Palatini quadratic gravity: spontaneous breaking of gauged scale symmetry and inflation

We study $R^2$ gravity with a dynamical connection ($\tilde\Gamma^\alpha_{\mu\nu}$) in the Palatini formalism where the metric and connection are independent. The action has a gauged scale symmetry (Weyl gauging) of gauge field $v_\mu\propto \tilde\Gamma_\mu-\Gamma_\mu$, with $\tilde\Gamma_\mu$ ($\Gamma_\mu$) the trace of the Palatini (Levi-Civita) connection, respectively. In this case the associated geometry is non-metric. We show that the gauge field becomes massive by a gravitational Stueckelberg mechanism by absorbing the derivative of the dilaton $\partial_\mu\ln\phi$ (where the scalar $\phi$"linearises"the $R^2$ term). Palatini quadratic gravity with dynamical $v_\mu$ is then a gauged scale invariant theory broken spontaneously. In the broken phase one finds the Einstein-Proca action of $v_\mu$ of mass near the Planck scale ($M$) with a positive cosmological constant. Below this scale $v_\mu$ decouples, the connection becomes Levi-Civita and metricity and Einstein gravity are recovered. The results remain valid in the presence of non-minimally coupled matter. This is similar to recent results by the author for Weyl quadratic gravity, up to different non-metricity effects. When coupled to a scalar field, Palatini quadratic gravity gives successful inflation and a specific prediction for the tensor-to-scalar ratio $0.007\leq r \leq 0.01$ for current spectral index $n_s$ (at $95\%$CL) and $N=60$ efolds. This value of $r$ is mildly larger than in inflation in Weyl gravity, due to different non-metricity. This establishes a connection between non-metricity and inflation predictions and enables us to test these theories by future CMB experiments.


Introduction
At a fundamental level gravity may be regarded as a theory of connections. An example is the "Palatini approach" to gravity, due to Einstein [1,2] (hereafter EP approach). In this case the "Palatini connection" (Γ) is apriori independent of the metric (g αβ ) and hence different from the Levi-Civita connection (Γ) of the metric formulation. SometimesΓ plays an auxiliary role only, with no dynamics. However, this is not general, so here we study gravity actions in the EP approach [3,4] in whichΓ defines dynamical degrees of freedom.
To address this question we study actions with local scale symmetry, hence we consider a quadratic gravity action. For a dynamicalΓ this symmetry is a gauged scale symmetry or Weyl gauge symmetry 1 [5][6][7]. The goal is: 1) to explain the breaking of this symmetry and the emergence of Einstein gravity and Planck scale (even in the absence of matter), 2) study the relation of such action to Weyl theory [5][6][7] of similar symmetry; 3) study inflation.
Consider R(Γ, g) 2 gravity in the EP approach which is local scale invariant, as reviewed in Section 2. The connection is conformally related to the Levi-Civita connection. When "fixing the gauge" of this symmetry, the "auxiliary" scalar field φ introduced to "linearise" the R 2 term decouples. As a result, one finds thatΓ = Γ and Einstein action is obtained.
Consider now R(Γ, g) 2 gravity in whichΓ is dynamical. By this we mean that this action also contains a gauge kinetic term (−1/4)F 2 µν for the vector field v µ ∼Γ α µα − Γ α µα which measures the trace of the deviation ofΓ from Γ. WithΓ independent of the metric rescaling, the local scale symmetry of this action becomes a gauged scale symmetry, with gauge field v µ (also known as Weyl field) 2 . This symmetry, known from Weyl gravity [5][6][7], is important for mass scales generation, hence our interest. In this work (Sections 3 and 4) we study this action, hereafter called "EP quadratic gravity". Due to the dynamical connection, the EP quadratic gravity is 3 non-metric i.e.∇ µ g αβ = 0, with the trace of lhs proportional to v µ . The equations of motion of the connectionΓ become now second-order differential equations. In this case the usual EP approach in f (R) theories to solve algebraically forΓ [4] does not work, due to local scale symmetry and non-metricity. Even so, with the connection a function of the dilaton, we show that the action obtained forΓ onshell is a ghost-free, 2-nd order gauged scale invariant theory with a dilaton field.
