Phenomenology of Gravitational Aether as a solution to the Old Cosmological Constant Problem
Siavash Aslanbeigi, Georg Robbers, Brendan Z. Foster, Kazunori Kohri, Niayesh Afshordi
aa r X i v : . [ a s t r o - ph . C O ] O c t Phenomenology of
Gravitational Aether as a solution to the
Old
Cosmological Constant Problem
Siavash Aslanbeigi,
1, 2, ∗ Georg Robbers, Brendan Z. Foster, Kazunori Kohri,
5, 6 and Niayesh Afshordi
1, 2, † Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5,Canada Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany Foundational Questions Institute, PO Box 3022, New York, NY 10163 Cosmophysics group, Theory Center, IPNS, KEK,and The University for Advanced Study (Sokendai), Tsukuba 305-0801, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan
One of the deepest and most long-standing mysteries in physics has been the huge discrepancybetween the observed vacuum density and our expectations from theories of high energy physics,which has been dubbed the
Old
Cosmological Constant problem. One proposal to address thispuzzle at the semi-classical level is to decouple quantum vacuum from space-time geometry via amodification of gravity that includes an incompressible fluid, known as
Gravitational Aether . In thispaper, we discuss classical predictions of this theory along with its compatibility with cosmologicaland experimental tests of gravity. We argue that deviations from General Relativity (GR) in thistheory are sourced by pressure or vorticity. In particular, the theory predicts that the gravita-tional constant for radiation is 33% larger than that of non-relativistic matter, which is preferredby (most) cosmic microwave background (CMB), Ly- α forest, and Li primordial abundance obser-vations, while being consistent with other cosmological tests at ∼ σ level. It is further shown thatall Parametrized Post-Newtonian (PPN) parameters have the standard GR values aside from theanomalous coupling to pressure ζ , which has not been directly measured. A more subtle predictionof this model (assuming irrotational aether) is that the (intrinsic) gravitomagnetic effect is 33%larger than GR prediction. This is consistent with current limits from LAGEOS and Gravity ProbeB at ∼ σ level. I. INTRODUCTION
The discovery of recent acceleration of cosmic expan-sion was one of the most surprising findings in mod-ern cosmology [1, 2]. The standard cosmological model(also known as the concordance model) drives this ex-pansion with a cosmological constant (CC). While theCC is consistent with (nearly) all current cosmologicalobservations, it requires an extreme fine-tuning of morethan 60 orders of magnitude, known as the cosmologicalconstant problem [3]. More precisely, a covariant regu-larization of the vacuum state energy of a Quantum FieldTheory(QFT), if it exists, acts just like the CC in linearorder, but has a value many orders of magnitude largerthan what is inferred from observations.If the QFT prediction of the cosmological constant isconsidered reasonable (and in lieu of an extreme fine-tuning), there is no choice but to abandon the ideathat vacuum energy should gravitate. This, however, re-quires modifying Einstein’s theory of gravity, in which allsources of energy gravitate. Attempts in this directionhave been proposed in the context of massive gravity [4],or braneworld models of extra dimensions such as cascad-ing gravity [5, 6], or supersymmetric large extra dimen-sions ( e.g., [7]). However, efforts to find explicit cosmo- ∗ Electronic address: [email protected] † Electronic address: [email protected] logical solutions that de-gravitate vacuum have proveddifficult ( e.g., [8, 9]).In [10], one of us proposed a novel approach to modifiedgravity in which the QFT vacuum quantum fluctuations(of linear order in the metric) are decoupled from grav-ity through the introduction of an incompressible perfectfluid called the Gravitational Aether. In this model, theright hand side of the Einstein field equation is modifiedas: (8 πG ′ ) − G µν = T µν − T αα g µν + T ′ µν (1) T ′ µν = p ′ ( u ′ µ u ′ ν + g µν ) , (2)where G ′ is the (only) constant of the theory and T ′ µν isthe aether fluid which has pressure p ′ and four-velocity u ′ . Our metric signature is ( − , + , + , +). Aether is con-strained by requiring the conservation of the energy-momentum tensor T µν , and the Bianchi identity: ∇ µ T ′ µν = 14 ∇ ν T, (3)which can be written in a similar form to the relativistichydrodynamic equations: p ′ ∇ · u ′ = −
14 ˙
