Linear Growth of Structure in the Symmetron Model

# Linear Growth of Structure in the Symmetron Model

Philippe Brax Institut de Physique Theorique, CEA, IPhT, CNRS, URA 2306, F-91191Gif/Yvette Cedex, France    Carsten van de Bruck School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, UK    Anne-Christine Davis DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK    Baojiu Li DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge CB3 0HA, UK    Benoit Schmauch Institut de Physique Theorique, CEA, IPhT, CNRS, URA 2306, F-91191Gif/Yvette Cedex, France    Douglas J. Shaw DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
July 13, 2019
###### Abstract

In the symmetron mechanism, the fifth force mediated by a coupled scalar field (the symmetron) is suppressed in high-density regions due to the restoration of symmetry in the symmetron potential. In this paper we study the background cosmology and large scale structure formation in the linear perturbation regime of the symmetron model. Analytic solutions to the symmetron in the cosmological background are found, which agree well with numerical results. We discuss the effect of the symmetron perturbation on the growth of matter perturbation, in particular the implications of the brief period of tachyonic instability caused by the negative mass-squared of the symmetron during symmetry breaking. Our analysis and numerical results show that this instability has only very small effects on the growth of structures on sub-horizon scales, and even at horizon scales its influence is not as drastic as naively expected. The symmetron fifth force in the non-tachyonic regime does affect the formation of structure in a nontrivial way which could be cosmologically observable.

## I Introduction

Scalar fields coupled to matter are generic predictions of many theories of high energy physics. In recent years, this idea has attracted a lot of attention in the context of dark energy cst2006 (), which is believed to be a scalar field or one of its variants Wang et al (2000); Armendariz-Picon et al (2000); Wetterich:1987fm (); Amendola (2000); Perrotta & Baccigalupi (1999). However, it is well known that if a scalar field couples to matter or curvature then a scalar fifth force and a modification to the gravitational law could result. Such new physics has been strongly constrained by local gravity experiments and solar system tests, so that fifth forces and modifications of gravity must been either very short-ranged or very weak, or both. If this is true, then the effect of the scalar field is mainly to drive the accelerating expansion of the Universe, such as in the quintessence model Wang et al (2000).

More cosmologically interesting models could be built if the fifth force or modification to the standard Einstein gravity is only weak and/or short ranged where local experiments are performed, such as in our Solar system where matter density is high and gravity is strong, but could become strong (of gravitational strength) and long-ranged elsewhere. Over the past few years, several models have been proposed to realise this, including the chameleon model kw2004a (); kw2004b (); Brax:2004qh (); ms2006 (); ms2007 (); lb2007 (); bbds2008 (), the environmentally-dependent dilaton model bbds2010 (); bbdls2011 (), the DGP model dgp2000 (), the galileon model nrt2009 (); dev2009 (); dde2009 (); dt2010 () and the symmetron model Hinterbichler & Khoury (2010); Hinterbichler et al. (2011); Winther et al. (2011).

Although all these models predict strong environmental dependence of the fifth force, they work in very different ways. In the chameleon model, for example, the fifth force is suppressed exponentially in high density regions where the scalar field acquires a heavy mass via its coupling to matter. The environmentally dependent dilaton model, on the other hand, drives the scalar field to some critical value in high density regions, which corresponds to a vanishing coupling and therefore vanishing fifth force (though the mass of the scalar field depends on the local matter density as well in this model).

The symmetron model, which is the topic of this work, relies on a similar mechanism to suppress the fifth force in high density regions. Here, when the matter density is high enough, the effective potential of the scalar field has a global minimum at the origin, which corresponds to a vanishing coupling to matter. When matter density drops below some critical value, the symmetry in the effective potential is broken and two local minima develop and move away from the origin, corresponding to a nonzero coupling to matter and thus a non-vanishing fifth force. If later the matter density inside a region becomes high again due to the structure formation process (such as in galaxies and galaxy clusters), the symmetry of the effective potential could be restored and the fifth force vanishes again for that region.

Local experiments could constrain the parameters of the symmetron model. Interestingly, in this constrained parameter space, there are still models which could deviate from the standard CDM cosmology on large scales. The cosmological observables, in particular those relevant for the large scale structure formation, can thus provide valuable information about the symmetron, such as whether it exists, what its observational signatures are, how to differentiate it from other models, etc. In this work, we shall concentrate on the structure formation of the symmetron model in the linear perturbation regime, and show how the symmetron can affect the large scale structure of the Universe.

