The Power of Cosmic Flexion in Testing Modified Matter and Gravity
aa r X i v : . [ a s t r o - ph . C O ] A p r The Power of Cosmic Flexion in Testing Modified Matter and Gravity
Stefano Camera ∗ and Antonaldo Diaferio † Dipartimenti di Fisica Generale e Teorica, Universit`a degli Studi di Torino,and INFN, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy Dipartimento di Fisica Generale “A. Avogadro”, Universit`a degli Studi di Torino; INFN,Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy,and Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA, USA (Dated: Received June 8, 2018; published – 00, 0000)Flexion is the weak lensing effect responsible for the weakly skewed and arc-like appearance oflensed galaxies. The flexion signal-to-noise ratio can be an order of magnitude larger than thatof shear. For the first time, we show how this makes flexion an invaluable tool for discriminatingbetween alternative cosmological models. We analyse a scalar field model of unified dark matter anddark energy, a brane-world cosmology and two f ( R ) modified-action theories. We show that thesemodels can be distinguished from ΛCDM at several standard deviations by measuring the powerspectrum of cosmic flexion. PACS numbers: 98.80.-k, 98.80.Es, 95.36.+d, 95.36.+x
Introduction.—
In the last decades, cosmologists pro-posed several models alternative to the concordance Λcold dark matter (ΛCDM) paradigm. These models at-tempt to find an agreement at least as good as that ofΛCDM with current cosmological datasets, such as thetemperature anisotropy pattern of the cosmic microwavebackground radiation [1], the dynamics of the large-scalestructure of the Universe [2] and the present-day cosmicaccelerated expansion [3]. However, in these theories,crucial topics such as the missing mass in galaxies andgalaxy clusters and the current Universe’s acceleratedexpansion are not explained by the usual dark matterand the cosmological constant Λ. On the contrary, thesemodels mainly rely on either a modification of the law ofgravity or the introduction of additional scalar or vectorfields in the Universe’s content.The family of alternative models with additional fields,also named modified matter models, includes dynam-ical dark energy or quintessence [4], but also modelswhich attempt to identify both the dark matter anddark energy effects with the properties of a single “darkfluid” [5, 6]. Conversely, the class of modified grav-ity includes a variety of approaches, which can how-ever be well represented by brane-world cosmologies[7] and modified-action theories [8]. Brane worlds de-scribe a four-dimensional “brane,” which is our own Uni-verse, embedded into a higher-dimensional spacetime, the“bulk.” In this scenario, Einstein’s general relativity isstill valid, but the higher-dimensional behaviour of grav-ity induces non-negligible signatures on the Universe’sevolution and the growth of cosmic structures on thebrane. Finally, modified-action theories directly modifythe law of gravity by generalising the Hilbert-Einstein La-grangian. Among all the possible theories, f ( R ) gravity,where Ricci’s scalar R is replaced by a generic function f ( R ), is probably the most investigated approach.In this Letter , we choose three models to explore the space of modified matter and gravity theories. Specif-ically, we consider a model of unified dark matter anddark energy [9], a phenomenological extension of the well-known DGP brane-world cosmology [10], and two f ( R )models able to pass the Solar system gravity tests [11, 12].All of them reproduce the ΛCDM expansion history, thusrepresenting viable alternatives for the description of thebackground evolution of the Universe. To be able to dis-criminate between them and ΛCDM it is therefore crucialto investigate the r´egime of cosmological perturbations.This analysis has been carried out using several observ-ables, for instance the power spectrum of density fluc-tuations and cosmic shear [13]. However, it is not rarethat the predicted signal is very similar to what expectedin ΛCDM. Here, we show that the degeneracy betweenmodels can be lifted by cosmic flexion, namely the flex-ion correlation function whose signal originates from thelarge-scale structure of the Universe. We will presentparameter forecasts and additional gravitational lensingstatistics elsewhere [14]. Cosmic Flexion.—
