# Constraining generalized non-local cosmology from Noether symmetries

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## Abstract

We study a generalized non-local theory of gravity which, in specific limits, can become either the curvature non-local or teleparallel non-local theory. Using the Noether symmetry approach, we find that the coupling functions coming from the non-local terms are constrained to be either exponential or linear in form. It is well known that in some non-local theories, a certain kind of exponential non-local couplings is needed in order to achieve a renormalizable theory. In this paper, we explicitly show that this kind of coupling does not need to be introduced by hand, instead, it appears naturally from the symmetries of the Lagrangian in flat Friedmann–Robertson–Walker cosmology. Finally, we find de Sitter and power-law cosmological solutions for different non-local theories. The symmetries for the generalized non-local theory are also found and some cosmological solutions are also achieved using the full theory.

## 1 Introduction

Apart from its remarkable success to interpret cosmological observations, the \(\Lambda \)-cold dark matter (\(\Lambda \)CDM) model still lacks according a satisfactory explanation to the issue why the energy density of the cosmological constant is so small if compared to the vacuum energy of the Standard Model (SM) of particle physics. Furthermore, the today observed equivalence, in order of magnitude, of dark matter and dark energy escapes any general explanation a part the introduction of a very strict fine tuning.

Starting from these facts, one cannot consider the cosmological constant fully responsible for the whole anti-gravity dynamics, like the incapability to find a convincing candidate for dark matter, and/or a quantum theory of gravity, many scientists started questioning whether the theory, i.e. general relativity (GR), needed to be changed, in order to explain the accelerating expansion and the large scale structure clustering without the introduction of “ad hoc” cosmological constant and new particles; see, for example, [1, 2, 3]. The most usual modifications consist in the introduction of new fields either in the matter sector (e.g. quintessence) or by modifying gravity (e.g. scalar–tensor theories, *f*(*R*), *f*(*T*), etc.). In some sense, the issue is related to adding new matter fields (dark matter, quintessence, etc.) or improving the geometry considering further degrees of freedom of the gravitational field.

*R*is the Ricci scalar,

*f*is an arbitrary function which depends on the retarded Green function evaluated at the Ricci scalar, \(L_{m}\) is any matter Lagrangian and \(\square \equiv \partial _{\rho }(e g^{\sigma \rho }\partial _{\sigma })/e\) is the scalar-wave operator, which can be written in terms of the Green function \(G(x,x')\) as

*f*in the action, which is called non-local distortion function. The interested reader is referred to the detailed review by Barvinsky [27], which summarizes the non-local aspects both from the quantum-field theory point of view and from the cosmological one.

*e*being \(e= \det (e^i{}_{\mu }) = \sqrt{-g}\) and

*T*is the torsion scalar, which is given by the contraction

*T*with an arbitrary function of this,

*f*(

*T*). This theory can present problems that are non-Lorentz invariant and because a covariant formulation of

*f*(

*T*) gravity is still not very well accepted since the spin connection is a field without dynamics. Nevertheless, it is always possible to give rise to the correct field equations choosing suitable tetrads (see the review of Ref. [29] for a detailed discussion of advantages and problems related to

*f*(

*T*) gravity).

The extra degrees of freedom introduced by *f* do not allow us to find an exact relation between *f*(*T*) and *f*(*R*), since now the boundary terms in (9) contribute to the field equations. These kinds of theories and their extensions are of great interest [31, 32, 33, 34, 35], since they provide a theoretical interpretation of the accelerating expansion of the Universe and also accommodate the radiation and matter dominated phases of it. In specific cases, one can also find inflationary solutions and avoid the Big Bang singularity with bouncing solutions.

*T*. In this theory, the action reads as follows [36]:

*f*depends on \(\square ^{-1}T\). The teleparallel equivalent of GR is recovered if \(f(\square ^{-1}T)=0\). It is possible to show [36] that this theory is consistent with the cosmological data by SNe Ia + BAO + CC + \(H_0\) observations. From (9), it is straightforward to notice that (1) and (10) correspond to different theories, where

*B*is the term connecting them.

