Cosmic Growth Signatures of Modified Gravitational Strength
aa r X i v : . [ a s t r o - ph . C O ] M a r Cosmic Growth Signatures of Modified Gravitational Strength
Mikhail Denissenya and Eric V. Linder , Energetic Cosmos Laboratory, Nazarbayev University, Astana, Kazakhstan 010000 Berkeley Center for Cosmological Physics & Berkeley Lab,University of California, Berkeley, CA 94720, USA (Dated: September 5, 2018)Cosmic growth of large scale structure probes the entire history of cosmic expansion and gravita-tional coupling. To get a clear picture of the effects of modification of gravity we consider a deviationin the coupling strength (effective Newton’s constant) at different redshifts, with different durationsand amplitudes. We derive, analytically and numerically, the impact on the growth rate and growthamplitude. Galaxy redshift surveys can measure a product of these through redshift space distor-tions and we connect the modified gravity to the observable in a way that may provide a usefulparametrization of the ability of future surveys to test gravity. In particular, modifications duringthe matter dominated era can be treated by a single parameter, the “area” of the modification, to anaccuracy of ∼ .
3% in the observables. We project constraints on both early and late time gravityfor the Dark Energy Spectroscopic Instrument and discuss what is needed for tightening tests ofgravity to better than 5% uncertainty.
I. INTRODUCTION
Future galaxy redshift surveys will measure cosmicstructure over an increasing volume of the universe, tohigher redshift. One particular cosmological probe com-ing from the surveys is redshift space distortions, the an-gular dependence of galaxy clustering viewed in redshiftspace, a direct probe of the growth, and growth rate, ofstructure. This, in terms of the relation between the den-sity and velocity fields, was identified as a test of cosmicgravity in an influential paper by Peebles [1].Measurements of redshift space distortion effects be-gan to place significant constraints on the matter density[2–6] and then were explicitly developed as tests of grav-ity [7]. Galaxy redshift surveys observational constraints[8–10] and further theoretical work treating the surveys[11–14] followed, as well as related techniques combin-ing redshift space distortions with other probes, e.g. [15].This is now a common and significant part of modernsurvey cosmology.Modified gravity is a major possibility for the originof current cosmic acceleration, and considerable effortis underway to understand how best to connect theo-retical ideas with observational measurements in a clearand accurate way. Theories of modified gravity oftenhave enough freedom that they can match a given cos-mic background expansion, leaving a main avenue for dis-tinguishing them from a cosmological constant or scalarfield in terms of the alteration of the cosmic growth his-tory. Within general relativity, the cosmic expansion de-termines the cosmic growth, but modified gravity allowsdeviations from this relation.On scales where density perturbations are linear, thescale dependence of modified gravity is generally neg-ligible, with the time dependence the key factor. Atthe theory level, many time dependent functions can en-ter the action but phenomenologically for the growth ofstructure these can effectively be condensed to a modifiedPoisson equation relating the metric perturbations to the density perturbations, involving a single factor giving thegravitational coupling strength G matter ( a ), where a is thecosmic expansion factor.Here our goal is to explore the connection between thedeviations of G matter ( a ) from the general relativity case,where it is simply Newton’s constant, and the observablesfrom redshift surveys. One aim is to enable clearer un-derstanding of the effects of modified gravity on growthmeasurements, without restriction to a particular the-ory. Another is to explore the possibility of a low orderparametrization that could fruitfully be used to fit obser-vational data to signatures of modification of gravity.In Sec. II we review the modified Poisson equation andits influence on the evolution equation of density pertur-bations. After predicting analytically the effects on thecosmic growth rate and amplitude in certain limits, weexplore the parameter space numerically in Sec. III. Weconsider late time modifications in Sec. IV, demonstrat-ing distinction between some broad classes, and deriveprojected constraints for future redshift surveys as probesof gravity in Sec. V. We conclude in Sec. VI. II. COSMIC STRUCTURE GROWTH ANDGRAVITY