This paper is arranged as follows: we first briefly overview the symmetron model and how the local tests constrain its parameters in Sects. II.1 and II.2, followed by analytical and numerical study of the background cosmology in the model (Sect. II.3) and a short description of the tachyonic period, during which the scalar field has a negative mass-squared (Sect. II.4). Then in Sect. III we study the general behaviour of the linear perturbations in this model, where we show that the fifth force essentially enhances gravity within its range (given by the comoving Compton length of the symmetron field) and keeps standard gravity unmodified beyond that range. We also show that the tachyonic instability does not affect the structure formation significantly on sub-horizon scales. In Sect. IV we give some numerical results of the large scale structure formation and show that significant deviations from the CDM paradigm can be found, which make the model cosmological interesting. We summarise and conclude in Sect. V. Throughout this paper we shall adopt and the metric convention .

## Ii Symmetron Dynamics

### ii.1 The symmetron

Scalar-tensor theories are characterised by their coupling to matter and their interaction potential. In such a context, the symmetron model was proposed in Hinterbichler & Khoury (2010); Hinterbichler et al. (2011) and is described by the action:

 S=∫√−gd4x[R2κ−12(∇ϕ)2−V(ϕ)]+Sm, (1)

where is the scalar field and its potential, the matter action with the matter fields which are minimally coupled to the Jordan frame metric ; is the Einstein frame metric, which is used to compute the Ricci scalar ; where is Newton’s constant and the reduced Planck mass.

For the symmetron model, the interaction potential and the coupling function are simply chosen such that

 V(ϕ) = V0−12μ2ϕ2+14λϕ4, A(ϕ) = 1+ϕ22M2, (2)

in which is a cosmological constant and has mass dimension 4, and have mass dimension one and is a dimensionless model parameter. We shall see that is needed to explain the recently observed accelerating expansion of the Universe, but being a constant it bears no influence on the dynamics of the symmetron field. Note that the potential has two nontrivial minima while the coupling function is monotonously increasing.

The field equations are obtained by varying the action with respect to the symmetron field , and we have

 □ϕ = V,ϕ(ϕ)−A,ϕ(ϕ)A3(ϕ)~T, ~T = ~gμν~Tμν, (3) ~Tμν = 2√−~gδSmδ~gμν, (4)

where we have defined the Jordan frame energy momentum tensor that is related to the Einstein frame one by . Note that we raise and lower the indices of using the Einstein frame metric, . With this definition, the field equation for and the Einstein equations become

 □ϕ = Veff,ϕ(ϕ;T), (5) Rμν−12Rgμν = κTμνtot, (6)

in which the symmetron field is governed by an effective potential

 Veff(ϕ;T) ≡ V(ϕ)−A(ϕ)T, (8)

and the total energy momentum is given by

 Tμνtot=A(ϕ)Tμν−gμνV(ϕ)+∇μϕ∇νϕ−12gμν(∇ϕ)2, (9)

which satisfies the following conservation equation

 ∇μTμν = A,ϕA(Tgμν−Tμν)∇μϕ. (10)

Note that this implies that for pressureless matter with with a 4-velocity (), we have and is conserved independently of :

 ∇μ(ρmuμ)=0. (11)

As is the usual practice, we define the effective mass of the symmetron field by

 m2(ϕ) = Veff,ϕϕ(ϕ;T) (12) = −μ2+3λϕ2+ρmM2,

where we have used the fact that, from the above equations, the effective potential can be rewritten as

 Veff(ϕ) = 12(ρmM2−μ2)ϕ2+14λϕ4. (13)

Hence, in the symmetron model, as long as is high enough, namely where

 ρ⋆ ≡ μ2M2, (14)

the minimum of the effective potential is at the origin (). In contrast, in vacuum, the symmetry is broken and the potential has two nonzero minima:

 ϕ⋆ = ±μ√λ. (15)

As long as the effective coupling to matter reads

 αϕ(ϕ) = mpl[lnA],ϕ ≈ mplϕM2, (16)

leading to the absence of modification of gravity in dense environments where the field vanishes. The modification of the growth of structure depends on

 α≡2α2ϕ, (17)

and the order of magnitude of in vacuum,

 α⋆=α(ϕ⋆), (18)

is crucial for the growth of the large scale structure. For values of the density lower than , gravity becomes modified in a way which could be tested cosmologically, and characterises the relative strength of the modification.