*T*is the torsion scalar,

*B*is a boundary term and \(f(\square ^{-1}T,\square ^{-1}B)\) is now an arbitrary function of the non-local torsion and the non-local boundary terms. The Greek letters \(\xi \) and \(\chi \) denote coupling constants. It is easily seen that by choosing \(\xi = -\chi =-1\) one obtains the standard Ricci scalar. From (2), we directly see that the following relation also holds true:

*f*by the so-called Noether symmetry approach [37]. There are a huge amount of articles in the literature which adopt the Noether symmetry approach to constraining the form of some classes of theories (see for example [1, 31, 38] and the references therein). In this way, one obtains models that, thanks to the existence of Noether symmetries, present integrals of motion that allow one to reduce the dynamics and then, in principle, to find exact solutions. Besides these technical points, the presence of symmetries fixes couplings and potentials with physical meaning [37]. In such a way, the approach can be considered a sort of criterion to “select” physically motivated theories [39]. Details of the approach will be given in Sect. 3.

The paper is organised as follows: in Sect. 2 we present details of the model, how to construct the action and its scalar–tensor analog with four auxiliary fields. At the end of this section, we present a diagram which shows the different theories that we can construct as subclasses of the general theory. In Sect. 3, we summarize the Noether symmetry approach, which we shall apply to three different cases: (i) the teleparallel non-local case (a coupling like \(Tf(\square ^{-1}T)\)), in Sect. 4; (ii) the curvature non-local gravity (a coupling like \(Rf(\square ^{-1}R)\)), in Sect. 5; and (iii) the generalized non-local case (given by the complete action (11)), in Sect. 6. In each case, after the study of the symmetries, we present a set of cosmological solutions. Discussion and conclusions are reported in Sect. 7. Appendix A is devoted to details of the conditions to select the Noether vector. Throughout the paper we adopt the signature \((+,-,-,-)\).

## 2 Generalized non-local cosmology

*R*and

*T*,

*B*are quantities defined in different connections, so mixed terms like \(Rf(\square ^{-1}T)\) are badly defined. The above half part of the figure represents different non-local teleparallel theories and the below part of it, the standard curvature counterpart. As is easy to see, only TEGR and GR dynamically coincide while this is not the case for other theories defined by

*T*,

*R*and

*B*. From a fundamental point of view, this fact is extremely relevant because the various representations of gravity can have different dynamical contents. For example, it is well known that

*f*(

*T*) gravity gives second order field equations, while

*f*(

*R*) gravity, in metric representation, is fourth order. These facts are strictly related to the dynamical roles of torsion and curvature and their discrimination at fundamental level could constitute important insight in really understanding the nature of the gravitational field (see [29] for a detailed discussion).

## 3 The Noether symmetry approach

*t*and \(x^{\mu }\). If there exists a function \(h=h(t,x^{\mu })\) such that

*X*[41], then the Euler–Lagrange equations remain invariant under these transformations. The generator is a Noether symmetry of the system described by \(\mathcal {L}\) and the relative integral of motion is given by

## 4 Noether’s symmetries in teleparallel non-local gravity with coupling \(Tf(\square ^{-1}T)\)

### 4.1 Finding Noether’s symmetries

*T*is coupled with a non-local function evaluated at the torsion scalar, that is, \(f(\square ^{-1}T)=f(\phi )\). For Noether symmetries, we need to consider

*h*,

*f*. It can be x seen v that the dependence on the coordinates of the Noether vector components is

*f*reads

### 4.2 Cosmological solutions

In the previous subsection we found that the form of the function *f* is constrained to be an exponential or a linear form of the non-local term (45). It can be shown that for the linear form, there are no power-law or de Sitter solutions. Here we will find solutions for the exponential form of the coupling function.

*p*is a constant, from (50) we directly find that

## 5 Noether’s symmetries in curvature non-local gravity with coupling \(Rf(\square ^{-1}R)\)

### 5.1 Finding Noether’s symmetries

*f*is

### 5.2 Cosmological solutions

## 6 Noether’s symmetries in the general case

### 6.1 Finding Noether’s symmetries

*c*are constants coming from the coefficients of the Noether vector. System (84)–(86) can easily be integrated but, depending on the vanishing or not of some constants, different solutions can be derived. Specifically, we obtain seven different symmetries described below. The Noether vectors and the function

*f*take the following forms.

