Physics Le e s B 826 (2022) 136907
Con en s lis s a ailable a ScienceDi ec
Physics Le e s B
www.else ie .com/loca e/physle b
Non-local cu a u e g a i y cosmology ia Noe he symme ies
Ad iano Acunzo a, F ancesco Baja di a,b, Sal a o e Capozziello a,b,c,d,e,∗
aDepa men o Physics “E. Pancini”, Uni e si y o Naples “Fede ico II”, Naples, I aly
bINFN Sez. di Napoli, Compl. Uni . di Mon e S. Angelo, Edificio G, Via Cin hia, I-80126, Naples, I aly
cScuola Supe io e Me idionale, La go San Ma cellino 10, I-80138, Naples, I aly
dLabo a o y o Theo e ical Cosmology, Tomsk S a e Uni e si y o Con ol Sys ems and Radioelec onics (TUSUR), 634050 Tomsk, Russia
eDepa men o Ma hema ics, Facul y o Ci il Enginee ing, VSB-Technical Uni e si y o Os a a, Lud ika Podes e 1875/17, 708 00 Os a a-Po uba, Czech Republic
a i c l e i n o a b s a c
A icle his o y:
Recei ed 14 No embe 2021
Accep ed 12 Janua y 2022
A ailable online 14 Janua y 2022
Edi o : N. Lambe
Keywo ds:
Non-local g a i y
Noe he symme ies
Cosmology
Exac solu ions
We conside ex ensions o Gene al Rela i i y based on he non-local unc ion (R, −1R), whe e Ris
he Ricci cu a u e scala and he non-locali y is due o he e m −1R. We ocus on cosmological
minisupe spaces and selec iable models by he Noe he Symme y App oach. Then we find iable exac
solu ions poin ing ou he ole o non-locali y in cosmology.
©2022 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license
(h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by SCOAP3.
1. In oduc ion
Eins ein’s Gene al Rela i i y (GR) gi es he bes desc ip ion o g a i a ional in e ac ion. I was p oposed in 1915 and gained a wide
success hanks o se e al expe imen s and obse a ions which con inuously confi m i s alidi y. These confi ma ions occu ed a se e al
scales o ene gy, anging om Sola Sys em up o cosmology. Fo ins ance, he expansion o he uni e se and he desc ip ion o la ge scale
s uc u e we e o midable p obes a cosmological le el. Fu he mo e, he de ec ion o g a i a ional wa es and he final e idences o black
holes exis ence p o ided excep ional es -beds o he heo y.
Ne e heless, se e al sho comings a ose bo h a ul a iole (UV) and in a ed (IR) scales ques ioning he alidi y o GR as he final
heo y o g a i y. In o he wo ds, hey sugges ha GR could be an e ec i e heo y no wo king a any ene gy scale.
A fi s ype o sho comings is ep esen ed by he space- ime singula i ies. Fo example, he K uskal-Szeke es maximal ex ension o
he Schwa zschild solu ion exhibi s a ue space- ime singula i y a he g a i a ional cen e , which canno be emo ed by coo dina e
ans o ma ions. A he as ophysical le el, he so called “Galaxy o a ion cu e p oblem” occu s [1,2]. I consis s o he disc epancy
be ween he heo e ical and he obse ed speed o s a s o bi ing a ound galaxies. In o de o p ope ly add ess his issue, Da k Ma e was
in oduced. I ep esen s a hypo he ical fluid supposed o accoun o he 26% o he Uni e se con en .
Simila p oblems a e su e ed by s anda d cosmology. Cu en obse a ions show an accele a ed expansion o he Uni e se a la e
imes. This e ec can be aken in o accoun by in oducing in he Eins ein-Hilbe ac ion he cosmological cons an . I can be physically
in e p e ed as an anomalous fluid wi h nega i e p essu e, dubbed Da k Ene gy, which should d i e he cosmological expansion [3,4].
Howe e , so a , he e is no final expe imen al e idence explaining Da k Ma e and Da k Ene gy a undamen al le el as u he pa icles
beyond he S anda d Model. The puzzling si ua ion is ha he CDM model is compa ible wi h p ecision cosmology obse a ions [5]bu
i su e s se e al concep ual sho comings a heo e ical le el. Fo ins ance, he obse ed alue o is 120 o de s o magni ude lowe
han he acuum ene gy densi y p edic ed by Quan um Field Theo y (QFT) in cu ed space- imes.
Rega ding UV scales, GR u ns ou o be eno malizable up o second-loop le el [6], which means ha incu able di e gences a ise in
he a emp o me ge GR wi h Quan um Mechanics. Fu he mo e, GR canno be ea ed unde he s anda d o Yang-Mills heo ies, so ha
i canno be unified o he o he undamen al in e ac ions. Fu he mo e, he absence o a Hilbe space and he lack o a p obabilis ic
*Co esponding au ho .
E-mail add esses: [email p o ec ed] (A. Acunzo),
ancesco.ba[email p o ec ed] (F. Baja di),
[email p o ec ed] (S. Capozziello).
h ps://doi.o g/10.1016/j.physle b.2022.136907
0370-2693/©2022 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by
SCOAP3.
