What is the relation between cobar duality and Feynman transform If $O$ is a cyclic operad, it can be regared as  a modular operad $P$ with $P(g,n)=0$, for $g >0$. So we have cobar dual $BO$ and Feynman transform $FP$(with trivial cocycle). Is there any relationship between them? I guess 
$Cyc(FP) = BO$
where $Cyc$ is the restricting or forgetting operator from modular operads to cyclic operads by keeping only its $g=0$ part.
Section 5.9 of Ginzburg-Kapranov has a related statement but there is a shift I don't understand.
Another related question is: what is the relationship if we replace cobar duality with dg duality? I am using the conventions of Getzler-Kapranov and Ginzburg-Kapranov.
 A: Your statement, up to the appropriate twists, is (5.9) in Getzler–Kapranov.
But let me take this opportunity to advertise the framework in Ben Ward's paper Six operations formalism for generalized operads that demonstrates a slightly more general result as a coherent part of a more general theory of passing among different types of operad-like structures. 
One part of Corollary 8.2 implies the statement that you made. This part of the Corollary (which is also Corollary 1.4) says that the modular envelope of $BO$ is naturally isomorphic to the Feynman transform of the ``extension by zero'' of $O$ to a modular operad (taking appropriate twists into account). Using Ward's notation, your $P$ is $L^!O$ and your $FP$ is $DL^!O$ while your $BO$ is $DO$. Ward's statement, $DL^!O\cong LDO$, is at the level of modular operads, but applying the truncation ($Cyc$ in your notation, $R$ in Ward's), yields 
$$
Cyc(FP)=RDL^!O\cong RLDO\cong DO=BO
$$
because $RL$ is naturally isomorphic to the identity.
Ward also gives a statement that he considers more obvious and less interesting, that $DR\cong RD$. This statement means, in your notation, that if you start with the modular operad $P$ instead of starting with the cyclic operad, that
$$
Cyc(FP)=RDP\cong DRP=B(Cyc(P)).
$$
The notation for $D$ (including twists) is discussed in Appendix A.2, using the notation of this paper by Kaufmann and Ward.
