# Effects of a Late Gravitational Transition on Gravitational Waves and Anticipated Constraints

^{*}

## Abstract

**:**

## 1. Introduction

- Early time models [12,13,14,15,16] (see also [17] for a review) that implement new degrees of freedom (e.g., Early Dark energy [12,13,14,18], New Early Dark Energy [15,16], etc.) to decrease the scale of the sound horizon at recombination. This scale can be used as a standard ruler to measure the Hubble parameter [19,20] after calibration from the CMB angular power spectrum peak locations [1]. These models, however, have two problems: 1. They require significant fine tuning in order to avoid conflict with cosmological observations after the time of recombination [21,22]. 2. They should also decrease the horizon scale at the time of equal time matter and radiation density, which can also be calibrated using the matter power spectrum [23,24].
- Late time models [25,26,27,28,29,30,31,32,33] that attempt to deform the Planck 18 best-fit $\mathsf{\Lambda}$CDM form of the Hubble free expansion rate $E\left(z\right)\equiv H\left(z\right)/{H}_{0}$ between the time of recombination and the present time so that the present-time value $H(z=0)$ becomes consistent with the SH0ES best-fit value [10]. The problem of this class of models is that the significant level of $E\left(z\right)$ deformation needed is not consistent with a wide range of other observations constraining the form of $E\left(z\right)$ like BAO and SnIa data [34,35,36,37].
- Ultra-late time models [36,38,39,40,41,42] that investigate the possible presence of either an unaccounted-for systematic effect and/or a change in the fundamental physics taking place during the last $150Myrs$ (redshift $z<0.01$) when the calibration of standard candles like SnIa is performed [11]. The problems of this class of models includes fine tuning since the change in physical laws should not only occur at very late times but also should be consistent with other local observations [43].

## 2. Evolving Gravitational Constant and Gravitational Waves in a Cosmological Background

#### 2.1. Effective Gravitational Strength in Scalar–Tensor Theories with a Gravitational Transition

#### 2.2. Luminosity Distance from Gravitational Waves in Modified Gravity

- Modifications to the cosmological expansion history: Some modified gravity theories predict a different cosmic expansion history compared to general relativity, which can lead to changes in the luminosity distance–redshift relationship. As a result, the measured luminosity distance using gravitational waves may differ in these theories [95].
- Additional gravitational-wave polarization modes: In some modified gravity theories, gravitational waves can have additional polarization modes [96], such as scalar or vector modes, in addition to the standard tensor modes found in general relativity. The presence of these additional modes can affect the amplitude and phase evolution of the gravitational-wave signal, leading to a different measured luminosity distance.
- Propagation effects: The propagation of gravitational waves in modified gravity theories can be affected by changes in the effective gravitational constant or the presence of additional fields. These effects can alter the gravitational-wave amplitude and, consequently, the inferred luminosity distance [93,97].

- The energy carried by gravitational waves is proportional to the square of the amplitude of the wave, which, in turn, is proportional to the time-varying effective gravitational constant ${G}_{\mathrm{eff}}\left(z\right)$.
- The energy carried by electromagnetic waves is not affected by the time-varying effective gravitational constant, as it is primarily determined by the electromagnetic interaction.

#### 2.3. Gravitational Waves in an Expanding Universe

- $w=0$: Dust
- $w=\frac{1}{3}$: Radiation
- $w=-\frac{1}{3}$: Curvature or cosmic strings
- $w=-\frac{2}{3}$: Domain walls
- $w\to -1$: Vacuum energy

