General Relativistic Corrections to the Weak Lensing Convergence Power Spectrum

John T. Giblin, Jr1,2, James B. Mertens2, Glenn D. Starkman2, and Andrew R. Zentner3 1Department of Physics, Kenyon College, 201 N College Rd, Gambier, OH 43022 2CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106 and 3Department of Physics and Astronomy, University of Pittsburgh, 100 Allen Hall, 3941 OHara´ St., Pittsburgh PA 15260

We compute the weak lensing convergence power spectrum, Cκκ(θ), in a dust-ﬁlled universe using fully non-linear general relativistic simulations. The spectrum is then compared to more standard, approximate calculations by computing the Bardeen (Newtonian) potentials in linearized gravity and utilizing the Born approximation. We ﬁnd corrections to the angular power spectrum amplitude of order ten percent at very large angular scales, ` ∼ 2−3, and percent-level corrections at intermediate angular scales of ` ∼ 20 − 30.

I. INTRODUCTION precise and very accurate so that the data are not misinterpreted (e.g., Refs. [3, 5, 6]). This continues to be one of the challenges to the exploitation of Weak lensing calculations rely on a number of as- weak lensing observations for cosmological analyses. sumptions in order to improve tractability of models. These include physical assumptions, such as Carefully examining the physical and perturbative the Born approximation, where physical arguments approximations made in the context of lensing mea- are used to justify neglecting sub-dominant effects. surements requires a number of subtle considerations Further assumptions are made when modeling the in order to compare to a fully relativistic treatment, gravitational physics of lensed systems, in partic- ranging from the gauge-dependent nature of vari- ular the assumption that a linearized gravitational ables used in calculations to the particular way in model provides a sufficiently accurate description of which averaged quantities are utilized. We attempt the dynamics of the evolution of the Universe as well to remain self-consistent in our treatment of this as the geodesic equations describing propagation of problem, and to explicitly define the quantities we light. Here, we explore the impact of these assump- consider and approximations we use. We begin by tions on weak lensing convergence calculations by defining angular diameter distances and convergence comparing standard, commonly used calculations to in terms of optical scalars, and describe the 3+1 a fully general relativistic treatment of the problem. framework we use to numerically integrate Einstein’s We find these approximations are accurate only to equations. We then compute the Bardeen (Newto- within a few percent on large angular scales. The rel- nian) gravitational potentials, use these potentials ative magnitude of corrections is found to increase to obtain the weak lensing convergence field in an on larger angular scales, and lessen on smaller angu- approximate setting, and compare the two models. lar scales. Past literature has explored the magnitude of cor- Such observations of weak lensing convergence rections to observables due to commonly made as- power spectra are among the primary science goals sumptions [7, 8], speculating that contributions from of the ongoing Dark Energy Survey (DES) and Hy- nonlinear gravitational effects can lead to approxi- per Suprime-Cam (HSC) Subaru Strategic Survey, mately percent-level corrections, contingent upon on as well as the forthcoming surveys of the Large the specific statistical measure being studied [9–12]. arXiv:1707.06640v1 [astro-ph.CO] 20 Jul 2017 Synoptic Survey Telescope (LSST), the Euclid mis- Here, we perform the first such study in a fully rel- sion, and Wide-Field InfraRed Survey Telescope ativistic setting, utilizing simulations of a universe (WFIRST). The primary driver behind these mea- containing a cosmologically-motivated spectrum of surements is their potential to use lensing power density fluctuations in a perfect, pressureless “dust” spectra to constrain cosmology, particularly the fluid. We find percent-level corrections to the con- cause of cosmic acceleration [1] and neutrino mass vergence power spectrum at ` ∼ 10 − 20. The rel- (for example, Refs. [2–4]). Observationally viable ative importance of corrections is found to increase models of dark energy and values of neutrino masses at smaller `, becoming of order ten percent at ` of induce only subtle alterations to lensing power spec- a few. At higher `, the relative importance of rel- tra on the order of a few percent. Consequently, it ativistic corrections is found to decrease – although is of critical importance to produce theoretical pre- perhaps a physical effect, this may also be a con- dictions for lensing power spectra that are both very sequence of the spectrum of perturbations that we 2

