Model-independent constraints on the cosmological anisotropic stress
Luca Amendola, Simone Fogli, Alejandro Guarnizo, Martin Kunz, Adrian Vollmer
MModel-independent constraints on the cosmological anisotropic stress
Luca Amendola, Simone Fogli,
1, 2
Alejandro Guarnizo, Martin Kunz,
3, 4 and Adrian Vollmer Institut Für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Dipartimento di Fisica e Astronomia, Università di Bologna, Via Irnerio 46, 40126 Bologna, Italy Département de Physique Théorique and Center for Astroparticle Physics,Université de Genéve, Quai E. Ansermet 24, CH-1211Genéve 4, Switzerland African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, 7945, South Africa (Dated: October 16, 2018)The effective anisotropic stress or gravitational slip η = − Φ / Ψ is a key variable in the charac-terisation of the physical origin of the dark energy, as it allows to test for a non-minimal couplingof the dark sector to gravity in the Jordan frame. It is however important to use a fully model-independent approach when measuring η to avoid introducing a theoretical bias into the results. Inthis paper we forecast the precision with which future large surveys can determine η in a way thatonly relies on directly observable quantities. In particular, we do not assume anything concerningthe initial spectrum of perturbations, nor on its evolution outside the observed redshift range, noron the galaxy bias. We first leave η free to vary in space and time and then we model it as suggestedin Horndeski models of dark energy. Among our results, we find that a future large scale lensingand clustering survey can constrain η to within 10% if k -independent, and to within 60% or betterat k = 0 . h/ Mpc if it is restricted to follow the Horndeski model.
I. INTRODUCTION
With the recent first results of the Planck satellite [1] we have definitely reached the era of precision cosmology:The Planck observations of the cosmic microwave background (CMB) are well described by the six-parameter flatΛCDM model, and most of those six parameters are determined to percent-level accuracy [2]. The most impressiveachievement is the measurement of the acoustic scale of the CMB with a precision of 0.06% by Planck, but also thephysical baryon and the matter densities have been determined to within an uncertainty of only 1 to 2%.But the conclusion from these measurements is that we live in an Universe where only 5% of today’s energy densityconsists of the kind of matter described by the standard model of particle physics. Another 27% appears to be matterthat is only interacting gravitationally with the visible world, and the remaining 68% is made up of a cosmologicalconstant.The physical nature of the dark sector is however completely unknown, and especially the cosmological constantsuffers from severe theoretical problems. For this reason it is of crucial importance to look beyond the perfectlyhomogeneous cosmological constant and to investigate general dark energy models, including also modifications ofEinstein’s theory of General Relativity (GR). When considering a general dark energy model however, high precisionis much harder to achieve, and it is important to understand first what can actually be observed, to avoid introducinga theoretical bias into the observational results. Coming from this angle, we determined in a recent paper [3] thatcosmological measurements at linear scales can determine, in addition to the expansion rate H ( z ), only three additionalvariables R , A and L , given by A = Gbδ m,0 , R = Gf δ m,0 , (1) L = Ω m,0 GY (1 + η ) δ m,0 . Denoting with k the norm of the wavenumber and with a the cosmic scale factor, we refer with G ( k, a ) to the lineargrowth function (normalized to unity today) with f = G /G to the growth rate, with b ( k, a ) to the galaxy bias withrespect to the dark matter density contrast and with δ m,0 ( k ) to the dark matter density contrast today. The functions η ( k, a ) and Y ( k, a ) describe the impact of the dark energy on the cosmological perturbations. Later on, we will alsoneed the quantities ¯ A ≡ A/δ t,0 , ¯ R ≡ R/δ t,0 , ¯ L ≡ L/δ t,0 with δ t,0 = δ m,0 /σ . If we write the line element describingthe perturbed Friedmann-Lemaître-Robertson-Walker metric as ds = − (1 + 2Ψ)d t + a ( t ) (1 + 2Φ) d x , (2)then η and Y are defined through [4, 5] η ( k, a ) ≡ − ΦΨ , Y ( k, a ) ≡ − k Ψ3Ω m δ m . (3) a r X i v : . [ a s t r o - ph . C O ] D ec We see that η corresponds to the gravitational slip, which is linked to the effective anisotropic stress of the dark energy,and Y describes the clustering of the dark energy. The function η is particularly important, as it is a key-variable todistinguish scalar-field type dark energy models from modifications of GR [6, 7].So far these are rather abstract considerations. An obviously important question is whether we can actuallymeasure these quantities with realistic surveys, and to what precision. In [3, 8] we showed that we can use the motionof light and of non-relativistic test-particles like galaxies to map out the metric functions Φ and Ψ in principle, andthat therefore η is an observable quantity. But Y depends on the dark matter distribution, which is not directlyobservable, and so also Y itself is in general not directly observable due to the dark degeneracy [9].In order to reconstruct η from A , R and L it is necessary to remove the dependence on δ t , (notice that ¯ A , ¯ R and¯ L are not observables), since it is an unknown quantity that does not depend on dark energy physics but rather oninflation or other primordial effects. This can be done by considering ratios like P = R/A , P = L/R and P = R /R .In terms of these model-independent ratios, the gravitational slip becomes [3, 8]1 + η = 3 P (1 + z ) E (cid:0) P + 2 + E E (cid:1) (4)where we also set E ( z ) ≡ H ( z ) /H .When constraining η later on, we will use an equivalent quantity which we call ¯ η , defined as¯ η ≡
21 + η = 2ΨΨ − Φ . (5)The reason is that even for large future surveys the expected error on P is substantial, especially when we want toallow for an unknown redshift and scale dependence. The large error makes the division by ( P + 2 + E /E ) in Eq.