To take benefit of the best prices you will need to prove you are not a business traveller, en route to the airport, so you will need to give either yr flight numbers or your 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 provide them on your travels from peterborough station to the airport.
Going to luton airport by train is great, you can sleep, read a book (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 the airport is that there must be at least two adults travelling together. But another great advantage is that children go free. All children under 5 go free, and then one child over 5 but under sixteen per grown up travelling by train to the airport.
make sure you leave enough time to get to the airport early enough to eat, shop and so on, so check the times of trains from the station before you go. If you end up with spare time at the airport you can always read a book or play travel scrabble again! As you have bought yr train ticket in advance, you don't need to worry about the opening hours of peterborough station.
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