Weighing the spatial and temporal fluctuations of the dark universe
A generic prediction of the standard cosmology, based on general relativity (GR), dark matter and the cosmological constant (and more generally, smooth dark energy), is that, the two gravitational potentials describing the spatial and temporal scalar perturbations of the universe are equivalent. Modifications in GR or dark energy clustering in general violate this relation. Thus this ratio serves as a smoking gun of the dark universe. We propose a method to extract this ratio at various cosmological scales and redshifts from a set of measurements, in a model independent way. The ratio measured by future surveys has strong discriminating power for a variety of dark universe scenarios.
Introduction.— Predictions based on general relativity (GR) plus the Standard Model of particle physics are at odds with a variety of independent astronomical observations on galactic and cosmological scales, implying failures in particle physics or GR. There are various astrophysical tools to probe this dark side of the physical universe (e.g. Albrecht et al. (2006); Jain07 ()). Combining them allows us to break parameter degenaricies, reduce statistical errors and diagnose possible systematics.
These multiple probes are also crucial to detect smoking guns of new physics. For example, combining probes of the expansion history of the universe and probes of the large scale structure, the relation between the expansion rate and structure growth rate can be checked for signs of deviation from GR CR (). Indeed, one of the key questions in physics today is whether new particles/fields, such as dark matter and dark energy, or modifications to GR are needed to explain the observations.
On large scales, two features of gravity can distinguish between a dark sector and modified gravity Zhang07 (); Amendola07 (); Caldwell07 (); Hu07 (); Bertschinger08 (). One is the effective Newton’s constant , which specifies the coupling between gravity and matter. In GR, is equal to Newton’s constant, but modified gravity models often predict deviations. The other is the relation between the two gravitational potentials and . Here, the two potentials are defined in the Newtonian gauge through where is the scale factor. The ratio weighs the relative ability of perturbations in matter-energy to distort the space-time.111Refer to other equivalent notations in Caldwell07 (); Amendola07 (); Bertschinger08 (). An analogy of is the PPN parameter (by forcing for point source). Solar system tests have revealed SST () and provided strong support of GR. Constraints at galactic size and sub-cluster scales are consistent with GR too gammaGC (); Caldwell07 (). The standard cosmology, based on GR, dark matter and the cosmological constant (and more generally, smooth dark energy), predicts . Modifications from GR or emergence of intrinsic viscosity in dark energy fluid generally lead to deviating from unity. Therefore, identifying observations, or sets of observations, that will measure and is of paramount importance Jain07 (); Zhang07 (); Caldwell07 (); Amendola07 (); Bertschinger08 (); Stabenau:2006td ().
In Zhang07 (), we showed how to isolate the first key feature, feasibly testing the Poisson equation at accuracy level by combining weak lensing with galaxy redshift distortion. In this paper, we will show that the same surveys allow us to directly measure , the second key feature, at cosmological scales. This can be done in a rather model independent manner.
Models with .— Here we consider three models which produce deviations from the standard prediction ().
Perturbations in the Dvali-Gabadadze-Porrati (DGP) model DGP () have been carefully studied DGPgamma (); Koyama06 (). For a flat DGP model, , where and is the Hubble expansion rate. Here is the cross-over scale beyond which higher dimensional effects become important. In a flat model with matter density , . Since , in this model and the deviation from unity can be significant (Fig. 1).
Another modified gravity model (which aims to eliminate dark matter, not dark energy) is TeVeS TeVeS (), a relativistic version of MONDMOND (). Besides the gravitational metric, TeVeS contains a scalar and a vector field. It has been shown Skordis06 (); Dodelson06 () that the TeVeS vector field can source the evolution of cosmological perturbations and compensate for the lack of dark matter in the model. To fit observations, the TeVeS parameter should be small, in which case the vector perturbations and become large. These vector perturbations then drive to deviate from unity Skordis06 (); Dodelson06 (); Schmidt07 (),
Here . Since the background value as imposed by nucleosynthesis bounds, the deviation of from unity is mainly driven by the vector perturbation (). A numerical evaluation of is shown in Fig. 1. For this figure we adopted a model with , , and no dark matter.
A final possibility is that gravity is still GR, but dark energy has non-negligible anisotropic stress and causes inequality in two potentials through Ma95 ()
Although quintessence models predict , there are some dark energy models that predict and Hu99 (); Koivisto06 (). As a specific example, we consider an extrinsic shear stress of the form , with with constant, following Caldwell07 (). In general, varies not only with time, but also with scale. Richer physics encoded in the scale dependence of would allow better discrimination between such dark energy model from other scenarios.
The estimator.— To measure , two independent measures of gravitational potentials are required. Both and source the particle acceleration. However, the contribution from is suppressed by a factor , where is the particle velocity. For this reason, non-relativistic particles such as galaxies only respond to . For the same reason, photons respond equally to both the potentials. Thus gravitational lensing measures the projected along the line of sight. We propose an estimator consisting of the cross-correlation of each (the lensing field and the velocity field) with the galaxy distribution.
The first cross-correlation is the lensing measurement with galaxy over-density in a narrow redshift binZhang07 (). We can then obtain the cross-power spectrum between and the galaxy number overdensity in the redshift bin associated with the galaxies.
