Approximate solutions of Jordan–Brans–Dicke (JBD) theory for perturbed scalar field and perturbed Robertson–Walker metric, were found for an era dominated by the scalar field. Solutions for the scale factor a(t) and the scalar field $$\phi (t)$$ ϕ(t) in unperturbed JBD theory are dependent on the $$\omega $$ ω parameter which determines how the scalar field is coupled to geometry of space–time. After adding metric perturbation $$h_{\mu \nu }(x)$$ hμν(x) to Robertson–Walker metric and perturbation $$\delta \phi (x)$$ δϕ(x) to the scalar field $$\phi (t)$$ ϕ(t) , we solved the JBD equations such that the scale factor and the scalar field solutions are $$a\propto t$$ a∝t and $$\phi \propto t^{-2}$$ ϕ∝t-2 with $$\omega =-3/2$$ ω=-3/2 . These results are necessary conditions for ordinary and scalar gravitational waves to exist in the scalar field-dominated era. Despite our result contradicts the value $$\omega >10^4$$ ω>104 which is favored by the current solar system environment observations, $$\omega =-3/2$$ ω=-3/2 makes JBD theory conformally invariant and fits recent supernovae type Ia data.