Measuring anisotropic stress with relativistic effects
aa r X i v : . [ a s t r o - ph . C O ] F e b Measuring anisotropic stress with relativistic effects
Daniel Sobral Blanco and Camille Bonvin
D´epartment de Physique Th´eorique and Center for Astroparticle Physics,Universit´e de Gen`eve, Quai E. Ansermet 24, CH-1211 Gen`eve 4, Switzerland (Dated: February 11, 2021)One of the main goal of large-scale structure surveys is to test the consistency of General Relativityat cosmological scales. In the ΛCDM model of cosmology, the relations between the fields describingthe geometry and the content of our Universe are uniquely determined. In particular, the twogravitational potentials –that describe the spatial and temporal fluctuations in the geometry– areequal. Whereas large classes of dark energy models preserve this equality, theories of modifiedgravity generally create a difference between the potentials, known as anisotropic stress. Eventhough measuring this anisotropic stress is one of the key goals of large-scale structure surveys, thereare currently no methods able to measure it directly. Current methods all rely on measurementsof galaxy peculiar velocities (through redshift-space distortions), from which the time componentof the metric is inferred, assuming that dark matter follows geodesics. If this is not the case, allthe proposed tests fail to measure the anisotropic stress. In this letter, we propose a novel testwhich directly measures anisotropic stress, without relying on any assumption about the unknowndark matter. Our method uses relativistic effects in the galaxy number counts to provide a directmeasurement of the time component of the metric. By comparing this with lensing observations ourtest provides a direct measurement of the anisotropic stress.
Introduction.
Testing the law of gravity at cosmologi-cal scales is one of the main science driver for the com-ing generation of large-scale structure surveys. At largescales and late time, our Universe can be consistently de-scribed by four fields: the two metric potentials Φ and Ψdescribing fluctuations in the geometry of the Universe ,the matter density fluctuation, δ , and the galaxy pecu-liar velocity, V . Testing the law of gravity requires thento test the relations between these four fields. Two ap-proaches can be used. The first one consists in assuminga specific model or class of models of gravity (e.g. Horn-deski models [1]), determine how the four fields dependon the parameters of the model, and use observations(which depend on the four fields) to constrain the pa-rameters. This approach has the obvious disadvantagethat it has to be performed separately for each model orclass of models.The second approach consists in building model-independent tests, that allow to probe directly the re-lations between the four fields Φ , Ψ , δ and V withoutassuming any model, see e.g. [2–10]. The outcome ofthese tests can then be used to determine the validity ofany theory of gravity. This second approach, which ismore powerful, is however suffering from an importantlimitation: the fact that our observables at late time aresensitive to only three combinations of the four fields,namely δ and V (through redshift-space distortions, seee.g. [11, 12]) and Φ + Ψ (through cosmic shear [13, 14],CMB lensing [15–18] or Integrated Sachs Wolfe [19]).This means that current observations are not able to testall relations between the four fields. The standard way of We use the metric convention ds = a [ − (1 + 2Ψ) dτ + (1 − d x ] where τ denotes conformal time. overcoming this problem is to assume that some of the re-lations between the four fields are known. Typically, oneusually assumes that the continuity equation for darkmatter holds: there is no exchange of energy betweendark matter and dark energy; and that Euler equation fordark matter holds: there is no fifth force acting on darkmatter, which consequently follows geodesics. Underthese conditions, a measurement of V can be translatedinto a measurement of Ψ, which can then be comparedto Φ+ Ψ to test if the two metric potentials are the same,i.e. to test for the presence of anisotropic stress [3, 20–22].This is a key test for modified theories of gravity since inGeneral Relativity (GR) and for large classes of dark en-ergy models, Φ = Ψ at late time, whereas very generallyin modified theories of gravity Φ = Ψ, see e.g. [23].In this letter we propose a novel model-independenttest for the anisotropic stress, which does not rely onany assumption for dark matter, i.e. which does not relyon the validity of the continuity or Euler equation. Tobuild this test we use the fact that galaxy number countsare affected by gravitational redshift, a relativistic effectthat is directly proportional to the field Ψ [24–27]. Wedevelop a method to isolate Ψ from galaxy number countsobservations. We then show how this can be comparedwith lensing observations, which are sensitive to Φ + Ψ,to directly measure the anisotropic stress. We finallycompare our method with the one presented in [3], whichassumes the validity of the continuity and Euler equationfor dark matter, and we show how this method breaksdown if these assumptions are not valid. Methodology.
