Adiabatic initial conditions for perturbations in interacting dark energy models Free

Abstract

1 INTERACTING DARK ENERGY

Dark energy (DE) and dark matter, the dominant sources in the ‘standard’ model for the evolution of the Universe, are currently only detected via their gravitational effects. This implies an inevitable degeneracy between them. A dark sector interaction could thus be consistent with current observational constraints. We look at such a model, assuming that the dark matter and DE can be treated as fluids whose interaction is proportional to the dark matter density. For interacting DE, the energy balance equations in the background are

1

2

where is the DE equation of state parameter, a prime indicates derivative with respect to conformal time τ, and Q_c is the rate of transfer of dark matter density due to the interaction. Various forms for Q_c have been investigated (see e.g. Wetterich 1995; Amendola 1999; Billyard & Coley 2000; Zimdahl & Pavon 2001; Chimento et al. 2003; Farrar & Peebles 2004; Koivisto 2005; Olivares, Atrio-Barandela & Pavon 2005; Sadjadi & Alimohammadi 2006; Guo, Ohta & Tsujikawa 2007; Bean et al. 2008a; Boehmer et al. 2008; Corasaniti 2008; He & Wang 2008; Quartin et al. 2008; Quercellini et al. 2008; Valiviita, Majerotto & Maartens 2008; Caldera-Cabral, Maartens & Urena-Lopez 2009b; Chongchitnan 2009; Gavela et al. 2009; He, Wang & Abdalla 2009a; Jackson, Taylor & Berera 2009; Pereira & Jesus 2009). We consider models where the interaction has the form of a decay of one species into another – as in simple models of reheating and curvaton decay (Malik, Wands & Ungarelli 2003; Sasaki, Valiviita & Wands 2006; Assadullahi, Valiviita & Wands 2007). Such a model was introduced by Boehmer et al. (2008) and Valiviita et al. (2008). It is not derived from a Lagrangian (in contrast with, e.g., Wetterich 1995; Amendola 1999), but it is motivated physically as a simple phenomenological model for decay of dark matter particles into DE. In this sense, it improves on most other phenomenological models, which are typically designed for mathematical simplicity, rather than as models of interaction. The methods that we use here and in the companion paper (Valiviita, Maartens & Majerotto 2009) may readily be extended to other interactions, including those based on a Lagrangian. We assume that in the background, the interaction takes the form (Boehmer et al. 2008; Valiviita et al. 2008)

3

where Γ is the constant rate of transfer of dark matter density. Positive Γ corresponds to the decay of dark matter into DE, while negative Γ indicates a transfer of energy from DE to dark matter.

In Valiviita et al. (2008), we considered the case of fluid DE with a constant equation of state parameter −1 < w_de≤−4/5 and found a serious large-scale non-adiabatic instability in the early radiation era. This instability grows stronger as w_de approaches −1. Phantom models, w_de < −1, do not suffer from this instability, but we consider them to be unphysical.

The instability is determined by the early-time value of w_de. We will show that the models are not affected by the large-scale non-adiabatic instability during early radiation domination if at early times w_de > −4/5. If we allow w_de to vary, such a large early w_de can be still consistent with Type Ia supernovae (SN) and baryon acoustic oscillation (BAO) observations, provided that at late times w_de∼−1. In this paper, we represent w_de via the parametrization w_de=w₀+w_a(1 −a) (Chevallier & Polarski 2001; Linder 2003), which we rewrite as

4

where w_e=w₀+w_a is the early-time value of w_de, while w₀ is the late-time value.

There are two critical features of the analysis of interacting models, which are not always properly accounted for in the literature.

• The background energy transfer rate Q_c does not in itself determine the interaction in the perturbed universe. One should also specify the momentum transfer rate, preferably via a physical assumption. We make the physical assumption that the momentum transfer vanishes in the dark matter rest frame; this requires that the energy–momentum transfer rate is given covariantly by (Kodama & Sasaki 1984; Valiviita et al. 2008)

  

  5

  where u^μ_c is the dark matter four-velocity and δ_c=δρ_c/ρ_c is the cold dark matter (CDM) density contrast.

