In certain circumstances a court can order support retroactive to the birth of the child. However if the court has already entered an order, it is very unlikely that they would go back.
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Given the facts expressed, and an assumption that you are not talking about arrears resulting from non- or under-payment between the time of the filing for the establishment or modification of support and the issuance of the support order, the short answer to your question is: no.
First, under Ohio law, a court can only order true retroactive support (e.g., reimbursement for birthing expenses, necessaries, etc.) at the same time time that paternity is established, and even then only under limited/qualifying circumstances.
Second, support can be "retroactive" in the sense that once a motion to establish or modify support is filed, the court can order the current support obligation "retroactive" to the date of filing. In such a situations, in addition to the amount of the current support obligation, courts order an amount specifically dedicated to liquidating the arrears, if there are any.
Under the scenario you've described, paternity has been established, a support order has issued, and you are asking about pursuing support, not for arrears between the time of requesting and establishing support, and not for non- or underpayment since the establishment of support, but rather for support prior to the effective date of the initial support order. I can't imagine a scenario where that would be possible.
Please note: my motto is "often wrong, never uncertain."