The Three Faces of \Omega_{m}: Testing Gravity with SN Ia Surveys

# The Three Faces of Ωm: Testing Gravity with Low and High Redshift SN Ia Surveys

Alexandra Abate and Ofer Lahav
Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK
###### Abstract

Peculiar velocities of galaxies hosting Type Ia supernovae generate a significant systematic effect in deriving the dark energy equation of state , at level of a few percent. Here we illustrate how the peculiar velocity effect in SN Ia data can be turned from a “systematic” into a probe of cosmological parameters. We assume a flat -Cold Dark Matter model () and use low and high redshift SN Ia data to derive simultaneously three distinct estimates of the matter density which appear in the problem: from the geometry, from the dynamics and from the shape of the matter power spectrum. We find that each of the three ’s agree with the canonical value to within , for reasonably assumed fluctuation amplitude and Hubble parameter. This is consistent with the standard cosmological scenario for both the geometry and the growth of structure. For fixed for all three ’s, we constrain in the growth factor , so we cannot currently distinguish between standard Einstein gravity and predictions from some modified gravity models. Future surveys of thousands of SN Ia, or inclusion of peculiar velocity data, could significantly improve the above tests.

###### keywords:
large-scale structure of universe – cosmological parameters – surveys – galaxies: kinematics and dynamics

## 1 Introduction

**footnotetext: E-mail: aabate@star.ucl.ac.ukfootnotetext: E-mail: lahav@star.ucl.ac.uk

The observed present acceleration of the universe was first confirmed a decade ago by two separate groups using Type 1a supernovae (SN Ia, Perlmutter et al., 1999; Riess et al., 1998). SN Ia are one of a number of probes needed to obtain tighter constraints on dark energy equation of state, including any possible time evolution. This will require surveys of thousands of supernovae out to high redshifts to accurately measure their luminosity distances from which parameters describing the dark energy can be inferred. To achieve the desired constraints on dark energy, in particular a few percent constraint on the equation of state parameter , the supernovae will have to be accurately calibrated. It is therefore vital that this calibration is done accurately, and it is the low redshift supernovae which are vital to achieve this, for details see Wood-Vasey et al. (2002). At low redshift the supernovae distances have little or no dependence on the cosmological parameters such as , and the dark energy equation of state . They do however put a tight constraint on a combination of what is essentially the calibrated magnitude zeropoint () and the Hubble constant , whereas for the high redshift supernovae there is a strong degeneracy between , and the cosmological parameters of interest. Figure 1 illustrates the importance of the low redshift supernovae in anchoring the Hubble diagram. It shows the gold sample from Riess et al. (2007) constraints on and with and without supernovae with redshifts less than . One can see that without the low redshift supernovae (blue/light contours, using 146 SN Ia) the constraints blow up significantly compared to the full gold sample (red/dark contours, using 182 SN Ia). There are several sources of systematic error which affect the calibration of the zeropoint, for example dust extinction, luminosity evolution, weak lensing, and Malmquist bias (see Kim et al., 2004, for more details). This type of error is not decreased by having a large number of supernovae and will necessarily come to dominate the error budget. The systematic errors mentioned above have long been discussed in the literature and are not considered in this Letter. There is a source of error which is unique in the fact that it affects only the low redshift “calibrating” supernovae, their peculiar motions relative to the Hubble flow.

Previous authors have set about using SN Ia to quantify the degradation of dark energy errors due to peculiar motions or use them to trace the peculiar velocity field itself in a variety of ways. Three distinct approaches to this have been discussed recently in the literature. In Neill, Hudson, & Conley (2007) different flow models based on the IRAS PSCz survey (Branchini et al., 1999) were used to “correct” the luminosity distances by the known peculiar velocities before fitting them for the cosmological parameters of interest. They find the potential systematic error in caused by ignoring peculiar velocities is of the order of 4 percent, i.e. quite significant.

