Cosmic inflation from broken conformal symmetry

Rong-Gen Cai,Yu-Shi Hao and Shao-Jiang Wang

1 CAS Key Laboratory of Theoretical Physics,Institute of Theoretical Physics,Chinese Academy of Sciences,Beijing 100190,China

2 School of Fundamental Physics and Mathematical Sciences,Hangzhou Institute for Advanced Study(HIAS),University of Chinese Academy of Sciences (UCAS),Hangzhou 310024,China

3 School of Physical Sciences,University of Chinese Academy of Sciences(UCAS),Beijing 100049,China

Abstract A period of rapidly accelerating expansion is expected in the early Universe implemented by a scalar field slowly rolling down along an asymptotically flat potential preferred by the current data.In this paper,we point out that this picture of the cosmic inflation with an asymptotically flat potential could emerge from the Palatini quadratic gravity by adding the matter field in such a way to break the local gauged conformal symmetry in both kinetic and potential terms.

Keywords: Palatini gravity,modified gravity,conformal symmetry,cosmic inflation

1.Introduction

A simultaneous resolution for the fine-tuned horizon problem,flatness problem,and monopole problem calls for a period of rapidly accelerating expansion of spacetime[1–8]in the early Universe at least prior to the big bang nucleosynthesis.This inflationary paradigm also provides the causal productions for the primordial cosmological perturbations with a nearly scale-invariant spectrum [9–16]responsible for the observed cosmic microwave background [17,18]and large-scale structures [19,20].The standard realization for such an inflationary period usually turns to a slow-roll scalar field along with some inflationary potential [8].The most recent constraint [21]on cosmic inflation still prefers a single-field slow-roll plateau-like potential.

There are two popular implements for such a plateau-like potential:the simplest one is the Starobinsky inflation[2]with an additional quadratic term for the Ricci scalar curvature R;the most economic one is the Higgs inflation [22]with the only known fundamental scalar field (Higgs boson) so far as the inflaton non-minimally coupled to R.It was realized in recent years that they could be all constructed in general from the cosmological attractors [23]to consist of the α-attractors[24–28](including the Starobinsky inflation as a special case[29]) and ξ-attractors [30](including the Higgs inflation and induced inflation [31–35]as special cases).

It is then intriguing to explore the theoretical origin of these asymptotically flat potentials.The current observational data merely reveals two clues: (i) a plateau-like potential is supposed to admit an approximate shift symmetry,which should be slightly broken to protect an asymptotically flat potential against quantum corrections.(ii) A nearly scaleinvariant spectrum of primordial perturbations suggests a slightly broken scale symmetry in the very early Universe from de Sitter (dS) to quasi-dS phases.An appealing understanding of cosmic inflation should explain the roles played by these two symmetries.

Motivated by the superconformal approach [36–38]to the Higgs-like inflation and Starobinsky inflation[39,40],the α-attractor approach is able to really appreciate the role played by the conformal(scale)symmetry.The starting point of this approach is an old observation that a single real conformal compensator(a scalar field called conformon)with the Lagrangianis equivalent to the pure Einstein gravity with a positive cosmological constant 9λ(thus a dS solution)after gauge-fixing the conformon field to some constant thanks to the local conformal symmetry of the Lagrangian.

Although the gauge-fixing for the conformon field eliminates the concern for the presence of ghost from the wrong-sign kinetic term,the conformon field cannot be gauge-fixed if one tries to construct any nontrivial structure(namely inflation with quasi-dS phase)by explicitly breaking the local conformal symmetry.Therefore,the α-attractor approach introduces an extra scalar field with a joint global symmetry [24,25,39,40]with the conformon field but still leaves the local conformal symmetry unbroken in order to fix the gauge of the would-be-ghost conformon field.After gauge-fixing,the local conformal symmetry is spontaneously broken,and the global-symmetry-breaking potential leads to an asymptotically flat potential.However,the global symmetry for a successful inflationary implementation is restricted due to the wrong-sign kinetic term required by the local conformal symmetry.

