jats:titleAbstract</jats:title>jats:pWe generalise Einstein’s formulation of the traceless Einstein equations to jats:italicf</jats:italic>(jats:italicR</jats:italic>) gravity theories. In the case of the vacuum traceless Einstein equations, we show that a non-constant Weyl tensor leads via a conformal transformation to a dimensionally homogeneous (‘no-scale’) theory in the conformal frame with a scalar field source that has an exponential potential. We then formulate the traceless version of jats:italicf</jats:italic>(jats:italicR</jats:italic>) gravity, and we find that a conformal transformation leads to a no-scale theory conformally equivalent to general relativity and a scalar field jats:inline-formulajats:alternativesjats:tex-math$$\phi $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miϕ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> with a potential given by the scale-invariant form: jats:inline-formulajats:alternativesjats:tex-math$$V(\phi )=\frac{D-2}{4D}Re^{-\phi }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miV</mml:mi> mml:mrow mml:mo(</mml:mo> mml:miϕ</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mo=</mml:mo> mml:mfrac mml:mrow mml:miD</mml:mi> mml:mo-</mml:mo> mml:mn2</mml:mn> </mml:mrow> mml:mrow mml:mn4</mml:mn> mml:miD</mml:mi> </mml:mrow> </mml:mfrac> mml:miR</mml:mi> mml:msup mml:mie</mml:mi> mml:mrow mml:mo-</mml:mo> mml:miϕ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, where jats:inline-formulajats:alternativesjats:tex-math$$\phi =[2/(D-2)]\ln f^{\prime }(R)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miϕ</mml:mi> mml:mo=</mml:mo> mml:mrow mml:mo[</mml:mo> mml:mn2</mml:mn> mml:mo/</mml:mo> mml:mrow mml:mo(</mml:mo> mml:miD</mml:mi> mml:mo-</mml:mo> mml:mn2</mml:mn> mml:mo)</mml:mo> </mml:mrow> mml:mo]</mml:mo> </mml:mrow> mml:moln</mml:mo> mml:msup mml:mif</mml:mi> mml:mo′</mml:mo> </mml:msup> mml:mrow mml:mo(</mml:mo> mml:miR</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. In this theory, the cosmological constant is a mere integration constant, statistically distributed in a multiverse of independent causal domains, the vacuum energy is another unrelated arbitrary constant, and the same is true of the height of the inflationary plateau present in a huge variety of potentials. Unlike in the conformal equivalent of full general relativity, flat potentials are found to be possible in all spacetime dimensions for polynomial lagrangians of all orders. Hence, we are led to a novel interpretation of the cosmological constant vacuum energy problem and have accelerated inflationary expansion in the very early universe with a very small cosmological constant at late times for a wide range of no-scale theories. Fine-tunings required in traceless general relativity or standard non-traceless jats:italicf</jats:italic>(jats:italicR</jats:italic>) theories of gravity are avoided. We show that the predictions of the scale-invariant conformal potential are completely consistent with microwave background observational data concerning the primordial tilt and the tensor-to-scalar ratio.</jats:p>