Why measuring the growth rate of cosmic structures?

We live in an universe in accelerated expansion!

Could it be due to a dark energy constant?

Could it be due to a dynamical dark energy field?

Credits: Pantheon+ Analysis Brout et al. 2022

Why measuring the growth rate of cosmic structures?

Alternative: GR is not valid at all scales and we need modified gravity models

Image credits: DALL.E

Luckily:

Those models have expansion history hard to distinguish from $Λ$ CDM...

...but they affect structures formation !

Image credits: arXiv:2204.06533

$f σ_{8}$ as a probe for general relativity

Structure evolution: Dark energy vs Gravity

Density contrast in linear theory:
$δ (a, x) = D (a) \tilde{δ} (x)$
where $D (a) \equiv$ growth factor

In measurement we use: $σ_{8} \propto D$

Velocities are linked to density through continuity equation:

$\nabla . v = - a H \frac{d δ}{d a} \propto - a H f σ_{8} δ$

where $f = \frac{d \ln D}{d \ln a} \equiv$ growth rate

In $Λ$ CDM + GR:
$f ≃ Ω_{m}^{γ}$ with $γ ≃ 0.55$

Image credits: Illustris TNG

How to measure $f σ_{8}$ ?

Velocities as probes of $f σ_{8}$ :
$\nabla . v \propto f σ_{8}$

Doppler effect on redshift:
$1 + z_{obs} = (1 + z_{\cos}) (1 + \frac{v_{p}}{c})$

Indirect method: Redshift Space Distorsions of the density field

Velocities leads to distorted positions: $r (z_{obs}) ≃ r (z_{\cos}) + \frac{1 + z_{\cos}}{H (z_{\cos})} v_{p}$

Direct method: velocity tracers

Galaxies Tully-Fisher and Fundamental Plane: $σ_{D} / D \sim 20 %$
Type Ia supernovae: $σ_{D} / D \sim 7 %$

Linear RSD

Non-linear RSD

2D corr. from Peacock et al. 2001

Tully-Fisher relation

Fundamental-Plane relation

RSD

Velocity tracers

Indirect measurement through distorsion of the density field from redshifts.

Direct measurement from the combination of redshifts and distance indicators.

Need high statistic

\Rightarrow

efficient at high-z

Limited by DI's precision

\Rightarrow

efficient at low-z

New player in the game: large low-z SNe Ia sample from the Zwicky Transient Facility !

Image credits: Illustris TNG

The Zwicky Transient Facility survey

The ZTF camera at Palomar observatory observes the full North hemisphere each night in 2 (g and r bands) + 1 (i band)

Already $\sim 8000$ classified supernovae
$\sim 5000$ SNe Ia at low redshift $z < 0.1$

Image credits: ZTF.Caltech

The Zwicky Transient Facility survey: spectroscopic follow-up

The Bright Transient Survey types every ZTF transients with $m a g < 17$

Typing efficiency is magnitude limited

No typing for $m a g > 19$

Image credits: ZTF.Caltech

The simulation and analysis pipeline

Use of ZTF existing observations as input for the simulation.
Includes observing strategy, filters, sky noise...

OuterRim N-body simulation:
We create 27 mocks of ${(1 Gpc . h^{- 1})}^{3} (z_{\max} ≃ 0.17)$
from the ${(3 Gpc . h^{- 1})}^{3}$ box at $z = 0$

Apply selection:

Photo-detection: 2 points with SNR $>$ 5

Spectro-typing selection from the ZTF Bright Transient Survey paper

Velocity estimator: velocities from Hubble diagram residuals

Standard candles

Standard candles + velocities

Standard candles + velocities + noise

The velocity estimator:
$\hat{v} = - \frac{\ln (10) c}{5} {(\frac{(1 + z) c}{H (z) r (z)} - 1)}^{- 1} Δ μ$

The velocity estimator error:
$σ_{\hat{v}} = - \frac{\ln (10) c}{5} {(\frac{(1 + z) c}{H (z) r (z)} - 1)}^{- 1} σ_{μ}$

The Maximum-Likelihood method (Johnson et al. 2014)

We can approximate the velocity field as a Gaussian random field

The correlation function depends of the Power Spectrum:

For densities:
$⟨ δ (x_{i}) δ (x_{i} + r) ⟩ \propto \int d k k^{2} P (k) W^{(δ)} (k; r) \propto σ_{8}^{2} \int d k k^{2} \tilde{P} (k) W^{(δ)} (k; r)$

For velocities:
$⟨ v (x_{i}) v (x_{j}) ⟩ \propto (f σ_{8})^{2} \int d k \tilde{P} (k) W^{(v)} (k; x_{i}, x_{j})$

It gives us a $f σ_{8}$ dependant model for our covariance matrix!
$C_{i j}^{v v} (f σ_{8}) = ⟨ v (x_{i}) v (x_{j}) ⟩$

Free parameters of the likelihood:

SNe Ia Hubble diagram parameters

p_{H D} = {α, β, M_{B}, σ_{M}}

Growth-rate related parameters

p = {f σ_{8}, σ_{u}, σ_{v}}

,

σ_{u} \equiv

RSD,

σ_{v} \equiv

non-linearities

The likelihood:

L (p, p_{H D}) \propto | C (p, p_{H D}) |^{- \frac{1}{2}} \times \exp [- \frac{1}{2} v^{T} (p_{H D}) C (p, p_{H D})^{- 1} v (p_{H D})]

C_{i j} (p, p_{H D}) = C_{i j}^{v v} (f σ_{8}, σ_{u}) + {σ_{v}}^{2} δ_{i j}^{K} + σ_{\hat{v}}^{2} (p_{H D}) δ_{i j}^{K}

C_{i j}^{v v} (f σ_{8}, σ_{u}) = ⟨ v (x_{i}) v (x_{j}) ⟩ \propto (f σ_{8})^{2} \int d k P (k) W (k) D (k; σ_{u})

Results: the selection bias

Magnitude limited selection $\Rightarrow$ Selection bias: when redshift increases, only brighter objects are detected

Bias on HD residuals

Bias on velocity estimates

$\hat{v} = - \frac{\ln (10) c}{5} {(\frac{(1 + z) c}{H (z) r (z)} - 1)}^{- 1} Δ μ$

Only the estimated velocities are biased !!!

Bias on $f σ_{8}$

Results: forecast for a ZTF 6-years complete sample

We can build a bias free sample by applying a redshift cut at $z = 0.06$ .

$⟨ N_{SN} ⟩ ≃ 1620 \sim$ half of the sample

Results for our 27 mocks :

Fit with $v_{true}$ : $\frac{⟨ f σ_{8} ⟩}{(f σ_{8})_{fid}} = 0.991 \pm 0.016$ , $\frac{\sqrt{⟨ σ_{f σ_{8}}^{2} ⟩}}{(f σ_{8})_{fid}} = 0.100$

Fit fixing $p_{HD}$ : $\frac{⟨ f σ_{8} ⟩}{(f σ_{8})_{fid}} = 1.036 \pm 0.031$ , $\frac{\sqrt{⟨ σ_{f σ_{8}}^{2} ⟩}}{(f σ_{8})_{fid}} = 0.185$

Fit with $p$ , $p_{HD}$ free: $\frac{⟨ f σ_{8} ⟩}{(f σ_{8})_{fid}} = 0.998 \pm 0.037$ , $\frac{\sqrt{⟨ σ_{f σ_{8}}^{2} ⟩}}{(f σ_{8})_{fid}} = 0.188$

Comparison with existing measurements

With only ~1600 SNe Ia, ZTF is at the same precision level than existing measurements with several thousands of galaxies !

How to improve the measurement?

Bias correction of the velocity estimates ?

How to improve the measurement? Bias correction of the velocities

Simulate a perfect correction of the bias:
$v_{debias, i} \sim N (v_{true}, σ_{\hat{v}, i})$

From a redshift limit of $z \sim 0.06$ to $z \sim 0.13$ , $N_{S N}$ is doubled but the error on $f σ_{8}$ only reduces from $17 %$ to $15 %$

Sampling drops rapidly after $z = 0.06$ and errors on ${\hat{v}}_{p}$ increase ~linearly with $z$

How to improve the measurement?

Bias correction of the velocity estimates ?

Does not improve the constraint on $f σ_{8}$

Use photo-typing to increase the redshift limit

DESC project led by Damiano Rosselli

Combination with RSD measurements (e.g. ZTF + DESI)

Ongoing work on this at CPPM by Julian Bautista

Future surveys

Peculiar velocities + RSD will put strong constraint on $f σ_{8}$ at low redshift

Forecast credits: DESI arxiv:1611.00036, LSST arxiv:1708.08236, EUCLID arXiv:1606.00180

Conclusion

We developed and ran a full detailed simulation and analysis pipeline to measure the growth rate of structure using ZTF SNe Ia

The spectroscopic selection causes a bias for $f σ_{8}$ above $z \sim 0.06$

We forecast that a 6-year ZTF SNe Ia spectro-identified sample cut at $z = 0.06$ will produce a $f σ_{8}$ measurement with a precision of $\sim 19 %$

Significant improvements are expected from future work on photometric typing analysis and combination with RSD

Thanks for your attention !!!

Backup slides

$f σ_{8}$ as a function of $z_{\max}$

Velocity estimators biases

{\hat{v}}_{1} = - \frac{\ln (10) c}{5} {(\frac{(1 + z) c}{H (z) r (z)} - 1)}^{- 1} Δ μ

{\hat{v}}_{2} = - \frac{\ln (10)}{5} \frac{H (z) r (z)}{(1 + z)} Δ μ

{\hat{v}}_{3} = - \frac{\ln (10) c}{5} {(\frac{1 + z}{z} - 1)}^{- 1} Δ μ

{\hat{v}}_{4} = - \frac{\ln (10) c}{5} \frac{z}{1 + z} Δ μ

Growth-rate measurement with type Ia supernovae

Bastien Carreres,

J. E. Bautista, F. Feinstein, D. Fouchez, B. Racine, M. Smith et al.