The deflecting gravitational field ofthe extended large-scale structure of the Universe – whichis simply the Newtonian potential Φ, in general relativ-ity – is responsible for deflection of light rays emittedby distant sources. This phenomenon is known as weakgravitational lensing. Therefore, photon paths from agalaxy located at θ on the sky are deflected by an angle α = ∂ψ, (1)where ∂ = ∂ + i∂ is the gradient with respect to di-rections perpendicular to the line of sight and ψ is theprojected deflecting potential. Unfortunately, the deflec-tion angle is not observable directly. This is because onedoes not know the true two-dimensional distribution ofthe sources on the sky. On the other hand, its gradient,the distortion matrix ∂ a ∂ b ψ , is measurable. In particular,the entries of the distortion matrix can be related to theeffects of convergence κ and (complex) shear γ occurringto the source image, i.e. κ = 12 ∂∂ ∗ ψ, γ = 12 ∂∂ψ. (2)If convergence and shear are effectively constant withina source galaxy image, the galaxy transformation is θ ′ a = A ab θ b , where A ab = δ ab − ∂ a ∂ b ψ and a, b = 1 , θ ′ at the second-order in the de-flection angle, it follows that θ ′ a ≃ A ab θ b + D abc θ b θ c / D abc ≡ ∂ c A ab . As the distortion matrix canbe decomposed into the convergence and the shear, it isusual to define a spin-1 and a spin-3 flexion, which read F = 12 ∂∂∂ ∗ ψ, G = 12 ∂∂∂ψ, (3)respectively. Since measurements of G are noisier than F [16], we will restrict our analysis to F only.To construct the flexion correlation function fromlarge-scale structure, we start from the definition of theprojected deflecting potential, ψ ( θ ) = Z d χ W ( χ )Φ( χ, θ ) , (4)where d χ = d z/H ( z ) is the radial comoving distance, H ( z ) is the expansion history of the Universe and W ( χ )is the weak lensing selection function [17]. W ( χ ) dependson the redshift distribution of the sources n [ χ ( z )], nor-malised such that R d χ n ( χ ) = 1.In the flat-sky approximation, we expand the flexionin its Fourier modes F ( ℓ ). Hence, from the definition ofangular power spectrum h F ( ℓ ) F ∗ ( ℓ ′ ) i = (2 π ) δ D ( ℓ − ℓ ′ ) C F ( ℓ ) , (5)which is the Fourier transform of the two-dimensionalcorrelation function, and from Eq. (3), we finally get [18] C F ( ℓ ) = ℓ Z d χ W ( χ ) χ P Φ (cid:18) ℓχ , χ (cid:19) . (6) Modified Matter/Gravity Models.—
We now briefly re-view the three models we use. We refer to them as: UDMfor the model of unified dark matter and dark energy;eDGP for the phenomenologically extended DGP braneworld; and St and HS for the two f ( R ) theories of [11]and [12], respectively.In the class of UDM models we use [9], a single scalarfield with a Born-Infeld kinetic term [19] mimics bothdark matter and dark energy. The energy density of the scalar field reads ρ UDM = ρ DM + ρ Λ , where ρ DM ∝ a − and ρ Λ = const . , that yields the ΛCDM Hubble pa-rameter exactly. However, there also is a pressure term p UDM = − ρ Λ which leads to a non-negligible speed ofsound for the perturbations of the scalar field itself. Thisis a common feature in modified matter models and ittypically causes an integrated Sachs-Wolfe effect incom-patible with current observations [20]. To solve this prob-lem, we use the technique outlined in [9], where the au-thors construct a UDM model able to reproduce boththe correct temperature power spectrum of the cosmicmicrowave background and the clustering properties ofthe large-scale structure we see today.The sound speed is parameterised by its late-time value c ∞ (in units of c = 1) and the growth of cosmic struc-tures strongly depends on it. Indeed, the presence of thesound speed produces an effective Jeans length λ J forthe Newtonian potential. Thus, its evolution is no morescale independent. Specifically, the Fourier modes Φ k are suppressed on scales k > /λ J and