T , (4) p ′ ˙ u ′ = −∇ ⊥ (cid:18) p ′ − T (cid:19) , (5)where ˙ ≡ u ′ · ∇ , and ∇ ⊥ is the gradient normal to u ′ four-vector. Notice that Eqs. (4-5) can be combined tofind a parabolic equation for the evolution of u ′ , whichgenerically has a well-defined initial value problem, atleast for a finite time [44].This modification of Einstein equations (1-2), if self-consistent and in agreement with other experimentalbounds on gravity, could potentially constitute a solutionto the old cosmological constant problem. We will showin this paper that none of these experimental bounds, asof yet, rule out this theory (at ∼ σ level) and that it issurprisingly similar to general relativity [45].Nevertheless, the new cosmological constant problem, i.e. the present-day acceleration of cosmic expansion isnot addressed by the original gravitational aether pro-posal. In [11, 12], it is argued that quantum gravityeffects in the presence of astrophysical black holes cannaturally explain this phenomenon. In this proposal, theformation of black holes leads to a negative aether pres-sure, that is set by the horizon temperature of the blackholes. However, in the present work we only focus onphenomenological implications of the classical gravita-tional aether scenario, and defer study of black hole-darkenergy connection, which could be potentially very im-portant on cosmological scales at late times. Instead, weuse a standard cosmological constant to model the late-time acceleration of cosmic expansion. Throughout thepaper we set the speed of light c=1. II. COSMOLOGICAL CONSTRAINTS ONGRAVITATIONAL AETHER
If the energy-momentum tensor of matter, T µν , can beapproximated by a perfect fluid with constant equationof state p = wρ and four-velocity u µ , direct substitutioninto Eq. (1) shows that if: u ′ µ = u µ , p ′ = (1+ w )(3 w − ρ ,then the solutions to the gravitational aether theory areidentical to those of General Relativity (GR) with arenormalized gravitational constant: G N → G eff = (1 + w ) G N , (6)where G N = 3 G ′ /
4. In other words, the gravitationalcoupling is not a constant anymore, and can change sig-nificantly for fluids with relativistic pressure. Not sur-prisingly, for vacuum equation of state w = − G eff = 0,which implies that vacuum does not gravitate.In particular, in the case of homogeneous FLRW cos-mology where the perfect fluid approximation is valid,this theory predicts that the effective G that relates ge-ometry to the matter density ρ in Friedmann equation isdifferent in the matter and radiation eras: G N G R ≡ G eff ( w = 0) G eff ( w = 1 /
3) = 34 . (7)This is the first cosmological prediction of this the-ory: radiation energy gravitates more strongly than non- relativistic matter. The expansion history in the radia-tion era depends on the product Gρ rad , and is constrainedthrough different observational probes. The constraintsare often described as the bound on the effective num-ber of neutrinos N ν, eff , which quantifies the total ra-diation density ρ rad . However, assuming only photons(that are constrained by CMB observation) and threeneutrino species, with no more light particles left overfrom the very early universe, we can translate the con-straints to those on G eff by requiring G eff ρ rad ( N ν = 3) = G N ρ rad ( N ν = 3 + ∆ N ν ). In particular, based on stan-dard thermal decoupling of neutrinos, Eq. (7) can betranslated to ∆ N ν = 2 .
5, at least for a homogeneousuniverse [46]. Using this correspondence, we can now dis-cuss cosmological constraints on the gravitational aetherscenario.FIG. 1: Allowed regions with 2 σ lines for D/H, Y p and Li / H are shown. The upper and lower horizontaldashed lines indicate GR and gravitational aetherpredictions, respectively. The thickness of Y p means theuncertainty in measurements of neutronlifetime [13, 14]. We can translate the vertical axis into∆ N ν by using a relation G N /G R ≃ / (1 + 0 . N ν ). A. Big Bang Nucleosynthesis
It has been known that the increase of the gravita-tional constant at around T = O (1) MeV epoch inducesearlier freezeout of the neutron to proton ratio becauseof a speed-up effect of the increased cosmic expansion.This raises the abundance of He sensitively and deu-terium (D) mildly, and can lower the abundance of Bethrough Be ( n, p ) Li( p, α ) He (Note that the second p isthermal proton). For a relatively large baryon to photonratio η > ∼ × − , the dominant mode to produce Liis the electron capture of Be at a later epoch through Be + e − → Li + ν e . Therefore, the decrease of Bemakes the fitting better because so far any observational Li abundances have been so low that they could nothave agreed with theoretical prediction in Standard BBN(SBBN) at better than 3 σ [15].In this study, we adopt the following observational lightelement abundances as primordial values: the mass frac-tion of He, Y p = 0 . ± . . ± . × − [17],and the Li to hydrogen ratio Log ( Li / H) = − . ± .