• Adiabatic initial conditions in the presence of a dark sector interaction require a very careful analysis of the early radiation solution, both in the background and in the perturbations. We derive these initial conditions by generalizing the methods of Doran et al. (2003) to the interacting case, thereby extending our previous results (Valiviita et al. 2008).

We give here the first systematic analysis of the initial conditions for perturbations in the interacting model given by equation (5)– and our methods can be adjusted to deal with other forms of interaction. In the companion paper Valiviita et al. (2009), we report the results of our full Monte Carlo Markov Chain likelihood scans for this model. Cosmological perturbations of other interacting models have been investigated, e.g., in Amendola et al. (2003); Koivisto (2005); Olivares, Atrio-Barandela & Pavon (2006); Mainini & Bonometto (2007); Bean, Flanagan & Trodden (2008b); Bean et al. (2008a); Corasaniti (2008); La Vacca & Colombo (2008); Pettorino & Baccigalupi (2008); Schäfer (2008); Schaefer, Caldera-Cabral & Maartens (2008); Caldera-Cabral, Maartens & Schaefer (2009a),Chongchitnan (2009); Gavela et al. (2009); He et al. (2009a); He, Wang & Jing (2009b); He, Wang & Zhang (2009c); Jackson et al. (2009); Koyama, Maartens & Song (2009); Kristiansen et al. (2009); La Vacca et al. (2009) and Vergani et al. (2009).

2 PERTURBATION EQUATIONS

The scalar perturbations of the spatially flat Friedmann–Robertson–Walker metric are given by

6

In the perturbed universe, the dark sector interaction involves a transfer of momentum as well as energy. The covariant form of energy–momentum transfer for a fluid component A is ∇_νT^μν_A=Q^μ_A, where Q^μ_c=a⁻¹(Q_c, 0) =−Q^μ_de in the background. The perturbed energy–momentum transfer four-vector can be split as (Valiviita et al. 2008)

7

where k is the comoving wavenumber, f_A is the intrinsic momentum transfer potential and is the total velocity perturbation (θ=−k²v). The evolution equations for density perturbations and velocity perturbations for a generic fluid are (Valiviita et al. 2008; Kodama & Sasaki 1984)

8

9

Here, c²_sA is the sound speed and π_A is the anisotropic stress. For our model π_de= 0, and we set c²_{s de}= 1, as in standard non-interacting quintessence models, in order to avoid adiabatic instabilities (see discussion in Valiviita et al. 2008).

For the interaction defined by equation (5), we find from equation (7) that

10

Then we can write the DE and CDM perturbation equations for our model:

11

12

13

14

3 BACKGROUND SOLUTION IN EARLY RADIATION ERA

The background solution in the early radiation era (ρ_tot≃ρ_r) is important for finding the initial conditions for the integration of cosmological perturbations. In what follows, we use occasionally the Hubble parameter instead of the conformal Hubble parameter ⁠. In the radiation era we have

15

where is the conformal Hubble parameter today and Ω_r0=ρ_r0/ρ_crit0≈ 2.47 × 10⁻⁵h⁻² is the radiation energy density parameter today. Here h is defined by H₀=h× 100 km s⁻¹ Mpc⁻¹, and as a₀= 1, we have ⁠. Furthermore, we have H= (2t)⁻¹ and where t is the cosmic time. By equation (15) we find

16

We define the ratio of DE to CDM density r=ρ_de/ρ_c. Then, employing equations (1) and (2),

17

where the dot indicates derivative with respect to cosmic time. At early enough times, ⁠, and we can neglect the term in square brackets, so that the solution is

18

where r_ref is an integration constant corresponding to ρ_de/ρ_c at some (early) reference time t_ref in the case where Γ= 0. From equation (18) we find that we have two regimes, depending on the value of the early-time equation of state parameter w_e. If w_e≤−2/3, then the second term dominates over the first as t→ 0, and we recover the solution of Valiviita et al. (2008):

19

If w_e > −2/3, then the first term in equation (18) dominates

20

For the background evolution of ρ_c in the radiation-dominated (RD) era

21

the second term in brackets is negligible relative to the first at times t≪ 3/(2Γ) or ⁠. For these times, ρ_c∝a⁻³. In typical models that provide a good fit to cosmic microwave background (CMB) data, ⁠, and the conformal time at matter-radiation equality is ⁠. If we demand that the evolution of ρ_c is effectively standard during the whole RD era, i.e. τ_switch > τ_eq, we require

