Lensed Cosmic Microwave Background Constraints on PostGeneral Relativity Parameters
Abstract
The constraints on departures from general relativity (GR) at cosmological length scales due to cosmic microwave background (CMB) data are discussed. The departure from GR is measured by the ratio, parameterized as , between the gravitational potentials conventionally appearing in the geodesic equation and the Poisson equation. Current CMB data indicate at the 2 confidence level, while remains unconstrained. The departure from GR affects the lensing conversion of Emode into Bmode polarization. Hence, the lensing measurements from a future CMBpol experiment should be able to improve the constraints to for a fiducial model and independent of .
pacs:
PACS number(s): 95.85.Sz 04.80.Nn, 97.10.VmIntroduction—The quest for the source of the cosmic acceleration has led to speculation that the proper theory for gravitation departs from general relativity (GR) on cosmological length scales (e.g. Ref. Uzan:2006mf ). There are numerous theoretical examples that introduce new gravitational degrees of freedom and that are capable of producing a latetime acceleration, with wideranging implications for observable phenomena (e.g. Refs. Dvali:2000hr ; Carroll:2003wy ). Given this possible abundance in new physics, it is important to identify tests that can distinguish between the effects of dark energy and those of modified gravity. Though latetime accelerated cosmic expansion is the principal indicator that a new “dark” physics is needed, it is not the only test such physics must satisfy. A successful cosmology must also agree with measurements related to the behavior of inhomogeneities as probed by the cosmic microwave background and largescale structure.
To understand the extent to which cosmological data support GR, we make use of an approach motivated by the postNewtonian parameterization of the gravitational field within the Solar system and introduce a postGR parameterization for cosmological perturbations. Such a parameterization is also motivated by the common feature within a broad range of gravity theories of a decoupling of the perturbed Newtoniangauge gravitational potentials and , defined by the perturbed RobertsonWalker lineelement
(1) 
using the notation and convention of Ref. Ma:1995ey .
Whereas GR predicts in the presence of nonrelativistic matter, a gravitational slip, defined as , occurs in modified gravity theories. For example, this inequality means that the gravitational potential of a galaxy cluster is not the same potential traced by the geodesic motion of the constituent galaxies. Hence, a new relation between these potentials is a launching point for investigations of cosmological manifestations of modified gravity Hu:2007pj ; Bertschinger:2008zb . For primordial cosmological perturbations, the potentials are not completely free, however, as there exists a constraint equation in the longwavelength limit Bertschinger:2006aw .
We consider an alternative theory of gravitation that predicts an expansion history indistinguishable from CDM, accompanied by postGR effects whereby
(2) 
following Refs. Caldwell:2007cw ; Daniel:2008et . If the new gravitational phenomena is to mimic the effects of by changing the amount of spacetime curvature produced by the cosmic matter density, then we expect to grow to order unity at late times on large scales. Looking for clues to such a scenario, CMB temperature anisotropies alone provide a weak constraint to as the departure from GR is primarily manifest in the integrated SachsWolfe effect Lue:2003ky ; Hu:2008zd , as illustrated in Fig. 1. However, CMB lensing is also sensitive to because the lensing deflection of CMB photons by foreground largescale structure depends on the sum of the potentials Huterer:2006mva ; Acquaviva:2004fv ; Schimd:2004nq ; Calabrese:2008rt . In this Letter, we show that the expected conversion of Emode to Bmode polarization through lensing Zaldarriaga:1998ar , shown in Fig. 1, allows a new probe of departures from GR that will be accessible to future CMB Bmode polarization experiments.
