Constraining Modified Gravity and Coupled Dark Energy with Future Observations Matteo Martinelli

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1 Coupled Dark University of Rome La Sapienza Roma, October 28th 2011

2 Outline

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4 Accelerated Expansion Cosmological data agree with an accelerated expansion of the Universe d L [Mpc] Hubble Law q 0 =0.5 q 0 =-0.5 q 0 = z This acceleration can not be explained if the Universe is composed only by matter and radiation q 0 = 1/H 2 0 (ä/a) t=t 0 = Ω m /2+Ω r Ω Λ

5 Nobel Prize 2011

6 Cosmological Standard Model This acceleration can be explained by a Cosmological Constant. Λ has a negative pressure (w = p/ρ = 1) and can produce the acceleration. Another component, Dark Matter, is necessary to achieve the total matter energy density needed to explain the growth of cosmological structure and other astrophysical phenomena (e.g. galaxies rotation curves, Bullet cluster...)

7 Problems of Λ The cosmological constant has two main problems: Why Now: the cosmological constant and dark matter energy densities are now of the same order of magnitude, but their ratio changes very rapidly Ω Λ /Ω m (t) a 3 There is no physical mechanism that predicts this equivalence at present time Fine Tuning: the energy associated with the cosmological constant can be tought as a vacuum energy because of its constancy property, but its value is much smaller than the energy predicted by particle physics ρ vac GeV 4 ρ Λ < ρ c = 3H 2 0 /8πG GeV 4 ρ Λ /ρ vac

8 The problems of the cosmological constant caused a burst of alternative models. To produce an accelerated expansion without a cosmological constant, it is necessary to modify the equations G µν = 8πGT µν The need of a modification arise because standard energy components (matter and radiation) produce a decelerated expansion. There are two ways to construct models alternative to the cosmological constant: to modify T µν introducing new energy components (eg. Quintessence). to modify the Einstein tensor G µν changing the Lagrangian of Relativity.

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10 Examples of models There are several ways to modify gravity, changing the Einstein-Hilbert lagrangian or modifying space-time properties. A few examples are: DGP Leaking of gravity in a 5 dimensional space-time. Scalar-Tensor theories Gravitation is not only given through the metric, but also through scalar fields. f(r) theories The gravity lagrangian depends in a more general way on the Ricci scalar.

11 Perturbations All the structures that we now see in the Universe evolve from small perturbations of the Friedmann metric, described by Ψ and Φ. ds 2 = [1+2Ψ( x,t)]dt 2 [1+2Φ( x,t)]d x 2 The evolution of these terms is determined by the gravitational theory. gravity changes the Newtonian and metric potentials Φ and Ψ with respect to the ΛCDM model. Dark matter clustering, as well as the evolution of the metric potentials, is changed and can be scale-dependent. Moreover, typically there might be an effective anisotropic stress and the two potentials appearing in the metric element are not necessarily equal, as is in the ΛCDM model.

12 We use a parametrization that takes into account the modified relation between the metric potentials given by modified gravity where k 2 Ψ = a2 2M P µ(a,k)ρ Φ Ψ = γ(a,k) µ(a,k) = 1+β 1λ 2 1 k2 a s 1+λ 2 1 k2 a s γ(a,k) = 1+β 2λ 2 2 k2 a s 1+λ 2 2 k2 a s This parametrization is implemented in the MGCAMB code, which only consider theories that mimic the ΛCDM expansion history G. Zhao et al., Phys. Rev. D 79 (2009)

13 f(r) gravity In the case of scalar-tensor theories the parameters are related as β 1 = λ2 2 λ 2 1 = 2 β 2 λ 2 2 λ 2 1 We focus on f(r) theories, thus specifying the couplings β i and the time evolution s of the length scale λ β 1 = 4 3 β 2 = 2 β 1 1 = 1 2 s 4 Thus the only free parameter for f(r) theories is the length scale λ of the modified force. When λ 2 1 = 0 we recover the cosmological constant model.