Cosmic structure growth proceeds through a competi-tion between gravitational instability – an overdensity ofmatter attracting more matter under gravity – and Hub-ble friction due to the cosmic expansion opposing growth.This gives the linear density perturbation evolution equa-tion ¨ δ + 2 H ˙ δ − H Ω m ( a ) G matter ( a ) δ = 0 , d (1)where δ = δρ m /ρ m is the matter overdensity, H = ˙ a/a is the Hubble parameter, where dot denotes a derivativewith respect to cosmic time, Ω m ( a ) = 8 πG N ρ m / (3 H )is the dimensionless matter density as a fraction of thecritical density, and G matter is the gravitational strengthin units of Newton’s constant G N .The source term of the gravitational instability, theterm proportional to δ , arises from the modified Poissonequation relating the Newtonian gravitational potential ψ to the density perturbation, ∇ ψ = 4 πG N G matter ρ m δ . (2)The growth equation as written assumes that the modi-fication is purely to the gravitational coupling strength,that there are no nonminimal couplings of the mattersector to other sectors. For the rest of this article weabbreviate G matter as G m .At high redshift, such as around last scattering of thecosmic microwave background (CMB), observations in-dicate that general relativity is an excellent descriptionof gravity and so the initial conditions for the growthequation are taken to be unchanged. In the high red-shift matter dominated universe, where Ω m = 1 and H = 2 / (3 t ), the solution for the growth is δ ∝ a . Thismakes it convenient to define a normalized growth factor g = ( δ/a ) / ( δ i /a i ), where a subscript i indicates an initialtime in that epoch.The growth equation can then be written g ′′ + (cid:20) d ln H d ln a (cid:21) g ′ a + (cid:20) d ln H d ln a −
32 Ω m ( a ) G m ( a ) (cid:21) ga = 0 , (3)where a prime denotes a derivative with respect to a .In order to focus on the impact of the modified G m , wetake the background expansion to be identical to that ofΛCDM, a flat matter plus cosmological constant universe.The mass fluctuation amplitude σ is proportional tothe growth factor g , but is difficult to extract from galaxyredshift surveys since the galaxy bias has a similar effect.The growth rate f = 1+ d ln g/d ln a is of particular inter-est since it gives a more instantaneous sensitivity to theconditions at a particular redshift than the integratedgrowth that enters the growth factor. The observablefrom redshift space distortions (RSD), at the linear level,is the product f σ , or f ga ∝ dδ/d ln a . We will exam-ine the impact of modified gravitational strength on f , g , and f σ .To build intuition for the physical interpretation of thelater numerical results, let us begin with an analytic in-vestigation. This can most fruitfully be done in terms ofthe growth rate equation, derived from the growth equa-tion to be dfd ln a + f + (cid:20) d ln H d ln a (cid:21) f −
32 Ω m ( a ) G m ( a ) = 0 . (4)Next consider the deviation in growth rate between themodel with modified gravity and that without, i.e. stan- dard ΛCDM: d ( f − f Λ ) d ln a + (cid:2) ( f − − ( f Λ − (cid:3) + (cid:20) d ln H d ln a (cid:21) ( f − f Λ ) = 32 Ω m ( a ) [ G m ( a ) − . (5)Until dark energy begins to dominate, f (and f Λ ) areclose to one so we could neglect the square bracket in-volving the difference of the ( f − factors. Integratingthe equation over ln a , we find Z a d ln a ′ (cid:26) dδfd ln a ′ + (cid:20) d ln H d ln a ′ (cid:21) δf (cid:27) = 32 Z a d ln a ′ Ω m ( a ′ ) [ G m ( a ′ ) − , (6)where δf = f − f Λ and δG m = G m −
1. The first termon the left is a total derivative and δf vanishes at earlytimes so the contribution is simply δf ( a ). Restrictingto the matter dominated epoch, where H ∝ a − andΩ m ( a ) = 1, yields Z a d ln a ′ δf ≈ Z a d ln a ′ δG m − δf ( a ) . (7)This is a very interesting expression because recall therelation of growth rate to growth factor: g = a − e R a d ln a ′ f ( a ′ ) , (8)and thus gg Λ = e R a d ln a ′ [ f ( a ′ ) − f Λ ( a ′ )] = e R a d ln a ′ δf ( a ′ ) . (9)If the deviations are small (recall that even for 10% devi-ations in growth the difference between e x and the firstorder expansion 1 + x is small, less than 0.5%) then wecan expand the exponential to get δgg Λ ≈ Z a d ln a ′ δf ( a ′ ) . (10)That is, the growth factor deviation is approximately thearea under the growth rate deviation curve, and Eq. (7)tells us this is closely related to the area under the grav-itational strength curve.In particular, if the growth rate change from the grav-itational modification has faded by the time at whichthe growth factor is evaluated, then we can neglect the(2 / δf ( a ) term in Eq. (7). Then the growth factorchange is indeed proportional to the area under δG m .We can be more precise by writing Eq. (5) under thesame assumptions of small deviations and matter domi-nation and using the integrating factor method for solu-tion. Then dδfd ln a + 52 δf ≈ δG m ( a ) (11)has the solution δf ( a ) ≈ a − / Z a d ln a ′ a ′ / δG m ( a ′ ) . (12)Substituting Eq. (12) into Eq. (7) and Eq. (9) gives δgg Λ ≈ Z a d ln a ′ δG m − δf ( a ) (13) ≈ Z a d ln a ′ δG m " − (cid:18) a ′ a (cid:19) / . (14)If the evaluation time a is much after the epoch when δG m ( a ′ ) is nonnegligible, then the square bracket quan-tity simply goes to one. In this situation the fractionalgrowth factor deviation is just (three-fifths) the area un-der the gravitational modification curve.For the modified gravitational strength G m we want touse a parametrization that is tractable in terms of hav-ing only a few parameters, but that is consistent withthe behavior of at least some theories of gravity. In par-ticular, it should vanish at high redshift. To explore thesignatures of modified gravity on growth, it is an ad-vantage if δG is also fairly localized so we can explorethe effect of deviations at different redshifts on growthduring the observable epoch of z ≈ −
3, where the red-shift z = a − −
