Gauging scale symmetry and inflation: Weyl versus Palatini gravity

We present a comparative study of inflation in two theories of quadratic gravity with {\it gauged} scale symmetry: 1) the original Weyl quadratic gravity and 2) the theory defined by a similar action but in the Palatini approach obtained by replacing the Weyl connection by its Palatini counterpart. These theories have different vectorial non-metricity induced by the gauge field ($w_\mu$) of this symmetry. Both theories have a novel spontaneous breaking of gauged scale symmetry, in the absence of matter, where the necessary scalar field is not added ad-hoc to this purpose but is of geometric origin and part of the quadratic action. The Einstein-Proca action (of $w_\mu$), Planck scale and metricity emerge in the broken phase after $w_\mu$ acquires mass (Stueckelberg mechanism), then decouples. In the presence of matter ($\phi_1$), non-minimally coupled, the scalar potential is similar in both theories up to couplings and field rescaling. For small field values the potential is Higgs-like while for large fields inflation is possible. Due to their $R^2$ term, both theories have a small tensor-to-scalar ratio ($r\sim 10^{-3}$), larger in Palatini case. For a fixed spectral index $n_s$, reducing the non-minimal coupling ($\xi_1$) increases $r$ which in Weyl theory is bounded from above by that of Starobinsky inflation. For a small enough $\xi_1\leq 10^{-3}$, unlike the Palatini version, Weyl theory gives a dependence $r(n_s)$ similar to that in Starobinsky inflation, while also protecting $r$ against higher dimensional operators corrections.


Motivation
In this work we present a comparative study of inflation in two theories of quadratic gravity that have a gauged scale symmetry also known as Weyl gauge symmetry. This symmetry was first present in the original Weyl quadratic gravity [1][2][3] (for a review [4]) that follows from an underlying Weyl conformal geometry. This is relevant in early cosmology when effective theories at short distances may become conformal. Due to their symmetry, these theories have no mass scales or dimensionful couplings -these must be generated by the vacuum expectations values (vev) of the fields and this is the view we adopt here.
The first theory is the original Weyl quadratic gravity revisited recently in [5,6] with new results. This was in fact the first gauge theory (of scale invariance) 1 . The second theory [7] has a similar action but in the Palatini formalism [8][9][10][11], which means replacing the Weyl connection by the Palatini connection. In the absence of matter the Lagrangian has the form whereΓ is the Weyl or Palatini connection, respectively and ξ 0 and α are constants. These terms involve the (scalar and tensor) curvatures R and R µν which are functions ofΓ; note thatΓ is not determined by the metric g µν . This is the minimal action with such gauge symmetry. More quadratic terms may be present in both cases, see later.
In both theories the connection (Γ) is Weyl gauge invariant. Hence this is not only a symmetry of the action, but also of the underlying geometry. Both theories have vectorial non-metricity which is due to the dynamics of the gauge field w µ of scale symmetry 2 ; w µ is dynamical since forΓ symmetric (which we assume to be the case) the term R 2 [µν] ∼ F 2 µν is just a gauge kinetic term of w µ . And if w µ is not dynamical it can easily be integrated out and both theories are Weyl integrable and metric (∇ µ g µν = 0), see e.g. [5,7]. In both theories the Weyl gauge field is related to the trace of non-metricity: w µ ∝ g αβ∇ µ g αβ wherẽ ∇ is computed with the Weyl or Palatini connection. The two theories have however a different non-metricity tensor; this leads to different inflation predictions that we discuss. We thus have a link between non-metricity and inflation predictions. Our study of these two theories with gauged scale symmetry is motivated by: a) In the absence of matter both theories of quadratic gravity have spontaneous breaking of this symmetry as it was shown for the first time in [5] for Weyl quadratic theory and in [7] for the Palatini theories. In both cases the Einstein-Proca action of w µ and the Planck scale emerge in the broken phase, after w µ becomes massive by "eating" the Stueckelberg field (would-be Goldstone/dilaton); this is the field that "linearises" R(Γ, g) 2 in the action, as we shall detail. After w µ decouples near the Planck scale M ∼ dilaton , the Einstein action is naturally obtained (together with metricity, see below) 3 . Thus, these theories provide a natural mass generation (Planck and w µ masses) via a symmetry breaking mechanism.
