Next Generation Redshift Surveys and the Origin of Cosmic Acceleration
Cosmologists are exploring two possible sets of explanations for the remarkable observation of cosmic acceleration: dark energy fills space or general relativity fails on cosmological scales. We define a null test parameter , where is the scale factor, is the growth rate of structure, is the matter density parameter, and is a simple function of redshift. We show that it can be expressed entirely in terms of the bias factor, , (measured from cross-correlations with CMB lensing) and the amplitude of redshift space distortions, . Measurements of the CMB power spectrum determine . If dark energy within GR is the solution to the cosmic acceleration problem, then the logarithmic growth rate of structure . Thus, on linear scales to better than 1%. We show that in the class of Modified Gravity models known as , the growth rate has a different dependence on scale and redshift. By combining measurements of the amplitude of and of the bias, , redshift surveys will be able to determine the logarithmic growth rate as a function of scale and redshift. We estimate the predicted sensitivity of the proposed SDSS III (BOSS) survey and the proposed ADEPT mission and find that they will test structure growth in General Relativity to the percent level.
In General Relativity (GR), there are four variables characterizing linear cosmic perturbations: the two gravitational potentials and , the anisotropic stress, , and the pressure perturbation, . All of these variables can depend on the wave number, , and the expansion factor . We focus initially on models with no dark energy clustering or pressure and discuss these effects later in the paper. We assume scalar linear perturbations around a flat FRW background in the Newtonian gauge,
and work in the quasi-static, linear approximation, which is valid for sub-horizon modes still in the linear regime.
The evolution of perturbations is described by the continuity, Euler and Poisson equations (i.e. Jain and Zhang (2007)):
With the assumption of no anisotropic stress, , these equations can be combined to derive the equation of motion for the growth factor , defined as :
where is the Hubble function, and a prime denotes derivative with respect to .
From Eq. (5) one can infer the two key features of GR with smooth dark energy. First, the growth factor is exactly determined once the Hubble function is known; and second, since none of the coefficients is a function of scale, the growth factor is scale-independent. Therefore, for a given expansion history, one can test GR in two ways: checking that the theoretical solution of Eq. (5) agrees with observations, and testing the hypothesis of scale-independence Bertschinger and Zukin (2008); Bertschinger (2006); Ishak et al. (2006); Daniel et al. (2008); Dore et al. (2007); Linder and Cahn (2007); Zhang et al. (2007); Wang (2007); Sahni and Starobinsky (2006); Yamamoto et al. (2008) . The growth rate of structure in GR is well approximated by , where the fitting function is accurate at the level Polarski and Gannouji (2007). We define a function that tests the growth rate of structure and can be directly related to observables:
The combination can be constrained via CMB measurements; it is currently known to within the 5% level fro the WMAP5 data Dunkley et al. (2008), with an expected gain of a factor from the upcoming satellite CMB mission Planck Pla (2006). The solid line in Fig. 1 shows in GR with dark energy: regardless of the details of the dark energy model, in the linear regime. By measuring this quantity, we can characterize deviations from General Relativity.
Ii f(R) theories in the PPF formalism
To quantify the expected deviations, we study a class of modified gravity (MoG) models known as theories, whose action is written as
is the action of standard matter fields. It has been noted long ago Carroll et al. (2004); Capozziello et al. (2003) that models where is an inverse power of the Ricci scalar can give rise to late-time acceleration; however, many of those models have been shown not to be cosmologically viable due to gravitational instability Dolgov and Kawasaki (2003).
Recently, Song, Hu and Sawicki Song et al. (2007) introduced an effective parametrization for theories which does not rely on any particular model and is able to discard models suffering from instabilities. The cosmological evolution is obtained by fixing the expansion history to match that of a dark energy model, for which we assume a constant equation of state :
Such requirement for translates into a second-order equation for , which can be solved numerically. Of the two initial conditions of this equation, one can be fixed requiring that at early times (i.e. for large ) in order to recover GR; the second defines a one-parameter family of curves which all generate the given . Such parameter is conveniently chosen as (also see Starobinsky (2007)):
Here and represent the first and second derivative of with respect to , respectively. GR is represented by the special case , so that effectively quantifies the deviation from GR at the present time. Furthermore, the gravitational stability condition is easily established as .
