To take advantage of the best prices you will need to prove you are a leisure traveller, on the way to heathrow airport, so you will need to provide either your flight details or yr holiday booking reference. I don't think you will need to give this when you arrive at the station but you may be asked to give them on your travels from the station to the airport.
Going to the airport on the train is great, you can take a nap, read (even at night), play travel scrabble, or just gaze out of the window... The other caveat to buying these cheap tickets from peterborough station to heathrow airport is that there must be at least two grown ups travelling together. But another great benefit is that children go free. All children under five go free, and then one child over 5 but under 16 per adult travelling by train to the airport.
be sure to leave enough time to get to the airport early enough to eat, shop and so on, so check the times of trains from peterborough station before you go. If you have time to kill at the airport you can always read a book or play travel scrabble again! As you have bought yr ticket in advance, you don't need to worry about the opening hours of the station.
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