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\section{Introduction}  Our knowledge about Neanderthals has greatly increased over the last two decades with more than two thousand research papers published in areas as diverse as chronology \cite{Gaudzinski_Windheuser_2014, Higham_2014}, ecology \cite{Finlayson_2007, Henry_2010}, population dynamics \cite{S_rensen_2011, Bocquet_Appel_2013}, adaptive traits \cite{S_rensen_2009, Rae_2011}, diet \cite{Henry_2010, Hardy_2011}, technology and technology,  cognition \cite{Fern_ndez_Peris_2012, and behavior \cite{Shaw2012, Fern_ndez_Peris_2012,  Riel_Salvatore_2010}, genetics \cite{Briggs_2009, Sanchez_Quinto_2014}, and their relationship with anatomically modern humans \cite{Gibbons_2014, Sankararaman_2012}. \cite{Sankararaman_2012}.  Still, we believe there is a lack of tangible information about the factors controlling the Neanderthals' distribution. For example, \citet{Wenzel_2007} describes the distribution of Neanderthals during MIS 5e across central Europe, and briefly discusses the influence of climatic conditions and habitat features over the general occupancy pattern, but does not establish any quantitative link between presence and environment. \citet{Banks_2008} analyzed the distribution of Neanderthals at the interphase of the Neanderthal/modern human transition applying palaeo species distribution models (PSDMs hereafter), assessed niche conservatism, and analyzed the importance of climatic predictors, concluding that temperature was the most important driver shaping the distribution Neanderthal at this time. In this paper we propose a hypothesis about how abiotic drivers may have shaped Neanderthals distribution that is rooted in the Grinnellian niche concept \cite{Grinnell_1917}, the hierarchical framework proposed by \citet{Pearson_2003}, and our current knowledge on the ecology of Neanderthals. According to \citet{Pearson_2003}, climate influences species distribution at global and continental scales, while the effect of topography is restricted to scales ranging from regional to local. Our hypothesis consist of three main points: 1) the northern edge of the Neanderthals range was limited by low winter temperatures; 2) the southern edge was shaped by a combination of high temperature and low water availability during the summer; 3) high topographic diversity combined with moderate slopes could have favored occupation at the local scale. Winter temperatures at the northern edge could have resulted in a lower availability of small and big game than in temperate areas \cite{Badgley_2000}, compromising the high caloric intake required by this species \cite{Steegmann_2002, S_rensen_2009}, and an increased cold stress accelerated by a low-caloric diet, that would have lead to reduced fertility \cite{Bocquet_Appel_2013} and a higher mortality rate \cite{Steegmann_2002}. Summer temperatures linked to continentality and the higher solar radiation of lower latitudes could have prevented the occupation of southern plains in the Mediterranean peninsulas due to increased heat stress, specially considering the low body surface area/volume ratio of this species \cite{Churchill2006}, which hampers heat dissipation. Also, high evapotranspiration could have reduced the diversity of plants, since under-storey and forest communities are less common under such climatic conditions, hampering the access to plant resources, and making these areas unsuitable during the warm season. Mediterranean coastal areas could have been suitable because of the buffering effect of the sea over temperature and the permanent availability of resources like shellfish \cite{Hardy_2011}. At the local scale, high topographic diversity, which fosters biodiversity by an increased availability of ecological niches \cite{Tews_2003}, could have provided the required abundance and diversity of prey and shelter \cite{Daujeard_2012}, but moderate to low slopes may have been important to reduce the high energetic expenditure of mobility in steep terrain. This limitation, combined with the lower abundance of animals at higher elevations \cite{Brown_2001}, could have prevented Neanderthal's presence in the higher elevations of the European mountain ranges.