THE GROWTH RATE OF COSMIC STRUCTURE FROM PECULIAR VELOCITIES AT LOW AND HIGH REDSHIFTS

In the standard cosmological model, the universe is dominated by cold dark matter combined with a cosmological constant or dark energy (ΛCDM). While the existence of dark matter is supported by dynamical tests, gravitational lensing, and fluctuations in the cosmic microwave background (CMB), the evidence for a cosmological constant is primarily through geometric constraints on the redshift–distance relation (standard candles and standard rulers) and hence on the expansion history of the universe (for example, Komatsu et al. 2011, hereafter WMAP7+BAO+H₀). Modified gravity theories, however, can mimic the expansion history of the ΛCDM model. Linder (2005) has emphasized that it is essential to measure the growth of structure as a function of cosmic time as this allows one to break this degeneracy. He also shows that for many models, the logarithmic derivative of the growth of structure can be parameterized as

$$\begin{equation} f (z)\equiv \frac{d \ln g}{d \ln a} = \mbox{$\Omega _{\rm m}$}(z)^{\gamma }, \end{equation} \tag{ 1 }$$

where z is the redshift, g is the linear perturbation growth factor, a = 1/(1 + z) is the expansion factor, and γ is 0.55 for ΛCDM (Wang & Steinhardt 1998). In contrast, for example, γ = 0.68 in the Dvali et al. (2000, hereafter DGP) brane world modified gravity model (Linder & Cahn 2007).

There are several ways to measure the amplitude of the dark matter power spectrum at redshifts lower than that of the CMB, including cosmic shear from weak gravitational lensing and the abundance of rich clusters. Another promising way to probe the growth rate of structure is via peculiar velocities (Guzzo et al. 2008; Kosowsky & Bhattacharya 2009). Peculiar velocities are directly proportional to the derivative of the growth factor, i.e., proportional to f.

There are two ways to measure peculiar velocities. The first method is statistical: given a galaxy redshift survey, the distortion of the power spectrum or correlation function in redshift space depends on β = f/b, where b is a galaxy bias parameter (Kaiser 1987). On large scales, we may assume that linear biasing holds, i.e., b = σ_{8, g}/σ₈, where σ₈ is the rms density contrast within an 8 Mpc h⁻¹ sphere, h is the Hubble parameter in units of 100 km s⁻¹ Mpc⁻¹, and the subscript "g" indicates the fluctuations in the galaxy density, whereas no subscript indicates fluctuations in the mass density contrast. Galaxy redshift surveys also allow one to measure σ_{8, g} directly, so one can combine the observables to obtain the combination fσ₈ = βσ_{8, g}. By combining redshift-space distortion (RSD) measurements of fσ₈ at different redshifts, one can study the growth of linear structures over a range of redshifts (Guzzo et al. 2008; Song & Percival 2009; Blake et al. 2011a; Samushia et al. 2012; Basilakos 2012; Basilakos & Pouri 2012).

A second method is to measure peculiar velocities directly, by measuring distances to individual galaxies (via standard candles or standard rulers), and comparing these distances to their redshifts. We refer to this method as "measured distance" (MD). The measured peculiar velocities can then be compared with the peculiar velocities predicted from a galaxy density field, $\delta _g(\mathbf {r})$, derived from an independent redshift survey, under the assumption that fluctuations in the mass are linearly related to those in the galaxy density. Specifically, we have (Peebles 1993)

$$\begin{equation} \mathbf {v}(\mathbf {r}) = \frac{H_0}{4 \pi }\frac{f}{b} \int _{0}^{\infty } d^3\mathbf {r}^{\prime }\delta _g(\mathbf {r}^{\prime }) \frac{\mathbf {r}^{\prime }-\mathbf {r}}{\mid \mathbf {r}^{\prime }-\mathbf {r}\mid ^{3}}\,. \end{equation} \tag{ 2 }$$

This comparison of the two measured quantities $\mathbf {v}(\mathbf {r})$ and $\delta _g(\mathbf {r})$ yields a measurement of the degenerate combination β = f/b, which can be converted into the combination fσ₈ in the same way as for RSDs. Note that because this is a velocity–density cross correlation, unlike RSDs, it is not affected by cosmic variance. Instead, the noise arises from the precision of the MDs themselves.

