# General Relativity and Cosmology: Unsolved Questions and Future Directions

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## Abstract

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## 1. Perspective

## 2. A Brief History

#### From Aristotle to Einstein

## 3. The Development of General Relativity

#### 3.1. From Special to General Relativity

#### 3.2. The Formalism of General Relativity

- The Strong Equivalence Principle: The laws of physics take the same form in a freely-falling reference frame as in Special Relativity
- The Weak Equivalence Principle: An observer in freefall should experience no gravitational field. That is to say, an observer cannot determine from a local experiment whether the his laboratory is being accelerated by a rocket of static at the surface of a gravitating body. Gravity is erased up to tidal forces, which are determined by the size of the laboratory and its distance to the centre of the gravitational attraction.

- it agrees with experiment
- it describes gravity entirely in terms of geometry
- it is free of any “prior geometry”

In 4 spacetime dimensions, the only divergence-free symmetric rank-2 tensor constructed solely from the metric g and its derivatives up to second differential order, and preserving diffeomorphism invariance, is the Einstein tensor plus a cosmological term.

#### 3.3. Newtonian Nostalgia: The First Wave of Alternative Theories

#### 3.4. Self-Consistency, Completeness, and Agreement with Experiment

#### 3.5. Metric Theories and Quantum Gravity

#### 3.6. The Gauge Approach and Non-Metric Theories

## 4. Why Consider Alternative Theories?

## 5. From General Relativity to Standard Cosmology

#### 5.1. Cosmological Expansion and Evolution Histories

#### 5.2. Matter (Dust)

#### 5.3. Radiation

## 6. The Components and Geometry of the Universe and Cosmic Expansion

## 7. The Hot Big Bang

#### 7.1. The Cosmic Microwave Background

#### 7.2. Matter-Radiation Equality

#### 7.3. Neutrinos

## 8. Inflation: The Second Wave of Alternative Theories

- The Horizon Problem
- The Flatness Problem
- The Monopole Problem

#### Hot Big Bang Plus Inflation

- The Horizon Problem. Solution: the entire universe evolved out of the same causally-connected region.
- The Flatness Problem. Solution: any initial curvature is diluted by the inflationary epoch and driven to zero.
- The Monopole Problem. Solution: the rapid expansion of the universe drastically reduces the predicted density of magnetic monopoles, if they exist.

## 9. The First Unknown Component: Dark Matter

## 10. The Second Unknown: Dark Energy and the New Wave Alternative Theories

## 11. The Evolution of Large-Scale Structure

#### 11.1. Evolution on Small Scales

#### 11.2. Growth oF Perturbations in the Presence of Dark Energy

#### 11.3. The Power Spectrum of Matter

#### 11.3.1. Nonlinear Evolution

#### 11.3.2. The Primordial Perturbations

## 12. How Do We Test General Relativity?

## 13. Cosmological Tests

- (1)
- A theory of gravitational interactions.
- (2)
- A description of the matter in the universe and the non-gravitational interactions such as electromagnetic emissions.
- (3)
- A hypothesis on the symmetry.
- (4)
- A hypothesis on the topology, or the global structure of the universe.

#### 13.1. Testing the Description of Matter and Non-Gravitational Interactions

#### 13.2. Testing the Assumption of Symmetry

#### 13.3. Testing the Gravitational Interactions

- Tests of the consistency between the expansion history and the growth of structure. A discrepancy in the equation of state parameter of dark energy w, inferred from the two approaches can indicate a breakdown of the GR-based smooth dark energy cosmological paradigm.
- Detailed measurements of the linear growth factor across different scales and redshifts.
- Comparison of the cosmological mass distribution inferred from different probes, especially redshift space distortions and lensing.

## 14. Possible Modifications of GR and Cosmological Implications

#### 14.1. Weak and Strong-Field Regimes

#### 14.2. Small and Large Distances

#### 14.3. Low and High Accelerations

#### 14.4. Low and High Curvature

#### 14.5. Cosmological Probes

## 15. The Nature of Dark Energy and the Implications for General Relativity

- (1)
- there is some new kind of component in the universe, or
- (2)
- there is some new property of gravity.

## 16. The Current Status of General Relativity

## 17. Future Developments

#### 17.1. Plausible Conclusions from Incomplete Information

- (1)
- Parameter inference (estimation). We assume that a model M is true, and we select a prior for the parameters
**θ**, or the $\mathrm{Prob}(\mathit{\theta}|M)$. - (2)
- Model comparison. There are several possible models ${M}_{i}$. We find the relative plausibility of each in the light of the data D, that is we calculate the ratio $\mathrm{Prob}(D|{M}_{i})/\mathrm{Prob}(D|{M}_{0})$.
- (3)
- Model averaging. There is no clear evidence for a best model. We find the inference on the parameters which accounts for the model uncertainty.

