Lensing and high-z supernova surveys

Gravitational lensing causes the distribution of observed brightnesses of standard candles at a given redshift to be highly non-gaussian. The distribution is strongly, and asymmetrically, peaked at a value less than the expected value in a homogeneous Robertson-Walker universe. Therefore, given any small sample of observations in an inhomogeneous universe, the most likely observed luminosity is at flux values less than the Robertson-Walker value. This paper explores the impact of this systematic error due to lensing upon surveys predicated on measuring standard candle brightnesses. We re-analyze recent results from the high-z supernova team (Riess et al. 1998), both when most of the matter in the universe is in the form of compact objects (represented by the empty-beam expression, corresponding to the maximal case of lensing), and when the matter is continuously distributed in galaxies. We find that the best-fit model remains unchanged (at Omega_m=0, Omega_Lambda=0.45), but the confidence contours change size and shape, becoming larger (and thus allowing a broader range of parameter space) and dropping towards higher values of matter density, Omega_m (or correspondingly, lower values of the cosmological constant, Omega_Lambda). These effects are slight when the matter is continuously distributed. However, the effects become considerably more important if most of the matter is in compact objects. For example, neglecting lensing, the Omega_m=0.5, Omega_Lambda=0.5 model is more than 2 sigma away from the best fit. In the empty-beam analysis, this cosmology is at 1 sigma.Comment: 11 pages, 3 ps figures. uses aaspp4.sty. accepted to ApJ Letters. includes analysis of lensing due to matter continuously distributed in galaxie

Localizing coalescing massive black hole binaries with gravitational waves

Massive black hole binary coalescences are prime targets for space-based gravitational wave (GW) observatories such as {\it LISA}. GW measurements can localize the position of a coalescing binary on the sky to an ellipse with a major axis of a few tens of arcminutes to a few degrees, depending on source redshift, and a minor axis which is $2 - 4$ times smaller. Neglecting weak gravitational lensing, the GWs would also determine the source's luminosity distance to better than percent accuracy for close sources, degrading to several percent for more distant sources. Weak lensing cannot, in fact, be neglected and is expected to limit the accuracy with which distances can be fixed to errors no less than a few percent. Assuming a well-measured cosmology, the source's redshift could be inferred with similar accuracy. GWs alone can thus pinpoint a binary to a three-dimensional ``pixel'' which can help guide searches for the hosts of these events. We examine the time evolution of this pixel, studying it at merger and at several intervals before merger. One day before merger, the major axis of the error ellipse is typically larger than its final value by a factor of $\sim 1.5-6$. The minor axis is larger by a factor of $\sim 2-9$, and, neglecting lensing, the error in the luminosity distance is larger by a factor of $\sim 1.5-7$. This large change over a short period of time is due to spin-induced precession, which is strongest in the final days before merger. The evolution is slower as we go back further in time. For $z = 1$, we find that GWs will localize a coalescing binary to within $\sim 10\ \mathrm{deg}^2$ as early as a month prior to merger and determine distance (and hence redshift) to several percent.Comment: 30 pages, 10 figures, 5 tables. Version published in Ap

Cosmology from supernova magnification maps

High-z Type Ia supernovae are expected to be gravitationally lensed by the foreground distribution of large-scale structure. The resulting magnification of supernovae is statistically measurable, and the angular correlation of the magnification pattern directly probes the integrated mass density along the line of sight. Measurements of cosmic magnification of supernovae therefore complements galaxy shear measurements in providing a direct measure of clustering of the dark matter. As the number of supernovae is typically much smaller than the number of sheared galaxies, the two-point correlation function of lensed Type Ia supernovae suffers from significantly increased shot noise. Neverthless, we find that the magnification map of a large sample of supernovae, such as that expected from next generation dedicated searches, will be easily measurable and provide an important cosmological tool. For example, a search over 20 sq. deg. over five years leading to a sample of ~ 10,000 supernovae would measure the angular power spectrum of cosmic magnification with a cumulative signal-to-noise ratio of ~20. This detection can be further improved once the supernova distance measurements are cross-correlated with measurements of the foreground galaxy distribution. The magnification maps made using supernovae can be used for important cross-checks with traditional lensing shear statistics obtained in the same fields, as well as help to control systematics. We discuss two applications of supernova magnification maps: the breaking of the mass-sheet degeneracy when estimating masses of shear-detected clusters, and constraining the second-order corrections to weak lensing observables.Comment: 4 pages, 2 figures, ApJL submitted; "Signal" discussed here is the extra covariance in astro-ph/050958