The two peculiar velocity probes are complementary: RSDs require large volumes, driving one to surveys at higher redshifts. MDs have errors which are a constant fraction of distance. Hence, the error in peculiar velocity in units of km s⁻¹ increases linearly with distance and so MD surveys are necessarily restricted to low redshifts. However, as we will show it is the lowest redshift data that have the most "lever arm" for constraining the cosmological parameters considered here. The important point is that by combining high- and low-redshift measurements of f(z)σ₈(z), we can break many degeneracies in the cosmological parameters.

An outline of this Letter is as follows. In Section 2, we present the data used in our analysis. Section 3 presents fits in the $\mbox{$\Omega _{\mathrm{m},0}$}$–σ_{8, 0} plane for the ΛCDM class of models. In Section 4, we allow γ to vary, but use CMB data to constrain the amplitude of σ₈(z_CMB). We discuss future prospects in Section 5 and summarize in Section 6.

We will combine measurements of fσ₈ from the two distinct methods discussed above. The majority of the data are from RSD measurements, many of which were previously compiled in Blake et al. (2011a), but adding recent results from the Baryon Oscillation Spectroscopic Survey (BOSS) (Reid et al. 2012) and 6dFGS (Beutler et al. 2012). These are summarized in Table 1.

As discussed above, it is also possible to measure fσ₈ from the comparison of peculiar velocity and density fields, where peculiar velocities are obtained directly from MDs. Most of the results from this method (up to 2005) were summarized in Table 3 of Pike & Hudson (2005, hereafter PH). They showed that results from estimates of fσ₈ were consistent² with 0.41 ± 0.03. However, since both the peculiar velocity and redshift survey data sets overlap somewhat, the measurements are not independent and it is therefore difficult to assess the uncertainties.

Its possible to obtain a value of fσ₈ using non-overlapping data sets: by comparing peculiar velocities from the "Composite" sample of Watkins et al. (2009) to predictions derived from the IRAS Point Source Catalog Redshift (PSCz) survey galaxy density field (Saunders et al. 2000), we find β_I = 0.49 ± 0.04 after marginalizing over residual bulk flows. This yields fσ₈ = 0.37 ± 0.04, in good agreement with the PH mean value quoted above. These uncertainties may underestimate possible systematics, however, since only one density field (PSCz) is used and, furthermore, the same comparison method is used for all peculiar velocity samples.

Therefore, in order to make a fair (albeit conservative) estimate of the uncertainties in the methods, in this Letter we focus on only two recent measurements of fσ₈ from MD surveys. The first of these is from Turnbull et al. (2012, hereafter THF), who compiled the "First Amendment" (hereafter A1) set of 245 peculiar velocities from Type Ia supernovae (SNe) with z < 0.067. The A1 peculiar velocities were compared to predictions from the IRAS PSCz galaxy density field, yielding fσ_{8, lin} = 0.40 ± 0.07 at a characteristic depth of z = 0.02. We note that the uncertainties on fσ₈ are marginalized over possible residual bulk flows. The results are also insensitive to details of, for example, corrections for extinction in the SN host.³ Further details about the uncertainties and light-curve fitters used for the A1 SNe can be found in THF.

The second low-z MD result is from Davis et al. (2011, hereafter DNM), who analyzed 2830 Tully–Fisher peculiar velocities at z < 0.033 and compared these to the predictions from the galaxy density field derived from the Two Micron All Sky Survey Redshift Survey (Huchra et al. 2012). The analysis differs from THF: rather than the simple point-by-point comparison of predicted and observed peculiar velocities, DNM applied a spherical harmonic decomposition to both fields and found fσ₈ = 0.31 ± 0.05. We note that a straight average of the THF and DNM results yields fσ₈ = 0.36 ± 0.04 in excellent agreement with the Composite versus PSCz result found above.

We stress the that these two distance measurement results are completely independent: they use different peculiar velocity samples, different density fields, and different reconstruction and comparison methods. The two measurements are consistent with each other: the difference in fσ₈ is 0.09 ± 0.08. The RSD and MD measurements are shown in Figure 1.

