PPrepared for submission
Actions of Effective Field TheoriesOctober 10, 2020
Pierros Ntelis, a, a Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, FranceE-mail: pntelis
Abstract.
In this document, we briefly introduce the effective field theor ies . We proposesome novel ideas in this manuscript. We introduce a novel formalism of the effective fieldtheories and we apply it to the effective field theories of large scale structures. The newformalism is based on functionals of the actions composing those theories. We discuss ourfindings in a
Cosmological Gravitology framework. We present with a cosmological inferenceapproach these results and we give a guideline of how we can disentangle the best candidatebetween those models with some latest understanding of model selection.
Keywords: cosmology; general relativity; gravity; effective field theory; large scale structure;dark energy a r X i v : . [ phy s i c s . g e n - ph ] O c t ontents The Standard Cosmological Model (SMC) (with the best description from the data called Λ CDM model) provides a satisfactory agreement with current observations [1]. Modifica-tions of Gravity (MG) is an important step of understanding models beyond the SMC [2–4]. Ithas been studied recently an Effective Dark Energy Theory within the Hordenksi frameworkand the modifications of gravity [5]. The Akrami et al. [6] have studied a doubly coupledBi-Gravity cosmology where they have constrained their model by the binary Neutron Grav-itational Wave detection. This theories have been studied within a framework which we call
Cosmological Gravitology .At the core of these theories underlies the most succesful theory of Gravity, GeneralRelativity[7] (GR). This theory, assumes a four dimensional pseudo-Riemannian manifoldwith local interacting metric background that satisfies Lorenzt invariance. The standardGravity action (or General Relativity action, that contains the Einstein Hilbert action) isgiven by: S GR = c (cid:90) d x √− g (cid:20) R πG N + L m ( g µν , ... ) (cid:21) (1.1)where c is the speed of light, g is the background metric determinant of the background metric g µν of a massless graviton, R is the Ricci Scalar, G N is the newton gravitational constantand L m is the lagrangian density that describes the matter content of our universe. This As the
Cosmological Gravitology term suggest (etymologically writting), it is the study of different gravi-tational theories within the framework of cosmology. – 1 –agragian defines the energy-momentum tensor via, T µν = − √− g δS m δg µν where S m [ g µν , ... ] = c (cid:82) d x L m [ g µν , ... ] .In this work, we are going to review some important Modified Gravity (MG) modelsin a general frame and present some interesting alternatives ways of thinking the actions ofEffective Field Theories (EFT hereafter) and in extend the Effective Field Theories of LargeScale Structures (EFTofLSS hearafter) [8].As Porto [3] reminds us, any theory of physics attempts to describe, if possible, allobserved phenomena in simple mathematical laws or mathematical relationships in a unifiedframe. However, this kind of theory or theories are under investigation. In particular, fieldtheories are all these theories which do not attempt to be valid at all scales, often namelyEffective Field Theories.Following Ezquiaga and Zumalacárregui [2], we describe the way we build the theoreticalframework of an effective field theory of Dark Energy (DE) and Modified Gravity (MG), usingthe equation: S DE , MG = c (cid:90) d x √− g (cid:20) f [ R ]16 πG N + l [ L m ( g µν , ... )] (cid:21) (1.2)where f [ ... ] is any functional of the argument ..., in this case f [ R ] is the functional of f ( R ) cosmologies. The l [ ... ] is any functional of the argument ... , in this case l [ L m ( g µν , ... )] is anyfunctional of the matter lagrangian density component of the standard model. The parameterspace of the functionals f [ ... ] and l [ ... ] is large and under investigation from most of the DarkEnergy and Gravity. Most theories were investigated by experimenting with the right-handside of Eq. 1.2. In this work we are going to experiment with the left hand-side of thesetheories, i.e. left hand side of Eq. 1.2.The main motivation of these theories is to explain the current physical phenomena, andpossibly produce explanations of current unknown issues in the standard paradigm. In extendthese theories usually suppose to produce a new observable that has to be tested with currentor future experiments so that can be experimentally confirmed. In these lines, we proposea novel idea. We note that these theories can be reformulated and possibly produce novelformulations which might lead to novel way of thinking towards the understanding of thecurrent paradigm and possibly see beyond this standard paradigm. The main idea discussedhere is the generalisation of the possible actions of effective field theory. Here we give a brief summary of Effective Field Theories which result to theories whichexplain the large scale structures. For a comprehensive summary of Effective Field theoriesand modified gravity in cosmology, see Ezquiaga and Zumalacárregui [2], Porto [3], Cliftonet al. [4]. We reintroduce most of these theories with a extended schematic diagram as shownin Fig. 1. Here, we reintroduce some of the concepts, and the interested reader can see thenotation in Ezquiaga and Zumalacárregui [2].The standard General Relativity model with a background metric g µν of a masslessgraviton [7]. This theory is denoted in Fig. 1, where we also show how this theory is extendedby several authors.With red we present most of the models that break most of these assumptions suchas compactification of dimensions [9, 10], Non-local , ekpyrotic models Θ CDM,
