Testing General Relativity with Current Cosmological Data
Abstract
Deviations from general relativity, such as could be responsible for the cosmic acceleration, would influence the growth of large scale structure and the deflection of light by that structure. We clarify the relations between several different model independent approaches to deviations from general relativity appearing in the literature, devising a translation table. We examine current constraints on such deviations, using weak gravitational lensing data of the CFHTLS and COSMOS surveys, cosmic microwave background radiation data of WMAP5, and supernova distance data of Union2. A Markov Chain Monte Carlo likelihood analysis of the parameters over various redshift ranges yields consistency with general relativity at the 95% confidence level.
I Introduction
The nature of gravitation across cosmological ages and distances remains a frontier of current knowledge as we try to understand the origin of the cosmic acceleration (1); (2). Newly refined observations of cosmic structure (3); (4) make it possible to test the predictions of general relativity (GR) for its influence on the growth of cosmic structure through gravitational instability and the gravitational lensing deflection of light by that structure. Indications of a deviation from GR would have profound consequences for cosmology, as well as for fundamental physics.
To explore for new gravitational phenomena, it is useful to parameterize the deviations from GR in the gravitational field equations. A common approach is to introduce two new parameters. The first parameter imposes a relation between the two gravitational potentials entering Newton’s gravitational law of acceleration and the Poisson equation. These are equal in GR in the absence of anisotropic stress but different in many theories of modified gravity. The second parameter establishes a new relation between the metric and matter through a modified PoissonNewton equation, which can be viewed as turning Newton’s gravitational constant into an effective function of time and space. Numerous realizations of these relations have been put forward in the literature (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16); (17); (18).
One motivation for our study is to attempt to relate these disparate, but closely related, approaches. Furthermore, many studies have focused on the ability of future measurements to discriminate among various models and to carry out parameter estimation (19); (20); (21); (22); (23); (24); (25); (26); (27); (28); (29); (30), however there is sufficient data at present to evaluate preliminary tests of GR (31); (32); (33); (34); (36); (37); (38). We concentrate here on current constraints, which also allows us to examine a recent claim of a possible departure from GR (39).
The main points of this article are thus to 1) clarify the relation between different parameterizations and what the degrees of freedom are in a consistent system of equations of motion, 2) confront the parameters encoding deviations from GR with current data to test the theory of gravity, and 3) discuss which features of the data have the most sensitivity to such a test and what astrophysical systematics may most easily mimic a deviation.
In Sec. II we lay out the gravitational field equations in terms of the metric potentials and matter perturbations and compare several forms of parameterizations, giving a “translation table” between them. We illustrate in Sec. III the influence of the parameters on the cosmic microwave background (CMB) temperature power spectrum, the matter growth and power spectrum, and the weak lensing shear statistics. Using Markov Chain Monte Carlo (MCMC) techniques, we then constrain the deviation parameters with current data in Sec. IV. We briefly discuss astrophysical systematics and future prospects in Sec. V.
Ii Systems of Parameterizing Gravity
The most accurate observations of the effects of gravity have been made in the local universe, e.g. within the solar system and in binary neutron star systems (40); (41); (42); (43). These observations can be used to distinguish between various theories of gravity through the parameterized postNewtonian (PPN) formalism (44); (45). The standard PPN formalism introduces a set of constant parameters that take on various values in different gravity theories. This, however, does not give a full description of possible deviations from General Relativity over cosmological scales.
Recent interest in modified gravity has concentrated on those theories that can serve as an alternative explanation for the current period of accelerated cosmic expansion. In order for modifications producing latetime acceleration on cosmic scales to agree with local tests of gravity they must contain length and/or time dependent modifications, which do not occur in the standard PPN formalism. Moreover, for some theories the natural arena for the PPN formalism – solar system and binary neutron star system observations – may be less discriminating than cosmological tests of gravity, given that the modifications are on large scales. This has led to efforts to establish a parameterized formalism that allows for meaningful comparison between modified gravity theories within a cosmological framework (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16); (17); (18), without assuming a specific model.
ii.1 Degrees of Freedom
Changes in the laws of gravitation affect the relationship between the metric and matter variables. Let us explore the degrees of freedom available to define this relation. Restricting our attention to scalar degrees of freedom of the gravitational field, the metric has only two physically relevant scalar functions, or potentials, given by the line element (in conformalNewtonian gauge, adopting the notation of (46))
(1) 
where is the scale factor, the conformal time, and the spatial coordinate. In addition to the metric potentials and , perturbations to a perfect fluid introduce four additional scalar functions: density perturbations , pressure perturbations , velocity (divergence) perturbations , and a possible nonzero anisotropic stress .