#### 2.4. The Imprints of a Sudden Cosmological Singularity

#### 2.5. Effects of a Gravitational Transition: A $\delta $-Function Impulse

#### 2.6. Numerical Solutions

## 3. Observational Constraints on the Transition Amplitude from Gravitational Waves and Other Cosmological Data

#### 3.1. Monte Carlo Data: Cosmological Parameters from the Einstein Telescope

#### 3.2. Real Data: BAO+CMB

#### 3.2.1. CMB Measurements

#### 3.2.2. BAO Measurements

#### 3.3. Real Data: Pantheon+

#### 3.4. Combined Data

#### 3.5. Results

## 4. Conclusions and Discussion

- Construction of a more detailed theoretical model that can induce the gravitational transition discussed in the present analysis. Such a model could be based on a scalar–tensor theory involving a false vacuum decay transition or a transition in time due to features of the potential determination of the dynamics of the scalar field.
- Consideration of additional astrophysical and/or cosmological data at redshifts $z<0.01$ (distances less than $40Mpc$) to impose constraints on such a gravitational transition or identify hints for its existence (see, e.g., [41]).
- The investigation of the effects of other types of singularities on the propagation of gravitational waves and on the luminosity distance as measured by either electromagnetic or gravitational waves. For instance, in the context of a generalized sudden cosmological singularity the $r=3$ derivative of the scale factor diverges. In this case, we have $\frac{\ddot{a}\left(t\right)}{a\left(t\right)}$∼$[1+\alpha \mathsf{\Theta}(t-{t}_{s})]$, while the scale factor a and its derivative $\dot{a}$ are continuous. Alternatively, a type III singularity can be constructed by admitting that $a\left(t\right)$∼$[1+\alpha \mathsf{\Theta}(t-{t}_{s})]$, or even a w-singularity by allowing that $w\left(t\right)=w[1+\alpha \delta (t-{t}_{s})]$.
- The consideration of alternative gravitational-wave mock data corresponding to other future gravitational-wave observatories including LISA [147,148] and the comparison of the constraints that can be imposed on the parameters of the sudden-leap transition model with those of the ET considered in the present analysis. LISA is projected to detect frequencies spanning from $0.1\phantom{\rule{0.166667em}{0ex}}mHz$ to ${10}^{-1}\phantom{\rule{0.166667em}{0ex}}Hz$, a markedly distinct frequency range from that of the ET. Within this spectrum of frequencies, the anticipation is to detect a myriad of GW sources [149], including Galactic binaries [150,151,152,153], binary systems hosting stellar-origin black holes [154], extreme-mass-ratio inspirals (EMRIs) [155], mergers of massive black hole binaries (MBHBs) even at high redshifts (up to z∼10) [145] and possibly stochastic GW backgrounds [156].

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Sudden-Leap Model (sLCDM)

**Figure A1.**Gray contours represent the $1-3\sigma $ confidence regions of the standard siren data for the corresponding best-fit values (Table 2).

**Figure A2.**Blue contours represent the projected $1-3\sigma $ confidence regions in $h-{z}_{s},\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Omega}}_{m,0}-{z}_{s}$ diagrams of the CMB+BAO data for the sudden-leap model (sLCDM); see Table 2.

**Figure A3.**Red contours represent the $1-3\sigma $ confidence regions of the ${\mathsf{\Omega}}_{m,0}-{z}_{s},\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Omega}}_{m,0}-M,\phantom{\rule{0.166667em}{0ex}}{z}_{s}-M,\phantom{\rule{0.166667em}{0ex}}{z}_{s}-h$ for the Pantheon+ data and their corresponding best-fit values.

**Figure A4.**Red contours represent the projected $1-3\sigma $ confidence regions in ${z}_{s}-h,\phantom{\rule{0.166667em}{0ex}}{z}_{s}-{\mathsf{\Omega}}_{m,0}$ diagrams of the ${\chi}^{2}={\chi}_{\mathrm{sirens}}^{2}+{\chi}_{\mathrm{BAO}}^{2}+{\chi}_{\mathrm{Panth}}^{2}+{\chi}_{\mathrm{CMB}}^{2}$ distribution, while the gray contours represent the ${\chi}^{2}={\chi}_{\mathrm{BAO}}^{2}+{\chi}_{\mathrm{Panth}}^{2}+{\chi}_{\mathrm{CMB}}^{2}$.