used, which contains only long-wavelength modes. variable, a contraction of a conformal Christoffel ¯i jk ¯i We begin by briefly detailing the methods we symbol, Γ =γ ¯ Γjk, is evolved to improve numeri- use to perform the fully relativistic calculation and cal stability properties of the system according to the approximate calculations we compare to. We ¯i ¯ij ¯i ¯jk 2 ij then present a quantitative comparison of simulated ∂tΓ = − 2A ∂jα + 2α ΓjkA − γ¯ ∂jK quantities using the two methods. We begin in Sec- 3 ij ¯ij j ¯i ¯j i tion II by describing the different formalisms we uti- − 8πγ¯ Sj + 6A ∂jφ + β ∂jΓ − Γ ∂jβ lize to perform numerical calculations. In Section III 2 1 + Γ¯i∂ βj + γ¯li∂ ∂ βj +γ ¯lj∂ ∂ βi , (6) we describe initial conditions for the toy universes 3 j 3 l j l j we utilize, and in Section IV we detail results from and is used when to computing the Ricci tensor and numerical simulations. scalar. Likewise, we can integrate the optical scalar equa- II. FORMALISM tions [22] using the full framework of general relativity. Optical integration through a spacetime involves tracking beam “areas” along photon geodesics. The A. Raytracing in the BSSNOK formulation cosmological observable we compute here is the angular diameter distance, DA ≡ `/Ω, for an object The field of numerical relativity has evolved over with some physical length ` that subtends an angle the past several decades to become a standard nu- Ω of an observer’s sky. The optical scalar equations merical tool in contemporary physics. The field has are valid in the limit of infinitesimal beams, or in progressed to the point where it can model physics the limit that both ` and Ω are small, although re- ranging from systems of strongly gravitating com- cent work may offer a way of working around this pact objects in a fully relativistic, cosmological set- limitation [23]. The optical scalar equations assume ting [13–15], to the dynamics of perfect fluids as they photons do not interact, or that the beam follows interact on cosmological length scales [16–18]. The photon geodesics and neither backreact nor interact BSSNOK formulation is a commonly used numerical with other matter in the universe. The optical scalar scheme that has been demonstrated capable of mod- equations are given by eling such systems with a high degree of accuracy, and importantly, numerical stability [19–21]. d2 ` = ` R − σ2 (7) The BSSNOK system of equations is a 3+1 con- dλ2 formal decomposition of the Einstein field equations. and In this language, the line element is d 2 2 ds2 = −α2dt2 + e4φγ¯ dxi + βidt dxj + βjdt , ` σ = ` W , (8) ij dλ (1) 4φ where e γ¯ij is the spatial metric, andγ ¯ij is a unit- for some affine parameter λ along a photon path, determinant matrix. The parameters α and βi are beam area `, shear rate σ, and Ricci and Weyl op- respectively known as the lapse and shift. Einstein’s tical scalars, R and W. Further details about these field equations can be written in terms of these vari- equations and our previous work numerically inte- ables as a system of first-order dynamical equations, grating these equations can be found in [24] and ref- erences therein. 1 i 1 i ∂tφ = − αK + β ∂iφ + ∂iβ (2) As a final point of potential interest, we remark 6 6 upon the computational complexity of the scheme ¯ k k ∂tγ¯ij = − 2αAij + β ∂kγ¯ij +γ ¯ik∂jβ described above. In synchronous gauge where α = 1, 2 used in this work, and indeed in the vast majority +γ ¯ ∂ βk − γ¯ ∂ βk (3) kj i 3 ij k of gauges typically used in numerical relativity, Ein- 1 stein’s equations are completely local, so calculations ∂ K = − γijD D α + α(A¯ A¯ij + K2) t j i ij 3 are O(N) for some number N of discretized elements of interest (grid points, particles, ...). This is in con- + 4πα(ρ + S) + βi∂ K (4) i trast to the use of a nonlocal gauge, where calcu- ¯ −4φ TF ∂tAij =e (−(DiDjα) + α(Rij − 8πSij)) lations typically scale as O(N log N). This penalty l k is incurred when using Newtonian gauge, common- + α(KA¯ij − 2A¯ilA¯ ) + β ∂kA¯ij j place in N-body simulations, and is encountered in k k 2 k + A¯ik∂jβ + A¯kj∂iβ − A¯ij∂kβ . (5) this work when we compute the Bardeen potentials. 3 The drawbacks of a fully relativistic calculation are The lapse and shift are considered gauge variables, due in part to the increased number of algebraic cal- and may be freely chosen. An additional auxiliary culations involved, but perhaps more important is 3