1. That is, we want to build up ourintuition and understanding of the connection betweengravitational modifications and observables.We adopt the form G m = 1 + δG e − [(ln a − ln a t ) / (2 σ )] , (15)where δG describes the amplitude of the deviation, a t the scale factor at which it peaks, and σ measures itsduration. This fulfills the desired characteristics above,and is Gaussian in e-folds of expansion, ln a . Such a peakgives similar results to the deviations seen in theories ofmodified gravity having multiple, competing terms in theLagrangian, such as the Horndeski class; see Fig. 5 of [16]for example.We emphasize that localization through use of a Gaus-sian is for clarity in interpretation; we derived above thatthe area under the gravitational modification curve wasa key parameter, so one could equally well treat multipleGaussians, or some other function, as long as it held tothe assumptions used above. We also stress that the ana-lytic arguments above were to guide intuition, and we donot assume matter domination at all redshifts, rather wetake the expansion history to be that of ΛCDM. We dis-cuss treatment of modifications at recent times in Sec. IVbut again our main aim is to achieve some insight in un-derstanding the signature of a deviation at a particularredshift on subsequent cosmic growth.Given a Gaussian, the area under the modificationcurve is easy to calculate and in particular if we are in-terested in the total growth factor to the present then wehave δg g Λ , ≈
35 Area ≈ √ πσ δG ≈ . σδG . (16) To summarize, our analytic understanding is that thegrowth rate deviation δf ( a ) should approximately trace δG m ( a ), with somewhat lower amplitude (e.g. at its peak,where df /d ln a = 0, δf ≈ (3 / δG m ), slightly shiftedto later times due to the integral, and skewed to latertimes due to the ( a ′ /a ) / factor (or alternately due tothat the magnitude of df /d ln a subtracts from the δf term in the growth rate equation before the peak butadds to it afterward). The growth factor itself is in turnan integral over f , and if δG m and so δf is sufficientlylocalized then at later times δg should go to a constantoffset proportional to the area under the gravitationalmodification curve, described by Eq. (16).In the next section we carry out a full numerical evo-lution of the cosmic growth and test our understandingof the signatures of this gravitational modification. III. SIGNATURES IN GROWTH EVOLUTIONA. Effects on observables
Taking a gravitational strength modification as aGaussian in the expansion e-fold scale on top of the gen-eral relativity behavior, i.e. Eq. (15), we solve numeri-cally the growth evolution equation to obtain the cosmicgrowth rate f , growth factor g , and redshift space dis-tortion amplitude f σ . Figure 1 shows the results for δG m = ( G m G N − G N ) /G N and δf /f Λ = ( f − f Λ ) /f Λ for the fiducial cosmology of a flat ΛCDM universe withpresent matter density Ω m = 0 . δf /f Λ is roughlyGaussian and slightly delayed from the gravitationalstrength perturbation. We can anticipate that if the red-shift of the gravitational modification is moved closer tothe present, or if its width is broadened, then the effecton f might overlap the present.Figure 2 shows the responses of all the growth quan-tities, for the same parameters as Fig. 1 but plotted ona scale linear in expansion factor. Again we see that theanalytic arguments hold fairly well: the growth factorapproaches a constant offset at a time much later thanthe impulse of nonzero δG m , basically when the delayed δf /f also restores to the standard cosmology. Deviationsin the RSD parameter f σ acts like δf /f at the begin-ning of the impulse, since δσ ∼ δg takes time to build(recall it is related to the area under the δf curve), andlike δg at late times as f restores to standard behavior.This shows that RSD are capable of testing gravity atall redshifts, from z = 0 out to the epoch at which themodification peaks.The growth quantities themselves, rather than the de-viations from the general relativistic cosmology, are plot-ted in Fig. 3. We see that the growth rate f indeedrestores to the standard evolution at recent times, and g and f σ suffer constant offsets. One subtlety is whetherthe growth amplitude is normalized at high redshift to FIG. 1. A modification of the gravitational strength δG m propagates into an alteration of the cosmic growth rate δf/f .Here the modification has the parameters δG = 0 . a t = 0 . σ = 0 .
5; we plot it in log a rather than ln a for simplicity.The response of the growth rate is a slightly delayed, some-what damped, near shadow of δG m , due to the physics of thegrowth equation discussed in Sec. II.FIG. 2. The deviations of all the growth quantities in re-sponse to a gravitational modification are plotted vs a forthe same parameters as in Fig. 1. In the recent universe thegrowth factor g and RSD parameter fσ go to constant offsetsfrom the standard cosmology. the same initial conditions, i.e. cosmic microwave back-ground power spectrum amplitude A s , or at redshift 0,i.e. σ , . We normalize to the CMB; if one instead nor-malized to σ , then the f σ curves should be shiftedvertically to agree at a = 1. FIG. 3. The growth quantities are plotted vs a for threecases: standard ΛCDM cosmology with general relativity(thin curves), with modified gravity with δG = 0 .
02 (mediumcurves), and with δG = 0 .
