We present an extensive comparison of models of structure formation with observations , based on linear and quasi-linear theory .
We assume a critical matter density , and study both cold dark matter models and cold plus hot dark matter models .
We explore a wide range of parameters , by varying the fraction of hot dark matter \Omega _ { \nu } , the Hubble parameter h and the spectral index of density perturbations n , and allowing for the possibility of gravitational waves from inflation influencing large-angle microwave background anisotropies .
New calculations are made of the transfer functions describing the linear power spectrum , with special emphasis on improving the accuracy on short scales where there are strong constraints .
For assessing early object formation , the transfer functions are explicitly evaluated at the appropriate redshift .
The observations considered are the four-year COBE observations of microwave background anisotropies , peculiar velocity flows , the galaxy correlation function , and the abundances of galaxy clusters , quasars and damped Lyman alpha systems .
Each observation is interpreted in terms of the power spectrum filtered by a top-hat window function .
We find that there remains a viable region of parameter space for critical-density models when all the dark matter is cold , though h must be less than 0.5 before any fit is found and n significantly below unity is preferred .
Once a hot dark matter component is invoked , a wide parameter space is acceptable , including n \simeq 1 .
The allowed region is characterized by \Omega _ { \nu } \la 0.35 and 0.60 \la n \la 1.25 , at 95 per cent confidence on at least one piece of data .
There is no useful lower bound on h , and for curious combinations of the other parameters it is possible to fit the data with h as high as 0.65 .

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