the need to resolve luminal propagation. However, We will also compute the radial coordinate using the these penalties should also be incurred by any code Born approximation, wishing to reliably integrate geodesics, resolve lu- Z z minally propagating phenomena, or resolve higher- 1 0 r(z) = 0 dz . (13) order gravitational effects, even within a framework 0 H(z ) of linearized gravity. This is the only time we use the Born approximation—the Bardeen potential and stress- B. Computing Convergence energy quantities are evaluated along a true geodesic, computed using the fully general rela- Convergence in weak lensing may be defined in tivistic expression. The derivative of the Bardeen terms of angular diameter distances as potential along the path of integration is also computed along this geodesic, with photon redshift D¯ − D rescaled to a coordinate expression using the Born κ = A A , (9) D¯ A approximation. The second derivative is then computed with respect to this radial coordinate. ¯ where DA is the angular diameter distance as de- Often, the radial derivative and peculiar velocity fined in a pure-FLRW universe [25]. Defined this contributions are neglected entirely; we do not in- way, convergence is meaningful in a fully relativistic clude these assumptions in our analysis. The pe- setting, ie. no perturbative assumptions need to be culiar velocity contribution can be accounted for made, and the above expression reduces to expres- by adding a term proportional to the fluid veloc- sions found in cosmological literature in a Newtonian ity components at the source and observer, ~vs and setting. ~vo along the line of sightn ˆ in Newtonian gauge [28], In a general relativistic setting, computing an- κ → κ + κv, where gular diameter distances, and thus convergence as a defined in Eq. 9, requires integration of the optical κ = − (~v − ~v ) · nˆ + ~v · nˆ . (14) scalar equations as detailed in Sec. II A. In Newto- v a˙ R dz/H s o s nian gauge (sometimes referred to as Poisson gauge or longitudinal gauge; here we follow the conven- The final approximate expression for convergence tions of [26]), this task is simplified after making we seek to integrate is thus several assumptions, both perturbative and phys- Z ical [11, 12, 27]. Typically, perturbative assump- r 2 tions enter by modeling spacetime and matter dy- κ = (rs − r) 4πa δρ rs namics within a linearized gravity framework, while + 12πa2(¯ρ +p ¯)δu − ∂2Φdr + κ . (15) the physical assumptions include assuming the be- nˆ v havior of the spacetime is sufficiently well-described For a final comparison, we compute this approximate by scalar degrees of freedom and perfect fluid compo- expression, Eq. 15, and compare to Eq. 9, a fully nents. Using these assumptions, this expression for general relativistic result. weak lensing convergence can be written in terms of As a final note, the precise magnitude of correc- the Bardeen (Newtonian) potential Φ, tions will depend upon the background cosmolog- Z r ical parameters that are chosen. To this end, we κ = (r − r) ∇2 Φdr . (10) ¯ s r ⊥ note that we compute H(z), a(z), and DA(z) us- s ing a background cosmology defined by the initial The gradient in this expression is transverse to the conditions in the simulation. direction of propagation,n ˆ, thus can alternatively be written in terms of a full Laplacian minus a com- ponent along this direction, C. Computing Bardeen potentials from a fully relativistic simulation 2 2 2 ∇⊥Φ = ∇ Φ − ∂nˆ Φ . (11) The full Laplacian may be evaluated using the The Bardeen potentials, or Newtonian potentials Hamiltonian constraint equation in Newtonian Φ and Ψ, may be computed from a known metric gauge linearized around an FLRW background in in an arbitrary gauge—here, we compute them us- the presence of a perfect fluid, or Poisson’s equation ing the synchronous gauge (geodesic slicing) metric. for gravity, We obtain the Bardeen potentials by first perform- ing a scalar-vector-tensor (SVT) decomposition of ∇2Φ = 4πa2δρ + 12πa2(¯ρ +p ¯)δu . (12) the metric linearized around a homogeneous FLRW 4