The introduction of the conformon field with wrong-sign kinetic term could be avoided if one dives into the Palatini formalism of gravity [41,42]where the metric and affine connection are treated as independent degrees of freedom.In the Palatini formalism,the conformon field with wrong-sign kinetic term naturally emerges as a geometric gauge degree of freedom from the R2term (see equation (14) below),which has been already derived but overlooked in [43].The focus there is mainly on the dynamical recovering of the metric Einstein gravity in the absence of matter field in the Palatini formalism of a general quadratic gravity with the local conformal symmetry.The metric Einstein gravity therefore emerges at the decoupling limit of the Weyl gauge field after eating up the dilaton field?μlnφ2with a shift symmetry inherited from the local gauged conformal symmetry of φ.See [44–46]for a similar realization in the Weyl quadratic gravity and a comparison to the Palatini quadratic gravity[47]as well as its concrete realizations in the standard model of particle physics [48]and cosmology [49].See also [50–55]for other trials.

However,to carry out an inflationary potential in the Palatini formalism in a conformally invariant manner,it seems that a global symmetry shared with an additional scalar field is still needed to be slightly broken[56,57]similar to the α-attractor approach.Nevertheless,we will point out in this paper that,in the Palatini quadratic gravity,the presence of an additional global symmetry is not necessary as also expected from the swampland conjecture [58–61]of no global symmetry in quantum gravity.Without introducing any global symmetry,a plateau-like inflationary potential is always implied when the matter field is included in such a way to appropriately break the local conformal symmetry.

The outline of this paper is as follows: In section 2,we review previous results on the emergence of metric Einstein gravity from Palatini quadratic gravity.In section 3,we show the emergence of non-plateau-like and plateau-like inflation models when adding the matter field differently in terms of the local conformal symmetry.We summarize our results and discuss possible future perspectives in section 4.The convention for metric gμνis(-,+,+,+),the Planck mass isMPl≡and quantities with an overbar symbol (like the Ricci scalarand covariant derivative) are always subjected to the Levi-Civita connectionThe Riemann tensor and its variation under the connection variationrespectively,where the torsion tensorTμρν=Γρμν-Γνρμwill beTμρν=Γρμν-Γνρμin teleparallel equivalent of general relativity simply set to zero hereafter for convenience due to the geometric trinity of gravity[62].We remind here that the geometric trinity of gravity is an equivalence among three different ways to describe gravity: the traditional way of using Riemann tensorRαβμνin general relativity describes a rotation of vector after transported in parallel along a closed curve,while the torsion tensor describes the non-closure of parallelograms formed by two vectors transported along each other,and the non-metricity tensorQλμν=?λ gμνin symmetric teleparallel equivalent of general relativity describes the dilation of the length of a vector when transported along a curve.This geometric trinity of gravity might be jeopardized when the matter field is added.We,therefore,leave the case with the presence of the torsion field for future work.

2.Palatini quadratic gravity

In this section,we review the Palatini quadratic gravity with a local conformal symmetry,which reduces to the metric Einstein gravity with a positive cosmological constant when fixing the gauge of the local conformal symmetry.Although most of the derivations in this section have been presented before in [43],we re-derive these results to set up our notations and conventions to be used later on.

2.1.Palatini R2 gravity

We start with the Palatini R2gravity with an action of a form

whose field equation for metric reads

Despite the trivial solution R=0,the non-trivial part of the field equation for metric is (differ from the usual GR case with an extra factor of 1/2 in front of the Ricci scalar)

whose trace is identically satisfied (unlike the usual GR case that the trace part of the Einstein field equation gives rise to the vacuum solution R=0).Therefore,despite the trivial solution R=0,the trace part of the field equation for metric puts no constraint on Ricci scalar R,and the only constraint on R comes from the equation of motion (EoM) for the symmetric connection,

which,after substituted with its trace in λ=ν by=0,becomes

Expanding the above equation as

by the non-metricity te nsorQλμν=?λ gμνand non-metricity vector Qλ=gμν?λgμνfollowed by contracted with gμν,one has

Note that the action(1)is actually a redundant description due to the local conformal symmetry,S[g,Γ]=under the local conformal transformations,

since the Ricci scalar-squareR(g,Γ)2=(gμν Rμν(Γ))2=compensates the contribution fromTo fix this gauge symmetry,one should fix one of the scalar degree of freedom,for example,gaugefixing R to some constant C ≠0.Then,the equation (7)reduces to the vanishing non-metricity with the metric compatible Levi-Civita connection,and the equation (3) reduces to the usual GR case of the Einstein field equation with a nonvanishing cosmological constant Λ=C/4.Note that the vacuum solution R=0 automatically satisfies the connection EOM(7),therefore,only in this case,it does not reduce to the metric Einstein gravity with a cosmological constant.In what follows,we will not consider the case with R=0.