14 CMB anisotropies f(r) theories modify the power spectra of CMB anisotropies T [ ] T Ψ+ˆr v+θ+ dη Ψ(x,η) Φ(x,η) [C l TT -Cl TT ( )]/Cl TT ( ) =0 Mpc 2 2 =10 3 Mpc 2 2 =10 4 Mpc 2 2 =10 5 Mpc 2 2 =10 6 Mpc Calabrese et al. Phys. Rev. D 80 (2009) l

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16 CMB data We analyzed how future CMB missions will improve modified gravity constraints. We took into account two experiments: Planck (2009) CMBpol (2020?) Channel FWHM T/T f sky = 0.85 Channel FWHM T/T f sky = 0.85 Using the experimental specifications of these satellite missions we can forecast future data with the right noise. In order to do this we have to assume a fiducial cosmological model (e.g. ΛCDM).

17 f(r) theories modify the lensing effect on the CMB, thus introducing a modification on lensing potential power spectrum. The modified gravitational lensing provides another way to constrain gravity theories. C l dd l(l+1)/2π 1.2e+06 1e l λ 1 =10 Mpc λ 1 =10 Mpc λ 1 2 =10 6 Mpc 2 λ 1 =10 Mpc ΛCDM

18 extraction The power spectrum of lensing deflection of CMB photons can be extracted from the other power spectra (temperature and polarization). We can therefore obtain Cl dd from Cl TT, Cl EE and Cl TE through the Okamoto and Hu method. T. Okamoto, W. Hu, Phys. Rev. D 67 (2003) e-05 1e-06 1e-07 1e-08 1e-09 1e l dd C l l (l+1)/2π Planck noise CMBpol noise

19 CMB Lensing results Adding the information given by this estimator it is possible to improve constraints on cosmological parameters. Ω m Planck no lens Planck lens λ 2 1 < Mpc 2 < Mpc 2 CMBpol no lens CMBpol lens λ 2 1 < Mpc 2 < Mpc λ 1 x 10 4 Ω m λ 1

20 Euclid Weak lensing measurements are able to map the growth of perturbations since they relate directly to the dark matter distribution and are not plagued by galaxy luminous bias. Since this growth is modified by alternative theories of gravity, we can use this probe to test GR on cosmic scales. P(k) k λ 1 2 =10 4 Mpc 2 λ 1 2 =10 5 Mpc 2 λ 1 =10 Mpc λ 1 =10 Mpc ΛCDM

21 Euclid results We forecasted Euclid data to study the ability of this mission to constrain modified gravity theories. n gal (arcmin 2 ) redshift f sky γ 2 rms 35 0 < z < This probe will dramatically improve the constraints on λ 1 : Planck : λ 2 1 < Mpc c.l. Planck+Euclid: λ 2 1 < Mpc 2 MM et al. Phys. Rev. D 83 (2011) c.l.

22 Euclid results H (Km s 1 Mpc 1 ) Ω m λ 1 (Mpc ) x λ (Mpc ) 1 x 10 4 Ω m H 0 (Km s 1 Mpc 1 ) λ (Mpc 2 ) λ 1 (Mpc )

23 Model Assumption We analyzed also an f(r) fiducial model with λ 2 1 = 300 Mpc2, but we assumed λ 1 = 0. We found a consistent bias in the recovered best fit values of the cosmological parameters. Planck+Euclid Planck+Euclid Fiducial values Model: λ 2 1 = 0 varying λ 2 1 Parameter Ω b h ± ± Ω ch ± ± θ s ± ± τ ± ± n s ± ± H ± ± Ω Λ 0.724± ± σ ± ±

24 Model assumption Even for small modifications to gravity, the best fit values recovered are more than 1σ away from the correct values. H Ω m σ n s Euclid and Planck will necessarily require to allow for possible deviations from general relativity, in order to not bias the best fit value

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26 Usually the dark components of the Universe are thought as two non interactive fluids. However some models take into account the possibility of an interaction between Dark Energy and Dark Matter. ρ dm +3H ρ dm = a Q ρ de +3H ρ de (1+w) = a Q where Q = ξh ρ de /a. The coupling doesn t modify only the background mean densities, but also the evolution of density perturbations. Thus, it it possible to constrain the coupling parameter ξ studying the effects that it has on CMB power spectra.