The above result is important since it shows a new mechanism of spontaneous breaking of scale symmetry (in the absence of matter) in which the necessary scalar field is not added ad-hoc to this purpose (as usually done); instead, the Stueckelberg field is here of geometric origin, being "extracted" from the R(Γ, g) 2 term. This situation is very different from previous studies that used instead e.g. modified versions of Weyl action that were linear-only in R and/or used additional matter field(s) to generate the Planck scale [16][17][18][19][20][21][22][23][24][25][26].
b) The breaking of Weyl gauge symmetry mentioned at a) is accompanied by a change of the underlying geometry (connection). For example in the Weyl theory after w µ becomes massive it decouples, the Weyl connection becomes Levi-Civita, thus the underlying Weyl geometry becomes Riemannian and the theory becomes metric. A similar change of the underlying geometry happens in the Palatini case. Hence, the breaking of the Weyl gauge symmetry shown in [5,7] is not the result of a mere choice of a gauge (as it happens in Weyl or conformal theories with no Weyl gauge field), but is more profound: it is accompanied by both a Stueckelberg mechanism (as mentioned) and by transformations at a geometric level. c) In both Weyl and Palatini theories w µ has a large mass (∼ M ) [5,7] so the associated non-metricity scale is very high; hence, non-metricity effects are suppressed by M . One thus avoids long-held criticisms [1] that had assumed a massless w µ (implying metricity violation at low scales or path dependence of clock's rates/rod's length, in contrast to experience [27]). d) If matter is present e.g. a Higgs-like scalar is non-minimally coupled to R(Γ, g), Weyl and Palatini theories have successful inflation, in addition to mass generation. The main goal of this work is to investigate comparatively their inflation predictions. We give new results in Section 3, such as the dependence r(n s ) of the tensor-to-scalar ratio r on the spectral index n s in Weyl and Palatini cases and their relation to Starobinsky inflation [28].
e) The Standard Model (SM) with a vanishing Higgs mass has a Weyl gauge symmetry. It is well-known that the fermions and gauge bosons do not couple to the gauge field w µ [29] but scalars (Higgs) have couplings to w µ . Having seen that w µ is massive [5,7] it is worth studying the SM in Weyl quadratic gravity or its Palatini version 4 . If the gauged scale symmetry is relevant for the mass hierarchy problem, it is intriguing that only the Higgs field couples directly to the gauge boson w µ of scale symmetry.
f ) w µ is a dark matter candidate [30] and, being part ofΓ, it could give a geometric solution to the dark matter problem. This brings together physics beyond SM and gravity.
g) The models with gauged scale symmetry do not have the unitarity issue (negative kinetic term) present in local scale invariant Lagrangians (without w µ ), when generating the Einstein action from such Lagrangians: . See [23] for a discussion on this issue in local scale invariant models 5 [31][32][33][34][35][36]. In a gauged scale invariant model this negative kinetic term is cancelled and φ is "eaten" by w µ which acquires mass [5,7]à la Stueckelberg [13,37] and decouples, to recover the Einstein action and gauge. h) In the gauged [5,7] and global [38][39][40] cases there is an associated non-zero conserved current, unlike in some local scale invariant models where this current is trivial [41,42].
i) A gauged scale symmetry seems stable under black-hole physics unlike a global one [43], so it is preferable when building models that include gravity. Global models are easily made gauged scale invariant by replacing their Levi-Civita connection by e.g. Weyl connection. The theories discussed can give a gauged scale invariant version of Agravity global model [44,45].
j) Another motivation to study theories with Weyl gauge symmetry is their geodesic completeness, as emphasized in [23] and summarised here. In conformal invariant theories geodesic completeness can be achieved without the Weyl vector presence, in the (metric) Riemannian universe; there, geodesic completeness or incompleteness is related to a specific gauge choice (with singularities due to an unphysical conformal frame) [46][47][48][49]. But Weyl gauge symmetry is more profound and complete: it is more than a symmetry of the action since, (unlike in conformal/Weyl invariant theory with no w µ ), it is also a symmetry of the underlying geometry (of connectionΓ). The geodesics are then determined by the affine structure and differential geometry demands the existence of the Weyl gauge field [50] for the construction of the affine connection, because this ensures that geodesics are invariant (as required on physical arguments). After the breaking, w µ decouples, see b) above, and we return to Riemannian geometry with geodesics given by extremal proper time condition 6 . The above arguments, a) to j), motivated our interest in theories beyond Standard Model (SM) with Weyl gauge symmetry. Section 2 reviews the two theories, showing their similarities and differences, see [5,7] for technical details. Section 3 studies comparatively their inflation predictions. The Appendix has technical details and an application to inflation.

