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We describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves, and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt).

Let A = (a 1 , . . . , a n ) be a vector of integers which sum to k(2g − 2 + n). The double ramification cycle DR g,A ∈ CH g (M g,n ) on the moduli space of curves is the virtual class of an Abel-Jacobi locus of pointed curves (C, x 1 , . . . , x n ) satisfyingThe Abel-Jacobi construction requires log blow-ups of M g,n to resolve the indeterminacies of the Abel-Jacobi map. Holmes [29] has shown that DR g,A admits a canonical lift logDR g,A ∈ logCH g (M g,n ) to the logarithmic Chow ring, which is the limit of the intersection theories of all such blow-ups.The main result of the paper is an explicit formula for logDR g,A which lifts Pixton's formula for DR g,A . The central idea is to study the universal Jacobian over the moduli space of curves (following Caporaso [12], Kass-Pagani [37], and Abreu-Pacini [3]) for certain stability conditions. Using the criterion of Holmes-Schwarz [34], the universal double ramification theory of Bae-Holmes-Pandharipande-Schmitt-Schwarz [5] applied to the universal line bundle determines the logarithmic double ramification cycle. The resulting formula, written in the language of piecewise polynomials, depends upon the stability condition (and admits a wall-crossing study). Several examples are computed.

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