#### 17.2. Experimental Progress

#### 17.3. Theoretical and Computational Progress

#### 17.4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

AU | Astronomical Unit |

CDM | Cold Dark Matter |

CMB | Cosmic Microwave Background |

EEP | Einstein Equivalence Principle |

ECKS | Einstein-Cartan-Kibble-Sciama |

eV | electronvolt |

FLRW | Friedmann-Lemaître-Robertson-Walker |

GR | General Relativity |

Gy | Gigayear (${10}^{9}$ years) |

ΛCDM | Λ Cold Dark Matter |

LLI | Local Lorentz invariance |

LLR | Lunar laser ranging |

LPI | Local position invariance |

Mpc | Megaparsec |

PPN | Parameterised post-Newtonian |

SR | Special Relativity |

TeV | teraelectronvolt (${10}^{12}$ electronvolts) |

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**Figure 1.**How we observe the universe. The lookback time is the difference between the age of the universe now, and the age of the universe when photons from an object were emitted. The more distant an object, the farther in its past we are observing its light. This distance in both space and time is expressed by the cosmological redshift z. We obtain most of our astrophysical information from the surface of our past light cone, because it is carried by photons. The only information from within the cone come from local experiments and observations, such as geological records. The green dotted line is the world-line of the atoms and nuclei providing the material for our geological data. Local experiments are carried out along this bundle of world-lines. They provide a useful test of physical constants. One example is the observation of the Oklo phenomenon [2]. The earliest information we have collected so far comes from the cosmic microwave background (CMB). Earlier than the CMB time-like slice is the cosmic neutrino background. We observe Big Bang nucleosynthesis (BBN) indirectly, through observations of the abundances of chemical elements.

**Figure 2.**The density evolution of the main components of the universe. The early universe was radiation-dominated, until the temperature dropped enough for matter density to being to dominate. The energy density of dark energy is constant if its equation of state parameter $w=1$. Because the matter energy density drops as the scale factor increased, dark energy began to dominate in the recent past. At the present time ($a(t)=1$), we live in a universe dominated by dark energy. For dark energy, the green band represents an equation of state parameter $w=-1\pm 0.2$, showing how a small change in the value of this parameter can give very different evolution histories for dark energy. If the Concordance Model is correct, the universe will be completely dominated by dark energy in future epochs (shown by the dashed lines). The matter density will keep decreasing as the universe expands. Our Milky Way will merge with the Andromeda Galaxy, and eventually, the entire Local Group will coalesce into one galaxy. The luminosities of galaxies will begin to decrease as the stars run out of fuel and the supply of gas for star formation is exhausted. In the very far future, this galaxy will be in the only one in our Hubble patch, as all the other galaxies will pass behind the cosmological horizon. The night sky, save for the stars in the Local Group, will be very dark indeed. Stellar remnants will either escape galaxies or fall into the central supermassive black hole. Eventually, baryonic matter may disappear altogether as all nucleons including protons decay, or all matter may decay into iron. In either scenario, the universe will end up being dominated by black holes, which will evaporate by Hawking radiation. The end result is a Dark Era with an almost empty universe, and the entire universe in an extremely low energy state, with a possible heat death as entropy production ceases (see, e.g., [149,150]) What happens after that is speculative.

**Figure 3.**How inflation solves the horizon problem. The light cones on the causal diagram of an inflationary FLRW model are at $\pm {45}^{\circ}$. The worldlines of comoving matter are vertical on this kind of diagram. The particle horizons are horizontal lines. Here we have shown the particle horizon for the CMB. Without inflation, conformal time would only go back to ${\tau}_{0}$, and different regions of the CMB which we observe today along our past light cone would never have been in causal contact. Because of inflation, conformal time is extended to the Big Bang singularity, so these regions would have been in causal contact at some point in our past light cone.

**Figure 4.**How the Concordance Model of Cosmology was developed. Theories and observations motivated the development of cosmological models, which were adjusted as new observations challenged the older models.