06 [18] [47]. Theoretical errors come from experimen-tal uncertainties in cross sections [15, 19, 20] and neutronlifetime [13, 14].Comparing theoretical prediction with the observa-tional light element aubndances provides a constrainton G N /G R . Fig. 1 shows the results of a comprehen-sive analysis for He, D, and Li. We also plotteda band for baryon to photon ratio, η which was re-ported from CMB observations by WMAP 7-year, η =(6 . ± . × − in case of G N /G R = 1 [21]. Thenwe can see that every light element agrees with the Grav-itational Aether theory within 2 σ . It is notable that Liin this theory fits the data better than that in SBBN.Performing χ fitting for three elements with three de-gree of freedom, however, the model is allowed only at99.7% (3 σ ) in total.However, notice that the main discrepancy is with deu-terium abundance observed in quasar absorption lines,which suffer from an unexplained scatter. Moreover, deu-terium could be depleted by absorption onto dust grainsthat would make its primordial value closer to our pre-diction (see [22] for a discussion). B. Power Spectrum of Cosmological Fluctuations
The Gravitational Aether theory can also be testedby considering the power spectrum of the CMB, justas a number of publications have recently investigatedthe apparent preference for extra relativistic degrees offreedom (see e.g., [23–25]). Using a modified version ofCmbeasy [26, 27], we compute constraints on G N /G R from scalar perturbations in a scenario with three mass-less neutrino species (details are discussed in AppendixA). The 7-year CMB data from WMAP [21] togetherwith small-scale observations from the Atacama Cosmol-ogy Telescope (ACT) [28] yield G N /G R = 0 . +0 . − . at 95%-confidence. Just like any additional relativisticcomponent can be compensated by a higher fraction ofdark matter in order to keep the time of matter-radiationequality constant, there is a high amount of degeneracybetween G N /G R and Ω m h and h (see Fig. 2). Recentdata from the South Pole Telescope (SPT), which mea- G N /G R WMAP+ACT 0 . +0 . − . WMAP+ACT+SPT 0 . +0 . − . WMAP+ACT+Hubble+BAO+Sne 0 . +0 . − . WMAP+ACT+SPT+Hubble+BAO+Sne 0 . +0 . − . WMAP+ACT+Sne+Ly- α (free Y p ) 0 . +0 . − . WMAP+ACT+SPT+Sne+Ly- α (free Y p ) 0 . +0 . − . TABLE I:
Summary of the constraints on G N /G R and the associated 95% confidence intervals fordifferent combinations of observational data. sured the CMB power spectrum in the multipole range650 < ℓ < . +0 . − . (for the combination of ACT and SPTdata we have adopted the SPT treatment of foregroundnuisance parameters). A similar effect can be seen whenadding Baryonic Acoustic Oscillations (BAO) [29] andconstraints on the Hubble rate. Here we adopt the valueof H = 73 . ± . − Mpc − [30]. Then, by break-ing the degeneracy between the matter content and h ,the combination WMAP+ACT+BAO+Sne+Hubble re-sult in G N /G R = 0 . +0 . − . . The supernovae data ofthe Union catalog [31] do not significantly contribute tothis constraint. We note that both cases, i.e. adding ei-ther SPT data or adding the Hubble constraints to thebasic WMAP+ACT set, move the gravitational Aethervalue of G N /G R = 0 .
75 to the border or just outside ofthe 95% confidence interval, while the General Relativityvalue of G N /G R = 1 . G N /G R to 0 . +0 . − . .In contrast, observational constraints at lower red-shifts, in particular data of the Ly- α forest [32] prefer theAether prediction. Furthermore, additional degeneracieswith e.g. the Helium mass fraction Y p might shift thepreferred values. Combining WMAP+ACT+Sne withthe Ly- α forest constraints yields, G N /G R = 0 . +0 . − . at 95% level, with Y p as a free parameter. However, weshould note that this result is more prone to systematicuncertainties due to the quasilinear nature of the Ly- α forest. Also, including the SPT data in this combinationchanges this result to the higher value of 0 . +0 . − . . Asummary of the constraints with different combinationsof data is provided in Table I.Future CMB observations by the Planck satellite, aswell as ground-based observatories are expected to im-prove this constraint dramatically over the next fiveyears, and thus confirm or rule out this prediction. III. PRECISION TESTS OF GRAVITY
Gravity on millimeter to solar system scales is welldescribed by General Relativity, which has passed many Ω m h G N / G R h G N / G R n G N / G R FIG. 2:
Constraints at the 95% confidence level for G N /G R from WMAP 7-year (background, green),WMAP+ACT+SPT (middle, blue) and WMAP+ACT+SPT+Sne+BAO+Hubble data (front, red). The white linesshow the 68% confidence levels. Note that the Gravitational Aether prediction is G N /G R = 0 .
75, while in GeneralRelativity G R = G N . Ω m h G N / G R h G N / G R Y p G N / G R FIG. 3:
Constraints at the 95% confidence level for G N /G R from WMAP+ACT+Sne+Ly- α (background, green) andWMAP+ACT+SPT+Sne+Ly- α (front, red). The white lines show the 68% confidence levels. Note that theGravitational Aether prediction is G N /G R = 0 .