oscillate aroundzero. The larger is the value of c ∞ , the earlier the Newto-nian potential starts decreasing (for a fixed scale) or at agreater scale (for a fixed epoch) [21]. Since the Newtonianpotential is responsible for light deflection, weak lensingis a powerful tool to constrain UDM models [21] and inparticular three-dimensional cosmic shear [22]. However,UDM models with c ∞ . − still produce a signal vir-tually indistinguishable from that of ΛCDM.In the eDGP model [10], the cross-over length r c , whichdefines the scale at which higher-dimensional gravita-tional effects become important, is tuned by a free pa-rameter α ∈ [0 , α = 0 and α = 1 / − Ψ holds inthe matter dominated era. Thus, when we study gravi-tational lensing we have to deal with the deflecting po-tential Υ ≡ (Ψ − Φ) /
2. Moreover, its Poisson equation,which relates it to the distribution of the matter overden-sities, is modified by the presence of an effective time- andscale-dependent gravitational constant.It is worth giving a final remark on the evolution ofmatter fluctuations. Unlike the linear growth of pertur-bations, that can be described analytically, the non-linearr´egime has to be explored numerically. Two approacheshave been followed and we refer to them as “KW” and“PPF.” The former generalises the halofit procedure[24] to the eDGP scenario according to recently per-formed N -body simulations [25], whilst the latter in-terpolates the eDGP non-linear matter power spectrumwith that of ΛCDM in order to reproduce general rel-ativity at small scales and be thus able to pass Solarsystem gravity tests [26]. The functional forms of thislast approach have been obtained by perturbation theory[27] and confirmed by N -body simulations [28]. Unfortu-nately, were PPF the correct non-linear prescription, wewould not be able to discriminate between the eDGP andΛCDM signals even with the present and next generationweak lensing surveys [23].Finally, we analyse the St and HS f ( R ) theories, whichalso are degenerate with ΛCDM at background level [29].Their functional form allow them to achieve the late-timeaccelerated expansion of the Universe with no formal cos-mological constant. On the contrary, they present threefree parameters, c , c and n . It has been shown thatthe growth of linear perturbations strongly depends onthe function f ( R ), which acts by generating a time- andscale-dependent gravitational constant, as well as an ef-fective anisotropic stress [30]. Regarding the non-linearevolution of perturbations, the PPF technique is stillvalid, as confirmed by N -boy simulations [31]. Cosmicshear studies on these models gave interesting results[32, 33], but the St signal is nonetheless almost com-pletely degenerate with ΛCDM [34]. Results and Discussion.—
Here, we present the cos-mic flexion power spectrum (6) expected in the al-ternative models outlined above and we compare itwith the ΛCDM prediction. For this, we use a fidu-cial flat Universe where the Hubble constant is H =100 h km s − Mpc − and h = 0 .
7. The matter density inunits of the critical density is Ω m ≡ Ω DM + Ω b = 0 . DM and Ω b = 2 . · − h − the dark matterand baryon fractions, respectively. The tilt of the pri-mordial matter power spectrum is n s = 0 .
96 and thedensity fluctuation rms on the scale of 8 h − Mpc is σ = 0 .
8. For the UDM model, we probe c ∞ = 5 · − and c ∞ = 10 − . The eDGP model parameters are α = 0 .
116 and r c H = 155 .
041 [23]. The St(HS) parame-ters read log c = 2 . . c = − . .
79) and n = 1 . .
67) [29].It is important to note that there currently is no linear-to-non-linear mapping in UDM models. Nevertheless,differences between ΛCDM and UDM models arise atscales smaller than the sound horizon. With a cross-overwavenumber k ≃ /λ J , if the sound speed is small enoughto guarantee that λ J is well within the non-linear regime,we can assume that the non-linear evolution of the UDMpower spectrum will be similar to that of ΛCDM [22].We use the specifics of the upcoming ESA Euclid satel-lite [35]. Euclid is one of the ESA Cosmic Vision 2010- ,
000 squaredegree, with a sky coverage f sky ≃ .