background. We perform this decomposition fol- in a cosmological setting as we expect the dynam- lowing Weinberg [26], writing the metric as a back- ics of the spacetime to be well-described by such a 2 ground plus perturbation, background, sog ¯µν = diag(−1, a δij). Fluctuations around this background are taken to be small so that gµν =g ¯µν + hµν . (16) quantities derived from hµν may be raised and low- ered using purely the background metric with terms An ambiguity exists in this deﬁnition in that the second-order in hµν dropped. The perturbed metric choice ofg ¯µν is arbitrary—any background metric is then decomposed as will suﬃce—however the FLRW metric is chosen

h00 hi0 −E a (∂iF + Gi) hµν = = 2 . (17) h0j hij a (∂jF + Gj) a (Aδij + ∂i∂jB + ∂iCj + ∂jCi + Dij)

The vector and scalar functions are transverse with with time-derivatives of B being computed using respect to the background metric, time derivatives of hij. An ambiguity in this defini- i i ij tion of B allows for the addition of an arbitrary time- ∂iC = ∂iG = ∂iD = 0 , (18) dependent function. To address this, we note that i we specify the zero-mode of the inverse Laplacian to and the tensor perturbation is trace-free, Di = 0. Given that the 3+1 metric in synchronous gauge is be zero. We numerically solve these equations for A written as and B in Fourier space. The remaining vector and tensor potentials may also be determined if desired, −1 0 g = , (19) however we do not do so here. At this point, we µν 0 γ ij have enough information to construct the Bardeen we immediately see that in synchronous gauge, a potentials. In terms of the synchronous gauge scalar number of potentials are zero, F = Gi = E = 0. We potentials, these are given by can also see that the metric perturbations and their a Φ = − 2˙aB˙ + aB¨ time derivatives (which will be needed to compute 2 the Bardeen potentials) can be written in terms of 1 Ψ = aa˙B˙ − A . (26) BSSNOK variables as 2 2 hij = γij − a δij (20) As mentioned before, there is one further minor am- ˙ biguity: the scale factor a can be chosen in several hij = −2Kij − 2aaδ˙ ij (21) different ways. For example, it can be chosen to cor- ¨ 2 2φ hij = −2 K˙ ij + a˙ + aa¨ δij , (22) respond to the average conformal factor e in a particular slicing, or the FLRW solution correspond- 4φ ¯ 1 where Kij = e Aij + 3 γ¯ijK is the extrinsic curva- ing to this value computed on the initial or final ture, and its time derivative can be written in terms slices. For this work, we opt to choose a scale fac- of BSSNOK variables, tor that coincides with the scale factor on the initial surface, and that evolves according to the standard ˙ 4φ 1 K˙ ij =4φe A¯ij + γ¯ijK matter-dominated Friedmann equations. 3 Converting density fluctuation amplitudes from 1 1 + e4φ A¯˙ + γ¯˙ K + γ¯ K˙ . (23) synchronous gauge to Newtonian should be per- ij 3 ij 3 ij formed as well, From this, we can reconstruct the SVT scalar field a2 δρN = δρS + B˙ ρ¯˙ , (27) A, 2 1 1 along with fluid 3-velocity velocity, δu = ∂ δu, A = h − ∂ ∂ h , (24) i i 2a2 ii ∇2 i j ij a2 δuN = δuS − B.˙ (28) and B, 2 1 h As a final note, although the Synchronous gauge B = ii − 3A , (25) ∇2 a2 metric is not uniquely determined in terms of the 5