Radburn-Smith, Lucey, & Hudson (2004) compared peculiar velocities from 98 local supernovae with the gravity field predicted from IRAS. In Haugbølle et al. (2007) an angular expansion of the radial velocity field was used to probe the local dipole and quadrupole of the velocity field at three different distances. They found that the dipole is consistent with galaxy surveys (e.g. Erdoğdu et al., 2006) at the same Hubble flow depths.

The third and somewhat different method is utlised by Hui & Greene (2006); Cooray & Caldwell (2006); Gordon, Land & Slosar (2007) who take a covariance matrix approach. From the fluctuation in the luminosity distance induced by the peculiar motions, (see Hui & Greene, 2006; Pyne & Birkinshaw, 2004, 1996; Sasaki, 1987, for derivations), a covariance matrix for the resulting errors in the luminosity distance (or similarly the apparent magnitude) can be calculated. The covariance matrix depends on cosmological parameters which describe the growth and distribution of structure. In addition to the peculiar velocity effect this is due to gravitational lensing effect, which is important for redshifts larger than 1, and we shall ignore it in this Letter.

Cooray & Caldwell (2006) found that peculiar velocities of the low redshift supernovae may prevent measurement of to better than 10 percent, and diminish the resolution of the time derivative of projected for planned surveys. Gordon, Land & Slosar (2007) used the covariance matrix approach on current data, showing the changing constraints on , and depending on the exact redshift range of the SN Ia sample and whether the full covariance was included or not. They also apply the analysis to forecasting constraints for future surveys.

Here we unify the analysis of SN Ia data to study simultaneously fits for the expansion of the universe and the growth of structure. There is plenty of discussion on the possibility that the accelerated expansion of the universe is caused by a modification of general relativity on large scales (e.g. Durrer & Maartens, 2008; Huterer & Linder, 2007, and references therein). By measuring the growth of structure, which directly effects the observed peculiar velocity field, information is gained to differentiate between the two scenarios.

The rest of the Letter is organised as follows. In Section 2 we describe the SN Ia sample used in this Letter. In Section 3 we describe the theory underlying SN Ia analysis in cosmology and the effect of the velocity field.

## 2 Data

We analyse both nearby supernovae () from Jha, Riess, & Kirshner (2007) and high redshift supernovae () from a sample compiled by Davis et al. (2007) which includes data from Riess et al. (2007) and Wood-Vasey et al. (2007). Davis et al. (2007) combined the data from the two samples by normalising to the low redshift supernovae they had in common. Following Jha, Riess, & Kirshner (2007) 9 supernovae are excluded from the low redshift set, those that are unsuitable due to bad lightcurve fits. This includes supernovae with their first observation more than 20 days after maximum light, those that are hosted in galaxies with excessive extinction ( mag) and one outlier (SN1999e), which appears to have an extremely large peculiar velocity. This leaves 124 supernovae from the Jha, Riess, & Kirshner (2007) data set in the redshift range , and median redshift . The overlapping SN Ia in the two data sets were used to estimate a small normalising offset to the magnitudes from the Davis et al. (2007) data set (the extra magnitude error is negligibly small). The same procedure was used by Davis et al. (2007) in normalising the two high redshift data sets. After eliminating duplicated SN Ia, our combined data set has 271 SNe with , and .

## 3 Methodology

We describe here how we utilise the SN Ia dataset described in Section 2 to estimate cosmological parameters by including the peculiar velocity covariance.

### 3.1 Covariance matrix approach

The luminosity distance is defined as

 F=L4πd2L (1)

where is the observed flux of the supernova and is its intrinsic luminosity. The apparent magnitude of a supernova at redshift depends on the luminosity distance as follows

 m(z)=5log10DL(z)−5log10(H0)+M+25 (2)

where is the magnitude zeropoint, and is defined without the Hubble constant as in kms. The equation above ignores the additional terms which involve applying dust corrections, K corrections etc. For a flat universe containing a matter component and a dark energy component with a constant equation of state, the luminosity distance can be written as

 DL(z) = c(1+z)∫z0dz′E(z′) (3) E(z′) = (Ωm(1+z′)3+(1−Ωm)(1+z′)−3(1+w))12 (4)

