Dark energy with non-adiabatic sound speed: initial conditions and detectability
aa r X i v : . [ a s t r o - ph . C O ] O c t CERN-PH-TH/2010-092LAPTH-017-10
Dark energy with non-adiabatic sound speed:initial conditions and detectability
Guillermo Ballesteros Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”. Piazza del Viminale 1, I-00184, Rome.Dipartimento di Fisica “G. Galilei”, Universit`a degli Studi di Padova, via Marzolo 8, I-35131 Padova, Italy.
INFN, Sezione di Padova, via Marzolo 8, I-35131 Padua, Italy.
Julien Lesgourgues CERN, Theory Division, CH-1211 Geneva 23, Switzerland.Institut de Th´eorie des Ph´enom`enes Physiques, EPFL, CH-1015 Lausanne, Switzerland.LAPTh (CNRS - Universit´e de Savoie), BP 110, F-74941 Annecy-le-Vieux Cedex, France.
Abstract
Assuming that the universe contains a dark energy fluid with a constant linear equa-tion of state and a constant sound speed, we study the prospects of detecting dark energyperturbations using CMB data from Planck, cross-correlated with galaxy distribution mapsfrom a survey like LSST. We update previous estimates by carrying a full exploration of themock data likelihood for key fiducial models. We find that it will only be possible to excludevalues of the sound speed very close to zero, while Planck data alone is not powerful enoughfor achieving any detection, even with lensing extraction. We also discuss the issue of initialconditions for dark energy perturbations in the radiation and matter epochs, generalizing theusual adiabatic conditions to include the sound speed effect. However, for most purposes,the existence of attractor solutions renders the perturbation evolution nearly independent ofthese initial conditions. [email protected] [email protected] Introduction
The biggest problem in cosmology today is the understanding of the accelerated expansion of theuniverse. Although one could try to attack this question leaving aside the cosmological principleor modifying Einstein’s gravity, the most classical approach consists of assuming a perturbedFLRW universe with a negative pressure component. The minimal model (in terms of numberof free parameters) compatible with the current data is a cosmological constant, which shouldbe perfectly homogeneous by definition. Other candidates (which may or may not alleviate thefine–tuning and coincidence problems of the cosmological constant) include, for instance, scalarfield models, or effective descriptions in terms of a fluid with free parameters yet to be measured.A canonically kinetic normalized scalar field would fluctuate, but since in that case the soundspeed c s ≡ δp/δρ (computed in the rest frame of the scalar field) is equal to one, local pressurewould prevent density contrasts to grow significantly. In an effective fluid description, the soundspeed is a free parameter, and dark energy clustering can be more efficient in the limit in whichoverdensities are not balanced by local pressure perturbations ( c s →
0) .Generally speaking, the study of small perturbations could be used as a tool for discriminat-ing between various models with a negative pressure component (cosmological constant, darkenergy fluid, quintessence or k–essence fields, coupled dark energy, etc.) or a modified theory ofgravity. One of the major difficulties comes from the fact that the expansion history predictedby a given Lagrangian theory of gravity can be reproduced in General Relativity by a dark fluidhaving an appropriate (time varying) equation of state, or by a scalar field with an adequatekinetic term and potential. Fortunately, a very precise measurement of clustering properties inour universe could at least help to discard some models in favor of others at the level of per-turbations [1, 2]. However, the spatial fluctuations of typical dark energy models are very muchsuppressed with respect to those of dark matter, and detecting their effect is a real challenge.The effects of quintessence perturbations (for which c s = 1) on the CMB and LSS powerspectra were discussed in [3]. For an (uncoupled) dark energy fluid, there have been severalstudies on the possibility to measure c s , but its value remains unconstrained with present data(see e.g. [4, 5, 6, 7, 8, 9, 10, 11]). In a recent attempt, the authors of [10] used present CosmicMicrowave Background (CMB), Large Scale Structure (LSS) and supernovae data (includingCMB × LSS cross–correlation), and showed that it is possible to see some weak preference for c s = 1 , but only for a certain kind of early dark energy model in which the equation of state is notconstant. We may still hope to discriminate between different values of c s using combinationsof future CMB data with 3–dimensional galaxy clustering data [12], with CMB × LSS cross–correlation data [13], or with results from a large neutral hydrogen survey such as that of theSKA project [14] (see also [15]).In this work, we focus on an effective description which has already been studied by severalauthors: namely, a dark energy fluid with a linear equation of state p = wρ , a constant equationof state parameter w close to − ≤ c s ≤ c s > w . We present for the first time theinitial conditions fulfilled by the dark energy fluid