The 31-year-old, who has recently been on the sidelines with a three-match ban, learned his fate during a scan this week.
"In the last couple of weeks Peter has been saying that he hasn't been able to function 100% and while he's not in a great deal of pain, he has been getting aches and soreness," physio Andy Thorpe told the club's official website.
"We sent him for a scan and it showed disruption in his Achilles. After seeing the surgeon, he has been told that he it will need up to six weeks to heal, which is obviously a blow for him and us.
"Taking time for it to heal is important because if he was to carry on then the disruption could develop into more and he could get a complete rupture. We don't want to go down that road, so we're taking the best possible option we can."
Rochdale currently occupy 13th spot in League Two.