Background material:
These are the parts that I can follow.
Previously Peskin & Schroeder have derived already the expression of the interaction ground state $|\Omega\rangle$ in terms of the free vacuum state $|0\rangle$:
$$|\Omega\rangle=\lim_{T\to \infty(1-i\epsilon)}\left(e^{-iE_0T}\langle\Omega|0\rangle\right)^{-1}e^{-iHT}|0\rangle\tag{4.27}$$
where $\epsilon $ is a positive real number.
We are postulating a time $t_0$ where the interaction field operator coincides with the free field operator, i.e. when the interaction was just about to be turned on:
$$\phi(t,\vec{x})=U^\dagger(t,t_0)\phi_I(t,\vec{x})U(t,t_0)\tag{4.16}$$
where $\phi_I$ is the free field operator, $U(t,t')$ is given by
$$U(t,t')=e^{iH_0(t-t_0)}e^{-iH(t-t')}e^{-iH_0(t'-t_0)}.\tag{4.25}$$
We are setting further that the free ground state energy is $0$. The interaction ground state energy is denoted $E_0$. With (4.25) then, we could move $T$ slightly to $T+t_0$ in (4.27) and obtain
$$|\Omega\rangle=\lim_{T\to\infty(1-i\epsilon)}\left(e^{-iE_0(t_0-(-T))}\langle\Omega|0\rangle\right)^{-1}U(t_0,-T)|0\rangle.\tag{4.28}$$
Problem to solve:
This is the part I cannot follow.
$$\langle\Omega|=\lim_{T\to\infty(1-i\epsilon)}\langle0|U(T,t_0)\left(e^{-iE_0(T-t_0)}\langle 0|\Omega\rangle\right)^{-1}.\tag{4.29}$$
Question I had when trying to solve the problem:
To obtain (4.29), it is certainly wrong to start from (4.28) because 1. we are sending $T$ to an imaginary number, so we need to take $T^*$ but because $\epsilon$ is so small this detour to imaginary part should not matter but even if this is the case 2. $e^{-iE_0(t_0+T)}$ should become $e^{iE_0(t_0+T)}$ which is NOT the expression in (4.29) and 3. $U^\dagger(t_0,-T)=U(-T,t_0)$.
So we better start from one step before (4.27), namely
$$e^{-iHT}|0\rangle=e^{-iE_0T}|\Omega\rangle\langle\Omega|0\rangle+\sum_{n\neq 0}e^{-iE_nT}|n\rangle \langle n|0\rangle.\tag{p.86}$$
Here when you move the ket state to its 'dual' in bra space you would get factor $e^{iE_nT}$. Now we need to take the same limit $T\to \infty(1-i\epsilon)$ for later use, but then the real part of power becomes $\mathcal{R}\to\epsilon E_n\infty$. This is not desirable because $\epsilon$ is real and positive and we would like to obtain something like $\mathcal{R}\to-\epsilon E_n\infty$ to ignore all terms containing $|n\rangle$.
So how did Peskin&Schroeder get to (4.29)? I am aware there are books that do differently from what they did, like Srednicki, and I know other approaches which give me Wick theorem but in this case I am just interested in how Peskin&Schroeder did it.
Best Answer
The key to derive the bra eq. (4.29) in P&S is to not just take the Hermitian conjugate of the ket eqs. (4.27)-(4.28) on p. 86 but also replace early time with late time $$-T~\to~ T.$$ In particular, the ket eq. before eq. (4.27) then tranform into the following bra eq.
$$\langle 0|e^{-iHT}~=~\langle 0|\Omega\rangle\langle\Omega|e^{-iE_0T} +\sum_{n\neq 0}\langle 0|n\rangle \langle n|e^{-iE_nT}.$$ Moreover, the $i\epsilon$ prescription then again leads to exponential suppression of excited states, as it should.