Does StatsModels' power.tt_ind_solve_power assume a single standard deviation despite two different means? Does StatsModels' power.tt_ind_solve_power (link) assume a single standard deviation despite two different means? I think so. Why is this a reasonable assumption?
I come from a primarily Bayesian background, so if I wanted to pool variances, I'd probably draw $\sigma_1$ and $\sigma_2$ from the same prior distribution.
I don't follow the logic that the means would be the different but the variances would be the same.
There is a ratio parameter (what % is population 1?) Perhaps this is used to decouple the variances?
 A: Yes, this function use's Cohen's d for the effect size which relies on a pooled standard deviation. Obviously, for certain distributions such as a binomial distribution where the variance is given by the mean, this doesn't make a lot of sense, but for a t-test, it's not unlike the pooled variance version of the t-test. The ratio parameter is not very directly related to this but rather to the fact that for optimal power, in most cases, both sample should be of equal size.
A: In many statistical software programs, 'power and sample size' procedures for a two-sample t test assume equal variances and sample sizes. There is a straightforward computation in such cases, which uses a noncentral t distribution.
For a Welch t test, the degrees of freedom $\nu$ depends on sample sizes and sample variances, so the approximate power for given sample size(s) is often found by simulation.
For example, the power of a Welch 2-sample test at the 5% level
for $\mu_1 = 50, \mu_2 = 55,$ $\sigma_1 = 5, \sigma_2=7,$ $n_1 = 30, n_2 = 25$ can be simulated
in R as about 83%.
set.seed(2022)
pv = replicate(10^5, t.test(rnorm(30,50,5),
                      rnorm(25,55,7))$p.val)
mean(pv <= 0.05)
[1] 0.8313

For a pooled test, a recent release of Minitab
gives the following exact results:
Power and Sample Size 

2-Sample t Test

Testing mean 1 = mean 2 (versus ≠)
Calculating power for mean 1 = mean 2 + difference
α = 0.05  Assumed standard deviation = 6

            Sample
Difference    Size     Power
         5      25  0.823010
         5      30  0.887557
The sample size is for each group.