The additional degrees of freedom of the gravity introduce modifications in the Poisson equation and in the relation between the two gravitational potentials Jain and Zhang (2007), which now read:
The equation for the growth factor in MoG takes the form:
Given , a MoG model is not completely defined unless , the effective Newton’s Constant, and the metric ratio , are known, contrary to the GR case. For the class of theories under study, and in the sub-horizon, linear regime, Hu and Sawicki Hu and Sawicki (2007a) have provided a fit to and as
where is the function appearing in Eq. (9), and is the super-horizon metric ratio (see Song et al. (2007); Hu and Sawicki (2007a) for details).
The key result is that the models all predict an enhancement in the growth rate of structure. In fact, any positive value of gives rise to a negative , so that the effective Newton constant is larger with respect to GR. Moreover, both terms in have a negative sign, which induce further enhancement of matter clustering, as can be seen from Eq. (12). Since does not depend on , the scale dependence of the growth factor can only arise from the second term of Eq. (13). On large scales, the dominant term in the metric ratio is , while for increasing values of , the second term in the expression for becomes important and tends to the constant value of for large . The scale of the transition from scale-free to scale-dependent growth factor is ; due to the asymptotic behavior of , the growth differs significantly from GR even for models with very small values of , on sufficiently small scales. A mechanism to restore GR on scales of the galaxy and smaller is discussed in Hu and Sawicki (2007b, a). Fig. 1 shows for a few different values of and . For the GR case, , with no scale dependence.
Iii Measuring the growth of structure
Peculiar velocities displace galaxies along the line of sight in redshift space and distort the power spectrum of galaxies observed in redshift space. This effect is known as linear redshift space distortion and was first derived by Kaiser Kaiser (1987). In redshift-space, the power spectrum is amplified by a factor over its real-space counterpart,
where and are redshift and real space power spectra, respectively, is the cosine of the angle between the wavevector and the line of sight , and is the linear redshift-space distortion parameter defined as
is the linear bias, which we assume to be independent of scale.
Galaxy redshift surveys can be used to directly measure , and, if the bias is known, the growth rate of perturbations. In fact, the redshift space power spectrum can be decomposed into harmonics, whose relative amplitude depend on the growth rate of structure through . The possibility of using such dependence in order to constrain dark energy properties has been explored in Guzzo et al. (2008); here we focus on the measurements of the scale dependence of as a smoking gun of Modified Gravity.
We assume that is obtained through the ratio of quadrupole to monopole moments of the redshift power spectrum Hamilton (1997)
and use the prescription in Feldman et al. (1994) to get the errors in the above quantities:
where is the mean galaxy density, is the weight function, is the volume of the shell in -space, and the index ”i” assumes the values 0 and 2 for monopole and quadrupole, respectively.
The linear bias for a population of large-scale structure tracers can be estimated by cross-correlating the line-of-sight projected density of the tracer with a convergence map reconstructed by CMB lensing techniques, and comparing the resulting signal with theory. The weak lensing potential responsible for lensing the CMB can be written as the line-of-sight integral Bartelmann and Schneider (2001),
where is the comoving angular diameter distance corresponding to the comoving distance , and is the comoving distance to the last scattering surface. A quadratic combination of the measured CMB temperature and polarization Hu and Okamoto (2002); Okamoto and Hu (2003); Hirata and Seljak (2003) provides an estimator of the convergence field, . In this study, we have used the prescription of Hu and Okamoto (2002) to compute the expected noise power spectrum, , corresponding to the reconstructed convergence field by cross-correlating with the projected fractional overdensity of the tracer,
where is the normalized tracer distribution function in comoving distance. We measure the cross-correlation spectrum:
where is the matter power spectrum at the comoving distance and we have related the wavenumber to the multipole via the Limber approximation Limber (1954). The signal-to-noise ratio for such a cross-correlation can be estimated as Peiris and Spergel (2000),
where is the fraction of sky over which the cross-correlation is performed. For tracer counts the noise is Poisson, and the power spectrum is given by, where is the number of tracer objects per steradian.
Since the signal is proportional to the bias, , the expected error on can be written as
We consider three present and forthcoming redshift galaxy surveys: the SDSS LRG sample Tegmark et al. (2006), its extension BOSS-LRG BOS (), which we divide in two redshift bins, labeled as BOSS1 and BOSS2, and the proposed survey ADEPT ADE (). Specifics of each experiment are listed in Table 1. In all cases we assume that and do not change significantly with redshift within a survey, so that the observed quantity is , where is roughly the central redshift of the survey. As a direct comparison the capabilities of the three galaxy surveys under examination, we show the real space matter power spectrum, normalized to the SDSS LRGs median redshift, with its errorbars in Fig. 2.