The dynamics of any particular theory are then specified when six independent relations between these six quantities are given. Further restricting attention to those gravity theories that maintain the conservation of stress energy, , the resulting generalized continuity and Euler equations give two scalar equations and the gravitational field equations supply the remaining four (46).
Since the cosmic expansion shifted from deceleration to acceleration only recently, since (47), gravity theories that account for this transition without any physical dark energy require a significant departure from GR at late times. Consequently, nonrelativistic matter is the dominant component of the cosmological fluid and so and . Hence, in these theories the dynamically important equations consist of two, as yet unspecified gravitational field equations and the two equations of stressenergy conservation applied to matter, which in Fourierspace are given by
(2)  
(3) 
In the above equations, with the homogeneous part of the matter density, , the dot denotes a derivative with respect to conformal time, and is the wavenumber. There still remains freedom in setting the two gravitational field equations to close the system, subject to the requirement that the theory approaches GR within the solar system.
The two field equations that can close the system in the case of GR are
(4)  
(5) 
where
(6) 
In a wide variety of alternative theories of gravitation, additional scalar degrees of freedom modify the strength of Newton’s constant, and enforce a new relationship between the potentials and . Therefore, one choice for the modified field equations in Fourierspace is
(7)  
(8) 
where and are constants, and and are functions of time and scale, which are still to be determined. As we will see, there are many other choices that can be made for the exact form of parameterization. These choices influence the constraints and the correlations between those constraints that particular observations give for a particular set of postGR parameters. We discuss some of the frameworks in the next subsections.
ii.2 Cdm
We refer to the equations of motion used in (6); (23); (34); (37) as CDM. In CDM, the equations of motion for cosmic perturbations are determined by enforcing the relation
(9) 
for the potentials arising from nonrelativistic matter, where the departure from GR is controlled by the parameter . In practice, this is carried out by adding a source to the offdiagonal spacespace Einstein equation in order to simulate a smooth transition from GR to modified gravity.
Next, requiring that the new gravitational phenomena do not introduce a preferred reference frame distinguished by a momentum flow, e.g. a that would be attributed to a dark fluid, the timespace Einstein equation is preserved, whereby
(10) 
As discussed in (37), preserving the timespace Einstein equation along with the modification in Eq. (9) still results in a correction to the GR Poisson equation. This can be thought of as a consequence of the conservation of stressenergy and the related Bianchi identity as applied to the modified gravitational field equations.
We now propose to extend the CDM model, to incorporate a new parameter that controls the modification to the Poisson equation, i.e.
(11) 
Procedurally, this equation replaces Eq. (10) for obtaining the evolution of the gravitational fields. We call this new parameterization CDM and note that in this parameterization the timespace Einstein equation is generally modified, as opposed to in the CDM parameterization. Note that setting does not reproduce the original CDM model since there itself modifies the Poisson equation as discussed above. This parametrization is consistent with the conservation of largescale curvature perturbations following an argument made in Ref. (21). They argue that superhorizon curvature perturbations are conserved, so long as the velocity perturbation is of order . This can be seen to be true from Eq. (24) presented in the appendix of this work.
ii.3 PPF Linear Theory
A parameterized postFriedmann (PPF) framework of linear fluctuations was introduced by (9); (13) to describe modified gravity models that yield cosmic acceleration without dark energy. It captures modifications of gravity on horizon, subhorizon, and nonlinear scales. Once the expansion history is fixed, the model is defined by three functions and one parameter, from which the dynamics are derived by conservation of energy and momentum and the Bianchi identities. Modifications to the relationship between the two metric perturbations are quantified by the metric ratio
(12) 
In the linearized Newtonian regime, a second function relates matter to metric perturbations via
(13) 
The corresponding quantity that defines this relationship on superhorizon scales is . The last quantity that needs to be defined is , which determines the transition scale from superhorizon to quasistatic behavior in the dynamical equations (see (9); (13) for details).
