20 C.F.R. § 404.359
You may be eligible for benefits as an equitably adopted child if the insured had agreed to adopt you as his or her child but the adoption did not occur. The agreement to adopt you must be one that would be recognized under State law so that you would be able to inherit a child's share of the insured's personal property if he or she were to die without leaving a will. The agreement must be in whatever form, and you must meet whatever requirements for performance under the agreement, that State law directs. If you apply for child's benefits after the insured's death, the law of the State where the insured had his or her permanent home at the time of his or her death will be followed. If you apply for child's benefits during the insured's life, the law of the State where the insured has his or her permanent home at the time or your application will be followed.
20 C.F.R. §404.359