Bardeen potentials, we do not transform variables The metric and matter ﬁelds on the initial surface from Newtonian to synchronous, and thus do not must satisfy the Hamiltonian and momentum con- encounter this issue. straint equations. In terms of BSSN variables, these are given by

III. INITIAL CONDITIONS eφ H = 0 =γ ¯ijD¯ D¯ eφ − R¯ i j 8 We set initial conditions by generating a ran- e5φ e5φ + A¯ijA¯ − K2 + 2πe5φρ (33) dom realization of a cosmologically-motivated power 8 ij 12 spectrum, similar to past work [29]. As the initial conditions we use are intended to mimic an inho- and mogeneous cosmology, we attempt to, at least ap- 2 Mi = 0 = D¯ e5φA¯ij − e6φD¯ iK − 8πe10φSi . proximately, match large-scale matter density fluc- j 3 tuations. At large scales, the power spectrum of den- (34) 1 sity fluctuations is expected to scale as Pδ ∝ k , and We choose A¯ij = 0, andγ ¯ij = δij, imposing the −3 at small scales as Pδ ∝ k . We choose a spectrum restriction that the 3-metric be conformally flat on that corresponds to these scalings, the initial slice. The momentum constraint can be trivially solved by choosing the extrinsic curvature 4 k/k∗ K to be constant and the momentum variable Si Pδδ = P∗ 4 × C(k, kc) , (29) 3 1 + (k/k∗) /3 to be zero, consistent with a fluid initially at rest. The Hamiltonian constraint equation can be solved with k∗ the peak frequency and P∗ the amplitude of by specifying the remaining metric components, and the power spectrum. The function C is included in solving for the corresponding density. For the con- order to introduce a short-wavelength cutoff scale, formally flat metric we have chosen, the confor- kc, to exclude small-scale modes that are not well- mal Ricci scalar is zero, R¯ = 0. The remaining resolved and can therefore lead to numerical instabil- metric term in the Hamiltonian constraint is the ity or inaccuracy. In practice, this means resolving ∇2eφ term, fluctuations of which will correspond all modes by O(5 − 10) grid points or more on the to fluctuations in ρ. In order to produce density initial surface. We choose C to be a logistic function, fluctuations described by the above cosmologically- 1 motivated power spectrum, we choose the conformal C(k, kc) = . (30) factor φ to be described by a related power spec- 10(k−kc) 1 + e trum, We additionally choose the initial peak frequency to P = k−4P . (35) correspond to a length scale of roughly 300 Mpc, and φφ δδ a power spectrum amplitude that corresponds to a We generate a Gaussian random realization of φ ac- realistic RMS amplitude of the density. Although cording to this prescription. The remaining metric 8 Mpc scales are not well-resolved, we still use a variable K, the local expansion rate, is chosen to power spectrum amplitude that corresponds to a σ8 correspond to a desired Hubble expansion rate. The value (RMS density fluctuation amplitude smoothed density is then fixed by the Hamiltonian constraint on 8 Mpc scales) of σ8 ∼ 0.8. We simulate half of equation. Further details on this method can be −1 a Hubble volume and include modes down to k = found in [29]. −1 1/40 H ∼ 100 Mpc. Smoothed on this scale, the Although the initial conditions we use are qual- expected RMS density amplitude is σ100 ∼ 0.07 at itatively similar to those found in a cosmological the time of observation [30]. We choose our power setting, the setup we use does not precisely corre- spectrum amplitude such that the amplitude of the spond to physical expectations. We therefore rec- conformal RMS density fluctuations, defined as ognize this as a toy model, rather than a precision s √ calculation. Nevertheless, we are hopeful that the R dV γ (¯ρ − ρ)2 effects we see here can provide a reliable indication σ = √ , (31) ρ R dV γ of the the order of magnitude of corrections to New- tonian calculations due to general relativistic effects, with the average density defined as and that they can provide a qualitative indication of the relevance of relativistic effects to observations. R √ dV γρ Generalizing these initial conditions to more closely ρ¯ = √ , (32) R dV γ correspond to expectations from Newtonian or linear theory, but in a relativistic setting, will be an approximately coincides with this value of σ100. important future task. 6

