Paizo published a solution for exactly this problem: the Boon Companion feat.

Prerequisites: Animal companion or familiar class feature.

Benefit: The abilities of your animal companion or familiar are calculated as though your class were 4 levels higher, to a maximum effective druid level equal to your character level. If you have more than one animal companion or familiar, choose one to receive this benefit. If you lose or dismiss an animal companion or familiar that has received this benefit, you may apply this feat to the replacement creature.

Special: You may select this feat more than once. The effects do not stack. Each time you take the feat, it applies to a different animal companion or familiar.

The Falconer page states:

The bird can be of any type of large hunting or scavenging bird (even a vulture). This ability functions like the druid animal companion ability (which is part of the nature bond class feature), but the falconer must take the bird animal companion, and that companion has only half the normal hit points.

Since the system doesn't specify exactly what qualifies as a bird of prey aside from telling you it can be any kind of large hunting or scavenging bird but that it does allow you to take a vulture, The likely course of action would be that the Ranger would have access to any Bird type animal companion. So you could take an Eagle, a Falcon, a Roc, or a vulture, provided it had the Bird sub-type.

At 4th level when Ranger would normally get an animal companion, instead your animal companion would level up and would have 100% of its HP instead of the half it starts off with.