An important result of this work is that the gauged scale invariance of the above action is broken by a gravitational Stueckelberg mechanism [62][63][64] (Section 3). The gauge field v µ becomes massive, with mass (m v ) near the Planck scale (M ), by "absorbing" the derivative of the dilaton field (∂ µ ln φ); when "gauge fixing" the Weyl gauge symmetry, near a scale M ∼ φ we obtain the Einstein-Proca action of v µ . Below the scale m v ∝ M , the field v µ decouples and we recover metricity, Levi-Civita connection and Einstein gravity. Then the Planck scale M is an emergent scale where this symmetry breaks. These results remain true if this theory has extra scalar fields (higgs, inflaton, etc) non-minimally coupled via Palatini connection, while respecting gauged scale invariance (Section 4). Briefly, the EP quadratic gravity with a dynamical connection is a gauged scale invariant theory brokenà la Stueckelberg to an Einstein-Proca action and a positive cosmological constant.
Another theory where the connection and the metric are independent is the original Weyl quadratic gravity of gauged scale invariance [5][6][7] (also [8]). Therefore, it is not too surprising that the above results are similar to those in [9][10][11] for Weyl theory. This theory came under early criticism from Einstein [5] for its non-metricity implying changes of atomic spectral lines, in contrast to experiment; however, if the Weyl "photon" (v µ ) of non-metricity is actually massive (mass ∼ M ), by the same Stueckelberg mechanism, metricity and Einstein gravity are recovered below its decoupling scale (∼ Planck scale). Non-metricity effects are then strongly suppressed by a large M (their current lower bound seems low [65,66]). Hence, criticisms that assumed v µ be massless are avoided and Weyl gravity is then viable [9][10][11]. As outlined, in this work we obtain similar results in Einstein-Palatini quadratic gravity, up to different non-metricity effects.
We also study inflation in EP quadratic gravity, with interesting results (Section 4). We consider this theory with an extra scalar field with non-minimal coupling and Palatini connection, that plays the role of the inflaton. We compute the inflaton potential after the gauged scale symmetry breaking. With the Planck scale as a phase transition scale, field values above M are natural. Interestingly, the inflaton potential is similar to that in Weyl quadratic gravity [11], up to couplings and field redefinitions due to a different nonmetricity. We show that inflation in EP quadratic gravity has a specific prediction for the tensor-to-scalar ratio 0.007 ≤ r ≤ 0.010 for the current spectral index n s (at 95% CL). This range is distinct from that in Weyl gravity theory [11,30] and will soon be reached by CMB experiments [67][68][69]. We thus establish an interesting connection between non-metricity and testable inflation predictions. Our Conclusions are in Section 5.
. This is the Palatini formulation of Einstein action; via eqs motion then∇µg αβ = 0 where∇µ is computed withΓ. HenceΓ is a Levi-Civita connection. However, this approach obscures the role of local scale symmetry, relevant later. 6 To see this, use that: with Levi-Civita connection Γ α µν = (1/2)g αλ (∂ µ g λν + ∂ ν g λµ − ∂ λ g µν ). From (10) one has with the Ricci scalar R(g) for g µν , and ∇ computed with the Levi-Civita connection (Γ). Using this in (3) of the same metric, then This is a second order theory with an additional ghost demanded by symmetry (4) and rather expected: using (11) in (1) then L 1 is a higher derivative action (forΓ onshell). Lagrangian (12) has a local scale symmetry so one may like to "fix the gauge". We choose the Einstein gauge, obtained after a particular transformation Ω 2 = ξ 0 φ 2 /(6M 2 ), taking φ to a constant (M is the Planck scale). Then L 1 becomes Hence Einstein action is recovered and the dilaton decoupled [72]. With φ "gauged fixed", then from (7) h µν ∝ g µν ,Γ = Γ and metricity is present 7 . Note that action (12) has a "fake" local scale symmetry [73][74][75] since its associated current is vanishing (this will change in Section 3), so it is not surprising φ trivially decoupled. This situation is unlike that in Riemannian case of R(g) 2 gravity [76] (eq.2.11), see also [77], where in the Einstein frame a kinetic term for φ remains present 8 .
