The Viability of Phantom Dark Energy: A Brief Review
aa r X i v : . [ a s t r o - ph . C O ] S e p September 26, 2017 0:27 WSPC/INSTRUCTION FILE ws-mpla
Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
The Viability of Phantom Dark Energy: A Review
Kevin J. Ludwick
Department of Chemistry and Physics, LaGrange College, 601 Broad StLaGrange, GA 30240, USA [email protected]
In this brief review, we examine the theoretical consistency and viability of phantom darkenergy. Almost all data sets from cosmological probes are compatible with dark energyof the phantom variety (i.e., equation-of-state parameter w < −
1) and may even favorevolving dark energy, and since we expect every physical entity to have some kind of fielddescription, we set out to examine the case for phantom dark energy as a field theory. Wediscuss the many attempts at frameworks that may mitigate and eliminate theoreticalpathologies associated with phantom dark energy. We also examine frameworks thatprovide an apparent measurement w < − Keywords : phantom dark energy; effective field theory; modified gravity.04.62.+v, 04.25.Nx, 11.10.Ef, 95.36.+x, 98.80.-k, 04.50.Kd
1. Introduction
A recent milestone in observational cosmology happened when the High-z SupernovaSearch Team in 1998 and the Supernova Cosmology Project in 1999 publishedobservations of the emission spectra of Type Ia supernovae indicating that the uni-verse’s rate of expansion is increasing. Galaxy surveys and the late-time integratedSachs-Wolfe effect also give evidence for the universe’s acceleration. Thus, ”dark en-ergy” was proposed as the pervasive energy in the universe necessary to produce theoutward force that causes this acceleration, which has been observationally testedand vetted since its discovery. The 2011 Nobel Prize in Physics was awarded toSchmidt, Riess, and Perlmutter for their pioneering work leading to the discoveryof dark energy. The present-day equation-of-state parameter w from the equationof state most frequently tested by cosmological probes, p = wρ with constant w ,assuming a flat universe and a perfect fluid representing dark energy, has been con-strained by Planck in early 2015 to be w = − . ± . and Planck’s 2013 valueis w = − . +0 . − . . The value from the Nine-Year Wilkinson Microwave AnisotropyProbe (WMAP9), combining data from WMAP, the cosmic microwave background(CMB), baryonic acoustic oscillations (BAO), supernova measurements, and H measurements, is w = − . ± . From these reported values, the prospectof w < − w does not always include the eptember 26, 2017 0:27 WSPC/INSTRUCTION FILE ws-mpla Kevin J. Ludwick value for the cosmological constant (CC) model, w = − w < − a little rip, a pseudo-rip, andseveral types of future signularities can occur. In a model with a constant w < − a will reachinfinity in a finite time from now. An energy source that continuously causes anincreasing acceleration rate seems unphysical, especially since it can lead to theripping apart of space-time itself in this way, which is at least unpalatable in somesense. More aspects detrimental to physicality are revealed when phantom darkenergy is examined as a field.All physical phenomena are expected to have a microscopic theory with a fielddescription, and phantom dark energy is no exception. In the following sections, wewill discuss the theoretical viability of phantom dark energy as a field. We will firstoutline the conventional approach to adding a scalar field for phantom dark energyin the standard cosmological metric, the flat Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) metric, and how the phantom field is incompatible with the metric. We willthen outline the methods around this obstacle, such as k-essence and scalar-tensortheories. We then discuss the pathologies that come along with these attemptsto ameliorate these difficulties. We then discuss attempts with conventional fieldmodels and frameworks to give an appearance of w < −
2. Scalar Field Phantom Dark Energy in Flat FLRW Space
