Modiﬁed gravity approaches to the cosmological constant problem Foundational Aspects of Dark Energy FADE Collaboration

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Modiﬁed gravity approaches to the cosmological

constant problem

Foundational Aspects of Dark Energy (FADE) Collaboration

Heliudson Bernardo,1∗Benjamin Bose,2,3,4†Guilherme Franzmann,4,5‡Steﬀen

Hagstotz,6,7§Yutong He,5,8¶Aliki Litsa,8‖and Florian Niedermann5∗∗

1Department of Physics, Ernest Rutherford Physics Building, McGill University,

3600 Rue Université, Montréal, Québec H3A 2T8, Canada

2Département de Physique Théorique, Université de Genève,

24 quai Ernest Ansermet, 1211 Genève 4, Switzerland

3Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edin-

burgh, EH9 3HJ, U.K

4Basic Research Community for Physics e.V., Mariannenstraße 89, Leipzig, Germany

5Nordita,

KTH Royal Institute of Technology and Stockholm University,

Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

6Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München,

Scheinerstraße 1, D-81679 München, Germany

7Excellence Cluster ORIGINS, Boltzmannstraße 2, D-85748 Garching, Germany

8The Oskar Klein Centre for Cosmoparticle Physics, Stockholm University

Roslagstullsbacken 21A, SE-106 91 Stockholm, Sweden

Invited review for Special Issue “Cosmological Constant” in Universe.

∗heliudson.deoliveirabernardo@mcgill.ca

†ben.bose@ed.ac.uk

‡guilherme.franzmann@su.se

§steﬀen.hagstotz@lmu.de

¶yutong.he@su.se

‖aliki.litsa@fysik.su.se

∗∗ﬂorian.niedermann@su.se

arXiv:2210.06810v1 [gr-qc] 13 Oct 2022

Abstract

The cosmological constant and its phenomenology remain among the greatest puzzles in

theoretical physics. We review how modiﬁcations of Einstein’s general relativity could alleviate

the diﬀerent problems associated with it that result from the interplay of classical gravity and

quantum ﬁeld theory. We introduce a modern and concise language to describe the problems

associated with its phenomenology, and inspect no-go theorems and their loopholes to motivate

the approaches discussed here. Constrained gravity approaches exploit minimal departures from

general relativity; massive gravity introduces mass to the graviton; Horndeski theories lead to

the breaking of translational invariance of the vacuum; and models with extra dimensions change

the symmetries of the vacuum. We also review screening mechanisms that have to be present

in some of these theories if they aim to recover the success of general relativity on small scales

as well. Finally, we summarise the statuses of these models in their attempt to solve the dif-

ferent cosmological constant problems while being able to account for current astrophysical and

cosmological observations.

Acronyms

CCP Cosmological Constant Problem

UV Ultraviolet

EFT Eﬀective Field Theory

GR General Relativity

EMT Energy-Momentum Tensor

CDM Cold Dark Matter

CC Cosmological Constant

DE Dark Energy

SM Standard Model of Particle Physics

QFT Quantum Field Theory

QCD Quantum Chromodynamics

DEP Dark Energy Problem

IR Infrared

DGP Dvali-Gabadadze-Porrati gravity

CMB Cosmic Microwave Background

CHC Cosmic History Constraint

AC Astrophysical Constraints

SLED Supersymmetric Large Extra

Dimension

Disclaimer: This review does not intend, in any way, to present an exhaustive collection of

modiﬁed gravity approaches. Instead, the models were chosen to exemplify the diﬀerent ways

one can bypass no-go theorems surrounding the cosmological constant problem, and they are

subjected to the authors’ personal biases and preferences on the subject.

Contents

1 Introduction 1

2 What is the problem after all? 3

3 How to modify gravity 5

3.1 Self-tuning 5

3.1.1 A failed start 6

3.1.2 Weinberg’s argument 8

3.1.3 Beyond Weinberg’s argument 10

3.1.4 Loopholes 11

3.2 Screening Mechanisms 13

3.2.1 Potential screening 15

3.2.2 Derivative screening 16

4 Modiﬁed Gravity Approaches 18

4.1 What do we want? 19

4.2 Constraining Gravity 19

4.2.1 The Global Vacuum Energy Sequestering 20

4.2.2 Local Sequestering 22

4.2.3 Non-local Approach 23

4.2.4 Unimodular Gravity 25

4.3 Massive gravity 26

4.3.1 Degravitation 26

4.3.2 Linear Massive Gravity 27

4.3.3 Nonlinear Theories 28

4.4 Self-tuning with Horndeski theories 29

4.4.1 The Fab-4 30

4.4.2 Well-tempered self-tuning 31

4.4.3 Outlook 32

4.5 Braneworld models 33

4.5.1 Our Universe as a cosmic string in six dimensions 34

4.5.2 Supersymmetric large extra dimensions 36

4.5.3 Outlook 38

5 Conclusions - You can’t always get what you want, or can you? 38

References 42

1 Introduction

The cosmological constant problem (CCP) is one of the most persistent puzzles in theoretical

physics. It appears at the interface between quantum mechanics and gravity, and it seemingly

contradicts one of the major building blocks of modern physics, which is that scales in nature

decouple. We hope that it is possible to understand physics at low energies without a detailed

knowledge of the ultraviolet (UV)-complete theory. Formally, this decoupling of scales is ex-

pressed through the enormously successful framework of eﬀective ﬁeld theories (EFT), which is

challenged by the CCP.

Viewed solely as a metric theory for spacetime, Einstein’s general relativity (GR) allows for

a cosmological constant Λthat is left unﬁxed by the principles and symmetries of the theory.