- 1.
- (a)For \(c_7 \ne 0 \) and \(c_3 \ne 0, c_4 \ne \frac{c_5}{c_7}(c_6+c_9)\), we haveand$$\begin{aligned} X= & {} (c_1 t + c_2)\partial _t + \frac{1}{3}(c_1 - c_3)a \partial _{a} \nonumber \\&+\, (c_4 + c_5(6 \ln a + \psi )) \partial _{\phi }\nonumber \\&+ \,(c_6 + c_7 (6 \ln a + \varphi )+ c_9) \partial _{\varphi } + c_3 \theta \partial _{\theta } \nonumber \\&+\, ((c_3 - c_7) \zeta - c_5 \theta + c_8 )\partial _\zeta \end{aligned}$$(87)$$\begin{aligned}&f(\phi ,\varphi ) = \frac{1}{\xi } + \frac{c_{11} \left( c_5 c_6-c_4 c_7+c_5 c_9\right) }{c_3}\nonumber \\&\quad \times \,\exp \left( {\frac{c_3}{c_5 c_6-c_4 c_7+c_5 c_9} \left( c_5 \varphi -c_7 \phi \right) } \right) . \end{aligned}$$(88)
- (b)For \(c_7 \ne 0 \) and \(c_3 = 0, c_4 = \frac{c_5}{c_7}(c_6+c_9)\), we haveand$$\begin{aligned} X= & {} (c_1 t + c_2)\partial _t + \frac{c_1}{3} a \partial _{a} \nonumber \\&+\, (c_4 + c_5(6 \ln a + \varphi )) \partial _{\phi }\nonumber \\&+\, (c_6 + c_7 (6 \ln a + \varphi )+ c_9) \partial _{\varphi }\nonumber \\&+\, (c_8-c_7 \zeta -c_5 \theta ) \partial _{\zeta } \end{aligned}$$(89)$$\begin{aligned} f(\phi ,\varphi ) = c_{11} + F(-c_7 \phi + c_5 \varphi ). \end{aligned}$$(90)

- (a)
- 2.
- (a)
- i.For \(c_7 = 0 \) and \(c_5 \ne 0\) and \(c_3 \ne 0 , c_5 \ne - c_6\), we haveand$$\begin{aligned} X= & {} (c_1 t + c_2)\partial _t + \frac{1}{3}(c_1 - c_3)a \partial _{a} \nonumber \\&+\, (c_4 + c_5(6 \ln a + \varphi )) \partial _{\phi } \nonumber \\&+\, (c_6 + c_9) \partial _{\varphi } + (c_{10}+c_3 \theta )\partial _{\theta }\nonumber \\&+\, (c_3 \zeta - c_5 \theta + c_8 )\partial _\zeta \end{aligned}$$(91)$$\begin{aligned} f(\phi ,\varphi ) = \frac{c_3-c_{10}}{\xi c_3 } + c_{11} e^{\frac{c_3}{c_6 c_9} \varphi }. \end{aligned}$$(92)
- ii.For \(c_7 = 0 \) and \(c_5 \ne 0\) and \(c_3 = 0 , c_5 = - c_6\), we haveand$$\begin{aligned} X= & {} (c_1 t + c_2)\partial _t + \frac{c_1}{3}a \partial _{a} \nonumber \\&+\, (c_4 + c_5(6 \ln a + \varphi )) \partial _{\phi } \nonumber \\&+\, (c_8 - c_5 \theta )\partial _\zeta \end{aligned}$$(93)$$\begin{aligned} f(\phi ,\varphi ) = c_{11} + F (\varphi ). \end{aligned}$$(94)