We can also introduce an auxiliary fieldφ22=F′(φ)=αφin the expansion ofF(R)=F(φ)+F′ (φ)(R-φ)for F(R)=(α/2)R2,and then arrive at an equivalent Jordan frame action

which reduces to(1)when putting φ-field on-shell by its EoM φ2/2=αR.This Jordan-frame action enjoys a local gauged conformal symmetry,S[g,Γ;φ]=under the local gauged conformal transformations

where φ is actually a gauge degree of freedom of the shift symmetry ln=lnφ-ln Ωcompensating the local conformal transformation (8).However,unlike in the metric formalism,the auxiliary field φ is not a dynamical degree of freedom.This could be seen after conformally transforming(9) into the Einstein-frame action as

with a specific conformal factor Ω(x)2=φ(x)2MP2l.Note that φ remains unchanged during the local conformal transformations (8) and it only transforms as=Ω-1φwhen testing for the local gauged conformal symmetry.It is easy to see that this Einstein-frame actionis equivalent to the Jordan-frame action S[g,Γ;φ]by directly gauge-fixing φ to MPlthanks to the local gauged conformal symmetry of φ.Now that the Einstein-frame action is minimally coupled,putting the connection on-shell reproduces the Levi-Civita connection,and the metric-affine geometry reduces to the Riemannian geometry.Hence the metric Einstein gravity is recovered in a gauge-fixing form but with an additional positive cosmological constant.

Equivalently [43],provides alternative treatment on the action (9) by first putting the connections on-shell before making either local conformal transformations (8) or gaugefixing φ to MPl.Note that the torsionless version of Stokes’theorem in Palatini formalism renders ∫ d4x?μ=0,one obtains the EoM of the connection,

which,after contracting ν=λ,gives rise to an equation=0that could be rewritten as=0 in terms of a metric-compatible auxiliary metric fμν≡φ2gμν.Therefore,the connection could be solved as the Levi-Civita connectionin terms of fμν,which,after expressed in terms of gμνexplicitly,becomes

with abbreviatingGμ≡?μlnφ2=lnφ2=?μlnφ2.Note that with on-shell connection,the Weyl gauge feildAμ≡=Gμis fxied and determined by Gμfeild alone,which is in fact related to the fact that the action (9) is invariant under the projective transformation=Γρμν+δρμξν(x)for an arbitrary vector feild ξμ(x)used for gauge-fxiing Aμ.Putting the connectionΓμρνon-shell(OS)with solution(13),the Ricci scalar readsR(g,ΓOS)=and the action (9) becomes

which is exactly the Lagrangian form with a wrong-sign kinetic term desired by the α-attractor approach in the first place.The on-shell action (14) also enjoys a local gauged conformal symmetry,S[g;φ]=under the local gauged conformal transformations

thanks to the plus sign of+3 (φ)2(namely conformon)that is crucial for exact cancellations with respect to the Ω-dependent terms inNow that φ is a gauge degree of freedom,one can either directly gaugefxi φ to MPlor choose a specifci conformal factorΩ2=φ2MP2lto conformally transform(14) viaS[gμν=Ω-2;φ]as

which is exactly the action (11) with on-shell connection.

In a short summary,the R2term in the Palatini formalism contributes an extra non-dynamical gauge degree of freedom φ of shift symmetry ln=lnφ2-ln Ω2under the local gauged conformal transformations (10) or (15).Therefore,lnφ2andGμ=?μlnφ2behave like the dilaton field and the would-be Goldstone field,respectively.After gauge-fixing φ to MPl,the metric Einstein gravity with a positive cosmological constant is recovered.