The second cross-correlation power spectrum can be obtained from the redshift distortions of the galaxy distribution in a spectroscopic survey Tegmark02 (); Zhang07 (); Jain07 (); Zhang08 (). Here, and is the comoving peculiar velocity. We show below that this cross-spectrum is directly related to , but first let us assume that this is so, that can be extracted from the - cross-correlation. In that case, the ratio of these two cross-spectra leads to an estimator for :
To see that is related to , recall that on large scales gravity is the only force accelerating galaxies, so , where is the proper motion. Taking the divergence of this leads to
Here, and is the growth factor of . The last relation holds in the linear regime where different modes decouple. We then have
the desired relation.
The proportionality factor relating the two cross-spectra in Eq. (5) requires knowledge of the expansion rate and the growth factor . We assume that the former can be measured by other means; indeed our goal is to distinguish dark sector models which produce identical expansion histories. No such assumption is needed for the growth factor, because the same survey that measures will also measure , which is proportional to and thus measurement of in multiple redshift bins can be used to recover (see the appendix for details). We adopt the minimum variance estimator to estimate errors in the reconstruction of and Zhang07 (); Zhang08 (). This reconstruction adopts no assumption on galaxy bias, so it is less affected by possible stochasticity or scale dependence in galaxy bias.
Application of the estimator in Eq. (3) relies on the condition of linear evolution such that Eq. (4) and therefore (5) hold. For this reason, we restrict our discussion to the linear regime. This approach is robust against several uncertainties: (1) It does not suffer uncertainty induced by the galaxy bias, whose effect cancels when taking the ratio in Eq. (3). (2) It is not susceptible to possible galaxy velocity bias, defined with respect to peculiar velocity of dark matter or dark energy, since we directly measure , instead of relying on a theory to calculate it. (3) It is applicable to general dark energy models and modified gravity models. It does not require dark energy to be smooth, nor gravity to be minimally coupled, nor scale-independent .
Forecast.—In order to measure in this way, the lensing and redshift surveys must be sufficiently deep and wide. The proposed spectroscopic galaxy survey ADEPT or 21cm survey HSHS Peterson06 (), combined with a lensing survey such as LSST, would be sufficient. Alternatively, SKA alone would be able to provide both suitable lensing, through cosmic shear Blake04 () and cosmic magnification Zhang05 (), and galaxy redshift measurements, as potentially would the Euclid222http://sci.esa.int/science-e/www/area/index.cfm?fareaid=102 mission. So we focus on SKA projections. can be measured by SKA at multiple bins of redshift and scale to impressive accuracy (Fig. 2). We then infer from the above measurements.
Projections for the errors on from SKA in a variety of bins are shown in Fig. 1. One example of the power of this measurement is in constraining the DGP model. The measurement proposed in Zhang07 () can only marginally distinguish the flat DGP model from CDM. Fig. 1 shows, though, that these models have significantly different predictions for ; The TeVeS model adopted has been shown to produce a good fit of CMB and LSS data Skordis06 (). However, with large deviation from , this model can be unambiguously distinguished from CDM. Thus and are highly complementary to probe the dark universe333Errors in and in are partly correlated. Future work should take this into account by fitting and simultaneously, while marginalizing all other parameters. ; Modifications in gravity or dark energy viscosity often lead to stronger scale dependence in than what is shown in Fig. 1. Our estimator could have stronger discriminating power for these models.
There is room to improve the measurement. (1) The fractional error in reconstructed from the redshift distortion is about 15 times larger than the cosmic variance limit Zhang08 (). Adopting the approximation of deterministic galaxy bias, the associated error will decrease by a factor of 3. However, to reach the cosmic variance limit, other velocity measurement techniques should be explored (e.g. Zhang08b ()). (2) The forecast outlined above only uses measurements from galaxies and thus limited the accuracy of measurement. Depending on the design and on the nature of 21cm emitting galaxies, SKA may allow measurements of at higher redshifts. Furthermore, at even higher redshifts () can be measured from redshift distortions of diffuse 21cm background. Improvement in the measurements of and at would result from the inclusion of such observations. (3) Furthermore, measurements of at can be made feasible by the inclusion of CMB lensing and 21cm background lensing.
We have shown that future precision imaging surveys of weak gravitational lensing and spectroscopic surveys of galaxy redshift distortions provide highly complementary methods to probe the dark universe. In combination they allow us to isolate two key features of the dark universe, the effective Newton’s constant and , from many astrophysical complexities, and distinguish competing scenarios of the dark universe robustly.
Acknowledgments.— PJZ is supported by the National Science Foundation of China grant 10533030, 10673022, CAS grant KJCX3-SYW-N2 and the 973 program grant No. 2007CB815401. RB’s work is supported by NASA ATP grant NNX08AH27G, NSF grants AST-0607018 and PHY-0555216 and Research Corporation. SD is supported by the US Department of Energy.
Appendix.—To infer from measured in limited redshift bins, a parametrization of is required. Since evolve smoothly, should not be strongly dependent on the precise form of the parametrization. In this paper, we extend a widely used parameterization for in standard gravity. For gravity models minimally coupled to matter, , where and is the linear density growth factor. One approximation adopted in the literature is (e.g. MMG ()). Here, is the normalized Hubble parameter. This approximation works well not only for CDM ( for and for ), but also for some modified gravity models such as DGP (, DGPgamma ()). We thus propose to fit a parameterization
Here, both and are parameters to be fitted for each bin. and are then obtained by the relation and .
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