A gauge-invariant approach to interactions in the dark sector
PPrepared for submission to JCAP
A gauge-invariant approach tointeractions in the dark sector
William J. Potter and Sirichai Chongchitnan
Oxford Astrophysics,Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, United KingdomE-mail: [email protected], [email protected]
Abstract.
We outline a gauge-invariant framework to calculate cosmological perturbationsin dark energy models consisting of a scalar field interacting with dark matter via energyand momentum exchanges. Focusing on three well-known models of quintessence and threecommon types of dark sector interactions, we calculate the matter and dark energy powerspectra as well as the Integrated Sachs-Wolfe (ISW) effect in these models. We show howthe presence of dark sector interactions can produce a large-scale enhancement in the matterpower spectrum and a boost in the low multipoles of the cosmic microwave backgroundanisotropies. Nevertheless, we find these enhancements to be much more subtle than thosefound by previous authors who model dark energy using simple ansatz for the equation ofstate. We also address issues of instabilities and emphasise the importance of momentumexchanges in the dark sector.
Keywords:
Cosmological perturbation theory, dark energy theory, cosmological simulations,integrated Sachs-Wolfe effect a r X i v : . [ a s t r o - ph . C O ] A ug ontents Observation of type Ia supernovae at high redshift, large scale structure surveys and ob-servations of the cosmic microwave background (CMB) all indicate that the Universe iscurrently undergoing accelerated expansion [1–4]. The most convincing explanation for thisphenomenon is that the Universe is filled in great abundance with a form of energy with neg-ative pressure. This so-called ‘dark energy’ (DE) can lead to an accelerated cosmic expansionif its equation of state w satisfies w < − /
3, although observations currently constrain w to be close to −
1. At present, the cosmological constant model in which w is exactly − w (e.g. w = constant). Thesesimplistic approaches may not accurately reproduce the dynamics of quintessence models assuggested by [29, 30].The primary aim of this paper is to investigate the prospects of detecting the observa-tional signatures of dark sector interactions. We shall consider three well-known models ofquintessence and study the effects of dark sector interactions in both the background andin the linear perturbations. In particular, we shall examine the effects of non-zero interac-tions on the linear matter power spectrum and the integrated Sachs-Wolfe (ISW) using thegauge-invariant approaches of Kodama and Sasaki [31]. Unlike previous works on this sub-ject, our approach takes into account dark energy perturbations, including the perturbativeeffects of energy and momentum transfer in the dark sector. Issues of instabilities previouslydiscovered in [14, 15, 27] will also be addressed. Quintessence refers to a scalar field, φ , evolving along a potential, V ( φ ), which becomessufficiently flat at late times, leading to cosmic acceleration (see e.g. [8]). At early times, thedark energy contribution to the overall energy density is small [32]. Some classes of potentialsexhibit tracking behaviour whereby the dark energy density tracks below that of matter andradiation and only comes to dominate the universe at late times [33]. Another interestingfeature of quintessence is the existence of attractor solutions [32] which greatly reduce theneed for the initial values of the dark energy parameters to be finely tuned.We will focus our analysis on three models of quintessence, namely, the Ratra-Peebles,SUGRA and double exponential potentials summarised below. From this point on, all equa-tions will be in natural units with c = (cid:126) = 1. We also write ¯ κ = 8 πG . Ratra and Peebles (RP) [9] showed that the class of inverse power law potentials of the form V ( φ ) = M α φ α , α > , (2.1)tracks the equation of state during matter and radiation-dominated eras, and dark energyonly becomes dominant at late times. This behaviour holds for a wide range of initial condi-tions. We will use M = 4 . × − m Pl and α = 0 . .2 SUGRA The SUGRA model with potential V ( φ ) = M α exp (cid:0) ¯ κ φ (cid:1) φ α , (2.2)is motivated by supersymmetry and differs from the Ratra-Peebles (RP) potential by anexponential ‘supergravity’ correction factor [35]. This factor can be shown to push theequation of state closer to − M = 1 . × − m Pl and α = 1 as indicated by [34]. Again, the SUGRA potential has attractor solutions and is viablefor a wide range of initial conditions. The double exponential (DExp) potential V ( φ ) = M e − λ φ + M e − λ φ , (2.3)allows dark energy to track the equation of state of radiation and matter and leads to acceler-ation at late times [36]. We shall use M = 10 − m Pl , M = 10 − m Pl , λ = 9 .