The PPF parameters can be directly related to the CDM parameters as follows:
(14)  
(15) 
ii.4 Gravitational Growth Index
Another way to close the system of equations is to specify the evolution of one of the perturbed fluid or metric variables. A standard choice is to determine a specific evolution for through the gravitational growth index introduced to parameterize deviations from general relativity in growth by (48). This was partly tied to the metric potentials in (49) but here we present a more complete relation.
From Eq. (23) of (49) we see the key quantity is the modification of the source term in the Poisson equation, there called . The second order equation for the evolution of the density perturbation arises from , and there is also a modification allowed in the gravitational coupling as in Eq. (11). In essence, . Thus . The relationship between , and the evolution of is presented rigorously here in Eq. (26) (also see Sec. III.2).
The gravitational growth index in Eq. (23) of (49) thus relates to the CDM formalism through
(16)  
(17) 
Note is an effective high redshift equation of state defined in terms of how the matter density in units of the critical density, , deviates from unity (specifically, ). In the last line of Eq. (17) we specialize to a CDM expansion history, as used throughout this article, so , and to the ansatz for and used later in Eqs. (18).
ii.5 Relating Parameterizations
The discussion above is by no means an exhaustive list of the parameterizations proposed in the literature to describe departures from GR. Many more exist, and while all of them have in common a relatively simple parameterization of the departure from , they all differ in how they close the system of equations. Some, like CDM, modify the Poisson equation directly. Others, like CDM, retain one of the Einstein equations.
Table 1 lists some of the most common parameterizations and presents a useful translation between their postGR parameters and CDM. With the possible exception of the parameterization from (39) (see next paragraph and its footnote), all of the parameterizations presented are presumed to leave the equations of stressenergy conservation unmodified.
parameter  closing  

parameterization  relating and  parameter  comments 
CDM (6); (34); (37)  Retains equation (10)  
Curvature (12)  Conserves curvature perturbations  Effectively retains Eq. (10).  
See appendix in Ref. (34)  
PPF (9); (13)  Includes scaledependent transition  
between super and subhorizon regimes  
MGCAMB (20); (21)  Modifies Poisson equation with  
cf. (7); (8); (10)  instead of in Eq. (11)  
Subhorizon (10)  
Growth index (48)  additional  Only defines not inverse  
Decoupled (39)  Overspecified (also enforces Poisson eqn). 
Since none of these modelindependent approaches start from an action,
one must be careful to trace the system of equations to make sure that
the phenomenological modifications do not under or overspecify the
system
Iii Influence of Gravity Modifications on Observations
The behavior of the CMB, weak lensing, and matter power spectrum in the CDM scenario have been discussed in (34); (37). The consequences are slightly different when we introduce in the CDM parameterization. In the case that and , both lead to an amplification of low CMB power; and both suppress it. This allows us to play the two parameters against each other, combining positive (negative) values of with smaller (larger) values of to generate nonGR power spectra that appear to be in better agreement with the data than those obtained within the confines of the CDM model. That either parameter can enhance or suppress power results in a degeneracy between and in any multiparameter exploration of the data. Observations that can break this degeneracy therefore become vital to diagnosing departures from GR.
For the purposes of the discussion in this section, we will assume the redshift dependences
(18) 
Note this form can be motivated by the scaling argument in (49), that the deviations in the expansion history should keep pace with the deviations in the growth history. Otherwise one tends to either violate GR at early times (causing difficulties for primordial nucleosynthesis and the CMB) or does not achieve acceleration by the present. In addition to the CMB, we also discuss the effects of our postGR parameters on the matter power spectrum and on weak lensing statistics.
iii.1 CMB Anisotropy Spectrum
We modified versions of the public Boltzmann codes CMBfast (51) and CAMB (52) to evolve the cosmological perturbations according to parameterization (18) and the equations of motion presented in Sec. II.2. We used these codes to generate examples of CMB anisotropy and matter power spectra for different values of and ; in order to focus on the nonGR effects, in this section all other cosmological parameters are set to their WMAP5 maximum likelihood values (53). Figure 1 shows the resulting CMB anisotropy spectra. As in (34) for CDM, negative values and extreme positive values of the postGR parameters amplify the power in the low multipoles. Moderate positive values suppress the low power. This is a manifestation of the integrated SachsWolfe (ISW) effect. The high power is unaffected.
