In the Genesis flood narrative, God set a rainbow in the clouds as the sign of the covenant.
After Noah left the Ark, Genesis 9 describes a covenant between God, Noah, his descendants, and every living creature. The rainbow in the clouds is presented as the sign of that covenant. Its stated purpose is to recall the promise that waters would not again become a flood destroying all flesh.
The rainbow is therefore not the same event as the dove's olive leaf. The leaf occurs while Noah is testing whether the waters have receded; the rainbow appears after the family and animals leave the Ark and Noah offers a sacrifice.
Genesis extends the covenant beyond Noah's household to living creatures generally. Later Jewish and Christian interpretation gives the rainbow additional theological meaning, while modern culture often uses it more broadly as a symbol of peace, hope, or inclusion. The narrative's specific covenant sign is the rainbow.