43 and λ = 1which, as shown in [37], should give rise to observables that are significantly different fromthe cosmological constant but are still broadly consistent with observational constraints. We assume an isotropic, homogeneous and spatially flat background as described by theFriedmann-Robertson-Walker (FRW) metric, ds = − dt + a ( t ) dx i dx i , (3.1)where a ( t ) is the scale factor as a function of cosmic time, t , and summation is impliedover i = 1 , ,
3. We assume that the background energy density comprises matter, radiationand dark energy (denoted by subcripts m , r and φ respectively). The total energy density, ρ = ρ m + ρ r + ρ φ , satisfies the Friedmann and acceleration equations H ≡ (cid:18) ˙ aa (cid:19) = ¯ κ ρ m + ρ r + ρ φ ) , (3.2)˙ H = − ¯ κ (cid:18) ρ m + 43 ρ r + ρ φ + p φ (cid:19) , (3.3)as well as the energy conservation equation˙ ρ + 3 H ( ρ + p ) = 0 . (3.4)Here overdots are derivatives with respect to t and p x is the pressure of component x relatedto its energy density via the equation of state w x = p x /ρ x . In the above, we have used w m = 0, w r = 1 / w φ defined below.The quintessence field evolves via the Klein-Gordan equation¨ φ + 3 H ˙ φ + V (cid:48) ( φ ) = 0 . (3.5)– 3 –ts energy density, pressure and equation of state are given by ρ φ = ˙ φ V ( φ ) , p φ = ˙ φ − V ( φ ) , (3.6) w φ = p φ ρ φ = ˙ φ / − V ( φ )˙ φ / V ( φ ) . (3.7)The case in which ˙ φ = 0 reduces to the cosmological constant with w Λ = −
1. Note that atpresent w φ is observationally constrained at low redshift to a value close to w φ = − ∼
10% accuracy [38, 39].We model interactions between dark matter and dark energy by including a sourceterm, Q x , in the evolution equations for the dark energy and dark matter background energydensities. ˙ ρ x = − Hρ x (1 + w x ) + Q x . (3.8)Non-zero Q thus represents the energy flow from one component to another. To conserve thetotal energy, we require (cid:88) x Q x = 0 . (3.9)If interactions are purely between the dark matter and dark energy (i.e. interactions only inthe dark sector) then Q m = − Q φ . (3.10) To evolve the background energy densities with time, we use the set of dimensionless ‘energy’variables defined in [33], x = ¯ κ ˙ φ √ H , y = ¯ κH (cid:114) V r = ¯ κH (cid:114) ρ r , m = ¯ κH (cid:114) ρ m , λ = − κV dVdφ . (3.11)Intuitively, x, y, r and m can be thought of as (the square-root of) the ratio of energy densityof each component to the total energy density ( ∼ H ). Converting the time variable inEquations 3 . , . . e -fold’ number, N = ln a , we obtain d ln HdN = − (cid:18) x − y + r (cid:19) , (3.12) dxdN = (cid:114) λy − x (cid:18) d ln HdN (cid:19) + κ H x Q φ , (3.13) dydN = − (cid:114) λxy − y d ln HdN , (3.14) drdN = − r (cid:18) d ln HdN (cid:19) , (3.15) dmdN = − m (cid:18)
32 + d ln HdN (cid:19) + κ H m Q m . (3.16)– 4 – -2 w φ zCCRPSUGRADExp -2 Ω x zMatter (CC)Radiation (CC)DE (CC)Matter (DExp)Radiation (DExp)DE (DExp) Figure 1 . Left:
The equation of state w φ ( z ) of dark energy for the cosmological constant (CC), Ratra-Peebles (RP), SUGRA and double exponential (DExp) models (see section 2). Right:
The evolutionof the density parameter Ω x = ρ x /ρ for the CC and DExp models. Apparently large deviations from w φ = − These variables obey the Friedmann constraint derived from dividing Equation 3 . H . x + y + r + m = 1 , (3.17)We solved these coupled differential equations by integrating them numerically fromredshift z = 10 until z = 0 (today). Initial values were chosen by requiring that the final( z = 0) values of the component energy densities agreed broadly with those measured today:Ω m = 0 .
