Medics in northern France fear month-old baby has suffered brain damage after allegedly being shaken and beaten by father

French police have charged a father with child abuse after he allegedly shook and beat his month-old baby because he could not bear the infant's crying – and then posted a photo of the baby's bruised face on Facebook "for a laugh".

In a case that has shocked France, the man reportedly admitted being violent towards the infant, who was taken to hospital where doctors diagnosed possible brain damage.

Police in the town of Tergnier in the Aisne department in Picardy, northern France, were alerted on Monday by one of the couple's friends who saw the child's picture on the father's Facebook account and was concerned at the baby's "abnormal appearance" and bruised face.

Officers summoned the mother for questioning. When she arrived detectives noted the infant's injuries were even worse than in the photograph and alerted the emergency services. The mother allegedly told detectives she was afraid to contact police for fear that her partner would leave her.

The father was arrested on Monday evening. Both parents have been formally put under investigation, the equivalent of being charged, for "violence against a minor causing permanent incapacity".

The baby is believed to have suffered bleeding on the brain. Medics told Le Parisien they feared the child had suffered neurological damage though it was "too soon to say".

A police officer told the newspaper the father had explained that he hit the child because he could not bear its crying and that he had published the photograph on Facebook "pour s'amuser" (for a laugh).

Both parents were described as young – in their early 20s – and unemployed. They were released on police bail. Their child remains in hospital. The identities of the parents and child have not been released.