Do you think my boyfriend has been cheating on me?

Okay we've been together 6 months and he has always been so compionate and acts like he cares so much about me. But he got a new job 2 weeks ago and he doesn't have a specific time he gets off. Sometimes he gets home at 8,9,10,11. But he's been acting like he just doesn't care and I saw that his ex girlfriend had sent him an email basically saying she hopes he's doing good and hopes he's happy and sry too hurting him she also said people change and I have. Well I've been asking him why he's acting different and he keeps saying I'm not ily toy have nothing to worry about. He also knows I know about the message she sent him but we never even talked about until today. I finally came out and said have u been talking to your ex? He was like um NO! I said alright if I find out you have and u lied about it I'm done. Andbi I said ill just ask her he was like ok. Well after I said that he's been hugging and kissing on me acting like hour cares. And lately all he's done is play xbox and not even talk to me. I was about to go to the store after I said all that and he was like I want to go baby. Do u guys think he's been talking to her and now he's scared ill find out so he's being nice and caring again?