48 and a sourcedistribution over redshifts [36] n ( z ) ∝ z e − (cid:16) zz (cid:17) . , (7)where z = z m / . z m = 0 . n = 35 arcmin − . Tocompute errorbars, we use∆ C F ( ℓ ) = s ℓ + 1) f sky (cid:2) C F ( ℓ ) + N F ℓ (cid:3) , (8)generalising thus the approach of [17]. This is because– unlike shear – flexion has a dimension of length − (orangle − ). This means that the effect by flexion dependson the source size. Recently, it has been shown that thenoise power spectrum N F ℓ for flexion is inversely propor-tional to the squared angular scale [37]; we therefore set N F ℓ = 4 π h F int2 i ℓ ¯ n , (9)with h F int2 i . ≃ .
03 arcsec − the galaxy-intrinsic flex-ion rms. FIG. 1. Power spectrum of cosmic flexion C F ( ℓ ) for the al-ternative cosmological models presented in the text. Fig. 1 shows the cosmic flexion power spectra C F ( ℓ )of ΛCDM (red, solid), eDGP (green) with both KW(dashed) and PPF (dot-dashed) linear-to-non-linearmappings and the f ( R ) models (red) of St (dashed) andHS (dot-dashed). As expected, the UDM signal is sup-pressed at small angular scales because of the presence ofthe scalar field sound speed. The eDGP model is still veryclose to the ΛCDM prediction, particularly the PPF non-linear power spectrum, for it being specifically designedto reproduce general relativity on small scales. On theother hand, the St and HS models clearly show the scaledependence of the Newtonian gravitational constant G .Indeed, in the so-called “scalar-tensor” r´egime it reachesthe value ∼ G/ σ -error region,whilst light-grey refers to errors six times larger. Flex-ion measurements are made on the shapes of the sourcegalaxies, exactly as in the cosmic shear analysis. There-fore, the source number density ¯ n is the same for thetwo observables and, with a space-based, wide-field sur-vey such as Euclid, we can collect a fairly large statistics.However, the intrinsic flexion rms h F int2 i . is an orderof magnitude smaller than the cosmic shear rms and thepower spectrum is thus significantly less noisy.We conclude that cosmic flexion is an excellent tool fortesting alternative cosmological models and discriminatebetween them. With realistic values for the mean galaxynumber density ¯ n and the flexion noise N F ℓ , which in-cludes its angular scale-dependence [37], expected fromthe upcoming Euclid mission, we find an admirable sep-aration between cosmic flexion power spectra C F ( ℓ ) ofviable models which are almost degenerate with ΛCDMwhen investigated with other observables, such as cosmicshear. We will provide a more detailed analysis of theseoutstanding results in [14].SC and AD acknowledge support from the INFN grantPD51 and the PRIN-MIUR-2008 grant 2008NR3EBK“Matter-antimatter asymmetry, dark matter and darkenergy in the LHC era.” This research has made useof NASA’s Astrophysics Data System. ∗ [email protected] † [email protected][1] E. Komatsu et al. (WMAP), Astrophys. J. Suppl. ,330 (2009), arXiv:0803.0547 [astro-ph].[2] A. G. Riess et al. , Astrophys. J. , 98 (2007),arXiv:astro-ph/0611572.[3] D. Larson et al. , arXiv:1001.4635(2010),arXiv:1001.4635 [astro-ph.CO].[4] L. Amendola and S. Tsujikawa, Dark Energy: Theory andObservations (2010) Cambridge University Press (2010).[5] D. Sapone, Int. J. Mod. Phys.
