© ESO 2020

1. Introduction

The growth of a large-scale structure is sensitive to the theory of gravity and its measurement is a powerful test of the standard and alternative models of cosmology. It is characterised at the most basic level by the rate of growth f = −dlnD/dln(1 + z), where D(z) is the growth function of the linear matter density contrast, δ(z, k) = D(z)δ(z_in, k)/D(z_in), given an initial redshift z_in. This rate governs the evolution of peculiar velocities, whose impact on the observed galaxy power spectrum is to introduce a redshift space distortion (RSD). The measurement of this anisotropy at redshift z delivers an estimate of f(z)σ₈(z), where σ₈ fixes the amplitude of the matter density fluctuations. The degeneracy between f and σ₈ echoes the degeneracy between the linear galaxy bias and σ₈, and it cannot be broken via RSD power spectrum measurements alone.

The degeneracy can be broken by using an alternative observable in the galaxy sample that involves σ₈ or f. For example, combining RSD power spectrum measurements with galaxy-galaxy lensing measurements has produced separate estimates of f and σ₈ (de la Torre et al. 2017; Shi et al. 2018; Jullo et al. 2019). There are currently only a handful of such estimates, but even with only three separated data pairs, constraints on cosmological models improve noticeably (Perenon et al. 2019). Another way to break the degeneracy is by combining RSD measurements in the power spectrum and bispectrum (Gil-Marín et al. 2017).

Breaking the growth degeneracy is expected to break degeneracies between certain cosmological and modified gravity parameters. Here we confirm this expectation by computing the improvement in precision when using future separated measurements of f and σ₈ as compared to using the usual combined measurements fσ₈. We make forecasts for the standard Λ cold dark matter (CDM) model and for scalar-tensor theories in the Horndeski class (Horndeski 1974), using the effective field theory (EFT) of dark energy (DE; Gubitosi et al. 2013; Bloomfield et al. 2013), see Frusciante & Perenon (2020) for a recent review and Gleyzes et al. (2016), Alonso et al. (2017), Leung & Huang (2017), Abazajian et al. (2016), Reischke et al. (2019), Spurio Mancini et al. (2018), Frusciante et al. (2019), Ballardini et al. (2019) for more general Horndeski forecasts).

2. Models

We consider two models to assess the constraining power of the different growth of structure quantities. The first is the standard cosmological model ΛCDM, whose free parameters are (Planck Collaboration V 2020)

$$\begin{array}{c}\hfill \{{\mathrm{\Omega }}_{\mathrm{b}}{h}^2,{\mathrm{\Omega }}_{\mathrm{c}}{h}^2,H_0,\tau ,A_{\mathrm{s}},n_{\mathrm{s}},\mathrm{\Sigma }{m}_{\nu }\},\end{array}$$(1)

where the total neutrino mass ∑m_ν is equally shared by the three degenerate species. For the second, we chose the popular benchmark for studies of alternative gravitational models (Frusciante & Perenon 2020) that are Horndeski theories (Horndeski 1974). They are the most general covariant scalar-tensor theories with direct second-order equations of motion. We use in particular their description of linear perturbations provided by the α-EFT basis Bellini & Sawicki (2014). Bellini & Sawicki (2014) provide complete details of the construction of the action.

Observations suggest that the speed of gravitational waves is equal to that of light (Abbott et al. 2017a,b). This reduces the number of redshift-dependent functions in the effective description that govern how modifications of gravity affect perturbations to three:

$$\begin{array}{cc}\hfill {\alpha }_M(z)& -\mathrm{evolution}\mathrm{of}\mathrm{the}\mathrm{effective}\mathrm{Planck}\mathrm{mass};\hfill \\ \hfill {\alpha }_B(z)& -\mathrm{mixing}\mathrm{between}\mathrm{the}\mathrm{metric}\mathrm{and}\mathrm{DE}\mathrm{field};\hfill \\ \hfill {\alpha }_K(z)& -\mathrm{kinetic}\mathrm{energy}\mathrm{of}\mathrm{scalar}\mathrm{perturbations}.\hfill \end{array}$$

Although α_K has virtually no effect on constraints from current data (Bellini et al. 2016; Frusciante et al. 2019), it needs to be included as a free parameter, since it regulates the propagation speed of DE perturbations. Setting it arbitrarily to zero could restrict the space of stable models and thus bias the constraints (Kreisch & Komatsu 2018; Frusciante et al. 2019).