3 Palatini quadratic gravity with gauged scale symmetry

The Lagrangian and its expression for onshellΓ
Consider now the EP quadratic gravity with a dynamical (trace of the) Palatini connection with , and Γ µ (g) does not contribute to the kinetic term. L 2 is quadratic in R but it resembles a second order theory. Unlike in the previous section,Γ µ ∼ v µ is now a dynamical field. The vector field v µ measures the trace of the deviation ofΓ from the Levi-Civita connection Γ(g).
As before, write L 2 in an equivalent form useful later on The equation of motion for φ has solution φ 2 = −R(Γ, g) which replaced in L 2 recovers (14). SinceΓ does not transform under (4) and with Γ µ (g) = ∂ µ ln √ g, then L 2 is invariant under (4) extended byv The invariance of L 2 under transformations (4), (17), is referred to as gauged scale invariance or Weyl gauge symmetry, with a (dilatation) group isomorphic to R + , as in Weyl gravity. Let us then compute the connectionΓ λ µν from its equation of motioñ Here∇ µ and ∇ µ are evaluated with the Palatini (Γ) and Levi-Civita (Γ) connections, respectively. Setting λ = ν and summing over gives (compare against (6)) Replacing (19) back in (18) leads tõ Notice that, had we not introduced φ in the first place to "linearise" the R 2 term in the action, (20) would have been similar but with φ 2 → −R; then (20) would be a second-order differential equation forΓ α µν , because ∂φ 2 ∼ ∂R ∼ ∂ 2Γ , with further complications. This shows the role of φ as an independent variable (no use of its equation of motion), in terms of which one easily computesΓ, as we do below. To find a solution, introduce [51] where V µ is some arbitrary vector field. To simplify notation, we use again our dimensionful metric h µν in terms of which eq.(20) is immediately re-expressed. Given its definition, h µν is invariant under (4) applied to φ and g µν . Replacing (21) in (20) the latter is indeed verified for arbitrary V µ . Hence we must findΓ from (21) 9 . One has found by multiplying (21) Hence, the theory is non-metric. From this we find the solution 10 with Levi-Civita Γ α µν (h) as in eq.(9). This will be used to compute the action forΓ onshell. As a remark, from (24) one hasΓ λ = Γ λ (h) + 2V λ ; then using (15), (22) As expected, v λ is the field of non-metricity Q λµν ≡∇ λ g µν , since the trace Q µ λµ = −4v λ . Non-metricity is here a consequence of a dynamical connection. Eq.(25) is similar to that in Weyl quadratic gravity, see e.g. [16]. Using this result for v λ , then solution (24) and also (23) are easily expressed in terms of v λ , φ and our metric g µν .
Using solution (24) we compute R µν (Γ) and then the scalar curvature R(Γ, h) or, in terms of our metric 11 g µν : R(h) and R(g) denote the usual Ricci scalars, related via the first three terms in the rhs. Finally, from (16) with (22), (25), (28) we find This Lagrangian hasΓ onshell and is gauged scale invariant, being invariant under (4), (17) for any Ω(x). L 2 is a second order ghost-free theory with a positive kinetic term for φ. 9 One cannot solve algebraically (21) as done in Palatini f (R) theories [3,4] due to non-vanishing rhs (dynamicalΓµ) and to the conformal symmetry of L2, absent in f (R) theories (see discussion in [78], p.5-6). 10 Use that∇ λ hµν = ∂ λ hµν −Γ ρ µλ hρν −Γ ρ νλ hµρ, for cyclic permutations of indices. 11 In terms of gµν one has for Rµν (which by contraction with g µν gives R(Γ, g) of (28)): This is relevant, since initial action (14) which was of second order is a four-derivative theory forΓ onshell, due to R(Γ, g) 2 term with (28); this has an equivalent second order formulation with additional φ in eq. (29). Lagrangian (29) is similar to that of Weyl quadratic gravity (up to a Weyl tensor-squared term [9,10]), but there non-metricity is assumed from the onset by the underlying Weyl conformal geometry, while here emerges after computingΓ from its equation of motion. If v µ = ∂ µ ln φ 2 (V µ = 0), v µ is "pure gauge", the situation is similar to Weyl integrable models and (29) recovers (12) of the previous section.

