TESTING G R COSMOLOGICAL S : E OF SPATIAL Cjdossett/files/AAS_2013.pdfMustapha Ishak∗ and Jason Dossett ... where H is the Hubble parameter and the eﬀect of the underlying gravity

TESTING GENERAL RELATIVITY AT COSMOLOGICAL SCALES: EFFECTS OF SPATIAL CURVATURE Jason Dossett

Advisor:Mustapha Ishak

MOTIVATIONS FOR TESTING GR?

¢ Cosmic acceleration �  Dark Energy �  Modification to gravity at

cosmological scales.

¢ Extend tests to other gravity theories. �  Are gravity models

proposed for quantizing gravity or unifying the four forces correct?

METHODS OF DISTINGUISHING BETWEEN GR AND MODIFICATIONS TO GRAVITY

¢ Looking for inconsistencies between expansion history and growth of structure �  The growth rate of large scale structure is coupled to

the expansion history via Einstein’s equations. These two effects must be consistent.

¢  “Trigger parameters”, γ. The logarithmic growth rate can be approximated by:

For different gravity models γ has a unique value.

¢ Gravitational Slip and Modifications to the Growth Eqns. arX

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Contiguous redshift parameterizations of the growth index

Mustapha Ishak! and Jason Dossett†

Department of Physics, The University of Texas at Dallas, Richardson, TX 75083, USA(Dated: August 10, 2009)

The growth rate of matter perturbations can be used to distinguish between di!erent gravitytheories and to distinguish between dark energy and modified gravity at cosmological scales as anexplanation to the observed cosmic acceleration. We suggest here parameterizations of the growthindex as functions of the redshift. The first one is given by !(a) = !(a) 1

1+(attc /a) +!early1

1+(a/attc )

that interpolates between a low/intermediate redshift parameterization !(a) = !late(a) = !0 + (1!

a)!a and a high redshift !early constant value. For example, our interpolated form !(a) can be usedwhen including the CMB to the rest of the data while the form !late(a) can be used otherwise. It isfound that the parameterizations proposed achieve a fit that is better than 0.004% for the growthrate in a "CDM model, better than 0.014% for Quintessence-Cold-Dark-Matter (QCDM) models,and better than 0.04% for the flat Dvali-Gabadadze-Porrati (DGP) model (with #0

m = 0.27) for theentire redshift range up to zCMB . We find that the growth index parameters (!0, !a) take distinctivevalues for dark energy models and modified gravity models, e.g. (0.5655,!0.02718) for the "CDMmodel and (0.6418, 0.06261) for the flat DGP model. This provides a means for future observationaldata to distinguish between the models.

PACS numbers: 95.36.+x;98.80.Es;04.50.-h

I. INTRODUCTION

Cosmic acceleration can be caused by a dark energy component in the universe or a modification to the Einsteinfield equations of General Relativity at cosmological scales. The growth rate of matter perturbations has been thesubject of much recent interest in the literature as a way to distinguish between one possibility or the other, see forexample [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] for a partial list. Indeed, distinct gravity theories may have degenerateexpansion histories but can be distinguished by their growth rate functions.

As usual, the large scale matter density perturbation ! = !"m/"m satisfies, to linear order, the di!erential equation

! + 2H ! ! 4#Geff"m! = 0, (1)

where H is the Hubble parameter and the e!ect of the underlying gravity theory is introduced via the expression forGeff . The distinct behavior of ! for di!erent gravity models can be seen in some of the aformentioned references suchas for example [9, 10]. Equation (1) can be written in terms of the logarithmic growth rate f = d ln !/d lna as

f " + f2 +

!

H

H2+ 2

"

f =3

2

Geff

G"m, (2)

where primes denote d/d ln a. Throughout this work we will use the numerically integrated solution to this equationnormalized at a = 0 (z = "). Next, the growth function f is usually approximated using the ansatz [14, 15, 16, 17]

f = "!m (3)

where $ is the growth index parameter. Reference [14] made an approximation that applies to matter dominated

models and proposed f(z = 0) = "0.6m0 and was followed by a more accurate approximation f(z = 0) = "4/7

m0 in [15, 16].Reference [17] considered dark energy models with slowly varying equation of state, w, and found an expression for $as function of "m and w. This has been discussed further in more recent references, see for example [3, 19], and alsoexpanded to models with curvature in [20] and [21].

The approaches of expanding the growth index around some asymptotic value or early, matter dominated timeswith "m # 1, or those considering specific redshift ranges to approximate $ do not cover other redshift ranges ofinterest where observational data is available and can constrain the growth parameters or break degeneracies betweenthem and other cosmological parameters.

! Electronic address: [email protected]† Electronic address: [email protected]

arX

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905.

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v2 [

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ug 2

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Contiguous redshift parameterizations of the growth index

Mustapha Ishak! and Jason Dossett†

Department of Physics, The University of Texas at Dallas, Richardson, TX 75083, USA(Dated: August 10, 2009)

The growth rate of matter perturbations can be used to distinguish between di!erent gravitytheories and to distinguish between dark energy and modified gravity at cosmological scales as anexplanation to the observed cosmic acceleration. We suggest here parameterizations of the growthindex as functions of the redshift. The first one is given by !(a) = !(a) 1

1+(attc /a) +!early1

1+(a/attc )

that interpolates between a low/intermediate redshift parameterization !(a) = !late(a) = !0 + (1!

a)!a and a high redshift !early constant value. For example, our interpolated form !(a) can be usedwhen including the CMB to the rest of the data while the form !late(a) can be used otherwise. It isfound that the parameterizations proposed achieve a fit that is better than 0.004% for the growthrate in a "CDM model, better than 0.014% for Quintessence-Cold-Dark-Matter (QCDM) models,and better than 0.04% for the flat Dvali-Gabadadze-Porrati (DGP) model (with #0

m = 0.27) for theentire redshift range up to zCMB . We find that the growth index parameters (!0, !a) take distinctivevalues for dark energy models and modified gravity models, e.g. (0.5655,!0.02718) for the "CDMmodel and (0.6418, 0.06261) for the flat DGP model. This provides a means for future observationaldata to distinguish between the models.

PACS numbers: 95.36.+x;98.80.Es;04.50.-h

I. INTRODUCTION

Cosmic acceleration can be caused by a dark energy component in the universe or a modification to the Einsteinfield equations of General Relativity at cosmological scales. The growth rate of matter perturbations has been thesubject of much recent interest in the literature as a way to distinguish between one possibility or the other, see forexample [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] for a partial list. Indeed, distinct gravity theories may have degenerateexpansion histories but can be distinguished by their growth rate functions.

As usual, the large scale matter density perturbation ! = !"m/"m satisfies, to linear order, the di!erential equation

! + 2H ! ! 4#Geff"m! = 0, (1)

where H is the Hubble parameter and the e!ect of the underlying gravity theory is introduced via the expression forGeff . The distinct behavior of ! for di!erent gravity models can be seen in some of the aformentioned references suchas for example [9, 10]. Equation (1) can be written in terms of the logarithmic growth rate f = d ln !/d lna as

f " + f2 +

!

H

H2+ 2

"

f =3

2

Geff

G"m, (2)

where primes denote d/d ln a. Throughout this work we will use the numerically integrated solution to this equationnormalized at a = 0 (z = "). Next, the growth function f is usually approximated using the ansatz [14, 15, 16, 17]

f = "!m (3)

where $ is the growth index parameter. Reference [14] made an approximation that applies to matter dominated

models and proposed f(z = 0) = "0.6m0 and was followed by a more accurate approximation f(z = 0) = "4/7

m0 in [15, 16].Reference [17] considered dark energy models with slowly varying equation of state, w, and found an expression for $as function of "m and w. This has been discussed further in more recent references, see for example [3, 19], and alsoexpanded to models with curvature in [20] and [21].

The approaches of expanding the growth index around some asymptotic value or early, matter dominated timeswith "m # 1, or those considering specific redshift ranges to approximate $ do not cover other redshift ranges ofinterest where observational data is available and can constrain the growth parameters or break degeneracies betweenthem and other cosmological parameters.