in the synchronous gauge (i.e., in the gaugeused by most Boltzmann codes), when all other fluids have adiabatic primordial perturbations.In Section 4 we study analytically the evolution of these perturbations during the matter epoch.We derive approximations for the attractor solutions followed by dark energy perturbations(both in the Newtonian and synchronous gauges). These new results can be used in the futurefor analytical estimates of the impact of dark energy on structure formation. In Section 5, inorder to update the analysis of [13], we carry a full Monte-Carlo exploration of the likelihoodof future mock CMB and LSS data, in order to infer the sensitivity of these data to the dark1nergy sound speed, and to investigate possible parameter degeneracies. Finally, in Section 6,we summarize our findings. At the level of inhomogeneities, the sound speed of a cosmological fluid plays a similar role tothat of the equation of state for the background cosmology, and relates the pressure and densityperturbations as: c s = δpδρ . (2.1)Defined in this way, the sound speed is gauge dependent. Indeed, the quantity that can beassumed to be a definite number (depending on the microscopic properties of the fluid) is theratio δp/δρ evaluated in the fluid rest frame, often denoted as ˆ c s . In another arbitrary frame, δp/δρ gets corrections related to the velocity of the fluid in that frame, such that in Fourierspace [4] ρ − δp = c s δ = ˆ c s δ + 3 H (1 + w ) (cid:0) ˆ c s − c a (cid:1) θk , (2.2)where δ = δρ/ρ is the relative density perturbation, θ the velocity divergence of the fluid, k the comoving wavenumber, H the conformal Hubble parameter d ln a/dτ , and c a the adiabaticsound speed of the fluid. The latter is defined as the (time dependent) proportionality coefficientbetween the time variations of the background pressure and energy density of the fluid,˙ p = c a ˙ ρ . (2.3)For non–interacting fluids one finds that˙ w = 3(1 + w ) (cid:0) w − c a (cid:1) H , (2.4)which implies that c a = w if w is constant and different from − w can indeed be approximated as a constant in time. Besides, wewill restrict our analysis to 0 ≤ ˆ c s ≤ c s for quintessence see [16]. For an expression relating the time evolution of w and c s throughthe intrinsic entropy contribution to the pressure perturbation see [33] (and also [34]). In order to compute the CMB and LSS power spectra, one needs to evolve cosmological per-turbations starting from initial conditions deep inside the radiation epoch and far outside theHubble radius. Initial conditions for photons, neutrinos, cold dark matter and baryons arereviewed in [17] in the synchronous and Newtonian gauges (see also [18]). Here, we want toextend this set of relations to dark energy perturbations, especially in the gauge used by mostBoltzmann codes: namely, the synchronous gauge. Surprinsingly, this issue has been overlookedin the literature, without a clear justification . In practical terms, initial conditions for dark There were however various studies closely related to this issue. For instance, in [19] various possible initialconditions for the quintessence field, and their impact on the CMB were discussed. In [20], the initial conditionsfor a dark energy fluid with quintessence-like perturbations were obtained in a gauge invariant formalism. In [21],the technique of [20] was extended to interacting dark energy models. In all these works, the dark energy soundspeed ˆ c s was kept fixed to one. i , the density ρ i ( τ, ~x ) can be written as ¯ ρ i ( τ + δτ ( ~x )) , where ¯ ρ i ( τ ) stands forthe background density, and δτ ( ~x ) is an initial time–shift function independent of i . Similarlythe pressure would read ¯ p i ( τ + δτ ( ~x )). Such conditions can be easily justified in all models inwhich primordial perturbations are generated from a single degree of freedom (like the inflaton),and/or in cases in which all components have been in thermal equilibrium in the early universewith a common temperature and no chemical potential. This form implies ρ i = ¯ ρ i + ˙¯ ρ i δτ ( ~x ) , (3.1) p i = ¯ p i + ˙¯ p i δτ ( ~x ) , (3.2)and hence δp i /δρ i = ˙¯ p i / ˙¯ ρ i , i.e. the sound speed of each species must be equal to its adiabaticsound speed. This generic assumption also implies that the total ratio [ P i δp i ] / [ P i δρ i ] isindependent of the purely spatial coordinates ~x . Finally, if all the components do not interact,we conclude that δ i ( τ, ~x )1 + w i = − H ( τ ) δτ ( ~x ) , ∀ i . (3.3)The fact that for all species the ratios δ i / (1 + w i ) are equal to each other is a well-knownproperty of adiabatic initial conditions. The meaning of such a relation is not so clear when oneintroduces a dark energy fluid, for which ˆ c s > c a . Hence, in most frames, one has c s = c a andthe fluid cannot obey simultaneously (3.1) and (3.2). This raises the issue of defining sensibleinitial conditions for a dark energy fluid. However, during radiation and matter domination,dark energy perturbations tend to fall inside the gravitational potential wells created by thedominant component and not much concern has been raised concerning their initial conditions.In other words, there is an attractor