Redshift surveys map the distributionof galaxies in the sky, providing a measurement of thegalaxy number counts fluctuations∆ ≡ N ( n , z ) − ¯ N ( z )¯ N ( z ) , (1)where N denotes the number of galaxies per pixel de-tected in direction n and at redshift z , and ¯ N is the av-erage number of galaxies per pixel at redshift z . At linearorder in perturbation theory, the dominant contributionsto ∆ are [24–27]∆( n , z ) = b δ − H ∂ r ( V · n ) (2)+ (5 s − Z r dr ′ ( r − r ′ ) r ′ r ∆ ⊥ (Φ + Ψ)( n , r ′ )+ − s + 5 s − r H − ˙ HH + f evol ! V · n + 1 H ˙ V · n + 1 H ∂ r Ψ , where H denotes the Hubble parameter in conformal time τ , r = r ( z ) is the comoving distance to redshift z , a dotdenotes derivative with respect to conformal time and∆ ⊥ is the Laplacian transverse to the photon direction n . The functions b ( z ), s ( z ) and f evol ( z ) are the galaxybias, the magnification bias and the evolution bias re-spectively. These functions depend on the population ofgalaxy which is observed as well as on the specificationsof the survey.The first contribution in Eq. (2), δ , is the mat-ter density fluctuation in comoving gauge. The sec-ond term, which depends on the galaxy peculiar veloc-ity, V , is the contribution from redshift-space distortion(RSD) [28, 29]. The second line contains the effect oflensing magnification [30, 31]. This contribution is sub-dominant with respect to density and RSD, except athigh redshift [32, 33]. From [10] we expect this term to benegligible for our test, at least below z = 1 .
5. In a forth-coming paper we will study this in more detail for specificsurveys. The last 2 lines in Eq. (2) contain the so-calledrelativistic effects, that depend on the galaxy peculiarvelocity, through Doppler effects, and on the metric po-tential Ψ, through gravitational redshift. These relativis-tic effects have the specificity to generate odd multipolesin the power spectrum and correlation function [34–37].As such they can be isolated from the dominant densityand RSD contributions, which generate even multipoles.In addition to these terms, ∆ contains other relativis-tic effects that contribute to the even multipoles. Theseterms are however suppressed by ( H /k ) with respect todensity and RSD and can therefore be safely neglected.The aim of our work is to isolate the contribution fromgravitational redshift given by the last term in Eq. (2), ∂ r Ψ / H , since it is directly proportional to the time com-ponent of the metric Ψ. The optimal way of targetingthis contribution is to cross-correlate two populations of galaxies with different luminosities, such that this termcontributes to odd multipoles [34–37]. In Fourier space,the galaxy number counts fluctuations for a populationof galaxies with luminosity L becomes (we use the con-vention f ( k , τ ) = R d x e i kx f ( x , τ ))∆ L ( k , z ) = b L δ ( k , z ) − k H (ˆ k · n ) V ( k , z ) (3)+ i (ˆ k · n ) (cid:20) α L V ( k , z ) + 1 H ˙ V ( k , z ) − k H Ψ( k , z ) (cid:21) , where α L ≡ − s L + 5 s L − r H − ˙ HH + f evolL , (4)and the velocity potential, V , is defined through V ( k , z ) = i ˆ k V ( k , z ). Eq. (3) is valid only in the flat-sky approximation, where n can be considered as fixed.We will study in a future work the validity of this ap-proximation for our test.The correlations between two populations of galaxieswith luminosity L and M are given by h ∆ L ( k , z )∆ M ( k ′ , z ) i = (2 π ) P LM ( k, µ, z ) δ D ( k + k ′ ) , where P LM = b L b M P δδ −
13 ( b L + b M ) k H P δV + 15 (cid:18) k H (cid:19) P V V + " −
23 ( b L + b M ) k H P δV + 47 (cid:18) k H (cid:19) P V V L ( µ )+ 835 (cid:18) k H (cid:19) P V V L ( µ ) + " ( b M α L − b L α M ) P δV (5)+ ( b M − b L ) 1 H P δ ˙ V + 35 ( α M − α L ) k H P V V iL ( µ )+ 25 ( α M − α L ) k H P V V iL ( µ ) + ( b L − b M ) k H P δψ iL ( µ ) . Here L ℓ denotes the Legendre polynomial of degree ℓ , theangle µ = ˆ k · n and the power spectra are defined through h X ( k , z ) Y ( k ′ , z ) i = (2 π ) P XY ( k, z ) δ D ( k + k ′ ) , (6)for X, Y = δ, V, ˙ V ,