! Electronic address: [email protected]† Electronic address: [email protected]

Modified Growth Equations

where

Perturbed FLRW Metric.

where and

GROWTH EQUATIONS

127

Finally we will explore what e!ect assuming a flat model when curvature is present will

have on the best fit MG parameters, to see if a bias is introduced by such an assumption

and discuss how this bias is related to the correlation coe"cients. We conclude in the last

section.

B. Growth Equations Including Spatial Curvature in General Relativity

The perturbed Friedmann-Lemaître-Robertson-Walker metric written in the general con-

formal Newtonian Gauge is given by:

ds2 = a(!)2[!(1 + 2")d! 2 + (1! 2#)$ijdxidxj], (147)

where # and " are scalar potentials describing the scalar mode of the metric perturbations,

! is conformal time, a(!) is the scale factor normalized to one today, and the xi’s are the

comoving coordinates. $ij is the 3-metric, which can be written in coordinates (x, y, z) as:

$ij = %ij

!

1 +K

4

"

x2 + y2 + z2#

$!2

, (148)

where K = H20 (!0 ! 1) is the spatial curvature, and H0 is the Hubble parameter today.

As discussed in Abbott & Schaefer (1986); Zaldarriaga et al. (1998), when working in

a non-flat universe the Fourier modes are generalized as eigenfunctions, G, of the Laplacian

operator such that:

"2G(&k, &x) = !k2G(&k, &x). (149)

In our analysis we expand perturbations in terms of G and its spatial covariant derivatives

(denoted by |) as seen in Zaldarriaga et al. (1998).

127

Finally we will explore what e!ect assuming a flat model when curvature is present will

have on the best fit MG parameters, to see if a bias is introduced by such an assumption

and discuss how this bias is related to the correlation coe"cients. We conclude in the last

section.

B. Growth Equations Including Spatial Curvature in General Relativity

The perturbed Friedmann-Lemaître-Robertson-Walker metric written in the general con-

formal Newtonian Gauge is given by:

ds2 = a(!)2[!(1 + 2")d! 2 + (1! 2#)$ijdxidxj], (147)

where # and " are scalar potentials describing the scalar mode of the metric perturbations,

! is conformal time, a(!) is the scale factor normalized to one today, and the xi’s are the

comoving coordinates. $ij is the 3-metric, which can be written in coordinates (x, y, z) as:

$ij = %ij

!

1 +K

4

"

x2 + y2 + z2#

$!2

, (148)

where K = H20 (!0 ! 1) is the spatial curvature, and H0 is the Hubble parameter today.

As discussed in Abbott & Schaefer (1986); Zaldarriaga et al. (1998), when working in

a non-flat universe the Fourier modes are generalized as eigenfunctions, G, of the Laplacian

operator such that:

"2G(&k, &x) = !k2G(&k, &x). (149)

In our analysis we expand perturbations in terms of G and its spatial covariant derivatives

(denoted by |) as seen in Zaldarriaga et al. (1998).

60

space-space component of these equations lets us relate the two potentials. Respectively, we

have:

k2! = !4"Ga2!

i

#i!i (69)

k2($ ! !) = !12"Ga2!

i

#i(1 + wi)"i2, (70)

where #i and "i are the density and the anisotropic stress, respectively, for matter species,

i. !i is the gauge-invariant, rest-frame overdensity for matter species, i, the evolution of

which describes the growth of inhomogeneities. It is defined by:

!i = %i + 3Hqik, (71)

where H = a/a is the Hubble factor in conformal time, and for species i, %i = %#i/# is the

fractional overdensity and qi is the heat flux and is related to the divergence of the peculiar

velocity, &i, by &i = k qi1+wi

. Enforcing the conservation of energy-momentum on the perturbed

matter fluid, these quantities for uncoupled fluid species or the mass-averaged quantities for

all the fluids evolve as described in Ma & Bertschinger (1995):

% = !kq + 3(1 + w)!+ 3H(w ! c2s)% (72)

q

k= !H(1! 3w)

q

k!

w

1 + w

q

k+ c2s% ! (1 + w)

"

2+ (1 + w)$. (73)

Above, w = p/# is the equation of state and c2s = %p/%# is the sound speed. Combining

3

B. Modified Gravity Growth Parameters

Parametrizing both modifications to Poisson’s equation, (2), as well as the ratio between the two metric potentials! and " in the perturbed FLRW metric (called gravitational slip by Caldwell et al. [11]) has recently been the subjectof a lot of the work on testing general relativity; see, for example, [11, 49–53]. The parameters we use in this paperto describe modifications to the growth [modified gravity MG parameters] are based upon those used in [50].The parametrized modifications to the growth equations proposed by [50] directly modify Eqs. (2) and (3) and

make no assumptions as to the time when a deviation from GR is allowed. These modifications are as follows:

k2! = !4#Ga2!

i

$i!i Q (8)

k2(" !R!) = !12#Ga2!

i

$i(1 + wi)%i Q, (9)

where Q and R are the MG parameters. The parameter Q represents a modification to the Poisson equation, whilethe parameter R quantifies the gravitational slip (at late times, when anisotropic stress is negligible, R = "/!). In ourcode, rather than using the parameter R which is degenerate with Q, we instead use the parameter D = Q(1 +R)/2as suggested in [50] (this parameter is equivalent to the parameter " in [45, 53] or G of [52]). Combining Eqs. (8)and (9), we arrive at the second modified growth equation used in this paper:

k2(" + !) = !8#Ga2!

i

$i!i D ! 12#Ga2!

i

$i(1 + wi)%i Q. (10)

So, the modified growth equations are (8) and (10), and Q and D are now the MG parameters. As discussed in ourprevious work [56], this approach of using the parameter D instead of R is also useful because observations of theweak-lensing and ISW are sensitive to the sum of the metric potentials ! + " and its time derivative respectively.Thus observations are able to give us direct measurements of this parameter.

C. Di!erent Approaches to Evolving Modified Growth Parameters in Time and Scale

To date, there have primarily been two approaches to evolving the MG parameters in time and scale; one usinga continuous functional form and the other based on binning. We implement in ISiTGR the two approaches and,additionally, a new hybrid approach, as we explain below.The first approach involves defining a functional form for each parameter that allows it to evolve monotonically

in both time and scale. This allows one to make no assumptions as to when a deviation from general relativity isallowed. Such an approach was taken in for example [50]. In that work the functional form,

X(k, a) ="

X0e!k/kc +X"(1! e!k/kc)! 1

#

as + 1, (11)

was assumed, where X denotes either Q or R in Eqs. (8) and (9). Thus a total of six model parameters are used totest GR: Q0, R0, Q", R", kc, and s. The parameters s and kc parametrize time and scale dependence respectively,with GR values s = 0 and kc = ". Q0 and R0 are the present-day superhorizon values while Q" and R" are thepresent-day subhorizon values of the Q(k, a) and R(k, a) , all taking GR values of 1.In the second approach, instead of evolving each of the parameters assuming some functional form, one can bin the

MG parameters. This approach allows the parameters to take on di#erent values in predefined redshift and scale bins.This technique was used in, for example, [52, 53]. In those works two redshift and two scale bins were defined and forredshifts above a certain critical redshift, GR was assumed to be valid. In each bin the parameters were allowed totake on di#erent values resulting in a total of eight model parameters used to test GR.The third approach that we propose here is a hybrid one where the evolution in redshift (or time) is binned into

two redshift bins, but the evolution in scale evolves monotonically in the same way as the functional form above. Ourmotivation for this hybrid binning approach is that it takes advantage of an evolution in scale that is not so abruptas that in the traditional binning method, while still taking advantage of a redshift (time) dependence expressed inthe form of bins, which was shown to be more robust than time functional forms [53, 56, 58]. For example, in [56], wefound that binning methods do not display the extent of tensions between the MG parameters (preferred by di#erentdata sets) as in the functional form method, where tensions are exacerbated by the chosen functional form. Also, inRef. [58], the author found similar to what we noticed and that the constraints on MG parameters depend stronglyon the parameter s (the scale factor exponent in the functional parametrization). They further looked at ways toremove this strong dependence on the parameter s suggesting and exploring binning as a solution.

129

universe can be fully described.

