295. Weak lensing signal in Unified Dark Matter models

This is a guest post by Stefano Camera (INFN and U. of Torino) about the work he has done in collaboration with D. Bertacca, A. Diaferio, N. Bartolo and S. Matarrese. Dmitry.

A particular description of DE is suggested by the Unified Dark Matter (UDM) models. While most of the models of DE rely on the potential energy of a scalar field to lead to the late time acceleration of the Universe, it is possible to have a situation where the accelerated expansion arises out of modifications to the kinetic energy of the scalar field. The major advantage of these models is that there is only one non-standard fluid, which can mimic both DM and DE. Thus, one of the main issues of these UDM models is to see whether the single dark fluid is able to cluster and produce the cosmic structures we observe in the Universe today. In fact, the effective speed of sound can be significantly different from zero at late times; the corresponding Jeans’ length (or sound horizon), below which the dark fluid cannot cluster, can be so large that the gravitational potential first strongly oscillates and then decays, thus preventing structure formation. Here we choose to investigate the class of scalar field Lagrangians with a non-canonical kinetic term that allow to obtain UDM models with a small effective sound speed.

Recently, it has been shown that the scalar field in UDM models can cluster, but it remains to be explored whether UDM models provide a good fit to the various sets of available data. Weak lensing is a powerful tool, indeed gravitational lens effects are due to the deflection of light occuring when photons travel near matter, i.e. in the presence of a non-neglegible gravitational field. The cosmic convergence and shear encapsulate information about both the source emitting light and the structures that photons cross before arriving at the telescope. Hence, weak lensing allows to explore both the basis of the cosmological model and LSS of the Universe, in other words it brings information about the geometry and the dynamics. Therefore the study of the power spectrum of weak lensing can be a crucial test.

The metric of the Universe is described by the standard Friedmann-Lematre-Robertson-Walker (FLRW) length element for an isotropic and homogeneous space filled with a perfect fluid, that, with the addition of scalar perturbations and in Newtonian gauge, takes the form

where is the conformal time, with the spatial metric

being the Kronecker delta, and the radial comoving distance is

where is the Hubble parameter and is the Hubble radius.

In LCDM, gravity is given by GR in a 4-dimensional Universe filled with a perfect fluid of photons, baryons, DM and DE (as cosmological constant). In linear theory of scalar perturbations the gravitational potential is given by

where the constant of integration is ,

with the matter transfer function, that describes the evolution of perturbations through the epochs of horizon crossing and radiaton-matter transition, the large-scale potential during the radiation dominated era.

UDM models use a scalar field that mimics both DM and DE. This can be achieved thanks to a non-canonical kinetic term, i.e. letting the kinetic energy be a generic function of the derivatives of the scalar field. The Lagrangian density can be written as

where . If is time-like, describes a perfect fluid. By requiring that on cosmological scales the background is identical to the background of LCDM and we easily get

where behaves like a cosmological constant “dark energy” component () and behaves like a “dark matter” component ().

The Newtonian potential is solution of the differential equation

where a prime denote a derivative with respect to the conformal time, is a function of the energy density and the pressure of the scalar field, and

Here, is the “speed of sound” relative to the pressure and energy density fluctuations of the scalar field. It accounts for the presence of intrinsic entropy perturbations of the fluid. Recently, 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

where is the value of the sound speed at late times.

On the Fig. above I present some gravitational potential Fourier’s components, normalized to unity at early times, at different scales and different values of for the models considered. By increasing the sound speed, the potential starts to decay earlier in time, oscillating then around zero. Moreover at small scales, if the sound speed is small enough UDM reproduces LCDM. This reflects the dependence of the gravitational potential on the effective Jeans’ length

.

Here I show , the sound horizon, for different values of

From GR we know that light beam paths are curved by the presence of matter. In the weak lensing framework the deflection of light is small and, consequently, we can use Born’s approximation, where lensing effects are evaluated on the null-geodesic of the unperturbed (unlensed) photon. All weak lensing observables may be expressed in terms of the projected potential

where

is the weight function of weak lensing, with representing the redshift distribution of sources, such that . The cosmic convergence is defined by

and its power spectrum is

where we introduced Limber’s approximation, in which the only Fourier modes that contribute to the integral are those with , and is the 3D power spectrum of the gravitational potential.

Now I will present the weak lensing power spectra for the CMB, background galaxy and high redshift proto-galaxy photons.

In UDM models, the background evolution of the Universe is the same as in LCDM, while the evolution of the gravitational potential and the growth of LSS suffer the non negligible sound speed, that increases with time. The discriminant is the effective Jeans’ length of the gravitational potential. The Newtonian potential in UDM models behave like in the LCDM model at scales much larger than the sound horizon, while at smaller scales it starts to decay and oscillate. Weak lensing observables, like cosmic convergence or shear, are an integral over the line-of-sight, hence they do not show directly these oscillations. However, high values of the multipole correspond to small scales, and thus cosmic convergence at high ‘s must show the decay of the deflecting potential.