75, while in General Relativity G R = G N . precision tests on these scales with flying colors (see e.g., [33] for a review). That is why it is hard to imaginehow an order unity change in the theory such as thatof Eq. (1) can be consistent with these tests, withoutintroducing any fine-tuned parameter. In this section,we argue that nearly all these tests are with gravitationalsources that have negligible pressure or vorticity , whichsource deviations from GR predictions in gravitationalaether theory. A. Parametrized Post-Newtonian (PPN) formalism
In Sec. II, we argued that for any perfect fluid withconstant equation of state, w , the solutions of gravita-tional aether theory are identical to those of GR witha renormalized gravitational constant ∝ (1 + w ). How-ever, for generic astrophysical applications, w is not con-stant except for pure radiation, or in the pressurelesslimit of w = 0. Focusing on the latter case, and giventhat pressure is 1st order in post-Newtonian expansion,we can quantify the gravitational aether theory throughthe Parametrized Post-Newtonian (PPN) formalism.The Parametrized Post-Newtonian (PPN) formalismis defined in a weak field, slow motion limit, and de-scribes the next-to-Newtonian order gravitational effectsin terms of a standardized set of potentials and ten pa-rameters. These PPN parameters will be determined by solving the field equations (1) order-by-order with a per-fect fluid source in a standard coordinate gauge. Theconventional introductory details of the formalism willbe skipped over (see [34] for a more detailed explanationof the procedure and the general PPN formalism).To be clear, though, we will assume a nearly globallyMinkowskian coordinate system and basis with respect towhich, at zeroth order, the metric is the Minkowski met-ric ( g µν = η µν ) and the fluid four-velocity u µ is purelytimelike ( u = 1 , u i = 0). The stress-energy tensor istaken to have the form T µν = ( ρ + ρ Π + p ) u µ u ν + pg µν where u µ , ρ , Π and p are the the unit four-velocity,rest-mass-energy density, internal energy density, andisotropic pressure of the fluid source, respectively. Thefluid variables are assigned orders of ρ ∼ Π ∼ pρ ∼ u i ∼ g µν = η µν + h µν .The components of the metric perturbations h µν withrespect to this basis will be assumed to be of orders: h ∼ h ij ∼ h i ∼ . u ′ µ will be assumed tobe of the same order as that of the matter fluid.Solving (3) to 1PN gives p ′ = − ρ/
4, which can be usedin (1) to solve for g and g ij to 1PN: h = 2 U (8) h ij = 2 U δ ij , (9)where U is the Newtonian potential and the follow-ing gauge condition is imposed: ∂ j h ij = ( ∂ i h jj − ∂ i h ). Comparing the continuity equations for matterand aether (i.e. the ”time” component of (3) to 1.5 PN),it can be shown that u ′ i − u i = t i , (10)where t i satisfies ∇ i ( t i ρ ) = 0. This implies that therotational component of aether is not fixed by matterwithin the PN expansion formalism. Here we will makethe assumption that t i = 0 so that aether is completelydragged by matter. We will discuss this choice further inSection (III B).Previously we mentioned that in this case, an exactsolution for u ′ µ and p ′ exists when matter has a constantequation of state. (It is worth noting that in the t i = 0case, higher PN equations appear to imply a nonstandardcondition on the pressure ∇ a ( u a p ) = 0, which is satisfiedfor a constant equation of state.) Using this solution andan additional gauge condition ∂ i h i = 3 ∂ U , the fieldequations can be solved for g i and g to 1.5PN and2PN, respectively: h i = − V i − W i , (11) h = 2 U − U + 4 φ + 4 φ + 2 φ + 6(1 + 13 ) φ , (12)where Appendix B includes the definition for all poten-tials. Collecting all the results (8), (9), (11), and (12) in-dicates that all metric components are as in standard GR,except for the term in g with the pressure-dependentpotential ζ . Consulting the parametrization rubric indi-cates that all PPN parameters have the standard valuesexcept ζ , which equals ζ = 13 , (13)which was already pointed out in [10]. Notice that ζ , i.e. the anomalous coupling of gravity to pressure is the onlyPPN parameter that is not measured experimentally, asone needs to probe the relationship between gravity andpressure of an object with near-relativistic pressures. Anotable exception is observation of neutron stars (or theirmergers, via gravitational wave observations), which canpotentially measure ζ , assuming that the uncertaintiesin nuclear equation of state are under control [35]. B. Gravitomagnetic Effect