Problems with Pencils: Lensing Covariance of Supernova Distance Measurements

While luminosity distances from Type Ia supernovae (SNe) provide a powerful probe of cosmological parameters, the accuracy with which these distances can be measured is limited by cosmic magnification due to gravitational lensing by the intervening large-scale structure. Spatial clustering of foreground mass fluctuations leads to correlated errors in distance estimates from SNe. By including the full covariance matrix of supernova distance measurements, we show that a future survey covering more than a few square degrees on the sky, and assuming a total of ~2000 SNe, will be largely unaffected by covariance noise. ``Pencil beam'' surveys with small fields of view, however, will be prone to the lensing covariance, leading to potentially significant degradations in cosmological parameter estimates. For a survey with 30 arcmin mean separation between SNe, lensing covariance leads to a ~45% increase in the expected errors in dark energy parameters compared to fully neglecting lensing, and a ~20% increase compared to including just the lensing variance. Given that the lensing covariance is cosmology dependent and cannot be mapped out sufficiently accurately with direct weak lensing observations, surveys with small mean SN separation must incorporate the effects of lensing covariance, including its dependence on the cosmological parameters.Comment: 4 pages, 2 figures, PRL submitted; "Noise" discussed here is the "signal" in astro-ph/050957

Model-Independent Constraints on Dark Energy Density from Flux-averaging Analysis of Type Ia Supernova Data

We reconstruct the dark energy density $\rho_X(z)$ as a free function from current type Ia supernova (SN Ia) data (Tonry et al. 2003; Barris et al. 2003; Knop et al. 2003), together with the Cosmic Microwave Background (CMB) shift parameter from CMB data (WMAP, CBI, and ACBAR), and the large scale structure (LSS) growth factor from 2dF galaxy survey data. We parametrize $\rho_X(z)$ as a continuous function, given by interpolating its amplitudes at equally spaced $z$ values in the redshift range covered by SN Ia data, and a constant at larger $z$ (where $\rho_X(z)$ is only weakly constrained by CMB data). We assume a flat universe, and use the Markov Chain Monte Carlo (MCMC) technique in our analysis. We find that the dark energy density $\rho_X(z)$ is constant for 0 \la z \la 0.5 and increases with redshift $z$ for 0.5 \la z \la 1 at 68.3% confidence level, but is consistent with a constant at 95% confidence level. For comparison, we also give constraints on a constant equation of state for the dark energy. Flux-averaging of SN Ia data is required to yield cosmological parameter constraints that are free of the bias induced by weak gravitational lensing \citep{Wang00b}. We set up a consistent framework for flux-averaging analysis of SN Ia data, based on \cite{Wang00b}. We find that flux-averaging of SN Ia data leads to slightly lower $\Omega_m$ and smaller time-variation in $\rho_X(z)$. This suggests that a significant increase in the number of SNe Ia from deep SN surveys on a dedicated telescope \citep{Wang00a} is needed to place a robust constraint on the time-dependence of the dark energy density.Comment: Slightly revised in presentation, ApJ accepted. One color figure shows rho_X(z) reconstructed from dat

Lensing and Supernovae: Quantifying The Bias on the Dark Energy Equation of State

The gravitational magnification and demagnification of Type Ia supernovae (SNe) modify their positions on the Hubble diagram, shifting the distance estimates from the underlying luminosity-distance relation. This can introduce a systematic uncertainty in the dark energy equation of state (EOS) estimated from SNe, although this systematic is expected to average away for sufficiently large data sets. Using mock SN samples over the redshift range $0 < z \leq 1.7$ we quantify the lensing bias. We find that the bias on the dark energy EOS is less than half a percent for large datasets ($\gtrsim$ 2,000 SNe). However, if highly magnified events (SNe deviating by more than 2.5$\sigma$) are systematically removed from the analysis, the bias increases to $\sim$ 0.8%. Given that the EOS parameters measured from such a sample have a 1$\sigma$ uncertainty of 10%, the systematic bias related to lensing in SN data out to $z \sim 1.7$ can be safely ignored in future cosmological measurements.Comment: 5 pages, 4 figures; one figure and references added; minor modifications to text; reflects version accepted for publication in Ap

Future Type Ia Supernova Data as Tests of Dark Energy from Modified Friedmann Equations

