= Solution
A positive <cosmological constant> contributes a constant <vacuum energy> density and negative pressure. In the time-unit convention of Question 2,
$$
\rho_\Lambda=\frac{\Lambda}{8\pi G},\qquad p_\Lambda=-\rho_\Lambda c^2,\qquad
\frac{\ddot R}{R}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^2}\right)+\frac{\Lambda}{3}.
$$
It therefore causes <accelerating expansion> once its negative-pressure effect outweighs the decelerating components. Its density stays constant while matter dilutes, so a small early contribution can dominate late. For an isolated positive-vacuum component the <scale factor (cosmology)> grows exponentially. The <cosmological constant> problem is the mismatch between the small inferred vacuum density and generic quantum-field vacuum-energy expectations, rather than a failure of the background solution itself.
The key direct expansion evidence came from <Type Ia supernova> distance-redshift measurements. After light-curve and colour standardization, distant events were fainter than expected in simple decelerating matter-only models, indicating larger <luminosity distances>. Riess and collaborators' 1998 analysis and Perlmutter and collaborators' independent 42-supernova study favoured an accelerating low-matter-density cosmology with a positive vacuum contribution. Dust, source evolution, selection effects, <photometric calibration> and <peculiar velocities> must be controlled: faintness alone is not a model-independent measurement of $\Lambda$.<Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant by Adam Riess and others>[abstract; distance fitting and systematic-error discussion]{ref}<Measurements of Omega and Lambda from 42 High-Redshift Supernovae by Saul Perlmutter and others>[abstract; cosmological likelihoods and systematic checks]{ref}
<cosmic microwave background> acoustic geometry and independent matter-density constraints help break the supernova parameter degeneracy: nearly flat geometry with substantially less than critical matter requires another component within the FLRW model. <large-scale structure of the universe> and <distance> measurements provide further consistency tests. \b[The combined evidence supports a dark-energy contribution compatible with a constant; <accelerating expansion> by itself does not prove exactly constant <vacuum energy> or rule out every alternative gravity model.]
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