**Figure 5.**The parameter space for quantifying the strength of a gravitational field. The horizontal axis measures the potential. The vertical axis measures the spacetime curvature of the gravitational field at a radius r away from a central object of mass M. Regions of this parameter space with potential greater than 1 represent distances from a gravitating object that are smaller than the event horizon radius and are therefore inaccessible to observers. The red vertical line on the right-hand side of the plot marks the horizon limit. This is a schematic plot, and in no way do we show an exhaustive list of objects and systems that have been used or could be used to test GR. The region of Solar System-scale tests is broadly bounded by the Moon, Gravity probe B [406,407], Mercury, and the Pioneer and New Horizons spacecraft. We have included the Voyager spacecraft in the diagram, even though, unlike the Pioneer probe, it was never suitable for tests of GR. The famous Hulse-Taylor binary pulsar [408], although a neutron star binary, is also roughly in the region of Solar System tests. Black holes and neutron stars are in the strong field regime, and the former are at at the limit of the event horizon boundary. Adapted from [409].

**Figure 6.**The parameter space for experiments. The horizontal axis is the typical length scale of the object in question. The vertical axis measures the gravitational potential. The red vertical line on the left-hand side marks the Planck scale. The vertical line on right-hand side of the plot marks radius of the observable universe, or Hubble radius. Experimental verification of GR is impossible beyond these limits. The radius of the surface of last scattering is only slightly smaller than our Hubble radius. Assuming GR implies assumptions far beyond the range that has been experimentally tests. For instance, if we define the Planck mass as ${m}_{\mathrm{Planck}}=\sqrt{\hslash}c/G$, we are assuming that the gravitational constant remains constant down to the Planck length. This extrapolates the inverse square law over a scale of more than ${10}^{30}$ from what ha s been tested. The green region is where Solar System tests have been carried out. Beyond ∼$100\phantom{\rule{0.166667em}{0ex}}\mathrm{Mpc}$, assumption of a homogeneous and isotropic metric becomes accurate enough to use in physical models. The blue region shows length scales at which the FLRW metric is valid. By way of comparison to this parameter space, the nonlinear regime for perturbation theory, which gives use the matter power spectrum, covers ∼${10}^{22}$ to ∼${10}^{23}$ m, the linear, quasi-static regime covers ∼${10}^{23}$ to ∼${10}^{25}$ m, and beyond that is the superhorizon regime. These length scales ranging from $1\phantom{\rule{0.166667em}{0ex}}\mathrm{Mpc}$ to above $1\phantom{\rule{0.166667em}{0ex}}\mathrm{Gpc}$ fit on a tiny part of the horizontal axis above, shown by the thick blue horizontal line.

**Figure 7.**The four main classes of dark energy theories, within the two broad strategies, classified as modifications of the General Relativistic action. Classes 1 and 2 assume the gravitational metric coupling of GR, whereas classes 3 and 4 modify this metric coupling, and are therefore modifications of gravity. In the upper line of classes (1 and 3), new fields dominate the matter content of the recent universe. Adapted from [431].

**Figure 8.**The growing data volume of experiments. The data volume of each experiment is shown as an order-of-magnitude multiple of the data volume of the 1965 Bells Labs experiment which detected the CMB. The labels show the year of ‘first light’ for each experiment. CMB surveys are marked by blue dots, while red dots show large-scale structure surveys. A first light date for CMBPol (light blue triangle) has not yet been fixed. Note that the vertical axis is logarithmic: the date volume increases about a thousandfold every ten years (grey dashed line). Note too that we plot CMB, large-scale structure, space and ground-based probes on the same graph. Ground-based probes will always tend to have a larger data volume than space-borne probes, due to the bandwidth limit on data transmission from spacecraft. Longer-running experiments will also have a larger data volume.

**Table 1.**A “comparative morphology” of some of the major alternatives to General Relativity, in approximate chronological order. We have only listed the theories of particular historical significance. The current landscape, in which cosmologists seek to explain Dark Matter, dark energy, and inflation, offers far more theories. It is generally easier to incorporate the non-gravitational laws of physics within metric theories, since other theories would result in greater complexity, rendering calculations difficult. The only way in which metric theories significantly differ from each other is in their laws for the generation of the metric. Abbreviations: Tensor (T), V (Vector), S (Scalar), P (Potential), Dy (Dynamic), Einstein Equivalence Principle (EEP), i.e., uniqueness of freefall, Local Lorentz Invariance (LLI), Local Position Invariance (LPI), param (Parameter), ftn (Function).