### ii.2 Gravity tests

In this section, we review some of the gravitational properties of the symmetronHinterbichler & Khoury (2010) The symmetron model is designed to induce modifications of gravity which could be tested in the near future, both gravitationally and cosmologically. Requiring that the energy density at which the curvature at the origin of the potential changes sign (roughly when gravitation undergoes a transition from standard to modified) is close to the current critical energy density (which implies that gravity is modified cosmologically recently), we have the estimate

 M2μ2 ∼ H20m2Pl

Moreover, the modification to gravity is detectable only if it is comparable to (or bigger than) standard gravity, or equivalently the effective coupling , which implies

 ϕ⋆M ∼ 1√A2,

where we have defined . These determine the vacuum mass

 m2(ϕ0) = 2μ2 ∼ O(m2PlM2)H20 = O(A2)H20 (19)

and correspondingly set the interaction range of the symmetron to be . This also determines the self coupling Hinterbichler & Khoury (2010)

 λ ∼ μ2m2plM4 ∼ H20m4PlM6 (20)

It is then crucial to have an estimate for . This follows from the study of solar system tests.

Let us consider a spherical object of density and radius . The static solutions for the field profile are obtained solving

 d2ϕdr2+2rdϕdr=(ρM2−μ2)ϕ+λϕ3 (21)

It is convenient to simplify this equation by separating two regions with different behaviours Hinterbichler & Khoury (2010)

 Veff(r

and

 Veff(r>R) = μ2(ϕ−ϕ⋆)2 (23)

where one assumes that the density vanishes at infinity and outside the body. The solutions read

 ϕ(r

and

 ϕ(r>R) = ϕout(r) = BRre−√2μr+ϕ⋆ (25)

Defining the modified Newton potential at the body’s surface

 ~α ≡ ρR2M2 = 6A2ΦN (26)

and the ratio of the size of the sphere to the range of the symmetron interaction

 ~β = μR (27)

we find that

 A=(1+√2~β)ϕ⋆√~α−~β2cosh√~α−~β2+√2~βsinh√~α−~β2 (28)

Notice that as long as . There are two types of solutions depending on the values of . When we have

 A ≈ ϕ⋆√~α(1−~α2) (29) B ≈ −~αϕ⋆3 (30)

The scalar force acting on a test mass outside the sphere is

 Fϕmass=~αϕmPl∂iϕ∼ϕ2⋆M2~α3Rr2 (31)

when the Newtonian force is implying that

 FϕFN∼2~αϕ2⋆M2m2PlρR2∼2~αM2ρR2=O(1). (32)

This implies that the scalar force is not screened. On the other hand when we have

 A ≈ 2ϕ0√~αe−√~α (33) B ≈ −ϕ0(1−1~α) (34)

The scalar force is now

 Fϕ=αϕmPl∂iϕ∼ϕ2⋆M2Rr2, (35)

hence

 FϕFN∼ϕ20M2m2PlρR2∼1~α≪1, (36)

leading to a large screening of the scalar force.

Phenomenologically, one must impose that in our galaxy . Imposing at least and upon using , one gets or equivalently

 A2≳106. (37)

This implies that .

The range of the symmetron in vacuum is given by

 μ−1≲103H−10∼ 1 Mpc (38)

which corresponds to relevant scales for astrophysics. We will come back to this point later. If then the scalar field is just about screened by the sun as . On the other hand, the earth is not screened as and . What matters then for solar system tests is the value of the field in the galaxy:

 ϕGM≈MmPlRG√αGRsexp(−RG−RsRG√αG). (39)

The most stringent constraint in the solar system is the Cassini bound on the Eddington parameter Bertotti:2003rm (). In the Einstein frame, the metric is expressed as

 g00=−(1+2ΦE) (40)
 gij=(1−2ΦE)δij (41)

while the one that particles follow is the Jordan frame metric

 ~g00=−(1+2ΦJ) (42)
 ~gij=(1−2γΦJ)δij, (43)

from which we have Hinterbichler & Khoury (2010)

 γ−1≈−ϕ2M2ϕ22M2+ΦE≈−ϕ2M2Φ. (44)

Denoting by the moment when the symmetron potential becomes unstable at the origin, we have

 μ2M2=3H20m2PlΩm0(1+z∗)3. (45)

Fixing for instance and while using for the galaxy we get a bound on admissible to evade the Cassini bound .