27, Ω r = 8 . × − , Ω φ = 1 − Ω m − Ω r and H = 70 . − Mpc − , whereΩ x ≡ ρ x /ρ. The evolution of w φ for the four models and the background energy densities for theCC and DExp models are shown in Figure 1, in which the eras of radiation, matter anddark energy domination are clearly seen in that order. The quintessence and CC models allsatisfy w φ (cid:39) − z ∼
1. Despite an apparently large deviation in w φ from − We consider only scalar perturbations to the FRW metric. Working in the Newtonian gaugeand ignoring anisotropic stresses, the perturbed metric takes the form ds = − (1 + 2Φ) dt + a ( t ) (1 − dx i dx i , (4.1)where Φ is the Newtonian potential (for details see [40]). The components of the energy-momentum tensor, T νµ are T = − ρ, T i = − k ( ρ + p ) v ,i , T ji = pδ ji , (4.2)where k is the Fourier wavenumber, v i is the peculiar velocity of the perturbation for species i and a comma denotes a partial derivative. Each energy density can be decomposed as ρ = ρ + δρ where barred quantities are the background values. We then define the variables:– 5 – m ≡ δρ m ρ m , δ r ≡ δρ r ρ r , δφ ≡ φ − φ .Physically, δ is a measure of density fluctuations (overdensities and underdensities). To avoidunphysical gauge artefacts, we shall consider the gauge-invariant overdensities ∆ x defined as[31] ∆ m ≡ δ m + 3 (cid:18) aHk (cid:19) (1 − q m ) v m , (4.3)∆ r ≡ δ r + 4 (cid:18) aHk (cid:19) (1 − q r ) v r , (4.4)∆ φ ≡ ˙ φδ ˙ φ + (3 H ˙ φ + V (cid:48) ( φ )) δφ − ˙ φ Φ ρ φ , (4.5)= δ φ + 3 aHk (1 + w φ ) v φ + 32 (1 + w φ ) (cid:18) aHk (cid:19) ∆ , (4.6)where v φ = kδφa ˙ φ , (4.7) q x = Q x Hρ x (1 + w x ) . (4.8)It will also be necessary to consider the perturbative effects arising from the dark sectorinteractions. We define (cid:15) to be the perturbation in the energy transfer Q x → (cid:101) Q x (cid:101) Q x = Q x (1 + (cid:15) x ) , (4.9)and note that (cid:15) x obeys the conservation equation (cid:88) x Q x (cid:15) x = 0 . (4.10)Similarly, the gauge-invariant variable for energy transfer, E x , can be constructed as [31] E x = (cid:15) x − a ˙ Q x kQ x v x . (4.11) We now describe how dark sector interactions can be covariantly formulated and incorporatedinto the gauge-invariant system (see [14, 15, 23] for previous attempts in this direction forsimple dark energy models).Dark sector interactions can be covariantly described by the conservation equation ∇ ν T µνx = Q µx . (4.12)The 4-vector Q µx can be constructed using the function Q x by Q ( x ) µ = Q ( x ) u ( a ) µ , (4.13)where u ( a ) µ is the 4-velocity of component a (which may be different from x ). By perturbing(4.13), we find (cid:101) Q ( x ) µ = (cid:101) Q ( x ) (cid:101) u µ + (cid:101) f ( x ) µ , (4.14)– 6 –here u µ is the average velocity. The perturbed average 4-velocity u µ → (cid:101) u µ and perturbed4-velocity of component a ( u ( a ) µ → (cid:101) u ( a ) µ ) can be expressed as (cid:101) u = (cid:101) u ( a )0 = − a (1 + Φ) , (cid:101) u j = a ¯ vY j , (cid:101) u ( a ) j = av a Y j . (4.15)where Y j is a basis vector constructed from harmonic functions . In 4.14, the vector (cid:101) f ( x ) µ can be interpreted as the momentum exchange with components given by (cid:101) f ( x )0 = 0 , (cid:101) f ( x ) j = aHρ x (1 + w x ) f x Y j , (4.16)where f x is an ordinary scalar representing the amplitude of the momentum exchange. Inaddition, f x satisfies the conservation of momentum (cid:88) x ρ x (1 + w x ) f x = 0 . (4.17)As with the energy exchange, we can similarly recast f x in a gauge-invariant form F x F x ≡ f x − Q x ( v x − ¯ v ) Hρ x (1 + w x ) . (4.18)All the ingredients introduced so far are now sufficient to allow us to study the evolutionof the perturbation variables { ∆ m , ∆ r , ∆ φ , v m , v r , v φ } . In the regime where perturbations aresmall, we find the following set of coupled differential equations [31] d ∆ m dN = 92 aHk (1 + w )(¯ v − v m ) − Q m ∆ m aHρ m − kaH v m + Q m E m ρ m H + F m , (4.19) d ∆ r dN = 6 aHk (1 + w )(¯ v − v r ) + ∆ r − k aH v m , (4.20) d ∆ φ dN = (cid:18) w φ − Q φ Hρ φ (cid:19) ∆ φ +2 x ρ φ (cid:20) aHk (1 + w )(¯ v − v φ ) − kaH v φ (cid:21) + Q φ E φ ρ φ H + (1 + w φ ) F φ , (4.21) dv m dN = − v m − aH k ¯∆ + F m , (4.22) dv r dN = − v r + k aH ∆ r − aH k ¯∆ , (4.23) dv φ dN = − v φ + k (1 + w φ ) aH ∆ φ − aH k ¯∆ + F φ , (4.24)where ¯ v = 11 + w (cid:18) x v φ + 43 r v r + m v m (cid:19) , (4.25)¯∆ = (cid:0) m ∆ m + r ∆ r + ( x + y )∆ φ (cid:1) + ak (cid:88) x Q x v x , (4.26) w = 1 ρ ( ρ r w r + ρ φ w φ ) = r x + y ) w φ . (4.27) see Appendix C of [31] for detail. We will not require the explicit form of Y j in the evolution equations. – 7 –ote that by setting Q x , E x and F x to 0, we recover the non-interacting case. Theinitial values of the perturbations are fixed using adiabatic initial conditions [41, 42] whichwe calculate to be ∆ m − q m = 34 ∆ r , ∆ φ = 3 ˙ φ ρ φ (cid:18) aHk (cid:19) ∆, q x = Q x aHρ x v m = k aH ∆ m − q m , v r = k ∆ r aH , v φ = 0 . (4.28)The only free variable is the initial matter density perturbation which is determined bynormalising the matter power spectrum at a pivot scale which we take to be k = 0 . − .The power spectrum of energy component x is calculated by comparing the amplitudes ofthe growing mode at early and late times P x ( k ) = k n s (cid:20) ∆ x ( z = 0)∆ x ( z = 10 ) (cid:21) , (4.29)where n s is the primordial scalar spectral index taken to be 0 . P m ( k ) ( h - M p c ) k (Mpc -1 ) CCRPSUGRADExp -1.5-1-0.500.511.5 0.0001 0.001 0.01 0.1 1 10 ∆ P m ( k ) / P m , CC x % k (Mpc -1 ) RPSUGRADExp
Figure 2 . Left:
If there is no interaction in the dark sector, the linear matter power spectra for allfour ‘best-fit’ dark energy models are essentially indistinguishable.
