the solid angle the experiment covers; or directly using the two-point correlation function C() (Scott et al. (1994)):
These methods may be applied to other experiments considering e.g. their scan strategy, appropriate solid angle and using (3.11) or computing the cosmic variance via (3.9) and then using the approximate relation (3.10). Both approaches may then be compared with the results obtained using method (2). This method involves producing Monte Carlo simulations of the temperature fluctuations according to specific characteristics of each experiment such as the beamwidth, the sky coverage, the switching pattern, the beamthrow, etc. then computing the correlation at zero lag, its distribution, variance and confidence intervals. Proceeding in this way one computes the overall theoretical error for a given experiment. These simulations follow different strategies according to angular scale and experimental configuration. For large angular scales one uses the standard spherical harmonics expansion of , considering the coefficients as stochastic variables with a Gaussian distribution, zero mean and as variance the angular power spectrum, , (as appropriate to Gaussian random fields). The interval of 's to be used for each experiment is found using its filter function, , and a standard criterion. These scales are contributed to by low order multipoles only, the relation between the angular scale and the multipole is given by , with in radians, used to compute the corresponding to the dispersion of the beam. This relation gives an idea of the significant values of contributing to the expansion. The 's are obtained using equations (2.6) or (2.7), the former suiting the aim of testing the primordial spectrum directly. This was applied to COBE and Tenerife for the BE models and simulations of other experiments are in progress. For small angular scales one considers the CMB radiation pattern as a Gaussian field in a flat 2D space (BE) with given by a Fourier expansion as follows: where L is the size of the small patch of the sky, , is a variable with a Gaussian distribution with zero mean and variance = with and with a random phase in the interval . is obtained by evaluating a Fast Fourier Transform (FFT) of . FFT's for L = on a grid were evaluated and grey scale pictures produced. For each experiment one must consider the sky coverage and use the selected points to do statistics according to experimental configuration. The 's used were computed using equation (2.7). Meanwhile other methods may be applied for instance the process used by Górsky et al. (1993) for the SP91 experiment. Simulations are in progress and its results will be published in the near future. These will be organized in a plot with versus FWHM with both the theoretical expected value theoretical error and the detected value instrumental error or observational upper limits as applied to most of the experiments and more relevant models.
We have considered methods to compute the theoretical predictions for different experimental strategies and presented results for the BE models for some experiments and a plot comparing these with the experimental results. This will be extended to other models and experiments. We have stressed the importance of assigning the theoretical uncertainties to the theoretical expected values in order to make a fair comparision of models and experiments. We have described processes to compute the cosmic and sample variances, proceeding with the Monte Carlo simulations. Once all these calculations are complete a proper plot comparing theory and observations can then be produced.