# Revisiting a Negative Cosmological Constant from Low-Redshift Data

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## Abstract

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## 1. Introduction

## 2. Overview of the Datasets Used

**BAO:**In the early Universe, the interplay between gravity and radiation pressure sets up acoustic oscillations, which produce a sharp feature in the two-point correlation function of luminous matter at a scale equal to the comoving size of the sound horizon at the drag epoch [262,263,264,265,266]:$${r}_{\mathrm{drag}}\equiv {r}_{s}\left({z}_{\mathrm{drag}}\right)\phantom{\rule{0.166667em}{0ex}},$$$$\begin{array}{ccc}\hfill \frac{{D}_{M}\left({z}_{\mathrm{eff}}\right)}{{r}_{\mathrm{drag}}}& \equiv & \frac{c}{{H}_{0}{r}_{\mathrm{drag}}}{\int}_{0}^{{z}_{\mathrm{eff}}}\frac{d{z}^{\prime}}{E\left({z}^{\prime}\right)}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \frac{{D}_{H}\left({z}_{\mathrm{eff}}\right)}{{r}_{\mathrm{drag}}}& \equiv & \frac{c}{{H}_{0}{r}_{\mathrm{drag}}}\phantom{\rule{0.166667em}{0ex}}\frac{1}{E\left({z}_{\mathrm{eff}}\right)}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\begin{array}{ccc}\hfill \frac{{D}_{V}\left({z}_{\mathrm{eff}}\right)}{{r}_{\mathrm{drag}}}& \equiv & \frac{c}{{H}_{0}\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{drag}}}\phantom{\rule{0.166667em}{0ex}}\text{exception 25:}3\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$See Appendix B of [268] for a more detailed discussion of BAO measurements.In our analysis, we consider anisotropic BAO measurements from the Baryon Oscillation Spectroscopic Survey (BOSS) collaboration [3] Data Release 12 (DR12) at the effective redshifts ${z}_{\mathrm{eff}}=0.38\phantom{\rule{0.166667em}{0ex}},0.51\phantom{\rule{0.166667em}{0ex}},0.61$ and from Lyman-$\alpha$ forest samples at ${z}_{\mathrm{eff}}=2.40$ [211,212]. We also include isotropic BAO measurements from the Six-degree Field Galaxy Survey (6dFGS) at ${z}_{\mathrm{eff}}=0.106$ [209], from the Sloan Digital Sky Survey Data (SDSS) Main Galaxy Sample (MGS) release at ${z}_{\mathrm{eff}}=0.15$ [210] and from the quasar sample of the extended Baryon Oscillation Spectroscopic Survey (eBOSS) at ${z}_{\mathrm{eff}}=1.52$ [213]. For completeness, we collect the BAO measurements used in this work in Table 1. We constructed the likelihood for the BAO data, $\mathcal{L}}_{\mathrm{BAO}$, using the data described and the correlation matrices provided by the collaborations.**SNeIa:**SNeIa catalogues report the corrected apparent magnitude of the $i\mathrm{th}$ supernova, which is given by the following:$${\mu}^{i}=M+5\phantom{\rule{0.166667em}{0ex}}{\mathrm{Log}}_{10}\left(\right)open="("\; close=")">{D}_{A}^{i}\left(z\right)/\mathrm{Mpc}$$$${\mu}^{i}=5.7+5\phantom{\rule{0.166667em}{0ex}}{\mathrm{Log}}_{10}\left(\right)open="("\; close=")">\frac{c(1+z)}{{H}_{0}{\ell}_{\mathrm{SN}}}{\int}_{0}^{z}\frac{d{z}^{\prime}}{E\left({z}^{\prime}\right)}$$$${\mathcal{L}}_{\mathrm{SN}}=exp\left(\right)open="["\; close="]">-\frac{1}{2}{\left(\right)}^{\mu}T$$**Anchors:**The cosmic distance ladder technique relies on measuring distances of extra-galactic objects, at distances beyond $\sim$100 Mpc, to map the Hubble flow. The expansion history of the Universe is mapped through Type 1a Supernovae (SNeIa) datasets [214,269,270,271,272,273,274,275] and BAO [265,267]. These distance scales need to be calibrated through anchors either at the high-redshift end (through $r}_{\mathrm{drag}$) or at the low-redshift end (through $H}_{0$) [276].For instance, BAO measurements constrain the combination $H}_{0}{r}_{\mathrm{drag}$. In fact, the tension between measurements at low and high redshifts of $H}_{0$ can be eased by modifying the value of $r}_{\mathrm{drag}$, which is the standard ruler providing the BAO length scale. The sole measurements of either $H}_{0$ or $r}_{\mathrm{drag}$ are respectively interpreted as anchoring the cosmic distance ladder or the inverse cosmic distance ladder. In any case, in order to interpret the BAO measurements, we have to anchor either $H}_{0$ or $r}_{\mathrm{drag}$ to an independent evaluation [51]. Here, we consider independent approaches, namely calibrating the cosmic distance ladder by means of the recent measurements of $H}_{0$ in [8] or calibrating the inverse distance ladder by using the CMB measurement of $r}_{\mathrm{drag}$ in [5]. It is worth remarking that much (but not all) of the information contained in the CMB temperature and anisotropy spectra resides in the position and height of the first acoustic peak, which accurately constrains $\theta}_{s$ given in Equation (1) and provides valuable information about the geometry and the content of the Universe. The position of the first peak is sensitive to early-time modifications through a change in $r}_{\mathrm{drag}$, as well as to late-time modifications through a change in ${D}_{A}\left({z}_{\star}\right)$.Here, we consider two scenarios using two different anchors, given by the following:**Sound horizon at drag epoch (Scenario I)**: The anchor at the CMB epoch is expressed by the comoving sound horizon at the end of the baryon drag epoch $r}_{\mathrm{drag}$. Measurements of CMB temperature and polarization anisotropies by the Planck collaboration (PlanckTTTEEE + lowP dataset) give $r}_{\mathrm{drag}}=\left(\right)open="("\; close=")">147.05\pm 0.30\phantom{\rule{0.166667em}{0ex}$Mpc at a 68% Confidence Level (C.L.). When anchoring data to the sound horizon $r}_{\mathrm{drag}$, we construct the likelihood function $\mathcal{L}}_{\mathrm{Anchor}}={\mathcal{L}}_{\mathrm{drag}$ as a Gaussian in $r}_{\mathrm{drag}$ using this measurement.**Hubble constant $H}_{0$ (Scenario II)**: When anchoring data to the present value of the Hubble rate, we use the latest result from the SH0ES program [8], which reports the local measurement $H}_{0}=\left(\right)open="("\; close=")">74.03\pm 1.42\phantom{\rule{0.166667em}{0ex}}\mathrm{km}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{Mpc}}^{-1$. This measurement is used to construct the likelihood function $\mathcal{L}}_{\mathrm{Anchor}}={\mathcal{L}}_{\mathrm{Ceph}$ as a Gaussian in $H}_{0$.