### ii.3 Cosmological evolution

We are interested in the symmetron evolution during the matter dominated era, when the symmetron potential is negligible compared to the matter density . This can be seen by evaluating the value of the potential at its absolute minimum in vacuum as the potential at the effective minimum is monotonously decreasing with

 V(ϕ⋆) = −μ44λ

which is always . Moreover, as , the coupling function is also approximately equal to one. The Friedmann equation is then to a high precision

 H2 = ρm3m2Pl (46)

in the matter dominated era.

A full picture of the time evolution of the reduced symmetron field is shown in Fig. 1, and we can see that after the symmetron breaking at the symmetron deviates from zero, and finally starts to oscillate around the moving minimum of the effective potential (). The detailed evolution of is of course more complicated, but still understandable analytically in certain limits, as we shall show now.

Let us denote by the instant when and the corresponding scale factor. The scalar field equation of motion reads now

 ¨ϕ+2t˙ϕ+μ2(t2⋆t2−1)ϕ+λϕ3=0 (47)

which can be cast in a dimensionless fashion by defining in addition to the and defined above, where we have the limit

 κ2 = μ2t2⋆ = 4μ29H2⋆ = 4m2Pl3M2 = 43A2 ≳ 106 (48)

The dimensionless version of the symmetron equation of motion is then given by

 φ′′(x)+2xφ′(x)+κ2(1x2−1)φ(x)+κ2φ(x)3=0, (49)

where .

Before the curvature at the origin changes its sign, the symmetron oscillates about the origin , which is then the minimum of the effective potential. Assuming that the initial amplitude is small, the symmetron equation of motion simplifies to

 x2φ′′+2xφ′−κ2(x2−1)φ ≈ 0 (50)

 φ(x) = 1√x[a1Iiι(κx)+b1Kiι(κx)] (51)

where , and are constants of integration which could be determined by the initial conditions. As long as we can expand

 ϕ(t) = (52)

The upper left panel of Fig. 2 shows a numerical example of the evolution of with respect to for (), from which we can see the oscillation around the global minimum of : .

After the change of curvature, the field rolls away from the origin and lags behind the minimum of the effective potential. The minima of the effective potential are at

 ϕ±(t) = ±1√λ√μ2−ρm(t)M2. (53)

Before reaching one of the minima and oscillating around it, the symmetron field will first linger around the origin before following the inflection point where the curvature of the effective potential vanishes and then settling down to that new minimum. At the inflection point the field is close to

 φ = 1√3√1−1x2 (54)

where , and in deriving this equation we have used the facts that (conservation of matter) and that in the matter dominated era . At this time, the first derivative of the symmetron effective potential becomes

 dVeff(ϕ)dϕ ≈ ∓2μ33√3λ(1−t2⋆t2)32.

The symmetron equation of motion becomes then

 φ′′+2xφ=2κ23√3(1−1x2)3/2 (55)

the solutions of which are

 φ(x)=±2κ23√3[1xarctan1√x2−1+73√x2−1x
 +16x√x2−1−32ln(x+√x2−1)−a2x+b2] (56)

where and are again constants of integration which depend on the initial conditions. The upper right panel of Fig. 2 shows the comparison of the numerical solution with the above analytic approximation (51) , where we can see that for small just over 1 the agreement is very good. In the lower left panel of the same figure, one shows the short period of time when the symmetron tracks the inflection point (56) before reaching the moving minimum.

In a third phase, the symmetron field catches up with the minimum of its effective potential. Let us define which satisfies

 ¨ψ+μ2(t2⋆t2−1)ψ+λψ3a3 = 0 (57)

Around the minimum of the effective potential, , one can expand to first order leading to

 ¨ψ(1)+2μ2(1−t2⋆t2)ψ(1) = −¨ψ+ (58) = μ√λa32⋆t2⋆(t2t2⋆−1)−32

This is a forced oscillation where the forcing term is negligible for and the mass-squared term satisfies The time variation of the mass compared to the period of the oscillation is given by:

 ˙mm2=1√2κ(t2t2⋆−1)−32

which is small when , a condition satisfied when , i.e., large enough . The solution for large enough is then given by

 ψ(1)H(t) = (t2t2−t2⋆)14[a3cosΩ(t)+b3sinΩ(t)], (59)

where

 Ω(t) ≡ √2κ(√t2t2⋆−1+tan−1t⋆√t2−t2⋆)

and and are constants of integration which depend on the initial conditions.