## Appendix B. Maximum Likelihood Method

$\mathsf{\Delta}{\chi}_{n-\sigma}^{2}$ | |||

$dim\left(M\right)$ | 68.27% | 95.45% | 99.73% |

3 | 3.52674 | 8.02488 | 14.1564 |

4 | 4.71947 | 9.71563 | 16.2513 |

5 | 5.8876 | 11.3139 | 18.2053 |

## Appendix C. ΛCDM Model

Sirens | CMB+BAO | Pantheon+ | Standrad Sirens+Pantheon+ | CMB+BAO+Pantheon+ | Sirens+CMB+BAO+Pantheon+ |
---|---|---|---|---|---|

1015.07 | 6.39 | 1522.98 | 2575.73 | 1570.97 | 2586.54 |

Data | $\mathit{M}$ | ${\mathsf{\Omega}}_{\mathit{m},\mathbf{0}}$ | $\mathit{h}$ | ||

Standard Sirens | — | 0.305 ±0.019 | 0.676 ± 0.006 | ||

CMB+BAO | — | 0.318 ± 0.006 | 0.672 ± 0.004 | ||

Pantheon+ | −19.25 ± 0.03 | 0.333 ± 0.018 | 0.734 ± 0.01 | ||

Standard Sirens+Pantheon+ | −19.41 ± 0.01 | 0.298 ± 0.012 | 0.683 ± 0.004 | ||

CMB+BAO+Pantheon+ | −19.43 ± 0.01 | 0.31 ± 0.005 | 0.677 ± 0.004 | ||

CMB+BAO+Pantheon++Standard Sirens | −19.43 ± 0.01 | 0.311 ± 0.004 | 0.676 ± 0.003 |

Data | $\frac{\mathsf{\Delta}\mathit{M}}{\mathit{M}}$ | $\frac{\mathsf{\Delta}{\mathsf{\Omega}}_{\mathit{m},0}}{{\mathsf{\Omega}}_{\mathit{m},0}}$ | $\frac{\mathsf{\Delta}\mathit{h}}{\mathit{h}}$ |
---|---|---|---|

Standard Sirens | — | 6.2% | 0.9% |

CMB+BAO | — | 1.9% | 0.6% |

Pantheon+ | 0.2% | 5.4% | 1.4% |

Standard Sirens+Pantheon+ | 0.1% | 4.0% | 0.6% |

CMB+BAO+Pantheon+ | 0.1% | 1.6% | 0.6% |

Standard Sirens+CMB+BAO+Pantheon+ | 0.1% | 1.6% | 0.4% |

## Notes

1 | There is also a completely analogous characterization with the corresponding Tipler integral [67]. |

2 | Some additional contour plots are presented in Appendix A (Figure A1). |

3 | Some additional contour plots are presented in Appendix A (Figure A2). |

4 | Some additional contour plots are presented in Appendix A (Figure A3). |

5 | Some additional contour plots are presented in Appendix A (Figure A4). |

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**Figure 1.**The presented figures illustrate a scenario involving a transition from a matter-dominated phase ($\tau <{\tau}_{s}$) with a fixed value of $\beta =\frac{2}{3}$ to a phase ($\tau >{\tau}_{s}$) with $\beta =\frac{2}{3}(1+\sigma )$. The dimensionless $\tau $ is plotted along the x-axis, and the following parameters for each case, ${\tau}_{s}=500,\phantom{\rule{0.166667em}{0ex}}\beta =\frac{2}{3},\phantom{\rule{0.166667em}{0ex}}\sigma =0.1,\phantom{\rule{0.166667em}{0ex}}{a}_{s}=0.995$, are fixed. We chose as initial conditions ${h}_{c}(\tau =200)=1$ and ${h}_{c}^{\prime}(\tau =200)=0$, just for illustration. The

**red**waveforms correspond to the gravitational waves after the singularity in the context of general relativity (${\alpha}_{M}=0$) and the

**blue**waveforms correspond to the propagation of gravitational waves with a modification that is incorporated by the friction term ${\alpha}_{M}$. The figures, denoted as 1 (

**a**) and (

**b**), depict an identical scenario, albeit corresponding to different ranges of the parameter $\tau \equiv kt$. It is also assumed that $\mathsf{\Theta}\left(0\right)=\frac{1}{2}$.