(4) badly behaved. ¯ η on the other hand is more stable, as we discuss in more detail in appendix A.Based on these results, we will use the Fisher matrix formalism in this paper to forecast the expected precision on¯ A , ¯ R and ¯ L , which are then projected onto the accuracy with which we can obtain P , P and P , and finally on¯ η , based on the expected performance of future large-scale galaxy and weak lensing surveys. We will also include asupernova survey to improve the constraints on the background expansion rate E ( z ), although we find that its impacton the final constraints on η is rather modest. In the final step we will assume four models for η :1. First, we assume that η is constant at all scales and at all redshifts (let us call this case the constant- η case).This occurs for instance in ΛCDM and in all models in which dark energy does not cluster and is decoupledfrom gravity.2. Second, we assume that η is constant in space but varies in redshift ( z -varying case). In other words, we assumethat η has a different arbitrary value for each redshift bin.3. Third, we assume η varies in both redshift and space ( z, k -varying case).4. Fourth, we take for η the quasi-static Horndeski result [3] η = h (cid:18) k h k h (cid:19) . (6)(Here we assume k to be measured in units of 0.1 h/ Mpc, so the h i functions are dimensionless). We denotethis model as the Horndeski case. The Horndeski Lagrangian is the most general Lagrangian for a single scalarfield leading to second-order equations of motion. The expression (6) arises in the quasi-static limit [5] wherethe time-derivative terms are sub-dominant, which implies that the scales of interest are inside the (sound-)horizon.In all cases the fiducial model will be chosen to be ΛCDM, for which η = ¯ η = 1. For the first two cases we need onlya binning in redshift, while for the third and fourth case we will bin both in redshift and in k -space. The fiducialvalues in the first Horndeski case are h = 1, h = h = 0.The outline of the paper is as follows: In sections III, IV and V we set up the Fisher matrix formalism for thegalaxy clustering, weak lensing, and SN-Ia observations. As already mentioned above, we will see that we need tocombine the different probes to obtain constraints on η , and we discuss the combination of the Fisher matrices inSec. VI before concluding in the final section. II. NOTATION AND GENERAL DEFINITIONS
In this section we complete the definition of our notation and provide definitions for quantities that are useful inseveral of the following sections. Our metric signature and the gravitational potentials are already defined in Eq. (2).In Eq. (3) we define the functions η and Y that parameterize the ‘dark energy perturbations’ (as the dark matterdoes not contribute to the anisotropic stress ). The function η assumes a central stage in this paper as it is observablewithout requiring further assumptions, see Eq. (4).Although the observables E , A , R and L can be measured in a fully model-independent way, the precision withwhich we can determine them depends also on the true nature of the Universe. When evaluating our forecasts, wewill use a flat ΛCDM fiducial model, characterized by the WMAP 7-year values, Ω m , h = 0 . b, h = 0 . n s = 0 . τ = 0 . h = 0 .
694 and Ω k = 0. The new WMAP 9-year and Planck results are not very different sothe results are not significantly affected by our choice. The dimensionless background expansion rate in the fiducialmodel and at low redshifts is given by E ( z ) = Ω m , (1 + z ) + (1 − Ω m , ) , (7)and we will often use the dimensionless angular diameter distance ˆ d A ( z ) = ˆ r ( z ) / (1 + z ) and the dimensionlessluminosity distance ˆ d L ( z ) = ˆ r ( z )(1 + z ), where in a flat FLRW Universeˆ r ( z ) = ˆ z d˜ zE (˜ z ) . (8)The usual distances are related to the dimensionless distances through ˆ r = H r and ˆ d = H d . In ΛCDM we havethat η = 1 and Y = 1. In the fiducial model, both G and f only depend on the scale factor, not on k .We will combine in the following the Fisher matrices for future galaxy clustering, weak lensing and supernovaesurveys. More specifically, we will take for galaxy clustering (GC) and weak lensing (WL) a stage IV kind of survey[11] like Euclid [12]. Notice that the survey specifications we use in this paper are meant only to be representativeof a future dark energy survey and do not necessarily reflect the actual Euclid configuration. For supernovae (SN) weassume a survey of 10 sources with magnitude errors similar to the currently achievable uncertainties, as expectedin the LSST survey [13]. III. GALAXY CLUSTERING
The galaxy power spectrum can be written as [14] P ( k, µ ) = ( A + Rµ ) e − k µ σ r = ( ¯ A + ¯ Rµ ) δ ( k ) e − k µ σ r , (9)where σ r = δz/H ( z ), δz being the absolute error on redshift measurement, and we explicitly use δ m,0 = σ δ t,0 , andwhere µ is the cosine of the angle between the line of sight and the wavevector. Notice that ¯ R is often denoted in theliterature as f σ ( z ).As already emphasized, we will ignore in the following the information contained in δ ( k ) since this depends oninitial conditions that are in general not known, and we cannot disentangle the initial conditions from the informationon the dark energy (we refer to [3] for a discussion about this point). Removing the information from the shape ofthe power spectrum of course reduces the amount of information available and so increases the error bars. This is theprice to pay if we want to stay fully model independent.The dependence on E is implicitly contained in µ and k through the Alcock-Paczyński effect [15]. However, we canonly take into account the µ dependence, since the k dependence occurs through the unknown function δ m,0 . TheFisher matrix for the parameter vector p α is in general [14] F GC αβ = 18 π ˆ − d µ ˆ k max k min k V eff D α D β d k , (10) Beyond first order in perturbation theory, the dark matter does in principle contribute to the pressure and anisotropic stress in theUniverse, but the contribution is very small and negligible for our purpose [10]. where D α ≡ d log Pdp α (cid:12)(cid:12)(cid:12)(cid:12) r (11)is the parameter derivative evaluated on the fiducial values (designated by the subscript ‘ r ’) and where V eff = (cid:18) ¯ nP ( k, µ )¯ nP ( k, µ ) + 1 (cid:19) V survey (12)is the effective volume of the survey, with ¯ n the galaxy number density in each bin (discussed later). The Fisher matrixis evaluated at the fiducial model. For this evaluation we will assume that the bias in ΛCDM is scale independentand equal to unity, which implies that the barred variables ¯ A and ¯ R also do not depend on k in the fiducial model(although of course in general they will be scale dependent).Our parameters are therefore p GC α = { ¯ A (¯ z ) , ¯ R (¯ z ) , E (¯ z ) , ¯ A (¯ z ) , ¯ R (¯ z ) , E (¯ z ) , . . . } , where the subscripts run overthe z bins. We could have used A, R directly as parameters as in Eq. (9), but we prefer to clearly distinguish betweenthe dark energy dependent parameters ¯ A, ¯ R and those that depend on different physics. Indices α or β always labelthe parameters in the Fisher matrix framework. From the definition of the galaxy clustering power spectrum, Eq. (9),(and without taking into account the correction from the error on redshift, as we will assume a spectroscopic surveywith negligible redshift errors) we find that D ¯ A = 2¯ A + ¯ Rµ , D ¯ R = 2 µ ¯ A + ¯ Rµ , (13)and using [16, p. 393] µ = Hµ r H r Q , (14)where Q = q E ˆ d A µ r − E r ˆ d Ar ( µ r − E r ˆ d A , (15)we get for the derivative with respect to the parameter ED E = 4 ¯ Rµ (1 − µ )( ¯ A + ¯ Rµ ) E r + 1ˆ d Ar ∂ ˆ d A ∂E ! . (16)Here we explicitly consider the dependence of the dimensionless angular diameter distance ˆ d A on E via Eq. (8). A. z binning We consider an Euclid-like survey [12] from z = 0 . − . z = 0 .