Theory | Metric | Other Fields | Free Elements | Status |
---|---|---|---|---|

Newton 1687 [20] | Nonmetric | P | None | Nonrelativistic, implicit action at a distance |

Poincaré 1890s–1900s [31,118] | Fails; does not mesh with electromagnetism | |||

Nordstrøm 1913 [57] | Minkowski | S | None | Fails to predict observed light detection |

General Relativity 1915 [46] | Dy | None | None | Viable |

Whitehead 1922 [59] | Violates LLI; contradiction by everyday observation of tides | |||

Cartan 1922–1925 [98] | ST | Still viable; introduces matter spin | ||

Kaluza-Klein 1920s [119,120] | T | S | Extra dimensions | Violates Equivalence Principle |

Birkhoff 1943 [61] | T | Fails Newtonian test; demands speed of sound equal to speed of light | ||

Milne 1948 [121] | Machian background | Incomplete; no gravitational redshift prediction; contradicts cosmological observations. | ||

Thiry 1948 [122] | ST | Unlikely; extremely constrained by results on ${\gamma}^{\mathrm{PPN}}$ | ||

Belifante-Swihart 1957 [123] | Nonmetric | T | K param | Violates EEP; contradicted by Dicke–Braginsky experiments |

Brans–Dicke 1961 [63] | Generic S | Dy | S | Viable for $\omega >500$ |

Ni 1972 [48] | Minkowski | T, V, S | 1 param, 3 ftns | Violates LPI; predicts preferred-frame effects |

Will-Nordtvedt 1972 [124] | Dy T | V | Viable but can only be significant at high energy regimes | |

Barker 1978 [125] | ST | Unlikely; severely constrained. | ||

Rosen 1973 [126,127] | Fixed | T | None | Contradicted by binary pulsar data |

Rastall 1976 [128] | Minkowski | S, V | None | Contradicted by gravitational wave data |

$f(R)$ models 1970s [129,130] | $n+1$ST | S | Free ftn | Consistent with Solar System tests; viable but severely constrained |

MOND 1983 [131,132,133] | Nonmetric | P | Free ftn | Nonrelativistic theory |

DGP 2000 [134] | ST/Quantum | Appears to be contradicted by BAOs, CMB and Supernovae Ia unless DE added | ||

TeVeS 2004 [135] | T,V,S | Dy S | Free ftn | Highly unstable [136]; ruled out by SDSS data [137] |

**Table 2.**The evolution of the various cosmological components. The quantities are the equation of state $w\equiv p/\rho {c}^{2}$, the density ρ, the pressure p, and the scale factor $a(t)$.

Component | w | $\mathit{\rho}={\mathit{a}}^{3(1+\mathit{w})}$ | $\mathit{a}(\mathit{t})={\mathit{t}}^{2/3(1+\mathit{w})}$ |
---|---|---|---|

Radiation (photons and relativistic neutrinos) | $1/3$ | ∼${a}^{-4}$ | ∼${t}^{1/2}$ |

Dust (includes CDM, baryons and non-relativistic neutrinos) | 0 | ∼${a}^{-3}$ | ∼${t}^{2/3}$ |

Curvature | $-1/3$ | ∼${a}^{-2}\to {a}^{-4}$ | t |

Cosmic strings | $-1/3$ | ∼${a}^{-2}\to {a}^{-4}$ | t |

Domain walls | $-2/3$ | ${a}^{-1}$ | ∼${t}^{2}$ |

Inflation | $\to -1$ | $\frac{1}{2}{\dot{\varphi}}^{2}+V(\varphi )$ | ∼${\mathrm{e}}^{Ht}$ |

Vacuum energy | $-1$ | constant | ∼${\mathrm{e}}^{Ht}$ |

Method | Constraint | Experiment |
---|---|---|

Shift of perihelion of Mercury | $|2{\gamma}^{\mathrm{PPN}}-{\beta}^{\mathrm{PPN}}-1|<3\times {10}^{-3}$ | Data to 1990 [333] |

Lunar laser ranging | $|4{\beta}^{\mathrm{PPN}}-{\gamma}^{\mathrm{PPN}}-3|=(4.4\pm 4.5)\times {10}^{-4}$ | Data to 2004 [334,335] |

Very long baseline interferometry | $|{\gamma}^{\mathrm{PPN}}-1|=4\times {10}^{-4}$ | Data from 1979 to 1999 [336] |

Time-delay variation | ${\gamma}^{\mathrm{PPN}}-1=(1.2\pm 2.3)\times {10}^{-5}$ | Cassini spacecraft [67] |

Planetary perihelion precessions | ${\beta}^{\mathrm{PPN}}-1=(-2\pm 3)\times {10}^{-5}$ ${\gamma}^{\mathrm{PPN}}-1=(4\pm 6)\times {10}^{-5}$ | Solar System ephemerides to 2013 [337] |

**Table 4.**What we are actually testing. Experimental tests of General Relativity (GR) often probe more than one assumption, and isolating the effects is a challenge in itself.