Stueckelberg breaking to Einstein-Proca action
Given L 2 in (29) with gauged scale symmetry we would like to "fix the gauge". We choose the Einstein gauge obtained from (29) by transformations (4), (17) of special Ω 2 = ξ 0 φ 2 /(6M 2 ), fixing φ to a constant. We find, after removing the hats (ˆ) on transformed g, v µ , etc, This is the Einstein-Proca action for the gauge field v µ with a positive cosmological constant, in which M was identified with the Planck scale. The initial gauged scale invariance is broken by a gravitational Stueckelberg mechanism [62][63][64]: the massless φ is not part of the action anymore, but v µ has become massive, after "absorbing" the derivative ∂ µ (ln φ) in eq.(29) of the Stueckelberg field (dilaton). Note the term ∂ µ (ln φ) is also the Goldstone of special conformal symmetry (this Goldstone is not independent but is determined by the derivative of the dilaton [79]). The number of degrees of freedom (dof) other than graviton is conserved in going from (29) to (30), as it should for spontaneous breaking: massless v µ and dynamical φ are replaced by massive v µ (dof=3). The mass of v µ is m 2 v = 6α 2 M 2 which is of the order of the Planck scale (unless one fine-tunes α ≪ 1).
Using the same transformation Ω, from (23) After the massive field v µ decouples, metricity is recovered below m v ,∇ λ g µν = 0 andΓ = Γ(g). To conclude, Einstein action is a "low energy" limit of the Einstein-Palatini quadratic gravity with dynamical connection and M is a phase transition scale (up to coupling α). For comparison, in Weyl quadratic gravity e.g. [9,10], non-metricity is different 12 Interestingly, the different non-metricity of these theories has phenomenological impact, see Section 4. In both theories the non-metricity scale m v ∼ Planck scale is large enough (above current bounds [65,66]) to suppresses unwanted effects e.g. atomic spectral lines spacing. Past critiques on non-metricity assumed a massless v µ .

Conserved current
Notice that, unlike in Section 2, eq. (19) shows there is now a non-trivial, conserved current due to a dynamical connection. In a similar context, for a Friedmann-Robertson-Walker metric with φ only t-dependent it was shown [43] that such current conservation naturally leads to φ=constant i.e. to a dynamical gauge fixing and a breaking of the symmetry. Then from (29) the Planck scale is identified to M 2 = ξ 0 φ 2 /6. Combining (33) with (21), one also has ∇ µ (V µ φ 2 ) = 0 which for constant φ is a condition similar to that for a Proca massive gauge field (leaving three degrees of freedom for v µ ).

Palatini quadratic gravity: additional fields and inflation
Consider now Palatini quadratic gravity coupled to a scalar field χ which may be the SM higgs or inflaton. We re-do the previous analysis in this case, then study inflation.

Adding matter
The starting Lagrangian of χ with gauged scale invariance, eqs.(4), (17), is then Under (4), (17) the Weyl-covariant derivative transforms asD µχ = (1/Ω)D µ χ. As in previous section, replace R(Γ, g) 2 → −2φ 2 R(Γ, g) − φ 4 , to find an equivalent L 3 where We replaced the scalar field φ by the radial direction field ρ 13 . The equation of motion forΓ α µν is similar to (18) but with a replacement φ → ρ and with an additional contribution from the kinetic term of χ. Following the same steps as in the previous section, we find This gives∇ where (40) one finds the Palatini connection with a result similar to (24) but with h µν ≡ ρ 2 g µν . We use this solution for the connection back in the action to find, forΓ onshell: This has a gauged scale symmetry and extents (29) in the presence of scalar fields. Finally, we choose the Einstein gauge by using transformation (4),(17) of a particular Ω = ρ/M , to new variables (with a hat) and find withD µχ = (∂ µ − 1/2v µ )χ and M identified with the Planck scale. As in the absence of matter, we obtained the Einstein-Proca action of a gauge field that became massive after Stueckelberg mechanism of "absorbing" the derivative term ∂ µ ln ρ. The mass of v µ is m 2 v = 6α 2 M 2 . A canonical kinetic term ofχ remains, since only one degree of freedom (ρ) is "eaten" by v µ . The potential becomes For a "standard" kinetic term forχ, similar to a "unitary gauge" in electroweak case, we remove the couplingv µ ∂ µχ from the covariant derivative in (42) by a field redefinition 13 ln ρ transforms as ln ρ → ln ρ − ln Ω and plays the role of the dilaton field.