2, and,in order to prevent accidental degeneracies due to low statistics, a single larger redshift bin between z = 1 . − . n B = 6). The lower boundaries of the z -bins are labeled as z a while the center of thebins are labeled as ¯ z a (latin indices a, b, . . . label the z -bins). The galaxy number densities in each bin are shown inTable II; for the bin between 1 . . . × − ( h/ Mpc) [17]. The error on themeasured redshift is assumed to be spectroscopic: δz = 0 . z ). The transfer function in the present matterpower spectrum ( δ ) is calculated using CAMB [18] for the ΛCDM cosmology defined in Sec. II. The limits on theintegration over k are taken as k min = 0 . h/ Mpc (but the results are very weakly dependent on this value) andthe values of k max are chosen to be well below the scale of non-linearity at the redshift of the bin , see Table I. The simplicity of the angular dependence of these expressions and the relative insensitivity of the effective volume, Eq. (12) to µ , meanthat the Fisher matrix (10) leads to a generic prediction for galaxy clustering surveys: The measurements of ¯ A and ¯ R will be slightlyanti-correlated, and galaxy clustering surveys can always measure ¯ A about 3.5 to 4.5 times better than ¯ R . The values of k max are calculated imposing σ ( R ) = 0 .
35, at the corresponding R = π/ k for each redshift, being R the radius ofspherical cells, see [14]. z A Galaxy Clustering (cid:72) n B (cid:61) (cid:76) z R Galaxy Clustering (cid:72) n B (cid:61) (cid:76) z H (cid:144) H Galaxy Clustering (cid:72) n B (cid:61) (cid:76) Figure 1: Errors on ¯ A , ¯ R and E from Galaxy Clustering in the z -binning case. Since the angular diameter distance can be approximated by the expressionˆ d A (¯ z a ) = 1(1 + ¯ z a ) b = a X b =0 ∆ z b E (¯ z b ) , (17)we have for the term ∂ ˆ d A ∂E in equation (16) ∂ ˆ d A (¯ z a ) ∂E (¯ z b ) = − ∆ z b (1 + ¯ z a ) E b δ ab , (18)where δ ab is a Kronecker delta symbol. Then we calculate the Fisher matrix block-wise with independent submatrices F GC αβ for each bin.The errors in the set of parameters p GC α are taken from the square root of the diagonal elements of the invertedFisher matrix, i.e. the errors are marginalized over all other parameters. In Table II we present the fiducial valuesfor ¯ A , ¯ R and E evaluated at the center of the bins (¯ z a ), and the respective errors, and in Fig. 1 we plot their fiducialvalues and errors.If we use a redshift dependent bias b ( z ) (for instance taking the values from the Euclid specifications, see [12, 19]),we get only slight deviations from the errors found for the previous case, as we can see in Table III. Thus, our choiceof a bias equal to unity does not impact the Fisher errors significantly. B. k binning For the third and fourth model we also need a binning in k -space. Since ultimately we would like to obtain errorestimates on three functions, h , h , h , we will need a minimum of three k -bins, which is the choice we make here.We denote with latin indexes a, b, c... the z bins and with indexes i, j, k... the k bins. So for the first z -bin wehave as parameters s = { ¯ A , ¯ R , E } , for the second s = { ¯ A , ¯ R , E } , and so forth, with ¯ A ai = ¯ A (¯ z a , ¯ k i ),¯ R ai = ¯ R (¯ z a , ¯ k i ), and E a = E (¯ z a ), where ¯ k i denote the centers of the k -bins. The set of parameters is therefore p GC α = { s , s , ... } . The Fisher matrix integration over k is split into three k -ranges between k max and k min which wechoose so that ∆ log k = const. The Fisher matrix becomes then F GC αβ = 18 π ˆ − dµ ˆ ∆ k k V eff D α D β d k , (19) ¯ z k min k k k max Table I: Values of k , k and k max for every redshift bin, in units of ( h/ Mpc). with ∆ k denoting the respective range of the integration. Denoting the entry F ¯ A ¯ R as ¯ A ¯ R , and so on, we can representthe structure of the matrix for every redshift bin as follows: ¯ A ¯ A ¯ A ¯ R A E ¯ R ¯ A ¯ R ¯ R R E A ¯ A ¯ A ¯ R A E R ¯ A ¯ R ¯ R R E A ¯ A ¯ A ¯ R ¯ A E R ¯ A ¯ R ¯ R ¯ R EE ¯ A E ¯ R E ¯ A E ¯ R E ¯ A E ¯ R EE , (20)In Table I we display the values for the integration limits at every redshift (the k -bins borders), and in Table IVwe present the errors for all ( z, k )-bins. Notice that the errors on E are not affected by the k -binning, as E does notdepend on k . IV. WEAK LENSING