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In the ΛCDM model, γ = 0.55. The model allows us to predict the value of $f(z) = \mbox{$\Omega _{\rm m}$}(z)^{\gamma }$ and σ₈(z) at any redshift, assuming their values at redshift z = 0 (denoted by subscript "0").

For the fits, we use a simple χ squared statistic with the following form:

$$\begin{equation} \mbox{$\chi ^2$}= \sum _{i} \frac{[f\sigma _{8}(\mathrm{meas})_i-f\sigma _8(\mathrm{pred})_i]^2}{\sigma _{i}^{2}} , \end{equation} \tag{ 3 }$$

where the first term is the measured value and the second term is the model, and σ²_i is the uncertainty for each measured value. For RSD measurements at z ≳ 0.1, fσ₈ is degenerate with the redshift–distance relation, due to the Alcock & Paczynski (1979, hereafter AP) effect. If we assume a flat universe, then the redshift–distance relation, and hence the AP distortion are all fixed by our choice of $\mbox{$\Omega _{\mathrm{m},0}$}$. With the AP distortion fixed, the appropriate fσ₈ values can then be calculated, but only for those surveys that have published the covariance matrix between the AP effect and fσ₈: WZ and BOSS. The peculiar velocity measurements (THF and DNM) and the low-z 6dF RSD measurement are negligibly affected by the AP distortion.

Figure 2 shows the results of the fit in the $\mbox{$\Omega _{\mathrm{m},0}$}$ versus σ_{8, 0} parameter plane with γ = 0.55. While the WZ+BOSS-only fit is quite degenerate in the $\mbox{$\Omega _{\mathrm{m},0}$}\hbox{--}\sigma _{8,0}$ plane, these degeneracies are broken when low-z data are included in the fit. Assuming a fixed γ = 0.55, the best fit is $\mbox{$\Omega _{\mathrm{m},0}$}= 0.259\pm 0.045$ and σ_{8, 0} = 0.748 ± 0.035 but the WMAP7+BAO+H₀ value is consistent with these results. Some ΛCDM variants, such as those with a non-standard effective number of relativistic species, are disfavored. The figure shows one such example (N_eff = 3.8), from Keisler et al. (2011). We stress that these results depend only on growth measurements and hence are independent of other determinations (CMB, SNe, BAO, etc.)

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As discussed above, it is also interesting to measure the growth rate index γ. However, once γ is included as a third parameter, the fits become very degenerate. These degeneracies can be broken using CMB data, for example, from WMAP7+BAO+H₀. Let us assume that the Friedman equation (and hence the expansion history) is the same as for the ΛCDM model, but treat γ as a phenomenological growth parameter which is allowed to differ from 0.55. At high redshifts (z ∼ 1000), $\mbox{$\Omega _{\rm m}$}= 1$ to high accuracy and so the growth of perturbations at early epochs is independent of γ. Therefore, we can use WMAP7+BAO+H₀ parameters⁴ Ω_{m, 0} and σ_{8, 0} to fix the amplitude of fluctuations at high redshift, as well as fixing the Friedman equation and expansion history. Note that the quoted WMAP7 σ_{8, 0} is extrapolated to z = 0 assuming ΛCDM. To allow for other values of γ we use ΛCDM to calculate σ₈(z_CMB), and then extrapolate the z_CMB predictions forward at later times, using different values of γ (following Samushia et al. 2012, Section 4.5). We then use the WMAP7+BAO+H₀ Monte Carlo Markov chains to marginalize over Ω_{m, 0} and σ_{8, 0}. The resulting fits are shown in Figure 1.

Table 2 gives the derived γ measurements for different combinations of the fσ₈ measurements. Also listed are the uncertainties in γ arising from fσ₈, from the CMB determinations of Ω_{m, 0} and σ_{8, 0}, and the total error. Note that the errors from WMAP+BAO+H₀ are not independent (between one fσ₈ measurement and another), are weakly dependent on z, and dominate the total error budget when all data are combined. Figure 3 shows the derived γ for all fσ₈ measurements assuming the WMAP7+BAO+H₀ parameters, and fixing σ₈(z_CMB). The results from the MD surveys are consistent with those from all of the RSD measurements. The hatched region shows the best fit to all data⁵ combined: 0.619 ± 0.054, consistent with ΛCDM.