2.2.Palatini R2+ gravity

which also enjoys a local gauged conformal symmetry,S[g,Γ;φ]=under the local gauged conformal transformations (10).Note that=Aμ-?μln Ω2does not transform independently from the local conformal transformations (8) but inherited from(g)=-2?μln Ω2under the local conformal transformations(8).It is easy to see that both(18)and(19)admit additional gauge shift symmetry under=Aμ-?μω2for an arbitrary gauge function ω(x),and hence Aμis actually a gauge degree of freedom.It is worth noting that this gauge shift symmetry of Aμis different from the gauge shift symmetry of φ since ω does not need to coincide with the local conformal transformation factor Ω.

Alternatively [43],provides another intriguing view on the action (19) by first putting the connection on-shell before making either local conformal transformations (8) or gaugefixing φ to MPl.The EoM of the connection is obtained as

Note that at this point Aμdoes not enjoy the arbitrary gauge shift symmetry underA?μ=Aμ-?μω2anymore.It seems that putting the connection on shell picks out a particular gauge choice ω=Ω for Aμwhen transformed coherently with the local gauged conformal transformations (15).Note also that,putting the connection on-shell does not fix all its components but leaves Aμundetermined since contracting ρ=ν in(21)simply reduces to a trivial identity.This is caused by the explicitly broken projective symmetry of (19) and (23) under the projective transformation=Γρμν+δρμξν(x)for an arbitrary vector field ξμ(x),which would otherwise fix the Weyl gauge field Aμ.This is different from the case in section 2.1 where Aμis fully determined byAμ=Gμ≡?μlnφ2since the projective symmetry is not broken there.

Finally,the on-shell action(23)still enjoys the local gauged conformal symmetry,S[g,A;φ]=under the local Ω2=to conformally transform (23) into the Einsteingauged conformal transformations (15),one can either directly gauge-fix φ to MPlor choose a specific conformal factor frame action byS[gμν=as

which is the Palatini Einstein gravity with a positive cosmological constant plus a Proca gauge field action.Fixing the gauge of φ breaks the local gauge conformal symmetry of (23),and the would-be Goldstone field Gμis therefore absorbed by Aμto render a massive gauge feild with a massmA2=6β2MP2l.When Aμis decoupled below mA,the metricity is deduced and the metric Einstein gravity with a positive cosmological constant is therefore recovered at this decoupling limit.

One can also arrive at the same result as(24)from(19)by putting the connection on-shell after making either local conformal transformations(8)or gauge-fixing φ to MPl.In specific,since the action (19) is locally gauged conformal invariant,we can fix the gauge of φ to some constant scale MPl,

which,after putting Γ on-shell,reduces to the same form as(24)(but without over-tilde symbols).We can also choose a specific conformal factorΩ2=φ2MP2lto conformally transform (19)into the Einstein-frame action byS[gμν=,Γρμν=;φ]as

3.Inclusion of matter field

Now that the Palatini quadratic gravity simply reproduces the metric Einstein gravity with a positive cosmological constant in a gauge-fixing form,we need to add matter field to the Palatini quadratic gravity in order to account for the inflaton field responsible for the cosmic inflation.There are two ways to add the matter field:either preserving or breaking the local gauged conformal symmetry.

3.1.Preserving the local conformal symmetry

3.1.1.Palatini R2gravity.We start with the Palatini R2gravity with the inclusion of a matter field h as

Note that at this point Aμis not an independent degree of freedom from the connectionΓμρν,thus one cannot separately vary the actions (27) or (29) with respect to AμfromΓμρν.In fact,Aμonly becomes an independent residual degree of freedom after putting the connection on-shell due to the explicit presence of Aμin the Dμterm that breaks the projective symmetry,

which would otherwise fix the Weyl gauge field Aμfrom gaugefixing the arbitrary vector field ξμ(x).To put the connection onshell,one first varies the action(29)with respect to theΓμρν,and then obtain the EoM for the connection as

This gauge symmetry allows us to fix one of the scalar degrees of freedom,for example,gauge-fixing ρ to the Planck scale MPl,and then the action (32) reduces to

