on a rubber rope model" one imagines an ant (idealized as pointlike) crawling at a constant speed on a perfectly elastic rope that is constantly stretching. If we stretch the rope in accordance with the ΛCDM scale factor and think of the ant's speed as the speed of light, then this analogy is numerically accurate – the ant's position over time will match the path of the red line on the embedding diagram above. In the "rubber sheet model" one replaces the rope with a flat two-dimensional rubber sheet that expands uniformly in all directions. The addition of a second spatial dimension raises the possibility of showing local perturbations of the spatial geometry by local curvature in the sheet. In the "balloon model" the flat sheet is replaced by a spherical balloon that is inflated from an initial size of zero (representing the big bang). A balloon has positive Gaussian curvature while observations suggest that the real universe is spatially flat, but this inconsistency can be eliminated by making the balloon very large so that it is locally flat to within the limits of observation. This analogy is potentially confusing since it wrongly suggests that the big bang took place at the center of the balloon. In fact points off the surface of the balloon have no meaning, even if they were occupied by the balloon at an earlier time. In the "raisin bread model" one imagines a loaf of raisin bread expanding in the oven. The loaf (space) expands as a whole, but the raisins (gravitationally bound objects) do not expand; they merely grow farther away from each other. Theoretical basis and first evidence Hubble's law The metric expansion of space is a feature of many solutions to the Einstein field equations of general relativity, such as the Friedmann–Lemaître–Robertson–Walker metric and the Lemaître–Tolman–Bondi metric, where distance is measured using the Lorentz interval. This explains observations that indicate that galaxies that are more distant from us are receding faster than galaxies that are closer to us (see Hubble's law). Cosmological constant and the Friedmann equations The first general relativistic models predicted that a universe that was dynamical and contained ordinary gravitational matter would contract rather than expand. Einstein's first proposal for a solution to this problem involved adding a cosmological constant into his theories to balance out the contraction, in order to obtain a static universe solution. But in 1922