C. Modifications to the Growth Equations For Spatially Curved Models

1. Modified Growth Equations in the Conformal Newtonian Gauge

Recently, one of the major routes to testing general relativity has been parameterizing

both the relation between the two metric potentials ! and " in the perturbed FLRW metric

(an inequality in this relation has been called gravitational slip by Caldwell et al. (2007))

as well as modifications to Poisson’s equation, Eq. (150). Examples of this approach can be

seen in Baker (2012); Bean & Tangmatitham (2010); Caldwell et al. (2007); Daniel et al.

(2009, 2010); Daniel & Linder (2010); Dossett et al. (2011a,b); Hojjati et al. (2011, 2012);

Laszlo et al. (2011); Lombriser (2011); Song et al. (2011); Toreno (2011); Zhao et al.

(2010) to name a few. To date, though, explorations of these modifications have only been

performed in flat spacetimes. We now focus on extending the modified growth equations

seen in, for example, Bean & Tangmatitham (2010); Dossett et al. (2011b) to non-flat

cases.

Extending the parameterized modifications of the growth equations, (150) and (151),

proposed by Bean & Tangmatitham (2010) to non-flat models gives:

!

k2 ! 3K"

! = !4#Ga2#

i

$i!i Q (154)

k2(" !R !) = !12#Ga2#

i

$i(1 + wi)%i Q, (155)

where Q and R are the modified growth parameters (MG parameters). A modification to the

Poisson equation is quantified by the parameter Q, while the gravitational slip is quantified

130

by the parameter R (at late times, when anisotropic stress is negligible, ! = R"). As

discussed in our earlier paper Dossett et al. (2011b) we use the parameter D = Q(1+R)/2

instead of R not only to avoid a strong degeneracy between Q and R, but also to have

a parameter which can be directly probed by observations. To obtain a modified growth

equation written in terms of only Q and D, we combine Eqs. (154) and (155) giving:

k2(! + ") =!8#Ga2

1! 3K/k2

!

i

$i!i D ! 12#Ga2!

i

$i(1 + wi)%i Q. (156)

2. Modified Equations in the Synchronous Gauge

In order to perform the tests, we must implement the MG framework into numerical

codes that allow comparisons to the data and calculations of parameter constraints. This is

done by an extended version, with the inclusion of spatial curvature, of the publicly available

package ISiTGR which is an integrated set of modified modules for the publicly available

codes CosmoMC Lewis & Bridle (2002) and CAMB Lewis et al. (2000). CAMB is used to

calculate the various CMB anisotropy spectra (CTT! , CTE

! , CEE! , CBB

! ) as well as the three-

dimensional matter power spectrum P"(k, z) all of which are very powerful in constraining

both the growth history of structure in the universe as well as the expansion history of the

universe, and thus MG parameters.

The package CAMB is written in the synchronous gauge, where the perturbed FLRW metric

is written as:

ds2 = a(&)2[!d& 2 + ('ij + hij)dxidxj ], (157)

Thus, instead of using the metric potentials " and ! of the conformal Newtonian gauge, it

uses the metric potentials h and ( consistent with the notation of Zaldarriaga et al. (1998).

K = �⌦kH20

EVOLVING THE MODIFIED GRAVITY PARAMETERS: BINNING METHODS

4

FIG. 1: MG parameter evolution in redshift and scale modeled using a new hybrid method. We plot here a 3D representationfor an example of the new hybrid binned evolution for the MG parameter Q(k, a) as given by our Eqs. (12) and (13) for theparameters Q(k, a) with Q1 = 1.20, Q2 = 1.15, Q3 = 1.05, Q4 = 1.10, zTGR = 2, and kc = 0.01. We can see along thez-axis how the binned aspect can allow for di!erent best fit values for the MG parameters in the redshift space while alongthe k-axis we can see the monotonic evolution in k evolving from some large scale (small k) value to a small scale (large k)value exponentially. The hybrid parametrization combines the z-binning method that was shown to be robust with a smoothevolution in k space.

To take advantage of all these techniques, we have developed two versions of our code. One version of the codeuses the functional form (11). It provides the option to apply the functional form (11) to either Q and R (as done in[50]) or Q and D. The other version of the code is based on binning methods. It provides the option of the secondapproach with traditional binning (described above), or alternatively the third, hybrid approach with two redshiftbins, but the evolution in scale evolves monotonically.In the binning version of the code, we evolve only Q and D. Transitions between the redshift bin are evolved

following [44, 53] and use a hyperbolic tangent function with a transition width ztw = 0.05. In this way the binningcan actually be written functionally as (with X representing Q or D)

X(k, a) =1 +Xz1(k)

2+

Xz2(k)!Xz1(k)

2tanh

z ! zdivztw

+1!Xz2(k)

2tanh

z ! zTGR

ztw, (12)

where zdiv is the redshift where the transition between the two redshift bins occurs and zTGR is the redshift belowwhich GR is to be tested. We hard code zTGR = 2zdiv to give us equally sized bins, but this of course is optional andcan easily be changed. Xzi(k) represents the binning method for k in the ith z bin. For the suggested hybrid methodit has the form

Xz1(k) = X1e!k/kc +X2(1 ! e!k/kc) (13)

Xz2(k) = X3e!k/kc +X4(1 ! e!k/kc),

while with traditional binning in principle evolves as

Xz1(k) =

!

X1 if k < kcX2 if k " kc,

(14)

Xz2(k) =

!

X3 if k < kcX4 if k " kc.

Here though, we have rather chosen to implement the traditional binning method with some control on the transitionas:

Xz1(k) =X2 +X1

2+

X2 !X1

2tanh

k ! kcktw

(15)

Xz2(k) =X4 +X3

2+

X4 !X3

2tanh

k ! kcktw

,

4




X(k, a) =1 +Xz1(k)

2+

Xz2(k)!Xz1(k)

2tanh

z ! zdivztw

+1!Xz2(k)

2tanh

z ! zTGR

ztw, (12)


Xz1(k) = X1e!k/kc +X2(1 ! e!k/kc) (13)

Xz2(k) = X3e!k/kc +X4(1 ! e!k/kc),


Xz1(k) =

!


(14)

Xz2(k) =

!



Xz1(k) =X2 +X1

2+

X2 !X1

2tanh

k ! kcktw

(15)

Xz2(k) =X4 +X3

2+

X4 !X3

2tanh

k ! kcktw

,

Both Traditional binning and Hybrid Method evolve in redshift as

Scale Dependence

Hybrid Method Traditional Binning Method

zdiv zTGRz

1Xz2!k"Xz1!k"

X!z"

kck

X2

X1

Xz1!k"

kck

X2

X1

Xz1!k"

4




X(k, z) =1 +Xz1(k)

2+

Xz2(k)!Xz1(k)

2tanh

z ! zdivztw

+1!Xz2(k)

2tanh

z ! zTGR

ztw, (12)


Xz1(k) = X1e!k/kc +X2(1 ! e!k/kc) (13)

Xz2(k) = X3e!k/kc +X4(1 ! e!k/kc),


Xz1(k) =

!


(14)

Xz2(k) =

!



Xz1(k) =X2 +X1

2+

X2 !X1

2tanh

k ! kcktw

(15)

Xz2(k) =X4 +X3

2+

X4 !X3

2tanh

k ! kcktw

,10−3 10−2 10−1 100102

103

104

105

k

P b (k

)

GRHybrid MethodTraditional Binning

4




X(k, z) =1 +Xz1(k)

2+

Xz2(k)!Xz1(k)

2tanh

z ! zdivztw

+1!Xz2(k)

2tanh

z ! zTGR

ztw, (12)


Xz1(k) = X1e!k/kc +X2(1 ! e!k/kc) (13)

Xz2(k) = X3e!k/kc +X4(1 ! e!k/kc),


Xz1(k) =

!


(14)

Xz2(k) =

!



Xz1(k) =X2 +X1

2+

X2 !X1

2tanh

k ! kcktw

(15)

Xz2(k) =X4 +X3

2+

X4 !X3

2tanh

k ! kcktw

,

EVOLVING THE MODIFIED GRAVITY PARAMETERS: BINNING METHODS

4




X(k, a) =1 +Xz1(k)

2+

Xz2(k)!Xz1(k)

2tanh

z ! zdivztw

+1!Xz2(k)

2tanh

z ! zTGR

ztw, (12)


Xz1(k) = X1e!k/kc +X2(1 ! e!k/kc) (13)

Xz2(k) = X3e!k/kc +X4(1 ! e!k/kc),


Xz1(k) =

!