22

where h∼ 0.7. As we study in this paper coupling strengths |Γ/H₀| ≲ 1, we can safely assume that the CDM evolution during radiation domination is completely the standard non-interacting one ρ_c=ρ^eq_c(a/a_eq)⁻³, where ρ^eq_c and a_eq are the dark matter energy density and the scale factor at matter-radiation equality, respectively. Noticing that the radiation energy density can be written as ρ_r=ρ^eq_r(a/a_eq)⁻⁴, and that ρ^eq_c=ρ^eq_r by definition, we find that in the RD era

23

where we used equation (15). In the non-interacting case we could continue by setting a_eq=Ω_r0/Ω_c0, but in the interacting case the dark matter evolution from τ_switch up to today (τ₀) differs from ∝a⁻³: by equation (21), it follows that at recent times, for a positive Γ, the dark matter density decreases faster and with a negative Γ it decreases slower than a⁻³. Therefore, we cannot do the ‘a⁻³ scaling’ all the way up to today but instead have to stop at some early enough reference time. Here, we choose the time of matter-radiation equality.

An upper limit to the early DE equation of state w_e could be set by requiring dark matter domination over DE at early times. Then equation (20) would set the constraint w_e < 0. However, if the DE equation of state is close to 0 at early times, it could well mimic the behaviour of CDM. On the other hand, if w_e is close to 1/3, the ‘DE’ component would behave like radiation at early times. So, for 0 ≤w_e < 1/3, we conclude that the fluid which at late times behaves like DE behaves at early times like a combination of matter and radiation. As this case cannot be ruled out, we set a conservative upper bound on w_e by demanding that in the early universe DE does not dominate over radiation, i.e. for τ→ 0, we have ρ_de/ρ_r→ 0. Using equations (20) and (23), we find

24

which implies, as expected, w_e < 1/3.

4 SUPERHORIZON INITIAL CONDITIONS FOR PERTURBATIONS

In order to solve numerically the perturbation equations, we need to specify initial conditions in the early radiation era. The wavelength of the relevant fluctuations is far outside the horizon during this period: kτ≪ 1. To compute the initial conditions, we start by writing the perturbation equations of each species and the perturbed Einstein equations in terms of the gauge-invariant variables developed by Bardeen (1980):

25

The general evolution equations for the density, velocity and anisotropic stress perturbations Δ_A, V_A and Π_A and the Einstein equations for the metric perturbations Φ and Ψ are given in Kodama & Sasaki (1984).

We use and generalize the systematic method of Doran et al. (2003) in order to analyse the initial conditions in the interacting DE model with time-varying w_e. The results are derived below, but let us summarize the key points before going into the details. The conclusion is that we can use adiabatic initial conditions for

26

For both of these intervals, the initial conditions for all non-DE quantities are the same as in the non-interacting case. For the second w_e interval, the initial DE density perturbation is the same as the standard one, Δ_de= 3(1 +w_e)Δ_γ/4, whereas for the first interval, we find a non-standard initial condition Δ_de=Δ_γ/4. The difference arises because of the different background evolution in the two cases (as given in the previous section). Note that for w_e < −1, it is also possible to have adiabatic initial conditions, but we consider this case to be unphysical. For −1 ≤w_e≤−4/5, we recover the non-adiabatic blow-up case of Valiviita et al. (2008).

Similar considerations could be extended to the early matter-dominated (MD) era. The key difference there is that the background behaves differently for the interval −1/2 < w_e < 1/3 than for w_e≤−1/2, where the interaction modifies the DE evolution. In the matter era, a non-adiabatic blowup may thus happen if

27

A detailed analysis shows, however, that in the interval

28

the ‘blow-up’ mode is in fact a decaying mode, and hence (non-standard) adiabatic evolution on super-Hubble scales is possible. In the interval −1 < w_e < −2/3, the non-adiabatic ‘blow-up’ mode is rapidly increasing and will dominate unless |Γ| is suitably small. Therefore, a blowup in the matter era will make large interaction models with −1 < w_e < −2/3 non-viable, while the blowup in the radiation era ruins all interacting models (no matter how weak) with −1 < w_e < −4/5. We summarize these results in Table 1.