If the universe was truly homogeneous and isotropic (FRW) this would be the end of the story, the observed would be described accurately by Eq. 3. However peculiar velocities have the effect of perturbing the luminosity distance

 δDLDL=vrc(1−c(1+z)2H(z)dL(z)) (5)

where is the radial peculiar velocity of the supernova and is the Hubble parameter. See Hui & Greene (2006); Bonvin, Durrer, & Gasparini (2006); Pyne & Birkinshaw (2004); Sasaki (1987) for a derivation. We emphasize that in the right hand side of the above equation is for an unperturbed FRW universe, derived at a perturbed redshift .

Therefore the covariance of the perturbation in , for a pair is given by

 CLij=⟨vrivrj⟩c2(1−c(1+z)2H(z)dL(z))i(1−c(1+z)2H(z)dL(z))j (6)

and

 ⟨vrivrj⟩=ξij=cosθicosθiΨ||(r)+sinθisinθjΨ⊥(r) (7)

is the linear theory radial peculiar velocity correlation function, (Gorski, 1988; Groth, Juszkiewicz, & Ostriker, 1989). The angles in Eq. 7 are defined by and the diagonal elements are given by Eq. 9 below. The and can be calculated from the matter power spectrum using linear theory

 Ψ||,⊥(r)=D′(zi)D′(zj)∫P(k)2π2B||,⊥(kr)dk (8)

where is the matter power spectrum, and and are the first and second derivative of the zeroth order spherical Bessel functions respectively and is the derivative of the growth function at redshift . is a function of the Hubble parameter and . See Section 3.3 for further discussion. The auto correlation is given by

 ξii=13D′2(zi)∫P(k)2π2dk. (9)

We can therefore calculate for a pair of supernovae at and respectively given a set of cosmological parameters.

In the above equations the growth factor is calculated exactly numerically. More insight to the dependence on is given by the commonly used approximation for the growth factor , where (Peebles, 1980), with little dependence on the cosmological constant (Lahav et al., 1991), and a slight dependence on (Wang & Steinhardt, 1998). Recent refined calculations predict for the concordance model, and (Linder & Cahn, 2007) for a particular modified gravity model, DGP braneworld gravity (Dvali, Gabadadze, & Porrati, 2000), though this is just an example of many possible modified gravity models. Below we shall constrain from the SN Ia data.

### 3.2 Likelihood analysis

To find the set of cosmological parameters that best fit the data we find the set that maximise the likelihood function. Assuming that the data and the observational errors are Gaussian random fields the likelihood function can be written as

 L=1√(2π)N|Σ|exp(−12N∑i,jDi(Σ−1)ijDj). (10)

where is defined as and is the covariance matrix including the observational noise. Following Gordon, Land & Slosar (2007) we write this as

 Σij=CLij+σ2iδij (11)

where is the standard uncorrelated error given by

 σ2i=(ln(10)5)2(σ2m+(μerri)2)+(1−c(1+z)2H(z)dL(z))2iσ2vc2 (12)

where is often set to 300kms and is included to account for nonlinear contributions to (which is derived only in linear theory, Silberman et al., 2001), and the velocity of the SN within the host galaxy. Here is the intrinsic magnitude scatter and is the error from the light curve fitting.

### 3.3 The Three Faces of Ωm

From the equations in Section 3.1 it can easily be seen that is a function of through

(i) : the geometry of the universe, from and also in Eq. 5. For a flat universe , with no dependence on other cosmological parameters. is most strongly constrained however by the high redshift SNe through Eqs. 3 and 4.

(ii) : the growth of structure in the universe, from in Eqs. 8 and 9 or the equivalent growth factor paramaterization .We note a strong degeneracy through the product where

 Ωm(z)=Ωm(z=0)(1+z)3Ωm(z=0)(1+z)3+1−Ωm(z=0) (13)

for a flat universe, our results can be scaled accordingly.