Weyl versus Palatini quadratic gravity 2.1 The symmetry
Consider a Weyl local scale transformation Ω(x) of the metric g µν and of a scalar field φ 1 To this geometric transformation one associates a Weyl gauge field w µ that transforms aŝ Eqs.
(2), (3) define a gauged scale transformation. The symmetry is a gauged dilation group isomorphic to R + (non-compact). It differs from internal gauge symmetries, since Ω is real. What is the relation of the Weyl field to the underlying geometry which is defined by g µν andΓ? One can define w µ via the non-metricity, but it is more intuitive to define w µ as a measure of the deviation of (the trace of)Γ from the Levi-Civita connection: with a notationΓ µ =Γ ν µν and Γ µ = Γ ν µν (g). Γ α µν (g) is the Levi-Civita connection for g µν whilẽ Γ α µν is the connection in either Weyl or Palatini gravity. We assume a symmetric connectioñ Γ α µν =Γ α νµ (no torsion). Note that w µ is a vector under coordinate transformation (Γ µ and Γ µ are not). Finally,Γ α µν and in particularΓ µ is invariant under (2), (3), in both Weyl and Palatini gravity (see also the Appendix). To check this invariance use (3) in (4) and that Γ µ (g) = ∂ µ ln √ g; then Γ µ (ĝ) = ∂ µ ln( √ g Ω 4 ). The change of the metric is compensated by that of w µ , leavingΓ µ invariant. 6 Since the Weyl gauge field brings in non-metricity, geodesic completeness is related to non-metricity. 7 Our conventions are those in the Appendix of [51] with metric (+,-,-,-), and g ≡ | det gµν|.

The Lagrangian: Weyl versus Palatini
Consider next a Lagrangian with gauged scale invariance for a scalar field with non-minimal coupling, in Weyl and Palatini quadratic gravity. The analysis being similar, we present simultaneously both Weyl and Palatini theories. The main difference between them is in the coefficientsΓ α µν which we do not need to specify right now. Consider then a (Higgs-like) scalar φ 1 with non-minimal coupling ξ 1 > 0: with a scalar curvature R(Γ, g) which depends on the Weyl or Palatini connectionΓ: With (7), the first and third term in L are invariant under (2), (3). Further, the second term in L is a gauge kinetic term of w µ and involves ∇ is defined byΓ and in the second step we used thatΓ is symmetric. From (8) F µν is invariant under (2), (3), and one verifies that the second term in L is also invariant under these transformations. Since Therefore φ 1 is charged under the Weyl gauge symmetry. With (9) one checks that the kinetic term of φ 1 is invariant under (2), (3). Finally, λ 1 φ 4 1 is the only potential term allowed by symmetry, so the entire L is invariant.
In the absence of matter (φ 1 ), L contains the first two terms only, giving the minimal action of the original Weyl quadratic gravity or its Palatini version; both actions have gauged scale symmetry and, after spontaneous breaking of this symmetry, one obtains the Einstein-Proca action for w µ , see [5,7]. If only the first term is present in L, both theories are Weyl integrable (metric) and Einstein action is obtained with a positive cosmological constant.