We move now to estimating the Fisher matrix for a future weak lensing survey. The lensing convergence powerspectrum from a survey divided into several redshift bins (same binning as in Sec. III) can be written as [20] P ij ( ‘ ) = H ˆ ∞ p ij ( z, ‘ )d z ≈ H X a ∆ z a E a K i K j ¯ L δ , (¯ z a , k ( ‘, ¯ z a )) , (21)with the integrand p ij ( z, ‘ ) = K i ( z ) K j ( z ) E ( z ) ¯ L ( z ) δ , ( z, k ( ‘, z )) , (22)where k ( ‘, z ) = ‘πr ( z ) and K i ( z ) = 32 (1 + z ) W i ( z ) , (23)and W i ( z ) is the weak lensing window function for the i -th bin W i ( z ) = H ˆ ∞ z (cid:18) − ˆ r ( z )ˆ r (˜ z ) (cid:19) n i (˜ z ) d˜ z . (24)Here, n i ( z ) equals the galaxy density n ( z ) if z lies inside the i -th redshift bin and zero otherwise. Note that n i ( z )d z = n i ( r ( z )) H ( z ) d r . (25)The overall galaxy density is modeled as n ( z ) ∝ z a exp( − ( z/z p ) b ) . (26) ¯ z ¯ n (¯ z ) × − ¯ A ∆ ¯ A ∆ ¯ A (%) ¯ R ∆ ¯ R ∆ ¯ R (%) E ∆ E ∆ E (%)0.6 3.56 0.612 0.0022 0.37 0.469 0.0092 2.0 1.37 0.12 8.50.8 2.42 0.558 0.0017 0.3 0.457 0.0068 1.5 1.53 0.073 4.81.0 1.81 0.511 0.0015 0.29 0.438 0.0056 1.3 1.72 0.058 3.41.2 1.44 0.47 0.0014 0.29 0.417 0.0049 1.2 1.92 0.05 2.61.4 0.99 0.434 0.0015 0.35 0.396 0.0047 1.2 2.14 0.051 2.41.8 0.33 0.377 0.0018 0.47 0.354 0.0039 1.1 2.62 0.061 2.3 Table II: Fiducial values and errors for ¯ A , ¯ R and E using six redshift bins. Units of galaxy number densities are ( h/ Mpc) . ¯ z ¯ A ∆ ¯ A ∆ ¯ A (%) ¯ R ∆ ¯ R ∆ ¯ R (%) E ∆ E ∆ E (%)0.6 0.645 0.0023 0.36 0.469 0.0094 2. 1.37 0.12 8.80.8 0.628 0.0018 0.28 0.457 0.0072 1.6 1.53 0.078 5.11.0 0.575 0.0015 0.26 0.438 0.0059 1.3 1.72 0.06 3.51.2 0.584 0.0014 0.24 0.417 0.0052 1.2 1.92 0.053 2.71.4 0.561 0.0015 0.27 0.396 0.005 1.3 2.14 0.053 2.51.8 0.561 0.0015 0.26 0.354 0.0038 1.1 2.62 0.056 2.1 Table III: Fiducial values and errors for ¯ A , ¯ R and E using six bins, considering a redshift dependent bias. We take a = 2, b = 3 / z p such that the median of the distribution is at z = 0 .
9, i.e. z p = 0 . / .
412 =0 . n i ( z ) (which are not to be confused with the ¯ n ( z ) from Galaxy Clustering) are then smoothedwith a Gaussian to account for the photometric redshift error (see [21]) and normalized such that ´ n i ( z )d z = 1.Following the Euclid specifications, we set the survey sky fraction f sky = 0 .
375 and the photometric redshift error to δz = 0 . z ).Including the noise due to intrinsic galaxy ellipticities we have C ij = P ij + γ ˆ n − i δ ij , (27)with the intrinsic ellipticity γ int = 0 .
22 and the number of all galaxies per steradian in the i -th bin, ˆ n i , which can bewritten as ˆ n i = n θ ´ z i +1 z i n ( z )d z ´ ∞ n ( z )d z , (28)where n θ is the areal galaxy density, an important parameter that defines the quality of a weak lensing experiment.We set it to n θ = 35 galaxies per square arc minute [12].For a weak lensing survey that covers a fraction of the sky f sky , the Fisher matrix is a sum over ‘ bins of size ∆ ‘ [22] F WL αβ = f sky X ‘ ∆ ‘ (2 ‘ + 1)2 ∂P ij ∂p α C − jm ∂P mn ∂p β C − ni , (29)and now the parameters are p α = { ¯ L (¯ z ) , E (¯ z ) , . . . } . Here, ‘ is being summed from 5 to ‘ max with ∆ log ‘ = 0 . ‘ max corresponds to the value listed in Table V for the redshift bin a or b — whichever is smaller. ¯ z i ¯ A ∆ ¯ A ∆ ¯ A (%) ¯ R ∆ ¯ R ∆ ¯ R (%) E ∆ E ∆ E (%)0.6 1 0.612 0.025 4. 0.469 0.07 15. 1.37 0.11 8.42 0.0058 0.94 0.017 3.63 0.0023 0.38 0.0097 2.10.8 1 0.558 0.018 3.2 0.457 0.05 11 1.53 0.074 4.82 0.0039 0.71 0.012 2.63 0.0018 0.32 0.0074 1.61.0 1 0.511 0.014 2.7 0.438 0.039 8.9 1.72 0.058 3.42 0.003 0.59 0.0089 2.3 0.0016 0.31 0.0062 1.41.2 1 0.47 0.011 2.4 0.417 0.032 7.7 1.92 0.051 2.62 0.0025 0.54 0.0072 1.73 0.0015 0.32 0.0055 1.31.4 1 0.434 0.01 2.3 0.396 0.028 7. 2.14 0.052 2.41 0.0024 0.55 0.0065 1.63 0.0018 0.41 0.0057 1.41.8 1 0.377 0.0063 1.7 0.354 0.015 4.3 2.62 0.059 2.32 0.0022 0.58 0.0047 1.33 0.0024 0.64 0.0061 1.7 Table IV: Relative errors for ¯ A , ¯ R and E at every redshift and every k -bin (labeled with the index i ). Since fiducial values for¯ A , ¯ R and E are independent of k , these are the same for the three k -bins. z p ii (cid:72) z , (cid:123) (cid:61) (cid:76) (cid:64) a u (cid:68) Figure 2: The integrand of Eq. (21). The curves from left to right correspond to p ii ( z, ‘ = 1000), where i = 1 , . . . ,
6. Thecontribution to the lensing signal is very broad in redshift and peaks at relatively low z even for the high-redshift bins. Themedian redshift for each curve is indicated by dashed lines. We give the median redshift for the lensing contribution in Table V. The value ‘ max is derived as follows. We start with the relationship ‘πr ( z med ( ‘, a )) = k, (30)where z med ( ‘, a ) is the median with respect to z of p aa ( z, ‘ ), which is defined in Eq. (22). For a given wave number k and a redshift bin a , we can solve for ‘ . To find ‘ max we use the following method:We begin with z med = 1, compute the k max for this redshift as before by imposing σ ( R ) = 0 .
