Is there a neutral word for the sense that something is ever-present and close by?

Like 'omnipresent' but not currently seen, heard etc. rather knowing the chance of the presence of something/someone could be near.

Examples---

I won't go swimming because of the ____ of a dangerous animal.

I feel safe because of the ____ of the police.

It's not a word I'd use in everyday conversation (except tongue-in-cheek) as it falutes too highly, but omnipresence has the desired meaning here, at least according to Merriam-Webster:

omnipresence: the quality or state of being omnipresent: ubiquity

omnipresent : present in all places at all times

Other dictionaries tend to be less explicit about the 'at all times' requirement. The M-W definition warrants an answer, I feel, though this word has been mentioned in comments [@FumbleFingers] and fits the title question but not the examples (far too formal).