A. Acunzo, F. Baja di and S. Capozziello Physics Le e s B 826 (2022) 136907
in e p e a ion o he wa e unc ion yield se e al obs acles so a unsol ed. In o he wo ds, a quan um heo y o g a i y is no ye a ailable
a he momen .
These p oblems sugges ha GR needs o be gene alized o a mo e comple e heo y capable o o e coming low and high-ene gy issues.
In his amewo k, ex ended heo ies o g a i y (ETGs) can be aken in o accoun [7]. They in oduce highe o de e ms in he cu a u e
in a ian s o non-minimal couplings be ween scala field and geome y in o he Eins ein-Hilbe ac ion. These addi ional e ms come om
he one-loop e ec i e ac ion p edic ed by he semi-classical app oach o QFT on cu ed space- ime [8–11]. In his app oach, me ic is
conside ed as a classical field, while ma e is ea ed as quan um fields. I comes ou ha he e ec i e Lag angian con ains geome ical
UV-di e gen e ms, p opo ional o R2, Rμν Rμν and o he cu a u e in a ian s [8]. See also [7,12–15].
These e ec i e co ec ions in ol e local fields, hus he ela ed heo ies a e desc ibed by local ac ions, acco ding o he p inciple o
locali y o he in e ac ions. I is impo an o dis inguish be ween kinema ical and dynamical locali y/non-locali y. The o me e e s o he
s a es o he heo y: classical heo ies a e kinema ically local while quan um heo ies a e kinema ically non-local. Ins ead, he la e e e s
o he in e ac ion, so ha any heo y can be dynamically local o non-local depending on he locali y/non-locali y o he ac ion.
To da e, no local ETG is a candida e o an ul ima e heo y o g a i y, mainly because he associa ed quan um heo ies a e no ully
eno malizable and uni a y.
An a emp o o e come GR sho comings is based on he b eaking o locali y p inciple by means o dynamically non-local ETGs. In
he las yea s non-local heo ies o g a i y sp ead ou in he con ex o Quan um G a i y, especially due o hei capabili y o p o ide a
quan um desc ip ion o he g a i a ional in e ac ion. I is a ma e o ac ha dynamical non-locali y is a p ope y sha ed by all he o he
undamen al in e ac ions when hei one-loop e ec i e ac ions a e conside ed [16]. In ac , eno malizable one-loop e ec i e Lag angians
a ise a e in eg a ing ou massi e e mions. This is he case o Quan um Elec odynamics. He e, non-locali y is due o he in insic non-
local na u e o in eg a ion, which can be ecas as he in e se o a di e en ial ope a o . Ano he significan example is p o ided by he
Yukawa heo y wi h a massi e scala field φ, whe e he non-locali y is due o he in eg al ope a o ( +m2)−1. See, e.g. [17].
Non-local ETGs b ing non-locali y in he g a i a ional in e ac ion, i.e. hey desc ibe g a i y by non-local e ec i e ac ions. In gene al,
wo classes o non-local ETGs can be conside ed: Infini e De i a i e Theo ies o G a i y (IDGs) and In eg al Ke nel Theo ies o G a i y
(IKGs).
IDGs conside analy ic anscenden al unc ions o some di e en ial ope a o , mainly exponen ial unc ions o he co a ian d’Alembe
ope a o . An example o such heo ies is p o ided by he ac ion [18]:
S=κ
2d4x√−gR−Gμν
eH(−s)−1
Rμν,(1)
whe e κ=c4
8πGNand H(−s)is an en i e analy ic unc ion o s≡/M2
s, being Msa mass/leng h scale. I is wo hwhile o no e ha
he in eg al ke nel 1/ ≡−1ope a o is employed only o dimensional easons since, a e a Taylo expansion o he exponen ial, he
denomina o cancels ou . This heo y can cu e classical Black Hole and Big Bang singula i ies as shown in [19,20].
IKGs gene ally employ in eg al ke nels o di e en ial ope a o s, bu mainly he in e se ope a o −1(see [21]). A s aigh o wa d
non-local ex ension o GR is
S=κ
2d4x√−gR1+F−1R+S(m).(2)
The e m −1Rcan accoun o he la e- ime cosmic expansion wi hou in oking any Da k Ene gy. F om a undamen al physics poin o
iew, IDGs eme ges in iew o ob ain ully eno malizable and uni a y quan um g a i y models [22]. On he o he hand, IKGs a e inspi ed
by IR quan um co ec ions coming om QFT on cu ed space- ime [16].
He e we will conside highe -o de cu a u e IKG models desc ibed by he gene al Lag angian densi y FR, −1R. This e ec i e
heo y is a gene aliza ion o he IKGs conside ed in li e a u e so a , e.g. in Re s. [23–26]. Since his model is a non-local ex ension o
F(R)-g a i y, i could accoun , in p inciple, o bo h UV and IR quan um co ec ions a once. Specifically, i could be use ul o achie ing
bo h infla iona y beha io (UV) and he oday cosmic accele a ion (IR).
In his pape , we will s udy cosmological solu ions o FR, −1Rg a i y using a spa ially fla F iedman-Lemaî e-Robe son-Walke
(FLRW) me ic as a cosmological backg ound. The main pu pose o he analysis is o selec physically ele an cosmological models. Nnon-
local g a i y models a e selec ed by he so called Noe he Symme y App oach, whose main aspec s a e ou lined e.g. in [27,28]. As shown
in li e a u e (see e.g. [29–37]), he app oach allows o educe dynamics by he exis ence o symme ies and o find ou , e en ually, exac
solu ions.