Right: the fractional percentagedifference between the matter power spectra of the quintessence models and the cosmological constant.The best-fit models all exhibit sub-percent differences from the cosmological constant.
Figure 2 shows the matter power spectrum for all four dark energy models as well asthe fractional difference in the matter power spectra for the quintessence models relative tothe cosmological constant expressed as a percentage. P m ( k ) model − P m ( k ) CC P m ( k ) CC × . Note that our calculations are not valid for scales k > . N − body simulation as in [34, 43] or a more sophisticatedperturbation theory [44]. – 8 – -25 -20 -15 -10 -5 P φ ( k ) ( h - M p c ) k (Mpc -1 ) CCRPSUGRADExp
Figure 3 . The linear power spectra for dark energy for the three quintessence models with nointeractions. Note the ‘turnover’ on very large scales close to the Hubble radius ( ∼ × − Mpc − )indicating the typical size of dark energy perturbations. Differences between the models are small with the largest difference ≈
1% observed forthe DExp potential. The quintessence models have an excess power for k > . − anda power deficit for k < . − . This correlates well with the ‘turnover’ to the darkenergy spectra shown in Figure 3. Nevertheless, these small differences of order 1% in matterpower spectra for the models are essentially unobservable.The power spectrum for dark energy perturbations, shown in Figure 3, are roughly fourorders of magnitude smaller than the matter power spectra. Thus, we do not expect to seelarge differences in the matter power spectra of the quintessence models due to dark energyclustering. We see that the large-scale clustering of dark energy enhances the clustering ofdark matter on large scales. The dark energy power spectra are approximately constant onvery large scales ( k < . − ), and decay exponentially on small scales. In fact, darkenergy clustering is significant mainly on large scales in all models in which the sound speedof dark energy equals the speed of light [45]. The decay of the power spectrum is split intotwo regions with different exponents changing near the turnover scale in the matter powerspectrum. The differences in the power spectra of the different quintessence models is dueto the different values of w φ at late times, since ∆ m ∝ (1 + w ) (see 4 .
19 and 4 . The Integrated Sachs-Wolfe effect (ISW) is an effect observed in the CMB for small multipolenumber (cid:96) caused by the blueshifting of photons as the gravitational potential wells becomeshallower due to the late-time cosmic acceleration [46]. The gravitational potential Φ can bedecomposed as: Φ = Φ m + Φ r + Φ φ . (5.1)The potential Φ is related to the perturbation variables by the cosmological Poisson’s equationΦ = − (cid:18) aHk (cid:19) ∆ . (5.2)– 9 – -1
1 2 3 4 5 6 7 8 9 ( ∆ C l / C l , CC ) x % Multipole number lCosmic VarianceRPSUGRADExp
Figure 4 . The percentage differences between C ISW (cid:96) for the three quintessence models and the CCmodel. The contributions from quintessence are overwhelmed by cosmic variance, hence the modelsare observationally indistinguishable via the ISW effect.
