  Plasma-based accelerators (PBAs), having demonstrated the production of GeV
electron beams in only centimetre scales, offer a path towards a new generation
of highly compact and cost-effective particle accelerators. However, achieving
the required beam quality, particularly on the energy spread for applications
such as free-electron lasers, remains a challenge. Here we investigate
fundamental sources of energy spread and bunch length in PBAs which arise from
the betatron motion of beam electrons. We present an analytical theory,
validated against particle-in-cell simulations, which accurately describes
these phenomena. Significant impact on the beam quality is predicted for
certain configurations, explaining previously observed limitations on the
achievable bunch length and energy spread. Guidelines for mitigating these
contributions towards high-quality beams are deduced.