05 (thick curves). We also plot thegravitational growth index γ . Also plotted is the gravitational growth index γ , de-fined through f ( a ) = Ω m ( a ) γ . This probes deviationsfrom general relativity in the growth of matter pertur-bations [17, 18] and we see its curves strongly pick upthe gravitational modifications. The quantity γ deviatesfrom its general relativity ΛCDM value of γ = 0 .
55 dur-ing the modification and then restores to it. Note thatin the standard case γ can be seen to be not perfectlyconstant at the value 0.55, but this is an excellent ap-proximation to its behavior, especially in the integratedsense in which γ enters the growth factor. During timesof strengthened gravity, γ gets smaller, i.e. the growthrate stays high even as the fractional matter density de-clines; when modified gravity increases the growth rate f >
1, then γ <
0. Thus γ contains considerable infor-mation about modifications of gravity. B. Numerical vs analytic results
Now let us investigate how the variation of parame-ters within this model impacts the behavior of the cos-mic growth variables, and conversely how sensitive thegrowth is at revealing characteristics of gravitationalmodification. Figure 4 varies the parameters of the modelone at a time. We change the fiducial value of the widthto σ = 0 .
25 since the narrower impulse gives clearer in-terpretation of the results.
FIG. 4. The deviation in the growth factor to thepresent, δg /g , is plotted vs the value of each model param-eter, varied one at a time around the fiducial { δG, σ, a t } = { . , . , . } . The deviation in the growth factor increases nearlylinearly with the amplitude δG , and the same holdsupon varying the duration σ of the modified gravita-tional strength. These are both just what is expected byEq. (16), having their origin in the physics of the growthequation and within the approximation that the growthrate has substantially restored to the standard behaviorby the present. The degree of linearity in the figure di-rectly tests the validity of approximation. For the timeof modification, a t , we see that the growth is fairly in-sensitive to this provided it occurs early enough. As themodification peaks closer to the present, the inertia ofthe effect on the growth rate f means that the influenceon the growth factor g diminishes toward zero.Let us pursue this further, assessing the analytic ap-proximation from Sec. II. This predicts that the growthfactor deviation should not only depend linearly on theamplitude and width of the modification, but that thekey quantity is the area under the curve showing thedeparture of the gravitational strength from general rel-ativity. We therefore plot in Fig. 5 the growth factordeviation vs the product σ δG .The linear fit to the growth behavior as a function ofamplitude times width, or area, appears to be an ex-cellent approximation, especially for the most observa-tional viable values of the growth deviation (i.e. less than FIG. 5. The deviation in the growth factor to the present, δg /g , is plotted vs the product δG × σ , proportional tothe area under the curve of gravitational modification. Alinear fit is plotted as the green, dot-dashed line. The solidblack curves show the results when varying the amplitude δG ,with σ = 0 .
25 and σ = 0 .
05 for the outer and inner curvesrespectively. The dashed blue curves show the results whenvarying the duration σ , with δG = 0 .
05 and δG = 0 . a t , showing how little dependence there is on a t over the range a t = [0 , . ∼ δG varies to larger values of devia-tion, the behavior is to curve slightly up from linearity,more so for larger values of the width σ . This arisesfrom the increasing importance of the ( f − term inthe growth rate Eq. (5), or alternately the higher orderexpansion of the exponential in Eq. (9), breaking linear-ity. Increasing σ further amplifies the effect of such highamplitudes on the growth factor.However, in the lower amplitude regime, increasing theduration σ causes the growth factor behavior to curveslightly down from linearity. This is due to the elon-gated persistence of the modification δG such that by thepresent where g is evaluated the growth does not feel thefull impact of the modification on δf , and hence δg . Es-sentially, one does not capture the entire area under the δf curve in Eq. (10). Moving the modification epoch a t earlier would ameliorate this effect, while moving it laterwould exacerbate the nonlinearity. Recall though thatlate modifications give smaller deviations, all else equal,so the nonlinearity is less important in this case.To quantify the excellent agreement of the numericalresults with the analytic prediction, to as high deviationsin the growth as it does, note that the analytic relationworks to 0.3% in g out to σ = 1, where the main effectis the modification persisting to the present. (Other pa-rameters are held at their fiducial values.) At σ = 0 . δG = 1.This close relation of the area of the modification tothe growth factor deviation suggests that this quantitymay play a useful role in parametrizing the gravitationalmodification and its signatures. We revisit this pointlater.The same relation holds for δf σ /f σ ( z = 0). Onecan readily see this analytically in that for small devia-tions this quantity is basically the sum of δf /f and δg/g .Since δf nearly vanishes by z = 0, the behavior is nearlythe same as for δg/g . In fact, the large σ deviationsfrom linearity seen in Fig. 5 for δg/g are suppressed for δf σ /f σ – the analytic area relation works better be-cause the suppression due to nonvanishing δf ( z = 0) inEq. (14), i.e. the correction factor in the square brackets,is counteracted by the δf /f contribution to δf σ /f σ .This is evident in Fig. 6. D e v i a t i on f /f vary G vary analytic 1.5 G x-axis is a t FIG. 6. As Fig. 5, but for δfσ /fσ ( z = 0). Again, the ana-lytic fit linear in area is an excellent approximation, especiallyfor viable deviations less than ∼ The RSD observable is accurately fit by the analyticarea formula to 0.4% at z = 0 and 0.6% at z=1, outto σ = 1. However for σ = 0 . f σ weakens to the3% level for an extreme δG = 1 (which would entail a nearly 40% deviation of f σ from ΛCDM).We can also illustrate the accuracy of the analytic ap-proximation by showing the isocontours of the deviationsin the growth factor and the redshift space distortion fac-tor in Fig. 7 and Fig. 8, respectively. The shape of the f σ contours is quite similar to those of the g contours,and their level as well, as expected by the above rea-soning. The dotted curves show the analytic, area ap-proximation; this works superbly for the viable range ofdeviations less than about 10%, and is quite reasonableeven out to ∼
30% deviations. . . . . . . . z=0 G FIG. 7. Isocontours of δg/g ( z = 0) are plotted in the σ - δG plane, for fixed a t = 0 .