Consider the CMB temperature anisotropy ∆
T /T ( n ) along the line-of-sight unit vector n . The CMB anisotropy power spectrum is given by (cid:28) ∆ TT ( n ) ∆ TT ( m ) (cid:29) n · m = µ = 14 π (cid:88) (cid:96) (2 (cid:96) + 1) C (cid:96) P (cid:96) ( µ ) , (5.3)where C (cid:96) is amplitude of the CMB angular power spectrum at the (cid:96) -th multipole and P (cid:96) isthe Legendre polynomial of order (cid:96) . By decomposing ∆ T /T into the spherical harmonicsand integrating over all directions we find [47] C ISW (cid:96) ∼ (cid:90) ∞ (cid:90) z dec k (cid:12)(cid:12)(cid:12)(cid:12) − Ha d Φ dz J (cid:96) (cid:18) k (cid:90) z (cid:20) H ( z (cid:48) )(1 + z (cid:48) ) (cid:21) dz (cid:48) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dz dk, (5.4)where z dec is the redshift at which photon decoupling occurs ( ∼ J (cid:96) ( x ) is thespherical Bessel function of order (cid:96) .Figure 4 shows the squared percentage differences between C ISW (cid:96) for the three quintessencemodels and the cosmological constant model (cid:32) C quint (cid:96) − C CC (cid:96) C CC (cid:96) (cid:33) × , in comparison with the cosmic variance( C var (cid:96) ) = 22 (cid:96) + 1 . (5.5)The relative differences in the ISW effect for quintessence models and ΛCDM are com-pletely overwhelmed by cosmic variance. They are also much smaller than indicated in otherinvestigations in which the equation of state for dark energy were parameterised by simpleansatz e.g. w φ = w or w φ = w + w z (where w and w are constants). This is perhaps– 10 –ot surprising since our ‘best-fit’ quintessence models can closely replicate ΛCDM dynamicswhereas these parametrizations can diverge from ΛCDM dynamics substantially.In summary, we find that the ISW is an ineffective discriminant between the cosmologicalconstant and models of quintessence that deviate from w = − In this section, we introduce three models of dark sector interactions and explain how theycan be rewritten in covariant forms.
We first consider a dark sector interaction of the form Q m = − Aρ m , (6.1)which represents the decay of dark matter into dark energy with a constant decay rate. Thedecay rate per unit energy density of dark matter is quantified by the positive constant A ,which we refer to as the interaction strength. This form of interaction has been previouslyinvestigated in [14].In the covariant formulation, the interaction term is given by Aρ m u m , where u m =( − a, ), since this is the rate at which 4-momentum is transferred to dark energy. Hence, Q ( m ) µ = − Aρ m u ( m ) µ . (6.2)We perturb the above equation to find (cid:101) Q ( m ) µ = Q m (1 + δ m ) (cid:101) u ( m ) µ , (cid:101) Q φ = Q φ (1 + δ m ) (cid:101) u ( m ) µ . (6.3)Using Equations 4 .
3, 4 .
9, 4 .
11, 4.15 and the above, we find the gauge-invariant energy transfer (cid:15) m = δ m , E m = ∆ m , E φ = ∆ m + 3 aHk (cid:18) − Q m Hρ m (cid:19) ( v φ − v m ) . (6.4)Similarly, using Equations 4 .
14, 4 .
16, 4 .
17 and 4 .
18, we find the momentum transfer f m = Q m ( v m − ¯ v ) Hρ m , F m = 0 , F φ = Q φ ( v m − v φ ) .Hρ φ (1 + w φ ) . (6.5) The second model represents the case in which dark energy decays into dark matter at aconstant decay rate [48]. This may be applicable, for instance, in the early dark energyscenario [49]. The interaction term has the form Q φ = − Aρ φ , Q m = Aρ φ , Q ( φ ) µ = − Aρ φ u ( φ ) µ . (6.6)Similarly, using Equations 4 . .
18, we find the gauge-invariant energy and momentum trans-fers E φ = ∆ φ − aHk (1 + w φ ) (cid:20) Q φ v φ Hρ φ (1 + w φ ) + aH k ∆ (cid:21) , (6.7)– 11 – m = ∆ φ + 3 aHk (1 + w φ ) (cid:20) v m − v φ − Q φ v m Hρ φ (1 + w φ ) − aH k ∆ (cid:21) . (6.8) F φ = 0 , F m = Q m ( v φ − v m ) Hρ φ . (6.9)Note that the momentum transfer F m and F φ are ‘orthogonal’ to those in the dark matterdecay model. Finally, let us consider a interaction term which appears in some scalar-tensor theories ofgravity [50] Q ( φ ) µ = Aρ m ∇ µ φ, Q ( m ) µ = − Aρ m ∇ µ φ. (6.10)Using Equation 4 .
14 we find (cid:101) Q φ = − (cid:101) u µ Aρ m (1 + δ m ) ∂ µ (cid:101) φ, Q φ = Aρ m ˙ φ (6.11)where u µ is the average velocity. To calculate the gauge-invariant interaction, we use thefollowing useful expressions for the perturbed scalar field ∂ (cid:101) φ∂t = ˙ φ − Φ ˙ φ + δ ˙ φ, δφ = av φ ˙ φk , (6.12) δ ˙ φ ˙ φ = ∆ φ w φ + Φ − (3 H ˙ φ + V (cid:48) ) δφ ˙ φ . (6.13)Using Equations 3 .
9, 4 .
15 and 4 .
10, we find the gauge-invariant energy transfer variables E φ = ∆ m + ∆ φ w φ + 3 aHk (cid:18) Q m Hρ m − (cid:19) v m + 32 (cid:18) aHk (cid:19) ∆ , (6.14) E m = ∆ m + ∆ φ w φ + 3 aHk (cid:18) Q m v m Hρ m − v φ (cid:19) + 32 (cid:18) aHk (cid:19) ∆ − aHk y λ √ x ( v m − v φ ) . (6.15)Similarly, from 4 .