## 3. Overview of Models and Model Comparison

- $\mathsf{\Lambda}$CDM model: Here, the normalized expansion rate is modelled as follows:$${E}_{\mathsf{\Lambda}\mathrm{CDM}}\left(z\right)\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{\left(\right)}^{{\mathsf{\Omega}}_{\mathsf{\Lambda}}}1/2$$
- $w$CDM model: Here, the normalized expansion rate is modelled as follows:$${E}_{w\mathrm{CDM}}\left(z\right)\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{\left(\right)}^{{\mathsf{\Omega}}_{\varphi}}1/2$$
- $c$CDM model: Here, the normalized expansion rate is modelled as follows:$$\begin{array}{c}\hfill {E}_{c\mathrm{CDM}}\left(z\right)={\left(\right)}^{{\mathsf{\Omega}}_{\mathrm{cc}}}1/2\phantom{\rule{0.166667em}{0ex}}.\end{array}$$We fix the value of $\mathsf{\Omega}}_{\varphi$ by demanding that ${E}_{w\mathrm{CDM}}\left(0\right)=1$, so that the parameter space associated with this model $\mathcal{S}}_{c\mathrm{CDM}}=\{{\mathsf{\Omega}}_{\mathrm{cc}},{\mathsf{\Omega}}_{M},{w}_{\varphi},{H}_{0},{r}_{\mathrm{drag}},{\ell}_{\mathrm{SN}}\$ is six-dimensional. We demand that $\mathsf{\Omega}}_{\mathrm{cc}$ be strictly negative by exploring the region of the parameter space ${\mathsf{\Omega}}_{\mathrm{cc}}<0$ when fitting the parameters in $\mathcal{S}}_{c\mathrm{CDM}$ against the data described in Section 2.