35, solve Eq. (30) for ‘ , and compute z med ( ‘, a ). We repeat this step until the value for z med converges with an accuracy of approximately1%. A list of the values for ‘ max as well as z med used in each redshift bin can be found in Table V. The integrandsalong with their median value are depicted in Fig. 2.To find the derivatives needed in Eq. (29), we divide the integral in Eq. (21) into n B integrals that each cover oneredshift bin. We could assume that ¯ L ( z ) is constant across any redshift bin to get an approximate expression for theintegral that depends on ¯ L in an analytical way, but the discrepancy between the actual integral and the approximateintegral (and consequently the discrepancy of the derivative) can be up to a factor of 2, which may not be sufficient.Assuming that the integrand is linear in z gives the same result (when using only the center of the bin as the samplingpoint), so the issue arises when the curvature of the integrand becomes large.As a solution, we take the actual value of the integral and simply assume that it depends quadratically on ¯ L (¯ z a ),such that the derivative can be written as ∂P ij ( ‘ ) ∂ ¯ L (¯ z a ) = 2¯ L (¯ z a ) ˆ z a +1 z a p ij ( z, ‘ )d z. (31)Since E appears in the comoving distance, it is more complicated for the derivatives of P ij with respect to E (¯ z a ). Wesubstitute the regular definition of E by an interpolating function that goes smoothly through all points (¯ z a , E (¯ z a ))and (0 , m it now depends on the values of all E (¯ z a ), and so do all functions that dependon E , in particular the comoving distance and consequently the window functions K i ( z ). The derivatives are then ¯ z ‘ max z med ¯ L ∆¯ L ∆¯ L (%) E ∆ E ∆ E (%)0.6 311 0.26 0.342 0.0044 1.3 1.37 0.0062 0.460.8 385 0.31 0.311 0.0044 1.4 1.53 0.0069 0.451.0 515 0.40 0.285 0.0059 2.1 1.72 0.017 0.961.2 609 0.45 0.262 0.0059 2.3 1.92 0.029 1.51.4 760 0.54 0.242 0.014 5.7 2.14 0.029 1.41.8 959 0.64 0.210 0.035 16 2.62 0.077 3.0 Table V: Errors on E and ¯ L from weak lensing only (with six redshift bins) and a list of the value ‘ max used at each redshifttogether with the corresponding z med value. z H (cid:144) H Weak Lensing (cid:72) n B (cid:61) (cid:76) z L Weak Lensing (cid:72) n B (cid:61) (cid:76) Figure 3: Errors on E (¯ z a ) (left) and ¯ L (¯ z a ) (right) from weak lensing. obtained by varying the fiducial values of E (¯ z a ) while keeping L = ¯ Lδ t,0 fixed so that we again do not include thederivative of δ with respect to k .It is instructive to consider the error on the spectrum itself for a particular pair ij . If we take as parameters p α = P ij we have a variance σ − = f sky X ‘ ∆ ‘ (2 ‘ + 1)2 C − ij C − ij , (32)(no sum over ij ) and neglecting the noise (appropriate for ‘ < C ij = P ij , this becomes, in a smallrange of ‘ from ‘ min to ‘ max so that we can approximate P ij with a constant, σ − P ij P ij = f sky X ‘ ∆ ‘ (2 ‘ + 1)2 = f sky ‘ − ‘ , (33)(for ‘ max,min (cid:29) ‘ min is much smaller than ‘ max this gives a relative error for every ijσP ij = ‘ − (cid:18) f sky (cid:19) − / ≈ . ‘ − , (34)so that for ‘ max = 300 we should get a minimum relative error of 0 . ‘ min .The resulting uncertainties on E (¯ z a ) and ¯ L (¯ z a ) can be found in Table V; they are visualized in Fig. 3. A. k binning To test the cases three and four of our models for η , we need to consider ¯ L as a function of k (although with thesame fiducial value for all k , as the fiducial model is ΛCDM), and we divide the full k -range again into the same threebins. The observables are then ¯ L an ≡ ¯ L (¯ z a , ¯ k n ), where ¯ k n denote the center of the k -bins, with n = 1 , ,