1. Dotted curves for the 0.05, 0.1, 0.15,and 0.2 level contours show the analytic, area prediction.
Since next generation galaxy redshift surveys aim tomeasure accurate redshift space distortions at z ≈
1, weillustrate in Fig. 9 how well the area approximation holdsfor δf σ /f σ ( z = 1). Since the δf ( a ) term is less neg-ligible at higher redshift, the integrand in Eq. (14) issuppressed somewhat from the pure area, but the ana-lytic form is still quite accurate for viable deviations lessthan ∼ g is good to 0.3%everywhere along the 10% deviation curve, and to 1%for g ( z = 1). However, if we restrict to σ . . f σ the accuracy is 0.7% at z = 0 and1% at z = 1, tightening to 0.4% and 0.6% for σ . . f σ of only 0.004, beyond even next generation surveyprecision.We find there is also little sensitivity to the value of thegravitational modification epoch a t , as long as a t . . . . . . . . . z=0 G FIG. 8. As Fig. 7 but for isocontours of δfσ /fσ ( z = 0). ( z t &
3) and the observational quantity deviations areviably small (less than ∼ . . . . . . . z=1 G FIG. 9. As Fig. 7 but for isocontours of δfσ /fσ ( z = 1). C. Extended modifications
While Sec. II showed analytically that the late timegrowth evolution should depend on the area of themodification, and we demonstrated this numerically inSec. III B for a localized modification of various ampli-tudes and widths, we now illustrate this for extendedearly modifications. Figure 10 plots the deviations in thegrowth factor g ( a ) and the RSD observable f σ ( a ) due tothree different forms for the gravitational modification: aGaussian as used in earlier plots, a box function with thesame peak amplitude but the width adjusted to matchthe same area, and a box function with half the ampli-tude but twice the width (in e-fold, i.e. ln a , units), so italso has the same area. FIG. 10. The deviation in the growth factor evolution, δg/g (dashed blue curves), and the RSD observable evolu-tion, δfσ /fσ (solid black curves), are plotted for three dif-ferent models. The Gaussian model, with δG = 0 . σ = 0 . a t = 0 . δG = 0 . a = 0 .
63 ending at ln a GR = ln a t + σ (mediumcurve), and a box model with half the amplitude and twicethe duration, ending at the same scale factor (thin curve), allhave the same area under the gravitational modification, andhence nearly the same growth evolution for a & . We see that indeed the quantities δg/g and δf σ /f σ are each nearly identical between models for a & . g ( a ) for the Gaussian modification and the box modifica-tion are less than 0.1% (0.25%) for a ≥ .
25 for the boxwith the same peak amplitude (half the peak amplitude,twice the duration). For the RSD observable f σ ( a ), thecorresponding deviations are 0.15%, 0.35%. This lendscredence to the concept that an acceptable parametriza-tion of matter dominated era gravitational modifications(such as are predicted by many theories involving multi-ple terms in the Lagrangian, e.g. in the Horndeski class ofgravity) is a single parameter corresponding to the areaof the modification, for matter growth observables.In the next section we explore late time modifications,where no such simplification is evident. IV. LATE TIME MODIFICATIONS
From the growth evolution equations, we see there is nophysical expectation that the area property should holdfor “late time” gravitational modifications once matterdomination wanes. Therefore a simple parametrizationof such gravitational modifications, and their effect oncosmic growth, is not obvious. To explore the diversityof behaviors, we take a phenomenological ansatz describ-ing three basic modifications during the recent universe:one constant with scale factor, one increasing, and onedecreasing.Specifically, we investigate δG ( a ) = δG c (17) δG ( a ) = δG r a s (18) δG ( a ) = δG f a − s , (19)over the range a = [0 . , R . d ln a δG ( a ) = 0 .
05 ln 4 corresponding tothe constant case with amplitude δG c = 0 .
05. Then δG r = 0 .
05 ln 4 s − − s (20) δG f = 0 .
05 ln 4 s s − . (21)We consider s = 3.Figure 11 exhibits the impact on the growth factor andgrowth rate evolution. Despite the gravitational modifi-cation areas being identical, the behaviors of the observ-ables are quite different, losing the immunity to variationof the modification parameters (under conserved area)found for the early time modifications, e.g. in Fig. 10.The late time modifications also give signatures in thegrowth observables distinct from that of early time mod-ifications. The thickest curve in Fig. 11 is for the usualearly Gaussian form with δG = 0 . a t = 0 .