solution for dark energy perturbations, and their initialvalues are almost irrelevant in practice, provided that for each Fourier mode the attractor isreached before dark energy comes to dominate (i.e. provided that initial conditions are imposedearly enough, and that initial dark energy perturbations are not set to dramatically large values).For this reason, in a Boltzmann code like camb [22], initial dark energy perturbations are setby default to zero.We will derive in the next subsections the attractor solution for a dark fluid with constant w and arbitrary ˆ c s , assuming that other quantitites obey the usual adiabatic initial conditions,and in two gauges: the Newtonian and the synchronous ones. For a detailed account of theconstruction of the two gauges and the relations among them we refer the reader to [17] . Inwhat follows, we will denote quantities corresponding to the conformal Newtonian gauge witha superscript. For instance, δ ( c ) x denotes the relative dark energy density in that gauge. Nosuperscript will be used for quantities in the synchronous gauge. The transformation equationsbetween the two gauges will be summarized below, later in this section. Early in the radiation era, the total energy density of the universe can be approximated by thesum of photon and neutrino densities, with a constant ratio R ν = ¯ ρ ν / ( ¯ ρ ν + ¯ ρ γ ) . In order tofind the perturbation evolution on super-Hubble scales and for adiabatic initial conditions, onecan combine the Einstein, photon and neutrino equations into a fourth order linear differentialequation for the trace part h of the metric perturbations in Fourier space [17]. The fastest grow-ing mode among the four possible solutions, h ∼ ( kτ ) , corresponds to the growing adiabaticmode. 3ince these conditions are established when the perturbations are still in the super–Hubbleregime, the product ( kτ ) ≪ − h = − C ( kτ ) = δ c = δ b = 34 δ ν = 34 δ γ , (3.4)where the subscripts refer to cold dark matter, baryons, neutrinos and photons; and C is aconstant. As usual, in order to fully fix the gauge, we impose not only synchronous metricperturbations, but also that dark matter particles have a vanishing velocity divergence θ c . Thecontinuity and Euler equations for the dark energy fluid read˙ δ x = − (1 + w ) θ + ˙ h ! −
3( ˆ c s − w ) H δ x − w )( ˆ c s − c a ) H θ x k , (3.5)˙ θ x = − (1 − c s ) H θ x + ˆ c s k w δ x − k σ x . (3.6)These equations are very general, since the only underlying assumption is that the fluid is non–interacting, and allow for the presence of shear stress σ x , non–adiabatic sound speed, and a timevarying w . From now on we assume that the fluid is shear free and has a constant equation ofstate. If the energy density of dark energy at early times is negligible, the solution for the metricperturbation h will not change. In order to find the attractor solution, we just need to replace˙ h in (3.5) according to (3.4), and solve equations (3.5), (3.6). As expected, we find that thesolution of the homogeneous equation becomes negligible with time, while δ x and θ x are drivento δ x = − C w ) 4 − c s − w + 3 ˆ c s ( kτ ) , (3.7) θ x = − C c s − w + 3 ˆ c s ( kτ ) k , (3.8)at lowest order in ( kτ ) .This attractor solution does not look like usual adiabatic initial conditions because, in gen-eral, c s is different from c a in the synchronous gauge. Hence, δp x /δρ x cannot be equal to ˙¯ p x / ˙¯ ρ x .However, this solution gives the correct behavior of dark energy perturbations when the othercomponents obey adiabatic initial conditions , once the attractor has been reached. Thereforethey could be called “generalized initial adiabatic conditions”. These conditions are valid notonly for dark energy fluids ( w < − /
3) but also for any other fluid with constant w and σ = 0 .For instance, one can easily check that the usual adiabatic initial conditions for matter andradiation can be recovered from (3.7, 3.8) by choosing w = ˆ c s .In a Boltzmann code like camb [22], the quantitites δ x and θ x are fixed to zero at initialtime for simplicity. For most practical applications, this arbitrary choice does not introduceany mistake in the final results, since δ x and θ x are quickly driven to the attractor solutions ofeqs. (3.7, 3.8). We illustrate this in Figure 1, for a very large wavelength mode. We suggesthowever to implement eqs. (3.7, 3.8) directly into camb ’s initial condition routine (as we did inSection 5 of this work), since this is completely straightforward, and since it offers a guaranteethat final results are independent of the early time at which initial conditions are defined.4 .2 Conformal Newtonian Gauge The equations that give the evolution of a generic fluid in the conformal Newtonian gauge are(see also [10])˙ δ ( c ) x = − (1 + w ) (cid:16) θ ( c ) x − φ (cid:17) − (cid:0) ˆ c s − w (cid:1) H δ ( c ) x − w )( ˆ c s − c a ) H θ ( c ) x k , (3.9)˙ θ ( c ) x = − (cid:0) − c s (cid:1) H θ ( c ) x + ˆ c s k w δ ( c ) x − k σ ( c ) x + k ψ . (3.10)The metric perturbations φ = η − α H , (3.11) ψ = ˙ α + α H , (3.12)can be obtained from those of the synchronous gauge using2 k α = ˙ h + 6 ˙ η , (3.13)where η is the traceless part of the metric scalar perturbation in the synchronous gauge inFourier space. Using these last equations one can immediately check that the product α H haszero time derivative, so that φ and ψ are time independent at lowest order in ( kτ ) . One caneither solve directly (3.9, 3.10), or use our results (3.7, 3.8) for the behavior of δ x and θ x in thesynchronous gauge, and perform the gauge transformations δ ( c ) x = − w ) α H + δ x , (3.14) θ ( c ) x = αk + θ x , (3.15) σ ( c ) x = σ x , (3.16)that are valid for non–interacting fluids. The two methods give δ ( c ) x = −
