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Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinite-dimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a one-dimensional problem of updating the means of the Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact …