Yes. The separated wavelengths of the refracted light still exist, regardless of an audience.

"The rainbow is an object that cannot be defined in any meaningful sense independently of the point of view from which it is seen. In this sense the rainbow, although it is constituted by a set of physical drops of water reflecting the light in a certain way, cannot be defined without knowing where and how it will be seen. For instance, it is not possible to fly under a rainbow or to sidestep it. The rainbow will move accordingly to the position of the observer. Furthermore it is possible to argue that the rainbow is a private physical phenomenon since two different observers always see two physically (even slightly) different rainbows. Since two separate observers occupy two different positions in space, they select different rays of light and thus different drops. Every observer sees a different rainbow. This can be verified by moving rapidly (for instance driving a car): the rainbow seems to be moving together with the observer. In reality by changing its position, the observer modifies the properties of his/her point of view and therefore continuously selects new sets of drops. A fundamental question arises at this point. Does the rainbow exist independently of the act of observation? We claim the answer is no. In fact, if there were no eyes looking at the rainbow (or a camera), that phenomenon that we call rainbow would not produce any effect. Furthermore the selection of a particular set of drops as constituting the rainbow cannot be done independently from the point of view. As we have seen, changing the position of the observer changes also the drops of water that are part of the rainbow. Even if it is possible to argue that an expert physicist knowing the position of each drop, of the sun and the point of view could calculate the projection of the rainbow on the observer’s retina, such a calculation requires as an essential element the knowledge about the point of view, which cannot be omitted. The rainbow is not there, as a bridge of stone. The rainbow occurs only when it is seen." Source and further information: http://www.lira.dist.unige.it/projects/adapt/files/D2.1.pdf 2) "When the sun is sufficienttly low on the horizon and projects its rays at an appropriate angle against a cloud, all the drops reflect the sunlight. If there were no observers, the rays would not produce an effect as a whole because they would continue their travel in space without interacting. Only those rays which have a particular geometrical relation to the observer are seen as a part of the rainbow. A given rainbow exists only when the observer is in a given position with respect to the external stimuli. If there were no observers, the rays will lose their opportunity to produce a joint effect. Therefore their cause (the supposed rainbow) would not have produced any effect and would not have existed as a cause. It could only have a theoretical existence. We assumed that there must have been a rainbow, but there wasn’t one. If an observer were there, the rays would have hit her/his photoreceptors and a complex chain of physical processes would have continued from the retina to the cortical areas up to a point where the recognition of the rainbow as a whole would have taken place. Thanks to the existence of the physical structure of the observer, the drops of water of the rainbow have been able to produce a joint effect. Until the whole process is concluded there is no actual rainbow. Something could happen at the very last moment in order to interfere with the completion of the process." Source and further information: http://emotion-research.net/ws/summerschool3/Tagliasco_2006.ppt 3) As a conclusion, it seems that the rainbow does not exist if nobody has the possibility to watch it, either as an observer, or because a picture would have been taken by a camera. It this last case, it would exist also without anybody looking at it, they would only watch its picture. 4) We can also ponder this classical philosophical question: “If a tree falls in the forest and no one is there to hear it, does it make a sound?”, a variation of "Can something exist without being perceived?" : http://www.answerbag.com/q_view/25568 Further information: "Observer dependent reality": http://forum.physorg.com/index.php?showtopic=9418&hl=

No. A rainbow is a spectra in a spatial sequence, defined by colors seen *by the observer*. If there are no observers, there are no colors therefore no rainbow...of course, there is still the spectra spatially separated. This is different to the tree falling in the forest, which does not require an observer to fall, make sound etc. Another clue is to consider if animals with only monochromatic or poly-chromatic vision see a rainbow as we see it?