Due to the absence of kinetic term for Aμ,it can be integrated out by putting it on-shell via its EoM(a constraint equation),

and then the action further reduces to

which,after normalizing the kinetic term by redefining

becomes

with the potential U(h(φ),MPl) abbreviated as W(φ) of form

If all effective potential terms of the matter field are absent(namely ξ=0 and λ=0),the final reduced theory is the metric Einstein gravity with a cosmological constant.Otherwise,the potential W(φ) does not admit an asymptotically flat potential since W(φ)is divergent at φ→∞limit.In fact,W(φ)supports a small-field tachyonic inflation at small φ limit for hierarchical couplings αλ ?ξ2?1 with the approximated potential

from which the slow-roll parameters can be expanded as

Therefore,the consistency relation r=-8ntis unchanged but the scalar spectral index ns=1+2η*-6∈* and the tensor-toscalar ratio r=16∈*evaluated at the horizon crossing of some pivot scale k*=a(t*)H(t*) is related by

3.1.2.Palatini R2+gravity.Parallel discussions also apply for PalatiniR2+R[2μν]gravity with an action of form

which,after replacing α2R2=φ2R-φ4/(4α),becomes

with ρ2≡φ2+ξh2and U(h,ρ)≡(λ/4)h4+(ρ2-ξh2)/(8α)as defined before.To put connection on-shell,solving the EoM of the connection from the action (45),

admits the same solution as(21),and the action(45)with onshell connection becomes

which still enjoys a local gauged conformal symmetry,S[g,A;h,ρ]=under the local gauged conformal transformation (33).Again,this allows us to gauge-fix ρ to MPl,and the reduced action reads

This is the same action proposed in [43]for the Palatini R2inflation with the same small-field tachyonic inflationary feature as (38).In a short summary,when the matter field is added in a way to preserve the local conformal symmetry(usually also break the projective symmetry at the same time),the asymptotically flat inflationary potential is not implied.

3.2.Breaking the local conformal symmetry

3.2.1.Palatini R2gravity.To break the local gauged conformal symmetry,we propose to replace the gauge covariant derivative Dμin (44) with a normal covariant derivative ?μ,namely.

As we will see shortly below that the cosmic inflation with an asymptotically flat potential is always obtained if one further breaks the local gauged conformal symmetry in the non-minimal coupling or matter potential by adding lower-than-quadratic terms beyond G(h)=ξh2or higher-than-quartic terms beyond V(h)=(λ/4)h4so that the ratioV(h) G(h)2is an increasing function of h at a large h limit.

Similar to the previous sections,we first extract the scalar degree of freedom in the R2term by replacing α2R2=φ2R-φ4/(4α),then we obtain the Jordan-frame action

If both the non-minimal coupling G=ξh2and the matter potential V=(λ/4)h4include no extra dimensional scales,then the effective potential W is merely a cosmological constant,

However,if G(h) or V(h) is amended with additional dimensional scales to break the local gauged conformal symmetry in such a way that G contains lower-than-quadratic terms,or V contains higher-than-quartic terms,

Note that the inflationary potential W is even more flattened when the potential V becomes very steep.Therefore,this k-inflation [63,68]but with an asymptotically flat potential largely emerges as a result of the broken local gauged conformal symmetry in both matter kinetic and potential terms(regarding the non-minimal coupling term as some kind of effective potential term induced by the background gravity).

respectively,from which the scalar/tensor spectral indexes and tensor-to-scalar ratio evaluated at the horizon crossing moment of some pivot scale k*=a(t*)H(t*)/cs(t*) are obtained as

It can be numerically checked that this approximation is sufficiently stable for model predictions,which are the functions of N* with input parameters λ,ξ,and α.In order to identify the parameter regions of observational interest,we can use the measured values of nsand Asto fix λ,ξ,

and then the tensor-to-scalar ratio reads

Requiring r to be smaller than the current upper bound r0.05＜0.036 [70],α should satisfy

Using the best-fit values ns=0.9649 and ln (1 010As)=3.045 from Planck 2018 TT,TE,EE+lowE constraints [18],we finally identify the parameter space of α as

The remaining freedom on N* can be traced back to different reheating histories.In general,as long as the above condition on α is satisfied,one can always find the parameter regions for λ and ξ to simultaneously meet the observational constraints ns=0.9649,ln (1010As)=3.045and r0.05＜0.036.