(14)

Xz2(k) =

!



Xz1(k) =X2 +X1

2+

X2 !X1

2tanh

k ! kcktw

(15)

Xz2(k) =X4 +X3

2+

X4 !X3

2tanh

k ! kcktw

,

4




X(k, a) =1 +Xz1(k)

2+

Xz2(k)!Xz1(k)

2tanh

z ! zdivztw

+1!Xz2(k)

2tanh

z ! zTGR

ztw, (12)


Xz1(k) = X1e!k/kc +X2(1 ! e!k/kc) (13)

Xz2(k) = X3e!k/kc +X4(1 ! e!k/kc),


Xz1(k) =

!


(14)

Xz2(k) =

!



Xz1(k) =X2 +X1

2+

X2 !X1

2tanh

k ! kcktw

(15)

Xz2(k) =X4 +X3

2+

X4 !X3

2tanh

k ! kcktw

,

Both Traditional binning and Hybrid Method evolve in redshift as

Scale Dependence

Hybrid Method Traditional Binning Method

zdiv zTGRz

1Xz2!k"Xz1!k"

X!z"

kck

X2

X1

Xz1!k"

kck

X2

X1

Xz1!k"

10−3 10−2 10−1 100102

103

104

105

k

P b (k

)

GRHybrid MethodTraditional Binning

EVOLVING THE MODIFIED GRAVITY PARAMETERS: FUNCTIONAL EVOLUTION

In this evolution method we assume scale independent evolution. The parameters evolve in terms of the scale factor as:

3

B. Modified Gravity Growth Parameters

Parametrizing both modifications to Poisson’s equation, (2), as well as the ratio between the two metric potentials! and " in the perturbed FLRW metric (called gravitational slip by Caldwell et al. [11]) has recently been the subjectof a lot of the work on testing general relativity; see, for example, [11, 49–53]. The parameters we use in this paperto describe modifications to the growth [modified gravity MG parameters] are based upon those used in [50].The parametrized modifications to the growth equations proposed by [50] directly modify Eqs. (2) and (3) and

make no assumptions as to the time when a deviation from GR is allowed. These modifications are as follows:

k2! = !4#Ga2!

i

$i!i Q (8)

k2(" !R!) = !12#Ga2!

i

$i(1 + wi)%i Q, (9)

where Q and R are the MG parameters. The parameter Q represents a modification to the Poisson equation, whilethe parameter R quantifies the gravitational slip (at late times, when anisotropic stress is negligible, R = "/!). In ourcode, rather than using the parameter R which is degenerate with Q, we instead use the parameter D = Q(1 +R)/2as suggested in [50] (this parameter is equivalent to the parameter " in [45, 53] or G of [52]). Combining Eqs. (8)and (9), we arrive at the second modified growth equation used in this paper:

k2(" + !) = !8#Ga2!

i

$i!i D ! 12#Ga2!

i

$i(1 + wi)%i Q. (10)

So, the modified growth equations are (8) and (10), and Q and D are now the MG parameters. As discussed in ourprevious work [56], this approach of using the parameter D instead of R is also useful because observations of theweak-lensing and ISW are sensitive to the sum of the metric potentials ! + " and its time derivative respectively.Thus observations are able to give us direct measurements of this parameter.

C. Di!erent Approaches to Evolving Modified Growth Parameters in Time and Scale

To date, there have primarily been two approaches to evolving the MG parameters in time and scale; one usinga continuous functional form and the other based on binning. We implement in ISiTGR the two approaches and,additionally, a new hybrid approach, as we explain below.The first approach involves defining a functional form for each parameter that allows it to evolve monotonically

in both time and scale. This allows one to make no assumptions as to when a deviation from general relativity isallowed. Such an approach was taken in for example [50]. In that work the functional form,

X(a) = (X0 ! 1) as + 1 (11)

was assumed, where X denotes either Q or R in Eqs. (8) and (9). Thus a total of six model parameters are used totest GR: Q0, R0, Q!, R!, kc, and s. The parameters s and kc parametrize time and scale dependence respectively,with GR values s = 0 and kc = ". Q0 and R0 are the present-day superhorizon values while Q! and R! are thepresent-day subhorizon values of the Q(k, a) and R(k, a) , all taking GR values of 1.In the second approach, instead of evolving each of the parameters assuming some functional form, one can bin the

MG parameters. This approach allows the parameters to take on di#erent values in predefined redshift and scale bins.This technique was used in, for example, [52, 53]. In those works two redshift and two scale bins were defined and forredshifts above a certain critical redshift, GR was assumed to be valid. In each bin the parameters were allowed totake on di#erent values resulting in a total of eight model parameters used to test GR.The third approach that we propose here is a hybrid one where the evolution in redshift (or time) is binned into

two redshift bins, but the evolution in scale evolves monotonically in the same way as the functional form above. Ourmotivation for this hybrid binning approach is that it takes advantage of an evolution in scale that is not so abruptas that in the traditional binning method, while still taking advantage of a redshift (time) dependence expressed inthe form of bins, which was shown to be more robust than time functional forms [53, 56, 58]. For example, in [56], wefound that binning methods do not display the extent of tensions between the MG parameters (preferred by di#erentdata sets) as in the functional form method, where tensions are exacerbated by the chosen functional form. Also, inRef. [58], the author found similar to what we noticed and that the constraints on MG parameters depend stronglyon the parameter s (the scale factor exponent in the functional parametrization). They further looked at ways toremove this strong dependence on the parameter s suggesting and exploring binning as a solution.

As a function of redshift with s = 3

1 2 3 z

1

X0

X!z"

CORRELATIONS WITH CURVATURE PARAMETER

136

D. Results

1. Modified Growth and Cosmological Parameters Used

For all results, in addition to the curvature parameter !k and the MG parameters we vary

the six core cosmological parameters: !bh2 and the !ch2, the baryon and cold-dark matter

physical density parameters, respectively; !, the ratio of the sound horizon to the angular

diameter distance of the surface of last scattering; "rei, the reionization optical depth; ns,

the spectral index; and ln 1010As, the amplitude of the primordial power spectrum.

We use three di!erent parameterizations of the MG parameters: two scale dependent

methods including a traditional binning method and a hybrid evolution method, both of

which were discussed in our previous work Dossett et al. (2011b), and a scale independent

method using a functional first introduced by introduced by Bean & Tangmatitham (2010):

1. For traditional binning both redshift, z, and scale, k, are binned in two bins creating

a total of four bins. The z-bins are 0 < z ! 1 and 1 < z ! 2 while for z > 2 GR is

assumed valid. The k-bins are simply k ! 0.01 and k > 0.01. This binning can be

described functionally as

Q(k, a) =1 +Qz1(k)

2+

Qz2(k)"Qz1(k)

2tanh

z " 1

0.05

+1"Qz2(k)

2tanh

z " 2

0.05, (175)

D(k, a) =1 +Dz1(k)

2+

Dz2(k)"Dz1(k)

2tanh

z " 1

0.05

+1"Dz2(k)

2tanh

z " 2

0.05,

¢ What can we predict analytically? �  We would expect the MG parameters to be positively

correlated with

¢ Use current data to explore correlations. �  WMAP 7 year temperature and polarization spectra �  Union 2 Supernovae Data �  BAO from Two-Degree Field, SDSS-DR7, and WiggleZ �  Matter Power Spectrum (MPK) from SDSS-DR7 �  ISW-galaxy cross-correlations (SDSS-LRG, 2MASS, NVSS) �  Refined HST COSMOS 3D weak lensing tomography.