In Sec. 2we o e iew he ounda ions o cu a u e based non-local heo ies o g a i y. Sec. 3is de o ed o he applica ion o he
Noe he Symme y App oach o non-local ETGs depending on unc ions o he scala cu a u e and he ope a o −1R. In Sec. 4, we find
analy ic cosmological solu ions coming om he unc ions selec ed by he Noe he symme ies. Sec. 5is de o ed o he discussion o he
esul s and he conclusions.
2. Non-local cu a u e based heo ies o g a i y
Le us in oduce now he main p ope ies o non-local heo ies o g a i y based on cu a u e in a ian s.1Le us s a om IKGs. The
mos gene al g a i a ional ac ion in ou dimensions, quad a ic in he cu a u e, which can be made ghos - ee, mus con ain infini e
co a ian de i a i es [22,38–40]. I eads:
S=κ
2d4x√−gR+αRF1(s)R+Rμν F2(s)Rμν +Rμνρσ F3(s)Rμνρσ ,(3)
1As discussed in [23], i is possible o o mula e non-local heo ies o g a i y based on o he geome ic in a ian s as he o sion scala conside ing a elepa allel app oach.
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A. Acunzo, F. Baja di and S. Capozziello Physics Le e s B 826 (2022) 136907
whe e α≡(Ms)−2is dimensional cons an , Msa mass/leng h scale and Fi(s) anscenden al en i e analy ic unc ions o he adimen-
sional co a ian d’Alembe ope a o s≡/M2
s. Being en i e unc ions, hey ha e no pole on he whole complex plane, p e en ing he
occu ence o ghos s. Fu he mo e, being analy ic unc ions, hey can be gene ally exp essed in e ms o Taylo expansion as:
Fi(s)=∞

n=0
i,n(s)n.(4)
An in e es ing class o IKGs can be ound in Re . [18]. The co esponding ac ion eads:
S=κ
2d4x√−gR−Gμν
eH(−s)−1
Rμν,(5)
whe e H(−s)is an en i e analy ic unc ion o s. The associa ed field equa ions a he o de O(R2)a e
Gμν +O(R2)=1
κe−H(−s)T(m)
μν (6)
and educe o GR equa ions a he ze o h o de . In a sphe ically symme ic space- ime, he field equa ions (6)yield egula black hole
solu ions wi hou singula i ies [19]. In cosmology, he heo y admi s bouncing solu ions [20]. The mechanism o esolu ion o he singu-
la i y is clea ly explained in [41,42]. I basically consis s o a non-local smea ing o he poin -like (Di ac del a) sou ce o he Schwa zschild
me ic induced by he infini e de i a i es. This non-local e ec implies ha he me ic is no longe a acuum solu ion. In ac , a om he
sou ce, g a i y is well desc ibed by GR, bu once app oaching he non-local egion <2/Ms he smea ing e ec s o he sou ce (induced
by he non-locali y) s a being ele an . In pa icula , all he cu a u e scala s u n ou o be non-singula in o his egion, so ha no
singula i y occu s e en a he g a i a ional cen e .
Ano he ele an example can be ound in [43], whe e he au ho s p o ide he maximal supe - eno malizable and uni a y UV-
comple ion o he S a obinsky model, whose ac ion is:
S=κ
2d4x√−gR−Gμν
V−1
2−1
Rμν +1
2RV−1
0−V−1
2
R,(7)
whe e
V−1
2≡eH2(−s)p(n2)(−s), V−1
0−V−1
2≡1
3eH0(−s)(1+s)−eH2(−s),(8)
wi h Hiand p(n2)being en i e analy ic unc ions ( o de ails see [43]). Any u he ex ensions o he ac ion (7)can be p o ed o be
non-uni a y.
IDGs a e he ul ima e consequence o he highe o de UV quan um co ec ions, as discussed in he in oduc ion. The local co ec ions
come om an expansion a ound s =0o a Schwinge p ope ime in eg al [8], which is he e o e alid o small imes. On he con a y,
o ge IR co ec ions, an expansion a ound s →∞is needed. Howe e , he Schwinge p ope ime in eg al is meaning ul only when he
masses o he ma e fields a e la ge han he po en ial and he space- ime cu a u e. In he massless limi , he p ope ime in eg a ion
becomes di e gen o la e imes (s →∞). This is due o he pe u ba i e na u e o he app oach, hus a non-pe u ba i e echnique
o calcula e he Schwinge p ope ime in eg al is necessa y o o e come he issue. This would allow o ake in o accoun bo h UV and
non-local IR e ec s, o any alue o po en ial, cu a u e and mass. Such echnique can be ound in [16] and p o ides he ollowing
non-pe u ba i e ac ion o some QFT in cu ed space- ime:
W0=−d4x√−gV(x)+V(x)(−V)−1V(x)+1
6,(9)
whe e V(x)is he po en ial o he heo y and is he ollowing su ace e m:
=d4x√−gR−Rμν −1Gμν +2−1R−1Rμν−1Rμν+
−Rμν−1Rμν−1R+−1Rαβ∇α−1R∇β−1R+
−2∇μ−1Rνα∇ν−1Rμα−1R+
−2−1Rμν∇μ−1Rαβ∇ν−1Rαβ+OR4
μν.