27 Effect on CMB anisotropies The existance of a s leaves an imprint on CMB anisotropies. This effect has a degeneracy with the one produced by dark matter abundance. C l TT l(l+1)/2π ξ=-0.2 ξ=-0.5 ξ=-0.8 ΛCDM C l TT l(l+1)/2π ξ=-0.17, Ω c =0.088 ξ=-0.17, Ω c = ξ=0, Ω c =0.088 ξ=0, Ω c = (ΛCDM fiducial model) l l

28 CMB Results Forecasting data from the future satellite missions Planck and CMBpol, we obtain Planck: ξ > c.l. CMBpol: ξ > c.l. Ω m ξ MM et al. Phys. Rev. D 81 (2010)

29 Results We forecasted Euclid data to study how the combination Planck+Euclid will constrain ξ, obtaining: Planck+Euclid: ξ > 0.04 Ω m c.l ξ F. De Bernardis, MM et al. Phys. Rev. D 84 (2011)

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31 Using extraction we can obtain constraints using only CMB data Planck: λ 2 1 < Mpc c.l. CMBpol: λ 2 1 < Mpc c.l. Adding the weak lensing probe to CMB data there is a huge improvement Planck : λ 2 1 < Mpc c.l. Planck+Euclid: λ 2 1 < Mpc c.l. CMB data can be also used to test the possibility of an interaction between the dark components of the Universe: Planck: ξ > c.l. CMBpol: ξ > c.l. Also constraints on ξ can be improved using weak lensing data: Planck : ξ > c.l. Planck+Euclid: ξ > c.l.

32 work To use other probes, in addition to CMB anisotropies and galaxy lensing, to constrain λ 1 (e.g. 21 cm). To study the degeneracy between modified gravity and neutrino mass in order to understand its effect on the possibility of neutrino mass detection, as the effect of f(r) gravity on the lensing potential power spectrum is opposite to the effect of neutrino mass. other modified gravity theories using the same parametrization.

33 List of publications 1 F. De Bernardis, M., A. Melchiorri, O. Mena, A. Cooray, Phys. Rev. D 84 (2011) M., L. L. Honorez, A. Melchiorri, O. Mena, Phys. Rev. D 81 (2010) M., E. Calabrese, F. De Bernardis, A. Melchiorri, L. Pagano, R. Scaramella, Phys. Rev. D 83 (2011) T. Giannantonio, M., A. Silvestri, A. Melchiorri, JCAP 1004 (2010) 30 5 E. Calabrese, A. Cooray, M., A. Melchiorri, L. Pagano, A. Slosar, G. F. Smoot, Phys. Rev. D 80 (2009) M., Nucl. Phys. Proc. Suppl. 194 (2009) M., A. Melchiorri, O. Mena, V. Salvatelli and Z. Girones, arxiv: [astro-ph.co] 8 M., A. Melchiorri, L. Amendola, Phys. Rev. D 79 (2009)

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35 Supernovae as standard candles Supernovae Ia can be used to determine d L because they are standard candles. This kind of supernovae are generated from White Dwarves in binary systems, where there is a mass transfer on the White Dwarf. When the mass becomes higher than the Chandrasekhar mass ( 1.4 M sun ) the dwarf explodes giving birth to a Supernova Ia. It is possible to calibrate the light curves of this kind of supernovae (Phillips law) obtaining the total luminosity emitted at the explosion. This luminosity is almost the same for every supernova.

