Inspired by nonlocal gravity theories, time-delayed cosmology proposes a delayed Friedmann equation that generically predicts an inflationary period with a natural end. The key parameter of this proposal is a time delay that is presumed to be very small in order for the model to evade potential astrophysical constraints. This work subjects this small-delay assumption to a test. We address the question of just how large a time delay can be accommodated within our current cosmological data. In order to do so, we do not restrict the model to the inflationary era and consider its possible operation in the late Universe as well, with an eye for any smoking-gun features that may indicate the presence of a time delay. We study the background evolution predicted by the delayed Friedmann equation and determine the growth of Newtonian perturbations in this delayed background. We show that a surprisingly large late-time cosmic delay is statistically consistent with Hubble expansion rate and growth data. Based on these observables, we also find that the standard $$\varLambda $$ Λ CDM model has no advantage over time-delayed cosmology in terms of the Bayes factor.

Motivated by the proposed time-delayed cosmology in the primordial inflationary era, we consider the application of the delayed Friedmann equation in the late-time Universe and explore some of its observable consequences. We study the background evolution predicted by the delayed Friedmann equation and determine the growth of Newtonian perturbations in this delayed background. We reveal smoking-gun imprints of time-delayed cosmology that can be traced to derivative discontinuities generic in delay differential equations. We show that a late-time cosmic delay is statistically consistent with Hubble expansion rate and growth data. Based on these observables, we compute a nonzero best estimate for the time delay parameter and find that the Bayesian evidence does not strongly rule out a late-time time delay but warrants the subject further study.