Constraints on $(\Omega_\mathsf{m},\Omega_\Lambda)$ using distributions of inclination angles for high redshift filaments

In this paper we present a scale free method to determine the cosmological parameters $(\Omega_{\rm m},\Omega_\Lambda)$. The method is based on the requirement of isotropy of the distribution of orientations of cosmological filaments. The current structure formation paradigm predicts that the first structures to form are voids and filaments, causing a web-like structure of the matter distribution at high redshifts. Recent observational evidence suggests that the threads, or filaments, of the cosmic web most easily are mapped in Ly α emission. We describe how such a 3D map can be used to constrain the cosmological parameters in a way which, contrary to most other cosmological tests, does not require the use of a standard rod or a standard candle. We perform detailed simulations in order to define the optimal survey parameters for the definition of an observing programme aimed to address this test, and to investigate how statistical and observational errors will influence the results. We conclude that observations should target filaments of comoving size $15{-}50$ Mpc in the redshift range $2{-}4$, and that each filament must be defined by at least four Ly α emitters. Detection of 20 filaments will be sufficient to obtain a result, while 50 filaments will make it possible to place significant new constraints on the values of $\Omega_{\rm m}$ and $\Omega_\Lambda$ permitted by the current supernova observations. In a future paper we study how robust these conclusions are to systematic velocities in the survey box.