Our analysis is made according to the linear theory of perturbations, and what we calculate is not correct for any . The multipole is related to the scale by a direct proportionality . We estimated the window of multipoles of validity of our approximations in the following way: the lower limit is , due to Limber’s approximation, but the upper one is floating, because at higher multipoles non-linear effects become more important. Weak lensing power spectra are made by integrating over the line-of-sight, thus over the wide range of the angular comoving distance. When we deal with high multipoles, wave numbers will appear in the integration. It is known that in linear theory is underestimated, but for now there is no linear to non-linear mapping in perturbation theory for UDM models. To estimate the upper limit of validity of our results, we proceed as follow: we made the integration in the convergence power spectrum, according to weak lensing theory, for , using only the linear power spectrum for the gravitational potential, obtaining obiouvsly a lensing signal lower than the real one, in which non-linear contributions to small scales are present. Then, we made the integral imposing a lower cut-off, setting it at . What we get in this way is a weak lensing power spectrum much more suppressed at large ‘s than the one obtained in the former integration, because in that case the linear power spectrum receives no contributions from non-linearities, but with this cut-off we set at zero, by hand, the integrand in the when it involves wave numbers larger than . With these two quantities, and , we fixed an arbitrary threshold

that enables us to estimate , under which our results are reliable, because most of the signal comes from the linear regime, and over which our ignorance on non-linear effects is too high and the real power spectra could be substantially different from those we obtaine.

For CMB light, the source is the last scattering surface at , and the distribution is

The upper panel shows the weak lensing power spectra of CMB light for LCDM and UDM. For UDM we present three curves, obtained for . In the lower panel every curve of the upper panel is divided by the convergence power spectrum of LCDM. As we can see, for small values of the sound speed (), we cannot distinguish the convergence of CMB photons in UDM models from the standard LCDM behaviour. By increasing , while at large scales the agreement is still good, at small enough scales is clearly suppressed.

Dealing with galaxy photons, the sources are spread over different redshifts, and the distribution is

that peaks at redshift , where , and are free parameters.

To better understand our results, it is useful to look at the weight functions for background galaxies. I present for the different choices of the source redshift distributions and we use in this work. The distribution of Kaiser (1999) has  and , that of Wittman et al. (2000) has and , and we also use a Dirac’s delta. As explained before, we consider UDM models which are able to reproduce the same expansion history as in LCDM. Peculiar dynamics of the scalar field become important starting from considering cosmological perturbations. The height of the peak of determines the order of magnitude of the weak lensing signal.

In the figures below,

the upper panels, we show the weak lensing power spectra of background galaxy light for LCDM and UDM. As for the CMB case, for UDM we present three curves, obtained for different values of the sound speed. As in the CMB case, for small sound speeds and large angular scales (), we cannot distinguish the convergence of background galaxy photons in UDM models from the standard LCDM behaviour. However, the agreement disappears at large and ‘s.

The redshift distribution of sources selects the peak redshift of the source emitting light. The power spectrum of the gravitational potential, through the Poisson’s equation, encodes the distribution of overdensities, thus the structures crossed by the photons. The weight function is a filter that selects mostly signals emitted at , following the information of . Consequently, for different values of the speed of sound, the weak lensing signal in UDM models is sensitive to the choice of . At the same time, has to be very small to let the scalar field cluster in order to form the LSS we observe today (while in the past, at high enough redshift, the gravitational potential is similar to that predicted by LCDM). However, at lower , sources emit light that feels strongly the decay and the oscillations of the Newtonian potential, because it is sensitive to the sound horizon , that increases with time, and to the presence of an effective , that plays the role of DE. It is easy to see that for small the differences between UDM models with different and between UDM and LCDM are very pronounced even at large angular scales, while for CMB and proto-galaxies (as I will show) the power spectra are less sensitive to the sound speed.

We also computed the convergence power spectra for proto-galaxy photons. The aim is to test UDM models at an intermediate redshift between background galaxies and the last scattering surface. We want to test a redshift distribution of sources more spread around the peak than that of background galaxies, but with no tail at small redshifts. We choose a peak redshift of and we use the source distribution with parameters and .

The shape is consistent with what we find for CMB photons and background galaxies.

Here, we have shown the lensing signal in linear theory as produced in LCDM and UDM; we have considered a number of sources: CMB photons, proto-galaxies at very high redshifts and background galaxies, whith different values of the peak redshift of the distribution, and different shapes of the redshift distribution of sources. For sound speed lower than , in the window of multipoles (Limber’s approximation) and (where our ignorance on non-linear effects due to small scales dynamics become relevant), the power spectra of the cosmic convergence in UDM and LCDM are not distinguishable. When the Jeans’ length increases, the Newtonian potential starts to decay earlier in time (for a fixed scale), or yet at greater scales (for a fixed epoch). This reflects on weak lensing by suppressing the convergence power spectra at high multipoles. We find that, for values of the sound speed between and , UDM models are still comparable with LCDM, while for higher values they are ruled out because of inhibition of structure formation. Moreover, we find that for low redshift sources the dependence of UDM weak lensing signal from the sound speed is much stronger.