136

D. Results















Q(k, a) =1 +Qz1(k)

2+

Qz2(k)"Qz1(k)

2tanh

z " 1

0.05

+1"Qz2(k)

2tanh

z " 2

0.05, (175)

D(k, a) =1 +Dz1(k)

2+

Dz2(k)"Dz1(k)

2tanh

z " 1

0.05

+1"Dz2(k)

2tanh

z " 2

0.05,

130

by the parameter R (at late times, when anisotropic stress is negligible, ! = R"). As

discussed in our earlier paper Dossett et al. (2011b) we use the parameter D = Q(1+R)/2

instead of R not only to avoid a strong degeneracy between Q and R, but also to have

a parameter which can be directly probed by observations. To obtain a modified growth

equation written in terms of only Q and D, we combine Eqs. (154) and (155) giving:

k2(! + ") =!8#Ga2

1! 3K/k2

!

i

$i!i D ! 12#Ga2!

i

$i(1 + wi)%i Q. (156)

2. Modified Equations in the Synchronous Gauge

In order to perform the tests, we must implement the MG framework into numerical

codes that allow comparisons to the data and calculations of parameter constraints. This is

done by an extended version, with the inclusion of spatial curvature, of the publicly available

package ISiTGR which is an integrated set of modified modules for the publicly available

codes CosmoMC Lewis & Bridle (2002) and CAMB Lewis et al. (2000). CAMB is used to

calculate the various CMB anisotropy spectra (CTT! , CTE

! , CEE! , CBB

! ) as well as the three-

dimensional matter power spectrum P"(k, z) all of which are very powerful in constraining

both the growth history of structure in the universe as well as the expansion history of the

universe, and thus MG parameters.

The package CAMB is written in the synchronous gauge, where the perturbed FLRW metric

is written as:

ds2 = a(&)2[!d& 2 + ('ij + hij)dxidxj ], (157)

Thus, instead of using the metric potentials " and ! of the conformal Newtonian gauge, it

uses the metric potentials h and ( consistent with the notation of Zaldarriaga et al. (1998).

K = �⌦kH20

CORRELATIONS WITH CURVATURE PARAMETER CONT’D

136

D. Results















Q(k, a) =1 +Qz1(k)

2+

Qz2(k)"Qz1(k)

2tanh

z " 1

0.05

+1"Qz2(k)

2tanh

z " 2

0.05, (175)

D(k, a) =1 +Dz1(k)

2+

Dz2(k)"Dz1(k)

2tanh

z " 1

0.05

+1"Dz2(k)

2tanh

z " 2

0.05,

¢ Can assuming a flat universe when the universe is actually curved affect MG parameter constraints? �  Generate simulated higher precision data to see.

Q1

1K

2 4

−0.03−0.02−0.01

0

Q2

0 2 4

−0.03−0.02−0.01

0

Q3

0.5 1.5 2.5

−0.03−0.02−0.01

0

Q4

0 2 4

−0.03−0.02−0.01

0

D1

1K

1 1.2

−0.03−0.02−0.01

0

D20.8 1.2 1.6

−0.03−0.02−0.01

0

D30.9 1 1.1

−0.03−0.02−0.01

0

D40.8 1 1.2

−0.03−0.02−0.01

0

Q1

1K

0 2 4

−0.03−0.02−0.01

00.01

Q2

0 2 4

−0.03−0.02−0.01

00.01

Q3

1 2 3

−0.03−0.02−0.01

00.01

Q4

0 1 2 3

−0.03−0.02−0.01

00.01

D1

1K

1 1.5

−0.03−0.02−0.01

00.01

D20.8 1.2 1.6

−0.03−0.02−0.01

00.01

D30.8 1 1.2

−0.03−0.02−0.01

00.01

D40.8 1 1.21.4

−0.03−0.02−0.01

00.01

Q0

1K

0.5 1.5 2.5

−0.02

−0.01

0

0.01

D00.8 1 1.2

−0.02

−0.01

0

0.01

R00 1 2

−0.02

−0.01

0

0.01

Traditional Binning Hybrid Binning

EFFECT OF CURVATURE ON MG PARAMETER CONSTRAINTS

Ωk=0.01 Ωk=-0.02

9

D1

Q1

1 1.2

1

2

3

D2

Q2

0.8 1 1.2

0.51

1.52

2.5

D3

Q3

0.9 1 1.1

0.51

1.52

D4

Q4

0.9 1 1.1

0.51

1.52

2.5

D1

Q1

1 1.5

1

2

3

4

D2

Q2

0.8 1 1.2

0.51

1.52

2.5

D3

Q3

1 1.2

1

2

3

D4

Q4

0.911.1

0.51

1.52

2.5

Q0

D0

0.5 1 1.5

0.9

1

1.1

Q0

R0

0.5 1 1.5

0.5

1

1.5

2

FIG. 4: Plotted are the 68% and 95% 2D confidence contours for the MG parameters when a spatially flat model is used butthe actual underlying universe has !k = 0.01. TOP: Confidence contours for the MG parameters Qi and Di, i = 1, ..4 when thetraditional binning method is used. All parameter values are pulled to smaller values. MIDDLE: Confidence contours for theMG parameters Qi and Di, i = 1, ..4 when the hybrid evolution method is used. Most of the parameter contours are pulled tosmaller values. BOTTOM: Confidence contours for the MG parameters Q0 and D0 and R0 when the functional form binningmethod is used. The Q0 !D0 contour is pulled noticeably toward smaller parameter values.

limits on !k. The constraints on the MG parameters for these data sets is plotted in Figs. 4 and 5 respectively.Already for these values of !k one can see that the GR point (shown where the two dot-dashed lines cross in eachfigure) is moving away from the best-fit point. In fact, in the case where !k = !0.02, for the functional form evolutionthere is very nearly a deviation from GR at the 95% level. This shows that indeed ignoring curvature can possiblyproduce a false positive for a deviation from general relativity when higher precision data is used.To further explore how large an impact assuming the universe is flat when in fact it is curved will have on constraints

on the MG parameters four more fiducial data sets were included with !k = ±0.05 and !k = ±0.1. Though thesevalues for the curvature parameter !k are well outside current constraints from all combined data, these constraintson !k were obtained by using a "CDM model, and while in the "CDM cosmological model observations requirea universe that is flat or very close to it, modified theories of gravity or inhomogeneous cosmological models mayrequire the universe to be significantly more curved to fit observations. Additionally, given the fact that our simulateddata sets do not represent the constraining power of future experiments, it is useful to use these larger values ofthe curvature parameter to illustrate how constraints from data sets with smaller uncertainties or more data will bea#ected. While the data sets we simulated and used may require these large curvature values for a significant apparentdeviation to arise, future data sets with more precise measurements may not. For most of these values of !k, we findfor every parameter evolution method (traditional binning, hybrid method, or functional form) there is at least oneMG parameter and in most cases more than one MG parameter that strongly deviates from its GR value. Almost allQ ! D 2D parameter contours for these values of !k show the GR, (1, 1), point outside their 95% confidence limitsconstraints. In Figs. 6 and 7 the 2D confidence contours for the !k = 0.05 and !k = !0.1 fiducial data sets are shownrespectively. Interestingly the negatively curved model (!k = 0.05) deviates from GR much more substantially thandoes the closed model for all evolution methods of the MG parameters. This could be due to the way the comovingangular diameter distance enters into both the weighting factor for the weak lensing as well as the way the wavenumber, k, is determined when calculating the lensing cross power spectrum.

10

D1

Q1

1 1.2

0.51

1.52

D2

Q2

1 1.2

0.5

1

1.5

2

D3

Q3

1 1.1

0.5

1

1.5

D4

Q4

0.9 1.1

0.5

1

1.5

2

D1

Q1

0.8 1 1.2

1

2

3

D2

Q2

1 1.2

0.5

1

1.5

2

D3

Q3

0.9 1.1

0.51

1.52

2.5

D4

Q4

0.9 1.1

0.5

1

1.5

2

Q0

D0

0.5 1 1.50.9

1

1.1

Q0

R0

0.5 1 1.50

1

2

FIG. 5: Plotted are the 68% and 95% 2D confidence contours for the MG parameters when a spatially flat model is used butthe actual underlying universe has !k = !0.02 TOP: Confidence contours for the MG parameters Qi and Di, i = 1, ..4 whenthe traditional binning method is used. All parameter values are pulled to larger values. MIDDLE: Confidence contours forthe MG parameters Qi and Di, i = 1, ..4 when the hybrid evolution method is used. Most of the parameter contours are pulledto larger values. BOTTOM: Confidence contours for the MG parameters Q0 and D0 and R0 when the functional form binningmethod is used. The Q0 !D0 contour is pulled noticeably toward larger parameter values.