The ﬁnal ingredient required in order to specify the sky seen by an observer is, of course, an observer. In this work, we lay down initial conditions and integrate the simulation forward to a desired time of observation. We then place an observer at the center of our simulation volume and integrate along geodesics away from this observer in Healpix [31] directions, from the observer’s spacetime point “backwards” in time. This observer, along with sources, are taken to be at rest in geodesic slicing, or to be co-moving with the local ﬂuid.

IV. RESULTS

Here we present results from a simulation in which photon geodesics are integrated from an observer back in time to a redshift of z = 0.25. In partic- FIG. 1: Observed skies: top left is a sky generated us- ular, we compare Eq. 9 to Eq. 10. In order to coming approximate theory (Eq. 10), top right a relativistic sky (Eq. 9), and bottom the difference between the two. pute the former of these we utilize the above 3+1 These skies are generated using a Healpix resolution of formulation of Einstein’s equations, thus integrating Nside = 32, and all maps have had the angular monopole through a fully general relativistic spacetime includ- and dipole contributions removed. ing no approximations or reductions to the Einstein field equations. The latter of these expressions orig- inates from a linearized, scalar gravity treatment, for which we additionally utilize the Born approximation in order to obtain a radial coordinate as described in Sec. II B. In order to compute the angular power spectrum, we decompose the convergence field on the sky into spherical harmonics. The convergence field is written as X κ(θ, φ) = almYlm(θ, φ) . (36) lm From this, the angular power spectrum is then defined, 1 X Cκκ = |a2 | . (37) l 2l + 1 lm FIG. 2: Power spectra of the simulated skies shown in In practice, we compute the angular power spec- Figure 1. The orange curve depicts results from a fully trum by integrating angular diameter distances— relativistic run, blue from approximate theory, and green and therefore convergences—in Healpix directions the power spectrum of the difference of the difference for an observer in our simulated universe. We map. then use standard Healpix routines to compute the power spectrum from the convergence maps we produce. Convergence maps are plotted in Figure 1, de- Richardson extrapolate continuum limit convergence picting the difference between relativistic simulation values using different pairs of runs, and use the dis- results and approximate results. The power spectra tribution of extrapolated values to provide us with that correspond to these images are shown in Fig- a measure of uncertainty in these convergence val- ure 2. ues. The values typically agree at one part in 104, In order to obtain meaningful results, we must or at a level significantly smaller than the difference also compute the numerical error for convergence between convergences computed using approximate values along each geodesic. We do so by perform- and relativistic methods. The uncertainty in extrap- ing runs using a set of four resolutions in our simu- olated power spectra is also found to be accurate at lation, N 3 = 1283, 1603, 1923, and 2563. We then this level. As an additional note, we compute power 7