Returning to L, we replace the first term in (5) by ξ 0 R(Γ, g) 2 → −ξ 0 (2 φ 2 0 R(Γ, g) + φ 4 0 ) where φ 0 is an auxiliary scalar; using the equation of motion of φ 0 (of solution φ 2 0 = −R) recovers onshell the term ξ 0 R 2 in (5). This gives a classically equivalent L, linear in R where We further replaced φ 0 by radial direction ρ in field space, so our new fields are now {ρ, φ 1 }. L has similarities to a global scale invariant Higgs-dilaton model, eqs. (2.9), (2.10) of [40] also [52,53]; φ 0 has a large coupling (ξ 0 > 1) to R since the R 2 term has a perturbative coupling 1/ √ ξ 0 < 1 and this corresponds to a Higgs of non-minimal coupling ξ h > 1 in [40]. The action in (10) depends onΓ through its first three terms. We have two cases: a). In Weyl quadratic gravity,Γ is determined by g µν and the gauge field w µ , see its expression in eq.(A-5) in the Appendix. Using this one replaces the scalar curvature in (10) in terms of the Ricci scalar of Riemannian geometry, eq.(A-11). The result is eq.(12) below. b). In Palatini gravity,Γ is simply determined by its equation of motion from the action in (10). After solving this equation [7], we obtain the connection shown in eq.(B-2) in the Appendix;Γ differs from that in Weyl case, due to different non-metricity (accounted for by γ in eq. (12)). With thisΓ, one computes the scalar curvature,as usually done eq.(B-5). Replacing this curvature back in action (10) one finds again L below (forΓ onshell): where γ = 1/4 for Weyl case; γ = 1 for Palatini case.
R(g) is the Ricci scalar for the metric g µν . This is a metric formulation equivalent to the initial Lagrangian eq.(5), invariant under transformations (2), (3); under these ln ρ transforms with a shift, ln ρ → ln ρ−ln Ω, so ln ρ acts like a would-be Goldstone ("dilaton"), see later.

Einstein-Proca action as a broken phase of Weyl or Palatini gravity
Since L has a gauged scale symmetry, we should "fix the gauge". We choose the Einstein gauge corresponding to constant ρ; this is obtained by using a transformation (2),(3) of a particular Ω = ρ/ ρ which is ρ−dependent and setsρ to a constantρ = ρ , and so introduces a mass scale. In terms of new variables (with a hat) eq.(12) becomes with R(ĝ) the Ricci scalar for metricĝ µν ,D µφ1 = (∂ µ − 1/2ŵ µ )φ 1 and with ∇ µŵ µ = 0; we denoted M = ρ which we identify with the Planck scale. The potential now depends on φ 1 only, see (11). This is the Einstein-Proca action forŵ µ : this field has become massive of mass m 2 w = 6α γ M 2 by absorbing the derivative of the Stueckelberg (would-be "dilaton") field ∂ µ ln ρ; then the radial direction in field space (ρ) is not present anymore in the action. This is a spontaneous breaking of Weyl gauge symmetry; the number n of degrees of freedom other than the graviton (n = 3) is conserved during this breaking: the initial massless scalar ρ and massless vector w µ are replaced by a massive gauge field w µ .
Note that in the absence of matter (φ 1 ), the Stueckelberg field needed for breaking becomes ln ρ ∝ ln φ 0 and has a pure geometric origin, being simply "extracted" from the quadratic curvature term R 2 (Γ, g) in the initial, symmetric action. Therefore, one does not need to add this scalar field ad-hoc as usually done to this purpose, and the breaking and mass generation (m w , Planck scale) takes place even in the absence of matter [5,7]. Finally, unless one is tuning the coupling α to small values, the mass ofŵ µ is near the Planck scale 8 .

Scalar potential
To obtain a standard kinetic term forφ 1 , similar to the "unitarity gauge" in the electroweak case, we remove the couplingŵ µ ∂ µφ1 from the term (D µφ1 ) 2 in (14) by a field redefinition In terms of the new fields eq. (14) becomes which is ghost-free and Lagrangian (16) describes Einstein gravity, a scalar field ϕ with canonical kinetic term and potential (17) that is γ-dependent, and a massive Proca field (ŵ µ ) that decouples near the Planck scale M . To make obvious the mass term of w µ in (16) use that cosh 2 x = 1 + sinh 2 x. Eqs. (16), (17) can be extended to more scalar fields, see second reference in [5] (eq.24). For small field values ϕ ≪ M , the potential in (17) becomes (recall that M = ρ ): In this case the potential is similar in Weyl and Palatini cases, up to a small γ-dependence of the quartic coupling, negligible for (ultra)weak couplings ξ 1 /ξ 0 ≪ 1; in this case also the quadratic coupling is suppressed (recall the perturbative couplings are 1/ √ ξ 0 < 1 and ξ 1 < 1).