`emcee`[294]. For each model $\mathcal{M}$ described in Section 3, we explore the associated parameter space $\mathcal{S}}_{\mathcal{M}$ by performing the analysis with a Metropolis–Hastings algorithm and assuming flat priors for all variables, except for the anchor for which we assume a Gaussian prior with mean and standard deviation given by the corresponding measurement. In more detail, we assume a flat prior for ${\mathsf{\Omega}}_{M}\in \left(\right)open="["\; close="]">0,1$ and $\ell}_{\mathrm{SN}}\in \left(\right)open="["\; close="]">0.9,1.2\phantom{\rule{0.166667em}{0ex}$Mpc for all models discussed, in addition to the DE EoS ${w}_{\varphi}\in \left(\right)open="["\; close="]">-2,-0.5$ for the $w$CDM and $c$CDM models and the cosmological constant ${\mathsf{\Omega}}_{\mathrm{cc}}\in \left(\right)open="["\; close="]">-30,0$ for the $c$CDM model. When anchoring the distance ladder to the comoving sound horizon at the drag epoch (Scenario I), we choose the flat prior $H}_{0}\in \left(\right)open="["\; close="]">60,80\phantom{\rule{0.166667em}{0ex}$km/s/Mpc and a Gaussian prior for $r}_{\mathrm{drag}$ based on the measurement by the Planck collaboration. When the present Hubble rate is used as the anchor (Scenario II), we choose the flat prior $r}_{\mathrm{drag}}\in \left(\right)open="["\; close="]">120,160\phantom{\rule{0.166667em}{0ex}$Mpc and a Gaussian prior for $H}_{0$ given by the SH0eS measurements [8]. We use the expression for the likelihood:

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AdS | anti-de Sitter |

AIC | Akaike Information Criterion |

BAO | Baryon Acoustic Oscillations |

BOSS | Baryon Oscillation Spectroscopic Survey |

CMB | Cosmic Microwave Background |

DE | Dark Energy |

DR12 | Data Release 12 |

dS | de Sitter |

eBOSS | Extended Baryon Oscillation Spectroscopic Survey |

EoS | Equation of State |

HST | Hubble Space Telescope |

MCMC | Markov Chain Monte Carlo |

MGS | Main Galaxy Sample |

SDSS | Sloan Digital Sky Survey |

SH0ES | Supernovae and H0 for the Dark Energy Equation of State |

SNeIa | Type Ia Supernovae |

6dFGS | Six-degree Field Galaxy Survey |

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**Table 1.**BAO scale measurements used in this work. For BOSS DR12, we set the fiducial scale $r}_{\mathrm{fid}}=147.78\phantom{\rule{0.166667em}{0ex}$Mpc [3].

Dataset | $\mathit{z}$ | Measurement | Reference |
---|---|---|---|

BOSS DR12 | $0.38$ | $H\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{r}_{\mathrm{drag}}/{r}_{\mathrm{fid}}=\left(\right)open="("\; close=")">81.2\pm 2.2\pm 1.0$ | [3] |

BOSS DR12 | $0.51$ | $H\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{r}_{\mathrm{drag}}/{r}_{\mathrm{fid}}=\left(\right)open="("\; close=")">90.9\pm 2.1\pm 1.1$ | [3] |

BOSS DR12 | $0.61$ | $H\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{r}_{\mathrm{drag}}/{r}_{\mathrm{fid}}=\left(\right)open="("\; close=")">99.0\pm 2.2\pm 1.2$ | [3] |

BOSS DR12 | $0.38$ | $D}_{M}/{r}_{\mathrm{drag}}=\left(\right)open="("\; close=")">1512\pm 22\pm 11\phantom{\rule{0.166667em}{0ex}}\mathrm{Mpc}/{r}_{\mathrm{fid}$ | [3] |

BOSS DR12 | $0.51$ | $D}_{M}/{r}_{\mathrm{drag}}=\left(\right)open="("\; close=")">1975\pm 27\pm 14\phantom{\rule{0.166667em}{0ex}}\mathrm{Mpc}/{r}_{\mathrm{fid}$ | [3] |

BOSS DR12 | $0.61$ | $D}_{M}/{r}_{\mathrm{drag}}=\left(\right)open="("\; close=")">2307\pm 33\pm 17\phantom{\rule{0.166667em}{0ex}}\mathrm{Mpc}/{r}_{\mathrm{fid}$ | [3] |

BOSS DR12 + Ly$\alpha$ | $2.40$ | ${D}_{M}/{r}_{\mathrm{drag}}=36.6\pm 1.2$ | [211] |

BOSS DR12 + Ly$\alpha$ | $2.40$ | ${D}_{H}/{r}_{\mathrm{drag}}=8.94\pm 0.22$ | [211] |

6dFGS | $0.106$ | ${D}_{V}/{r}_{\mathrm{drag}}=2.98\pm 0.13$ | [209] |

SDSS-MGS | $0.15$ | ${D}_{V}/{r}_{\mathrm{drag}}=4.47\pm 0.17$ | [210] |

eBOSS quasars | $1.52$ | ${D}_{V}/{r}_{\mathrm{drag}}=26.1\pm 1.1$ | [213] |

**Table 2.**Left column: corner plots showing the 1D marginalized and 2D joint posterior distributions for the parameters of the $\mathsf{\Lambda}$CDM (top panel), $w$CDM (middle panel), and $c$CDM models (bottom panel), when anchoring BAO measurements to $r}_{\mathrm{drag}$ as measured Planck [5]. Right column: same as the left column, but anchoring BAO measurements to $H}_{0$ as measured by the SH0ES program [8].