- i.
- (b)
- i.For \(c_7 = 0 \) and \(c_5 = 0\) and \(c_3 \ne 0 , c_4 \ne 0\), we haveand$$\begin{aligned} X= & {} (c_1 t + c_2)\partial _t + \frac{1}{3}(c_1 - c_3)a \partial _{a} \nonumber \\&+\, c_4\partial _{\phi } + (c_6 + c_9) \partial _{\varphi } + (c_{10}+c_3 \theta )\partial _{\theta }\nonumber \\&+\, (c_8 + c_3 \zeta )\partial _\zeta \end{aligned}$$(95)$$\begin{aligned} f(\phi ,\varphi ) = \frac{c_3-c_{10}}{\xi c_3 } + F\left( -\frac{c_6+c_9}{c_4}\phi +\varphi \right) e^{\frac{c_3}{c_4} \phi }. \end{aligned}$$(96)
- ii.
- A.For \(c_7 = 0 \) and \(c_5 = 0\) and \(c_3 = 0 , c_4 = 0\) and \(c_6 \ne -c_7\), we haveand$$\begin{aligned} X=(c_1 t + c_2)\partial _t + \frac{c_1}{3}a \partial _{a} + (c_6 + c_9) \partial _{\varphi } + c_{10}\partial _{\theta } + c_8\partial _\zeta \end{aligned}$$(97)$$\begin{aligned} f(\phi ,\varphi ) = \frac{c_{10}}{(c_6+c_9)\xi }\varphi + F(\phi ). \end{aligned}$$(98)
- B.For \(c_7 = 0 \) and \(c_5 = 0\) and \(c_3 = 0 , c_4 = 0\) and \(c_6 = -c_7\), we haveand the equations are satisfied for any$$\begin{aligned} X=(c_1 t + c_2)\partial _t + \frac{c_1}{3}a \partial _{a} + c_8 \partial _\zeta , \end{aligned}$$(99)
*f*.

- A.

- i.

- (a)

### 6.2 Cosmological solutions

Let us now find cosmological solutions for the generalized Lagrangian (78). In principle, it is possible to find cosmological solutions for each of the above cases depending on the coupling functions. Due to the physical importance of the exponential couplings, we will present cosmological solutions for the coupling function given by (88). However, the procedure for the other cases is the same.

In the case (88), we have the constraint given by the integration constants, that is, \(c_7 \ne 0 , c_3 \ne 0, c_4 \ne \frac{c_5}{c_7}(c_6+c_9)\). Hence, the Euler–Lagrange equations obtained by (78), together with the energy condition, give a system of six differential equations for \(a(t),\phi (t),\varphi (t),\theta (t)\,\text {and}\,\zeta (t)\).

*f*becomes

*f*becomes

## 7 Discussion and conclusions

Motivated by an increasing amount of studies related to non-local theories, here we proposed a new generalized non-local theory of gravity including curvature and teleparallel terms. These kinds of theories were introduced motivated by loop quantum effects and they have attracted a lot of interest since some of them are renormalizable [14]. In suitable limits, the general action that we proposed can represent either curvature non-local theories with \(Rf(\square ^{-1}R)\) based on [4] or teleparallel non-local theories \(Tf(\square ^{-1}T)\) based on [36]. Since the theory is highly non-linear, it is possible to introduce four auxiliary scalar fields in order to rewrite the action in an easier way. Then, for a flat FRW cosmology, using the Noether symmetry approach, the coupling functions can be selected directly from the symmetries for the various models derived from the general theory. It is obvious that the theory (11) can give several models, depending on the values of the constants \(\xi \) and \(\chi \) and on the form of the distortion function. We prove that, in most physically interesting cases, the only forms of the distortion function selected by the Noether symmetries are the exponential and the linear ones. According to Refs. [22, 40], this is an important result, because, up to now, these kinds of couplings were chosen by hand in order to find cosmological solutions, while, in our case, they result from a first principle. In addition, there is a specific class of exponentials non-local gravity models which are renormalizable [16, 43]. This means that the Noether symmetries dictate the form of the action and one may choose an exponential form for the distortion function. As discussed in [39], the existence of Noether symmetries is a selection criterion for physically motivated models. Finally, from models selected by symmetries, it is easy to find cosmological solutions like de Sitter and power-law ones. The integrability of the dynamics is guaranteed by the existence of first integrals. In forthcoming studies, the cosmological analysis will be improved in view of the observational data.

## Notes

### Acknowledgements

SB is supported by the Comisión Nacional de Investigación Científica y Tecnológica (Becas Chile Grant no. 72150066). SC and KFD are supported in part by the INFN sezione di Napoli, *iniziative specifiche* TEONGRAV and QGSKY. The article is also based upon work from COST action CA15117 (CANTATA), supported by COST (European Cooperation in Science and Technology).

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