(iii) : the matter power spectrum in Eqs. 8 and 9. It is wel known that the shape of the power spectrum depend on the product , with some degeneracy with e.g. the spectral index , and baryon and neutrino mass densities and .
Please note that Eq.’s 8 and 9 contain all of the low redshift terms. If the CDM model of the universe is correct then when varying each of these “faces” of separately the results should be consistent with each other. If not this suggests that the CDM model is inconsistent and the data may favour a model which changes the theory of general relativity on large scales or other dark energy models.

## 4 Results

In the following analysis we assume a flat CDM universe with a dark energy equation of state . To gain an insight for the effect of varying the other cosmological parameters, namely , , and the “nuisance” parameters and we do not marginalise over them but present the results at some choice values for these parameters. The effect of marginalising over and degrades the error on by about 10 percent, the error on changes negligibly and the error on degrades by about 30 percent. The larger degradation of the error on is because of its strong degeneracy with . Other parameters are fixed as follows; , and . For clarity most contours have only the 1 confidence level, and where appropriate the assumed values of other parameters are stated.

Table 1 shows a set of results for each “face” of under different parameter combinations. This table shows that the different ’s are consistent with the canonical value of to within 1-. One can see the degeneracy direction of the errors (, ) and , discussed below. See table caption for an explanation of the columns.

Figure 2 shows the constraints on and the parameter , described at the end of Section 3.1. The contours shown are the 1 and 2 . For the 68 percent confidence constraint on is consistent with both Einstein gravity () and DGP gravity ().

Figure 3 shows the effect of and on the - contours. As expected there is only a strong influence on . One can also see the degeneracy direction of and in the power spectrum, shown by the indistinguishable differences between the red (dashed) contours in both panels. The positions of the red (dashed) contours show that decreasing by roughly 10 percent is equivalent to increasing also by roughly 10 percent. This is also shown by the blue (light) contours. The contours for - and - are not shown here since the same combinations of and as plotted in Figure 3 have a nearly negligible effect on the contour positions. Under different permutations of and the contours only shift in either the or direction and even then there is only a maximum shift of of .

Finally Figure 4 shows the effect of on all the contour pairs. It has the largest effect on and very little effect on . This is because the diagonal elements of (see Eq. 6) are approximately proportional to and to . Therefore the best-fit decreases as increases.

## 5 Conclusions

We have presented in this paper a unified approach for probing both the expansion of the universe and the growth of structure with SN Ia data and to test the consistency of the CDM model. We utilised the SNIa data to derive three distinct estimates of the matter density which appear in the problem: from the geometry, from the dynamics and from the shape of the matter power spectrum. We found that each of them agrees with canonical value to within 1. We note we are restricting our discussion to CDM, if we allow to vary our constraints on will weaken. We also constrained in the growth factor and found for , . This value of is consistent with both concordance and some proposed modified gravity models.

Current and future SN Ia surveys such as SN Factory, GAIA and Skymapper (for low redshift), SDSS-II (for intermediate redshift) and DES, Pan-STARRS, LSST, DUNE and SNAP (for high redshift) will generate samples of thousands of SN Ia (e.g. Albrecht et al., 2006; Peacock et al., 2006, for overviews). Large samples of low redshift SNe will greatly improve our constraints on and and the high redshift SNe on . Utilising galaxy peculiar velocity data (using and Tully-Fisher distance indicators) will also provide improvement on the and constraints. Our approach can also be generalised for a range of other cosmological parameters and exotic models of dark energy and gravity.

## 6 Acknowledgments

We are grateful to Kate Land for providing us with information about SN Ia data sets, and to Sarah Bridle, Josh Frieman, Chris Gordon, Saurabh Jha, John Marriner and Jochen Weller for useful conversations. AA acknowledges the receipt of a STFC studentship and OL acknowledges a Royal Society Wolfson Research Merit Award. Both AA and OL thank Fermilab and the Kavli Institute for Cosmological Physics at Chicago University for their hospitality. We thank the anonymous referee for their detailed comments.

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