32 (1 + w ) ψ + δ x , (3.17) θ ( c ) x = 12 ψ ( kτ ) k + θ x , (3.18)where ψ = 2015 + 4 R ν C (3.19)and φ = (1 + 25 R ν ) ψ − R ν R ν ) C ( kτ ) . (3.20)Notice that the leading contributions to the velocity and density perturbations in the conformalNewtonian gauge are independent of the speed of sound and such that δ ( c ) x w = − ψ = δ ( c ) c = δ ( c ) b = 34 δ ( c ) ν = 34 δ ( c ) γ . (3.21)Therefore, the usual adiabatic conditions are recovered, and ˆ c s enters only in the next ordercorrections . Indeed, the Newtonian gauge is the one in which, beyond the Hubble scale, c s Note that the gauge transformation law (3.14) implies that δ ( c ) x w − δ ( c ) j w j = δ x w − δ j w j , (3.22)
5s equal to c a at leading order, even when ˆ c s = c a . This can be checked by keeping thedominant terms in (3.14, 3.15), and replacing δ x and θ x by these values in (2.2): one gets c s δ ( c ) x = c a δ ( c ) x + O ( kτ ) . So, in the Newtonian gauge and on super-horizon scale, δ ¯ p/δ ¯ ρ is equalto p/ρ for any fluid with contant w , and the common intuition according to which super-Hubblefluctuations behave in the same way as background quantitites is recovered. In the previous section, we found the attractor solutions for dark energy perturbations duringradiation domination. Here we will derive similar solutions during matter domination. Theseresults can be used to provide initial conditions for dark energy perturbations in problems inwhich following the behavior of cosmological perturbations during radiation domination is notrelevant.If we assume that the energy density of photons and (massless) neutrinos is negligible deepinside matter domination, the perturbations can be studied using a two–fluid approximation.One of the fluids is formed by baryons and cold dark matter (which cannot be distinguishedfrom each other) and the other one is dark energy. This description in terms of two componentscan be accurately used to study the growth of matter perturbations up to the present day.Mathematically, the problem consists of a system of six independent equations with six variables:the density and velocity perturbations of the two fluids plus the scalar metric perturbations. Wewill now find the relevant growing modes of the perturbations in the two gauges and concludethis section with some remarks concerning the initial conditions for dark energy.
In the synchronous gauge, since we consider that cold dark matter and dark energy are shearfree, the Einstein equations imply that the two metric degrees of freedom η and h are related toeach other. Eliminating η in terms of h , and expressing h in terms of δ c (the continuity equationfor cold dark matter gives ˙ h = − δ c ) , we obtain a system of two reasonably short second orderdifferential equations that describe the evolution of density fluctuations [23] :¨ δ c + H ˙ δ c − H Ω c δ c = 32 H Ω x (cid:20)(cid:0) c s (cid:1) δ x + 9 (1 + w ) H (cid:0) ˆ c s − w (cid:1) θ x k (cid:21) , (4.1)¨ δ x + (cid:2) (cid:0) ˆ c s − w (cid:1) H − F ( k, H ) (cid:3) ˙ δ x − (cid:0) ˆ c s − w (cid:1) (cid:20)(cid:0) w Ω x − c s (cid:1) H + 2 F ( k, H ) H −
23 ˆ c s ˆ c s − w k (cid:21) δ x = (1 + w ) ¨ δ c − (1 + w ) F ( k, H ) ˙ δ c , (4.2) for any fluid j with constant equation of state w j . Since the usual adiabatic conditions hold at leading orderin the Newtonian gauge, one may naively infer from the above equality that they hold also in the synchronousgauge. This is not correct since on the left-hand side, the two terms are dominated by order zero terms in a ( kτ )expansion, while on the right-hand side the leading terms are of order two. Assuming that the fluid j has anadiabatic sound speed (like cold dark matter or photons), a full order-two calculation of all the terms leads to δ ( c ) x w − δ ( c ) j w j = δ x w − δ j w j = C (cid:18) c s − w )4 − w + 3ˆ c s (cid:19) ( kτ ) . (3.23)The righ-hand side does not vanish since (ˆ c s − w ) is by assumption strictly positive. Being of order two, thisdifference contributes to the solutions at leading order in the synchronous gauge, but only at next-to-leading orderin the Newtonian gauge. w ) θ x k = 1 D ( k, H ) (cid:16) − ˙ δ x + (1 + w ) ˙ δ c − (cid:0) ˆ c s − w (cid:1) H δ x (cid:17) , (4.3)and D ( k, H ) = k + 9( ˆ c s − w ) H , (4.4) F ( k, H ) = − w Ω x ) ˆ c s − w D ( k, H ) H − (1 − c s ) H . (4.5)In the purely matter dominated epoch (Ω c = 1, Ω x = 0) the equation that describes theevolution of matter perturbations is the classical growth formula:¨ δ c + H ˙ δ c − H Ω c δ c = 0 . (4.6)Its general solution is a linear combination of a growing mode ( δ c ∼ a ) and a decaying one (cid:0) δ c ∼ a − / (cid:1) . In order to find the relevant attractor solution for dark energy perturbations, oneshould keep the first of these two solutions δ c ∝ ( kτ ) . (4.7)The behavior of cosmological perturbations depends on their wavelength as compared to thecharacteristic scales of the problem. In our case, and in terms of comoving scales, there are tworelevant quantitites: the comoving Hubble scale H − (which gives the order of magnitude ofthe causal horizon associated to any process starting after inflation), and the comoving soundhorizon H − s = ˆ c s H − . Analytical approximations for the evolution of dark energy perturbationscan be obtained in the three regimes defined by these two scales.For super–Hubble perturbations with k ≪ H , one can approximate (4.5) as F ( k, H ) ≃ (cid:0) c s − (cid:1) H , and the homogeneous part of (4.2) gives two decaying modes for δ x . The growingsolution, sourced by the dark matter perturbations, is δ x = (1 + w ) 5 − c s − w + 9 ˆ c s δ c , k ≪ H . (4.8)If instead H s ≫ k ≫ H , the perturbations are below the Hubble scale but above the soundhorizon, and F ( k, H ) ≃ (cid:0) c s − (cid:1) H . As in the previous case, the growing mode for the densityperturbations of dark energy comes from the one of dark matter: δ x = (1 + w ) 1 − c s − w + ˆ c s δ c , H s ≫ k ≫ H . (4.9)These formulas agree with the results of [24] (when transformed into the synchronous gauge)if we take the sound speed to be zero. The equations (4.8) and (4.9) can be used to defineinitial conditions for a dark energy fluid during matter domination, provided that at the initialtime all relevant wavelengths are still larger than the sound horizon. Below the sound horizon,it becomes more difficult to obtain simple analytic approximations. When H s ≪ k , we canstill approximate F ( k, H ) as in the previous case, but the term proportional to ˆ c s k in (4.2)becomes dominant. One gets:¨ δ x + (1 − w ) H ˙ δ x + ˆ c s k δ x = 32 (1 + w )(1 − c s ) H δ c , H s ≪ k . (4.10)The general solution of the homogeneous equation goes as τ w − / with a multiplicative factorthat is a linear combination of the Bessel functions J w − / ( c s kτ ) and Y w − / ( c s kτ ) . The7articular solution for the complete equation including δ c can also be written in terms of non–elementary functions. Dark energy perturbations are anyway suppressed with respect to darkmatter ones in this regime, since below the sound horizon (i.e below the Jeans length of the fluid)the pressure perturbation can resist the gravitational infall. Indeed, equation (4.10) implies thatin the limit k → ∞ , the dark energy density contrast δ x should vanish. In the conformal Newtonian gauge, the complete equations for the evolution of density pertur-bations, equivalent to (4.1) and (4.2), become longer, because metric perturbations cannot betrivially replaced in terms of δ c . Having obtained the solutions in the synchronous gauge, it issimpler to apply (3.14) rather than solving the conformal Newtonian equations directly. In thelimit of pure matter domination, i.e. Ω x = 0 , one can easily prove that˙ h = 2 k α , (4.11)and therefore δ ( c ) c = (cid:18) H k (cid:19) δ c , (4.12) δ ( c ) x = δ x + 3(1 + w ) H k δ c . (4.13)These equations show that outside the Hubble radius, the solution is driven as usual to δ ( c ) x = (1 + w ) δ ( c ) c . (4.14)The same equations also imply that in the regime H s ≫ k ≫ H , (4.9) remains valid in theconformal Newtonian gauge (relations valid inside the Hubble radius are expected to be gaugeindependent). An important feature in the evolution of dark energy perturbations in the two gauges, whichis common to the three regions we have studied, is that initial conditions for the dark energyperturbations are almost irrelevant for the evolution in the purely matter dominated epoch.During this period, dark energy fluctuations track matter inhomogeneities. Above the soundhorizon, as soon as the attractor solution of (4.8) or (4.9) is reached, the ratio δ x /δ c only dependson the sound speed and equation of state of the dark energy fluid. Notice that in the newtoniangauge the sound speed dependence actually disappears outside the Hubble scale.In the dark energy dominated period (Ω x → δ c that determines δ x in the matter epoch, the initial conditionsfor the evolution in the dark energy period are in reality given only by δ c . In accordance withthe results of section 3 , this argument can be extended back into the radiation era. Besides,from the previous reasonings, it is clear that the velocity perturbations are also unimportant. Inconclusion, the amount of dark energy perturbations today can be well estimated just by knowingthe dark matter perturbations at some initial time in the radiation or the matter epoch. Weillustrate this behavior in Figure 1 for a very large wavelength mode.8 -2 -3 -4 -5 -6 δ x / δ γ k τ (for k=2.3 10 -6 Mpc -1 )w=-0.9, cˆ s2 =0.1 RD equality MD DED(1+w)/(1+w γ )[(1+w)/(1+w γ )] [(4-3cˆ s2 )/(4-6w+3cˆ s2 )][(1+w)/(1+w γ )] [(5-6cˆ s2 )/(5-15w+9cˆ s2 )] Figure 1: The three thick lines show the evolution of the ratio δ x /δ γ in the synchronous gauge,obtained numerically with camb , in a model with: w = − .