Ψ.Our aim is to isolate the last term in Eq. (5), which isproportional to P δψ . The anisotropic stress can then bedirectly measured by dividing this contribution with theso-called galaxy-galaxy lensing correlation, which is pro-portional to P δ (Φ+Ψ) [38]. To isolate P δψ , we first extractthe dipole of P LM which is proportional to P δψ , P δV , P δ ˙ V and P V V . We then look for combinations of the othermultipoles of P LM in order to cancel the P δV , P δ ˙ V and P V V contributions.The multipole ℓ of P LM can be extracted by weightingit with the Legendre polynomial of degree ℓ and integrat-ing over µP ( ℓ )LM ( k, z ) = 2 ℓ + 12 Z − dµL ℓ ( µ ) P LM ( k, µ, z ) . (7)For our test, we split the galaxies into a bright, B, andfaint, F, populations. We then measure the monopoleand quadrupole of these two populations, the hexade-capole of the whole population and the dipole and oc-tupole of the cross-correlation between bright and faint: P (0)L = b P δδ − b L k H P δV + 15 (cid:18) k H (cid:19) P V V , (8) P (2)L = − b L k H P δV + 47 (cid:18) k H (cid:19) P V V , (9) P (4) = 835 (cid:18) k H (cid:19) P V V , (10) P (1)BF = − i " ( b B α F − b F α B ) P δV + ( b B − b F ) 1 H P δ ˙ V (11) −
35 ( α F − α B ) k H P V V + i ( b B − b F ) k H P δψ ,P (3)BF = i
25 ( α F − α B ) k H P V V , (12)with L=B, F. From these observed multipoles we con-struct the following observables: O δδ L ( k, z ) ≡ P (0)L − P (2)L + 38 P (4)L = b P δδ ( k, z ) , (13) O δV L ( k, z ) ≡ P (2)L − P (4) = − b L k H P δV ( k, z ) , (14) O δ ˙ V L ( k, z ) ≡ − (1 + z ) q O δδ L ( k, z ) ddz O δV L ( k, z ) q O δδ L ( k, z ) = b L k H " ˙ HH P δV ( k, z ) − H P δ ˙ V ( k, z ) . (15)To obtain the last line in Eq. (15) we use the fact that,in any theory of gravity, the density and velocity fieldscan be written as δ ( k , z ) = D ( k, z ) δ ( k ,
0) and V ( k , z ) = G ( k, z ) δ ( k , δ ( k ,
0) is a constant and denotes thepresent dark matter density, while D ( k, z ) and G ( k, z )are functions of k and z mapping δ ( k ,
0) into the past.With this we can easily verify that p P δδ ( k, z ) ddτ P δV ( k, z ) p P δδ ( k, z ) ! = D ( k, z ) ˙ G ( k, z ) P δδ ( k, P δ ˙ V ( k, z ) , (16)which gives rise to expression (15). We are now able to isolate P δψ with the following combination O δ Ψ ( k, z ) ≡ i H k (cid:20) P (3)BF − P (1)BF (cid:21) − (cid:18) H k (cid:19) h O δ ˙ V B ( k, z ) − O δ ˙ V F ( k, z ) i − (cid:18) H k (cid:19) (cid:20) − r H − s F (cid:18) − r H (cid:19) + f evolF (cid:21) O δV B ( k, z )+ (cid:18) H k (cid:19) (cid:20) − r H − s B (cid:18) − r H (cid:19) + f evolB (cid:21) O δV F ( k, z )= ( b B − b F ) P δψ ( k, z ) . (17)We see that O δ Ψ ( k, z ) can be measured from the galaxynumber counts, without making any assumption on thetheory of gravity. It depends indeed on • The multipoles of the power spectrum, which areobservable. • The background quantities H /k and r H . Thesetwo combinations can be inferred from backgroundobservations. For example, observations of typeIa supernovae provide a measurement of the lumi-nosity distance, up to a multiplicative constant :ˆ d L ≡ d L H , from which one can infer the ratio H ( z ) / H . We then have r H = ˆ d L z HH , and H k = HH k , (18)where ˆ k ≡ k/ H is independent of h for k in unitsMpc − h . • The magnification bias, s , and evolution bias, f evol ,of the bright and faint populations. These quanti-ties can be directly measured from the two popula-tions of galaxies. The magnification bias requires ameasurement of the number of galaxies as a func-tion of luminosity [30], whereas the evolution biasrequires a measurement of the