Correlation coe!cients between !k and the MG parameters

MG parameters evolved using traditional binningQ1 Q2 Q3 Q4 D1 D2 D3 D4

0.3783 0.1289 0.1201 0.0074 0.3135 0.0492 0.0748 -0.0102

MG parameters evolved using hybrid binningQ1 Q2 Q3 Q4 D1 D2 D3 D4

0.4591 0.1997 -0.0489 0.1584 0.4244 0.0982 0.0968 0.0278

MG parameters evolved using the functional formQ0 D0 R0

0.0289 0.0969 -0.0095

TABLE II: We list the correlation coe"cients between !k and the various MG parameters for a universe with !k = 0.05.In contrast to those obtained from current observational data, these correlation coe"cients are almost all consistent with thetrends observed in Fig. 8 when a flat universe is assumed but curved simulated future data is used.

One other feature to notice in all four sets of plots is the directions which the constraint contours move in theparameter space. Looking at Eqs. (8) and (10), one would expect that for a universe with a positive value of !k andthus a negative K value the assumption of a flat model would demand smaller values for the parameters Q and Dand vice versa. This is indeed the trend we see for the most part in Figs. 4, 5, 6, and 7. To further show how fiducialmodels with higher values of !k have lower best-fit MG parameters when a flat universe is assumed, in Fig. 8 we plotthe best-fit MG parameter values versus !k for the various evolution methods. As was discussed above most of theparameters exhibit a negatively sloping trend. This seems to suggest that the observed behavior where most of theQ and D parameters move the same direction in the parameter space could act as a signature of a false positive in

EFFECT OF CURVATURE ON MG PARAMETER CONSTRAINTS CONT’D

11

D1

Q1

0.9 10.85

0.9

0.95

D2

Q2

0.95 10.85

0.9

0.95

1

D3

Q3

0.9 10.7

0.8

0.9

1

D4Q4

0.9 1.10.6

0.8

1

D1

Q1

0.9 1

0.85

0.9

0.95

D2

Q2

0.9 1

0.85

0.9

0.95

D3

Q3

0.9 1.1

0.81

1.21.4

D4

Q4

0.8 10.4

0.6

0.8

1

Q0

D0

0.4 0.6

0.65

0.7

0.75

Q0

R0

0.4 0.6

1.6

1.7

1.8

1.9

FIG. 6: Plotted are the 68% and 95% 2D confidence contours for the MG parameters when a spatially flat model is used butthe actual underlying universe has !k = 0.05. TOP: Confidence contours for the MG parameters Qi and Di, i = 1, ..4 whenthe traditional binning method is used. All parameter values are pulled to smaller values and indicate a deviation from generalrelativity. MIDDLE: Confidence contours for the MG parameters Qi and Di, i = 1, ..4 when the hybrid evolution method isused. Most of the parameter contours are pulled to smaller values and indicate a deviation from general relativity. BOTTOM:Confidence contours for the MG parameters Q0 and D0 and R0 when the functional form binning method is used. A strongdeviation (due to the assumption of spatial flatness) from reneral Relativity is present in both contours.

the future if flatness is assumed and a deviation from GR is detected.One can also notice that the trends discussed above do not match the behavior expected from the correlation coef-

ficients obtained when using the current data. For the Q and D parameters the behavior described above correspondsto positive correlation because by assuming flatness (for example, in the case of a positive !k) we are forcing a lowervalue for !k and thus would expect a lower value for our MG parameters. We do not observe this behavior with thecurrent data. This is most likely due to the large uncertainties in this data which allows a large number of models tofit quite well, and thus makes the calculation of the correlation coe"cients less accurate. A more accurate descriptionof the correlation coe"cients can be obtained by using some that were calculated when running the ”null” tests onour fiducial models. We show the correlation coe"cients for the case when !k = 0.05 in Table II. These correlationcoe"cients are almost all consistent with the trends we see in the constraints as well as the best-fits.

V. CONCLUSION

In this work we extended previous studies and the framework of modified growth (MG) parameters to test generalrelativity (GR) at cosmological scales in order to include spatial curvature in the models, as while current data whenanalyzed using the #CDM model points to a universe that is flat or very close to it, this constraint may not hold inmodified theories of gravity or inhomogenous cosmological models. Using the latest cosmological data sets we exploredthe correlations between MG parameters and the curvature parameter !k finding that indeed there are non-negligiblecorrelations. We next used future simulated data to explore whether assuming a spatially flat model on a spatiallycurved universe would a$ect the MG parameter constraints. We found that indeed, for our simulated data sets, suchan assumption of flatness can cause tension with GR when using an !k as little as 0.02 away from the flat caseand significant apparent deviations for |!k| ! 0.05 . Models with larger departures from flatness cause more MGparameters to deviate from GR with even larger discrepancies. We also found that negatively curved models deviated

Ωk=0.05

12

D1

Q1

1 1.11.121.141.161.181.21.22

D2

Q2

1 1.051

1.021.041.061.08

D3

Q3

1 1.11.05

1.1

1.15

1.2

D4

Q4

1 1.11.1

1.15

1.2

1.25

D1

Q1

1 1.1

1.15

1.2

1.25

D2

Q2

1 1.11.081.11.121.141.161.18

D3

Q3

0.9 10.8

0.9

1

1.1

D4

Q4

1 1.1

1.351.41.451.51.55

Q0

D0

1 2

1.1

1.2

1.3

Q0

R0

1 2

0.51

1.52

2.5

FIG. 7: Plotted are the 68% and 95% 2D confidence contours for the MG parameters when a spatially flat model is used butthe actual underlying universe has !k = !0.1 TOP: Confidence contours for the MG parameters Qi and Di, i = 1, ..4 whenthe traditional binning method is used. All parameter values are pulled to larger values and indicate a deviation from generalrelativity. MIDDLE: Confidence contours for the MG parameters Qi and Di, i = 1, ..4 when the hybrid evolution method isused. Most of the parameter contours are pulled to larger values and indicate a deviation from general relativity. BOTTOM:Confidence contours for the MG parameters Q0 and D0 and R0 when the functional form binning method is used. The Q0!D0

shows a deviation from general relativity, while the Q0 !R0 contour shows some tension with the GR point.

more quickly and more significantly from GR when the assumption of flatness is made.As expected from the derived modified growth equations, for all approaches and for most of the bins, we find

positive correlation coe!cients between the MG parameters and "k from using simulated future data which is muchmore precise than what is currently available. The trends for the best-fit MG parameters versus fiducial values of "k

used are found to be consistent with these correlations. Though the values of the correlation coe!cients found are ofcourse dependent upon the parametrization used, the signs of the correlations coe!cients for the most part remainconsistent across parametrizations.The results obtained in this analysis show that when using high-precision data from future experiments in order

to test general relativity at cosmological scales or look for deviations from it, one must take into account the e#ectof curvature. This point may also be relevant when trying to test other, alternative theories of gravity. Indeed,our results indicate that the assumption of a spatially flat universe when performing these tests can bias the MGparameter constraints, leading to apparent deviations from general relativity.

Acknowledgments

We would like to thank Dr. Lucia Popa and Ana Carmete for pointing out a bug in the binning version of the ISiTGRcode. This bug a#ected the numerical results of this paper, but not its main conclusions. M.I. acknowledges that thismaterial is based upon work supported by the Department of Energy (DOE) under grant DE-FG02-10ER41310, NASAunder grant NNX09AJ55G, and that part of the calculations for this work have been performed on the CosmologyComputer Cluster funded by the Hoblitzelle Foundation. J.D. acknowledges that this research was supported in partby the DOE O!ce of Science Graduate Fellowship Program (SCGF), made possible in part by the American Recoveryand Reinvestment Act of 2009, administered by ORISE-ORAU under contract no. DE-AC05-06OR23100.