Such a parameter may compete with the energy-momentum tensor (EMT) of local sources in

the equations of motion to produce certain solutions, as was the case of Einstein’s cosmological

static solution [1,2]. In fact, the weak strength of the gravitational interaction is such that all

local tests of GR are compatible with Λ=0, singling out cosmological observations to constrain

Λ.

The current cosmological concordance model, ΛCDM, is fully based on classical GR and

employs a cosmological constant (CC) to describe the present period of accelerated expansion

necessary to ﬁt both early- [3] and late-time (e.g. [4–10], and see [11] for a recent review)

observational data. Although there are other cosmological models in which the driver of the

current epoch of accelerated expansion, herein referred to as dark energy (DE), is not described

by a cosmological constant, standard GR with a CC is still the preferred model once many data

sets are analysed together [11].

However, problems appear once we consider the standard model of particle physics (SM)

together with GR. The SM is based on quantum ﬁeld theory (QFT) in Minkowski spacetime.

Within the QFT paradigm the local energy-momentum tensor of ﬁeld systems might be non-

zero even in the vacuum state, in which case they have the same structure as the EMT of a

CC term [12]. Hence, the existence of the SM ﬁelds, inferred from local experiments, implies

that there are quantum-matter induced contributions to the cosmological constant. This means

that if DE is a CC, then the Λ-ﬁtted value (a.k.a. eﬀective cosmological constant) also includes

contributions from the SM quantum ﬁelds, generically referred to as the vacuum energy density,

ρvac.

Typically, any vacuum energy contribution is enormous and would completely dominate the

gravitational dynamics of the Universe. This was ﬁrst noted by Nernst in 1916 [13], and famously

Pauli quipped that “the Universe would not even reach to the moon” [14]2given the large vacuum

energy from quantum ﬁelds. The problem was formally analysed in a cosmological context by

Zel’dovich in the 70s [19,20]. By the late 80s, due to the perturbative UV incompleteness of

GR [21,22], it was expected that a quantum version of the theory would ﬁx Λ. Prior to the

discovery of the accelerated cosmological expansion, popular attempts of doing so were based on

2Lenz [15] was the ﬁrst to make such a comparison (see e.g. [16–18] and references therein for historical notes).

supergravity theories, which typically do not allow a positive cosmological constant. The seminal

review [23] summarises the status of the problem before the early 90s.

Perhaps the most straightforward procedure for solving the mismatch is to postulate a bare,

purely classical contribution inherent to GR to the cosmological constant, Λb, that would cancel

part of ρvac/M2

Pl. However, the modern view of QFT within the eﬀective ﬁeld theory approach

is that such a proposal is radiatively unstable or extremely ﬁne-tuned, as discussed later. There

are many reviews on other approaches (see, for instance, [24,25]), and the present work aims at

reviewing modiﬁed gravity theories to alleviate or solve the cosmological constant problem. As

we will discuss in this review, the problem actually has diﬀerent facets, and it is a priori unclear

whether all of them can be addressed at once or not.

Since the problem occurs when trying to calculate the gravitational eﬀect of QFT vacuum

energies, one approach to tackle the issue is to modify gravity itself. As shown in Sec. 2and 3

there is little space to solve the CCP within classical GR. Setting aside anthropic arguments and

discussions about ﬁne-tuning [26–33], modifying GR not only oﬀers a new view on the CCP but

also provides new phenomenological signatures. Generally speaking, modiﬁed gravity theories

introduce extra ﬁelds and/or constraints on GR such that the coupling of matter to the metric

is modiﬁed and/or there are extra universal couplings with the extra ﬁelds in a way that is

consistent with observations. It is natural then to ask if and how the CCP manifests itself in

such theories, and this is one of the motivations for developing them.

Another motivation for studying modiﬁed gravity approaches is the fact that some of these

theories are typical examples of how to evade Weinberg’s no-go theorem, as discussed in Sec. 3.1.2,

and its complement, reviewed in Sec. 3.1.3. We highlight possible loopholes in the assumptions of

the no-go theorems in Sec. 3.1.4 and use them as a systematic guiding principle for the modiﬁed

gravity theories considered throughout the review. The idea of this review is not to cover all

modiﬁed gravity literature. Instead, we make a narrow choice of models that we deem promising

and exemplify the discussed loopholes with an emphasis on how the CCP is addressed in each

one of them. For other reviews on modiﬁed gravity and the cosmological constant issues, see

[23,24,34–41].

We start by carefully introducing the diﬀerent contributions and aspects of the cosmological

constant problem in Sec. 2. Any possible solution to the cosmological constant problem is severely

constrained by powerful no-go theorems, which we recap in Sec. 3together with potential ways

to circumvent the theorems while taking into consideration various observational constraints. In

Sec. 4.1, we summarise the task at hand from both a theoretical and phenomenological point of

view before diving into various approaches in Sec. 4. We discuss the extent to which the various

modiﬁed gravity theories can solve parts of the problem in Sec. 5.

Conventions: Unless otherwise stated, we set c=~= 1 and use the metric signature (−+++).

The gravitational coupling is denoted by M2

Pl in all sections, apart from Sec. 4.2.1 and 4.2.2 on

sequestering, where κ2and κ2(x)are used to indicate the promotion of the Planck mass to a

variable. We use Lmor Smfor a generic matter component, and only specify its content as L(...)

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ModiedgravityapproachestothecosmologicalconstantproblemFoundationalAspectsofDarkEnergy(FADE)CollaborationHeliudsonBernardo,1*BenjaminBose,2;3;4GuilhermeFranzmann,4;5SteenHagstotz,6;7YutongHe,5;8¶AlikiLitsa,8andFlorianNiedermann5**1DepartmentofPhysics,ErnestRutherfordPhysicsBuilding,McGillUnive...

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Modiﬁed gravity approaches to the cosmological constant problem Foundational Aspects of Dark Energy FADE Collaboration

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