number of galaxiesas a function of redshift [26, 39].The observable O δ Ψ is, on its own, a very interestingquantity since it probes directly the correlations betweendensity and gravitational potential Ψ. It provides there-fore a way of measuring these correlations at cosmologicalscales, for the first time.To extract the anisotropic stress from O δ Ψ , we needin addition a measurement of P δ (Φ+Ψ) . This can be ob-tained by correlating gravitational lensing with galaxy This is due to the fact that the absolute intrinsic luminosity of su-pernovae is unknown, so that only ratios of luminosity distancesat different redshifts are independent of normalisation. number counts, called galaxy-galaxy lensing [38]. Obser-vations of galaxy shapes provide a measurement of theconvergence field κκ ( n , z ) = Z r ( z )0 ds ( r − s ) s r ∆ ⊥ (Φ + Ψ)( n , s ) . (19)The Fourier transform of κ ( n , τ ) cannot be calculatedin a straightforward way. It contains indeed an integralof κ ( n , τ ) on a hypersurface of constant time. However,the integral in Eq. (19) is only meaningful on the pastlight-cone of the observer. Therefore, we first define thecorrelation function in configuration space, and then ex-tract the power spectrum from this well-defined quantity.The observable O δ Ψ depends on the bias difference be-tween the bright and faint populations, b B − b F . In or-der to cancel this dependence we consider the followinggalaxy-galaxy lensing correlation ξ BF∆ κ ≡ h ∆ B ( n , z ) κ ( n ′ , z ′ ) i − h ∆ F ( n , z ) κ ( n ′ , z ′ ) i (20)= ( b B − b F ) Z r ′ ds ( r ′ − s ) s r ′ h δ ( n , z )∆ ⊥ (Φ + Ψ)( n ′ , s ) i , where r ′ ≡ r ( z ′ ). Fourier transforming δ and Φ + Ψ andusing Limber approximation [40, 41], we obtain ξ BF∆ κ = − ( b B − b F ) ( r ′ − r ) r r ′ Θ( r ′ − r ) (21) × π Z ∞ dk ⊥ k ⊥ P δ (Φ+Ψ) ( k ⊥ , z ) J ( k ⊥ ∆ x ⊥ ) . Here J is the Bessel function of order zero, k ⊥ = | k ⊥ | isthe amplitude of the wave-number contribution perpen-dicular to the line-of-sight, ∆ x ⊥ = | ∆ x ⊥ | denotes theamplitude of the vector joining the pixel in which ∆ L ismeasured and the pixel in which κ is measured, projectedin the plane orthogonal to the line-of-sight, and Θ is theHeaviside function, accounting for the fact that the cor-relation between ∆ L and κ is non-zero only if κ is behind∆ L . Note that h ∆ L κ i contains also a lensing-lensing con-tribution, due to the second line in Eq. (2) [42]. Howeverthis contribution does not depend on the galaxy popula-tion and vanishes therefore in ξ BF∆ κ . Moreover, the corre-lation between κ and the velocity contributions in Eq. (2)exactly vanishes in the Limber approximation.Since ξ BF∆ κ depends on P δ (Φ+Ψ) , on the bias difference b B − b F and on the observable quantities r and r ′ , we coulddirectly compare it with O δ Ψ to extract the anisotropicstress. However, to build a more direct test, it is conve-nient to Fourier transform the correlation function anddefine O δ (Φ+Ψ) ( k ⊥ , z ) ≡ (22) − π ∆ rk Z ∞ d ∆ x ⊥ ∆ x ⊥ ξ BF∆ κ (∆ r, ∆ x ⊥ , z ) J ( k ⊥ ∆ x ⊥ )= ( b B − b F ) P δ (Φ+Ψ) ( k ⊥ , z ) . Here the correlation function is expressed in terms of thetransverse separation, ∆ x ⊥ , and the radial separation,∆ r , between ∆ L and κ . To obtain the second equalityin Eq. (22) we have used the orthogonality relation for J . Eq. (22) contains an integral over ∆ x ⊥ going from0 to ∞ . In practice, since the correlation function andthe Bessel function go to zero at large separation, theintegral can be cut at some maximum transverse sep-aration ∆ x max ⊥ . The observable O δ (Φ+Ψ) depends onlyon the transverse wave number k ⊥ (in the Limber ap-proximation, the radial modes do not contribute to thecorrelation function ξ BF∆ κ ). Therefore, to compare with O δ Ψ , we have to evaluate O δ (Φ+Ψ) at k = k ⊥ . Anisotropic stress estimator.