If we identify ϕ with the Higgs field, we have electroweak symmetry breaking, since ξ 1 > 0. For a classical hierarchy ξ 1 /ξ 0 ≪ 1 one may be able to tune the mass of ϕ near the electroweak scale m 2 = (ξ 1 /ξ 0 ) ρ 2 . Gravitational corrections to λ 1 may be negative but there is no instability: the exact form of V (ϕ) is positive, even if the self-coupling λ 1 = 0! For large ϕ the potential is different in Weyl and Palatini cases due to a different γ. This potential changed from initial (5) to (17) following two steps: the "linearisation" of the R 2 term by φ 0 that induced the φ 4 0 term, then transformation (15) which decoupled the (trace of) the connection from ∂ µ φ 1 and brought the presence of γ i.e. non-metricity dependence.

Weyl versus Palatini
We can now use Lagrangian (16) and potential V (ϕ) of (17) to study inflation with ϕ as the inflaton and compare its predictions for the Weyl (γ = 1/4) and Palatini (γ = 1) cases. For a previous study of inflation in the Weyl case, see 9 [6,69]. Lagrangian (16) describes a single scalar field in Einstein gravity and the usual formalism for a single-field inflation can be used. However, notice there exists a coupling of ϕ to the Weyl fieldŵ µ , the second term in (16). Hence, we must first show that this coupling andŵ µ do not affect inflation by ϕ.
Firstly, we do not consider here the possibility of the Weyl vector field itself as the inflaton 10 since it could induce a substantial large-scale anisotropy [72] which would be in conflict with CMB isotropy. The anisotropy is obvious in the stress-energy tensor contribution ofŵ µ which is not diagonal. This issue can be avoided if one considers a large number of randomly oriented vector fields or a triplet of mutually orthogonal vector fields [72], however this is not possible in the current fixed setup.
Secondly, one may ask if the Weyl field could play the role of a curvaton with ϕ as the inflaton. The scenario of a vector field as a curvaton was discussed in detail in [73,74]; in such scenario the vector field does not drive inflation (to avoid large scale anisotropy) but becomes important after inflation when it may dominate the Universe and imprint its perturbation spectrum. A scale invariant spectrum can be generated byŵ µ provided that during inflation the mass-squared ofŵ µ is negative and large in absolute value (∼ H 2 ) while after inflation is positive and the vector field engages in oscillations and behaves as pressureless matter; this means it does not lead to large-scale anisotropy when it dominates [73,74]. This scenario cannot apply here since m 2 w is always positive. Indeed, the second term in (16) has f (ϕ) > 0, for any value of ϕ and the effective mass-squared ofŵ µ is always positive.
, the vector field background compatible with the metric isŵ µ (t) = (ŵ 0 (t), 0, 0, 0). However, from the equation of motion ofŵ µ one immediately sees thatŵ µ (t) = 0, (see also eq.(C-13) for details). In this case ∆L is vanishing. Therefore, we are left with potential (17) and the usual formalism of single-field inflation in Einstein gravity applies, with ϕ as inflaton.
One may ask what happens at the perturbations level? One easily sees that perturbations δϕ of ϕ do not mix with perturbations δŵ µ (of longitudinal mode/Stueckelberg field ρ) of massiveŵ µ . Such mixing is in principle possible, with potential impact on inflation predictions, but it vanishes since it is proportional toŵ µ (t)(= 0), as seen from expanding ∆L to quadratic level in perturbations: ∆L ∝ŵ µ (t) δϕ δŵ µ + · · · 11 . As a result, the coupling ∆L does not affect δϕ and the predictions of inflation by ϕ. For further discussion on perturbations δϕ and δŵ µ see Appendix C which supports these results.