$\mathit{r}}_{\mathbf{drag}$ as Anchor (Scenario I) | $\mathit{H}}_{0$ as Anchor (Scenario II) |
---|---|

**Table 3.**Constraints on the parameters of the three models considered in this work, obtained using the two different choices of anchor (specified in the first column). We report 68% C.L. intervals on all parameters, except for $\mathsf{\Omega}}_{c$, for which we do not have a detection, and hence report the 95% C.L. lower bound. The last column reports $\Delta \mathrm{AIC}$, defined in Equation (15) and calculated with respect to the $\mathsf{\Lambda}$CDM model. A positive value of $\Delta \mathrm{AIC}$ indicates a preference for $\mathsf{\Lambda}$CDM.

Anchor | Model | $\mathsf{\Omega}}_{\mathit{c}$ | $\mathit{w}}_{\mathit{\varphi}$ | $\mathsf{\Omega}}_{\mathit{M}$ | $\mathit{H}}_{0$ km/s/Mpc | $\mathit{r}}_{\mathit{s}$ (Mpc) | $\mathit{\ell}}_{\mathbf{SN}$ (Mpc) | $\mathsf{\Delta}\mathbf{AIC}$ |
---|---|---|---|---|---|---|---|---|

$r}_{\mathrm{drag}$ | $\mathsf{\Lambda}$CDM | $0.31\pm 0.02$ | $68.53}_{-0.78}^{+0.79$ | $147.05}_{-0.21}^{+0.22$ | $1.04\pm 0.01$ | 0 | ||

$r}_{\mathrm{drag}$ | $w$CDM | $-{1.17}_{-0.19}^{+0.17}$ | $0.36}_{-0.05}^{+0.04$ | $68.82}_{-0.86}^{+0.86$ | $147.06}_{-0.21}^{+0.21$ | $1.04\pm 0.01$ | 1.9 | |

$r}_{\mathrm{drag}$ | $c$CDM | >$-13.88$ | $-{1.02}_{-0.09}^{+0.02}$ | $0.36}_{-0.05}^{+0.05$ | $68.83}_{-0.85}^{+0.86$ | $147.05}_{-0.31}^{+0.30$ | $1.04\pm 0.01$ | 3.6 |

$H}_{0$ | $\mathsf{\Lambda}$CDM | $0.31\pm 0.02$ | $74.03}_{-1.01}^{+1.03$ | $136.16}_{-2.42}^{+2.43$ | $0.97\pm 0.01$ | 0 | ||

$H}_{0$ | $w$CDM | $-{1.18}_{-0.19}^{+0.17}$ | $0.36}_{-0.05}^{+0.04$ | $74.04}_{-1.01}^{+0.99$ | $136.72}_{-2.43}^{+2.52$ | $0.97\pm 0.01$ | 1.3 | |

$H}_{0$ | $c$CDM | >$-14.48$ | $-{1.02}_{-0.09}^{+0.02}$ | $0.36}_{-0.05}^{+0.05$ | $73.96}_{-1.45}^{+1.41$ | $136.86}_{-3.04}^{+3.22$ | $0.97\pm 0.02$ | 4.8 |

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Visinelli, L.; Vagnozzi, S.; Danielsson, U.
Revisiting a Negative Cosmological Constant from Low-Redshift Data. *Symmetry* **2019**, *11*, 1035.
https://doi.org/10.3390/sym11081035

**AMA Style**

Visinelli L, Vagnozzi S, Danielsson U.
Revisiting a Negative Cosmological Constant from Low-Redshift Data. *Symmetry*. 2019; 11(8):1035.
https://doi.org/10.3390/sym11081035

**Chicago/Turabian Style**

Visinelli, Luca, Sunny Vagnozzi, and Ulf Danielsson.
2019. "Revisiting a Negative Cosmological Constant from Low-Redshift Data" *Symmetry* 11, no. 8: 1035.
https://doi.org/10.3390/sym11081035