Assuming tight coupling between photons and baryons, so that V_b=V_γ, passing from conformal time τ to the time variable x=kτ, using a rescaled velocity and a rescaled anisotropic stress ⁠, as in Doran et al. (2003), we obtain the following evolution equations:

29

30

31

32

33

34

35

36

37

38

Here ⁠, and we used the Einstein equations

39

40

41

to eliminate Φ in favour of Ψ.

Using equations (39) and (41), we have

42

The total velocity appearing in equations (29) and (37) is

43

Recalling that and ⁠, we then see that equations (29)–(38) form a set of 10 linear differential equations for 10 perturbation variables and ⁠. (Note that Π_A= 0 for A≠ν.)

Since we are interested in the early radiation era, we make the approximation Ω_A=ρ_A/ρ≃ρ_A/ρ_r. Using equations (15), (19), (23) and (24), and the (standard non-interacting) background evolution of photons, baryons and neutrinos, we obtain

44

The next step of the method proposed in Doran et al. (2003) consists in writing the system of differential equations (29)–(38) in a matrix form:

45

where

46

and the matrix can be read from equations (29)–(38) after substituting equations (42)–(44) and the background evolution of ρ_c/ρ_de from (19) or (20), depending on the value of w_e.

The initial conditions are specified for modes well outside the horizon, i.e. for x≪ 1. There will be several independent solution vectors to equation (45), that we write as ⁠. If no term of diverges for x→ 0, then we can approximate by a constant matrix ⁠. If we require more accuracy, we can expand up to a desired order in x. For example, up to order x³ the matrix contains the constant term as well as terms proportional to x, x² and x³ in the case where w_e≤−2/3. However, in the case contains in addition to integer powers of x also non-integer powers 1 − 3w_e, 2 − 3w_e, 3 − 3w_e, etc. and 2 + 3w_e, 3 + 3w_e, 4 + 3w_e, etc. The listed ones and possibly their multiples can fall in the range (0, 3). For a given w_e, however, one should drop those which turn out to be of higher order than x³.

Thus going beyond zeroth order, up to the order of x³, we can expand and each solution as

47

48

where and are constant (not depending on the time variable x) matrices and U⁽ⁱ⁾_j are constant vectors. Note that for w_e≤−2/3 all and U⁽ⁱ⁾_Cj terms vanish. For simplicity, we demonstrate this case below, which leads to only integer powers. Substituting equations (47) and (48) into the evolution equation (45) and equating order by order, we obtain

49

50

51

52

Now the general solution to the differential equation (45) is a linear combination of solutions ⁠:

53

where c_i are dimensionless constants. If we define an initial reference time t_init, then the constants represent the initial contribution of the vector U⁽ⁱ⁾ to the total perturbation vector U(x_init). The imaginary part of λ_i represents oscillations of ⁠, while the real part gives its power-law behaviour: ⁠. The contribution corresponding to the eigenvalue(s) with largest real part, Re(λ_i), will dominate as time goes by, while initial contributions from eigenvectors corresponding to λ_i with smaller real part will become negligible compared to the dominant mode. Hence, to set initial conditions deep in the radiation era but well after inflation, it is sufficient to specify the contribution coming from mode(s) with largest Re(λ_i).

From now on, we divide the treatment into two cases −2/3 < w_e < 1/3 and w_e≤−2/3. Before proceeding, we should point out that the matrix method presented in Doran et al. (2003) and applied to non-interacting constant-w_de DE, represents a systematic and efficient approach for finding initial conditions. Once the matrix has been read from the set of first order differential equations (29)–(38), one can feed it into a symbolic mathematical program such as maple or mathematica and easily extract the constant part as well as the other parts (such as ⁠, …, ⁠) up to any desired order. Then it is simple linear algebra to find the eigenvalues λ_i and eigenvectors U⁽ⁱ⁾₀ of and, if higher order solutions in kτ are needed, to substitute these step by step into equations (50)–(52) etc. in order to find the solutions ⁠.