A25 , 5253 (2010),arXiv:1006.5694 [astro-ph.CO].[6] D. Bertacca, N. Bartolo, and S. Matarrese, Advances inAstronomy (2010), arXiv:1008.0614 [astro-ph.CO].[7] R. Maartens and K. Koyama, Living Rev. Rel. , 5(2010), arXiv:arXiv:1004.3962 [hep-th].[8] A. De Felice and S. Tsujikawa, Living Rev. Rel. , 3 (2010), arXiv:1002.4928 [gr-qc].[9] D. Bertacca, N. Bartolo, A. Diaferio, and S. Matarrese,JCAP , 023 (2008), arXiv:0807.1020 [astro-ph].[10] N. Afshordi, G. Geshnizjani, and J. Khoury, JCAP ,030 (2009), arXiv:arXiv:0812.2244 [astro-ph].[11] A. A. Starobinsky, JETP Lett. , 157 (2007),arXiv:0706.2041 [astro-ph].[12] W. Hu and I. Sawicki, Phys. Rev. D76 , 064004 (2007),arXiv:0705.1158 [astro-ph].[13] B. Jain and J. Khoury, Annals Phys. , 1479 (2010),arXiv:1004.3294 [astro-ph.CO].[14] S. Camera et al. (2011, in prep.).[15] D. M. Goldberg and D. J. Bacon, Astrophys.J. , 741(2005), arXiv:astro-ph/0406376 [astro-ph].[16] Y. Okura, K. Umetsu, and T. Futamase, Astrophys.J. , 995 (2007), arXiv:astro-ph/0607288 [astro-ph].[17] N. Kaiser, Astrophys. J. , 272 (1992).[18] D. J. Bacon, D. M. Goldberg, B. T. P. Rowe,and A. N. Taylor, Mon. Not. R. Astron. Soc. , 414(2006), arXiv:astro-ph/0504478.[19] M. Born and L. Infeld, Proc. Roy. Soc. Lond.
A144 , 425(1934).[20] D. Bertacca and N. Bartolo, JCAP , 026 (2007),arXiv:0707.4247 [astro-ph].[21] S. Camera, D. Bertacca, A. Diaferio, N. Bartolo,and S. Matarrese, Mon. Not. R. Astron. Soc. , 1995(2009), arXiv:0902.4204 [astro-ph.CO].[22] S. Camera, T. D. Kitching, A. F. Heavens, D. Bertacca,and A. Diaferio, Mon. Not. R. Astron. Soc.(2011, inpress), arXiv:1002.4740 [astro-ph.CO].[23] S. Camera, A. Diaferio, and V. F. Cardone, JCAP ,029 (2011), arXiv:1101.2560 [astro-ph.CO].[24] R. E. Smith et al. (The Virgo Consortium),Mon. Not. R. Astron. Soc. , 1311 (2003),arXiv:astro-ph/0207664.[25] J. Khoury and M. Wyman, Phys. Rev.
D80 , 064023(2009), arXiv:0903.1292 [astro-ph.CO].[26] W. Hu and I. Sawicki, Phys. Rev.
D76 , 104043 (2007),arXiv:0708.1190 [astro-ph].[27] K. Koyama, A. Taruya, and T. Hiramatsu, Phys. Rev.
D79 , 123512 (2009), arXiv:0902.0618 [astro-ph.CO].[28] F. Schmidt, Phys. Rev.
D80 , 043001 (2009),arXiv:0905.0858 [astro-ph.CO].[29] V. F. Cardone, S. Camera, and A. Diaferio, Phys. Rev. D (2010, submitted).[30] S. Tsujikawa, Phys. Rev. D76 , 023514 (2007),arXiv:arXiv:0705.1032 [astro-ph].[31] H. Oyaizu, M. Lima, and W. Hu, Phys. Rev.
D78 ,123524 (2008), arXiv:0807.2462 [astro-ph].[32] E. Beynon, D. J. Bacon, and K. Koyama,Mon. Not. R. Astron. Soc. , 353 (2010),arXiv:0910.1480 [astro-ph.CO].[33] S. A. Thomas, S. A. Appleby, and J. Weller, JCAP ,036 (2011), arXiv:1101.0295 [astro-ph.CO].[34] S. Camera, A. Diaferio, and V. F. Cardone(2011),arXiv:1104.2740 [astro-ph.CO].[35] A. Refregier et al. (2010), arXiv:1001.0061 [astro-ph.IM].[36] I. Smail, R. S. Ellis, and M. J. Fitchett, Mon. Not. R. As-tron. Soc. , 245 (Sep. 1994), arXiv:astro-ph/9402048.[37] S. Pires and A. Amara, Astrophys.J.723