(10)
Fu he de ails can be ound in [16]. I is wo h s essing ha he la e- ime e ec i e ac ion s ongly depends on he in eg al ope a o −1,
which is hus able o g asp la e- ime quan um co ec ions o GR. As men ioned abo e, i was sugges ed in [21]. The conside ed ac ion is
S=κ
2d4x√−gR1+F−1R+S(m),(11)
wi h F−1Rbeing an a bi a y unc ion o −1R. The associa ed field equa ions a e
Gμν +Gμν =1
κT(m)
μν ,(12)
whe e
3
A. Acunzo, F. Baja di and S. Capozziello Physics Le e s B 826 (2022) 136907
Gμν =Gμν +gμν −∇μ∇νF+−1RF
+δ(ρ
μδσ)
ν−1
2gμν gρσ ∂ρ−1R∂σ−1RF
,(13)
wi h he defini ions F≡F−1Rand F≡∂F
∂−1R.
I can be showed ha he ope a o −1can na u ally igge he cu en la e- ime cosmic accele a ion. The e o e he non-local quan i y
−1Rgene a es he la ge numbe s equi ed by he cu en cosmic accele a ion a oiding he fine uning o pa ame e s. No e ha hese
co ec ions only occu a la e- imes: du ing he adia ion domina ed e a he Ricci scala anishes and he non-local e ec s a e hus
negligible. In gene al, non-local ETGs a e conside ed in li e a u e o add ess sho comings o GR in cosmological [24,44–46] and sphe ically
symme ic [47,48] backg ounds. Specifically, in he la e e e ences, he au ho s ake in o accoun non-local co ec ions o he New onian
po en ial and check how hese addi ional e ms can be de ec ed a Galac ic scales [47]o a scales o galaxy clus e s. Also g a i a ional
wa es, coming om non-local g a i y, ha e been conside ed [49,50].
3. Noe he symme y app oach o FR, −1Rg a i y
Le us now in oduce a class o non-local IKG models which we wan o selec by he exis ence o Noe he symme ies. The app oach
can be seen as a physical c i e ion o selec iable models. Fo a discussion see [35].
The s a ing ac ion is:
S=d4x√−gFR,−1R.(14)
This e ec i e heo y is a gene aliza ion o F(R)-g a i y including non-local e ms. A poin -like Lag angian, use ul o cosmological consid-
e a ions, can be cons uc ed by he auxilia y local scala field φ, defined as:
φ≡−1Ro R≡φ. (15)
In such a way, he ac ion can be “localized” and he s a ing model can be ecas in e ms o a scala - enso heo y, desc ibed by he
ac ion
S=d4x√−gF(R,φ). (16)
Using he Lag ange mul iplie s me hod in a FLRW backg ound (wi h Lag ange Mul iplie s λ1and λ2) bo h wi h he cosmological exp es-
sions o he Ricci scala and he highe -o de e m φ, i is possible o ecas he ac ion as
S=2π2d a3F(R,φ)−λ1(R−¨
φ−3H˙
φ) −λ2R+6¨
a
a+˙
a
a2.(17)
By a ying he ac ion wi h espec o Rwe immedia ely find
λ2=∂F(R,φ)
∂R−λ1,(18)
hus, by p omo ing λ1 o a scala field and se ing λ1≡λ( ), Eq. (17)can be ecas as:
S=d a3F(R,φ)−λ(R−¨
φ−3H˙
φ) −∂F(R,φ)
∂R−λR+6¨
a
a+˙
a
a2.(19)
A e in eg a ing ou he second de i a i es, he cosmological poin -like Lag angian in he minisupe space Q ≡{a, R, φ, λ} eads as
L=a3F−a3˙
φ˙
λ−a3R∂RF+6a˙
a2∂RF−6a˙
a2λ+6a2˙
a˙
R∂RRF+6a2˙
a˙
φ∂
RφF−6a2˙
a˙
λ, (20)
whe e F≡F(R, φ) and he subsc ip Rdeno es he de i a i e wi h espec o R. Eq. (20)is he poin -like Lag angian ha will be aken
in o accoun o he applica ion o he Noe he app oach discussed in de ails in he Appendices.