3. They ¯ z ‘ ‘ ‘ ‘ Table VI: Borders of the ‘ -bins for each redshift bin converted from the k -bins according to Eq. (30). ¯ z ¯ L a ∆¯ L a ∆¯ L a (%) ¯ L a ∆¯ L a ∆¯ L a (%) ¯ L a ∆¯ L a ∆¯ L a (%) E a ∆ E a ∆ E a (%)0.6 0.342 0.025 7.4 0.342 0.0076 2.2 0.342 0.0050 1.5 1.37 0.0069 0.510.8 0.311 0.025 7.9 0.311 0.0064 2.1 0.311 0.0053 1.7 1.53 0.0074 0.481.0 0.285 0.022 7.8 0.285 0.0074 2.6 0.285 0.0062 2.2 1.72 0.017 0.971.2 0.262 0.024 9.1 0.262 0.0080 3.0 0.262 0.0073 2.8 1.92 0.030 1.61.4 0.242 0.041 17. 0.242 0.019 7.7 0.242 0.015 6.1 2.14 0.030 1.41.8 0.210 0.098 46. 0.210 0.048 23 0.210 0.037 17 2.62 0.079 3.0 Table VII: Errors of ¯ L ai and E using weak lensing only with their fiducial values. are defined as in Sec. III, and are given explicitly in Table I. The k -bins fix the ranges for ‘ via the relation used inEq. (30). We label the center of the ‘ -bins accordingly as ‘ n . See Table VI for a list of the ‘ -bins. The derivativesneeded for the Fisher matrix will be evaluated at the center of these ‘ -bins.They can be computed similarly as in Eq. (31). We find (using Kronecker deltas, no summation): ∂P ij ( ‘ ) ∂ ¯ L (¯ z a , k n ) = 2 δ an ¯ L ( z a ) ˆ z a +1 z a p ij ( z, ‘ )d z × ( ‘ n − < ‘ < ‘ n . (35)The derivatives with respect to E (¯ z a ) are computed the same way as before. We can then define the parameter vector p α = { ¯ L , E , ¯ L , E , ¯ L , E , ¯ L , E , ... } and evaluate the Fisher matrix formally as before. The structure of theFisher matrix can be schematically represented as follows: ¯ L ¯ L L E L ¯ L L E L ¯ L ¯ L E ¯ L E ¯ L E ¯ L E EE (36)The uncertainties placed on the observables by weak lensing only can be found in Table VII.
V. SUPERNOVAE
We consider now the forecasts for a supernovae survey. The likelihood function for the supernovae after marginal-ization of the offset is [16] L = − log L = 12 (cid:18) S − S S (cid:19) , (37)where S n = X i ( m i − µ i ) n σ i , (38)and µ i = 5 log ˆ d L , where ˆ d L is the dimensionless luminosity distance, see Eq. (8). This can be written as L = 12 X i M ij X j , (39) ¯ z σ data ,a n a E (¯ z ) ∆ E ∆ E (%)0.6 0.287 46429 1.37 0.0026 0.190.8 0.285 25000 1.53 0.0041 0.271.0 0.329 16071 1.72 0.0086 0.501.2 0.327 7143 1.92 0.016 0.831.4 0.258 5357 2.14 0.028 1.3 Table VIII: Redshift uncertainties, number of supernovae, fiducial value of E and errors for each bin. z H (cid:144) H Supernovae (cid:72) n B (cid:61) (cid:76) Figure 4: Errors on E from Supernovae. where X i = m i − µ i and M ij = s i s j δ ij − s i s j S , (40)(no sum) where s i = 1 /σ i . The Fisher matrix can be written as F SN αβ = (cid:28) ∂ L ∂p α ∂ L ∂p β (cid:29) , (41)where now the parameters are p SN α a = E (¯ z a ). Similarly to section III we can writeˆ d L (¯ z a ) = (1 + ¯ z a ) b = a X b =0 ∆ z b E (¯ z b ) , (42)so that ∂ ˆ d L ( z a ) ∂E (¯ z b ) = − ∆ z b E b (1 + z a ) δ ab (43)where δ ab is a Kronecker symbol. The Fisher matrix for the parameter vector p α = { E ( z a ) } with a running over the z -bins is then F SN αβ = (cid:28)(cid:18) ∂µ i ∂p α a M ij X j (cid:19) (cid:18) ∂µ i ∂p β b M ij X j (cid:19)(cid:29) = 25 Y iα M ij Y jβ . (44)where Y iα ≡ ∂ log ˆ d L (¯ z i ) ∂p α = 1ˆ d L (¯ z i ) ∂ ˆ d L (¯ z i ) ∂E (¯ z α ) = − d L ( z i ) ∆¯ z α E α (1 + ¯ z i ) δ iα . (45)We have to make a choice to define the redshifts z i and the uncertainties σ i for the supernovae of the simulatedfuture experiment. We take the Union 2.1 catalog as a reference (580 SNIa in the range 0 < z (cid:46) . . < z < .
