Outlook

Concluding Remarks & Outlook

Double Field Theory (DFT) has developed into a self-sustaining autonomous theory of gravity, offering a compelling alternative to General Relativity (GR). Governed by the \(\ODD\) symmetry principle rooted in string theory, DFT uniquely fixes the self-interaction of the gravitational sector, \(\{\cH_{AB}, d\}\) or the Riemannian trio \(\{g, B, \phi\}\), as well as its minimal coupling to other sectors or additional matter, placing it in a distinct position among various modified theories of gravity. The rigorously defined structure of DFT makes it inherently falsifiable, a critical feature for endowing theoretical physics with value.

The spherically symmetric solution (3.1) has been evaluated against solar system experiments using the Parametrised Post-Newtonian (PPN) formalism. Unlike General Relativity (GR), where Birkhoff's theorem holds and a single component of the energy-momentum tensor determines the geometry, Double Field Theory incorporates multiple components (3.13), enabling the exploration of more complex phenomena. For example, studying the geometry outside compact objects or stars may offer insights into their generalised equation of state (3.15) or the generalised gravitational form factor within hadrons, expressed in terms of \(K_{\mu\nu}\) and \(\To\) rather than the traditional \(T_{\mu\nu}\). Notably, recent experimental findings, such as the discovery of immense pressure inside protons [Burkert:2018bqq], challenge traditional assumptions of matter being "cold" and may underscore the relevance of DFT's multi-component energy-momentum tensors. While these possibilities are intriguing, solar system tests of DFT remain inconclusive, necessitating further investigations, including insights from nuclear physics.

On cosmological scales, the open-Universe solution given through (3.16 - 3.19) captures accelerating expansion and demonstrates strong consistency with late-time cosmological observations, including type Ia supernovae and quasars. However, the theory's applicability to the early Universe remains uncertain. The non-convergent phase of the string dilaton during the early Universe suggests a nontrivial coupling to electromagnetism (2.152), (3.24), which may influence light propagation or null geodesics, thereby affecting measurements of luminosity distance. Further corroboration, incorporating diverse data sources such as the cosmic microwave background [Planck:2018vyg], is essential to clarify DFT's role in explaining the early Universe.

In terms of symmetry, DFT distinguishes itself by being approximately twice as symmetrical as General Relativity. This profound structure invites speculation about whether Nature and the Universe might harness such symmetry. As a falsifiable prediction of string theory, DFT stands as a promising candidate awaiting experimental verification.

Beyond its phenomenological significance, DFT, as the gravitational framework of string theory, faces the critical challenge of addressing extra dimensions and higher-derivative corrections. A compelling and still largely unexplored possibility regarding the former is whether these extra dimensions could take a non-Riemannian form [Morand:2017fnv, Cho:2018alk]. Regarding the latter, some pioneering contributions include [Hohm:2013jaa, Bedoya:2014pma, Hohm:2014xsa, Marques:2015vua].

Lastly, the quantisation of Double Field Theory in an \(\ODD\)-symmetric framework remains a vital and promising avenue for further investigation.

While the aforementioned topics represent potential future works within the limitations of the author's imagination, the over 100-year legacy of GR suggests that DFT could unlock far broader and richer gravitational-related research directions. The development of DFT is an ongoing process, with immense potential to deepen our understanding of gravity.