The above arguments justify our use below of single-field slow-roll formulae 12 The number of e-folds is with the last step in (20), (21), (22) valid in the leading approximation λ 1 ξ 0 ≪ ξ 2 1 needed for a deep enough minimum for inflation; ϕ e is determined by ǫ(ϕ e ) = 1 and ϕ * is the initial value of the scalar field. Further, the scalar spectral index With the tensor-to-scalar ratio r = 16ǫ * , then from (20), (21), (23) The non-minimal coupling is reducing r, for fixed n s . If we ignore the term ∝ ξ 2 1 and higher orders, then the Palatini case (γ = 1) has a larger r than Weyl theory (γ = 1/4), for the same n s . This is confirmed by exact numerical results, see later. From (22), we also find r ≈ 48γ with N ≈ N + 9 and γ = 1/4 in the Weyl case and N ≈ N + 28 and γ = 1 for the Palatini case. Eqs. (25) are only an approximation and ignore some ξ 1 dependence in N , but give an idea of the exact behaviour (see later, Figure 2). There is an additional constraint on the parameters space of Weyl/Palatini models, from the normalization of the CMB anisotropy V 0 /(24π 2 M 4 ǫ * ) = κ 0 , κ 0 = 2.1 × 10 −9 and with r < 0.07 [77] then ξ 0 = 1/(π 2 rκ) ≥ 6.89 × 10 8 . With this bound, condition λ 1 ξ 0 ≪ ξ 2 1 is respected for any perturbative ξ 1 , 1/ξ 0 , by choosing an ultraweak λ 1 ≪ ξ 2 1 /ξ 0 . Let us compare eq. (24) to that in the Starobinsky model of 11 The absence of such mixing is also due to the FRW metric and to the fact that ρ (radial direction) and ϕ ∼φ1 were orthogonal directions in field space (that do not mix) and similar for their perturbations. 12 With M ∼ ρ a simple phase transition scale, values of the field ϕ ≥ M are natural.
The case of Starobinsky model for N = 60 corresponds to the upper limit of r (0.003) of the Weyl model (top curve in figure 1 and highest r in figure 2 for N = 60), while in the Palatini case a larger r is allowed for the same n s , N .
While the plots in figure 1 have λ 1 ξ 0 = 10 −8 , they are actually more general. In the extreme case of λ 1 ∼ 0, corresponding to a simplified potential (without the last term in (17)), the same range of values for (n s , r) shown in this figure remains valid. However, if we increase λ 1 ξ 0 to λ 1 ξ 0 ≈ ξ 2 1 , the last term in (17) becomes relatively large, the rightmost curves of V (of smallest ξ 1 ) have their minimum lifted and the range for (n s , r) in (28) to (31) is reduced: the smaller values ξ 1 ∼ 10 −3 in Figure 1, cannot then have successful inflation.
The main results of this work are summarised in figure 2; in this figure the dependence r(n s ) is shown for different curves of constant N , that respect the required parametric constraint λ 1 ξ 0 ≤ ξ 2 1 . The curves r(n s ) give a numerically exact representation of the dependence in eq.(24); they are extended even outside the 95% CL range for n s . In all cases, the Palatini case has r larger than in the Weyl case. This aspect and the different slope of the curves r(n s ) can be used to distinguish these models from each other and from other models in future experiments.
The small r predicted by both Weyl and Palatini gravity models may be reached by the next generation of CMB experiments: CMB-S4, LiteBIRD, PICO, PIXIE [78][79][80][81][82][83] that will reach a precision for r of ∼ 5 × 10 −4 . Therefore they will be able to test these two inflation models. Compared to another model with Weyl gauge symmetry [26] (figure 2) which is linear in R(Γ, g) and had r ∼ 0.04 − 0.06, we see that the presence of the R 2 (Γ, g) term in the Weyl theory reduced r significantly (for a fixed n s ). Reducing r by an R 2 term also exists in the Palatini models without Weyl gauge symmetry [63]. Therefore, a small measured r ∼ 10 −3 may indicate a preference for quadratic gravity models of inflation. The Weyl inflation case of ξ 1 = 10 −3 or smaller is similar to the Starobinsky model. Here we have an additional scalar field 13 , with ϕ playing the role of the inflaton 14 . The other scalar in the Weyl theory (radial direction ρ in the φ 0 , φ 1 space) is used to generate the Planck scale and the mass of w µ . Briefly, Weyl gravity gives a relation r(n s ) similar to that in the Starobinsky model, with similar, large ξ 0 , while also providing protection against corrections to r from higher dimensional operators; these are forbidden since their effective scale violates the symmetry; the Stueckelberg field cannot play the role of this scale since it was eaten by the Weyl field to all orders. Another benefit for Weyl inflation is the minimal approach: one only needs to consider the SM Higgs field in the Weyl conformal geometry; the underlying geometry provides the spontaneous breaking of the Weyl quadratic gravity action to the Einstein action and the Planck scale generation.