The functional forms of α_I(z), I = M, B, K, are not given by the effective description. For simplicity, we use the effective DE parametrisation (Piazza et al. 2014; Bellini & Sawicki 2014) common in the literature,

$$\begin{array}{c}\hfill {\alpha }_I(z)=a_I\frac{{\mathrm{\Omega }}_{\mathrm{x}}(z)}{{\mathrm{\Omega }}_{\mathrm{x},0}}·\end{array}$$(2)

We also allow for deviations from a ΛCDM background by using the Chevallier-Polarski-Linder (CPL; Chevallier & Polarski 2001; Linder 2003) parametrisation for the effective DE equation of state of the Horndeski models:

$$\begin{array}{c}\hfill w_{\mathrm{x}}(z)=w_0+w_a\frac{z}{1+z}·\end{array}$$(3)

In summary, the Horndeki model we consider contains five additional free parameters with respect to ΛCDM:

$$\begin{array}{c}\hfill \{{\mathrm{\Omega }}_{\mathrm{b}}{h}^2,{\mathrm{\Omega }}_{\mathrm{c}}{h}^2,H_0,\tau ,A_{\mathrm{s}},n_{\mathrm{s}},\mathrm{\Sigma }{m}_{\nu },w_0,w_a,a_M,a_B,a_K\}.\end{array}$$(4)

The ΛCDM model is recovered for w₀ = −1 and w_a = a_M = a_B = a_K = 0.

3. Methodology

The cosmological evolution of the models was computed using the Boltzmann code¹ CLASS (Blas et al. 2011), and its modified version² hi_class (Zumalacarregui et al. 2017; Bellini et al. 2020). The cosmological data – hereafter referred to as the “baseline” – contain the SDSS-II/SNLS3 Joint Light-curve Analysis (JLA) sample of type Ia supernova (SNIa; Betoule et al. 2014), the Baryon Oscillation Spectroscopic Survey (BOSS) baryon acoustic oscillation (BAO) measurements (Beutler et al. 2011; Anderson et al. 2014; Ross et al. 2015), and the low- and high-multipole temperature and polarisation of Planck 2018 cosmic microwave background (CMB) data (Planck Collaboration V 2020). We chose not to include CMB lensing data to avoid inconsistencies related to potential ΛCDM-dependent assumptions made during the lensing reconstruction.

Our aim was to focus on the gain from breaking growth degeneracy, rather than making realistic mocks and forecasts. In order to compare the constraining power of separated measurements of f and σ₈ with the combined measurements fσ₈, we simulated data for a nominal future galaxy sample that delivered a one percent precision for f, σ₈, and fσ₈. We assumed a redshift range containing ten measurements at z = 0.1, 0.2, …, 1. The effects of extending the redshift range are studied in Sect. 4.3. We anticipate that a Stage IV experiment conducting a spectroscopic galaxy count survey together with a weak lensing survey, such as Euclid (Amendola et al. 2018), should be able to achieve close to 1% precision on fσ₈, f, and σ₈, using Planck priors on standard cosmological parameters.

Whenever needed, the growth quantities are computed with CLASS or hi_class. In order to compare the constraints on the same footing and avoid non-linear model dependencies, we computed the growth quantities with the linear power spectrum only. The values of σ₈ were obtained via the usual weighted integral of the linear power spectrum and f was computed as the log derivative f = −(1 + z)dlnσ₈/dlnz for simplicity.

As fiducial parameters we used the best-fit values obtained from the baseline constraints for the ΛCDM and Horndeski models. Then we created three sets of mocks for both models (for f, σ₈, and fσ₈), each exactly centred on their fiducial, meaning no random variance added to the data. The ΛCDM model has been shown to lie in a corner of the parameter space of stable Horndeski models (Piazza et al. 2014), which are ghost- and gradient-free models. When performing forecasts using Markov chain Monte Carlo (MCMC) methods, the stability priors can lead to a disfavouring of models lying close to the corner, purely due to volume effects and independently of their actual likelihood. Such considerations may have a significant effect on our results. This is hinted at for example by the highly irregular posteriors in the baseline case in Fig. 3 (grey contours) and the mismatch between their maximum and the best-fit model (dotted lines), characteristic of non-negligible prior effects. We can however expect those effects to be mitigated when additional data is added to the analysis, due to the fact that our Horndeski fiducial model (derived from the baseline best fit and used to produce our mocks) lies noticeably away from the “ΛCDM corner”. Even if our MCMC explorations were impacted by such priors, this should not affect our conclusions since we always make statements regarding relative improvements.