In the previous section, we showed that rotation ofaether is not fixed by matter in the non-relativisticregime. We further assumed that aether rotates with matter. Here we will argue that observational bounds onthe gravitomagnetic effect provide a mild prefernce forthis assumption.Space-time around a rotating object with a weak grav-itational field, like Earth, can be described in terms ofa set of potentials. With appropriate definitions, thesepotentials satisfy equations analogous to Maxwell’s equa-tions [36]. The gravitomagnetic effect describes the drag-ging of spacetime around a rotating object and can bequantified by a gravitomagnetic field ~B defined as: ~B = − ~r ( ~r · ~S ) − ~Sr r , (14) S i = 2 G ′ Z ǫ ijk x ′ j T k eff d x ′ . (15)where ~r is the position vector measured from the centerof the object, ǫ ijk is the three-dimensional Levi-Civitatensor, and T µν eff is the RHS of the field equations (1) [36].The gravitomagnetic field causes the precession of theorbital angular momentum of a free falling test particle.The angular velocity of this precession is [36] ~ Ω = − ~B . (16)If aether is irrotational, T k eff = T k to within the accuracyof linearized theory and since G ′ = G N , we have: ~ Ω aether = 43 ~ Ω GR . (17)Gravity Probe B (GP-B) is an experiment that mea-sures the precession rate < Ω > of four gyroscopes or-biting the Earth. Recently, GP-B reported a frame-dragging drift rate of − . ± . − . −
15% ( e.g., [38]), and more conservative estimatesas large as 20% −
30% ( e.g., [39]).Therefore, we conclude that even though perfect co-rotation of aether by matter is preferred by current testsof intrinsic gravitomagnetic effect, an irrotational aetheris still consistent with present constraints at 2 σ level. IV. CONCLUSIONS AND DISCUSSIONS
In the current work, we studied the phenomenologicalimplications of the gravitational aether theory, a modi-fication of Einstein’s gravity that solves the old cosmo-logical constant problem at a semi-classical level. Weshowed that the deviations from General Relativity canonly be significant in situations with relativistic pres-sure, or (potentially) relativistic vorticity. The mostprominent prediction of this theory is then that gravityshould be 33% stronger in the cosmological radiation erathan GR predictions. We showed that many cosmologi-cal observations, including CMB (with the exception ofSPT), Ly- α forest, and Li primordial abundance preferthis prediction, while deuterium may prefer GR values.We then examined the implications for precision tests ofgravity using the PPN formalism, and showed that theonly PPN parameter that deviates from its GR value is ζ , the anomalous coupling to pressure, that has neverbeen tested experimentally. Finally, we argued that cur-rent tests of Earth’s gravitomagnetic effect mildly prefera co-rotation of aether with matter, although they areconsistent with an irrotational aether at 2 σ level.In our opinion, the fact that gravitational aether has the same number of free parameters as GR , and is yet(to our knowledge) consistent with all cosmological andprecision tests of gravity at 2 σ level, indicates that thistheory could be a strong contender for Einstein’s theoryof gravity.Future observations are expected to sharpen these dis-tinctions. In particular, the most clear test will comefrom the Planck CMB anisotropy power spectrum thatis expected to be released in early 2013. Judging bythe predictions for constraints on the effective numberof neutrinos, Planck should be able to distinguish GRand Aether at close to 10 σ level [23].Another interesting implication of this theory is for thecosmic baryon fraction. As we increase the gravity dueto radiation (or effective number of neutrinos), we needto increase the dark matter density to keep the redshiftof equality constant, since it is well constrained by CMBpower spectrum (see e.g., [21]). This implies that thetotal matter density should be bigger by a factor of 4 / /
4, i.e. from 17% [21] to 13%. This couldpotentially resolve the missing baryon problem in galaxyclusters [40], as well as the deficit in observed Sunyaev-Zel’dovich power spectra, in comparison with theoreticalpredictions [28, 41].
Acknowledgments
We would like to thank Tom Giblin, Ted Jacobson,Justin Khoury, Maxim Pospelov, Josef Pradler, BobScherrer, and Kris Sigurdson for useful discussions andcomments throughout the course of this project. GRthanks the Perimeter Institute for hospitality. SA andNA are supported by the University of Waterloo and thePerimeter Institute for Theoretical Physics. Research atPerimeter Institute is supported by the Government ofCanada through Industry Canada and by the Provinceof Ontario through the Ministry of Research & Innova-tion. K.K. was partly supported by the Grant-in-Aid forthe Ministry of Education, Culture, Sports, Science, andTechnology, Government of Japan, No. 18071001, No. 22244030, No. 21111006, and No. 23540327, and by theCenter for the Promotion of Integrated Sciences (CPIS)of Sokendai.