The lensing of the CMB affects temperature perturbations at the level of a few percent at arcminute angular scales, which is on the damping tail of CMB anisotropies Lewis:2006fu . Using temperature anisotropy data from WMAP Komatsu:2008hk and ACBAR Reichardt:2008ay we can only put weak constraints on the postGR parameterization at present. On the other hand, Bmodes at tens of arcminute angular scales are mainly due to the lensing conversion from Emodes. Using the combination of E and Bmodes one can reconstruct the lensing signal in CMB data by using quadratic statistics Hu:2001kj ; Cooray:2002py ; Kesden:2003cc and likelihood methods Hirata:2002jy . The projected lensing potential power spectrum out to the last scattering surface can then be used to extract . As we find, upcoming high sensitivity CMB polarization experiments, such as CMBpol Baumann:2008aj ; Bock:2008ww of NASA’s Beyond Einstein program, have a significant role to play in constraining GR at cosmological length scales.
Calculational Method—The treatment of cosmological perturbations under modified GR follows from Ref. Daniel:2008et . The metric perturbation variables in the synchronous and conformal Newtonian gauges are related as , , where , are synchronousgauge metric variables, and the dot indicates the derivative with respect to the conformal time Ma:1995ey . In GR (), the perturbed Einstein equations,
(3)  
(4)  
(5) 
are used to evolve the metric variables, where Ma:1995ey .
In our postGR description, we assume the stressenergy tensor is conserved and that there is no preferred reference frame introduced by the new gravitational effects. Consequently, Eq. (4) remains valid but Eqns. (3, 5) do not. Because gravitational slip is degenerate with a cosmological fluid component with shear, Eq. (2) becomes
(6) 
This modification preserves the consistency condition for long wavelength cosmological perturbations Bertschinger:2006aw ; Daniel:2008et .
In our study we restrict attention to a homogeneous model of gravitational slip,
(7) 
and seek to constrain the postGR parameters and . By allowing the redshift dependence of the modified gravity parameter to be a free parameter, this relationship is more general than the one introduced in Ref. Caldwell:2007cw . (Note that we have also removed a prefactor .)
The lensing deflection of CMB photons by foreground largescale structure depends upon gradients in the total gravitational potential transverse to the line of sight to the last scattering surface Lewis:2006fu . The evolution of the lensing potential is separated from the primordial curvature perturbation using a transfer function , whereby . Hence, the power spectrum of the lensing potential is
(8) 
Here is the primordial power spectrum, is the conformal time at which a given photon was at the position , and the lensing source is given by:
(9) 
where we have made use of the postGR relation between and to simplify the expression in terms of the transfer function of . To evaluate the lensing source and angular power spectrum, we use Eqns. (4, 6) to evolve and , from which is obtained. The lensing potential for different values of the postGR parameters is shown in Fig. 1. In the case of temperature, lensing modifies the damping tail. The Bmode polarization signal due to lensing that peak at tens of arcminute angular scales is directly proportional to the lensing power spectrum. We ignore nonlinear corrections to the lensing calculation as nonlinearities are responsible for less than a 6% change to the Bmodes Lewis:2006fu and we only consider parameter constraints out to when using .





The E to Bconversion is on an angular scale where it is not contaminated by primordial gravitational wave signal in the Bmodes, which are relevant at larger angular scales, if at all. And although the implicitly assumed theory of gravitation should introduce new degrees of freedom, the scalarvectortensor decomposition of perturbations in linear theory ensures us that no further sources of Bmode polarization should arise. We further caution that viable models must satisfy within the Solar System, with a transition taking place near the outskirts of the galaxy. Rather than implying a scaledependence for postGR effects, this suggests that a viable model for must display a nonlocal or environmental dependence on the density field, with vanishing within a few tens of kpc of a galactic core. CMB photons are weakly lensed by Mpcscale density pertubations, but should not experience postGR effects while passing so near to galactic cores. On the celestial sphere, kpc radii subtend angular scales well below the angular scales of interest for nextgeneration polarization experiments. Thus, we tentatively ignore the positiondependence of the postGR effects introduced by Eq. 2.