3.2.2.Palatini R2+gravity.Parallel discussions also apply for PalatiniR2+R[2μν]gravity with an action of form

which,after replacing α2R2=φ2R-φ4/(4α),becomes

Putting the connection on-shell with the same solution (21)gives rise to an action of a form

which,after conformally transformed into Einstein frame viawith a specific conformal factorΩ2=,is reduced into

When Aμis decoupled below the scale 6βMPl,we return back to (52) that immediately leads to the K-essence theory(54) and hence an asymptotically flat inflationary potential is similarly obtained.In a short summary,the asymptotically flat potential emerges as a result of breaking the local conformal symmetry appropriately for both scalar kinetic and effective potential terms,and is independent of the presence or absence of the projective symmetry as shown for (49) or (91),respectively.

4.Conclusions and discussions

Cosmic inflation is the standard pillar for the standard model of modern cosmology,describing a period of nearly exponential expansion of spacetime in the very early Universe to solve several fine-tuning problems of the standard hot big bang scenario and generate nearly scale-invariant primordial perturbations observed in the cosmic microwave background and large scale structures.The current observational data prefers a single-field slow-roll plateau-like inflationary potential,which could be theoretically motivated from the cosmological attractor approach.A conformon field with a wrong-sign kinetic term is introduced to respect the local conformal symmetry and a second scalar field is added in such a way to impose an additional global symmetry jointed with the conformon field,which is broken by the potential term but with the local conformal symmetry intact.After fixing the gauge of conformon field,the potential term with broken global symmetry gives rise to the exponentially flattened inflationary potential.

However,this approach introduces the wrong-sign conformon field at the price of introducing an additional global symmetry for inflationary model buildings.Nevertheless,the wrong-sign conformon field could naturally arise in the Palatini quadratic gravity,though an additional global symmetry is also adopted for inflationary model buildings.In this paper,we point out that,in Palatini quadratic gravity,such an encumbrance of an additional global symmetry is needless.Appropriately breaking the local conformal symmetry alone for both kinetic and potential terms of a matter field is sufficient to produce an asymptotically flat inflationary potential regardless of the high steepness of original matter potential.

For future perspectives,it is still mysterious what position should we find for the Palatini quadratic gravity in approaching the underlying quantum gravity.A related question is that,for Palatini quadratic gravity without matter field or with conformally invariant matter field,since the local conformal symmetry is a gauge symmetry,then what causes this redundancy or what is the origin for this local conformal symmetry?This is a profound question[71,72]on how gauge symmetry emerges from more physical symmetry [73,74].

The next question concerns the transition from the local conformally symmetric matter phase to the locally conformalsymmetry broken matter phase.Breaking the local conformal symmetry in matter potential is easy by quantum corrections or renormalization group flow.However,the reduction of a gauge covariant derivative term into a normal covariant derivative term is unclear.A dynamical mechanism for triggering such a broken conformal symmetry in the kinetic term is desirable.

The last question runs into the initial conditions for the cosmic inflation,which is usually the realm of the quantum cosmology [75]for the no-boundary [76,77]and tunneling[78–82]proposals.As far as we know,there is currently no study on quantum cosmology starting from the Palatini quadratic gravity,which might be related to the recent new result [83]in presence of non-minimal coupling compared to the case of absence [84,85].

Acknowledgments

We thank Li Li,Run-Qiu Yang,Shan-Ming Ruan for helpful discussions.This work is supported by the National Key Research and Development Program of China Grant No.2020YFC2201501,the National Natural Science Foundation of China Grants No.12105344,No.11 647 601,No.11821505,No.11851302,No.12047503,No.11991052,No.12075297 and No.12 047 558,the Key Research Program of the CAS Grant No.XDPB15,the Key Research Program of Frontier Sciences of CAS,and the Science Research Grants from the China Manned Space Project with NO.CMS-CSST-2021-B01.