4. Constraints

The sampling of all the considered likelihoods, as well as the computation of best-fit parameters, are performed using the publicly available³ suite of codes ECLAIR (Ilić et al. 2020). It uses as its main sampling algorithm the affine-invariant ensemble method of Foreman-Mackey et al. (2013) and contains a novel and robust maximiser with reliable convergence towards the global maximum of the posterior.

4.1. ΛCDM

Marginalised posterior distributions are shown in Fig. 1. The corresponding means and 68% confidence intervals are given in Table 1, while Table 2 shows the gain in precision relative to the baseline (first two columns) and for the separated growth measurements f + σ₈ relative to the standard fσ₈ measurements (last column). We define the precision as the inverse width of the 68% marginalised confidence interval rather than using relative errors, since the latter can become misleading when the mean values are close to zero (e.g., in the case of Σm_ν). In addition, comparing relative errors would also be biased when the mean values shift, as happens for the Horndeski models (see below).

Fig. 1. One-dimensional and two-dimensional marginalised posterior distributions for ΛCDM parameters derived from the baseline only (grey), baseline with mock on fσ8 (blue), and baseline with mocks on f and σ8 (red). The dotted lines indicate the parameter values for the fiducial model (corresponding to the baseline best fit) used when generating mocks.

Next-generation surveys are forecast to deliver improved constraints from high-precision RSD fσ₈ data (see e.g. Amendola et al. 2018; Bacon et al. 2020). The triangle plots and the tables confirm this. Table 2 (first column) shows that the gain in precision ranges from ∼10% for Ω_bh² up to more than ∼50% for Ω_ch², H₀ and Σm_ν, when considering the addition of the mock data on fσ₈, with 1% relative error combined to current cosmological data sets.

As expected the constraints improve further with the split mock data on f and σ₈, each with a 1% relative error. This combination performs from 6% to almost 50% better. In particular, the precision on Ω_ch² and H₀ is more than doubled relative to the baseline data alone.

The improvement obtained from the split f and σ₈ data over fσ₈ (as quantified by the third column of Table 2) does not lead to an equal increase in precision on all the parameters that were already well constrained with fσ₈ RSD data. As an example, we can compare Σm_ν and H₀. Adding fσ₈ data yields almost a 50% gain on Σm_ν, while the split f + σ₈ data further increases the precision by 13%. By contrast, H₀ precision first increases by 55% followed by another 46% with the splitting.

The growth probes f, σ₈, and fσ₈ have different sensitivities to each cosmological parameter, which explains the range of changes in precision. One way to examine those sensitivities is to start with the baseline-only constraints. Figure 2 shows the posterior distributions of f, σ₈, and fσ₈ at redshift z = 0.1 as derived parameters versus the cosmological parameters⁴. Each posterior thus illustrates how a change in a given cosmological parameter impacts the values of the derived growth quantities, taking into account (i.e. marginalising over) the remaining cosmological parameters and how their values need to change to keep a decent fit to the data.

Fig. 2. One-dimensional marginalised posterior distributions (top row) for ΛCDM parameters, from baseline only (grey), baseline + mock on f (green), and baseline + mock on σ8 (purple). Rows below show 2D posteriors of cosmological parameters against derived parameters f, σ8, and fσ8 at z = 0.1.

On the other hand, adding constraints on the growth quantities amounts to convolving their posteriors with a Gaussian distribution (with a width equal to 1% of the central value). This in turn may reduce the width of the posterior on cosmological parameters, depending on the amount of correlation between the two. It is thus expected that cosmological parameters that are highly correlated (i.e. thin tilted ellipses) with a given growth quantity in the baseline case, will show the best improvements after including measurements of that growth quantity.