9, ˆ c s = 0 .
1, standard values ofthe other cosmological parameters, and adiabatic initial conditions for photons, neutrinos, cdm,baryons. We choose the case of a very long wavelength mode ( k = 2 . × − Mpc − ) whichremains outside the Hubble radius during all relevant stages: radiation domination (RD), matterdomination (MD) and dark energy domination (DED). We integrated this mode starting either: (i) from the “usual initial condition” δ x /δ γ = (1 + w ) / (1 + w γ ) with w γ = 1 / (upper thinhorizonal line) , which has no physical justification in the synchronous gauge in this context; (ii) from eq. (3.7) (middle thin horizontal line) and (3.8); (iii) from δ x = 0, like in the publicversion of camb . In each case, the solution quickly evolves in such way to fulfill eq. (3.7) (middlethin horizontal line) during radiation domination, and then eq. (4.8) (lower thin horizontal line) during matter domination. The detectability of the sound speed of dark energy has already been studied by various authors,under different assumptions and for various datasets. For example, for a model with constant w and c s (identical to the one we consider in this work), the authors of [10] showed that the combi-nation of present CMB, LSS and supernovae data is not sensitive at all to the dark energy soundspeed. Since dark energy perturbations would change the growth rate of matter inhomogeneitieson intermediate scales (between the Hubble radius and the dark energy sound horizon), theycan affect the CMB photon temperature through the Late Integrated Sachs-Wolfe (LISW) effect.Small variations in the LISW effect are difficult to detect in the CMB temperature spectrum,due to cosmic variance and to the fact that we only see the sum of primary anisotropies andLISW corrections. However, the LISW contribution can be separated from primary anisotropiesby computing the statistical correlation between CMB and projected LSS maps of the sky. In[10], most of the presently available cross–correlation data were included in the analysis, butcurrent statistical and systematic error bars are far too large for probing sub–dominant darkenergy clustering effects.One could think of constraining the dark energy sound speed in the future, either by mea-suring with better accuracy the auto–correlation function of matter distribution tracers (galaxy9urveys, cluster surveys, lensing surveys, . . . ) or, again, by studying their cross–correlation withCMB maps in order to extract the LISW contribution. The first option was studied for instancein [12], and the second in [13]. In the latter, the authors focus on the cross–correlation of aCMB experiment similar to Planck with a survey comparable to the Large Synoptic SurveyTelescope (LSST) project. The author of [12] considered the combination of Planck with fu-ture galaxy redshift surveys, not including any cross–correlation information, but using the fullthree–dimensional power spectrum of galaxies instead of their angular power spectrum sampledin a few redshift bins (the loss of information on the LISW effect can then be compensatedby more statistics on the auto-correlation function). Both works reached the same qualitativeconclusion that the next generation of LSS surveys could discriminate between quintessence–likemodels with c s = 1 and alternative models with a sound speed c s ≪
1, and estimated underwhich threshold in c s the discrimination would be significant.The works of [12] and [13] are based on analytic estimates of the measurement error for eachcosmological parameter, using the Fisher matrix of each future data set. This is the second–orderTaylor expansion with respect to the cosmological parameters of the data likelihood around itsmaximum, i.e. in the vicinity of a putative fiducial model. This method should be taken with agrain of salt whenever the dependence of the observable spectrum on a given parameter cannotbe approximated at the linear level within the range over which this parameter gives good fitsto the data. This is precisely the case for the model at hand. The dark energy sound speed willnot be pinned down with great accuracy using the Fisher matrix approach. Its variation withinthe error bar has a non–trivial effect on the total matter power spectrum which, although small,cannot be captured at the linear level, since it amplifies perturbations over a range of scalesdepending on c s itself. This means that Fisher matrix estimates for the error ∆ c s should onlybe considered as a first–order approximation. This caveat was already emphasized in SectionIV of [13]. Hence, it is worth checking the results of [13] with a full exploration of the likelihoodof mock Planck + LSST data. We generated such data for some fiducial models including adark energy fluid, sampled the data likelihood with a Monte Carlo Markhov Chain (MCMC)approach, and inferred the marginalized posterior probability distribution of the dark energysound speed. Unlike the Fisher matrix approach, this method takes into account the preciseeffect of each parameter throughout the parameter space of the model. Hence, it can assessrealistically whether the data can resolve non–trivial parameter degeneracies. As in [12], weincluded the total neutrino mass in the list of parameters to measure, in order to check whethersome confusion between the effect of dark energy clustering and massive neutrino free–streamingcould arise when the dark energy sound horizon is comparable to the neutrino free–streamingscale. Our mock data sets were generated and fitted with two modified versions of the public CosmoMC package [26] which were already developed and described by the authors of [27]and [28].