1, and σ = 0 . σ chosen to match the areaconstraint.The differences in the shapes, i.e. the evolution, of theobservables indicate that galaxy survey measurementshave the potential to distinguish between these classes ofrising/constant/falling modifications, and moreover be-tween late and early modifications, if the measurements FIG. 11. The deviation in the growth factor evolution, δg/g (dashed blue curves), and the RSD observable evolu-tion, δfσ /fσ (solid black curves), are plotted for three latetime modification models, as well as one early time modifica-tion model. All models have the same area under the grav-itational modification. The thin/medium/thick curves cor-respond to the late time rising/constant/falling models withpower law index s = 3, 0, − a = 0 . δG = 0 .
05 and σ = 0 .
553 to match the area. Note thatunlike in Fig. 10, curves with identical late time modificationareas can be readily distinguished. extend beyond z ≈ .
5. The greatest similarity betweenearly and late time variations is for the falling class,as expected since this gives the greatest modification atsmaller a like the early time class. Even so, by a . . f σ ( a ) is significantly different between thetwo. In the growth factor this is even clearer, for a . . f σ ( a ) through redshift space distortions.First, though, let us combine the results of the twosections on early and late gravitational modifications. Ageneral modification could be viewed as the sum of thesetwo, giving a more arbitrary G m ( a ). This could well benonmonotonic, as is common in theories of gravity withmultiple terms, such as the Horndeski class, and seen inFig. 5 of [16] for example. We are particularly interestedin how the sum of early and late modifications translatesinto the impact on cosmic growth quantities such as thegrowth factor and RSD observable.We choose a nonmonotonic deviation δG m ( a ) given bythe sum of two Gaussians, one early ( a t = 0 . a t = 0 . σ = 0 . δg/g , δf /f , and δf σ /f σ . Theoverall effect is very close to the sum of the effects fromthe individual contributions to the modification. FIG. 12. The deviations of all the growth quantities in re-sponse to a nonmonotonic gravitational modification consist-ing of two overlapping Gaussians are plotted vs a . The thickcurves show the results, while the thin curves show the be-haviors for each individual Gaussian modification, and theirsum. The full results are very close to those for the sum ofthe individual contributions to G m ( a ). We can understand this analytically. Consider Eq. (6),with or without the integrals. The right hand side can bewritten as the sum of individual contributions to G m ( a ),and if we write δf on the left hand side as the sum of thecorresponding δf i , then we see that the equation holdsboth for each contribution and for their sum. The reasonthis works is that we linearized the full Equation (5) forthe growth rate. For small (i.e. observationally viable)deviations of f from f Λ – one does not require f close to1 as in the matter dominated regime, just that f is closeto f Λ – then the term [( f − − ( f Λ − ] has negligibleeffect.This sum rule propagates to the growth factor throughEq. (9), and again to linear order in the growth deviationsthe deviation in the sum of modification contributions issimply the sum of the deviations, δg = δg + δg . For f σ the situation is slightly more involved as the productof δf and δg enters, but the summation still works well.These are all evident from Fig. 12 where the summed δf curve cannot be distinguished from the full result, the δg curves are barely distinguishable and the f σ curves areclose. The maximum deviations are 0.00003, 0.0005, and0.001 for the three growth quantities, well below obser- vational uncertainty.The property that gravitational modifications can betreated as sums of their contributions, say early and latemodifications, with respect to the growth observables,has a significant and useful implication. It indicates thatwe can partially solve the parametrization problem: fora nearly arbitrary gravitational modification history wecan accurately treat the gravitational modifications dur-ing the matter dominated epoch as in Sec. III, with asingle parameter corresponding to area, and we are thenleft with how to parametrize the late time modifications,from z = 0 −
3, say. One possibility is to try to at least dis-tinguish rising/constant/falling behaviors over this morerestricted range, which adds two more parameters. Weexplore the possibility of such identification through theirdistinct signatures on observational quantities in the nextsection.