The ISW effect arises when time evolving and potentials cause a net energy shift in CMB photons. The CMB ISW power is sourced as
(19) 
As was discussed in (37), the evolution of and potentials in the universe is a competition between gravitational collapse trying to deepen the potentials and cosmic expansion trying to dilute them. Under GR with a cosmological constant, the expansion wins and the source term for the ISW (note , ). By weakening gravity, or tilts the competition even more towards cosmic expansion, hastening the dilution of and , causing to be even larger, and amplifying the ISW effect. Positive or amplifies gravity – either by directly deepening the Newtonian potential so that mass is more attractive ( case) or by causing to source a deeper potential through the modified Poisson equation ( case) – so that the dilution due to cosmic expansion is slowed, leading to a weaker ISW effect. In the case of extremely positive or the ISW deepening is so pronounced that the sign of is reversed, but since the ISW effect in the power spectrum depends on the square, the ISW effect is again amplified. High power is unaffected because the ISW is a subdominant effect on those scales.
Figure 2 more clearly illustrates this bimodal behavior by plotting the change in quadrupole power relative to GR as a function of the postGR parameter, varying one at a time (compare Fig. 4 in (37)). The blue, dotdashed curve is generated by varying and holding fixed . The red, dashed curve is generated by varying and holding fixed . Note that the CMB appears to be more sensitive to differing values of than of . The black, solid curve is generated by varying and compensating for this by setting . This choice is motivated by the alternative definition of the unmodified Poisson equation
(20) 
(see further discussion in the next section). We see that, for a wide range of values of , complementary ( and or viceversa) values of cancel out much of the latetime ISW effect found in Fig. 1, as alluded to in the introduction to this section.
iii.2 Matter Power Spectrum and Weak Lensing Statistics
We investigate the power spectrum of the matter perturbations as a function of wavenumber in Fig. 3 for the same set of models. Again the most dramatic postGR effects occur at large scales. This is not due to any scale dependence in the modifications (we took and to be independent of ), but simply from the factor in the modified Poisson equation (11).
For the weak lensing shear correlation function, as for many other observables, we need to know how overdensities grow with scale factor. In the case of GR and CDM, this is a relatively simple proposition since the growth of overdensities is scaleindependent after decoupling. As just discussed, this no longer holds for CDM. It is possible, using energy conservation and Eqs. (9) and (11), to derive a secondorder differential equation for the evolution of . We show the derivation and result in the Appendix, and focus here on the parameter dependence.
With the exception of one term on the middle line of Eq. (26), all of the terms containing metric potential modifications to general relativity (the and terms) are multiplied by a factor of . Hence, we expect that the strongest departures from GR predictions occur for small values of . Since the most important aspect for comparing modifications against observations is the change in shape of the power spectrum, rather than its normalization, in Figure 3 we normalize the power spectrum to agree with at large . The strongest deviation in shape indeed occurs for . The one exceptional term in Eq. (26) is precisely the term discussed in Sec. II.4 entering the gravitational growth index formalism, and this will dominate for large values of , giving a scaleindependent enhancement (suppression) for positive (negative) or .
Figure 4 plots , the E mode of the weak lensing shear twopoint correlation function (Eq. (8) of Ref. (4)), normalized to the value under GR as a function of angular separation on the sky. For the angular scales of interest, the effects of changing and principally manifest themselves as a renormalization of . This is because the scales plotted are much smaller than the scales () at which shapechanging effects manifested themselves in Figure 3. Nonlinear power is treated using the usual subroutine halofit based on the semianalytic fitting scheme presented in Ref. (35). While we acknowledge that this is not strictly appropriate for modified gravity, we have no reason to think that the effect will be substantial for reasonable values of and . Furthermore, the constraints presented below in Section IV appear to principally derive from effects at the low, rather than the high, limit.
Iv Constraints on Deviations from GR
We now examine constraints imposed by current data on deviations from GR, allowing a large set of cosmological parameters to vary simultaneously. The investigation includes two different functional dependences for the gravitational modification parameters and . The first model for the postGR parameter form does not assume a particular redshift dependence but allows and to take independent values in each of three redshift bins. (In fact, we slightly smooth the transitions so as to avoid infinities in the derivatives entering the ISW effect, with a transition modeled by an arctan form of width .) That is, and for . We assume that and are scaleindependent. For we assume that differences from GR are negligible so and .