**Figure 2.**The long-term effects on the amplitude of the gravitational wave due to friction, as opposed to the initial gravitational wave (gray waveform). This behavior is due to a transition from a matter-dominated phase ($\tau <{\tau}_{s}$) with a fixed value of $\beta =\frac{2}{3}$ to a phase ($\tau >{\tau}_{s}$) with $\beta =\frac{2}{3}(1+\sigma )$. The dimensionless parameter $\tau $ is plotted along the x-axis, and the following parameters are fixed for each case: ${\tau}_{s}=500,\phantom{\rule{0.166667em}{0ex}}\beta =\frac{2}{3},\phantom{\rule{0.166667em}{0ex}}\sigma =0.1,\phantom{\rule{0.166667em}{0ex}}{a}_{s}=0.995$. To illustrate this behavior, we selected the initial conditions ${h}_{c}(\tau =200)=1$ and ${h}_{c}^{\prime}(\tau =200)=0$. The

**red**waveforms represent the gravitational waves after the singularity within the context of general relativity (${\alpha}_{M}=0$), while the

**blue**waveforms depict the propagation of gravitational waves with a modification, which is incorporated by the extra friction term ${\alpha}_{M}$ through a gravitational transition. Additionally, we assume that $\mathsf{\Theta}\left(0\right)=\frac{1}{2}$.

**Figure 3.**Let ${N}_{0}$ be the normalization constant, which is approximately equal to $1.42198\times {10}^{-9}$, and let $f\left(z\right)$ be the number density function (see Figure 8 in [80] or Figure 2 in [77]). We define the non-standard cumulative distribution function $N\left(z\right)$. To generate a list of non-uniformly distributed ${z}_{i}$ values, we solve the equation $N\left({z}_{i}\right)=i$ for $i=1,2,\dots ,1000$, where i is an integer.

**Figure 4.**The cosmology model assumed is $\mathsf{\Lambda}$CDM with ${\mathsf{\Omega}}_{m,0}=0.3166$ and ${H}_{0}=67.27\frac{km}{s}Mp{c}^{-1}$, the blue curve corresponds to the mean value or else ${\overline{d}}_{L}\left(z\right)=(1+z){\int}_{0}^{z}\frac{c}{{H}_{0}\sqrt{{\mathsf{\Omega}}_{m,0}{(1+{z}^{\prime})}^{3}+1-{\mathsf{\Omega}}_{m,0}}}d{z}^{\prime}$, while the yellow and red curves correspond to ${\overline{d}}_{L}\left(z\right)\pm \sigma \left(z\right)$, respectively (see Figure 7 [80]). The dots constructed are the 1000 standard sirens, which are distributed through the relation (43).

**Figure 5.**The black contours on the $h-\alpha $, ${\mathsf{\Omega}}_{m,0}-h$ and ${\mathsf{\Omega}}_{m,0}-h$ diagrams represent the projected confidence regions, at $1-3\sigma $, of the standard sirens using mock data. These contours correspond to a transition that occurs at the best-fit value of ${a}_{s}=0.401\pm 0.397$ and for the corresponding best-fit values each time, ${\mathsf{\Omega}}_{m,0},h,\alpha $, which can be found in Table 2.

**Figure 6.**The blue contours in the $h-\alpha $, ${\mathsf{\Omega}}_{m,0}-h$ and ${\mathsf{\Omega}}_{m,0}-h$ diagrams represent the projected $1-3\sigma $ confidence regions based on the CMB+BAO data for the sudden-leap model (sLCDM). The best-fit value for the transition occurring at ${a}_{s}=0.299\pm 0.048$, along with the corresponding best-fit values of ${\mathsf{\Omega}}_{m,0}$, h and $\alpha $, can be found in Table 2.

**Figure 7.**The red contours in the $h-\alpha $, ${\mathsf{\Omega}}_{m,0}-h$ and ${\mathsf{\Omega}}_{m,0}-h$ diagrams represent the projected $1-3\sigma $ confidence regions obtained from analyzing the Pantheon+ data within the sudden-leap model (sLCDM). The best-fit values for the transition occurring at ${a}_{s}=0.995\pm 0.004$ and $M=19.24\pm 0.03$, along with the corresponding best-fit values of ${\mathsf{\Omega}}_{m,0}$, h and $\alpha $, can be found in Table 2.

**Figure 8.**A combined plot including the contours of Figure 5 (black contours, which correspond to the confidence regions of the mock data of the standard sirens for the sudden-leap model); Figure 6 (blue contours, which correspond to the confidence regions of the assemblage of $CMB,\phantom{\rule{0.166667em}{0ex}}BAO$ data of the sudden-leap model); and Figure 7 (red contours, which correspond to the confidence regions of the Pantheon+ data of the sudden-leap model), in contrast to one another.