From Fig. 2 we find that Ω_b, Ω_c, H₀, and n_s are better constrained by adding the f mock (green) to the baseline, while τ, A_s, and Σm_ν are better constrained by adding the σ₈ mock (purple). This may appear counter to the common expectation that σ₈ is more sensitive to parameters affecting the power spectrum amplitude, while f is more sensitive to parameters affecting its shape. It is the correlations induced by the baseline constraints that are the decisive factor.

Let us consider an illustrative example from Fig. 2: the 2D posterior of {f(0.1),H₀} exhibits a high correlation (thin tilted ellipse), while that of {σ₈(0.1),H₀} is relatively irregular and close to an uncorrelated case. As a result, the addition of the f mock improves the H₀ constraint significantly more relative to the baseline (see the 1D posterior of H₀ in the top row of Fig. 2).

These correlations can even lead to improved constraints on parameters that f and σ₈ should not depend on. An example is the tight constraint on the reionisation parameter τ produced by the mock on σ₈, which originates in the tight constraint on A_s from σ₈, combined with the underlying high correlation between A_s and τ, as shown in Fig. 1. A tight constraint on τ is obtained even though it does not play a role in the value of σ₈.

4.2. Horndeski

The Horndeski parameter space is extended to include modifications in the background (w₀, w_a) and in the perturbations (α_M, α_K, α_B). Marginalised posterior distributions with the baseline and mock data sets are displayed in Fig. 3, with the corresponding means and 68% confidence intervals in Table 3. We observe that the maximum of the posterior distribution for the extension parameters shifts significantly towards the best-fit model (dotted lines), while the contours assume a much more regular, ellipsoidal shape compared to the baseline case. This is expected in a transition from a regime where priors still play a significant role (as discussed at the end of Sect. 3), to a situation where data dominate the posterior. Interestingly, these results also show that if the true underlying cosmology is indeed close to the Horndeski best-fit fiducial, then growth data with 1% relative precision (over the redshift range considered) could lead to the detection of this deviation from ΛCDM with strong significance (more than 5σ).

Fig. 3. One-dimensional and two-dimensional marginalised posterior distributions for Horndeski parameters derived from the baseline only (grey), baseline with mock on fσ8 (blue), and baseline with mocks on f and σ8 (red). The dotted lines indicate the parameter values for the fiducial model (corresponding to the baseline best fit) used when generating mocks.

Table 4 shows the gain in precision relative to the baseline (first two columns) and for the separated growth measurements f + σ₈ relative to the standard fσ₈ measurements (last column). As pointed out earlier, the kineticity coupling α_K is not constrained by the data and is therefore not included in the figure and tables, but a_K is included as a free parameter in the analysis. The accuracy that was gained on the cosmological parameters in ΛCDM is largely lost. Adding the mock on fσ₈ only delivers a precision gain of up to ∼20% (see Table 4). This can be attributed to the addition of new, poorly constrained degrees of freedom, which naturally leads to larger errors on all the original parameters via correlations, as both sets may have similar and degenerate effects on the growth of structure. For example, Fig. 3 shows how a_M and a_B are relatively degenerate with other parameters when using the baseline data only.

However, there is significant improvement for the extension parameters: adding future fσ₈ data yields a 230% improvement for the running of the effective Planck mass α_M and a remarkable ∼50% gain for Σm_ν. Even though fσ₈ is a probe of the perturbations, adding its mock to the baseline achieves a surprising ∼30% and ∼60% gain in precision for w₀ and w_a respectively.

The additional gain from disentangling f and σ₈ measurements is also subject to the effects of opening up the parameter space. The standard parameters see little improvement (< 15%) over the fσ₈ case. By contrast, w_a, Σm_ν, and α_M precisions jump by a further ∼20%, ∼ 65%, and ∼80% respectively.

The underlying reason why growth data provide such an enhancement in precision for the Horndeski parameters is rooted in the modification of gravitational dynamics (e.g. the Poisson equation) by α_I. As discussed in Perenon et al. (2019), these modifications produce two opposing contributions:

⁎ a fifth force, enhancing growth;

⁎ a higher effective Planck mass, suppressing growth.

The effective Planck mass is controlled solely by α_M for the models we consider. As a result, growth data strongly constrains a_M and also a_B. Table 4 shows that the splitting of fσ₈ into f and σ₈ is very effective to further constrain a_M, thereby disentangling the fifth force and effective Planck mass contributions. This feature was seen even with current split data in Perenon et al. (2019).

