God established the rainbow with Noah after the flood as a sign of the promise never again to destroy Earth by flood. Genesis 9:13–15 describes the rainbow as the sign of this covenant.
The covenant is made not only with Noah and his descendants but also with every living creature affected by the flood. Genesis presents the rainbow as appearing in the clouds, reminding God of the everlasting covenant between God and Earth. The promise concerns a worldwide destruction by flood in the form described by the narrative.
The olive leaf belongs to an earlier scene: a dove brings it to Noah as evidence that the waters are receding. It is therefore related to the end of the flood but is not the covenant sign specified in Genesis 9.
The rainbow has also become a widely used symbol of hope and peace in later religious, artistic, and cultural traditions. In the Genesis account, however, its stated role is specifically covenantal and connected to the future of all life.