**Figure 9.**The red contours observed in the $h-\alpha $, ${\mathsf{\Omega}}_{m,0}-\alpha $ and ${\mathsf{\Omega}}_{m,0}-h$ diagrams represent the projected $1-3\sigma $ confidence regions obtained through the analysis of the combined ${\chi}^{2}$ function within the sudden-leap model (sLCDM). This combined ${\chi}^{2}$ function comprises contributions from ${\chi}_{\mathrm{sirens}}^{2}$, ${\chi}_{\mathrm{BAO}}^{2}$, ${\chi}_{\mathrm{Panth}}^{2}$ and ${\chi}_{\mathrm{CMB}}^{2}$. The best-fit values for the transition occurring at ${a}_{s}=0.812\pm 0.2$ and the absolute magnitude $M=-19.43\pm 0.015$, as well as the corresponding best-fit values for ${\mathsf{\Omega}}_{m,0}$, h, and $\alpha $, are provided in Table 2. Additionally, the gray contours reflect the contributions from ${\chi}_{\mathrm{BAO}}^{2}$, ${\chi}_{\mathrm{CMB}}^{2}$ and ${\chi}_{\mathrm{Panth}}^{2}$ for the corresponding best-fit values of ${a}_{s}=0.789\pm 0.2$ and $M=-19.43\pm 0.015$.

**Figure 10.**The resulting red contours, which correspond to the confidence regions of the Pantheon+ data, the blue contours to the CMB+BAO data, and the black contours to the SS mock data of the sudden-leap model, in contrast to the corresponding confidence regions of the (red hue) Pantheon+, the (blue hue) CMB+BAO and the (dashed) $SS$ mock data of the $\mathsf{\Lambda}$CDM model (see Appendix C).

**Figure 11.**The resulting contours that correspond to the confidence regions sketched around the minimum of ${\chi}_{Pant}^{2}+{\chi}_{\mathrm{BAO}}^{2}+{\chi}_{\mathrm{CMB}}^{2}$ (gray) and ${\chi}_{Pant}^{2}+{\chi}_{\mathrm{BAO}}^{2}+{\chi}_{\mathrm{CMB}}^{2}+{\chi}_{\mathrm{sirens}}^{2}$ (red) of the sudden-leap model in contrast to the corresponding confidence regions sketched around the minimum of ${\chi}_{Pant}^{2}+{\chi}_{\mathrm{BAO}}^{2}+{\chi}_{\mathrm{CMB}}^{2}$ (gray hue) and ${\chi}_{Pant}^{2}+{\chi}_{\mathrm{BAO}}^{2}+{\chi}_{\mathrm{CMB}}^{2}+{\chi}_{\mathrm{sirens}}^{2}$ (red hue) of the $\mathsf{\Lambda}$CDM model (see Appendix C).

**Table 1.**Constraints on the evolution of the gravitational constant. Methods with star (*) constrain ${G}_{*}$ (Planck-mass-related), while the rest constrain ${G}_{\mathrm{eff}}$ (see Table 1 in [41]).

Method | $|\frac{\mathbf{\Delta}{\mathit{G}}_{\mathbf{eff}}}{{\mathit{G}}_{\mathbf{eff}}}{|}_{\mathit{max}}$ | Time Scale (Yr) | References |
---|---|---|---|

Hubble diagram SnIa—$1\sigma $ confidence level | 0.1 | ∼${10}^{8}$ | [69] |

Gravitational waves | 8 | $5\times {10}^{-8}$ | [70] |

Paleontology | $0.1$ | $2\times {10}^{-11}$ | [71] |

Big Bang Nucleosynthesis—$2\sigma $ confidence level * | $0.05$ | $1.4\times {10}^{10}$ | [72,73] |

Anisotropies in CMB—$2\sigma $ confidence level * | $0.095$ | $1.4\times {10}^{10}$ | [74] |

**Table 2.**The ${\chi}^{2}$ distributions and the best-fit parameters accompanied by their corresponding errors can be seen below. The parameters are M (absolute magnitude), ${\mathsf{\Omega}}_{m,0}$ (density parameter for matter), h (parameter related to the Hubble constant ${H}_{0}$), $\alpha $ (the transition amplitude) and ${a}_{s}$ (the value of the scale factor at the event of transition).