In ΛCDM and for largeclasses of dark energy models, the two metric potentialsare the same at late time, and we have therefore O δ (Φ+Ψ) O δ Ψ = 2 . (23)On the other hand, in theories of modified gravity, themetric potentials are generally different. This difference,that can be parameterized by the variable η throughΦ( k , z ) = η ( k, z )Ψ( k , z ), leads to O δ (Φ+Ψ) O δ Ψ = 1 + η . (24)The observable O δ Ψ provides therefore a direct way ofmeasuring η . In particular, if the ratio in Eq. (24) isdifferent from 2 at any redshift or scale k , then ΛCDMis ruled out, as well as all classes of dark energy modelswith no anisotropic stress.To emphasise the robustness of our test compared tostandard methods, let us consider the following model,widely used in the literature: we parameterize deviationsfrom GR by two functions, η ( k, z ) (introduced above) and Y ( k, z ), which encodes modifications to Poisson equa-tion [3] − k Ψ( k , z ) = 32 H Ω m ( z ) Y ( k, z ) δ ( k , z ) , (25)where Ω m ( z ) is the matter density parameter at redshift z . The function Y ( k, z ) (sometimes called µ in the liter-ature) reduces to 1 in GR. In addition to these functions,we allow for another departure from GR by modifyingEuler equation˙ V ( k , z ) + H V ( k , z ) − k Ψ( k , z ) = ˆ E break ( k , z ) , (26)where ˆ E break is a generic function encoding deviationsfrom geodesic motion for dark matter. For example, inmodels where dark matter experiences a fifth force due toa non-minimal coupling to a scalar field, ˆ E break takes theform ˆ E break = k Γ( z )Ψ( k , z ), where Γ is the amplitude ofthe fifth force [8].We now apply the test developed in [3] to this particu-lar model. By doing this we clearly use the test outside ofits domain of validity, since in [3] it is explicitly assumedthat Euler equation is valid. However, since in practicewe do not know if dark matter obeys Euler equation ornot, it is relevant to see what happens in this case. Theevolution equation for the density contrast becomes δ ′′ + (cid:18) H ′ H (cid:19) δ ′ = − k ( aH ) Ψ − k ( aH ) ˆ E break , (27)where H ( z ) = H ( z )(1 + z ) and a prime denotes a deriva-tive with respect to N = ln a . The combination of ob-servables proposed in [3] to measure the anisotropic stressbecomes then3(1 + z ) P E (cid:0) P + 2 + E ′ E (cid:1) − η + k ( aH ) (1 + η ) f ′ + f + (cid:0) E ′ E (cid:1) f ˆ E break δ = η . Here P and P are the ratios of observables defined in [3](see their Eqs. (14) and (15)), f is the growth rate and E ≡ H/H .From Eq. (28) we see that the test developed in [3]is not a measurement of η when Euler equation is notvalid. In other words, a non-trivial outcome of this testcan either mean that the anisotropic stress is non-zero,or that dark matter does not obey Euler equation. Conclusion.
In this letter, we constructed an observ-able, O δ Ψ , which is directly proportional to the time com-ponent of the metric Ψ. This observable is constructedfrom the multipoles of the galaxy number counts, ∆, andit relies only on observable quantities. We then showedhow this novel observable can be used to measure directlythe anisotropic stress, i.e. the difference between the twometric potentials Φ and Ψ.This test has the strong advantage that it does notrely on any assumption about the theory of gravity, apartfrom the fact that photons propagate on null geodesics.In particular, this test does not assume that dark matterobeys the continuity or Euler equation. This differs fromstandard measurement of the anisotropic stress, whichrely on the validity of the continuity and Euler equa-tions. As an example, we showed how the test proposedin [3] will fail if Euler equation is not valid: instead ofmeasuring directly η , the combination of observables de-fined in [3] contains an additional term proportional tothe deviation from Euler equation. This limitation sim-ply follows from the fact that standard observables areinsensitive to Ψ. The only way to test the relation be-twen Ψ and Φ + Ψ is then to translate a measurement of V into a measurement of Ψ assuming that dark matterobeys Euler equation.The test proposed in this letter overcomes this limita-tion by using an observable sensitive to relativistic effects,which allows a direct measurement of Ψ. In a forthcom-ing paper we will study in more detail the sensitivity of this test for the coming generation of large-scale struc-ture surveys, like DESI, Euclid and the SKA. Acknowledgements.
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