36 Λ as Dark Energy or as modified gravity The cosmological constant can be thought as an additional energy component that does not vary during the expansion of the Universe: G µν = 8πG T µν = 8πGT µν +Λg µν It can be also thought as a modification to the standard gravity lagrangian: S = 1 d 4 x g[r 2Λ] 16πG Thus both dark energy and modified gravity models contains the cosmological standard model as a limit case.

37 f(r) theories The simplest way to modify the gravity lagrangian brings to f(r) theories. These models are constructed by simply introducing a general function of the Ricci scalar into the gravity Lagrangian S GR = d 4 x [ ] R g 16πG +L m S MG = d 4 x [ ] R+f(R) g 16πG +L m In order to produce the accelerated expansion at the proper time, modifications must be important at late times, thus at low curvature f(r) = R n

38 Criteria for f(r) theories Cosmological viability: gravity changes also the expansion before the accelerated phase, for example in the matter dominated era a m t 2 3 a m t k The matter epoch is well described by general relativity, thus the new gravity theory must not change too much this expansion phase. Local tests: The new gravity theory must be in agreement with local gravity tests, performed in the solar system. These tests are in very good agreement with Relativity.

39 Cosmological viability It is possible to obtain condition on f(r) function for a viable cosmological evolution. For a standard matter epoch m(r 1) 0, m (r 1) > 1 while to have an accelerated expansion after the matter era, we need 0 m(r 2) 1 where m(r) = R f RR 1+f R, r = R 1+f R R+f(R), f R = df dr, f RR = d2 f dr 2 L. Amendola et al., Phys. Rev. D 75 (2007)

40 Local tests The modified gravity lagrangian can be expressed in the Einstein frame as a general relativity lagrangian plus a scalar field L = g[r+f(r)] L = g[ R ] g µν φ,µ φ,ν 2V(φ) Thus, we have general relativity plus a fifth force that couples with matter as gravity. This fifth force must be suppressed in order to satisfy local tests; it must have a very short range or a very small intensity. However it must produce the accelerated expansion of the whole Universe.

41 extraction This estimator does not include contribution from the BB spectrum, as this method works only if the lensing contribution is negligible compared to the primary anisotropy. Different estimation methods can bring to better results (e.g. C. Hirata, U. Seljak, Phys. Rev. D 68 (2003) ) Convergence power per mode, C κκ L Convergence power per mode, C κκ L 1e-07 1e-08 1e-09 1e-07 1e-08 1e-09 Error in lens reconstruction: quadratic vs. iterative estimators Ref. Expt. A Ref. Expt. B Raw C κκ L Raw C κκ L Quad. est. (sim.) Quad. est. (sim.) Quad. est. (theor.) Iter. est. (sim.) Fisher limit 1e-07 1e-08 1e Multipole number, L Ref. Expt. D Raw C κκ L Quad. est. (sim.) Quad. est. (theor.) 1e-07 Iter. est. (sim.) Fisher limit 1e-08 1e-09 Quad. est. (theor.) Iter. est. (sim.) Fisher limit Multipole number, L Ref. Expt. E Raw C κκ L Quad. est. (sim.) Quad. est. (theor.) Iter. est. (sim.) Fisher limit 1e-07 1e-08 1e-09 1e-07 1e-08 1e-09 Ref. Expt. C Raw C κκ L Quad. est. (sim.) Quad. est. (theor.) Iter. est. (sim.) Fisher limit Multipole number, L Ref. Expt. F Raw C κκ L Quad. est. (sim.) Quad. est. (theor.) Iter. est. (sim.) Fisher limit Multipole number, L Multipole number, L Multipole number, L

42 Tomographic survey With Euclid we will be also able to perform a tomographic survey, splitting the galaxy distribution in three redshift bins. This way we obtain a 30% improvement on constraints, confirming the importance of tomography for future data analysis. λ Ω m In this analysis we are not including systematic effects.