Standard Sirens | CMB+BAO | Pantheon+ | Standard Sirens+Pantheon+ | CMB+BAO+Pantheon+ | Standard Sirens+CMB+BAO+Pantheon+ |
---|---|---|---|---|---|

1013.01 | 5.58 | 1521.64 | 2575.55 | 1570.69 | 2586.44 |

Data | $\mathit{M}$ | ${\mathsf{\Omega}}_{\mathit{m},\mathbf{0}}$ | $\mathit{h}$ | $\mathit{\alpha}$ | ${\mathbf{a}}_{s}$ |

Standard Sirens | — | 0.323 ± 0.024 | 0.673 ± 0.008 | 0.052 ± 0.123 | 0.401 ± 0.397 |

CMB+BAO | — | 0.323 ± 0.009 | 0.671 ± 0.004 | 0.008 ± 0.01 | 0.299 ± 0.048 |

Pantheon+ | −19.24 ± 0.03 | 0.325 ± 0.02 | 0.718 ± 0.023 | 0.057 ± 0.068 | 0.995 ± 0.004 |

Standard Sirens+Pantheon+ | −19.42 ± 0.01 | 0.321 ± 0.015 | 0.678 ± 0.004 | 0.039 ± 0.025 | 0.498 ± 0.015 |

CMB+BAO+Pantheon+ | −19.43 ± 0.01 | 0.308 ± 0.009 | 0.678 ± 0.006 | −0.004 ± 0.007 | 0.789 ± 0.2 |

Standard Sirens+CMB+BAO+Pantheon+ | −19.43± 0.01 | 0.31 ± 0.005 | 0.677 ± 0.003 | −0.001 ± 0.005 | 0.812 ± 0.2 |

Data | $\frac{\mathsf{\Delta}\mathit{M}}{\mathit{M}}$ | $\frac{\mathsf{\Delta}{\mathsf{\Omega}}_{\mathit{m},0}}{{\mathsf{\Omega}}_{\mathit{m},0}}$ | $\frac{\mathsf{\Delta}\mathit{h}}{\mathit{h}}$ |
---|---|---|---|

Standard Sirens | — | 7.4% | 1.2% |

CMB+BAO | — | 2.8% | 0.6% |

Pantheon+ | 0.2% | 6.2% | 3.2% |

Standard Sirens+Pantheon+ | 0.1% | 4.6% | 0.6% |

CMB+BAO+Pantheon+ | 0.1% | 2.9% | 0.9% |

Standard Sirens+CMB+BAO+Pantheon+ | 0.1% | 1.6% | 0.4% |

Data | $\mathsf{\Delta}\mathbf{=}{\mathit{AIC}}_{\mathit{sLCDM}}\mathbf{-}{\mathit{AIC}}_{\mathsf{\Lambda}\mathit{CDM}}$ |
---|---|

Standard Sirens | 1.94 |

CMB+BAO | 3.19 |

Pantheon+ | 2.66 |

Standard Sirens+Pantheon+ | 3.82 |

CMB+BAO+Pantheon+ | 3.72 |

Standard Sirens+CMB+BAO+Pantheon+ | 3.9 |

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## Share and Cite

**MDPI and ACS Style**

Paraskevas, E.A.; Perivolaropoulos, L.
Effects of a Late Gravitational Transition on Gravitational Waves and Anticipated Constraints. *Universe* **2023**, *9*, 317.
https://doi.org/10.3390/universe9070317

**AMA Style**

Paraskevas EA, Perivolaropoulos L.
Effects of a Late Gravitational Transition on Gravitational Waves and Anticipated Constraints. *Universe*. 2023; 9(7):317.
https://doi.org/10.3390/universe9070317

**Chicago/Turabian Style**

Paraskevas, Evangelos Achilleas, and Leandros Perivolaropoulos.
2023. "Effects of a Late Gravitational Transition on Gravitational Waves and Anticipated Constraints" *Universe* 9, no. 7: 317.
https://doi.org/10.3390/universe9070317