The modified background parameters w₀ and w_a also contribute to the growth of structure through Hubble friction. Their effects on growth are therefore degenerate with those of α_I. We see in Fig. 4 that w₀,  w_a,  a_B, and a_M display some degeneracies in their 2D marginalised posteriors.

Fig. 4. One-dimensional marginalised posterior distributions (top row) for Horndeski parameters, from baseline only (grey), baseline + mock on f (green), and baseline + mock on σ8 (purple). Rows below show 2D posteriors of cosmological parameters against derived parameters f, σ8, and fσ8 computed at z = 0.1.

Following the arguments for ΛCDM, we can understand the separate improvements from f and σ₈ by analysing their posterior distributions versus cosmological parameters, shown in Fig. 4. We note that the stability requirements for the Horndeski models induce highly non-Gaussian posterior distributions, which makes the analysis more subtle. Figure 4 shows that f correlates more strongly with a_B and a_M than σ₈, so that adding f measurements results in a larger increase in precision for these parameters. Since these two parameters control the strength of the fifth force, this could be expected, given that σ₈ is an integrated function of f, which tends to wash out the effects of the fifth force. The fifth force is an effect occurring at low redshifts as opposed to the effect of Hubble friction or neutrinos. A chain of correlations – seen in the baseline constraints – shows that σ₈ brings a larger gain in precision for w₀,  w_a, and Σm_ν. This signals therefore a higher sensitivity of σ₈ to modifications of gravity spanning longer periods.

It is in fact expected that the effect of neutrinos is partially degenerate with that of modified gravity (see e.g. Wright et al. 2019; Ballardini et al. 2020). Massive neutrinos suppress the growth of structure on small scales, which can either oppose or reinforce modified gravity, depending on whether the fifth force or the Planck mass running is favoured. Horndeski models compatible with current RSD fσ₈ constraints produce a suppression of growth at late times (Perenon et al. 2019).

The baseline constraints in Fig. 3 show that the 2D posteriors of Σm_ν with a_B and a_M have a fairly irregular shape, while those with w₀ and w_a are more correlated. More surprisingly, as noted above, Σm_ν has almost a 50% gain with the addition of the fσ₈ mock data, as in the case of ΛCDM. The splitting improves constraints by a further 70% as opposed to 7% in ΛCDM. It is therefore clear that these growth mocks break the neutrino-modified gravity degeneracy by efficiently constraining Σm_ν and the extension parameters. Figure 4 tells us that this is rooted in the correlation of Σm_ν with σ₈ in the baseline.

On the other hand, we also see that all the intricate degeneracies between the extension parameters and standard model parameters render the baseline constraints for the latter much less correlated than in the case of ΛCDM. This explains why the improvements from the splitting are not as great in the case of Horndeski for the other standard parameters. We note that when the background evolution is fixed to that of ΛCDM, Σm_ν displays a correlation with α_B (Bellomo et al. 2017). Here, the freedom that arises from varying w₀ and w_a lessens that correlation.

4.3. Extending the redshift range

Having understood better the influence of each mock data set on the constraints, we now assess the effect of extending the redshift coverage of the mocks. More specifically, we examine the respective merits of adding fσ₈ or f + σ₈ measurements, when extending the maximum redshift of each mock. Table 5 shows that the combined data fσ₈ with z_max = 2 (first column) performs no better than f + σ₈ data with half the redshift range (z_max = 1, see Tables 1 and 3). We find that extending the redshift range further improves the precision up to 30% with respect to z_max = 1 in the case of the combined mock fσ₈ for ΛCDM and Horndeski models, and respectively 20% and 15% in the case of f and σ₈ mocks.

5. Conclusion

Upcoming galaxy surveys such as Euclid (Amendola et al. 2018) and Square Kilometre Array (SKA; Bacon et al. 2020) with their unprecedented precision is a call to sharpen our tools for constraining gravity. One cosmological probe well suited for that task is the growth of structure. This toolbox is further complemented by the releases of measurements on f and σ₈ (de la Torre et al. 2017; Shi et al. 2018; Jullo et al. 2019; Gil-Marín et al. 2017).