A better solution can be obtained by taking into account the forcing term whose characteristic time is . The forcing term is slowly varying when , which reads

 1mτ ∼ 3κt⋆t(1−t2⋆t2)−32

and is indeed small when the period is slowly varying. In this case, a better approximation is given by

 ψ(1)(t) = −¨ψ+m2+ψ(1)H (60)

where the forcing term rapidly becomes negligible. The lower right panel of Fig. 2 shows the analytic and numerical solutions for this stage; note the drift in the period of the oscillation due to the fact that close to the ratio is not negligible.

### ii.4 Tachyonic instability

The mass-squared of the symmetron field is briefly negative when the symmetron lags behind the minimum of its effective potential. Defining

 Δt≡tf−t⋆,

in which is the time when the symmetron field settles down to the effective potential minimum, one could estimate . In the case for , a good approximation for the time spent in the tachyonic regime can be numerically fitted as

 Δtt⋆ ≈ −2×10−4log2(φ(teq))−9.7×10−3log(φ(teq)) +2×10−2.

As it stands, this tachyonic period is extremely short (see the discussion below). We will see shortly that the effect of the negative mass-squared is only relevant on large enough scales for cosmological perturbations, and even for those large scales it is quite insignificant.

## Iii Linear Perturbations

### iii.1 Growth of structure

We will now be interested in the growth of linear perturbations in the matter dominated era, and for that we work in the Newtonian gauge where the perturbed metric in the absence of any anisotropic stress is given by

 ds2 = −(1+2ΦN)dt2+a2(t)(1−2ΦN)dxidxi (61)

where represents the Newtonian potential. For simplicity let us assume that matter comprises a single fluid of pressureless particles with the energy momentum tensor , being the conserved energy density and the four-velocity of the matter particles. In general the conservation equation reads

 dρdτ+3hρ = 0 (62)

where is the proper time along the particle trajectories. The local Hubble expansion rate is defined as The perturbed conservation equation is then

 ˙δ = −θ+3˙ΦN (63)

The Euler equation in terms of the divergence of the velocity field becomes here

 ˙θ+2Hθ+ΔΦNa2+αϕmPl(Δδϕa2+˙ϕθ) = 0, (64)

which is modified by the presence of the symmetron field as indicated by the last term on the left-hand side.

Similarly, the modified Poisson equation now involves the perturbation of the Einstein frame matter (and scalar field) density and reads

 →∇2ΦN = 12m2PlA(ϕ)ρ(δ+αϕmPlδϕ) (65)

which becomes in Fourier space when

 −k2a2ΦN=32H2(δ+αϕmPlδϕ) (66)

The scalar field equation of motion is expressed as

 ¨δϕ+3H˙δϕ+(k2a2+m2(ϕ))δϕ (67) = −2ΦNV′eff(ϕ)+4˙ΦN˙ϕ−¯ρϕM2δ

From the above equations one can derive a second order differential equation for , with the coupling to the scalar field taken into account:

 (1−92a2H2k2)¨δ+[2H+β˙ϕ(1+92a2H2k2)]˙δ−(32H2+92a2H2k2[βH˙ϕ−12H2)]δ (68) = −92a2H2k2β¨δϕ−92a2H2k2(2˙β+β2˙ϕ)˙δϕ+[32βH2−k2a2β−92a2H2k2(¨β+12βH2+β˙β˙ϕ−β2H˙ϕ)]δϕ,

where have defined .

We have seen above that the symmetron field lags behind the minimum of its effective potential for a very brief period when the mass-squared . In such a tachyonic phase, the perturbation grows exponentially for modes such that Winther et al. (2011) as can be seen from Eq. (67). At the time of symmetry breaking, the effective mass-squared vanishes and then decreases to a fraction of before increasing to its value at the effective potential minimum. Clearly larger-scale modes, for which is smaller, are easily in the tachyonic regime; as is much larger than the Hubble expansion rate today, some of the tachyonic modes could indeed be sub-horizon (big ) and this could possibly influence the growth of large-scale structure on sub-horizon scales.