Experiment | Assumption Tested |
---|---|

Solar System tests | Metric coupling |

Quadrupolar shift of nuclear energy levels | Isotropy [340,341,342] |

Lunar laser ranging and orbiting gyroscopes | Universality of freefall [343] and structure of metric |

Space-borne clocks | Gravitational redshift [344] |

Shift in perihelion of planets | Structure of metric [345] |

Time invariance of physical constants | Metric coupling [346] |

Detection of gravitational waves | Lorenz gauge condition and inhomogeneous wave equation |

**Table 5.**Bridging the length scales to test the cosmological model. Experimental tests of gravity and dark sector couplings, at their typical length scales. Massive gravity (MG) screening mechanisms would show up at short ranges, while smooth dark energy manifests itself at cosmological scales. We give the experimental accuracy from current and future experiments planned over the next decade. The comparison between growth and expansion history comes from combined BAO, supernova, weak lensing, redshift distortion, cosmic microwave background (CMB) lensing, and cluster data. Lensing effects and dynamical mass comparisons can be carried out over a range of scales: inside galaxies, using strong lensing and stellar velocities, and on cosmological scales, using cross-correlations. This is a route to testing screening effects in alternative theories. Laboratory and Solar System tests can also probe dark sector couplings besides short-range effects, but many of the constraints obtained depend on the cosmological model.

Test | Length Scale | Theories Probed | Current Status (and Future) |
---|---|---|---|

Growth vs. expansion history | $100\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Mpc}-1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Gpc}$ | GR with smooth dark energy | $10\%$ accuracy ($2\%$–$4\%$) |

Lensing vs. Dynamical mass | $0.01$ – $100\phantom{\rule{0.166667em}{0ex}}\mathrm{Mpc}$ | Test of GR | $20\%$ accuracy ($5\%$) |

Astrophysical tests | $0.01\phantom{\rule{3.33333pt}{0ex}}\mathrm{AU}$ – $1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Mpc}$ | MG screening mechanisms | ∼$10\%$ (Up to 10 times improvement) |

Laboratory and Solar System tests | $1\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ – $1\phantom{\rule{3.33333pt}{0ex}}\mathrm{AU}$ | PPN parameters in MG | Constraints are model dependent (Up to a tenfold improvement) |

Effect | Milestones | Current, Future and Proposed Experiments | Status |
---|---|---|---|

Mass equivalence | Galileo [366,473] Eötvös [367] | Eöt-Wash Group [371] Lunar Laser Ranging [343] MICROSCOPE [474] STEP [475] Galileo Galilei satellite [476] | Confirmed |

Gravitational time dilation [478] | Eddington solar eclipse [4] Pound-Rebka experiment [479] Space-borne hydrogen masers [481] | Quantum interference of atoms [477] ACES [480] Galileo 5 and 6 satellites [344] Einstein Gravity Explorer [482] | Confirmed |

Precession of orbits | Orbit of Mercury (Einstein [3]) | Binary pulsar observations [483,484,485] Solar System and extrasolar planets [337,402] | Confirmed |

De Sitter precession | Gravity Probe B [486] Lunar laser ranging [488,489,490] Binary pulsars [493,494] | Binary pulsars [487] Improved lunar laser ranging [491,492] | Confirmed |

Lense-Thirring precession | Gravity Probe B [486] LARES [420,421,496] | LARASE [495] Laboratory tests [497] Solar System bodies [339] Binary pulsars [420,421] Black holes [478,496,498,499] | Confirmed |

Gravitational waves | LIGO [80] | Advanced LIGO [500] eLISA https://www.elisascience.org | Recently confirmed in two events [80,501] |

Strong field effects | PSR J0348 + 0432 [483] | Black holes [411,502] Binary pulsars [117] | Recently confirmed |

Orbital precession due tooblateness of central body | Low-orbit satellites [503,504,505,506] Stars orbiting black holes [506] Juno spacecraft around Jupiter [507] | Not yet observed |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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Debono, I.; Smoot, G.F. General Relativity and Cosmology: Unsolved Questions and Future Directions. *Universe* **2016**, *2*, 23.
https://doi.org/10.3390/universe2040023

**AMA Style**

Debono I, Smoot GF. General Relativity and Cosmology: Unsolved Questions and Future Directions. *Universe*. 2016; 2(4):23.
https://doi.org/10.3390/universe2040023

**Chicago/Turabian Style**

Debono, Ivan, and George F. Smoot. 2016. "General Relativity and Cosmology: Unsolved Questions and Future Directions" *Universe* 2, no. 4: 23.
https://doi.org/10.3390/universe2040023