Ωk=-0.1

EFFECT OF CURVATURE ON MG PARAMETER CONSTRAINTS CONT’D

13

Traditional Binning Evolution Hybrid Evolution Functional Form Evolution

−0.1 −0.05 0 0.05 0.1

0.6

0.8

1

1.2

Ωk

Qi

i=1i=2i=3i=4

−0.1 −0.05 −0.02 0.01 0.05 0.1

0.3

0.7

1.1

1.5

Ωk

Qi

i=1i=2i=3i=4

−0.1 −0.05 −0.02 0.01 0.05 0.10.4

0.6

0.8

1

1.2

1.4

Ωk

Q0

−0.1 −0.05−0.02 0.01 0.05 0.1

0.92

0.99

1.06

1.13

Ωk

Di

i=1i=2i=3i=4

−0.1 −0.05−0.02 0.01 0.05 0.1

0.85

0.95

1.05

1.15

Ωk

Di

i=1i=2i=3i=4

−0.1 −0.05 −0.02 0.01 0.05 0.1

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Ωk

D0

FIG. 8: Here we plot the best-fit points for the parameters Q and D for the various evolution methods versus the underlying!k fiducial values while a spatially flat universe is assumed. Most of the trends have a negative slope which is expected fromlooking at Eqs (8) and (10) (note that this trend is consistent with the positive correlations between the MG parameters andthe curvature parameter as explained in Sec. IV-C). TOP LEFT: Plots for the parameters Qi, i = 1, 2, 3, 4 for the traditionalbinning method. TOP MIDDLE: Plots for the parameters Qi, i = 1, 2, 3, 4 for the hybrid evolution method. TOP RIGHT: Plotsfor the parameters Q0 for the functional form evolution method. BOTTOM LEFT: Plots for the parameters Di, i = 1, 2, 3, 4for the traditional binning method. BOTTOM MIDDLE: Plots for the parameters Di, i = 1, 2, 3, 4 for the hybrid evolutionmethod. BOTTOM RIGHT: Plots for the parameters D0 for the functional form evolution method.

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CONCLUSIONS

¢ Curvature is positively correlated with the MG parameters Q and D.

¢  Ignoring curvature can cause an apparent deviation from GR.

¢ Negatively curved models deviate more significantly than do positively curved models.

¢ Must include Ωk in parameter analysis along with MG and other cosmological parameters when using future data.

ACKNOWLEDGMENTS

¢ DOE Office of Science Graduate Fellowship. ¢ DOE Grant DE-FG02-10ER41310. ¢ Part of the calculations for this work were

performed on Cosmology Computer Cluster funded by Hoblitzelle Foundation.

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16







16







16







6

where px, py are the parameters, Cov(px, py) is the covariance of the two parameters, and !(px) and !(py) are theirrespective standard deviations. We see that indeed there exists a correlation between these parameters indicatingthat ignoring curvature may possibly be able to mimic a deviation to GR in the MG parameter space. To explorethis possibility further, in the next section, we generate future cosmological data with di!erent curvature values andsee how much the MG parameter constraints are a!ected by (wrongly) assuming a flat background.

C. Future constraints

V. CONCLUSION

The impact of curvature on tests of general relativity is an important area of study in an era of precision cosmologywhere data allows us to test if the cause of cosmic acceleration is due to dark energy or an extension to generalrelativity. To this point studies using modified growth parameters have done so assuming a flat background. In thiswork we extended the framework established in [13] to allow for curved models. Then using the latest cosmologicaldata sets we explored the correlations between MG parameters and "k finding that indeed there is correlation. Finallyto see how much an a!ect assuming a flat model when the background was curved would have on the MG parameterconstraints, we generated curved fiducial data and fit it assuming a flat model. We find ...

Acknowledgments

We thank T. Schrabback for providing the refined HST COSMOS data. We thank A. Lewis for useful comments.It should be noted that this paper has been prepared as a follow up paper to our previous work [16]. During thepreparation of this manuscript, a new version of the code MGCAMB was released with an accompanying paper [86] witha similar aim to ours but a separate code. ISiTGR is an integrated set of modified modules for CosmoMC and CAMBcontaining a variety of changes to both and incorporating a modified version of the ISW-galaxy cross correlationmodule of [69, 70], a new likelihood module for the refined HST COSMOS weak-lensing tomography of [71], and ageneralized baryon acoustic oscillation likelihood code that includes recently released WiggleZ BAO measurementsfrom [72]. M.I. acknowledges that this material is based upon work supported by Department of Energy (DOE) undergrant DE-FG02-10ER41310, NASA under grant NNX09AJ55G, and that part of the calculations for this work havebeen performed on the Cosmology Computer Cluster funded by the Hoblitzelle Foundation. J.D. acknowledges thatthis research was supported in part by the DOE O#ce of Science Graduate Fellowship Program (SCGF). The DOESCGF Program was made possible in part by the American Recovery and Reinvestment Act of 2009. The DOE SCGFprogram is administered by the Oak Ridge Institute for Science and Education for the DOE. ORISE is managed byOak Ridge Associated Universities (ORAU) under DOE contract number DE-AC05-06OR23100.

[1] J.R. Bond, G. Efsthatiou, and M. Tegmark, Mon. Not. R. Astron. Soc.291, L33 (1997).[2] M. Zaldarriaga, D.N. Spergel, and U. Seljak, Astrophys. J. 488, 1 (1997).[3] G. Efsthatiou and J.R. Bond, Mon. Not. R. Astron. Soc.304, 75 (1999).[4] G. Huey et al., Phys. Rev. D.59, 063005 (1999).[5] E. Komatsu et al., Astrophys. J. S. 180, 330 (2009).[6] J.M.Virey et al., (2008), arXiv:0802.4407.[7] R. Bean and M. Tangmatitham, Phys. Rev. D.81, 083534 (2010), arXiv:1002.4197.[8] S. Daniel and E. Linder Phys. Rev. D.82, 103523 (2010), arXiv:1008.0397.[9] G. Zhao et. al. Phys. Rev. D.81, 103510 (2010), arXiv:1003.0001.

[10] L. Lombriser, Phys. Rev. D.83, 063519 (2011), arXiv:1101.0594.[11] I. Toreno, E. Semboloni, and T. Schrabback, arXiv:1012.5854.[12] Y. S. Song, G. B. Zhao, D. Bacon, K. Koyama, R. C. Nichol, L. Pogosian, arXiv:1011.2106.[13] J. Dossett, M. Ishak, and J. Moldenhauer, Phys. Rev. D.84, 123001 (2011), arXiv:1109.4583[14] R. Caldwell, A. Cooray, and A. Melchiorri, Phys. Rev. D.76, 023507 (2007), arXiv:astro-ph/0703375.[15] S. Daniel, E. Linder, T. Smith, R. Caldwell, A. Corray, A Leauthaud, and L. Lombriser, Phys. Rev. D 80, 123508 (2010),

arXiv:1002.1962.[16] J. Dossett, J. Moldenhauer, and M. Ishak, Phys. Rev. D.84, 023012, (2011), arXiv:1103.1195.[17] A. Lue, R. Scoccimarro, and G. Starkman, Phys. Rev. D.69, 124015 (2004).[18] Y. S. Song, Phys. Rev. D.71, 024026 (2005).[19] L. Knox, Y.S. Song, and J. A. Tyson, Phys. Rev. D.74, 023512 (2006), arXiv:astro-ph/0503644.