4.1 Case − 2/3 < w_e < 1/3

We substitute Ψ from equation (42), from equation (43) and the energy density parameters from equation (44) into equations (29)–(38). Then, using equation (20) for ρ_c/ρ_de in the last two of them, and taking the limit x→ 0, we find the matrix:

54

where ⁠.

The eigenvalues of are

55

where

56

For the range 0 < R_ν < 0.405 and −2/3 < w_e < 1/3, all eigenvalues have a non-positive real part. In (56), the term inside the square root, −20 + 12w_e+ 9w_e², falls between −24 and −15, and hence Re(λ^±_d) =−1 + 3w_e/2 falls between −2 and −1/2.

As explained in Doran et al. (2003), since the eigenvalue with largest real part, λ= 0, is four-fold degenerate, it is possible to choose a basis from the subspace spanned by the eigenvectors with eigenvalue λ= 0, so that physically meaningful choices can be made for the initial condition vector. One can form four independent linear combinations from the four vectors with λ= 0. The physical choices are an adiabatic mode and three isocurvature modes. Here we choose adiabatic initial conditions, specified by the condition that the gauge-invariant entropy perturbations S_AB of every A, B =γ, ν, c, b vanish, where (Malik et al. 2003)

57

We will show later the interesting new result that demanding adiabaticity between the standard constituents automatically guarantees adiabaticity with respect to DE.

We should remind the reader that for the interacting constituents the coupling appears in the continuity equation, and we should not use blindly the standard result

58

where the 1 +w_A factors result from applying the continuity equation to ρ′_A/ρ_A. Indeed, for CDM in the early radiation era we find, using equations (1), (3) and (15),

59

where w_c= 0. For DE, we find using equations (2), (3), (15) and (20)

60

At zeroth order in x=kτ, we incidentally regain the standard non-interacting result (58). From S_AB= 0 it then follows that

61

Imposing this condition on a linear combination of the four eigenvectors with eigenvalue λ= 0, we obtain

62

where and C₁ is a dimensionless normalization constant corresponding to, e.g., Δ_γ and Δ_ν at the initial time.

The vector (62) is identical to the standard adiabatic initial condition vector (see Doran et al. 2003). In particular, it should be noted that although we did not require adiabaticity of DE, (62) automatically satisfies the condition S_de,A= 0 for all A=γ, ν, c, b. In Doran et al. (2003), this result was found for non-interacting DE. Here we have now shown that also interacting DE is automatically adiabatic, once CDM, baryons, photons and neutrinos are set to be adiabatic.

Finally, since all components of the vector (62) are different from zero, it is not necessary to compute terms to higher order in x, and we use equation (62) as our adiabatic initial condition for the computation of the CMB power spectrum for models with −2/3 < w_e < 1/3.

Lee, Liu & Ng (2006) have reported that the quintessence isocurvature mode decays away (in an interacting quintessence model which is quite similar to our setup). After our systematic derivation of initial conditions, this decay can be tracked down to the fact that Re(λ^±_d) in equation (55) are negative. The reason for this is that in quintessence models the early-time equation of state parameter is typically larger than −2/3, indeed positive [but in Lee et al. (2006) less than +2/3]. So Re(λ^±_d) is negative, and hence the isocurvature mode decays. In the next subsection, we will see that also in the range −4/5 ≤w_e≤−2/3 (or w_e < −1) the DE isocurvature mode decays (although in this case the interaction affects the evolution of DE perturbations), whereas in the range −1 < w_e < −4/5 the DE isocurvature mode is a rapidly growing mode as recently realized by Valiviita et al. (2008).

4.2 Case w_e≤−2/3

Since at early times the equation of state can be approximated as a constant, w_e=w₀+w_a, this case has already been studied in Valiviita et al. (2008), where a constant −1 < w_de≤−2/3 was analysed. A serious non-adiabatic large-scale instability that excludes these models was found whenever −1 < w_de < −4/5, no matter how weak the interaction was. However, we notice that there is a limited region of parameter space, −4/5 ≤w_de≤−2/3, where the instability can possibly be avoided. In the case of a constant DE equation of state parameter, this range would be observationally disfavoured, since, for example, supernova data require that w_de is closer to −1 at recent times. In the case of time-varying w_de(a), we do not have this problem as w₀ can be close to −1 while −4/5 < w_e≤−2/3. In the following, we repeat the analysis of initial conditions done in Valiviita et al. (2008), but using the matrix method of Doran et al. (2003), extended to include the interaction, and give the conditions for a viable cosmology.