We test this theory against the data using a modified version of the public MCMC code COSMOMC (52); (54); (55) with a module (first presented in (56)) to incorporate the COSMOS weak lensing tomography data (3) and data from the CFHTLS survey (4). We also include WMAP5 CMB data (57); (58); (59) and Union2 supernova distance data (60). In all cases, we use the full covariance matrix (including systematics in the Union2 case) provided by the group who collected and initially analyzed the data. In addition to the postGR parameters, the parameter set includes , , (the ratio of the sound horizon to the angular diameter distance to last scattering), (the optical depth to reionization), , the amplitude of the SZ effect, and the amplitude of primordial scalar perturbations. We assume that for our effective dark energy, that , and that there are no massive neutrinos contributing to dark matter. Each weak lensing data set requires 3 nuisance parameters. Thus, we integrate over a total of up to 16 parameters, depending on the data sets used and the parametrization of and chosen. Under the binned parametrization, we vary or but not both simultaneously, which would require 19 independent parameters. This choice was made both for convenience and to reproduce the analysis of Ref. (39). Additional MCMC calculations done in which both and were allowed to vary in all three bins also returned results consistent with GR in the presence of a cosmological constant.
Figure 5 shows the marginalized probabilities on the parameters for runs in which , so that and the Poisson equation defined as in Eq. (4) remains valid at all redshifts. Figure 6 shows similar constraints on in the case where . Our results in all cases are consistent with GR within the 95% confidence limit, although they do allow the possibility of departures from GR with or .
Constraints on the usual cosmological parameters are largely unaffected by the introduction of and . Mean values shift by less than and marginalized uncertainties are comparable between GR and nonGR MCMC runs. The only notable exceptions are and (the physical density of cold dark matter in the universe), whose marginalized uncertainties increase by up to a factor 2.3 upon the introduction of postGR parameters. This is consistent with the observation that and principally modify the growth history of cosmological perturbations.
Figure 7 plots the 2dimensional confidence contours for the postGR parameters , in the case of redshift dependence as in Eqs. (18). Note that since this parameterization has the strongest effect at low redshift, the greater sky area of CFHTLS has more leverage in constraining the parameters than the greater depth of COSMOS. For the binned parametrization, the constraints from MCMC runs with WMAP5+Union2+CFHTLS (no COSMOS) were indistinguishable from those including COSMOS as well, supporting the supposition that the sky coverage of CFHTLS is, for current data, more important than the redshift depth of COSMOS in constraining the postGR parameters.
Table 2 presents the 95% constraints on our postGR parameters for all of the MCMC calculations considered in Figures 57. All of the results are consistent with GR.
Binned , :  Binned , :  Parameterization (18):  

COSMOS  +CFHTLS  +CFHTLS  COSMOS  +CFHTLS 
We also note that in Figure 7 the contours exhibit the same degeneracy implied by Figure 2. Apparently, the probe of growth provided by current weak lensing data is not able to add much more leverage to the CMB data. This can also be seen in the lack of significant change in the width of the probability distributions in Figure 5 when adding weak lensing.
The degeneracy illustrated in Figure 2 is plotted as the black, solid curve in Figure 7. The agreement with the likelihood contours is quite interesting, calling to mind the discussion in Sec. III.1 about parameter covariances. This arose from the observation that an unmodified Poisson equation (20) that relates the sum of the two metric potentials to the underlying density fluctuations leaves the largescale CMB predictions nearly unchanged when varying the ratio of the metric potentials, i.e. . That degeneracy is due to the fact that the largescale CMB predictions depend on the sum of the two metric potentials (cf. Eq. 19). If this sum is directly related to the underlying density perturbation then the only effect can have on the largescale CMB is through its effect on the evolution of ; by contrast, if the Poisson equation is of the form of Eq. (11), where only one potential is related to , then also appears in a multiplicative factor. Thus the specific approaches to modifying gravity give distinct relations between the parameters and the observables.