In he abo e minisupe space Qo configu a ions, he fi s p olonga ion o he Noe he ec o eads
X[1]=α∂
∂a+β∂
∂R+γ∂
∂φ +δ∂
∂λ +(˙
α−˙
ξ˙
a)∂
∂˙
a+(˙
β−˙
ξ˙
R)∂
∂˙
R+(˙
γ−˙
ξ˙
φ) ∂
∂˙
φ+(˙
δ−˙
ξ˙
λ) ∂
∂˙
λ.(21)
Imposing he exis ence o Noe he symme y
X[1]L+L˙
ξ=˙
g,(22)
we ob ain a sys em o 28 PDEs, lis ed in App. B. Neglec ing linea combina ions, he sys em educes o six di e en ial equa ions:
α∂RF−αλ+aβ∂
RRF+aγ∂RφF−aδ+2a∂RF∂aα−2aλ∂
aα+
+a2∂RRF∂aβ+a2∂RφF∂aγ−a2∂aδ−a∂RF∂ ξ+aλ∂
ξ=0
2α∂RRF+aβ∂
RRRF+aγ∂RRφF+a∂aα∂RRF+
+a∂Rβ∂
RRF−a∂ ξ∂
RRF=0
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A. Acunzo, F. Baja di and S. Capozziello Physics Le e s B 826 (2022) 136907
12α∂RφF+6aβ∂
RRφF+6aγ∂Rφφ F+6a∂aα∂RφF+
+6a∂φβ∂
RRF+6a∂φγ∂RφF−a2∂aδ−6a∂RφF∂ ξ=0
−12α−6a∂aα+6a∂λβ∂
RRF−a2∂aγ−6a∂λδ+6a∂ ξ=0
−3α−a∂φγ−a∂λδ+a∂ ξ=0
3αF−3αR∂RF−aRβ∂
RRF+aγ∂φF−aRγ∂RφF+
+aF ∂ ξ−aR∂RF∂ ξ=0.
(23)
The abo e sys em admi s he ollowing solu ion o he infini esimal gene a o
ξ( )=(3˜
k1+˜
c3) +k2,α(a)=˜
k1a,β=−2(˜
c3+3˜
k1)R,γ=c2,δ(λ)=˜
c3λ+˜
c3˜
c1,(24)
wi h ˜
ci,
˜
kicons an s. Mo eo e , i u ns ou ha wo di e en unc ions F(R, φ) a e selec ed by he Noe he symme ies and bo h o hem
co espond o he abo e gene a o . They ead:
FI(R,φ)=−˜
c1R+[2(˜
c3+3˜
k1)R]1−˜
c3
2(˜
c3+3˜
k1)Fφ+c2log[2(˜
c3+3˜
k1)R]
2(˜
c3+3˜
k1),(25)
FII(R,φ)=−˜
c1R+G(R)e˜
c3
c2φ(26)
whe e Fφ+c2log[2(˜
c3+3˜
k1)R]
2(˜
c3+3˜
k1)is an a bi a y in eg a ion unc ion which depends on he a gumen φ+c2log[2(˜
c3+3˜
k1)R]
2(˜
c3+3˜
k1). The
fi s unc ion is a solu ion o he sys em (23)i and only i he condi ion ˜
c3+3˜
k1= 0 holds, while he second elies on he condi ion
˜
c3+3˜
k1=0. These esul s show, in a s aigh o wa d way, how he Noe he symme ies selec models.
Fo he sake o simplici y, le us se m ≡2(˜
c3+3˜
k1), so ha FIcan be ew i en as
FI(R,φ)=−˜
c1R+m1−˜
c3
mR1−˜
c3
mFφ+c2log(mR)
m.(27)
In o de o ge exac cosmological solu ions, he ye unknown unc ion Fmus be ca e ully chosen. Conside ing ha all hose unc ions
which depend on he a gumen φ+c2log[2(˜
c3+3˜
k1)R]
2(˜
c3+3˜
k1)admi Noe he symme ies, we conside he simples choice, namely:
F1φ+c2log(mR)
m≡φ+c2log(mR)
m+k,
wi h kbeing any a bi a y cons an . Unde his assump ion, we ge
F1(R,φ)=−˜
c1R+km1−˜
c3
mR1−˜
c3
m+m1−˜
c3
mR1−˜
c3
mφ+m1−˜
c3
mR1−˜
c3
mc2log(mR)
m(28)
The model F1is pa icula ly in e es ing because, by se ing ˜
c3/m =−1, i educes o
F1(R,φ)
˜
c3
m=−1=−˜
c1R+km2R2+m2R2φ+c2mR
2log(mR), (29)
which ep esen s a non-local ex ension o he S a obinsky cosmological model.
Rega ding he second unc ion FII, by se ing G(R) =kRn, wi h k, n eal cons an s, we ob ain he s model
F2(R,φ)=−˜
c1R+kRne˜
c3
c2φ,(30)
which is a sligh gene aliza ion o he model conside ed in [23]. The model F2indeed, con ains he exponen ial non-local ac o ap-
pea ing in he supe - eno malizable and uni a y IDGs discussed in Re s. [18,43]. In o he wo ds, he exis ence o symme ies selec s
supe - eno malizable models.
No ice ha bo h F1(R, φ) and F2(R, φ) con ain highe o de cu a u e in a ian s and local scala fields, which can igge , in p inciple,
he ea ly- ime infla ion and he la e- ime cosmic accele a ion by means o ex a geome ic e ms.
4. Cosmological solu ions
We w i e now he cosmological Eule -Lag ange equa ions associa ed o a gene al F(R, φ) model. Then we eplace in o he sys em he
unc ions selec ed by he Noe he symme y and find ou he co esponding exac cosmological solu ions. The equa ions o mo ion coming
om he Lag angian (20)yield a sys em o fi e di e en ial equa ions:
5

A. Acunzo, F. Baja di and S. Capozziello Physics Le e s B 826 (2022) 136907
1
2F(R,φ)−1
2˙
φ˙
λ+(˙
H+3H2)∂RF+(2˙
H+3H2)λ +2Hd
d +d2
d 2(−∂RF+λ) =0(31.1)
R=−6(2H2+˙
H)(31.2)
¨
φ+3H˙
φ+12H2+6˙
H=0(31.3)
¨
λ+3H˙
λ+∂φF=0(31.4)
1
2F(R,φ)+1
2˙
φ˙
λ+3H2+3Hd
d λ+3(˙
H+H2)−3Hd
d ∂RF=0(31.5)
The las equa ion is he ene gy condi ion EL≡˙
qi∂L
∂qi−L, which co esponds o he (0,0) componen o he field equa ions. Eqs. (31.2),
(31.3) and (31.4)a e he cosmological exp ession o Rand he wo Klein-Go don equa ions o he scala fields φand λ, espec i ely. The
Eule -Lag ange equa ion wi h espec o he scale ac o is he cosmological F iedmann equa ion.