The simplest field is a scalar field. Consider the Einstein-Hilbert action for generalrelativity with a complex scalar field ( c = 1): S = Z d x √− g (cid:20) R πG − g µν ∇ µ φ ∗ ∇ ν φ − V ( | φ | ) (cid:21) + S m , (1)where the first term is the usual contribution to the Einstein tensor, the second andthird terms are the contribution to the scalar field dark energy, and S m is the actionfor the rest of the components of the energy-momentum tensor T µν . Minimizing theaction leads to Einstein’s equation, R µν − Rg µν = 8 πG ( T µν [ φ ] + T µν [ m ]) , (2)where T µν [ φ ] = − δ L φ δg µν + g µν L φ .Assuming dark energy is spatially homogeneous as a perfect fluid, the density ρ φ and pressure P φ for the scalar field are ρ φ = ˙ | φ | a + V ( | φ | ) , P φ = ˙ | φ | a − V ( | φ | ) . (3)eptember 26, 2017 0:27 WSPC/INSTRUCTION FILE ws-mpla The Viability of Phantom Dark Energy: A Brief Review We used the flat FLRW metric ds = a ( τ ) (cid:2) − dτ + dx i dx i (cid:3) , (4)and · represents differentiation with respect to τ .The kinetic energy for the scalar field from the Lagrangian density L φ is − g µν ∇ µ φ ∗ ∇ ν φ = ˙ | φ | a . The equation-of-state parameter w = Pρ for dark energy is w φ = ˙ | φ | a − V ( | φ | ) ˙ | φ | a + V ( | φ | ) , (5)and one can see that w φ < − ρ φ ≥ ρ φ + P φ = ˙ | φ | a = 2 KE φ <
0, which mathematically cannot be true for a complexor real scalar field.
3. Wrong-Sign Kinetic Term
What is usually done to allow for compatibility of the scalar field with flat FLRWspace is to flip the sign in front of the kinetic energy term in the Lagrangian density.Then the ratio for w φ becomes w φ = − ˙ | φ | a − V ( | φ | ) − ˙ | φ | a + V ( | φ | ) , (6)and w φ < −
4. K-Essence
The wrong-sign kinetic term approach is a specific instance of k-essence, which ingeneral is the approach that replaces the kinetic term in the Lagrangian density witha function of the kinetic term and the field: F ( X, φ ), where X = − g µν ∂ µ φ∂ ν φ isthe kinetic term. In this approach, a negative kinetic term is generally required inorder to have w < − A two-field model is used in the quintom approach, onea canonical field and one a phantom field.In general, k-essence theories are plagued with caustics in the non-linear regimeso that the field is not single-valued and second derivatives of the field are diver-gent.
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One possible way around this problem is a dynamical metric that providesbackreaction that prevents the formation of caustics, and the introduction of acomplex scalar field prevents the divergence from developing in real (as opposed toimaginary) time. And two exceptions to this generic development of caustics arethe Born-Infeld theory and Sen’s Lagrangian for the tachyon. However, k-essencefield theories suffer from other pathologies, as we discuss below.eptember 26, 2017 0:27 WSPC/INSTRUCTION FILE ws-mpla Kevin J. Ludwick
5. Problems with Wrong-Sign Field Theories and PhantomPathologies
Ultimately, these field theories with wrong-sign kinetic terms (called ”ghost” fieldtheories) are unstable when coupled to matter in any way, and the dark energyfield must at least be coupled gravitationally. An infinite decay rate of the vacuumis a consequence of this coupling, regardless of whether or not the mass is abovea cut-off scale, and this infinite decay rate is clearly not observed, despite sometheoretical connection between k-essence and the effective field theory of a super-conducting membrane. Either the phantom ghost field has positive density andviolates unitarity, rendering it unphysical, or unitarity is satisfied and the densityis negative, which leads to vacuum instability and unbounded decay of the vacuumvia the ghost field. Null Energy Condition
The null energy condition (NEC) is a constraint on the energy-momentum tensor: T µν n µ n ν ≥
0, where n µ is a null vector. Violation of the NEC is often used asan indicator of phantom behavior of a field. For causal, Lorentz-invariant scalartheories, the NEC is sufficient to determine the classical stability of a theory. (Ef-fective field theories that violate the NEC while remaining stable must lack isotropyand have superluminal modes. ) Buniy et al show that for causal, Lorentz invari-ant theories (minimally or non-minimally coupled scalar and gauge field theorieswith second-order equations of motion), NEC violation implies classical instabilitywith respect to the formation of gradients, and violation of the quantum averagedNEC, h α | T µν n µ n ν | α i ≥
0, involving the bare energy-momentum tensor implies localinstability of the quantum state. If dark energy is modeled as a perfect fluid, NEC violation implies a complexspeed of fluid propagation or a clumping instability of the fluid. And for an isolatedsystem that is homogeneous and isotropic, violation of the NEC implies a negativetemperature, or an entropy which decreases with energy. This implies negative ki-netic energy in order for the partition function of the system to converge. Fieldtheories with negative kinetic energy will roll up a potential hill instead of rollingdown a potential well.