Corrections and other models
Despite this similarity of the Weyl and the Starobinsky models, it is possible to distinguish between them; it may happen that a curve r(n s ) corresponding to ξ 1 > 10 −3 is preferred by data (see r(n s ) curves in figure 1), in which case it is shifted below that of the Starobinsky model for the same n s -the two models are distinguishable. Also the Weyl model has an additional coupling, see ∆L in (19). While ∆L does not mix linear perturbations of δw µ and of δϕ, it can lead however to cubic interactions of the form f ′ (ϕ)δϕδw µ δw µ . These can result in different predictions for the inflationary bispectrum compared to the pure single-field case. This can be used to further distinguish the Weyl case from the Starobinsky R 2 inflation (for ξ 1 ∼ 10 −3 ). The analysis of non-Gaussianity is thus interesting for further research.
The above results are subject to corrections from other operators of d = 4 that may exist and are Weyl gauge invariant, as we discuss below.
In the Weyl case the Weyl-tensor-squared operator of Weyl geometry may be present (1/ζ)C 2 µνρσ . This can be re-written in a metric description as the Weyl-tensor-squared term of Riemannian geometry (1/ζ) C 2 µνρσ plus a gauge kinetic term of w µ which gives a threshold correction to our coupling α. The Weyl tensor term is invariant under Weyl gauge transfor-mations performed to reach the Einstein-Proca action, hence one simply adds it to the final action, eq. (14). This operator has an impact on the value of r that we found numerically and in eq.(25) with γ = 1/4 for Weyl case. The overall impact of the Weyl tensor term is essentially a rescaling of r into 15 r c = r (1 + 8/(ζ ξ 0 )) 1/2 [6]. Since our ξ 0 is large, only a low |ζ| ∼ 1/ξ 0 can increase r and this comes with an instability since the mass of the associated spin-two ghost (or tachyonic) state that this operator brings is m 2 ∼ ζ M 2 , where M is the Planck scale. Therefore, a stable Weyl gravity model up to the Planck scale will not modify the value of r. Other operators in Weyl gravity are topological and do not affect r (classically).
In the Palatini case one should consider the remaining quadratic operators of d = 4 [84] that are Weyl gauge invariant and have a symmetric connection. They modify the equation of motion ofΓ and the vectorial non-metricity (B-6); unfortunately, it does not seem possible to find in this case an analytical solution to this equation due to its modified, complex structure and new states present (ghosts, etc). Additional simplifying assumptions would be needed, making the analysis model dependent. We only mention here the interesting possibility that for a symmetricΓ, the solutionΓ may become equal to that in Weyl-geometry (A-5); if so, the Palatini approach would provide an "offshell" version of Weyl quadratic gravity that is recovered forΓ onshell.

Conclusions
We made a comparative study of the action and inflation in two theories of quadratic gravity with Weyl gauge symmetry: the original Weyl gravity action and the Palatini version of the same action, obtained by replacing the Weyl connection by Palatini connection. The actions of these theories are non-minimally coupled to a (Higgs-like) field φ 1 .
Given the symmetry, there is no scale in these theories. Mass scales are generated by an elegant spontaneous breaking of gauged scale symmetry that happens even in the absence of matter: the necessary scalar field (Stueckelberg field φ 0 ) is not added ad-hoc as usually done to this purpose, but is of geometric origin and is "extracted" from the R(Γ, g) 2 term in the action. If matter (φ 1 ) is present, the Stueckelberg field is actually the radial direction (ρ) in the field space of φ 0 and φ 1 ; the field ρ is then eaten by the Weyl gauge field w µ which acquires mass m w ∼ ρ near the Planck scale. The breaking conserves the number of degrees of freedom and generates in the broken phase the Einstein-Proca action for w µ . In both theories, below the mass of w µ the connection becomes Levi-Civita and Einstein gravity is recovered, with an "emergent" Planck scale M ∼ ρ and a scalar potential (of the remaining, angular-variable field ϕ).