16

[41] J. Dossett, M. Ishak, J. Moldenhauer, Y. Gong, A. Wang, J. Cosmol. Astropart. Phys.04 (2010) 022, arXiv:1004.3086.[42] P. Ferreira and C. Skordis, Phys. Rev. D.81, 104020 (2010), arXiv:1003.4231.[43] J. Jing and S. Chen, Phys. Lett. B 685, 185 (2010), arXiv:0908.4379.[44] L. Pogosian, A. Sivestri,K. Koyama, and G. Zhao, Phys. Rev. D.81, 104023 (2010), arXiv:1002.2382.[45] Y. S. Song, L. Hollenstein, G. Caldera-Cabral, and K. Koyama, J. Cosmol. Astropart. Phys.04 (2010) 018, arXiv:1001.0969.[46] S. Basilakos, M. Plionis, and A. Pouri, Phys. Rev. D.83, 123525 (2011), arXiv:1106.1183.[47] S. Wang, L. Hui, M. May, and Z. Haiman, Phys. Rev. D.76, 063503 (2007), arXiv:0705.0165.[48] M. Ishak, Foundation of Physics Journal 37, 1470 (2007), astro-ph/0504416.[49] S. Daniel, R. Caldwell, A. Cooray, P. Serra, and A. Melchiorri, Phys. Rev. D.80, 023532 (2009), arXiv:0901.0919.[50] R. Bean and M. Tangmatitham, Phys. Rev. D.81, 083534 (2010), arXiv:1002.4197.[51] S. Daniel, E. Linder, T. Smith, R. Caldwell, A. Corray, A Leauthaud, and L. Lombriser, Phys. Rev. D 81, 123508 (2010),

arXiv:1002.1962.[52] S. Daniel and E. Linder Phys. Rev. D.82, 103523 (2010), arXiv:1008.0397.[53] G. Zhao et. al. Phys. Rev. D.81, 103510 (2010), arXiv:1003.0001.[54] L. Lombriser, Phys. Rev. D.83, 063519 (2011), arXiv:1101.0594.[55] I. Toreno, E. Semboloni, and T. Schrabback, Astron. Astrophys. 530, A68 (2011), arXiv:1012.5854.[56] J. Dossett, J. Moldenhauer, and M. Ishak, Phys. Rev. D.84, 023012, (2011), arXiv:1103.1195.[57] A. Hojjati, L. Pogosian, G. Zhao, J. Cosmol. Astropart. Phys.08 (2011) 005, arXiv:1106.4543.[58] Y. S. Song, G. B. Zhao, D. Bacon, K. Koyama, R. C. Nichol, and L. Pogosian, Phys. Rev. D.84, 083523 (2011),

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[76] B. A. Reid et al. Mon. Not. R. Astron. Soc.Volume 404, 60 (2010), arXiv:0907.1659.[77] W. J. Percival et al. Mon. Not. R. Astron. Soc.Volume 401, 2148 (2010), arXiv:0907.1660.

6

where px, py are the parameters, Cov(px, py) is the covariance of the two parameters, and !(px) and !(py) are theirrespective standard deviations. We see that indeed there exists a correlation between these parameters indicatingthat ignoring curvature may possibly be able to mimic a deviation to GR in the MG parameter space. To explorethis possibility further, in the next section, we generate future cosmological data with di!erent curvature values andsee how much the MG parameter constraints are a!ected by (wrongly) assuming a flat background.

C. Future constraints

V. CONCLUSION

The impact of curvature on tests of general relativity is an important area of study in an era of precision cosmologywhere data allows us to test if the cause of cosmic acceleration is due to dark energy or an extension to generalrelativity. To this point studies using modified growth parameters have done so assuming a flat background. In thiswork we extended the framework established in [13] to allow for curved models. Then using the latest cosmologicaldata sets we explored the correlations between MG parameters and "k finding that indeed there is correlation. Finallyto see how much an a!ect assuming a flat model when the background was curved would have on the MG parameterconstraints, we generated curved fiducial data and fit it assuming a flat model. We find ...

Acknowledgments

We thank T. Schrabback for providing the refined HST COSMOS data. We thank A. Lewis for useful comments.It should be noted that this paper has been prepared as a follow up paper to our previous work [16]. During thepreparation of this manuscript, a new version of the code MGCAMB was released with an accompanying paper [86] witha similar aim to ours but a separate code. ISiTGR is an integrated set of modified modules for CosmoMC and CAMBcontaining a variety of changes to both and incorporating a modified version of the ISW-galaxy cross correlationmodule of [69, 70], a new likelihood module for the refined HST COSMOS weak-lensing tomography of [71], and ageneralized baryon acoustic oscillation likelihood code that includes recently released WiggleZ BAO measurementsfrom [72]. M.I. acknowledges that this material is based upon work supported by Department of Energy (DOE) undergrant DE-FG02-10ER41310, NASA under grant NNX09AJ55G, and that part of the calculations for this work havebeen performed on the Cosmology Computer Cluster funded by the Hoblitzelle Foundation. J.D. acknowledges thatthis research was supported in part by the DOE O#ce of Science Graduate Fellowship Program (SCGF). The DOESCGF Program was made possible in part by the American Recovery and Reinvestment Act of 2009. The DOE SCGFprogram is administered by the Oak Ridge Institute for Science and Education for the DOE. ORISE is managed byOak Ridge Associated Universities (ORAU) under DOE contract number DE-AC05-06OR23100.

[1] J.R. Bond, G. Efsthatiou, and M. Tegmark, Mon. Not. R. Astron. Soc.291, L33 (1997).[2] M. Zaldarriaga, D.N. Spergel, and U. Seljak, Astrophys. J. 488, 1 (1997).[3] G. Efsthatiou and J.R. Bond, Mon. Not. R. Astron. Soc.304, 75 (1999).[4] G. Huey et al., Phys. Rev. D.59, 063005 (1999).[5] E. Komatsu et al., Astrophys. J. S. 180, 330 (2009).[6] J.M.Virey et al., (2008), arXiv:0802.4407.[7] R. Bean and M. Tangmatitham, Phys. Rev. D.81, 083534 (2010), arXiv:1002.4197.[8] S. Daniel and E. Linder Phys. Rev. D.82, 103523 (2010), arXiv:1008.0397.[9] G. Zhao et. al. Phys. Rev. D.81, 103510 (2010), arXiv:1003.0001.

[10] L. Lombriser, Phys. Rev. D.83, 063519 (2011), arXiv:1101.0594.[11] I. Toreno, E. Semboloni, and T. Schrabback, arXiv:1012.5854.[12] Y. S. Song, G. B. Zhao, D. Bacon, K. Koyama, R. C. Nichol, L. Pogosian, arXiv:1011.2106.[13] J. Dossett, M. Ishak, and J. Moldenhauer, Phys. Rev. D.84, 123001 (2011), arXiv:1109.4583[14] R. Caldwell, A. Cooray, and A. Melchiorri, Phys. Rev. D.76, 023507 (2007), arXiv:astro-ph/0703375.[15] S. Daniel, E. Linder, T. Smith, R. Caldwell, A. Corray, A Leauthaud, and L. Lombriser, Phys. Rev. D 80, 123508 (2010),

arXiv:1002.1962.[16] J. Dossett, J. Moldenhauer, and M. Ishak, Phys. Rev. D.84, 023012, (2011), arXiv:1103.1195.[17] A. Lue, R. Scoccimarro, and G. Starkman, Phys. Rev. D.69, 124015 (2004).[18] Y. S. Song, Phys. Rev. D.71, 024026 (2005).[19] L. Knox, Y.S. Song, and J. A. Tyson, Phys. Rev. D.74, 023512 (2006), arXiv:astro-ph/0503644.

15

managed by Oak Ridge Associated Universities (ORAU) under DOE Contract No. DE-AC05-06OR23100.

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arXiv:1002.1962.[52] S. Daniel and E. Linder Phys. Rev. D.82, 103523 (2010), arXiv:1008.0397.[53] G. Zhao et. al. Phys. Rev. D.81, 103510 (2010), arXiv:1003.0001.[54] L. Lombriser, Phys. Rev. D.83, 063519 (2011), arXiv:1101.0594.[55] I. Toreno, E. Semboloni, and T. Schrabback, Astron. Astrophys. 530, A68 (2011), arXiv:1012.5854.[56] J. Dossett, J. Moldenhauer, and M. Ishak, Phys. Rev. D.84, 023012, (2011), arXiv:1103.1195.[57] A. Hojjati, L. Pogosian, G. Zhao, J. Cosmol. Astropart. Phys.08 (2011) 005, arXiv:1106.4543.

ISITGR!

¢ ISiTGR is publicly available at: http://www.utdallas.edu/~jdossett/isitgr

¢  J. Dossett, M. Ishak, and J. Moldenhauer, Phys. Rev. D 84, 123001 (2011), arXiv:1109.4583

¢  J. Dossett, M. Ishak, Phys. Rev. D 86, 103008, (2012), arXiv:1205.2422

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TESTING G R COSMOLOGICAL S : E OF SPATIAL Cjdossett/files/AAS_2013.pdfMustapha Ishak∗ and Jason Dossett ... where H is the Hubble parameter and the eﬀect of the underlying gravity