Extra Dimen-sions [11, 12],
Lorentz Violations [13],
Einstein Aether models,
Horava models [14]. The DGPmodels[15, 16] are within this category, studying a 4D gravity in a 5D Minkowski space[15]– 2 – igure 1 . Cosmological Gravitology: In this simplified version scheme, we are trying to capturemost of the interconnections of the Effective Field Theories and their main constraints from differentobservables. Diagram extended and remodified from Ezquiaga and Zumalacárregui [2] [ See section 2]
Some of these models are constrained by the GW oscillations denoted by the blue and thecharacteristic black border which is shown in parts of the red contours of the diagram.Brown contours present the massive gravity models, m g > . With pink we show thedRGT models of resummation of massive gravity [17] .Blue contours show all the models which we add an additional field. This field canby a Tensor, T µν , and models such Bigravity and Multigravity are constrained from GWOscillations.Light blue contours present the models where we add an additional field which is avector, V µ , and models such as Proca m V > , General Proca and TeVeS (MOND) models.With Green contours, we show the models where we add an additional Field which isa scalar, φ . Most simple extensions of this models have been describe by Hordenski, whichwe re-introduce in section 2.1. These models include Love-Lock, Quintesence, Brans-Dicke, f ( R ) , KGB, Gauss-Bonnet, Galileon and the Galileon of unified Dark Energy and Dark Matter(UDEDM) models. The Galileon UDEDM where recently constrained by SuperNovae-Type Ia– 3 –s it is denoted from the diagram by green colors [18]. Most of these models are constrainedby Large Scale Structure observables as indicated by the characteristic black border. TheBeyond Hordenski models can by classified as the ones that have different metric within theHordenski framework, such as C ( X ) , D ( X ) , E ( X ) , which are modifications of the metric,which are not yet tested against observational data.Carrasco et al. [8] have introduced the notion of the Effective Field Theory of the Cos-mological Large Scale Structure (EFTofLSS for short). Theses theories have been successfulon predicting some aspects of the large scale structures [19, 20].Last but not least, we introduce to this diagram, the newly proposed ideas. These ideasare some basic manipulations of the action. These manipulations potentially may produceseveral novel actions and potentially novel theories. The name we give for these theories isthe Actions of Effective Field Theories. A possible application are the Actions of EFT, whichwe represent it with the orange color in the diagram Fig. 1, and we are going to describe itbelow in section 2.2. In 1974 Horndeski [21] has formulated a Generalized D ( = 3space + 1time ) theory of gravity.Hordenski’s theory find several applications in physics in general. From explaination of Grav-itational waves[22] to black hole models[22–24]. Recently there are several efforts to explainthe large scale structures. This theory was reformulate to contain several other paradigms,such as that of inflation, as Kobayashi et al. [25] have showed.Modern Theories of Hordenski are built using the action principle: S H = (cid:90) d x √− g L H (2.1)where S H is the hordenski action, g is the determinant of g µν , where g µν is the Jordan framemetric.Recently, Charmousis [23], Kobayashi et al. [25], Ezquiaga and Zumalacárregui [26] re-formulated theses theories (at 2nd order EoM) as: S H [ g µν , φ, ψ M ] = (cid:90) d x √− g (cid:34) (cid:88) i =2 πG N L i [ g µν , φ ] + L m [ g µν , ψ M ] (cid:35) (2.2)with the lagrangian densities given by:• L = G ( φ, X ) • L = − G ( φ, X ) (cid:3) φ • L = G ( φ, X ) R + G ,X ( φ, X ) (cid:2) ( (cid:3) φ ) − φ ; µν φ ; µν (cid:3) • L = G ( φ, X ) G µν φ ; µν − G ,X ( φ, X ) (cid:2) ( (cid:3) φ ) + 2 φ ; µν φ ; να φ ; µ ; α − φ ; µν φ ; µν (cid:3) φ (cid:3) .Here G N is the Newton’s constant, L m represent the matter lagrangian where ψ m are thematter fields, G i , i ∈ [2 − are generic functions of scalar field φ , and the kinetic term, X = g µν φ ; µ φ ; ν . R, G µν are the Ricci and Einstein tensors. The repeated indices aresummed over following Einstein’s convention. Here, semicolon, ";", denotes the usual co-variant derivative[27] φ ; µ = ∇ µ φ , and comma ",", indicate partial derivatives (cid:3) φ = g µν φ ; µν .The free parameters of this theory, in particular from L and L are strongly constrained by– 4 –irect measurements of the speed of gravitational waves, following Lombriser and Taylor [28].Note that ψ M is some matter field.The generic functions quantify different modern Hordenski models, symbolically, as fol-lows: L H − L m ∝ G − G (cid:3) φ + G R − G ,X {∇∇ φ } + G G µν φ ; µν − G ,X {∇∇ φ } (2.3)• G : quitessence, k-essence (minimal coupling)• G : kinetic gravity braiding (derivative interactions)• G : generalized Brans-Dicke, f ( R ) (non-minimal coupling)• G ,X : covariant Galileon (non-minimal derivative coupling)• G : Gauss-Bonnet (non-minimal 2nd derivative coupling)These models are constraint by currented observations such as using observables fromthe large scale structure surveys as well as gravitational waves observables.There were several beyond hordenski schemes, among which the Gleyzes et al. [29], haveadded some more lagrangians to the aforementioned system and Ezquiaga and Zumalacárregui[2] have added modifications to the metric using the components, C ( X ) , D ( X ) , E ( X ) . Here we touch one of the fundamentals of the theoretical arguments that most of effective fieldtheories are built upon. Instead of formulating the right-hand side of Eq. 1.2, we constructone basic action of EFT, by reformulating the left hand side of Eq. 1.2 as: S EFT ⊃ (cid:90) Ω S dS (cid:48) , (2.4)where (cid:82) Ω S dS (cid:48) is an integral over all possible space of these kind of actions, Ω S . Now, we canimagine some simple functionals that this S EFT can have, such that of a simple integral: S EFT ⊃ (cid:90) Ω S dS (cid:48) S (cid:48) , (2.5)Now, we play a bit further and we assume that there are different functionals of these actionsas S EFT ⊃ (cid:90) Ω S dS (cid:48) (cid:8) F [ S (cid:48) ] + L [ S (cid:48) ] (cid:9) . (2.6)where F [ S (cid:48) ] and L [ S (cid:48) ] are some generic functionals of the elements S (cid:48) of the action space, Ω S .Some of these terms might also include terms of the form of: S EFT ⊃ (cid:90) Ω S dS (cid:48) S (cid:48) µνρσ S (cid:48) µνρσ (2.7)where S (cid:48) µνρσ is an action which has been promoted to a Remaniann tensor of some actionelement S (cid:48) . Note that this kind of modelling includes the actions which model the cosmicinflation [30, 31]. – 5 –ow we simplify one of the actions as expressed by Eq. 2.6-action as: S SimplifiedEFT = (cid:90) Ω S dS (cid:48) (cid:110) α ( S (cid:48) ) S (cid:48) + β ( S (cid:48) ) + S (cid:48) + γ ( S (cid:48) ) ( S (cid:48) ) (cid:111) . (2.8)where α ( S (cid:48) ) , β ( S (cid:48) ) , γ ( S (cid:48) ) is a parametrisation of one of the functionals, i.e. the functional F, eachone attached to the corresponding element S (cid:48) . We omit the functional L for simplification.Now we imagine that the space of actions can be simplified in having only two actions, S and S . In this case we can write a more simplified action as : S Simplified (cid:48)
EFT = (cid:110) α ( S ) S + β ( S ) + S + γ ( S ) ( S ) (cid:111) + (cid:110) α ( S ) S + β ( S ) + S + γ ( S ) ( S ) (cid:111) . (2.9)In the following subsection (i.e. sections 2.2.1, 2.2.3), we discuss the limits of one ofthese theories, i.e. the theories governed by Eq. 2.4. Note that in the limit of the integral actions become a sum of some action, we recover thesimple theory of General Relativity. In particular, we have that: Eq. 2.4 reduces to Eq. 1.1,if we assume that the Integral of possible actions, (cid:82) Ω S dS (cid:48) reduces to a simple sum of actions (cid:80) i =1 , S i , and therefore we write: S EFT ⊃ (cid:90) Ω S dS (cid:48) → (cid:90) So dS (cid:48) GR limit −− → (cid:88) i =1 , S i = S R + S m = c (cid:90) d x √− g R πG N + c (cid:90) d x √− g L m ( g µν , ... ) S GR ≡ c (cid:90) d x √− g (cid:20) R πG N + L m ( g µν , ... ) (cid:21) . (2.10) In a similar limit where the integral of actions becomes a sum of some actions, we can alsorecover the simple theory of Hordenski. In particular, we have that Eq. 2.4 reduces to theEq. 2.2. This is shown simply as: S EFT ⊃ (cid:90) Ω S dS (cid:48) → (cid:90) So dS (cid:48) Hordenski limit − − −− → (cid:88) i =1 S i = (cid:32) (cid:88) i =2 S i (cid:33) + S m S H ≡ (cid:90) d x √− g (cid:34) (cid:88) i =2 πG N L i [ g µν , φ ] + L m [ g µν , ψ M ] (cid:35) . (2.11) In a similar limit where the integral of actions becomes a sum of some actions, we can alsorecover the simplest inflationary paradigm which is described generally via the inflation action,– 6 – I , [32, 33]. In particular, we have that Eq. 2.4 reduces to : S EFT ⊃ (cid:90) Ω S dS (cid:48) → (cid:90) So dS (cid:48) Inflation limit − − −− → (cid:88) i =1 S i = S + S (2.12) S I ≡ (cid:90) d x √− g πG N (cid:20) R + R M (cid:21) . Here we give an example of how these theories contain the Effective Field Theories of LargeScale Structures as expressed in Baumann et al. [19]. Baumann et al. [19] have shown thatthe EFTofLSS is basically expressed via "effective stress-energy via Einstein - deductive ap-proach.". Using this approach one defines the UV-IR coupling of cosmological fluctuationsto rise from a reorganisation of the Einstein Field Equations (EFE). In particular, the Ein-stein tensor is decomposed in a homogeneous background (denoted by a bar) and terms thatare Linear (L) and Non Linear (NL) in the metric perturbations, collectively denoted by δX ( t, (cid:126)x ) = X L ( t, (cid:126)x ) − ¯ X ( t ) . The EFE are rewritten as: ¯ G µν (cid:2) ¯ X (cid:3) + ( G µν ) L [ δX ] + ( G µν ) NL (cid:2) δX (cid:3) = 8 πG N c T µν . (2.13)Note that the linear background equation, i.e. ¯ G µν (cid:2) ¯ X (cid:3) = πG N c ¯ T µν and the linearised EFE, ( G µν ) L [ δX ] = πG N c ( T µν ) L are defined in the standard way. While the non-linear EFE canbe written in a form which is similar to the linear EFE, i.e.: ¯ G µν (cid:2) ¯ X (cid:3) = 8 πG N c (cid:0) τ µν − ¯ T µν (cid:1) , (2.14)where they have defined the effective stress-energy pseudo-tensor: τ µν ≡ T µν − c ( G µν ) NL πG N (2.15)Therefore, we can deduce the above formalism in a modification of the Einstein-Hilbert actionas: S EFTofLSS = c (cid:90) d x √− g (cid:20) R πG N + f L ( R ) + f NL ( R ) + L m ( g µν , ... ) (cid:21) (2.16)From the Eq. 2.16, one can perform a variational principle, from which we have the followingmathematical correspondence:1. R πG N corresponds to ¯ G µν (cid:2) ¯ X (cid:3) f L ( R ) corresponds to ( G µν ) L [ δX ] f NL ( R ) corresponds to ( G µν ) NL (cid:2) δX (cid:3) L m ( g µν , ... ) corresponds to πG N c T µν .Therefore, our new formalism basically contains the EFTofLSS in the following way. S EFT ⊃ S
EFTofLSS = (cid:88) i =1 , , , S i (2.17) = S R + S f L ( R ) + S f NL ( R ) + S m (2.18) = c (cid:90) d x √− g (cid:20) R πG N + f L ( R ) + f NL ( R ) + L m ( g µν , ... ) (cid:21) (2.19)– 7 – .3 Strings limits Polyakov [34] have studied the action of string theory dynamics [35, 36], and successfullyquantized string theory. Here we show that the actions of Effective Field Theory is alsoreduced to the one of the actions of the String theory, simply as: S EFT ⊃ (cid:90) Ω S dS (cid:48) → (cid:90) So dS (cid:48) Strings limit − − −− → S String = T (cid:90) d σ √− hh ab g µν ( x ) ∂ a x µ ( σ ) ∂ b x ν ( σ ) (2.20)where T is the string tension, g µν is the metric of any targeted manifold of a D-dimensionalspace and x µ ( σ ) is the coordinate of the targeted manifold. Moreover, h ab is the worldsheetmetric, ( h ab is its inverse), and h is as usual the determinant of the h ab . The signature of themetrics is chosen so that the timelike directions are positive, + , while the spacelike directionsare negative, − . The spacelike coordinate is denoted with σ , while the timelike coordinate isdenoted with τ . In order to properly and systematically study these theories, one would proceed as follows:1. Compute the Equations of Motion using the standard variational principle approach, δ S = 0 , from these theories(a) Express the corresponding Friedman background equations analogues(b) Express Gravitational waves observables analogues(c) or also express the Gravitational Waves on these Friedmann background equationsanalogues2. To confront it to the observational data, using the above methodology, one would ex-press:(a) the Large Scale Structure Clustering statistics such as the n-order correlations func-tions or their corresponding Fourier power spectra, bi-spectra, ...) from angle po-sitions of the tracers and their corresponding redshifts ( ˆ θ, ˆ φ, ˆ z ). Also relations thatexpress the angular distance, motion distance, volume distance would also be neces-sary to be computed in the analogues of these theories, ∝ { d A ( z ) , d M ( z ) , d V ( z ) , H ( z ) } ,or Modification of Gravity Observables analogues such as the modifications of pois-son equation, anisotropic stress or lensing potentials ∝ (cid:8) µ ( z, k ) , η ( z, k ) (cid:39) ΨΦ , Σ( z, k ) (cid:9) [1]. Telescopes that can be used here are the SDSS[37], DESI[38], Euclid[39].(b) SNIa luminocity distance diagrams, ∝ d L ( z ) [18](c) Angular correlation functions that summarise the Cosmic Microwave Backgroundmaps, from telescopes such as the Planck [1].(d) Standard sirens observables from LIGO/VIRGO [40], LISA/Einstein Telescope [41]To study those things systematically, it would be necessary to build simulations of theseobservables for specific surveys, test these observables in those simulations. Then apply theseobservables to data, with current state-of-the-art model selection methods.– 8 –n this cosmological inference analysis, we simplify things and we proceed as follows.We start by the basics of cosmological statistical inference[42], which lies from the basics ofa generalised Baye’s Theorem: P ( θ | d, M ) = P ( d | θ, M ) P ( θ | M ) P ( d ) (2.21)where P ( θ | d, M ) is the posterior probability, likelihood of the parameters of physical param-eters of interest, θ , given some d , and some theory M. P ( d | θ, M ) is the probability of thedata, d , given some parameters, θ and some model M . P ( θ, M ) is the prior probability of theparameters of interest, θ , of the theory M , and P ( d ) is the bayesian evidence of the data.Note that the normalising constant, namely Bayesian evidence is defined as: P ( d ) = (cid:90) M (cid:90) θ P ( d | θ, M ) = (cid:90) M · · · (cid:90) M n (cid:90) (cid:126)θ · · · (cid:90) (cid:126)θ n p ( d | M ( (cid:126)θ ) . . . M n ( (cid:126)θ n )) (2.22)where note that we have adopted the notation (cid:82) x f ( x ) = (cid:82) X f ( x ) dx which implies the usualRemannian integration. The Bayesian evidence is irrelevant for parameter inference. Usually,the set of parameters θ can be divided in some physically interesting quantities φ and a set ofnuisance parameters n . The posterior obtained by eq. 2.21 is the joint posterior for θ = ( φ, n ) .The marginal posterior for the parameters of interest