In this paper, we considered the performance that a future nominal galaxy sample can deliver with a ∼1% relative error on f and σ₈ separately and on the combination fσ₈. We compared the constraints from the separated data with those from the combination data. We assumed ten measurements per growth quantity equally spread over the redshift range z = 0.1,  0.2,  …, 1.0. For the case of ΛCDM, the improvements in precision range over ∼5–50%. For modified gravity described by Horndeski models, the improvements on these standard model parameters reduce to ∼0–15%.

However, the splitting of f and σ₈ stands out as very effective in breaking the neutrino – modified gravity degeneracy, with the sum of neutrino masses enjoying an improvement of 65% over the case with only fσ₈ data. We also find a significant increase in the precision on the background and perturbation Horndeski parameters, with an additional gain of ∼20% for the varying effective DE equation of state parameter w_a and ∼80% for the evolution of the effective Planck mass a_M. Extending the redshift of the mocks up to z_max = 2 shows that the constraints provided by the combined fσ₈ data are already matched by the split data f and σ₈ with z_max = 1.

Our results highlight that growth data, whether split or combined and with 1% relative error could lead to the detection of deviations from ΛCDM with strong significance (more than 5σ), should the underlying cosmology be close to the current Horndeski best-fit fiducial. The splitting of growth data on fσ₈ into data on f and σ₈ with galaxy-galaxy lensing (de la Torre et al. 2017; Shi et al. 2018; Jullo et al. 2019) or by combinations with the bispectrum (Gil-Marín et al. 2017) emerges clearly from this work as both a powerful complementary probe for the standard model and a stringent probe to detect departures from it. The latter could prove crucial in the era of future surveys, given the current tensions within the standard model and the emergence of alternative models of gravity favoured by Bayesian evidence (Peirone et al. 2019; Solà Peracaula et al. 2019).

Acknowledgments

We are grateful to Julien Bel for helpful exchanges and feedback on the draft. We thank Matteo Martinelli for useful discussions. The authors acknowledge the Sciama High Performance Compute Cluster, which is supported by the ICG at the University of Portsmouth, and the CHPC the Centre for High Performance Computing (CHPC), South Africa, for providing computational resources for this research project. LP and RM are supported by the South African Radio Astronomy Observatory (SARAO) and the National Research Foundation (Grant No. 75415). RM is also supported by the UK STFC Consolidated Grant ST/N000668/1. SI was supported by the European Structural and Investment Fund and the Czech Ministry of Education, Youth and Sports (Project CoGraDS – CZ.02.1.01/0.0/0.0/15_003/0000437). AdlCD acknowledges financial support from NRF Grants No.120390, Reference: BSFP190416431035 and No.120396, Reference: CSRP190405427545, Project No. FPA2014-53375-C2-1-P from the Spanish Ministry of Economy and Science, Project No. FIS2016-78859-P from the European Regional Development Fund and Spanish Research Agency (AEI), and support from Projects Nos. CA15117 and CA16104 from COST Actions EU Framework Programme Horizon 2020.

References

All Tables

All Figures

	Fig. 1. One-dimensional and two-dimensional marginalised posterior distributions for ΛCDM parameters derived from the baseline only (grey), baseline with mock on fσ8 (blue), and baseline with mocks on f and σ8 (red). The dotted lines indicate the parameter values for the fiducial model (corresponding to the baseline best fit) used when generating mocks.
In the text

	Fig. 2. One-dimensional marginalised posterior distributions (top row) for ΛCDM parameters, from baseline only (grey), baseline + mock on f (green), and baseline + mock on σ8 (purple). Rows below show 2D posteriors of cosmological parameters against derived parameters f, σ8, and fσ8 at z = 0.1.
In the text

	Fig. 3. One-dimensional and two-dimensional marginalised posterior distributions for Horndeski parameters derived from the baseline only (grey), baseline with mock on fσ8 (blue), and baseline with mocks on f and σ8 (red). The dotted lines indicate the parameter values for the fiducial model (corresponding to the baseline best fit) used when generating mocks.
In the text

	Fig. 4. One-dimensional marginalised posterior distributions (top row) for Horndeski parameters, from baseline only (grey), baseline + mock on f (green), and baseline + mock on σ8 (purple). Rows below show 2D posteriors of cosmological parameters against derived parameters f, σ8, and fσ8 computed at z = 0.1.
In the text