In a first step, we focus on the potential of the Planck experiment alone, assuming Blue Booksensitivities for the most relevant frequency channels, and a sky coverage of 65% (as summarizedin [29]). Given that small variations in the LISW effect cannot be probed due to cosmic variance,the only hope to detect dark energy clustering with Planck data alone is through lensing extrac-tion. A lensing estimator can in principle measure the deflection of CMB anisotropies causedby gravitational lenses (e.g. galaxy clusters). This techniques allows to reconstruct the powerspectrum of the gravitational potential projected along the line of sight, and to disentangle afraction of the total LISW contribution to temperature anisotropies (by cross–correlating thelensing map with the temperature map). Like in [27], we ran
CosmoMC in order to fit somemock Planck data, including reconstructed lensing data with a noise level corresponding to the10minimum variance quadratic estimator” of [30]. The mock data generator and all modifica-tions to the
CosmoMC package are publicly available . Our results show that Planck alone iscompletely insensitive to the dark energy sound speed for any constant and reasonable value of w . Indeed, when assuming a flat prior on log [ c s ] in the range [ − , P (log [ c s ]). We checked this results with several assumptionson the fiducial values of w (bewteen − . c s (between 10 − and 1) and Σ m ν (between0.05 eV and 0.2 eV). This negative result can be attributed, on the one hand, to the limitedprecision of lensing extraction with Planck, and on the other hand, to the fact that the projectedgravitational potential probed by CMB lensing gives more weight to high redshifts (of the orderof z ∼
3) than to the redshifts at which dark energy comes to dominate and eventually affectsthe growth of total matter perturbations ( z < w ). In the rest ofthis section, we will not include Planck lensing extraction data anymore. In a second step, we performed a combined analysis of mock Planck and LSST data, includinginformation on the angular auto–correlation function of LSST galaxy maps (in 65% of the skyand in 6 redshift bins spanning the range from z = 0 to z ∼ P m ν = 0 . w = − . c s = 1 or ˆ c s = 10 − . For each model we ransixteen chains with flat priors on the usual CosmoMC parameter basis ( ω b , ω c , θ , τ , log[ A s ], n s , f ν , w ) plus log [ ˆ c s ], imposing a restriction log [ c s ] ≤
0. After reaching a convergencecriterion R − < .
01, we obtained the marginalized posterior probability of each of theseparameters, and of the derived parameters P m ν (total neutrino mass) and H . We show inTable 1 the 68% standard error for each of these. In the first plot of Figure 2, we display themarginalized distribution of log [ ˆ c s ] in the two models. The correlation between log [ ˆ c s ] andother parameters appears to be small, except for w , f ν (or P m ν ), and θ (or H ). In Figure 2,we also show the two–dimensional 68% and 95% confidence levels contours for these pairs ofparameters.We learn from these runs that correlated CMB+galaxy data from Planck+LSST may dis-criminate between various sound speed values with a standard deviation ∆ log [ ˆ c s ] of order 1 .In other words, such a data combination can give an indication on the order of magnitude of ˆ c s .With a fiducial value ˆ c s fiducial = 1 , the lower bound of the 68% (resp. 95%) confidence intervalis log [ ˆ c s ] > − . − .
8) . With a fiducial value ˆ c s fiducial = 10 − , these numbers becomelog [ ˆ c s ] > − . − .
