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Let $X$ be a simply connected CW-complex of finite type and $\mathbb{K}$ anyfield. A first known lower bound of LS-category $cat(X)$ is the Toomerinvariant $e_{\mathbb{K}} (X)$ (\cite{Too}). In $1980$'s F\'elix et al.introduced the concept of {\it depth} in algebraic topology and proved thedepth theorem: $depth (H_*(\Omega X, \mathbb{K})) \leq cat(X)$.  In this paper, we use the Eilenberg-Moore spectral sequence of $X$ tointroduce a new numerical invariant, denoted by $\textsc{r}(X, \mathbb{K})$,and show that it has the same properties as those of $e_{\mathbb{K}} (X)$.  When the evaluation map (\cite{FHT88}) is non-trivial and$char(\mathbb{K})\not = 2$, we prove that $\textsc{r}(X, \mathbb{K})$interpolates $depth(H_*(\Omega X, \mathbb{K}))$ and $e_{\mathbb{K}} (X)$.Hence, we obtain an improvement of L. Bisiaux theorem (\cite{Bis99}) and thenof the depth theorem.  Motivated by these results, we associate to any commutative differentialgraded algebra $(A,d)$, a purely algebraic invariant $\textsc{r}(A,d)$ and, viathe theory of minimal models, we relate it with our previous topologicalresults.  In particular, if $(\Lambda V,d)$ is a Sullivan minimal algebra such that$d=\sum_{i\geq k}d_i$ and $d_i(V)\subseteq \Lambda ^iV$, a greater lower boundis obtained, namely $e_0(\Lambda V, d)\geq \textsc{r}(\Lambda V, d) + (k-2)$.

A new invariant that's a lower bound of LS-category