Various Phantom Field Theories
Many theories of phantom dark energy which avoid most pathologies are possible(some among them being theories of vector dark energy, Dvali-Gabadadze-Porrati(DGP) branes, Dirac-Born-Infeld (DBI), galileon, kinetic braiding, and other scalar-tensor varieties ). They usually feature at least one of either ghosts, superlumi-nal modes, Lorentz violation, non-locality, or instability to quantum corrections.It is possible for a k-essence or F ( X, φ ) theory to obey unitarity at tree level andto violate the NEC while remaining ghost-free and free of gradient instabilities ateptember 26, 2017 0:27 WSPC/INSTRUCTION FILE ws-mpla
The Viability of Phantom Dark Energy: A Brief Review the expense of a fine-tuned theory that necessarily invokes higher-order irrelevantoperators and imposes either a shift symmetry or technically unnatural small opera-tor coefficients in the low-energy effective theory. We note that Lorentz symmetryis not maintained in this model, however. F ( X, φ ) theories with ghost condensation can avoid superluminal modes whilestill violating the NEC by the inclusion of higher-derivative spatial gradient termsto stabilize the dispersion relation. However, there is no Lorentz-invariant vacuumin this theory. DGP brane and kinetic braiding theories are also generally knownto violate Lorentz symmetry.The conformal galileon theory violates the NEC while remaining stable againstperturbations and quantum corrections. However, similar to the ghost condensate,there is no Lorentz-invariant vacuum.The DBI galileon theory violates the NEC and is stable against radiativecorrections, and the 2 → However, perturbations of its Poincar´e-invariant vacuumresult in superluminal perturbation propagation.An attempt to overcome these issues involves a theory that violates the NECwhile maintaining a Poincar´e-invariant vacuum with stable, sub-luminal perturba-tions. However, because the theory breaks dilation-invariance, the authors of thiswork suspect the theory to be unstable to quantum corrections.In Lorentz-invariant theories, vacuum stability demands the positivity of notonly the kinetic term but also the mass terms and self-interaction terms. Effectivefield theories of any variety (not restricted to scalar field theories) which are Lorentz-invariant must have kinetic terms and leading interaction terms which are positivein order to ensure that fluctuations around translationally invariant backgroundsdo not propagate superluminally. These superluminal propagations do not allowfor a Lorentz-invariant notion of causality, and such field theories turn out to benon-local and do not meet S -matrix analyticity requirements. Even theories withpositive kinetic terms but negative higher-order derivative interactions suffer fromthese pathologies, and they are flawed in general in both the ultraviolet (UV) andinfrared (IR) scales. For a ghost theory obeying Lorentz invariance, superluminal modes propagateat all scales. If one is willing to allow for Lorentz violation, it is possible quarantinethese problematic modes below some low scale with the use of multiple kinetic terms,and the modes will grow for a very short time before becoming super-horizon. Andit is possible to quarantine instabilities to such early times that are unobservable,for example, by making the Planck mass of the auxiliary metric small in a bimet-ric massive gravity theory. For a low-energy effective field theory, assuming thephantom field interacts at least gravitationally, a strangely low Lorentz-violatingultraviolet cut-off of 3 MeV or below is needed to push instabilities to unobserv-able scales, and Lorentz-conserving cut-offs are experimentally excluded completelybecause of the implied modifications to gravity that would be incompatible withexperiment. eptember 26, 2017 0:27 WSPC/INSTRUCTION FILE ws-mpla Kevin J. Ludwick
In order for a Lorentz-violating theory with such a low cut-off to be viable,some low-energy unknown sector must be responsible for the ghosts, which seemsimprobable. Perhaps one can concoct a fine-tuned theory in which low-energy ef-fective ghosts appear below a very low scale while still reproducing the ghost-freestandard sector below the TeV scale, but it is not clear that such a theory is fea-sible, especially since Lorentz-violation may communicate between energy sectorvia graviton loops. Lorentz violation is not consistent with general covariance,which is at the heart of the well-tested theory of general relativity. Also, there areextremely stringent experimental constraints on Lorentz violation in the StandardModel.