V. FUTURE CONSTRAINTS ON GRAVITY
To estimate constraints on the gravitational modifica-tions from measurements of the redshift space distortionparameter f σ ( a ) from future galaxy redshift surveys, weemploy the Fisher information matrix formalism. Thislooks at the sensitivity of the observable to each modelparameter and takes into account similarities in the re-sponse, i.e. covariances, to translate a given precision indata to a parameter constraint.For the future RSD measurements, we adopt the pre-cisions given for the Dark Energy Spectroscopic Instru-ment (DESI) in Tables 2.3 and 2.5 of [19] for f σ ( a )between z = 0 . k max = 0 . h /Mpc, since the impact of gravitationalmodification, and in particular its scale dependence, on f σ beyond this is not clearly known. We will show inone case that assuming linear theory results hold out to k max = 0 . h /Mpc does not yield significant improvementbecause of covariances; a robust treatment of scale de-pendence may well break this impasse. Also, we do notinclude the growth factor g ( a ) since it is degenerate withgalaxy bias in the linear regime. Again, improvementsmay be made with a robust treatment of bias at higherwavenumbers.Within a flat ΛCDM background, the parameters af-fecting f σ are the matter density Ω m , the mass fluctu-ation amplitude σ , and the gravitational modificationsin the matter dominated era, δG hi , and more recently, δG lo . We have shown, both analytically and numerically,that gravitational modifications during the matter dom-inated era have an influence on f σ at later times wellapproximated by a fractional offset, i.e. a multiplicativefactor, proportional to the area under the modificationcurve. Thus, the area (or δG hi × σ hi ) is degenerate with σ , , the present value of the mass fluctuation amplitude.That is, one could obtain the same amount of structurewith an intrinsically low amplitude and extra growth or ahigher amplitude and less growth. Therefore we combine0these into a parameter S = [1 + (3 / hi ] σ , .For lower redshift gravitational modifications, we in-vestigate the three classes discussed in Sec. IV. Thesehave two parameters, an amplitude and power law in-dex. The fiducial values for the calculation are Ω m = 0 . S = 0 .
85 (e.g. σ , = 0 . δG hi = 0 . σ hi = 0 . δG lo = 0 .
05 when s = 0, i.e. constant, and otherwisegiven by Eqs. (18)–(21) for s = 3.Figure 13 shows the results for the falling case, wherethe modification is decreasing from higher redshifts.From Fig. 11 we see that this model has the highest am-plitude effect on f σ , but also a fairly constant amplitudefor z .
2, which could lead to covariances. Indeed, thatis what the results show. An overall diagonally orientedcovariance between dg lo and S is seen, with the thicknessof the confidence contour sensitive to the uncertainty inΩ m . Recall that the source term in the growth equa-tion (1) involves the product Ω m Gδ ∼ Ω m Gσ , . FIG. 13. Future constraints on the low redshift gravitationalmodification δG lo and the mass fluctuation amplitude S com-bining σ , and the high redshift modification are plotted asjoint 68% confidence contours, marginalizing over the otherparameters. This uses the falling class δG ∝ a − and we cansee the strong covariance with S . The dashed blue contourapplies an external prior σ ( S ) = 0 .
05, the solid black contouradds a prior σ (Ω m ) = 0 .
01, and the dotted red contour sharp-ens the measurement precision σ ( fσ ) by roughly a factor twoby extending to k max = 0 . h /Mpc (see [19]). We can view the priors on Ω m and S as coming frominformation in the galaxy survey besides the growth ratemeasurements, or from other experiments. Due to thediagonally oriented covariance, the rule of thumb is thatthe uncertainty on the low redshift gravitational mod-ification will be of order the uncertainty on S , i.e. the overall mass fluctuation amplitude, σ ( δG lo ) ∼ σ ( S ).Improving the precision of the f σ measurements fromthis level, e.g. by going to higher k max – assuming no newscale dependence to bring degeneracies, does not signifi-cantly tighten the constraints due to this covariance. Thedegeneracy needs to be broken, by direct measurementof the mass fluctuation amplitude (e.g. by the CMB athigh redshift and by galaxy clusters or weak gravitationallensing at lower redshift, or by further information withinthe galaxy survey itself).We compare the different classes of low redshift grav-itational modification in Fig. 14. The results show aninteresting interplay between the shapes and amplitudesof the f σ deviations exhibited in Fig. 11. Recall allthree cases have same integrated gravitational modifica-tion, i.e. “area”. FIG. 14. As Fig. 13 but contrasting the constraint for thethree classes of falling, constant, and rising modified gravita-tional strength. The outer solid black contour corresponds tothat from Fig. 13 for the falling class. Inner contours for eachclass use a tightened external prior of σ ( S ) = 0 .