spectra using `max ∼ 2.4max(`) in Healpix in order gesting the system is evolving away from the lin- to obtain more accurate results. The resulting nu- earized constraint equations typically enforced in a merical error in the spectra we present in this paper cosmological setting. Further exploration will be re- is then expected to be better than a part in 104. quired to precisely characterize the physics at play There is, in addition, sampling error—or cosmic here, and to determine how both physical and per- variance—resulting from the limited number of sim- turbative approximations are breaking down. ulations we run. In order to address this, we simulate twenty skies in total, and average the power spectra together. The resulting spectra are shown in Fig- V. DISCUSSION ure IV, in which we find an `-dependent increase in the amplitude of the approximate power spectrum In this manuscript, we have described the pos- compared to the fully relativistic spectrum. Some sibility of percent-level corrections to lensing power spectrum predictions due to a fully relativistic treatment of gravitational lensing by large-scale structure. This suggests circumspection in the utilization of weak lensing measurements to constrain cosmological parameters. However, a direct comparison of our work to prior literature on lensing cosmology is not possible at the present time. Due to computational limitations, we work within a toy, inhomogeneous Einstein-de Sitter cosmology, explore only large angular scales (` ∼ 10), and only consider lensing out to a redshift of z ∼ 0.25. By way of contrast, lensing by ongoing and forthcoming observational facilities is dominated by structure at significantly higher redshifts (z ∼ 0.6 − 1), and the majority of the cosmological information is contained in lensing correlations on considerably smaller scales (significantly less than a degree, multipoles of ` & 300). FIG. 3: Shown are averages of 20 power spectra ob- We also do not currently have a reliable method tained from independent simulations. The orange curve shows the fully relativistic power spectrum, and blue of extrapolating our results to the more practical the approximate spectrum. The green curve is the aver- case of small-scale lensing correlations induced by age power spectrum of the difference maps, and the red high-redshift, large-scale structure in a dark energy- power spectrum the average of a direct subtraction of dominated universe. However, it is interesting to the GR and approximate power spectra. speculate on the possible importance of our work in this context. Using the methods of [6] and [3] remaining cosmic variance can be seen as ripples or it is straightforward to estimate the potential im- wiggles in the power spectra; such effects may be ex- pact of the systematic errors that we explore on the pected to diminish as an increasing number of sim- program to constrain cosmology using weak gravita- ulations are averaged over. tional lensing correlations. For an LSST- or Euclid- Finally, we remark on the potential origins of the like survey, we estimate that a one-percent system- discrepancies we see: are these due to nonlinear atic offset in the lensing power spectrum corresponds physics, the Born approximation, or merely artifacts to a systematic error on the inferred dark energy of the gauge transformations we have performed? equation of state parameter, w, that is roughly twice The amplitude of the Newtonian potentials, and am- the statistical error with which this parameter may plitude of the components of the gauge transforma- be measured. We estimate a similar level of error tion are not large, being of order a part in 105 on for the neutrino mass. We argue that this is strong these large scales. Fluctuations in the synchronous motivation to pursue fully relativistic lensing stud- gauge metric itself, σφ/φ, are closer to a part in ies further. However, we emphasize that these es- 104. One may therefore expect ambiguities due to timates remain speculative as we do not yet under- gauge and nonlinear effects to be smaller than the stand the cosmology dependence, scale dependence, observed percent-level corrections, indicating phys- or redshift dependence of the effects we describe, and ical approximations made may be breaking down. all of those factors can significantly alter these esti- However, we also find that the Newtonian potentials mates. Φ and Ψ evolve towards percent-level disagreement, Future studies may also wish to examine the be- or that a significant gravitational slip develops, sug- havior of specific dark energy or dark matter mod- 8 els in a fully relativistic context. Important effects National Science Foundation, PHY-1414479; JBM have been considered using approximate treatments and GDS are supported by a Department of En- in the past, including baryonic physics [32], radiation ergy grant de-sc0009946 to CWRU; and ARZ is [33], interactions of propagating light with contents supported in part by the DoE through Grant de- of the Universe [34], and a more complete picture sc0007914 and by the Pittsburgh Particle physics of phase space dynamics [35]. Incorporating such Astrophysics and Cosmology Center (Pitt PACC) phenomenology into a fully general relativistic sim- at the University of Pittsburgh. The simulations in ulation has not yet been performed in a cosmological this work made use of the High Performance Com- setting, and will be an important task for relativistic puting Resource in the Core Facility for Advanced simulations in the coming years. Research Computing at Case Western Reserve Uni- versity, and of hardware provided by the National Science Foundation and the Kenyon College Depart- VI. ACKNOWLEDGMENTS ment of Physics.

We would like to thank Marcio O’Dwyer for valuable conversations. JTG is supported by the

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