In the Cardassian model, dark energy density arises from modifications to the Friedmann equation, which becomes H^2 = g(\rhom), where g(\rhom) is a new function of the energy density. The universe is flat, matter dominated, and accelerating. The distance redshift relation predictions of generalized Cardassian models can be very different from generic quintessence models, and can be differentiated with data from upcoming pencil beam surveys of Type Ia Supernovae such as SNAP. We have found the interesting result that, once $\Omega_m$ is known to 10% accuracy, SNAP will be able to determine the sign of the time dependence of the dark energy density. Knowledge of this sign (which is related to the weak energy condition) will provide a first discrimination between various cosmological models that fit the current observational data (cosmological constant, quintessence, Cardassian expansion). Further, we have performed Monte Carlo simulations to illustrate how well one can reproduce the form of the dark energy density with SNAP. To be concrete we study a class of two parameter ($n$,$q$) generalized Cardassian models that includes the original Cardassian model (parametrized by $n$ only) as a special case. Examples are given of MP Cardassian models that fit current supernovae and CMB data, and prospects for differentiating between MP Cardassian and other models in future data are discussed. We also note that some Cardassian models can satisfy the weak energy condition $w>-1$ even with a dark energy component that has an effective equation of state $w_X < -1$.Comment: revised version accepted by Ap

Reduced convergence and the local smoothness parameter: bridging two different descriptions of weak lensing amplification

Weak gravitational lensing due to the inhomogeneous matter distribution in the universe is an important systematic uncertainty in the use of standard candles in cosmology. There are two different descriptions of weak lensing amplification, one uses a local smoothness parameter $\tilde{\alpha}$, the other uses reduced convergence $\eta= 1+ \kappa/|\kappa_{min}|$ (where $\kappa$ is convergence). The $\tilde{\alpha}$ description involves Dyer-Roeder distance $D_A(\tilde{\alpha}|z)$ ($\tilde{\alpha}=1$ corresponds to a smooth universe); it is simple and convenient, and has been used by the community to illustrate the effect of weak lensing on point sources such as type Ia supernovae. Wang (1999) has shown that the $\tilde{\alpha}$ description can be made realistic by allowing $\tilde{\alpha}$ to be a local variable, the local smoothness parameter. The $\eta$ description has been used by Wang, Holz, & Munshi (2002) to derive a universal probability distribution (UPDF) for weak lensing amplification. In this paper, we bridge the two different descriptions of weak lensing amplification by relating the reduced convergence $\eta$ and the local smoothness parameter $\tilde{\alpha}$. We give the variance of $\tilde{\alpha}$ in terms of the matter power spectrum, thus providing a quantitative guidance to the use of Dyer-Roeder distances in illustrating the effect of weak lensing. The by-products of this work include a corrected definition of the reduced convergence, and simple and accurate analytical expressions for $D_A(\tilde{\alpha}|z)$. Our results should be very useful in studying the weak lensing of standard candles.Comment: Revised and expanded version. ApJ accepte

Large-Scale Bulk Motions Complicate the Hubble Diagram

We investigate the extent to which correlated distortions of the luminosity distance-redshift relation due to large-scale bulk flows limit the precision with which cosmological parameters can be measured. In particular, peculiar velocities of type 1a supernovae at low redshifts may prevent a sufficient calibration of the Hubble diagram necessary to measure the dark energy equation of state to better than 10%, and diminish the resolution of the equation of state time-derivative projected for planned surveys. We consider similar distortions of the angular-diameter distance, as well as the Hubble constant. We show that the measurement of correlations in the large-scale bulk flow at low redshifts using these distance indicators may be possible with a cumulative signal-to-noise ratio of order 7 in a survey of 300 type 1a supernovae spread over 20,000 square degrees.Comment: 6 pages; 4 figure

A new measure of σ8\sigma_8 using the lensing dispersion in high-zz type Ia SNe

The gravitational lensing magnification or demagnification due to large-scale structures induces a scatter in peak magnitudes of high redshift type Ia supernovae (SNe Ia). The amplitude of the lensing dispersion strongly depends on that of density fluctuations characterized by the $\sigma_8$ parameter. Therefore the value of $\sigma_8$ is constrained by measuring the dispersion in the peak magnitudes. We examine how well SN Ia data will provide a constraint on the value of $\sigma_8$ using a likelihood analysis method. It is found that the number and quality of SN Ia data needed for placing a useful constraint on $\sigma_8$ is attainable with Next Generation Space Telescope.Comment: 9 pages, 3 figures. Accepted for publication in The Astrophysical Journa