By eplacing he fi s unc ion F1(R, φ) in o eqs (31.1)-(31.5), i is possible o find he ollowing se s o solu ions:
I):
a( )=a0e R( )=−122φ( )=−1
3(40 +3k)−4 λ( )=576m35 −C3e−3
3−˜
c1,
wi h
=−1
12me (m<0), ˜
c3=−2m.
II)
a( )=a0 1
2R( )=0φ( )=C2λ( )=−˜
c1−2C3
√ .
In his case, he equa ions o mo ion p o ide u he cons ain s o he o m o he unc ion F1(R, φ), which u ns ou o be educed wi h
espec o Eq. (28). One possible solu ion is gi en by GR minimally coupled o a scala field, namely
F1(R,φ)=−˜
c1R+φ,
while, in he o he case, he ee pa ame e s a e cons ained such ha he unc ion akes he o m
F1(R,φ)=−˜
c1R+4√2(−˜
c3)5
4R5
4(φ +k). (32)
III) Finally, we ha e
a( )=a0 −10 R( )∼ −2φ( )∼C2+log( )λ( )=−˜
c1+C3 31 +C4m3 −4.
Specifically, i u ns ou ha he only unc ion associa ed o solu ions I) and III), which con ains symme ies and is compa ible wi h he
sys em in Eqs. (31.1)-(31.5) eads
¯
F1(R,φ)=−˜
c1R+km3R3+m3R3φ+m3R3c2log(mR)
m.(33)
Replacing he second unc ion (gi en by Eq. (30)) in o he sys em (31.1)-(31.5), bo h exponen ial and powe -law solu ions occu . As
be o e, depending on he solu ion conside ed, he unc ion F2 u ns ou o be u he cons ained by he equa ions o mo ion, wi h he
esul ha he in eg a ion cons an s a e cons ained acco ding o gi en addi ional ela ions (see App. C). The se o solu ions eads as:
I)
a( )=a0e
R( )=−122
φ( )=−4 +C2,
λ( )=−3n4n−1c2k−2n−1e
C2˜
c3
c2−4¯
c3
c2
3c2−4˜
c3−C3e−3
3−˜
c1,
n=3c2
4˜
c3+c2
,
II)
a( )=a0 p,
R( )=−6−p
2+2p2
2,
φ( )=C2−6p(2p−1)log( )
3p−1,
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A. Acunzo, F. Baja di and S. Capozziello Physics Le e s B 826 (2022) 136907
λ( )=
6n(3p−1)kseC2s(1−2p)p
2n
2−6p(2p−1)s
3p−1
−2n−6p(2p−1)s
3p−1+2n(6p−2)+3p2(4s−3)−6ps +1+C3 1−3p
1−3p−˜
c1
s=˜
c3
˜
c2
.
In he la e case, he pa ame e s n, p, sa e dependen acco ding o he ela ions p o ided in App. C.
As an example, by se ing n =1 om he beginning, he wo solu ions ake he o m:
I)n=1
a( )=a0e
R( )=−122
φ( )=−4 +C2
λ( )=−3ke
C2
2−2 −C3e−3
3−˜
c1
co esponding o he model:
¯
F3(R,φ)=−˜
c1R+kReφ
2.(34)
II)n=1
a( )=a p
R( )=−6−p
2+2p2
2
φ( )=C2−6p(2p−1)log( )
3p−1
λ( )=−˜
c1−k(3p−1)e
C2(3p−1)
6p−3 −2p
p−1+C3 1−3p
1−3p
s=3p−1
3(2p−1)=0,2
3,
co esponding o he model
¯
F3(R,φ)=−˜
c1R+kRe
3p−1
3(2p−1)φ.(35)
I is wo h no icing ha all physically in e es ing cosmological beha io s can be eco e ed, in pa icula accele a ing beha io s. They a e
s ic ly ela ed o he exis ence o he symme y ha allows o educe dynamics selec ing he o m o he in e ac ing Lag angian.
5. Discussion and conclusions
We conside ed e ec i e highe -o de IKG models desc ibed by he unc ion F(R, −1R), a non-local s aigh o wa d ex ension o F(R)
g a i y, showing ha hese models can po en ially allow bo h ea ly and la e- ime accele a ed expansion. In pa icula , infla ion would be
igge ed by highe -o de cu a u e in a ian s, while la e- ime expansion by non-local scala field φ≡−1R. This p esc ip ion allows o
in e p e da k ene gy as a geome ic con ibu ion, wi hou in oducing any exo ic fluid which, o da e, has ne e been obse ed di ec ly.