5, and divide that interval in bins of fixedwidth ∆ z = 0 . n SN = 100000 in that range, as expectedfor the LSST survey [13]. We further assume that the supernovae of the future survey will be distributed uniformlyin each bin, respecting the proportions of the data of the catalog Union 2.1 and with the same average magnitudeerror. The values of σ data ,a and n a for the bins centered in ¯ z a are summarized in Table VIII.Finally, the corresponding errors on E from supernovae are shown in Fig. 4 and in Table VIII. In Table XI wecompare the errors on E from the three different probes with each other. We notice that the supernova constraintsare the most stringent ones among the three probes and improve the WL+GC constraints by almost a factor of two.All this of course assumes that systematic errors can be kept below statistical errors.2 z P n B (cid:61) z P n B (cid:61) (cid:45) z P n B (cid:61) Figure 5: Errors on P , P and P in the z -varying case. ¯ z P ∆ P ∆ P (%) P ∆ P ∆ P (%) P ∆ P ∆ P (%) ( E /E ) ∆ E /E ∆ E /E (%) ¯ η ∆¯ η ∆¯ η (%)0.6 0.766 0.012 1.6 0.729 0.013 1.8 0.134 0.13 99 -0.920 0.022 2.4 1 0.11 110.8 0.819 0.010 1.2 0.682 0.011 1.6 0.317 0.12 38 -1.04 0.046 4.4 1 0.091 9.11.0 0.859 0.0093 1.1 0.650 0.011 1.7 0.460 0.12 26 -1.13 0.099 8.7 1 0.090 9.01.2 0.888 0.0092 1.0 0.628 0.014 2.3 0.569 0.13 23 -1.21 0.12 10 1 0.097 9.71.4 0.911 0.010 1.1 0.613 0.020 3.3 0.654 0.11 16 -1.26 0.09 7.1 1 0.073 7.3 Table IX: Fiducial values and errors for the parameters P , P , P , E /E and ¯ η for every bin. The last bin has been omittedsince R is not defined there. VI. COMBINING THE MATRICES
Once we have the three Fisher matrices for galaxy clustering, weak lensing and supernovae, we insert them block-wise into a (4 n B ) × (4 n B ) matrix for the full parameter vector p α = { ¯ A, ¯ R, ¯ L, E } × n B , (46)Notice that we need also ¯ R = − (1 + z )[ ¯ R ( z + ∆ z ) − ¯ R ( z )] / ∆ z and E = − (1 + z )[ E ( z + ∆ z ) − E ( z )] / ∆ z . The fullschematic structure for every bin will be: ¯ A ¯ A ¯ A ¯ R AE ¯ A ¯ R ¯ R ¯ R RE L ¯ L ¯ LE ¯ AE ¯ RE ¯ LE ( EE ) Σ , (47)with ( EE ) Σ = ( EE ) GC + ( EE ) WL + ( EE ) SN . This matrix must then be projected onto ¯ η . It is however interestingto produce two intermediate steps, namely the matrix for q α = { P , P , P , E } where P = R/A , P = L/R and P = R /R , as well as the matrix for q α = { P , P , P , E /E } . They are given by F ( q ) αβ = F ( p ) γδ ∂p γ ∂q α ∂p δ ∂q β . (48)We then project onto { P , P , ¯ η, E } . In Table IX we present the fiducial values for the parameters P , P , P , definedin Sec. I; In Fig. 5 we plot their fiducial values and errors. Let us call this the basic Fisher matrix.As we mentioned in the introduction, we decided to consider four models for ¯ η : constant, variable only in redshift,variable both in space and redshift, and the Horndeski model. For the constant ¯ η case we project the basic FisherMatrix for P , P , ¯ η, E onto a single constant value for ¯ η . The resulting uncertainty for ¯ η is 0.010. ¯ z P ∆ P ∆ P (%) P ∆ P ∆ P (%) P ∆ P ∆ P (%) ( E /E ) ∆ E /E ∆ E /E (%) ¯ η ∆¯ η ∆¯ η (%)0.7 0.794 0.0079 0.99 0.703 0.0074 1.0 0.231 0.042 18 -0.983 0.023 2.3 1 0.031 3.11.1 0.875 0.0067 0.77 0.638 0.0072 1.1 0.518 0.050 9.7 -1.17 0.044 3.7 1 0.037 3.71.5 0.920 0.0099 1.1 0.607 0.010 1.7 0.688 0.048 7.0 -1.29 0.060 4.6 1 0.032 3.2 Table X: Same as Table IX, but with four redshift bins. The last bin has again been omitted. k (cid:64) h (cid:144) Mpc (cid:68) Η Figure 6: Constraints on η ( k ) in the Horndeski case for z = 0 . z = 1 . For the z -variable case we project on five ¯ η parameters, one for each bin. The results are in Table IX. We see thatthe error on ¯ η rises to around 10%. Without the SN data, the final constraints on η would weaken only by roughly1%. If we collect the data into only three wider z bins, the error reduces to about 3%.For the z, k varying case, we consider the k -binning of Sec. III B. Now the information is distributed over manymore bins, so the errors obviously degrade (see Table XII). We find errors from 10% to more than 100%.Finally, for the Horndeski case, Table XIII gives the absolute errors on h , h (measuring k in units of 0.1 h/ Mpc).Here we are forced to fix h to its fiducial value (i.e. to zero) due to the degeneracy between h and h when thefiducial model is such that h = h , as in the ΛCDM case. This means we are only able to measure the difference h − h rather than the two functions separately. The absolute errors on h , h are in the range 0.2-0.6. This resultimplies for instance that, at a scale of 0.1 h/ Mpc and in a redshift bin 0.5-0.7, a Euclid-like mission can detect thepresence of a k behavior in η if it is larger than 60% than the k -independent trend (see Fig. 6 for a visualization ofthe constraints on η ). VII. CONCLUSIONS
In this paper we study the precision with which a future large survey of galaxy clustering and weak lensing likeEuclid can determine the anisotropic stress of the dark sector with the help of the model-independent cosmologicalobservables introduced in [3], when augmented with a supernova survey.We find that galaxy clustering and weak lensing will achieve precise measurements of the expansion rate E ( z ) = H ( z ) /H , with errors of less than a percent in redshift bins of ∆ z = 0 . z = 1 .