Outside the tachyonic regime, the evolution equation of the symmetron perturbation is sourced by the matter perturbation. Assuming that the symmetron field tracks the effective potential minimum, i.e., for non-tachyonic modes and neglecting the short period during which the symmetron field lags behind the minimum, its equation of motion in the sub-horizon limit becomes

 (k2a2+m2(ϕ))δϕ ≈ −ρϕM2δ (69)

where the time derivatives are much smaller than the spatial derivatives in the sub-horizon limit and thus are neglected. From this equation the symmetron field perturbation can be solved as

 δϕ ≈ −a2k2ϕM2ρ11+a2m2(ϕ)k2δ, (70)

which shows that the symmetron field perturbations tracks that of the matter density. This in turn implies that matter perturbation well within the horizon grows according to

 ¨δ+(2H+β˙ϕ)˙δ−32H2δ ≈ −k2a2βδϕ

or equivalently

 ¨δ+2H˙δ−32H2⎡⎢ ⎢⎣1+α(ϕ)11+a2m2(ϕ)k2⎤⎥ ⎥⎦δ = 0 (71)

Therefore, in this adiabatic approximation valid for non-tachyonic modes and neglecting the interval of time when the symmetron lags behind the minimum, we find that gravity is modified according to the comoving Compton radius with an amplitude depending on . Structures on scales outside the Compton radius grow as in GR

 δ ∼ a

while those on scales inside the Compton radius have a modified growth due to the renormalised Newton constant

 Geff = GN(1+α). (72)

We will see in the following that this result is hardly modified by the tachyonic instablity.

### iii.3 Tachyonic instability

As mentioned earlier, if the mass-squared of the symmetron field becomes negative (see Fig. 3), then the perturbation will undergo an unstable growth which could be problematic in some cases. We have also seen that such tachyonic instability problem is most likely to plague the large-scale (small ) modes. The purpose of this subsection is to assess how big the impact it could have on the growth of matter perturbations.

We are interested in modes which enter the horizon after matter-radiation equality and before the tachyonic instability happens. Normalising , this corresponds to

 kH⋆ ⩾ 1 (73)

and

 kH⋆ ⩽ √1+zeq1+z⋆ ≲ 60 (74)

The tachyonic modes can be conveniently studied using the reduced symmetron perturbation

 δφ ≡ δϕϕ⋆

and the parameters

 ω ≡ ϕ⋆M≲10−3

and

 ξk ≡ kt⋆ = 2k3H⋆.

Keeping terms in (or ), neglecting terms in (or ), and considering that the variations in both and are very rapid in the tachyonic period, the growth equation simplifies to

 δ′′+43xδ′−23x2δ = −2ω2ξ2kx23(φδφ′′+2φ′δφ′+φ′′δφ)

in the sub-horizon limit for the tachyonic modes. Notice that in the absence of the rapid variation of due to the tachyonic instability, this equation reduces to the growth equation in GR.

Defining , the growth equation becomes

 ~δ′′−49x2~δ ≈ −2ω2ξ2k(φδφ′′+2φ′δφ′+φ′′δφ), (75)

the solutions of which are

 ~δ = α1x43+β1x13+6ω25ξ2k∫xxk⎛⎝s43x13−x43s13⎞⎠(φδφ)′′(s)ds,

where and are integration constants, and is the value of when the -mode under consideration enters the horizon. As a result,

 δ(x) = α1x23+β1x+6ω25ξ2k∫xxk⎛⎝s43x−x23s13⎞⎠(φδφ)′′(s)ds.

Integrating by parts

 ∫xxk⎛⎝s43x−x23s13⎞⎠(φδφ)′′(s)ds (76) = ⎡⎣⎛⎝s43x−x23s13⎞⎠(φδφ)′(s)⎤⎦xxk −∫xxk⎛⎝43s13x+13x23s43⎞⎠(φδφ)′(s)ds.

At the horizon crossing the symmetron perturbations are taken to vanish, as during inflation the mass of the symmetron is much larger than the Hubble expansion rate. This implies that the initial conditions for the symmetron perturbation are

 δϕtk ≈ 0, ˙δϕtk ≈ 0 (77)

where is the horizon-entry time for the mode

 tk = 827k3t2⋆ (78)

Using the fact that the tachyonic growth starts at , we have

 −∫xxk⎡⎣43s13x+13x23s43⎤⎦(φδφ)′(s)ds (79) ≈ −∫x1⎡⎣43s13x+13x23s43⎤⎦(φδφ)′(s)ds

The variation of is much faster than the other terms in this integral, so that the terms in the brackets could be absorbed into the derivative with respect to , and the integrand becomes a total derivative. This implies that the growth factor behaves like

 δ(x) ≈ ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