can now be written as (marginalisingover the nuisance parameters). L ( φ | d, M ) ≡ P ( φ | d, M ) ∝ (cid:90) P ( d | φ, n, M ) P ( φ, n | M ) dn . (2.23)This pdf is the final inference on φ from the joint posterior likelihood. The followingstep, to apprehend and exploit this information, is to explore the posterior. In this work, we are firstly interested to constrain the action models, and not select amongthe best candidate. So if we apply the cosmological inference method, Eq. 2.23, for someinteresting parameters, such as the physical parameters of the model S EFT , we can disentagledifferent models which build the building blocks of nature for Large Scale Structures.We could also consider that a physical quantity is the actual action, φ = S (cid:48) , (or usingany general formulation of an action Eq. 2.4 i.e. φ = S EF T ) using the method describedin Eq. 2.23, which is basically model selecting among the different scenarios of actions thatwe have built. In practice, this is somewhat a large computational problem and somewhatabstract. In order to simplify things and direct ourselves to a more realistic approach, wesimplify things here, and we proceed as follows:We start by using Eq. 2.8 simplified a bit further as: S Simplified , = β ( S ) S + α ( S ) + S (3.1)We make an assumption here, that S has some simple polynomial form: S ( x ; φ , n ) = e − . φ x − . /n ) (3.2)and S ( x ; φ , n ) = φ x + n (3.3)– 9 –here x is some variable and φ i , n i are some physical and nuisance parameter resepctivelyfor the model S i , where i = 1 , . Now we consider that α S , β S , α S , are the variables of themodel. All this parameters we assume that they are in real space (cid:8) φ i , n i , β ( S ) , α ( S ) ∈ R (cid:9) .These means that this model can be written in the form: S Simplified , = β ( S ) e − . φ x − . /n ) + α ( S ) + φ x + n (3.4) We make clear that the theoretical model is described by: m th ( x ; Θ ActionEFT ) = β ( S ) e − . φ x − . /n ) + α ( S ) + φ x + n (3.5)where: Θ ActionEFT = ( β ( S ) , φ , n , α ( S ) , φ , n ) (3.6)The simulated data are defined as: d sim ( x ; Θ ActionEFTsim ) = e − . x − . + 1 + 2 x + 0 (3.7)where notice that we have assumed that Θ ActionEFTsim = ( β ( S ) , φ , n , α ( S ) , φ , n ) = (1 , , , , , (3.8)Theoretical simulated uncertainty is described by a gaussian approximation variance as: σ sim ( x ) = d sim ( x ; Θ ActionEFTsim ∗ . (3.9) We use a simple likelihood within the gaussian approximation limit with a diagonal covarianceas: − L = (cid:88) i (cid:104) d sim ( x i ) − m th ( x i ; α ( S ) , φ , n , β ( S ) , φ , n ) (cid:105) σ − ( x i ) . (3.10)Note that this is a simplification, and an interested reader can used more complex likelihoodsthat the aforementioned one. To sample the aforementioned likelihood, we use a modifiedversion of PYMC , as it is integrated in the COSMOPIT . Table 1 . Prior information on the parametrisation Θ ActionEFT . [See section 3.1, section 3.2 ]
Parameter name [min,max] µ θ i σ θ i Type β ( S ) [0.0,1.9] - - Uniform φ [0.0,4] - - Uniform n [-1,1.9] 1.3 0.1 Gaussian α ( S ) [-1.0,1.0] - - Uniform φ [1.1,2.9] - - Uniform n [1,3.0] 2.0 0.1 Gaussian Open source software
PYMC https://pymc-devs.github.io/pymc/ – 10 –e assume for the sampled parameters, some prior information as expressed in thetable 1. This table presents this information as follows. The first column is the Parametername, the second column shows the minimum and maximum of the parameter, the thirdcolumn shows the mean of the Gaussian prior of this parameter if any, µ θ i , the forth columnshows the 1 standard deviation of this parameter, σ θ i and the last column shows the type ofthe parameter, which is either uniform or a gaussian one. x S S i m p li f i e d , E F T ( x ) Simulated dataModel2 sim sim Figure 2 . The simplified action Effective Field Theory (EFT) model and some simulated data, asthey described in section 3.1 and 3.3. in this section we present the numerical results of section 3.1 and section 3, respectively.In Fig. 2 we show the comparison of the model with the simulated data, assuming the Θ ActionEFTsim . We show that this is an interesting way to constrain these kind of models.In Fig. 3, we present the results of a Monte Carlo Markov Chains (MCMC) samplingusing the χ = − L described in section 3. The figure is a corner plot of the MCMC outputthe parametrisation of model and the likelihood L as described in section 3. In the upperright corner-plot panel, we show the L as estimated of the MCMC output, as well as the cor-responding number of degrees of freedom, ndf and the corresponding error of the ndf , namely √ ndf . The diagonal of the corner matrix plot is the marginalised probability distribution ofeach parameter. At the right top legend of of diagonal element of the corner matrix plot, wepresent the MCMC output result of each parameter with the mean, standard deviation esti-mation ( (cid:39) C.L. ), as well as the mode, i.e. the value of the parameter which corresponds– 11 – .8 1.0 1.20123 ( ( S ) ) / m a x =-0.03 2 30123 2.655 ± 0.253, mode=2.640.8 1.0 1.21.01.5 n =-0.04 2 31.01.5 =0.87 1.00 1.25 1.50 1.750123 1.304 ± 0.100, mode=1.320.8 1.0 1.20.50.00.5 ( S ) =-0.25 2 30.50.00.5 =0.19 1.00 1.25 1.50 1.750.50.00.5 =0.02 0.5 0.0 0.50123 0.006 ± 0.114, mode=0.020.8 1.0 1.21.92.02.1 =-0.62 2 31.92.02.1 =0.24 1.00 1.25 1.50 1.751.92.02.1 =0.05 0.5 0.0 0.51.92.02.1 =0.45 1.9 2.0 2.10123 2.005 ± 0.033, mode=2.000.8 1.0 1.2 ( S ) n =0.01 2 3 n ( S ) n ± 2 ndf =11.26±13.71 ,ndf=94.00 Figure 3 . MCMC corner plot of the simplified action Effective Field Theory of Large ScaleStructures (Action EFT) model and some simulated data. The model described here is a sim-ple parametrisation of Action EFT models, described by Eq. 3.4 with the parameters described by Θ ActionEFT = ( β ( S ) , φ , n , α ( S ) , φ , n ) . The parameters are described with JPDF of 68% (95%)shaded (lighted) area contours. [see section 3.1, 3.3 and section 3.2]. to the maximum value of the corresponding probability distribution function. In the off di-agonal panels of each corner plot, we present the Joint Probability Distributions Functions(JPDF) of the combinations of two parameters for each parametrisation case. These JPDFsare described with (shaded area) and (lighted area) contours. In the legend of theoff diagonal panels of the corner plot we show the correlation coefficient for the combinationof the two parameters, ρ = C kl / √ C kk C ll . Note that these action theories, modelled by S EFT are already constrained by standard observ-ables, such as (cid:8) d A ( z ) , d M ( z ) , d V ( z ) , H ( z ) , d L ( z ) , µ ( z, k ) , η ( z, k ) (cid:39) ΨΦ , Σ( z, k ) (cid:9) as were revisedin section 2.4, since these observables directly give constrains on some EFT theories which areincluded in the reformulation of the Action of EFT. Therefore we expect that the extendedmodels that the Action of EFT provides are now contrained by these observables. Examplesof these theories can be tracked by the EFT Theories, such as modifications of standard Gen-eral Relativity Theories, S EFT ⊃ S EH or the hordenski theories, S EFT ⊃ S H , as it is shownby Ezquiaga and Zumalacárregui [2] – 12 – Conclusions
In this work, we briefly summarise the Effective Field Theory (EFT). We reintroduce someof the models that represent these models with emphasis given in the Hordenski models.We propose novel ideas of reformulating the Action principle. This is a way of describingthe Effective Field Theory and in extend it can be applied to the Effective Field Theory ofLarge Scale Structures with arguments that rise from the Action principle, namely Actionof EFT and possibly extend these theories, in a more concrete mathematical framework.We provide a guide of constraining these models systematically with the latest cosmologicalinferences. We express how some simple classes of these models can be constrained using aGaussian approximated likelihood analysis with some numerically simulated data and errors.Our theories may be able to offer a viable alternative or complementarity to Λ CDM,or modifications of gravity or more generally the aforementioned Effective Field Theories ofLarge Scale Structures. To confirm this statement a more detailed theoretical and numericalanalysis is required. We plan to present such an analysis in a future work. Another possibleroute would be to reconsider what is beyond the variational principle, or the principle of leastaction, for e.g. S [ δ S ] . O . E . D . AKNOWLEDGEMENTS
PN is funded by ’Centre National d’Études Spatiales’ (CNES).PN would like to thank Federico Piazza for his inspirational phrase: "Here, we are tryingto open new possibilities!".We acknowledge open libraries support
IPython [43],
Matplotlib [44],
NUMPY [45]
SciPy1.0 [46]
COSMOPIT [47, 48].
A The matter fields
As [49, 50] have shown, the matter fields has one main component, the higgs fields, whichgives mass to the other field-particles of the standard model of particle physics. In particularthe Action for the Higgs field is the following: S m , Higgs = c (cid:90) d x √ η L Higgs (cid:104) η µν , (cid:126)φ ( η ) , A µ (cid:105) (A.1)where η is the determinant of the Minkovski metric, η µν , which is taken as, –+++, (cid:126)φ ( η ) is a vector of real scalar fields and A µ is a real vector field used for the interactions. Thelagrangian density is composed as: L Higgs (cid:104) η µν , (cid:126)φ ( η ) , A µ (cid:105) = −
12 ( ∇ φ ) −
12 ( ∇ φ ) − V (cid:0) φ + φ (cid:1) − F µν F µν (A.2)where φ i ≡ φ i ( η µν ) , i = 1 , are the two real scalar fields which interact with the A µ fieldand ( ∇ φ i ) ≡ ∇ µ φ i ∇ µ φ i ≡ η µν ∇ µ φ i ∇ ν φ i . Note that this ∇ µ is different than the ∇ µ insection 2.1. Here this ∇ µ is defined as: ∇ µ φ = ∂ µ φ − eA µ φ (A.3) ∇ µ φ = ∂ µ φ − eA µ φ (A.4) F µν = ∂ µ A ν − ∂ ν A µ (A.5)– 13 –here e is a dimensionless coupling constant. Note that L Higgs (cid:104) η µν , (cid:126)φ ( η ) , A µ (cid:105) is invariantunder simultaneous gauge transformation of the first kind on φ ± iφ and of the second kindon A µ . In the case where V (cid:48) ( φ ) = 0 and V (cid:48)(cid:48) ( φ ) > , where φ is the ground state of either φ i , then spontenous breakdown of U(1) symmetry occurs. References [1] Aghanim, N., Y. Akrami, M. Ashdown, et al. Planck 2018 results. vi. cosmologicalparameters. arXiv preprint arXiv:1807.06209 , 2018.[2] Ezquiaga, J. M. and M. Zumalacárregui. Dark Energy in light of Multi-MessengerGravitational-Wave astronomy.