We used as a c i e ion o selec iable models he exis ence o Noe he symme ies o poin -like Lag angians desc ibing cosmological
dynamics. The Eule -Lag ange equa ions, associa ed o he symme ies, yield fi s in eg als o mo ion which allow o educe dynamics and,
e en ually, o find ou exac solu ions.
A key s ep owa ds he sea ch o symme ies in non-local heo ies, is he in oduc ion o he auxilia y local scala field φ≡−1R. I
implemen s a o mal “localiza ion p ocess” o he field −1R, so ha he heo y can be ecas in e ms o a local scala - enso heo y
F(R, φ), wi h he cons ain φ=R.
Noe he Symme y App oach can be applied o he poin -like Lag angian, by using he exis ence condi ion (A.12), which leads o a
sys em o 28 PDEs. We selec ed wo di e en models con aining symme ies and s udied he associa ed cosmological beha io s. Bo h
exponen ial and powe -law solu ions occu o he scale ac o and he applicabili y o ealis ic cosmological beha io s depends on he
ene gy anges ela ed o he pa ame e s.
As a gene al ema k, i is clea ha local and non-local con ibu ions wo k a di e en scales and his could be a conside able inpu
o add ess pa ame e ensions in cosmological beha io ecen ly epo ed, in pa icula he H0 ension, namely he disc epancy in he
alue o he Hubble pa ame e as ob ained om Cosmic Mic owa e Backg ound da a and kinema ic measu emen s ela ed o Cepheids
and Supe no ae. This issue, no due o sys ema ic expe imen al e o s, can ep esen a eal weakness o he CDM model. A possible
explana ion o such an incompa ibili y is o include u he deg ees o eedom in he g a i a ional sec o . These u he deg ees o
eedom, coming om local modifica ions o GR (see e.g. [51–53]) o non-local modifica ions [54], can alle ia e he H0and he o he
ensions. In pa icula , he combina ion o local and non-local co ec ions could add ess UV and IR beha io s o cosmic his o y.
Specifically, he wo gene al models (28) and (30)can be seen as iable e ec i e non-local ETGs ma ching he di e en cosmological
beha io s. In a o hcoming s udy, he abo e esul s will be ma ched wi h obse a ions.
7
A. Acunzo, F. Baja di and S. Capozziello Physics Le e s B 826 (2022) 136907
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing financial in e es s o pe sonal ela ionships ha could ha e appea ed o
influence he wo k epo ed in his pape .
Acknowledgemen s
F.B. and S.C. acknowledge he suppo o Is i u o Nazionale di Fisica Nuclea e (INFN) (inizia i e specifiche GINGER, MOONLIGHT2, and
QGSKY).
Appendix A. Noe he symme y app oach
Noe he symme ies o he Lag angian a e use ul o educe dynamics and analy ically sol e sys ems o di e en ial equa ions. In wha
ollows we b iefly in oduce he o mula ion o he Noe he Symme y App oach used in Sec. 3.
Le us conside he poin ans o ma ion (x, y) →(¯
x,
¯
y)and le εbe an a bi a y eal pa ame e such ha
¯
x=¯
x(x,y;ε), ¯
y=¯
y(x,y;ε). (A.1)
The fi s -o de Taylo expansion o he infini esimal ans o ma ion (A.1)a ound ε=0yields
¯
x(x,y;ε)=x+ε∂¯
x
∂εε=0=x+εξ(x,y)(A.2)
¯
y(x,y;ε)=y+ε∂¯
y
∂εε=0=y+εη(x,y). (A.3)
The unc ions ξ(x, y), η(x, y)a e he componen s o he angen ec o X o he o bi o he ans o ma ion a he poin (x, y), i.e.
X=ξ(x,y)∂
∂x+η(x,y)∂
∂y.(A.4)
Since (x, y)is an a bi a y poin , Eq. (A.4) p o ides he angen ec o field o he g oup o bi s, he so called infini esimal gene a o o he
one-pa ame e g oup o poin ans o ma ions. The p olonga ion o he angen ec o , in ol ing he n − h de i a i es, can be compu ed
by means o he ollowing ela ions:
¯
y≡d¯
y(x,y;ε)
d¯
x(x,y;ε)=y(∂ ¯
y/∂ y)+(∂ ¯
y/∂x)
y(∂ ¯
x/∂ y)+(∂ ¯
x/∂x)=¯
y(x,y,y;ε), (A.5)
¯
y ≡d¯
y
d¯
x=¯
y(x,y,y,y;ε), (A.6)
...
The p olonga ions o he gene a o Xcan be ob ained h ough a fi s -o de Taylo expansion a ound ε=0. Replacing Eqs. (A.2) and (A.3)
in o Eqs. (A.5) and (A.6), he n h de i a i es o he ans o med coo dina es (up o he fi s o de ) ead:
¯
y=y+εdη
dx −ydξ
dx =y+εη[1],(A.7)
.
.
.