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Table 2. Measurements of γ from Combinations of the Data

Sample	γ	σCMB	σtot
WZ	0.666+0.077 − 0.073	0.053	0.092
LRG	0.625+0.072 − 0.077	0.046	0.088
All RSD	0.607+0.038 − 0.040	0.046	0.060
THF+DNM	0.653+0.073 − 0.064	0.035	0.077
All	$\mathbf {0.619^{+0.033}_{-0.035}}$	0.042	0.054

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Although for purposes of illustration we have focused on a constant scale-independent γ, peculiar velocity data also allow one to test more complicated modified gravity scenarios (see, e.g., the review by Clifton et al. 2012). One example is the DGP brane world model. Figure 1 shows the predicted values of fσ₈ for a self-accelerating flat DGP model, with parameters chosen to match the expansion history and CMB constraints (Lombriser et al. 2009), and with a perturbation amplitude at early times chosen to match the CMB. Although the expansion history of the DGP model is similar to ΛCDM, the Friedmann equation, and hence Ω_m(z), differs. Furthermore, its γ = 0.68 also differs considerably from the ΛCDM value. These effects lead to lower growth and much lower values of fσ₈ at z < 1. This model, already disfavored at the 5σ level from other data (Lombriser et al. 2009), is excluded at the ∼8σ level using only the fσ₈ measurements discussed here. An open DGP model, which is a better fit to the redshift–distance relation, is excluded at the ∼10σ level by growth measurements alone. This suggests that alternative modified gravity models will be strongly constrained by the requirement of simultaneously matching the ΛCDM expansion history and the ΛCDM growth history.

The prospects for improving the measurements of γ are excellent. At present, the error budget is dominated by the WMAP7+BAO+H₀ estimates of the parameters at high redshift. Planck will reduce the CMB uncertainties so that these become subdominant.

Peculiar velocity measurements will continue to improve, leading to a reduction in the observational errors in fσ₈. The BOSS measurement will improve with further data releases. RSD measurements are also being made at higher redshifts (Bielby et al. 2012). However, the statistical power all of the RSD measurements combined (which are based on 800,000 redshifts) is similar to that of MD (based only on ∼3000 peculiar velocity measurements), the subsamples having uncertainties of 0.060 and 0.077, respectively, in γ (Table 2). It is therefore clearly of great interest to improve the statistics of MDs. At low redshift, supernovae will continue to accumulate. We can also look forward to fundamental plane peculiar velocities from 6dFGS (Springob et al. 2012), and later an order-of-magnitude increase in the number of Tully–Fisher peculiar velocities from WALLABY.⁶ Finally, the kinetic Sunyaev–Zel'dovich effect will be used to measure the velocity field of clusters at intermediate redshift (Hand et al. 2012). The redshift survey data used to construct the density field and hence the predicted peculiar velocities is also improving (Lavaux & Hudson 2011). It remains to better understand systematics by comparing measurements using the same sets of peculiar velocity and density data but different methods.

We have shown that by combining measurements of fσ₈(z) at different redshifts, and in particular by including results at z ∼ 0 from MD surveys, we can break the degeneracy between $\mbox{$\Omega _{\mathrm{m},0}$}$ and σ_{8, 0} and obtain $\mbox{$\Omega _{\mathrm{m},0}$}= 0.259\pm 0.045$ and σ_{8, 0} = 0.748 ± 0.035, consistent with independent determinations from WMAP7+BAO+H₀.

We can also constrain the growth index γ by comparing measurements of fσ₈(z) at low z, after fixing their values at z_CMB. The strongest leverage on γ arises from peculiar velocity measurements at the lowest redshifts. By including these measurements, we derive γ = 0.619 ± 0.054, consistent with ΛCDM. The Planck mission plus upcoming peculiar velocity and redshift surveys will tighten these constraints further.

We thank Gigi Guzzo, Florian Beutler, and David Parkinson for useful comments. M.J.H. and S.J.T. acknowledge the financial support of NSERC and OGS, respectively.