Most textbooks and research papers define the comoving distance between comoving observers to be a fixed unchanging quantity independent of time, while calling the dynamic, changing distance between them "proper distance". On this usage, comoving and proper distances are numerically equal at the current age of the universe, but will differ in the past and in the future; if the comoving distance to a galaxy is denoted χ χ , the proper distance d ( t ) at an arbitrary time t is simply given by d ( t ) = a ( t ) χ χ where a ( t ) is the scale factor (e.g. Davis & Lineweaver 2004). The proper distance d ( t ) between two galaxies at time t is just the distance that would be measured by rulers between them at that time.