For observations that depend on the combination there will be a degeneracy along the curve (see Eqs. 13 and 15)
(21) 
We find numerically that this degeneracy applies approximately to both largescale CMB as well as weak lensing observations, even though both measurements have a further dependence on and through the growth factor (cf. Eq. 17). The relation in Eq. (21) gives the black, solid curve in Figure 7 and indeed is quite close to a degeneracy in the constraints.
V Discussion
Testing general relativity on cosmological length scales is an exciting prospect enabled by improvements in data. To interpret such a test requires an approach to parameterizing modifications from GR, similar to the PPN method for tests within the solar system and using binary pulsars, but appropriate for cosmic scales. Numerous parameterizations have been suggested and we compare, and in some cases, unify them through a “translation” table. These approaches can effectively be interpreted within one formalism with two parameters and (an extension of the previous CDM scenario).
In this generalized CDM model, even if the two parameters characterizing modifications to gravity are scale independent we find effects that are visible in the largescale structure matter power spectrum, and thus in weak lensing shear correlations, that depend on scale. We give quantitative results for the effects of the modifications on the cosmic microwave background temperature power spectrum, the growth of matter density perturbations and the density power spectrum, and the weak lensing statistics, along with analysis of the physical basis of the effects. On large scales in the density power spectrum, values of or above their GR values cause suppression of power while leading to enhancement on smaller scales.
We confront the modifications to GR with current cosmological observations, analyzing CMB (WMAP 5year), supernovae (Union2), and weak lensing (CFHTLS and COSMOS) data. Employing two different forms of dependence of the modifications on redshift, we find no evidence at 95% confidence level for such extensions to GR, regardless of the combinations of data used. Note that this holds for both the data employed by (39) (which used an overspecified system of equations in that analysis), and a more comprehensive set of observations.
We also verify the tradeoff between and predicted analytically. Such covariance leads to an interesting degeneracy for measurements depending on the sum of the metric potentials, although growth measurements depend on a different combination. Since large scale CMB and weak lensing depend on the sum of the metric potentials, one could consider the Poisson equation for the sum, and here the key parameter is the effective NewtonÕs constant . The matter density growth factor is primarily sensitive to extensions beyond GR in terms of the factor . These parameters still appear to have covariance, however, in our initial explorations. Overall, this suggests that exploration of gravity through cosmological measurements requires a sufficiently flexible theory space and a diverse set of observations.
As seen from Figures 24, robust identification of deviations from GR will require measurement over a large range of scales. Well below the Hubble scale, the modifications we have examined become scale independent and so can become confused with shifts in the fiducial amplitude (), galaxy bias, or normalization errors from photometric redshift estimation of weak lensing source densities. These will need to be addressed to have confidence in claims of any detected deviation, as will allowance for expansion histories different from CDM.
Finally, future data, including observations sensitive to growth and the growth rate, and those sensitive to the expansion history, will be essential to providing true tests of the framework of gravity on cosmic scales.
Acknowledgements.
We are extremely grateful to the Supernova Cosmology Project for permission to use the Union2 supernova data before publication. We acknowledge the sue of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. We thank Rachel Bean for helpful discussions on Ref. (39). This work has been supported by the World Class University grant R322008000101300 (SD, EL). EL has been supported in part by the Director, Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DEAC0205CH11231. AC acknowledges support from NSF CAREER AST0645427. RC acknowledges support from NSF CAREER AST0349213. LL was supported by the Swiss National Science Foundation under Contract No. 2000 124835 1.Appendix A Density Perturbation Growth
We here obtain the analog of the GR second order differential equation for matter density perturbation evolution, working in the conformal Newtonian gauge. After matterradiation decoupling, conservation of energy gives
(22)  
(23) 
assuming , i.e. there is no pressure, no pressure perturbation, and no anisotropic shear. Rearranging Eq. (22) and substituting Eq. (23), we can write
(24)  
We can use Eqs. (11) and (9) to write in terms of , , , and background quantities; similarly we can use the time derivative of Eq. (11) to write . This gives
(25)  
where the second equality comes from using Eqs. (22) and (23). Substituting Eq. (24) into (25) would just return the truism . However, if we take the first conformal time derivative of Eq. (25) before substituting, we find a second order differential equation describing the evolution of for arbitrary and . We omit the explicit copious algebra and show the result:
Footnotes
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