The potential V (ϕ) is controlled by the symmetry of the theory together with effects from the non-trivial connectionΓ, different in the two theories. For small field values, V is similar in both theories; the scalar field can act as the Higgs field, in which case the potential displays electroweak symmetry breaking. For large field values, the potential has the same form in Weyl and Palatini theories up to couplings and field rescaling (due to different non-metricity) and gives successful inflation.
Our main results, comparing inflation predictions in the two theories and summarised in Figure 2, showed how a different non-metricity impacts on inflation predictions. In Weyl gravity the scalar-to-tensor ratio 0.00257 ≤ r ≤ 0.00303, which is smaller than in Palatini case, 0.00794 ≤ r ≤ 0.01002, for measured n s at 68% CL and N = 60 e-folds. Similar results exist for n s at 95%CL or mildly different N , etc. Such values of r will be measured by new CMB experiments that can then test and distinguish Weyl and Palatini quadratic gravity.
There are similarities of inflation in Weyl and Palatini cases to Starobinsky inflation (R + ξ 0 R 2 ). In Weyl and Palatini theories one also has an R 2 term with a large ξ 0 that reduces r, but there is also a non-minimally coupled scalar field (φ 1 ); one combination of fields is acting as the inflaton while the other (radial) combination enabled the breaking of the gauged scale symmetry and the generation of mass scales (Planck, w µ mass). In both Weyl and Palatini theory, for a fixed n s , reducing the non-minimal coupling (ξ 1 ) increases r which in Weyl theory is bounded from above by that of Starobinsky inflation. Unlike in the Palatini theory, Weyl gravity for ξ 1 ≤ 10 −3 gives a dependence r(n s ) essentially similar to that in Starobinsky inflation, while also protecting r against higher dimensional operators corrections. ---------

C Inflation: perturbations to the scalar and vector fields
We discuss in detail the scalar (δφ) and vector (δw µ ) fields perturbations in a FRW universe g µν = (1, −a(t) 2 , −a(t) 2 , −a(t) 2 ) and show that ∆L of (19) does not affect inflation by ϕ. To simplify notation hereafter we remove the 'hat' (ˆ) on w µ , g µν when we refer to action (16).
• We saw above that the perturbations δϕ do not mix with those of w µ and ϕ-inflation decouples from w µ in a FRW universe. While somewhat beyond the purpose of this work, we also examine below the vector field perturbations, following [73,74], in the approximation H ∼ constant. Compatibility with the FRW metric demands computing the perturbations about a background w µ (t) = 0 as seen earlier. In fact we may take a more general background, if initially the vector field contribution to the stress-energy tensor is negligible relative to that of the scalar, in an isotropic universe; we shall then consider a quasi-homogeneous field ∂ i w α = 0. Our FRW case is always restored by setting anywhere below w µ (t) = 0. Then from (C-13) for µ = 0 and µ = i, respectively w 0 (t) = 0 andẅ i (t) + Hẇ i (t) + f (ϕ) w i (t) = 0.
Similar considerations apply to the perturbations to the parallel (longitudinal) mode of w µ . For our FRW-compatible background q z (t) = 0 (w µ (t) = 0), hence θ 2 = 0. Therefore, there is no mixing of δq z and δϕ k perturbations in (C-26), in agreement with the earlier similar finding, see discussion around eq.(C-12). Note also that if k 2 ≪ a 2 f (ϕ), δq z has an equation similar to the transverse modes, with θ 1 ∼ 0 (with H ∼constant,φ 2 ∼ −2Ḣ 2 M 2 ).
Similar to the transverse case, the effective mass m 2 = 2H 2 + Hθ 1 + f (ϕ(t)) of δq z is again larger than H and its generation is exponentially suppressed. We see again that in the FRW case one can ignore the effect of δw µ and of coupling of w µ − ϕ on δφ k .
In a general background case q z (t) = 0, then θ 2 = 0; then a mixing of perturbations of ϕ and of longitudinal mode of w µ exists in (C-26) due to coupling f (ϕ)w µ w µ , eq. (19). However, even in this case, q z is suppressed by the scale factor, due to eq.(C- 16), and thus the same is true for the mixing.