which replacesχ → σ. We find In (45) one finally rescalesv ′ µ → αv ′ µ for a canonical gauge kinetic term. For small field values, σ ≪ M , the potential is Higgs-like, see also (43), and for ξ 1 > 0 has spontaneous breaking of the symmetry carried by σ e.g. electroweak symmetry if σ is the Higgs; this is triggered by the non-minimal coupling to gravity (ξ 1 = 0) and Stueckelberg mechanism. The negative mass term originates in (38) due to the φ 4 term (itself induced bỹ R 2 ). The mass m 2 σ ∝ ξ 1 M 2 /ξ 0 may be small enough by tuning ξ 1 ≪ ξ 0 . It may be interesting to study if the gauged scale symmetry brings some "protection" to m σ at quantum level.
This Lagrangian is similar to that in Weyl quadratic gravity with a non-minimally coupled scalar field [9][10][11], up to a rescaling of the couplings (ξ 1 , λ 1 ) and fields (σ). This difference is due to the different non-metricity of the two theories, eqs.(31), (32). Both cases give a gauged scale invariant theory of quadratic gravity coupled to matter, and recover Einstein gravity and metricity in their broken phase below the scale m v ∼ αM (α ≤ 1). This result may be more general and may apply to other theories with this symmetry.

Palatini R 2 inflation
For large field values, the potential in (46) can be used for inflation (hereafter Palatini R 2 inflation), with σ as the inflaton 14 . Since M is just a phase transition scale, field values σ ≥ M are natural here.V(σ) is similar to that studied in Weyl R 2 -inflation, see [11] for details 15 but, as mentioned, its couplings and field normalization differ (for similar initial couplings in the action); hence the spectral index n s and tensor-to-scalar ratio r are different, too, and need to be analyzed.
The potential is shown in Figure 1 for perturbative values of the couplings relevant for successful inflation. This demands λ 1 ξ 0 ≪ ξ 2 1 ≪ 1, with the first relation from demanding that the initial energy be larger than at the end of inflationV 0 >V min , respected by choosing a small enough λ 1 for given ξ 0,1 . Therefore, we shall work in the leading order in (λ 1 ξ 0 ).
The slow-roll parameters are Then where σ * is the value of σ at the horizon exit. With r = 16ǫ * we have 16 The contribution of ǫ is subleading for small ξ 1 considered here. The slope of the curves in the plane (n s , r), shown in leading order in (50), is steeper than in Weyl R 2 inflation [11] (or Starobinsky model) where r = 3(1 − n s ) 2 + O(ξ 2 1 ). The exact numerical results for (n s , r) in our model, for different e-folds number N , are shown in Figure 1. From experimental data n s = 0.9670 ± 0.0037 (68% CL) and r < 0.07 (95% CL) from Planck 2018 (TT, TE, EE + low E + lensing + BK14 + BAO) [89]. Using this data, Figure 1 (right plot) shows that a specific, small range for r is predicted in our model for the current range for n s at 95% CL: 16 There is an additional constraint on the parametric space from the normalization of the CMB anisotropy V0/(24π 2 M 4 ǫ * ) = κ0, κ0 = 2.1 × 10 −9 and r = 16ǫ * with r < 0.07 [89] then ξ0 = 1/(π 2 rκ) ≥ 6.89 × 10 8 . Condition λ1ξ0 ≪ ξ 2 1 in the text is then respected for any perturbative ξ1, 1/ξ0 by choosing small λ1 ≪ ξ 2 1 /ξ0.