40, 41
Much work has been done on Lorentz-violating models from which thescientific community has learned much, and perhaps Lorentz invariance is brokenbeyond an unobservable scale that allows for the viability of such theories; this isyet to be seen.For a theory that is Lorentz-invariant but non-local above a certain scale, itmay be possible for causality to be maintained, and such a theory can have aLorentz-invariant cut-off of (1 . − .
6) meV, which is technically consistent withlimits on small modifications to general relativity. However, such a theory withnon-local interactions may not satisfy normal S -matrix analyticity constraints. Ifwe observe macroscopic non-locality, it would be very surprising and would overturnbasic assumptions we have of the physical nature of our universe.
6. Canonical-Sign Dark Energy and Other Frameworks withApparent w < − Wrong-sign field models and NEC-violating theories are clearly fraught with diffi-culties. Now we examine canonical-sign field models. We have already shown thatcanonical scalar field theories in flat FLRW space cannot have w < −
1, so the the-ory must be modified in some way to allow for at least an appearance of w < − w consistent with the formulation of the canonical fieldtheory. And we are interested in real scalar fields because complex scalar fields, ingeneral, imply a complex dark energy density, and we expect the energy density tobe completely real.If a field or microscopic description for dark energy were not necessary, darkenergy modeled as a fluid with w < − However, we expect a field description to be funda-mental, and we have already discussed the problems with a scalar field model with w < − w φ < − w = p φ /ρ φ ,dark energy is a perfect fluid, and the FLRW metric. Under these assumptions, theNEC implies P φ + ρ φ = ρ φ (1 + w ) ≥ ρ φ ≥ KE φ ≥ The Viability of Phantom Dark Energy: A Brief Review can be framed as a perfect fluid. Energy-momentum tensors containing terms withsecond derivatives cannot be modeled as perfect fluids. For an imperfect fluid,or a theory that lacks perfect homogeneity and isotropy, the NEC will manifestas a different inequality, and adherence to the NEC may not automatically implypositivity of the kinetic energy term in the Lagrangian. However, in general, theNEC is tied to the sub-luminality of a field theory. In theory, it is possible for a model to lead to an apparent measured valueof w < − Photon-Axion Conversion with Apparent w < − Cs´aki et al ,
52, 53 show how the magnitudes of supernovae may be dimmed by photon-axion conversion enough to result in an inference from the data of a rate of accel-eration faster than the actual one. They show that an inferred value for w fromsupernovae data would be w ≃ − − (2 . m + 0 . L dec H ) − , (7)where L dec is the decay length. A cosmological constant with a sufficient rate ofphoton-axion conversion, consistent with all observational constraints on axions,can lead to a value of w as low as − . Weakening Gravity in the Infrared to Achieve an Apparent w < − Modifying gravity in the IR is another avenue that offers an apparent w < − et al investigate whether scalar-tensor theories can result in an apparent w < − G . The Friedmannequations are modified due to the the modified gravity Lagrangian, so an apparentmeasured value of w < − uses a class of braneworld modelsin which the scalar curvature of the induced brane metric contributes to the braneaction. The spatially flat braneworld can exhibit acceleration while still satisfyingeptember 26, 2017 0:27 WSPC/INSTRUCTION FILE ws-mpla Kevin J. Ludwick the Randall-Sundrum constraint on the brane. Their model leads to modified Fried-mann equations, which allow for w < − w < − Quintessence Field with Apparent w < − It is possible to achieve an apparent w < − et al demonstrate that a quintessence fieldriding up a mild uphill section of its potential, after having gained kinetic energyfrom riding down a slope, can lead to a measured constant value of w φ < − The standard determination of the equation-of-state parameter w φ involves theFLRW luminosity distance in the magnitudes of the supernovae, and fitting thedata assuming a constant w φ gives a different equation of state compared to a non-constant assumption. They show that for a quintessence potential, a decrease in w φ from -0.73 for redshift z > .