Substituting Ψ from equation (42), from equation (43) and the energy density parameters from equation (44) into equations (29)–(38), as well as ρ_c/ρ_de from equation (19) into the last two of them, and taking the limit x→ 0, we find the matrix, which is very similar to our previous result (equation 54). This happens because everything remains unchanged, except that we must replace in equations (37) and (38) the evolution of ρ_c/ρ_de with equation (19), and whenever Ω_de appears we must now substitute the ∝x³ behaviour from (44), instead of the behaviour. Therefore, only the last two rows in (54) are modified, and will now read

63

The eigenvalues of are

64

where

65

The first eight eigenvalues coincide with the previous case (equation 55). Of those, four have a negative real part, corresponding thus to modes that will decay away quickly and that we can neglect. The last two eigenvalues, λ^±_g, are instead very different from the previous case and depend on the value of w_e. The eigenvalue with the largest real part, λ⁺_g, is real and positive for ⁠. In addition to this, Re(λ⁺_g) is positive also in the small range ⁠. Therefore, Re(λ⁺_g) > 0 for −1 < w_e < −4/5. This corresponds to the blow-up solution found in Valiviita et al. (2008); λ⁺_g is larger, the closer w_e is to −1. There is no blowup of perturbations for −4/5 ≤w_e≤−2/3, because then the largest Re(λ_i) are zero.

4.2.1 Case −1 < w_e < −4/5; non-adiabatic blowup

We now calculate the initial condition vector corresponding to the fastest growing mode, λ⁺_g. At zeroth order in x, it is given by

66

In this case, since only the last two components of the vector are different from zero, we need to compute higher order corrections. It turns out that an expansion up to x³ is necessary and as explained both before and after equations (47) and (48), the expansion contains only integer powers of x, when w_e≤−2/3. Therefore, we have

67

68

By substituting and into the evolution equation (45) and equating order by order, we obtain

69

70

71

Using these formulas, we find corrections to equation (66). Keeping for each perturbation variable only the leading order (in x) terms, we obtain the following initial condition vector:

72

where and are and ⁠. This solution coincides with equations (63)–(70) of Valiviita et al. (2008), after substituting n_ψ=λ⁺_g+ 3, J= 1 − 16R_ν[5(n_ψ+ 2)(n_ψ+ 1) + 8R_ν]⁻¹, converting into Newtonian gauge (by using equations (25) with B=E= 0) and conveniently renormalizing the vector. Equation (72) is the initial condition vector for the case −1 < w_e≤−4/5, when the dominant eigenvector is that corresponding to λ⁺_g.

The initial condition (72) is trivially adiabatic with respect to γ, ν, c and b, but not with respect to DE. Indeed, for DE we find using equations (2), (3) and (19)

73

Therefore

74

for any A=γ, ν, c or b. Even if we were able to set the initial conditions at τ= 0 and demanded adiabaticity there, after a short time the solution would not be adiabatic. Thus, equation (72) represents the non-adiabatic ‘blow-up’ solution (Valiviita et al. 2008) for the case −1 < w_e < −4/5.

4.2.2 Cases w_e < −1 or −4/5 ≤ w_e≤−2/3; adiabatic initial conditions

In the range −4/5 < w_e≤−2/3, as well as for w_e < −1, we have Re(λ^±_g) < 0, so that the largest eigenvalue is the four-fold degenerate λ= 0. [If w_e=−4/5, then λ= 0 is four-fold degenerate, and there are also two oscillating solutions with Re(λ^±_g) = 0.] We look for a linear combination of the four eigenvectors (corresponding to λ= 0) that satisfies adiabaticity (see equation (61)) of photons, neutrinos, baryons and CDM. The resulting eigenvector is

75

where ⁠. This corresponds to equations (59)–(61) of Valiviita et al. (2008). All components except Δ_de are equal to the initial conditions for −2/3 < w_e < 1/3 (equation (62)). However, as pointed out in Valiviita et al. (2008), Δ_de=Δ_γ/4 corresponds exactly to the adiabaticity condition for DE: S_{de A}= 0. Namely, substituting the result (73) into definition (57), we find

76

Thus, equation (75) is an adiabatic initial condition vector for the cases −4/5 ≤w_e≤−2/3 or w_e < −1.