We also consider three possible CMB experiments: a PLANCK-like CMB experiment with sky coverage and temperature and polarization sensitivities of 28 K-arcmin and 57 K-arcmin, respectively; a next generation CMB survey based on using a camera similar to that on ACT or SPT with a polarimeter and a years observing program (labeled PACT) with sky coverage and temperature and polarization sensitivities of 13 K-arcmin and 18 K-arcmin and an ideal polarization experiment (labeled IDEAL), with sky coverage and temperature and polarization sensitivities of 1 K-arcmin and 1.4 K-arcmin, respectively. The expected results from cross-correlation with the ADEPT and BOSS surveys, and the SDSS LRG are displayed in Table 1.
We show our main results in Fig. 3. Errorbars are computed using Planck as the complementary CMB lensing survey for SDSS LRG and BOSS, and PACT for ADEPT.
We summarize the current status of and bias measurements in Table 2, and add the expected errorbars on at two different scales, = 0.05 and = 0.2 , from our analysis, for comparison.
The errors on bias are scale-independent and vary from 17 for SDSS LRG to 1 for ADEPT. Errors on depend on scale; we bin our simulated data in bins in space of width = , and see that for all surveys errorbars decrease as we go to smaller scales. The corresponding constraints on are as strong as a few percent for BOSS, and of the order of 1% for ADEPT.
The combined effect of smaller errors and of the asymptotic behavior of the growth factor, which induces a large deviation of from its GR value on small scales, is that redshift galaxy surveys are more sensitive to the small-scale modification of gravity than to the large-scale one. Ultimately, the smallest observable value of will not be set by the capabilities of the survey, but by the breakdown of the linear regime assumption. Assuming Mpc as an upper limit, and for values of corresponding to redshifts between 0 and 2, the smallest inducing scale-dependent growth is . Such value is within reach of ADEPT; future experiments that detect redshifted 21 centimeter emission could probe even larger values of in the linear regime.
V Conclusions and discussion
We have built a null test parameter for General Relativity, , based on the consistency between expansion history and structure growth expected in GR. Such parameter can be expressed in terms of the combination , probed by the CMB experiments, the linear matter perturbations growth factor, probed by redshift galaxy surveys, and the linear bias, probed by cross-correlation of the two.
We have predicted the achievable precision in the measurement of for three redshift galaxy surveys, SDSS LRG, BOSS and ADEPT, together with Planck and a possible future CMB experiment, PACT. We have interpreted such result in the context of a one-parameter family of modified gravity theories, known as , which can give rise to cosmic acceleration. In such models, the matter clustering is enhanced on all scales with respect to the GR case, and the enhancement is largest on small scales. We concluded that the peculiar signatures of the theories will be definitely detectable with a survey like ADEPT.
More generally, any detection of deviation of from zero that was not due to some observational systematic would be a signature of truly novel physics with enhanced growth, pointing either to non-GR physics or to unexpected properties of dark energy: dark energy models with a non-zero sound speed are characterized by an oscillatory behavior of the growth DeDeo et al. (2003), and scalar field dark energy suppresses growth on large scales Unnikrishnan et al. (2008). Similarly, massive neutrinos suppress on scales below the neutrino free streaming scale (see Lesgourgues and Pastor (2006) for review).
We warmly thank E. Aubourg, C. Hirata, W. Hu, R. H. Lupton, M. A. Strauss and L. Verde for useful suggestions. DNS thanks the APC in Paris for its hospitality. This work was supported by NSF grant AST-0707731, the NSF PIRE program and the NASA LTSA program.
|0.15||Hawkins et al. (2003),Verde et al. (2002)||0.3||22.0||10.1||BOSS1 + Planck|
|0.35||Tegmark et al. (2006)||0.31||39.5||21.0||SDSS LRG + Planck|
|0.55||Ross et al. (2006)||0.5||9.3||5.5||BOSS + Planck|
|0.77||Guzzo et al. (2008)||0.6||10.6||6.5||BOSS2 + Planck|
|1.4||da Angela et al. (2006)||1.35||2.1||1.1||ADEPT + PACT|
|3.0||McDonald et al. (2005)|
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