Limits on the Primordial Fluctuation Spectrum Void Sizes and CMBR Anisotropy

We suggest to use the common appearance of voids with a scale of $5000 \kms$ in the galaxy distribution to estimate the power spectrum on this scale. We use a simple model for a gravitational formation of voids and we compare the results with the matter fluctuations as constrained by the CMBR observations of COBE. We find that a power spectrum $P(k)\propto k^n$ with $n \approx 1.25$ is compatible both with COBE and with gravitational growth of large voids in an $\Omega=1$ universe. A Harrison-Zel'dovich spectrum, $n=1$, normalized to produce the observed CMBR fluctuations, does not have enough power for gravitational growth of voids with a diameter of $5000 \kms$ in an $\Omega=1$ Universe. Such a spectrum would be compatible if (i) void diameters are smaller ($3500 \kms$), the voids are shallower or if the voids are rare (we assume that the universe is void filled); (ii) $\Omega < 1$ i.e. the Universe is open; (iii) galaxies do not trace matter on very large scale or if (iv) the voids do not grow gravitationally.