02 to showthe impact the covariance of this parameter with δG lo has onthe gravity constraint. As stated, the falling case has the greatest amplitudeof f σ deviation during the bulk of the redshift rangeobserved, but with a shape not well distinguished from aconstant offset such as S gives. Therefore it has a diag-onally oriented covariance. The rising case has a shapedistinct from the standard cosmological parameters, andthus has relatively little covariance with them, but also alow amplitude of deviation during the important redshiftrange, giving weaker constraints on δG lo : i.e. a more hor-izontal, and broader confidence contour. The constantcase is somewhat in between, with less covariance but1also a low amplitude. We also investigate the case of theGaussian low redshift modification of Fig. 12 and as ex-pected it also has little covariance with S (not shown),but is over a restricted redshift range; its constraints fallbetween the constant and rising cases. The 5-10% con-straints on gravitational modifications we find are compa-rable to those of [20], which used G m piecewise constantin redshift bins.We can understand the 5-10% limit, at least within afactor of a few, by considering the expressions from Sec. IIrelating δG and the measurement precision δf σ /f σ .The quantity δf σ /f σ involves δg/g and δf /f , both ofwhich are integrals over δG ( a ). Basically there is a lin-ear functional relation between δG and δf σ /f σ , so forunmarginalized uncertainties σ ( δG ) ∼ σ ( δf σ /f σ ). Anexperiment with a precision of 2% in δf σ /f σ (as DESIachieves over a certain redshift range) should deliver anunmarginalized constraint on δG of the same order.In a bit more detail, the integral over δG ( a )[cf. Eq. (14)] outside the matter dominated era isweighted by the matter density Ω m ( a ) and a dilutionfactor of ( a H ) − (which gives the a − / in the matterdominated era). Furthermore, there are multiple mea-surements of f σ at various redshifts, which reduces theuncertainty on δG . We can incorporate all these effectsinto the following illustrative approximation, σ ( δG ) ≈ σ (cid:18) δf σ f σ (cid:19) N eff , (22)where N eff is the effective, weighted number of e-foldsgoing into the integral over δG ( a ). The time when δG ( a )is significant gives a tradeoff between the weighting fac-tors and the persistence (that an early deviation has alasting effect in f σ ) such that N eff is largest for thefalling case and smallest for the rising case. This alsointerplays with the redshift dependence of the measure-ment precision σ ( δf σ /f σ ), though DESI has 2-3% pre-cision over a substantial redshift range. The unmarginal-ized uncertainties σ ( δG ) = 0 . S ; since the falling case has moredegeneracy and the rising case has little, the final resultsall end up in the 5-10% constraint range.In all cases the power law of the modification scale fac-tor dependence is poorly determined, of order σ ( s ) ≈ G matter ;however, they depend on the gravitational coupling forlight deflection, G light , as well so an analogous formalismor parametrization scheme is required for this quantity.One particularly interesting future prospect is the kineticSunyaev-Zel’dovich effect used to measure the velocityfield, probing Hf σ (see, e.g., [21]). Understanding of galaxy bias and scale dependent growth will also be use-ful if we aim to go beyond 5% tests of gravity. VI. DISCUSSION AND CONCLUSIONS
A new generation of galaxy redshift surveys will vastlyincrease the volume and depth of the universe over whichwe measure the cosmic growth history. This brings withit the ability to test the foundations of gravity and lookfor modifications of general relativity, i.e. to confront pos-sible extensions with observations. This can be donemodel by model, or one can seek general signatures inthe observations that point to properties of the unknowntheory of gravitation. We took the latter approach, con-sidering the effective Newton’s constant – the gravita-tional strength entering into the Poisson equation for thegrowth of structure, called G matter .This governs the growth rate, amplitude, and the prod-uct of these that enters the redshift space distortion ob-servable. Remarkably, we found that a modification tak-ing place at any time during the matter dominated era,i.e. z &
3, could be parametrized in terms of a singlenumber – the area under the deviation curve δG m ( a ) withrespect to e-fold ln a . We derived this analytically anddemonstrated it numerically. Whether the deviation islocalized, extended, or nonmonotonic, the area approx-imation reproduces the growth observables to . . . .
6% in the RSD quantity f σ ( z ≈
1) in most cases, better than the measurementprecision of next generation surveys.Such an accurate, derived parametrization dramati-cally simplifies the task of comparing gravitational mod-ifications to cosmic growth observations. Recall thatmany gravity theories, in particular much of the Horn-deski class, predict such matter era variations. Further-more, we demonstrated that the full gravitational modi-fication history could be accurately treated by the sum ofthe separate matter dominated era (“early”) and late im-pacts on the growth quantities. For example, this sum re-produces the exact RSD observable f σ to within 0.001,well below the statistical uncertainty. Combined with theprevious result, this reduces the treatment of the entiremodified gravity history to one number (the area fromthe early modification) plus a description at z . z = 0–3. We showed that these had distinct effectson the growth observables. However, covariances withother cosmological parameters needed to be taken intoaccount, so we performed a Fisher information analysisusing the measurement precisions on f σ ( a ) baselined forthe upcoming DESI galaxy redshift survey over the range z = 0–1.9.The projected constraint analysis showed that DESIcould achieve gravitational modification amplitude esti-mation at the 5-10% level, with the limiting factor being2the covariances, particularly with the mass fluctuationamplitude σ , and also the matter density Ω m . Also,the rising/constant/falling classes could not be reliablydistinguished.In order to obtain a significant improvement, futuregalaxy surveys would need to strengthen its measurementprecision on f σ ( a ) to below 2%, or additional probes ofgravity (such as lensing, CMB, and galaxy clusters) ortighter external priors on covariant parameters need to beimplemented. Even 1% measurements of f σ across theentire z = 0–1.9 range give 2.6%, 3.0%, 4.4% constraintsof gravity for the three classes.Testing gravity experimentally, and connecting the ob-servations to theory, is a challenging subject. In onesense, this work has “solved” the problem for z = 3–1000 and only left the last 1.5 e-folds of cosmic history lacking a clear connection. That is less than satisfactorilyenlightening, however, and the remaining work on how toeffectively and practically parametrize the late time grav-itational modifications is substantial. Other aspects ofgravity, such as how to characterize G light for light prop-agation and other modifications affecting gravitationalwave propagation, also require future work. ACKNOWLEDGMENTS