The Ferraris-Kijowski purely affine Lagrangian for the electromagnetic field, that has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the metric Einstein-Maxwell Lagrangian, except the zero-field limit, for which the metric tensor is not well-defined. This feature indicates that, for the Ferraris-Kijowski model to be physical, there must exist a background field that depends on the Ricci tensor. The simplest possibility, supported by recent astronomical observations, is the cosmological constant, generated in the purely affine formulation of gravity by the Eddington Lagrangian. In this paper we combine the electromagnetic field and the cosmological constant in the purely affine formulation. We show that the sum of the two affine (Eddington and Ferraris-Kijowski) Lagrangians is dynamically inequivalent to the sum of the analogous (ΛCDM and Einstein-Maxwell) Lagrangians in the metric-affine/metric formulation. We also show that such a construction is valid, like the affine Einstein-Born-Infeld formulation, only for weak electromagnetic fields, on the order of the magnetic field in outer space of the Solar System. Therefore the purely affine formulation that combines gravity, electromagnetism and cosmological constant cannot be a simple sum of affine terms corresponding separately to these fields. A quite complicated form of the affine equivalent of the metric Einstein-Maxwell-Λ Lagrangian suggests that Nature can be described by a simpler affine Lagrangian, leading to modifications of the Einstein-Maxwell-ΛCDM theory for electromagnetic fields that contribute to the spacetime curvature on the same order as the cosmological constant.

Gravitation, Electromagnetism and Cosmological Constant in Purely Affine Gravity

Affine gravity is a connection-based formulation of gravity that does not involve a metric. After a review of basic properties of affine gravity, we compute the tree-level scattering amplitude of scalar particles interacting gravitationally via the connection in a curved spacetime. We find that, while classically equivalent to general relativity, affine gravity differs from metric quantum gravity.

Scattering amplitudes in affine gravity

The main aim of this thesis is to reveal some interesting aspects of the purely affine theory of gravity and its cosmological implication. A particular attention will be devoted to its consequences when applied to cosmological inflation. Primarily, affine spacetime, composed of geodesics with no notion of length and angle, accommodates gravity but not matter. The thesis study is expected to reveal salient properties of matter dynamics in affine spacetime and may reveal an intimate connection between vacuum state and metrical gravity. An interesting application of the framework is the inflationary regime, where it is shown that affine gravity prefers only a unique metric tensor such that the transition from nonminimal to minimal coupling of the inflaton is performed only via redefinition of the latter. This allows us to avoid the use of the so called conformal frames. In fact, unlike metric gravity, the metric tensor in affine gravity is generated and not postulated a priori, thus this tensor is absent in the actions and conformal transformation does not make sense. Last but not least, we try to show how metric gravity can be induced through a simple structure that contains only affine connection and scalar fields. General relativity arises classically only at the vacuum, and this view of gravity may be considered as a new way to inducing metric elasticity of space, not through quantum corrections as in standard induced gravity, but only classically. The thesis is concluded by analyzing affine gravity in a particular higher-dimensional manifold (product of two spaces) in an attempt to understand both, the cosmological constant and matter dynamically.

/pdf/cosmological-implications-of-affine-gravity-245zg6ce8q.pdf

Cosmological Implications of Affine Gravity

Here we concisely review the nonminimal coupling dynamics of a single scalar field in the context of purely affine gravity and extend the study to multifield dynamics The coupling is performed via an affine connection and its associated curvature without referring to any metric tensor The latter arises a posteriori and it may gain an emergent character like the scale of gravity What is remarkable in affine gravity is the transition from nonminimal to minimal couplings which is realized by only field redefinition of the scalar fields Consequently, the inflationary models gain a unique description in this context where the observed parameters, like the scalar tilt and the tensor-to-scalar ratio, are invariant under field reparametrization Overall, gravity in its affine approach is expected to reveal interesting and rich phenomenology in cosmology and astroparticle physics

Are there really conformal frames? Uniqueness of affine inflation

The standard model of elementary particles (SM) suffers from various problems, such as power-law ultraviolet (UV) sensitivity, exclusion of general relativity (GR), and absence of a dark matter candidate. The LHC experiments, according to which the TeV domain appears to be empty of new particles, started sidelining TeV-scale SUSY and other known cures of the UV sensitivity. In search for a remedy, in this work, it is revealed that affine curvature can emerge in a way restoring gauge symmetries explicitly broken by the UV cutoff. This emergent curvature cures the UV sensitivity and incorporates GR as symmetry-restoring emergent gravity ({\it symmergent gravity}, in brief) if a new physics sector (NP) exists to generate the Planck scale and if SM+NP is fermi-bose balanced. This setup, carrying fingerprints of trans-Planckian SUSY, predicts that gravity is Einstein (no higher-curvature terms), cosmic/gamma rays can originate from heavy NP scalars, and the UV cutoff might take right value to suppress the cosmological constant (alleviating fine-tuning with SUSY). The NP does not have to couple to the SM. In fact, NP-SM coupling can take any value from zero to $\Lambda^2_{SM}/\Lambda^2_{NP}$ if the SM is not to jump from $\Lambda_{SM}\approx 500\, {\rm GeV}$ to the NP scale $\Lambda_{NP}$. The zero coupling, certifying an undetectable NP, agrees with all the collider and dark matter bounds at present. The {\it seesawic} bound $\Lambda^2_{SM}/\Lambda^2_{NP}$, directly verifiable at colliders, implies that: {\it (i)} dark matter must have a mass $\lesssim \Lambda_{SM}$, {\it (ii)} Higgs-curvature coupling must be $\approx 1.3\%$, {\it (iii)} the SM RGEs must remain nearly as in the SM, and {\it (iv)} right-handed neutrinos must have a mass $\lesssim 1000\, {\rm TeV}$. These signatures serve as a concise testbed for symmergence.

/pdf/symmergent-gravity-seesawic-new-physics-and-their-2zfrjxwu0w.pdf

Symmergent Gravity, Seesawic New Physics, and their Experimental Signatures

Universality of affine formulation in general relativity

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Universality of affine formulation in general relativity