Appendix A: Aether perturbations through equality
Here we present a consistent treatment of cosmologi-cal scalar perturbation theory for Gravitational Aether(GA). As we argued in Section II, when matter is ap-proximated by a perfect fluid with density ρ , pressure p = wρ ( w constant), and four velocity u µ = dx µ √− ds (i.e. T µν = (1 + w ) ρu µ u ν + wρg µν ), u ′ µ = u µ and p ′ = (1+ w )(3 w − ρ solve (4) and (5) and the GA fieldequation (1) becomes(8 π ) − G µν = G N (1 + w ) T µν . (A1)In cosmology, therefore, if the constituents of the uni-verse are matter and radiation and they are separatelyconserved , the GA field equations become(8 π ) − G µν = G N T mµν + 43 G N T rµν , (A2)where m and r stand for matter and radiation respec-tively. This approximation, of course, is false when in-homogeneities are considered since baryons and photonsinteract strongly. Therefore, we shall perturb about thisexact solution and do a consistent treatment of cosmo-logical scalar perturbation theory.In what follows, b , dm , m , and r stand for baryon, darkmatter, matter, and radiation respectively, and all barredquantities are unperturbed. Following [42], we will usethe Conformal Newtonian Gauge: ds = a ( τ ) {− [1 + 2 ψ ( τ, ~x )] dτ + [1 − φ ( τ, ~x )] dx i dx i } . (A3)To linear order in perturbation theory, the matter energy-momentum tensor takes the form T = − (¯ ρ + δρ ) (A4) T i = (¯ ρ + ¯ p ) δu i a (A5) T ij = (¯ p + δp ) δ ij + Σ ij , (A6)where Σ ij is the traceless anisotropic shear stress pertur-bation and δρ = ρ − ¯ ρ ; δp = p i − ¯ p ; δu iµ = u iµ − ¯ u µ , (A7)where i = { dm, b, r } . In our coordinate system ¯ u = a ,¯ u = − a , and ¯ u i = ¯ u i = 0. The Aether pressure andfour-velocity perturbations are defined as follows: p ′ = − ρ m δp ′ , u ′ µ = u dmµ + δu µ . (A8)Dark matter only interacts gravitationally and is sepa-rately conserved. We assume that there is negligible en-ergy transfer between baryons and relativistic particles(i.e. ∇ µ ( ρ b u bµ ) = 0). Then, to first order in perturbationtheory (4) and (5) give:3 ˙ aa δp ′ = ¯ ρ m ∂ i ( δu i + ¯ ρ b ¯ ρ m δw i ) (A9) ∂ i δp ′ = a ¯ ρ m δ ˙ u i + 2 ˙ aa δu i ) , (A10)where δw i = δu idm − δu ib = a − ( δu dmi − δu bi ) and δu i = a − δu i . Taking the comoving divergence of (A10) andapplying the comoving Laplacian to (A9), we can elimi-nate δp ′ and get an equation for Ω ≡ ∂ i δu i :3 ˙ aa ∂ τ ( a Ω) − ∇ Ω = ¯ ρ b a ¯ ρ m ∇ ( ˙ δ b − ˙ δ dm ) , (A11)where δ dm = δρ dm ¯ ρ dm , δ b = δρ b ¯ ρ b , and we have used thefact that ∂ i δw i = a ( ˙ δ b − ˙ δ dm ). In Fourier space, thisequation can be numerically integrated for modes of dif-ferent wavelength, given the equations that govern δ dm and δ b . Once Ω is known, (A9) can be used to find δp ′ . In the Conformal Newtonian Gauge, only scalar per-turbations are treated and we can ignore the rotationalpart of δu i . This can also be physically motivated: let δu i = ∂ i u S + ∂δu iV where ∂ i δu iV = 0. Taking the curlof (A10), it follows that ∇ × δ~u V ∝ a . As a result, therotational part of the Aether fluid decays and it doesn’tplay a major role in cosmology. As a result, given Ω wecan find δu i in Fourier space ( ∂ j → ik j ): δu j = − i k j k Ω , (A12)where k = δ ij k i k j . Similarly, δw j = δu jdm − δu jb = i k j ak ( ˙ δ dm − ˙ δ b ) . (A13)To first order in perturbation theory, the GA field equa-tions now take the form(8 πG N ) − G µν = T mµν + 43 T rµν + ǫ µν (A14)with ǫ = 0, ǫ i = a ¯ ρ m (cid:0) δu i + ¯ ρ b ¯ ρ m δw i (cid:1) , and ǫ ij = a δp ′ ¯ g ij .Having both the left and right hand sides of thisequation, we can now solve for the scalar perturbations.However, this does not provide an obvious way ofchecking the prediction of this theory, namely G R G N = .This can be easily accommodated for by having fieldequations that contain G R as a constant, and reduceto General Relativity and GA for G R = G N and G R = G N respectively. Consider the field equations(which we will refer to as the Generalized