In the case of temperature anisotropies, at small angular scales where the lensing effect is present, confusion from other secondary signals, most notably the SunyaevZel’dovich effect Cooray:2000ge ; Komatsu:2002wc and clustering of unresolved extragalactic point sources Serra:2008ge , must also be considered. When fitting to WMAP and ACBAR data, we take into account the contribution from clustered point sources on the angular power spectrum in order to avoid a bias in the determination of the cosmological parameters. We do this by writing the total CMB anisotropy spectrum as and allowing the amplitudes of both the SZ contribution and clustered point sources (with varying amplitudes for the two different experiments Serra:2008ge ) to vary as free parameters when fitting for the combined cosmological and postGR parameters.
We make use of the publicly available Markov Chain Monte Carlo (MCMC) package CosmoMC Lewis:2002ah with a convergence diagnostic based on the Gelman and Rubin statistic gelman to model fit the data. The postGR parameters are allowed to take values and . In addition, we implement the flat CDM cosmological model with six standard parameters: baryon density ; dark matter density ; reionization optical depth ; ratio of the sound horizon to the angular diameter distance at the decoupling measured by ; amplitude of the curvature perturbation (with flat prior on ; spectral index . These two last parameters are defined with respect to a pivot scale at h/Mpc, as in Ref. Dunkley:2008ie .
Results—We first use the combination of WMAP 5year Komatsu:2008hk and ACBAR data Reichardt:2008ay (both temperature and temperaturepolarization crosscorrelation). To avoid complications due to overlapping of the datasets, we use WMAP data out to and then ACBAR data in the range . The constraint on is at the 2 confidence level, but remains unconstrained. As shown in the upper panel of Fig. 2, there is a clear correlation between and : when goes to only very small values of are allowed and when , values of are allowed at the confidence level.
By including the postGR parameterization, we find that cosmological parameters from WMAP+ACBAR change by less than 1; for example, with postGR effects, and while and Komatsu:2008hk without postGR effects.
To study the extent to which future CMB data improve these constraints, we create mock datasets for both Planck and CMBpol. For Planck we create a mock temperature and polarization dataset with noise properties consistent with a combination of Planck at , , and GHz channels of HFI bluebook . We assume the best fit WMAP5ACBAR parameters without modified gravity Komatsu:2008hk as the underlying cosmological model. We use the fullsky likelihood function given in Ref. Lewis:2005tp when fitting the data. The upper and lower limits with Planck don’t show improvement compared to the case with WMAP and ACBAR; is still unconstrained and (2), mostly due to degeneracies with other cosmological parameters. While we include polarization information, Planck does not probe the lensed Bmode spectrum with adequate signaltonoise ratio, as seen in the middle panel of Fig. 1.
To study how improved polarization measurements, and thereby a measurement of the lensing potential power spectrum, improve the parameter constraints, we also make mock datasets for CMBpol using GHz to GHz for the 2meter version of the EPIC concept study Bock:2008ww . We also make a mock dataset of the lensing potential power spectrum under the same cosmological model by creating the noise spectrum for the lensing construction with using same experimental noise as above concept with the reconstruction calculated using quadratic statistics Hu:2001kj . To avoid any biases from nonlinearities, we consider only multipoles out to probed by CMBpol. The projected upper limit with CMBpol is (2), showing significant improvement compared to the presentday CMB constraints; the parameter remains unconstrained as we find the same strong correlation with the amplitude .
Previous studies have shown that the combination of Planck and a probe of the largescale structure weak lensing such as from NASA/DOE JDEM or the ESAbased Euclid can improve constraints of modified gravity. With fixed, the Planck and future weak lensing combination constrains at the 95% confidence level Daniel . In comparison with an experiment such as CMBpol using lensing information from Bmodes, we have considered the constraints on both the amplitude and the redshiftdependence of the postGR effects. If we fix , then for CMBpol we find (2), which is comparable to the projections for Planck and a halfsky spacebased weak lensing survey with Euclid. While competitive, the advantage with the CMB constraint is that it comes from a single dataset and avoids issues related to systematics that can impact weak lensing observations.
AC and PS acknowledge support from NSF CAREER AST0645427. PS thanks Alex Amblard for useful discussions. RC and SD are supported by NSF AST0349213 at Dartmouth.
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