7, 4 .
16, 4 .
18 and 6 .
10, we find the gauge-invariant momentum transfervariables to be f φ = Aρ m ˙ φ ( v φ − ¯ v ) Hρ φ (1 + w φ ) , F φ = 0 , F m = Q φ ( v m − v φ ) Hρ m . (6.16) We are now ready to calculate the observables for the different interaction models. In particu-lar, we would like to establish if dark sector interactions could give rise to observable imprintsin the power spectrum or the ISW effect whilst satisfying the constraint that w is very closeto −
1. We use the SUGRA potential for the dark energy decay and scalar-tensor modelbecause, at the background level, it behaves most similarly to the cosmological constant (seeFigures 1 and 2). It is computationally more challenging to implement the SUGRA potentialwith the dark matter decay model due to the non-monotonic nature of the potential and theRP potential was used instead (where an analytic solution for φ can be calculated given thevalue of V ( φ )). In all cases, however, we did not find a sensitive dependence on the choice ofpotential. – 12 – x ln(a) A=10 -70.569 Figure 5 . Evolution of the variable x ∝ ˙ φ in the decaying dark energy model using the SUGRApotential. x passes through zero and becomes negative at ln a = − .
336 causing an instability in thepower spectra. This behaviour occurs for interaction strengths larger than ∼ − m Pl . Our computation shows that the decaying dark energy model was found to suffer from aninstability in the perturbations. From Equation 3 .
13, we see that significant values of in-teraction strength act to drive the value of x towards 0, causing dx/dN and v φ to becomeinfinite. Below the critical value of interaction strength associated with this instability, thereis no significant effect on the power spectra or ISW effect. An example of this instability isshown in Figure 5 in which the interaction strength A is chosen to be just above the crit-ical value. Starting at z = 10 , we see that x decreases to 0 at ln a = − . The primary effect of the dark matter decay model is to alter the time at which dark energycomes to dominate the Universe. This is because dark matter decay is most effective at earlytimes when matter dominates over dark energy. This means that the background evolutionof our ‘best-fit’ models is sensitive to the interaction strength. For instance, an interactionstrength A = 10 − m Pl changes the dark energy density at z = 0 by roughly 1%, but leavesthe matter power spectrum and ISW effect virtually unchanged.Figure 6 shows the evolution of the x variable for interaction strengths in the range10 − − − m Pl . We see that the dark energy density is sharply boosted at early times, as– 13 – -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 x ln(a) A=10 -60 A=10 -61
A=10 -62
A=10 -63
A=0
Figure 6 . Evolution of the energy variable x ∝ ˙ φ in the decaying dark matter model using the RPpotential. We see that increasing the interaction strength A boosts the value of x at early times whichcan lead to an increase in late time dark energy densities. the energy is immediately transferred from dark matter to dark energy. In Figure 7, we showthe evolution of w φ over the same range of interaction strengths. We see the correspondingsharp rise in w φ towards 1 at early times. This has the effect of redshifting away the excessdark energy at a rate ρ φ ∝ a − . This rapid redshifting means that for a wide range of V ( φ ), dark energy does not come to dominate the universe too soon and so this model ofinteractions can indeed be reconciled with observations.A number of previous investigations of this interaction have used simple parameteri-sations of w φ . However, we see from Figure 6 that the complex redshifting behaviour ofdark energy would be extremely difficult to reproduce with a simple analytic form of w φ .In fact, an instability in this form of interaction was found in [14] when considering darkenergy parametrized by w φ = constant. We can see that in a w =constant model where w <
0, the early boost in the dark energy density will not be redshifted away relative to thedark matter energy density and dark energy will come to dominate the Universe too soon forany appreciable interaction strength. It is interesting that due to the redshifting behaviourdescribed above, our quintessence models do not encounter such an instability.
The scalar-tensor model exhibits the most interesting behaviour of the three interactionmodels considered in this work. The sign of the interaction can be changed to represent theflow of energy from one dark component to the other since the interaction depends both ondark matter and dark energy. If we choose Q φ = Aρ m ˙ φ where A is negative, we encounterthe same instability in the x variable which we found in the dark energy decay model.From Equation 3 .
