Ann. Phys. (Berlin) 19, No. 3 – 5, 328 – 331 (2010) / DOI 10.1002/andp.201010439 Constraining unified dark matter models with weak lensing Stefano Camera1,2,∗ 1 2 Dipartimento di Fisica Generale “Amedeo Avogadro”, Università degli Studi di Torino, Torino, Italy Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Torino, Italy Received 1 October 2009, accepted 4 January 2010 Published online 26 February 2010 Key words Cosmology, uniﬁed models, dark matter, dark energy, weak lensing, large-scale structure. Uniﬁed Dark Matter (UDM) models provide an intriguing alternative to Dark Matter (DM) and Dark Energy (DE) through only one exotic component, i.e. a classical scalar ﬁeld ϕ(t, x). Thanks to a non–canonical kinetic term, this scalar ﬁeld can mimic both the behaviour of the matter–dominated era at earlier times, as DM do, and the outcoming late–time acceleration, as a cosmological constant DE. Thus, it has been shown that these models can reproduce the same expansion history of the ΛCDM concordance model. In this work I review the ﬁrst prediction of a physical observable, the power spectrum of the weak lensing cosmic convergence (shear). I present the weak lensing signal as predicted by the standard ΛCDM model and by a family of viable UDM models parameterized by the late–time sound speed c∞ of the scalar ﬁeld.last– scattering surface and a series of background galaxies peaked at different redshifts and spread over different redshifts as described by a functional form of their distribution of sources. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Cosmological perturbations in UDM models In Uniﬁed Dark Matter (UDM) models [1–6], the Universe is ﬁlled with a perfect ﬂuid of radiation, baryons and a scalar ﬁeld ϕ(t). Thanks to a non–canonical kinetic term in its Lagrangian, the energy density of the scalar ﬁeld of the family of UDM models I use in this work [7, 8] presents two terms ρ(t) = ρDM (t) + ρΛ , where ρDM behaves like a DM component (ρDM ∝ a−3 ) and ρΛ like a cosmological constant DE component (ρΛ = const.). Consequently, ΩDM = 8πGρDM (a = 1)/(3H0 2 ) and ΩΛ = 8πGρΛ /(3H0 2 ) are the density parameters of DM and DE today, and so the Hubble parameter in these UDM models is the same as in ΛCDM, 3 (1) H(z) = H0 Ωm (1 + z) + ΩΛ with H0 = 100 h km s−1 Mpc−1 and Ωm = Ωb + ΩDM . Now I introduce small inhomogeneities of the scalar ﬁeld δϕ(t, x), and in the linear theory of cosmological perturbations and in the Newtonian gauge with no anisotropic stress, the line element for a spatially ﬂat [9] Universe takes the form (in units such that c = 1 and signature {−, +, +, +}) ds2 = −(1 + 2Φ)dt2 + a2 (t)(1 − 2Φ)dx2 . (2) The evolution of Fourier’s modes of the Newtonian potential Φk (a) are described by [10] v − cs 2 ∇2 v − θ v = 0, θ (3) where a prime denote a derivative with respect to the conformal time dτ = dt/a, ∗ E-mail: stefano.camera@ph.unito.it, Phone: +39 011 670 7066 c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Ann. Phys. (Berlin) 19, No. 3 – 5 (2010) 1 θ= a Φ , v= √ ρ+p 329 − 12 p 1+ ρ (4) and cs (a) is the speed of sound of the homogeneous scalar ﬁeld condensate. In [11] it has been proposed a technique to construct UDM models where the scalar ﬁeld can have a sound speed small enough to allow structure formation and to avoid a strong integrated Sachs-Wolfe effect in the CMB anisotropies, which tipically plague UDM models. The parametric form for the sound speed is cs 2 (a) = ΩΛ c∞ 2 ΩΛ + (1 − c∞ 2 )ΩDM a−3 (5) where c∞ is the value of the speed of sound when a → ∞. Fig. 1 (online colour at: www.ann-phys.org) Normalized potentials Φk (a)/Φk (0) are shown for ΛCDM (solid) and UDM (dot-dashed). The lower panel shows potentials at k = 0.001 h Mpc−1 , the medium panel at k = 0.01 h Mpc−1 and the upper panel at k = 0.1 h Mpc−1 . UDM curves are for c∞ 2 = 10−6 , 10−4 , 10−2 from top to bottom, respectively. At small c∞ , the ΛCDM and UDM curves are indistinguishable. In Fig. 1 [8] I present some Fourier’s components Φk (a) of the gravitational potential, normalized to unity at early times. In the case of the UDM models there are two simple but important aspects: ﬁrst, the ﬂuid which triggers the accelerated expansion at late times is also the one which has to cluster in order to produce the structures we see today. Secondly, from the last scattering to the present epoch, the energy density of the Universe is dominated by a single dark ﬂuid, and therefore the gravitational potential evolution is determined by the background and perturbation evolution of this ﬂuid alone. As a result, the general trend is that the possible appearance of a sound speed signiﬁcantly different from zero at late times corresponds to the appearance of a Jeans’ length under which the dark ﬂuid does not cluster any more, causing a strong evolution in time of the gravitational potential. By increasing the sound speed, the potential starts to decay earlier in time, oscillating around zero afterwards. Moreover at small scales, if the sound speed is small enough, UDM reproduces ΛCDM. This reﬂects the dependence of the gravitational potential on effective Jeans’ length λJ 2 (τ ) = cs 2 |θ/θ | [12]. 