47 to w φ = − z < .
47 with a dark energyfraction of the total energy density Ω DE = 0 .
80 very closely mimics dark energywith constant w φ < −
1. Such a potential providing this change in w φ would notbe anymore fine-tuned than other potentials in the literature, and the effectivetheory would be perfectly normal. This change in w φ represents an increase in theacceleration of the universe, a phenomenon usually associated with phantom darkenergy, while maintaining w φ > − Considering Small Deviations to FLRW Space to Achieve anApparent w < − We now examine attempts within the confines of general relativity that assumesmall deviations from the standard FLRW metric, since we know that our physicaluniverse is not perfectly homogeneous throughout. The author of this review ex-amined a quintessence field φ in 1st-order perturbed FLRW space. The goal wasto manifest an apparent w < − w φ ≥ − w < − w φ ≥ −
1. For some selectedmodels with non-constant apparent w < −
1, we were able to show that this maybe satisfied with w φ ≥ − w φ ≥ − w ≥ − w φ < − The Viability of Phantom Dark Energy: A Brief Review metric for certain time and length scales, indicating that a naively quintessence fieldtheory in FLRW space would actually have phantom pathologies in the 1st-orderFLRW framework. Similarly, Onemli and Woodard show that a non-phantomscalar field in classical FLRW space (de Sitter, specifically) can violate the NEC oncosmological scales when quantum corrections via renormalization were taken intoaccount, making it subject to phantom pathologies. G¨umr¨uk¸c¨uo˘glu et al examinea non-phantom scalar field model with an anisotropic Bianchi I background andshow that there are superluminal modes present in the IR range, another exhibitionof phantom pathology on the large scale. However, they show that these modes maybe squarely identifiable with Jeans instability, a classical phenomenon, as opposedto a quantum instability. They point out that the spatial gradient in the metric waskey in the formation of these superluminal modes.Since there were hints found of an apparent w < − Another deviation from the FLRW assumptions of isotropy and homogeneitycomes from gravitational backreaction. The universe may have started out fairlyuniform according to the cosmic microwave background (CMB), but we know thatit is not perfectly homogeneous due to the structures that have formed and arecontinuing to form. So if we assume an inhomogeneous space and take the spatiallyaverage of quantities in Einstein’s equations, we can take into account the localeffects on space-time expansion due to inhomogeneities in the universe. However, itdoes not seem that backreaction effects can account for a present-day apparent valueof w < −
63, 64
A very recent constraint on the present-day apparent decelerationparameter in timescape cosmology is q = − . +0 . − . .
7. Conclusion
Dark energy with w < − w φ < − w φ < −
1. However, this kind of theory is plagued withan unstable vacuum with a divergent decay rate. The generalization to k-essence,allowing the Lagrangian density to be written as a function of the kinetic term F ( X, φ ), and modifications to general relativity are fraught with some subset of thefollowing pathologies: ghosts, superluminal modes, Lorentz violation, non-locality,and instability to quantum corrections. Perhaps a stable effective field theory that isonly Lorentz-violating or non-local at certain unobservable scales, as we discussed, isphysically acceptable, but this is not clear at the present time and seems improbable.eptember 26, 2017 0:27 WSPC/INSTRUCTION FILE ws-mpla Kevin J. Ludwick
The alternative to a field theory that is fundamentally phantom is one that is notphantom but still accords with a measurement of w < −
1. Since this observed w < − w < − w < −
1, and perhaps a fieldtheory in a cosmological background that is more physically accurate than FLRWspace is as well. We plan to investigate a full quantum treatment of a dark energyfield in a perturbed FLRW space in future work. More simply, a non-phantom scalarfield theory that is rolling up its potential can also exhibit w < − w < − Acknowledgments
I am thankful to acknowledge support from the LaGrange College Summer ResearchGrant Award.
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