GravitationalAether (GGA) field equations)(8 π ) − G µν [ g µν ] = G R T µν + ( G N − G R ) T αα g µν + ˜ T µν ˜ T µν = ˜ p (˜ u µ ˜ u ν + g µν ) . (A15) Conservation of G µν and T µν implies ∇ µ ˜ T µν = ( G R − G N ) ∇ ν T. (A16)Defining p ′ = ˜ p G R − G N ) and making the obvious iden-tification ˜ u µ = u ′ µ , we see that this equation becomesexactly (3). Therefore, all of our solutions before canbe used here after a trivial rescaling of the pressure.For example, if T µν is a perfect fluid with equation ofstate w , exact solutions are obtained by ˜ u µ = u µ and˜ p = ( G R − G N )(1 + w )(3 w − ρ , which again just renor-malizes Newton’s constant: G N −→ G eff ( w ) = G N (cid:8) w G R G N + (1 − w ) (cid:9) . (A17)Note that G eff ( w = 0) = G N and G eff ( w = 1 /
3) = G R .Again, if matter and radiation are separately conservedin a cosmological setting, (A15) becomes(8 π ) − G µν = G N T mµν + G R T rµν . (A18)More importantly, when G R = G N , these field equationsreduce to those of General Relativity (GR) (this istrue in the cosmological case because ∇ µ ˜ u µ = 0, whichmeans that the conservation of Aether implies that itspressure vanishes identically). Also when G R = G N ,the GAA field equations reduce to those of GA, with theappropriate rescaling T ′ µν = G N ˜ T µν . Therefore, fittingthis theory to data, we will be able to make a likelihoodplot of G R G N and see how far away the best fit is from theGA and GR predictions.Because of the similarity of the underlying equa-tions, the linear perturbation theory of the GGA fieldequations is very close to those of GA, which we alreadydescribed. We treat all matter perturbations as beforeand perturb ˜ T µν similarly:˜ p = ( G N − G R ) ρ m + δ ˜ p, ˜ u µ = u dmµ + δu µ . (A19)The equations of interest are (in Fourier space):3 H∂ τ ( a Ω) + a ( τ ) k Ω = k ¯ ρ b ¯ ρ m ( ˙ δ dm − ˙ δ b ) (A20) δ ˜ p = ( G R − G N )¯ ρ m H (cid:2) Ω + ¯ ρ b a ¯ ρ m ( ˙ δ b − ˙ δ dm ) (cid:3) (A21) δ ˜ u j = − i k j k Ω , (A22)where H = ˙ aa and we have recognized that ¯ ρ b ¯ ρ m = ¯ ρ b ¯ ρ m is fixed by the values at the present time. Once (A20)is solved for Ω, δ ˜ p and δ ˜ u j are determined by (A21) and(A22), respectively. At long last, to linear order in per-turbation theory, the GAA field equations read(8 π ) − G µν = G N T mµν + G R T rµν + ˜ ǫ µν , (A23)where˜ ǫ = 0 (A24)˜ ǫ j = i k j k ( G N − G R )( a ¯ ρ m ) (cid:2) Ω + ¯ ρ b a ¯ ρ m ( ˙ δ b − ˙ δ dm ) (cid:3) (A25)˜ ǫ ij = ( G R − G N ) ¯ ρ m a H (cid:2) Ω + ¯ ρ b a ¯ ρ m ( ˙ δ b − ˙ δ dm ) (cid:3) δ ij . (A26)Having both the left and right hand sides of (A23), thescalar perturbations can now be consistently solved for. Appendix B: PPN notations
The metric components are in terms of particular po-tential functions, thus defining the PPN parameters: g = − U − βU − ξφ W + (2 γ + 2 + α + ζ − ξ ) φ + 2(3 γ − β + 1 + ζ + ξ ) φ + 2(1 + ζ ) φ + 2(3 γ + 3 ζ − ξ ) φ − ( ζ − ξ ) A (B1) g ij = (1 + 2 γU ) δ ij (B2) g i = −
12 (4 γ + 3 + α − α + ζ − ξ ) V i −
12 (1 + α − ζ + 2 ξ ) W i (B3)The potentials are all of the form F ( x ) = G N Z d y ρ ( y ) f | x − y | (B4)and the correspondences F : f are given by U : 1 φ : u i u j φ : U φ : Π φ : p/ρφ W : Z d z ρ ( z ) ( x − y ) j | x − y | h ( y − z ) j | x − z | − ( x − z ) j | y − z | i A : ( v i ( x − y ) i ) | x − y | V i : u i W i : u j ( x j − y j )( x i − y i ) | x − y | . (B5) [1] A. G. Riess et al. (Supernova Search Team), Astron. J. , 1009 (1998), astro-ph/9805201.[2] S. Perlmutter et al. (Supernova Cosmology Project), As-trophys. J. , 565 (1999), astro-ph/9812133.[3] S. Weinberg, Rev. Mod. Phys. , 1 (1989).[4] G. Dvali, S. Hofmann, and J. Khoury, Phys. Rev. D76 ,084006 (2007), hep-th/0703027.[5] C. de Rham et al., Phys. Rev. Lett. , 251603 (2008),0711.2072.[6] C. de Rham, S. Hofmann, J. Khoury, and A. J. Tolley,JCAP , 011 (2008), 0712.2821.[7] C. P. Burgess and D. Hoover, Nucl. Phys.
B772 , 175(2007), hep-th/0504004.[8] N. Agarwal, R. Bean, J. Khoury, and M. Trodden, Phys.Rev.