2 Results and discussion 2.1 UDM weak lensing signal The weak lensing power spectrum for cosmic convergence [13] C κκ (l) = www.ann-phys.org l4 4 0 ∞ dχ W 2 (χ) Φ P χ6 l ,χ χ (6) c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 330 S. Camera: Constraining UDM models with weak lensing Fig. 2 (online colour at: www.ann-phys.org) Upper panels: power spectra l(l + 1)C κκ (l)/(2π) of CMB (a) and background galaxy (b) photons for non–linear ΛCDM (thick, solid), linear (thin, solid) and UDM (dot-dashed), for c∞ 2 = 10−6 , 10−5 , 10−4 from top to κκ bottom. Lower panels: ratio C κκ (l)/Clin. ΛCDM (l) for ΛCDM and 2 UDM, one for each value of c∞ plotted in the upper panels. The vertical lines denote lε , the upper limit estimated for the reliability of results in the linear approximation, with two assumptions for the threshold ε = 10%, 20%. The boxes show 1σ errors on the ΛCDM power spectrum according to a EUCLID-like 20, 000 sq degree survey. The source redshift ditribution in (a) peaks at zp = 1000, in (b), in the left panel peaks at zp = 1, in the middle panel at zp = 2 and in the right panel at zp = 3. contains information about both geometry, through the radial comoving distance χ(z) and the window function W [(χ, n(χ)] where n(z) is the redshift distribution of sources, and dynamics, because the power spectrum of the Newtonian potential P Φ (k, z) is linked to the matter power spectrum thanks to Poisson’s equation. In Fig. 2 [8], upper panels, I show the weak lensing convergence power spectra l(l + 1)C κκ (l)/(2π) of CMB (a) and background galaxy (b) light for the ΛCDM and UDM models. For ΛCDM I show both the linear and non–linear power spectrum, and for UDM I present three curves, obtained for c∞ 2 = 10−6 , 10−5 , 10−4 . In the lower panels the UDM curves of the upper panels are divided by the linear convergence power spectrum of ΛCDM. As we can see, for small sound speeds (c∞ 2 = 10−6 ) and large angular scales (l 100), we cannot distinguish the convergence of background galaxy photons in UDM models from the standard ΛCDM behaviour. However, the agreement disappears at large c∞ and l’s. For background galaxy light UDM features are more signiﬁcant than in the CMB case, and an EUCLID–like survey (error boxes shown) will enable us to distinguish between UDM and ΛCDM if c∞ 2 10−4 . It is easy to see that, for small zp , the differences between UDM models with different c∞ and between UDM and ΛCDM are very pronounced even at large angular scales, because photons emitted at late times feel much more the decay of the Newtonian potential, shown in Fig. 1. 2.2 Uncertanties of cosmological parameters on the ΛCDM weak lensing signal In the ΛCDM model, we compute the convergence power spectra for different values of Ωm and σ8 within the 68% region of their uncertainties, as derived by [14] with WMAP5 data (their Fig. 3). In Fig. 3 (left panel) the top and bottom solid lines represent the upper and lower limits of the power spectra due to these uncertainties. We notice that the CMB is not a good source to constrain UDM models with weak lensing c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 19, No. 3 – 5 (2010) 331 Fig. 3 (online colour at: www.ann-phys.org) Role of the 68% uncertainties of Ωm and σ8 on the ΛCDM signal. The top and bottom solid lines represent the upper and lower limits of the non–linear convergence power spectrum in the ΛCDM model due to these uncertenties; the dot-dashed curves are UDM power spectra with c∞ 2 = 10−6 , 10−5 , 10−4 , from top to bottom. In the left panel, the source is the CMB, in the right panel the sources are background galaxies peaked at zp = 1 and distributed over a range of angular diameter distances according to the redshift distribution of sources of [13]. because, even if the errors on the signal are very small (see panel (a) of Fig. 2), the UDM convergence power spectrum lies within the ΛCDM strip. On the contrary, if we use the convergence power spectrum of background galaxies (right panel), we are in principle still able to distinguish between the signals of ΛCDM and UDM models. Finally, we ﬁnd that it is not possible to reproduce a UDM power spectrum with a ﬁxed c∞ = 0 by properly choosing Ωm and σ8 in the ΛCDM model, even if the former lies within the uncertainty of the latter. In fact, the dependence of the Newtonian potential in UDM models on scale and redshift is not factorizable in a scale-dependent transfer function T (k) and a redshift-dependent growth factor D(z), but the two dependences are linked together by the sound horizon λJ (z). Acknowledgements I thank my collaborators for allowing me to show results from a common project. I also gratefully aknowledge professor M. P. Dabrowski and all the organizing commettee for the stimulating and fruitful Grassmannian Conference in Fundamental Cosmology1 (Szczecin, Poland, 14–19/09/2009). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] 1 A. Y. Kamenshchik, U. Moschella, and V. Pasquier, Phys. Lett. B 511, 265–268 (2001). M. C. Bento, O. Bertolami, and A. A. Sen, Phys. Rev. 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