Frontiers in Astronomy and Space Sciences , 5:44, 2018. arXiv:astro-ph.CO/1807.09241 .[3] Porto, R. A. The effective field theorist’s approach to gravitational dynamics.
PhysicsReports , 633:1–104, 2016. arXiv:hep-th/1601.04914 .[4] Clifton, T., P. G. Ferreira, A. Padilla, et al. Modified gravity and cosmology.
PhysicsReports , 513:1–189, 2012. arXiv:astro-ph.CO/1106.2476 .[5] Perenon, L., C. Marinoni, and F. Piazza. Diagnostic of horndeski theories.
Journal ofCosmology and Astroparticle Physics , 2017:035, 2017.[6] Akrami, Y., P. Brax, A.-C. Davis, et al. Neutron star merger gw170817 strongly con-strains doubly coupled bigravity.
Physical Review D , 97:124010, 2018.[7] Einstein, A. Kosmologische und relativitatstheorie.
SPA der Wissenschaften , 142, 1917.[8] Carrasco, J. J. M., M. P. Hertzberg, and L. Senatore. The effective field theory ofcosmological large scale structures.
Journal of High Energy Physics , 2012:82, 2012.[9] Bailin, D. and A. Love. Kaluza-klein theories.
Reports on Progress in Physics , 50:1087,1987.[10] Overduin, J. M. and P. S. Wesson. Kaluza-Klein gravity.
Physics Reports , 283:303–378,1997. arXiv:gr-qc/gr-qc/9805018 .[11] Antoniadis, I., N. Arkani-Hamed, S. Dimopoulos, et al. New dimensions at a mil-limeter to a fermi and superstrings at a TeV.
Physics Letters B , 436:257–263, 1998. arXiv:hep-ph/hep-ph/9804398 .[12] Randall, L. and R. Sundrum. Large Mass Hierarchy from a Small Extra Dimension.
Physical Review Letters , 83:3370–3373, 1999. arXiv:hep-ph/hep-ph/9905221 .[13] Blas, D. and E. Lim. Phenomenology of theories of gravity without Lorentz invariance:The preferred frame case.
International Journal of Modern Physics D , 23:1443009, 2014. arXiv:gr-qc/1412.4828 .[14] Hořava, P. Quantum gravity at a Lifshitz point.
Physical Review D , 79:084008, 2009. arXiv:hep-th/0901.3775 . – 14 –15] Dvali, G., G. Gabadadze, and M. Porrati. 4D gravity on a brane in 5D Minkowski space.
Physics Letters B , 485:208–214, 2000. arXiv:hep-th/hep-th/0005016 .[16] Nicolis, A. and R. Rattazzi. Classical and Quantum Consistency of the DGP Model.
Journal of High Energy Physics , 2004:059, 2004. arXiv:hep-th/hep-th/0404159 .[17] de Rham, C., G. Gabadadze, and A. J. Tolley. Resummation of Massive Gravity.
PhysicalReview Letters , 106:231101, 2011. arXiv:hep-th/1011.1232 .[18] Koutsoumbas, G., K. Ntrekis, E. Papantonopoulos, et al. Unification of dark matter-darkenergy in generalized Galileon theories.
Journal of Cosmology and Astroparticle Physics ,2018:003, 2018. arXiv:gr-qc/1704.08640 .[19] Baumann, D., A. Nicolis, L. Senatore, et al. Cosmological non-linearities as aneffective fluid.
Journal of Cosmology and Astroparticle Physics , 2012:051, 2012. arXiv:astro-ph.CO/1004.2488 .[20] Pajer, E. and M. Zaldarriaga. On the renormalization of the effective field theory oflarge scale structures.
Journal of Cosmology and Astroparticle Physics , 2013:037, 2013. arXiv:astro-ph.CO/1301.7182 .[21] Horndeski, G. W. Second-order scalar-tensor field equations in a four-dimensionalspace.
International Journal of Theoretical Physics , 10:363–384, 1974. https://link.springer.com/article/10.1007/BF01807638 .[22] Babichev, E., C. Charmousis, G. Esposito-Farese, et al. Stability of a black holeand the speed of gravity waves within self-tuning cosmological models. arXiv preprintarXiv:1712.04398 , 2017.[23] Charmousis, C. From lovelock to horndeski?s generalized scalar tensor theory. In
Modifi-cations of Einstein’s Theory of Gravity at Large Distances , pages 25–56. Springer, 2015.[24] Babichev, E., C. Charmousis, G. Esposito-Farèse, et al. Hamiltonian vs stability andapplication to horndeski theory. arXiv preprint arXiv:1803.11444 , 2018.[25] Kobayashi, T., M. Yamaguchi, and J. Yokoyama. Generalized g-inflation: ?inflationwith the most general second-order field equations?