D81 , 084020 (2010), 0912.3798.[9] C. de Rham, G. Gabadadze, L. Heisenberg, and D. Pirt-skhalava, Phys. Rev.
D83 , 103516 (2011), 1010.1780.[10] N. Afshordi (2008), 0807.2639.[11] C. Prescod-Weinstein, N. Afshordi, and M. L. Balogh,Phys. Rev.
D80 , 043513 (2009), 0905.3551.[12] N. Afshordi (2010), 1003.4811.[13] K. Nakamura et al. (Particle Data Group), J. Phys.
G37 ,075021 (2010).[14] A. Serebrov et al., Phys. Lett.
B605 , 72 (2005), nucl-ex/0408009.[15] R. H. Cyburt, B. D. Fields, and K. A. Olive, JCAP ,012 (2008), 0808.2818.[16] E. Aver, K. A. Olive, and E. D. Skillman, JCAP , 003 (2010), 1001.5218.[17] M. Pettini, B. J. Zych, M. T. Murphy, A. Lewis, andC. C. Steidel (2008), 0805.0594.[18] J. Melendez and I. Ramirez, Astrophys. J. , L33(2004), astro-ph/0409383.[19] M. S. Smith, L. H. Kawano, and R. A. Malaney, Astro-phys. J. Suppl. , 219 (1993).[20] R. H. Cyburt, B. D. Fields, and K. A. Olive, New Astron. , 215 (2001), astro-ph/0102179.[21] E. Komatsu et al. (WMAP), Astrophys. J. Suppl. ,18 (2011), 1001.4538.[22] M. Pospelov and J. Pradler, Ann. Rev. Nucl. Part. Sci. , 539 (2010), 1011.1054.[23] J. Hamann, S. Hannestad, G. G. Raffelt, I. Tamborra,and Y. Y. Y. Wong, Phys. Rev. Lett. , 181301 (2010),1006.5276.[24] Z. Hou, R. Keisler, L. Knox, M. Millea, and C. Reichardt(2011), * Temporary entry *, 1104.2333.[25] E. Calabrese, D. Huterer, E. V. Linder, A. Melchiorri,and L. Pagano (2011), * Temporary entry *, 1103.4132.[26] M. Doran, JCAP , 011 (2005), astro-ph/0302138.[27] M. Doran and C. M. Mueller, JCAP , 003 (2004),astro-ph/0311311.[28] J. Dunkley, R. Hlozek, J. Sievers, V. Acquaviva, P. Ade,et al. (2010), * Temporary entry *, 1009.0866.[29] B. A. Reid et al. (SDSS Collaboration),Mon.Not.Roy.Astron.Soc. , 2148 (2010), 0907.1660.[30] A. G. Riess, L. Macri, S. Casertano, H. Lampeitl, H. C. Ferguson, et al., Astrophys.J. , 119 (2011), * Tempo-rary entry *, 1103.2976.[31] R. Amanullah et al., Astrophys. J. , 712 (2010),1004.1711.[32] U. Seljak, A. Slosar, and P. McDonald, JCAP , 014(2006), astro-ph/0604335.[33] C. M. Will, Living Rev. Rel. , 3 (2005), gr-qc/0510072.[34] B. Z. Foster and T. Jacobson, Phys. Rev. D73 , 064015(2006), gr-qc/0509083.[35] F. Kamiab and N. Afshordi (2011), 1104.5704.[36] M. L. Ruggiero and A. Tartaglia, Nuovo Cim. , 743(2002), gr-qc/0207065.[37] C. W. F. Everitt et al. (2011), 1105.3456.[38] I. Ciufolini and E. C. Pavlis, Nature , 958 (2004).[39] L. Iorio, Space Sci. Rev. , 363 (2009), 0809.1373.[40] N. Afshordi, Y.-T. Lin, D. Nagai, and A. J. R. Sander-son, Mon. Not. Roy. Astron. Soc. , 293 (2007), astro-ph/0612700.[41] M. Lueker et al., Astrophys. J. , 1045 (2010), 0912.4317.[42] C.-P. Ma and E. Bertschinger, The Astrophysical Journal , 7 (1995).[43] P. Bonifacio et al. (2006), astro-ph/0610245.[44] See Appendix A for an example of explicit solutions atlinearized level. While aether singularities may developin the vicinity (or inside the horizon) of black holes, aswe demonstrate throughout the paper, solutions exist forall other situations of physical relevance.[45] However, we should note that there is no known actionprinciple that could lead to Eqs. (1-3).[46] Requiring G eff ρ rad ( N ν = 3) = G N ρ rad ( N ν = 3 + ∆ N ν ),we can determine ∆ N ν in terms of G R G N = by using ρ rad = π g ∗ T rad where g ∗ ≈ . N ν . Solving for∆ N ν gives ∆ N ν ≈ . ( Li / H) = − . ± ..