The dominant linear contribution to cosmic microwave background ( CMB ) fluctuations at small angular scales ( \lesssim 1 ^ { \prime } ) is a second-order contribution known as the Vishniac or Ostriker-Vishniac effect .
This effect is caused by the scattering of CMB photons off free electrons after the universe has been reionized , and is dominated by linear perturbations near the R _ { V } = 2 Mpc/ ( h \Gamma / 0.2 ) scale in the Cold Dark Matter cosmogony .
As the reionization of the universe requires that nonlinear objects exist on some scale , however , one can compare the scale responsible for reionization to R _ { V } and ask if a linear treatment is even feasible in different scenarios of reionization .
For an \Omega _ { 0 } = 1 cosmology normalized to cluster abundances , only \sim 65 \% of the linear integral is valid if reionization is due to quasars in halos of mass \sim 10 ^ { 9 } M _ { \odot } , while \sim 75 \% of the integral is valid if reionization was caused by stars in halos of \sim 10 ^ { 6 } M _ { \odot } .
In \Lambda or open cosmologies , both the redshift of reionization and z _ { V } are pushed further back , but still only \sim 75 \% to \sim 85 \% of the linear integral is valid , independent of the ionization scenario .
We point out that all odd higher-order moments from Vishniac fluctuations are zero while even moments are non-zero , regardless of the gaussianity of the density perturbations .
This provides a defining characteristic of the Vishniac effect that differentiates it from other secondary perturbations and may be helpful in separating them .