Progress of Theoretical Physics ,126:511–529, 2011.[26] Ezquiaga, J. M. and M. Zumalacárregui. Dark energy after gw170817: dead ends andthe road ahead.
Physical review letters , 119:251304, 2017.[27] Wikipedia contributors. Covariant derivative — Wikipedia, the free encyclopedia,2018. URL https://en.wikipedia.org/w/index.php?title=Covariant_derivative&oldid=842130744 . [Online; accessed 16-August-2018].[28] Lombriser, L. and A. Taylor. Breaking a dark degeneracy with gravitational waves.
Journal of Cosmology and Astroparticle Physics , 2016:031, 2016.[29] Gleyzes, J., D. Langlois, F. Piazza, et al. Exploring gravitational theories be-yond Horndeski.
Journal of Cosmology and Astroparticle Physics , 2015:018, 2015. arXiv:astro-ph.CO/1408.1952 . – 15 –30] Starobinsky, A. A. Dynamics of phase transition in the new inflationary universe scenarioand generation of perturbations.
Physics Letters B , 117:175–178, 1982.[31] Guth, A. H. and S.-Y. Pi. Fluctuations in the new inflationary universe.
Phys. Rev.Lett. , 49:1110–1113, 1982.[32] Starobinsky, A. A new type of isotropic cosmological models without singularity.
PhysicsLetters B , 91:99 – 102, 1980.[33] Planck Collaboration, Y. Akrami, F. Arroja, et al. Planck 2018 results. X. Constraints oninflation. arXiv e-prints , page arXiv:1807.06211, 2018. arXiv:astro-ph.CO/1807.06211 .[34] Polyakov, A. M. Quantum geometry of bosonic strings.
Physics Letters B , 103:207–210,1981.[35] Deser, S. and B. Zumino. Consistent Supergravity.
Phys. Lett. B , 62:335, 1976.[36] Brink, L., P. Di Vecchia, and P. S. Howe. A Locally Supersymmetric and Reparametriza-tion Invariant Action for the Spinning String.
Phys. Lett. B , 65:471–474, 1976.[37] Eisenstein, D. J., I. Zehavi, D. W. Hogg, et al. Detection of the Baryon Acoustic Peakin the Large-Scale Correlation Function of SDSS Luminous Red Galaxies.
AstrophysicalJournal , 633:560–574, 2005. arXiv:astro-ph/astro-ph/0501171 .[38] Aghamousa, A., J. Aguilar, S. Ahlen, et al. The desi experiment part i: Science, targeting,and survey design. arXiv preprint arXiv:1611.00036 , 2016.[39] Amendola, L., S. Appleby, A. Avgoustidis, et al. Cosmology and Fundamental Physicswith the Euclid Satellite.
ArXiv e-prints , 2016. arXiv:1606.00180 .[40] Abbott, B. P., R. Abbott, T. D. Abbott, et al. Prospects for observing and localiz-ing gravitational-wave transients with advanced ligo, advanced virgo and kagra.
LivingReviews in Relativity , 21:3, 2018. arXiv:gr-qc/1304.0670 .[41] Caprini, C. and D. G. Figueroa. Cosmological backgrounds of gravitational waves.
Clas-sical and Quantum Gravity , 35:163001, 2018. arXiv:astro-ph.CO/1801.04268 .[42] Leclercq, F. Bayesian large-scale structure inference and cosmic web analysis. arXive-prints , page arXiv:1512.04985, 2015. arXiv:astro-ph.CO/1512.04985 .[43] Perez, F. and B. E. Granger. Ipython: A system for interactive scientific computing.
Computing in Science Engineering , 9:21–29, 2007.[44] Hunter, J. D. Matplotlib: A 2d graphics environment.
Computing in Science & Engi-neering , 9:90–95, 2007.[45] Walt, S. v. d., S. C. Colbert, and G. Varoquaux. The numpy array: A structure forefficient numerical computation.
Computing in Science and Engg. , 13:22–30, 2011.[46] Virtanen, P., R. Gommers, T. E. Oliphant, et al. SciPy 1.0–Fundamental Algorithmsfor Scientific Computing in Python. arXiv e-prints , page arXiv:1907.10121, 2019. arXiv:cs.MS/1907.10121 . – 16 –47] Ntelis, P., J.-C. Hamilton, J.-M. Le Goff, et al. Exploring cosmic homogeneity with theBOSS DR12 galaxy sample.
Journal of Cosmology and Astroparticle Physics , 2017:019,2017. arXiv:astro-ph.CO/1702.02159 .[48] Ntelis, P., A. Ealet, S. Escoffier, et al. The scale of cosmic homogeneity as astandard ruler.
Journal of Cosmology and Astroparticle Physics , 2018:014, 2018. arXiv:astro-ph.CO/1810.09362 .[49] Higgs, P. W. Broken symmetries and the masses of gauge bosons.
Phys. Rev. Lett. , 13:508–509, 1964.[50] Higgs, P. W. Broken symmetries, massless particles and gauge fields.