Similar values for r can be read from Figure 1 for N between 55 to 65. The lower bound on r comes from that for n s while the upper one corresponds to a saturation limit (ξ 1 → 0). This range for r does not overlap with that in Weyl R 2 inflation (same n s , 95% CL) [11,30] The different predicted range for r is important since it enables us to distinguish these two models; this is ultimately due to their different non-metricity 17 . Such values for r will soon be reached by CMB experiments [67][68][69]. We thus established an interesting connection between non-metricity and testable inflation predictions. Similar values were found in other inflation models in Palatini R 2 gravity [87,88] which are not gauged scale invariant. Unlike in other successful models (e.g. Starobinsky model [92]) where higher curvature operators (R 4 , etc) of unknown coefficients bring corrections to r [93], such operators and corrections are not allowed here. They must be suppressed by some scale whose presence here would violate the gauged scale invariance. The only scale of the theory, from the dilaton field vev, is not available since this field was already "eaten" to all orders by the gauge field v µ . Also, given the gauged scale symmetry, Palatini R 2 inflation is allowed by black-hole physics (similar for Weyl R 2 inflation), in contrast to models of inflation with global scale symmetry 18 .

Conclusions
At a fundamental level gravity may be regarded as a theory of connections. An example is the Einstein-Palatini (EP) approach to gravity where the connection (Γ) is apriori independent of the metric. For simple actionsΓ plays an auxiliary role only (no dynamics), but this is not general. In this work we considered R(Γ, g) 2 gravity with a dynamical connection (Γ). This means that the action contains an additional gauge kinetic term for the vector field v µ defined by the trace of the difference between Palatini (Γ) and Levi-Civita (Γ) connections. This theory has a gauged scale symmetry, with v µ as the Weyl field ("photon"). Our motivation was: a) to study the breaking of this symmetry and emergence of Einstein gravity and Planck scale; b) to compare this theory to Weyl gravity of similar symmetry; c) to study inflation.
Due to the dynamical connection, the theory is non-metric, hence∇ µ g αβ = 0. While this theory is of second order, after solving it for the connection (Γ onshell) becomes a higher derivative theory (as in the metric case). This action is equivalent to a second order theory with an extra scalar field (dilaton), while preserving onshell the gauged scale symmetry.
One important result is that the gauged scale invariance of the action is broken by a gravitational Stueckelberg mechanism. The derivative of the dilaton field, ∂ µ ln φ is "eaten" by v µ which becomes massive. One obtains the Einstein-Proca action for the gauge field and a positive cosmological constant. This is a "low-energy" broken phase of the initial action with gauged scale symmetry. The gauge field v µ has a mass near the Planck scale (M ), m v ∼ αM (unless one is tuning the coupling α to α ≪ 1). Below the decoupling scale of v µ , metricity and the Einstein-Hilbert action are recovered. Non-metricity effects are strongly suppressed by a large scale ∝ M , which is relevant for the theory to be viable.
The above results remain valid in the presence of matter (higgs, inflaton, etc) nonminimally coupled to this theory, with a Palatini connection; in such case and following the Stueckelberg mechanism, the scalar potential can also have a breaking of the symmetry under which the scalar is charged. To conclude, Einstein-Palatini R(Γ, g) 2 gravity with dynamicalΓ µ is a gauged theory of scale invariance that is spontaneously broken to the Einstein-Proca action for the Weyl field, with positive cosmological constant.
This situation is very similar to a recent analysis by the author for the original Weyl quadratic gravity theory, despite the different non-metricity of these two theories. This is not too surprising, since in both cases the connection and the metric are varied independently, except that in Weyl gravity non-metricity is present from the onset (due to underlying Weyl conformal geometry) while here it emerges forΓ onshell. It would be interesting to study further the relation of these theories with such symmetry.
There are also interesting results for inflation. With the Planck scale M a simple phase transition scale, field values above M are natural. The inflaton potential is similar to that in Weyl quadratic gravity, up to couplings and field redefinitions due to the different non-metricity of the two theories. We find a specific prediction 0.007 ≤ r ≤ 0.01 for the tensor-to-scalar ratio, for the current value of the spectral index at 95% CL. This value of r is mildly larger than that predicted by Weyl R 2 inflation. This result enables us to distinguish and test these two theories by future CMB experiments. It also establishes an interesting connection between non-metricity and testable inflation predictions.