13, we can easily deduce whether a model with Q φ ∝ x n would developan instability or not. When n <
1, the interaction term becomes increasingly negative as x – 14 – w φ ln(a)A=10 -60 A=10 -61
A=10 -62
A=10 -63
A=0
Figure 7 . Evolution of w φ for the dark matter decay models presented in Figure 6. At early times w φ responds to the sharp increase in x by rising suddenly to a value of 1. See the text for furtherdiscussion. decreases, hence leading to an instability. For models where n >
1, however, the negativeconstant tends to 0 as x tends to 0 and so an instability is avoided. In this model n = 1and to avoid an instability we require that the sign of the interaction term represent flow ofenergy from dark matter to dark energy, i.e. Q φ = Aρ m ˙ φ with A >
0. This supports theconclusion in [27] which found no background instabilities in this model.We find that this form of interaction does not change the energy density of dark energysignificantly at early times since Q φ ∝ ˙ φ and ˙ φ is small at early times. The energy transferbecomes significant only at late times when dark energy becomes dominant. As we increasethe interaction strength we see an enhancement in the matter power spectrum on large scales,as shown in Figure 8. The power spectrum on smaller scales k > . − decreases by ∼
6% relative to ΛCDM for the largest interaction strength shown.We have shown that a scalar-tensor interaction can lead to an enhancement in thematter power spectrum on large scales. In order to detect this intrinsic enhancement wemust be able to distinguish it from other artificial and observational effects such as gaugeartifacts, the scale dependence of the galaxy bias and redshift space distortions which canalso cause enhancements to the matter power spectrum on large scales and mimic our result(see [51], [52], [53] and [54]). We will now address these effects in more detail and explain howto break the degeneracy between an intrinsic large-scale enhancement due to a scalar-tensorinteraction and a similar enhancement caused by the effects above.The work by [51] showed that calculating matter power spectra using gauge-dependentquantities can introduce gauge modes which can cause artificial large-scale enhancement inthe power spectrum. We have used gauge-invariant quantities in our calculations and sothe large-scale enhancement we have found is not due to such a gauge artifact. The linearbias is a linear relation between fractional overdensities of galaxies and matter ( δ g ∝ δ m )– 15 – P m ( k )( h - M p c ) k (Mpc -1 ) A=0A=10 -3 A=10 -2.5
A=10 -2.2
A=10 -2 A=10 -1.9
A=10 -1.8
A=10 -1.75
Figure 8 . The linear matter power spectra in the SUGRA quintessence model with dark sectorinteraction of the ‘scalar-tensor’ form Q φ = Aρ m ˙ φ . Large values of A cause a significant enhance-ment to the power spectrum on very large scales. The interaction also leads to a smaller ∼ k > . − . which is used to infer the total matter power spectrum from a galaxy survey. Work by [52]showed that the linear bias relation is scale-independent only in the comoving-synchronousgauge and using perturbations defined in a different gauge leads to the linear bias having ascale-dependence. They showed that if the linear bias is used as a scale-independent relationin a different gauge it can lead to an artificial large-scale enhancement in the matter powerspectrum [52]. Failure to use the correct form of the linear bias relation in a particular gaugecould artificially replicate the signal from a scalar-tensor interaction.Redshift space distortions are caused by the peculiar velocities of observed luminousmatter. These small peculiar velocities produce an additional doppler shift in the observedphoton frequency relative to the average redshift caused by the background expansion velocityof the region. For large-scale overdensities the surrounding infalling matter will have anadditional component of their peculiar velocities directed towards the overdensity due togravitational attraction. This leads to the region appearing compacted in redshift space,which causes an apparent enhancement to these large-scale overdensities (since redshift isused as a proxy for distance).Recent work by [51], [53] and [54] calculated the effects of redshift space distortions,magnification by gravitational lensing and distortions to the luminosity distances of sourceson measured quantities such as the matter power spectrum. They calculate photon geodesics– 16 – -4 -3 -2 -1
1 2 3 4 5 6 7 8 9 ( ∆ C l / C l , A = ) x % Multipole number lCosmic VarianceA=10 -1.75
A=10 -1.8
A=10 -1.9
A=10 -2 Figure 9 . The fractional difference in the ISW effect in an interacting model relative to no interactionusing the SUGRA potential. Only the (cid:96) = 1 dipole moment of the largest two interaction strengths areobservable above the cosmic variance. The largest interaction strength leads to an order of magnitudeenhancement to the CMB dipole moment relative to ΛCDM. in a perturbed universe using linear perturbation theory. These geodesics take into accountthe deflection and redshifting of observed photons through interactions with overdensities.They then use the perturbed geodesics of the observed photons to relate gauge-invariant ob-servables from galaxy surveys (like the matter power spectrum) to the cosmological quantitiescalculated in theoretical models such as ours. Using the results of [51], [52], [53] and [54] wecan break the degeneracy between an intrinsic large-scale enhancement due to interactionsin the dark sector and enhancements due to the artificial and observational effects discussedabove.Recently [16] investigated the effect of an interaction of the form (6 .