5 CONCLUSION

We have presented, for the first time, a systematic derivation of initial conditions for perturbations in a model of interacting dark matter–DE fluids, in the early radiation era. These initial conditions are essential for studying the further evolution of perturbations up to today's observables. They are the initial values for perturbations in any Boltzmann integrator which solves the multipole hierarchy and produces the theoretical predictions for the CMB temperature and polarization angular power spectrum, as well as the matter power spectrum. We have focused on the interaction Q^μ_c=−Γρ_c(1 +δ_c)u^μ_c, where Γ is a constant rate of energy density transfer [see equations (1) and (2)]. Generalizing a previous result for non-interacting DE in Doran et al. (2003), we find that, in our interacting model, requiring adiabaticity between all the other constituents (photons, neutrinos, baryons and CDM) leads automatically also to DE adiabaticity, if its early-time equation of state parameter is w_e < −1 or −4/5 ≤w_e≤ 1/3. In our previous work (Valiviita et al. 2008), we showed that if the equation of state parameter for DE is −1 < w_de < −4/5 in the radiation or matter eras, the model suffers from a serious non-adiabatic instability on large scales. In this paper, the systematic derivation of initial conditions confirms that result. However, in this paper we have shown that the instability can easily be avoided, if we allow for suitably time-varying DE equation of state. The main results are verbally summarized in Table 1.

In the companion paper (Valiviita et al. 2009), we modified the Code for Anisotropies in the Microwave Background (CAMB) Boltzmann integrator1 (Lewis, Challinor & Lasenby 2000), using the adiabatic initial conditions derived here for the interacting model, and performed full Monte Carlo Markov Chain likelihood scans for this model as well as for the non-interacting (Γ= 0) model for a reference, with various combinations of publicly available data sets: WMAP (Komatsu et al. 2009), WMAP and Arcminute Cosmology Bolometer Array Receiver (ACBAR; Reichardt et al. 2009), SN (Kowalski et al. 2008), BAO (Percival et al. 2007), WMAP and SN, WMAP and BAO, WMAP and SN and BAO.

With the parametrization w_de=w₀a+w_e(1 −a), viable interacting cosmologies result for w₀ close to −1 and w_e < −1 or −4/5 < w_e≤ 1/3, as long as w₀+ 1 and w_e+ 1 have the same sign (Valiviita et al. 2009). These particular conclusions apply exclusively to the interaction model we considered in this paper.

However, the method can be easily adapted for studying different interactions: one only needs to modify the background evolution and interaction terms in equations (29), (30), (37) and (38), before reading a new matrix from them. Based on section IV of Valiviita et al. (2008), the other interacting fluid models [ or ⁠, where α, β≲ 1 are dimensionless constants], that are common in the literature, behave in a very similar way to the model studied here, i.e. for −1 < w_e < w_crit the models are not viable due to the early-time large-scale blowup of perturbations, for w_crit < w_e < w_adi the models can be viable and non-standard adiabatic initial conditions may be found and for w_e > w_adi (or w_e < −1) the models are viable and standard (non-interacting) adiabatic initial conditions can be found. The critical value w_crit is determined by demanding that the ‘blow-up’ mode is actually a decaying mode and the fastest ‘growing’ curvature perturbation mode is a constant, i.e. the largest real part of the eigenvalues λ_i is zero, which with the notation of Valiviita et al. (2008) is guaranteed whenever Re(n₊) ≤ 3. In our model the critical value, w_crit=−4/5, is independent of the strength of interaction, but in the above-mentioned models it depends on α or β, as indicated by equations (85) and (98) in Valiviita et al. (2008). In general, our results show that the (early-time) DE equation of state plays, together with the interaction model, an important role in the (in)stability of perturbations.

JV and RM are supported by STFC. During this work JV received support also from the Academy of Finland.

REFERENCES