5, and with less than 4% out to z = 2, see Table XI.They will also be able to measure P = f /b to about a percent precision over the full redshift range (in the samebins), and achieve a comparable precision on P = Ω m , Σ /f , except at z > . P = f + f /f , is constrained much less precisely, only to about 30%, because it involves an explicitderivative. The detailed results are given in Tables IX and X.We then considered four different models for η = − Φ / Ψ: WL GC SN WL+GC WL+GC+SN¯ z E ∆ E ∆ E (%) ∆ E ∆ E (%) ∆ E ∆ E (%) ∆ E ∆ E (%) ∆ E ∆ E (%)0.6 1.37 0.0062 0.46 0.12 8.5 0.0026 0.19 0.0062 0.45 0.0023 0.160.8 1.53 0.0069 0.45 0.073 4.8 0.0041 0.27 0.0068 0.44 0.0029 0.191.0 1.72 0.017 0.96 0.058 3.4 0.0086 0.50 0.016 0.91 0.0067 0.391.2 1.92 0.029 1.5 0.050 2.6 0.016 0.83 0.024 1.2 0.012 0.651.4 2.14 0.029 1.4 0.051 2.4 0.028 1.3 0.022 1.0 0.017 0.781.8 2.62 0.077 3.0 0.061 2.3 - - 0.046 1.8 0.043 1.7 Table XI: Errors on E from the three probes. ¯ z i P ∆ P ∆ P (%) P ∆ P ∆ P (%) P ∆ P ∆ P (%) ¯ η ∆¯ η ∆¯ η (%)0.6 1 0.766 0.14 18 0.729 0.12 17 0.134 1.4 1100 1 1.1 1202 0.032 4.1 0.030 4.1 0.33 240 0.26 263 0.013 1.7 0.015 2.0 0.15 110 0.12 120.8 1 0.819 0.11 13 0.682 0.092 13 0.317 1.2 380 1 0.93 932 0.024 2.9 0.021 3.1 0.26 83 0.2 203 0.011 1.4 0.013 1.9 0.14 43 0.1 101.0 1 0.859 0.093 11 0.65 0.076 12 0.46 1.1 240 1 0.82 822 0.020 2.3 0.019 2.9 0.23 51 0.17 173 0.011 1.2 0.012 1.8 0.14 31 0.1 111.2 1 0.888 0.084 9.4 0.628 0.074 12 0.569 1.1 190 1 0.78 782 0.017 2.0 0.021 3.3 0.23 40 0.16 163 0.011 1.2 0.017 2.7 0.17 29 0.12 121.4 1 0.911 0.079 8.7 0.613 0.084 14 0.654 0.79 120 1 0.55 552 0.017 1.9 0.027 4.4 0.17 26 0.12 123 0.013 1.4 0.023 3.8 0.14 21 0.094 9.4 Table XII: Here, the errors on P , P , P and η are listed for the z, k -varying case with a similar structure as Table IV.
1. A constant η : In this case we find that we can determine the derived quantity ¯ η with a precision of about 1%.2. η varying with redshift, but not with scale: For bins with a size of ∆ z = 0 .
2, we find a precision on ¯ η of about10% out to z = 1 . η varying both in z and in k : the errors vary considerably across the z, k range, from 10% to more than 100%.4. The Horndeski case: now the absolute errors on h , h are in the range 0.2-0.6We stress again that in this paper we used only directly observable quantities without any further assumptions aboutthe initial power spectrum, the dark matter, the dark energy model (beyond the behaviour of η in the last step) orthe bias, as such assumptions may be unwarranted in a general dark energy or modified gravity context. On the otherhand, we do assume that a window between non-linear scales and sub-sound-horizon scales exists and is wide enoughto cover all the wavelengths we have been employing in our forecasts. Acknowledgments
We thank M. Motta, I. Saltas and I. Sawicki for useful discussions. L.A., A.G. and A.V. acknowledge supportfrom DGF through the project TRR33 “The Dark Universe”. A.G. also acknowledges financial support from DAADthrough program “Forschungsstipendium für Doktoranden und Nachwuchswissenschaftler”. M.K. acknowledges finan-cial support from the Swiss NSF.
Appendix A: Sampling vs Fisher matrix analysis
In order to check whether the Fisher matrix analysis is appropriate for the non-linear parameter combinations thatmake up the P i and η we also use an alternative approach. We assume that the Fisher matrix forecast for the errorson ¯ A , ¯ R , ¯ L and E is sufficiently accurate (i.e. that the joint posterior of these variables can be described by a Gaussianprobability distribution function with the covariance matrix given by the inverse of the Fisher matrix), which should ¯ z ∆ h ∆ h Table XIII: Absolute errors on h and h . Because of the degeneracy between h and h , h has been fixed. The fiducial valuesare h = 1 and h = 0. . . . . . . . . . . . . . . η /η = (1 + η ) / Fisher matrix − − . . . . . . η /η = (1 + η ) / Fisher matrix
Figure 7: The probability density function (pdf) for (1 + η ) / η (red solid line) based on sampling fromthe Fisher matrix for { ¯ A, ¯ R, ¯ L, E } , compared to the Gaussian pdf from the Fisher matrix projection on ¯ η (black dotted line) inthe z - and k -binning case. The left panel shows the second k -bin for ¯ z = 1, and the right panel the first k -bin for ¯ z = 1 .
2. Weuse (1 + η ) / η because it has the same pdf shape as η and (to lowest order) the same variance as ¯ η . We see thateven when the standard deviation of ¯ η is well below 1 as in the left panel, the pdf of η is significantly less Gaussian than thepdf of ¯ η . For large standard deviation (right panel) the pdf of ¯ η is still well behaved and close to Gaussian, while the one of η is strongly distorted and exhibits large tails (not shown in the figure) due to a division by zero problem in the expression (4). be a reasonable assumption given how precise the surveys that we consider here are. We then draw random samplesfrom the multivariate Gaussian distribution defined by those Fisher matrices.For each sample we compute P , P and P at the corresponding values of z and k . We compute the derivativesof E and ¯ R by fitting a cubic spline through each realisation of E ( z ) and ¯ R ( z ) and calculating the derivative of thespline. This procedure allows us to obtain estimates of the derivatives in all bins, but at the price of having to chooseboundary conditions for the splines (we use the “natural spline” convention that the second derivative vanishes at theboundary).Overall we find good agreement, and even excellent agreement when using the derivative at the points in betweenthe bins (which agrees better with the finite difference method used for the Fisher forecasts). The agreement becomesmuch worse for η , as already mentioned in the introduction. This is however no surprise, as the posterior distributionof η becomes very non-Gaussian for the survey specifications considered here (while the posterior distributions of the P i remain close to Gaussian). We observe however that ¯ η retains a normal posterior, which makes it much bettersuited for the Fisher forecast approach, see Fig. 7. The same holds true for Markov-Chain Monte Carlo approacheswhich tend to have difficulties with sampling from curved, “banana-shaped” posteriors, and so we recommend quitegenerally to use ¯ η rather than η in data analysis. We finally note that when η is well-constrained and has a pdf closeto Gaussian, then its standard deviation should be about twice that of ¯ η . [1] Planck Collaboration
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