10) on the back-ground evolution, matter power spectrum and halo mass function using quintessence models.They found that significant negative values of the interaction strength A could lead to anenhancement in the matter power spectrum relative to ΛCDM, in agreement with our find-ings. The authors did not use fully gauge-invariant perturbation variables or include theeffects of momentum transfer in their investigation, although they did include the effect ofbaryons and non-linearities. They did not find the instability in the x variable when A < k corresponding to small (cid:96) ). Onlythe (cid:96) = 1 moment of the two strongest interaction strengths plotted ( A = 10 − . m − and A = 10 − . m − ) is observable above cosmic variance. The largest interaction strength leadsto an order of magnitude enhancement to the CMB dipole moment relative to ΛCDM.The recent high redshift survey by [56] found evidence of an excess clustering on largescales ( k < . h Mpc − ) at 4 σ significance from ΛCDM. Using photometric redshifts from asample of SDSS galaxies they measured the angular power spectrum and found an excess inthe lowest multipoles of the angular power spectrum. This is a promising result, since it canbe interpreted as the effect of the dark sector interaction shown in Figures 8 and 9. Afterrigorous analysis of their data and biases they concluded that the large-scale enhancementwas a real effect which hinted at an exotic form of dark energy.We find the recent evidence for enhanced power in P ( k ) on large scales to be encouragingin light of our findings. However, we can see from Figures 8 and 9 that even if there weresignificant increments in the matter power spectrum at these near-horizon scales, they areonly detectable in the dipole moment of the CMB, a measurement which is dominated bythe Doppler shift caused by our own peculiar velocity. Unfortunately, the enhancements tothe matter power spectrum on scales k < . − are well beyond the scales probed incurrent galaxy surveys ( k ∼ . − ), so we do not expect that a direct detection of thislarge scale enhancement via galaxy counts will be possible in the near future.Using the cross-correlation of the CMB and large-scale structure from galaxy surveys islikely to be one of the most effective observational probes of our results. This technique allowsus to isolate the ISW effect from other contributions to the CMB at low multipole momentssince perturbations at the surface of last scattering were small compared to the observedlarge scale structure today. Large scale structure is correlated with the ISW effect since theISW effect is caused by the evolution of gravitational potentials. Using the cross-correlationallows us to measure the ISW effect and compare this to the enhancement predicted in thescalar-tensor model for large interaction strengths (Figure 9). The work by [57] calculatedthe cross-correlation using WMAP 7 and several galaxy surveys and found an excess ISWcross-correlation of 1 σ from ΛCDM. This excess could be explained by a scalar-tensor darksector interaction. In this paper, we have calculated the background evolution, matter and dark energy powerspectra and ISW effect for three quintessence models with and without dark sector interac-tions. Our calculations are based on gauge-invariant perturbation theory originally developedto calculate inflationary perturbations. We feel that our calculations are more reliable thanthose obtained when dynamical dark energy is parameterised by phenomenological ansatz for– 18 – , since in our approach the clustering of dark energy can be directly linked to the underlyingscalar-field perturbations.We found that without interactions our quintessence models differ from ΛCDM by atmost a few percent in observable quantities, meaning that any differences are essentiallyunobservable.We have shown how a covariant treatment of dark sector interactions can give rise tomomentum exchanges in the dark sector, which many authors have overlooked. Our approachcomplements and extends the work of [14], who examined momentum exchanges in constant w models using the dark matter decay interaction. We demonstrate our techniques on threemodel of interactions, namely, the dark energy decay, dark matter decay and scalar-tensortype of interaction.The dark energy decay model was found to have an instability in the background andperturbation equations, in agreement with previous works on this interaction, and thus canbe ruled out. In the decaying dark matter model, we found that its primary effect was tochange the evolution of the background energy densities, with no observable differences inthe power spectra or ISW effect. We explained why w =constant models, when associatedwith this interaction, will lead to an instability, which is absent when field dynamics aretaken into account.The scalar-tensor model of interactions appears to be the most interesting and robustof all three. We showed that it is stable if energy flows from dark matter to dark energy.For this case, we found that the interaction enhances the clustering of dark energy and darkmatter on very large scales ( k < . − ) and produced an enhanced ISW effect whichexceeds cosmic variance only in the lowest multipoles. We also found that the interactionproduces a significant enhancement in the matter power spectra on very large scales.We find the recent indications of enhanced large scale clustering via peculiar velocitymeasurements [55] and enhancement to the lowest multipoles of the angular power spectrum[56] encouraging, although, measurements on these scales are very difficult to make and areoften dominated by cosmic variance and our own peculiar velocity.Overall, the prospects of constraining dark sector interactions via the matter powerspectrum and ISW effect appear daunting. The enhancement to the matter power spectrumon scales k < . − is too large to be probed by current galaxy surveys k ∼ . − .There are certainly other observational probes, however, which may reveal the presence ofinteractions in the dark sector, including i) cross-correlation of the ISW with galaxy andquasar distributions [57–59], ii) the growth rates of large-scale structures at high redshift,[60–62], iii) effects of large super-horizon perturbations on the CMB [63, 64], iv) gravitationallensing of the CMB using EPIC [65]. We envisage that our calculation techniques can beadapted to explore these issues at least in the linear regime.Finally, there will almost certainly be interesting non-linear effects on the matter powerspectrum and CMB anisotropies arising from dark sector interactions. However, analyticprogress in this regime is extremely challenging (though not impossible [44]). 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