¯
y(n)=y(n)+εdη(n−1)
dx −y(n)dξ
dx =y(n)+εη[n],(A.8)
whe e
η[n]≡dη(n−1)
dx −y(n)dξ
dx =dn
dxn(η−yξ)+y(n+1)ξ(A.9)
is he n h p olonga ion unc ion o η. The e o e, he n h p olonga ion o Noe he ’s ec o can be w i en as:
X[n]=X+η[1]∂y+... +η[n]∂y(n)(A.10)
S a ing om Eq. (A.10) and conside ing a poin -like Lag angian L =L
( ,q( ), ˙
q( )), wi h qibeing he coo dina es and he ime, he fi s
p olonga ion o Noe he ’s ec o can be w i en as
X[1]=ξ( ,q)∂
∂ +ηi( ,q)∂
∂qi+(˙
ηi−˙
ξ˙
qi)∂
∂˙
qi.(A.11)
The fi s Noe he Theo em s a es ha he one-pa ame e g oup o poin ans o ma ions gene a ed by Xis a one-pa ame e g oup o
Noe he poin symme ies o he dynamical sys em desc ibed by L, i and only i he e exis s a unc ion g , q( )such ha
8
A. Acunzo, F. Baja di and S. Capozziello Physics Le e s B 826 (2022) 136907
X[1]L+˙
ξL=˙
g,(A.12)
whose associa ed fi s in eg al o mo ion is:
I( ,q,˙
q)=ξ˙
qi∂L
∂˙
qi−L−ηi∂L
∂˙
qi+g.(A.13)
Appendix B. Sys em o di e en ial equa ions coming om Noe he ’s symme y exis ence condi ion
The sys em o di e en ial equa ions coming om he symme y exis ence condi ion is:
12a∂RF∂ α−12aλ∂
α+6a2∂RRF∂ β+6a2∂RφF∂ γ−6a2∂ δ+a3F∂aξ−a3R∂RF∂aξ=∂ag(B.1.1)
6a2∂RRF∂ α+a3F∂Rξ−a3R∂RF∂Rξ=∂Rg(B.1.2)
6a2∂RφF∂ α−a3∂ δ+a3F∂φξ−a3R∂RF∂φξ=∂φg(B.1.3)
−6a2∂ α−a3∂ γ+a3F∂λξ−a3R∂RF∂λξ=∂λg(B.1.4)
α∂RF−αλ+aβ∂
RRF+aγ∂RφF−aδ+2a∂RF∂aα−2aλ∂
aα+
+a2∂RRF∂aβ+a2∂RφF∂aγ−a2∂aδ−a∂RF∂ ξ+aλ∂
ξ=0(B.1.5)
6a2∂RRF∂Rα=0(B.1.6)
6a2∂RφF∂φα−a3∂φδ=0(B.1.7)
6a2∂λα+a3∂λγ=0(B.1.8)
6a∂RF∂aξ−6aλ∂
aξ=0(B.1.9)
12a∂RRFα+6a2∂RRRFβ+6a2∂RRφFγ+6a2∂RRF∂aα+12a∂RF∂Rα+
−12aλ∂Rα+6a2∂RRF∂Rβ+6a2∂RφF∂Rγ−6a2∂Rδ−6a2∂RRF∂ ξ=0(B.1.10)
12a∂RφFα+6a2∂RRφFβ+6a2∂Rφφ Fγ+6a2∂RφF∂aα+12a∂RF∂φα+
−12aλ∂
φα+6a2∂RRF∂φβ+6a2∂RφF∂φγ−a3∂aδ−6a2∂φδ−6a2∂RφF∂ ξ=0(B.1.11)
−12aα−6a2∂aα+12a∂RF∂λα−12aλ∂
λα+6a2∂RRF∂λβ+
−a3∂aγ+6a2∂RφF∂λγ−6a2∂λδ+6a2∂ ξ=0(B.1.12)
6a2∂RφF∂Rα+6a2∂RRF∂φα−a3∂Rδ=0(B.1.13)
−6a2∂Rα+6a2∂RRF∂λα−a3∂Rγ=0(B.1.14)
−3a2α−6a2∂φα+6a2∂RφF∂λα−a3∂φγ−a3∂λδ+a3∂ ξ=0 (B.1.15)
6a3∂aξ−6a2∂RφF∂λξ+6a2∂φξ=0(B.1.16)
−6a2∂RRF∂λξ+6a2∂Rξ=0(B.1.17)
−6a2∂RRF∂φξ−6a2∂RφF∂Rξ=0(B.1.18)
a3∂Rξ=0(B.1.19)
a3∂φξ=0(B.1.20)
a3∂λξ=0(B.1.21)
6a2∂λξ=0(B.1.22)
−6a2∂RφF∂φξ=0(B.1.23)
−6a2∂RRF∂Rξ=0(B.1.24)
−6a∂RF∂Rξ+6aλ∂Rξ−6a2∂RRF∂aξ=0(B.1.25)
−6a∂RF∂φξ+6aλ∂
φξ−6a2∂RφF∂aξ=0(B.1.26)
−6a∂RF∂λξ+6aλ∂
λξ+6a2∂aξ=0(B.1.27)
3αF−3Rα∂RF−aRβ∂
RRF+aγ∂φF−aRγ∂RφF+aF ∂ ξ−aR∂RF∂ ξ=a−2∂ g(B.1.28)
Appendix C. Rela ions among ee pa ame e s in he powe -low solu ion o he model F2(R, )
The wo equa ions which de e mine he ela ion among he h ee ee pa ame e s n, s, p, co esponding o he model F2o sec ion 3,
a e:
9