Every index has criteria, usually somewhat unique, or there would be no marketable difference between them?

> What do you mean "better profitability criteria"? I don't want an index to exclude companies on the basis of profitability. I want it to hold the market.

OK, then put your money in VTI/VTSAX instead of a S&P 500 fund. I own some VTI and also some FXAIX, you can do the same thing and choose which index to buy.