3) . The 68% confidence intervals always include log [ ˆ c s ] = 0, al-though the case of ˆ c s fiducial = 10 − appears to be at the threshold below which the data wouldimpose a negative 68% upper bound on log [ c s ]. These estimates are not as optimistic as thosein [13], most probably because of our more accurate technique for error forecast (based on amarginalization of the actual likelihood over the full parameter space). They are also less dra-matic than those of [12], inferred from the Fisher matrix of a different dataset. Still, they showin a robust way that with Planck+LSST one can exclude dark energy models with maximumclustering (i.e. ˆ c s → http://lesgourg.web.cern.ch/lesgourg/codes.html [c s ] log [c s ] w −3 −2 −1 0−0.95−0.9−0.85−0.8log [c s ] Σ m ν ( e V ) −3 −2 −1 00.10.150.20.25 log [c s ] H −3 −2 −1 064666870 Figure 2: (Top left)
Marginalized posterior probability distribution of log [ c s ] for Planck+LSSTand the two fiducial models described in the text and in Table 1, in which c s = 1 (red lines)or c s = 10 − (black lines). The dotted lines show the mean likelihood for comparison. (Otherplots) For the same models and mock data, joint 68% and 95% confidence contours for log [ c s ]and the three parameters most correlated to the dark energy sound speed.to parameter degeneracies, since log [ ˆ c s ] is not particularly correlated with other parameters.Of course, the higher is w , the longer will be the dark energy domination and the stronger willbe the effect of ˆ c s . This explains the correlation between log [ ˆ c s ] = 0 and w seen in Figure 2:if w were found to be closer to − . −
1, the measurement of c s would be considerablyeasier. This is in agreement with the c s dependence of the bounds on w found in [25]. Thejoint (log [ ˆ c s ], w ) likelihood contours actually suggest that even with w fiducial = − .
85 , a valuelike ˆ c s = 10 − could be pinned down with a fairly good accuracy. Like in [12], we find that thedegeneracy between the dark energy sound speed and the neutrino mass is not significant: thiscan be understood from Figure 5. of [12], which shows that dark energy clustering and neutrinofree–streaming can produce a step in the matter power spectrum on similar scales, but withdifferent shape. The step induced by dark energy is sharper because this effect occurs during abrief period of time compared to non–relativistic neutrino free–streaming. In sections 3 and 4, we have studied analytically the behavior of dark energy perturbationsin the radiation and matter epochs in the synchronous and conformal Newtonian gauges. Wehave written down the formulas that generalize the usual adiabatic initial conditions for smallinhomogeneities when a dark energy fluid with constant equation of state and speed of soundis considered. Our equations (3.7, 3.8) can be readily implemented in the initial conditions ofe.g. camb , which applies to early radiation domination. Instead, equations (4.8, 4.9) – or theircounterpart (4.14, 4.9) in the Newtonian gauge – provide correct initial conditions above thesound horizon for codes simulating only the growth of matter fluctuations during radiation anddark energy domination. The fact that there is an attractor behavior for the evolution of the12tandard errorsˆ c s fiducial = 1 ˆ c s fiducial = 10 − Ω b h c h θ τ [ c s ] > -1.1 > -1.9 f ν w n s A s ] 0.0095 0.0090Σ m ν (eV) 0.024 0.023 H (km/s/Mpc) 0.64 0.87Table 1: Standard deviation on each cosmological parameter for correlated CMB+galaxy datafrom Planck+LSST for two fiducial models with standard values of the six ΛCDM parameters,a total neutrino mass P m ν = 0 . w = − . c s = 1 or c s = 10 − . The last two lines show derived parameters. For log [ ˆ c s ], instead of quoting thestandard error, we provide the lower bound of the 68% confidence interval (the upper boundbeing imposed by the prior, log [ ˆ c s ] ≤ w and constant sound speedˆ c s . We have performed a full exploration of the likelihood of our mock data using a Monte Carloapproach, in order to cross-check previous Fisher matrix estimates in a more robust way. Wefind that the confidence interval for ˆ c s inferred from the data will potentially allow us to puta lower bound on ˆ c s , and to constrain at least its order of magnitude (although with a slightlysmaller significance than in earlier analytic estimates). Our results build upon previous workson the same topic [4, 13, 12, 10] and complement other researches were different models havebeen studied (see for instance [10] for early dark energy and [32] for the case in which there isa coupling to dark matter). We thank Wessel Valkenburg for his large contribution to the numerical module [28] that wehave employed in our analysis. We wish to thank Juan Garcia-Bellido, Katrine Skovbo andToni Riotto for very useful discussions. Numerical computations were performed on the MUSTcluster at LAPP (CNRS & Universit´e de Savoie). GB thanks the CERN TH unit for hospitalityand support and INFN for support. JL acknowledges support from the EU 6th FrameworkMarie Curie Research and Training network ‘UniverseNet’ (MRTN-CT-2006-035863).13 eferences [1] E. Bertschinger and P. Zukin, “Distinguishing Modified Gravity from Dark Energy,” 2008,0801.2431.[2] M. Kunz and D. Sapone, “Dark energy versus modified gravity,”
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