Neither federal law nor the law of most states requires that the landlord must prorate the rental amount if you start the tenancy in the middle of the monthly period (assuming rent paid monthly). Under the common law the basic rule for a periodic tenancy was that the tenant owes the rent for the full period (e.g. a month in a month-to-month rental) if the tenant occupies the premises for any part of that month.
California has a more extensive landlord-tenant law than most states, and you can read your various rights from the California Consumer Affairs Department here: California Tenants Guide. But as detailed as those rules are, there appears to be no requirement that the landlord prorate the rent if you move into the place in the middle of the month. So California may retain the old common law rule on this. But that rule may be altered by the agreement of the parties. This is why I said it depends on what the lease says. This is a point you want to negotiate with the landlord prior to entering into the lease. If you get a clear agreement on this then everyone knows what is expected and you significantly reduce the chances of conflict over it. If your rental is in a city in California that imposes rent control then it is possible that the rent control ordinance would require prorated rent, though.

