In this study, we investigate the role of bulk viscosity in the evolution of a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe dominated by dust. By parameterizing the bulk viscosity as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\tilde{p}= -3(l + m (H'(t) + H^2) + n H^2) H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>p</mi> <mo stretchy="false">~</mo> </mover> <mo>=</mo> <mo>-</mo> <mn>3</mn> <mrow> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mi>m</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>H</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>n</mi> <msup> <mi>H</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation>, we derived the modified Einstein field equations and reformulate them in terms of the redshift z. Using the Hubble datasets with 46 data points and Pantheon+ compilation with 1701 supernova measurements, we estimated the model parameters (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, l, m, n) through <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\chi ^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>χ</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> minimization and refined them using Markov chain Monte Carlo (MCMC) simulations. Our analysis reveals a transition redshift <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$z_t = 0.585$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>z</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.585</mn> </mrow> </math></EquationSource> </InlineEquation>, marking the universe’s shift from decelerated to accelerated expansion, and a current deceleration parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$q_0 =- 0.705$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>q</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>-</mo> <mn>0.705</mn> </mrow> </math></EquationSource> </InlineEquation> consistent with the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>CDM model. The current age of the universe obtained from our model was <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$t_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>t</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> = 14.5734 Gyrs. Additionally, we estimate the present Hubble constant, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$H_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, to be 68 km/s/Mpc based on the Hubble datasets and approximately 73 km/s/Mpc using the Pantheon+ datasets. The best-fit cosmological parameters indicate a recent transition to accelerated expansion at <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$z_{t} \approx 0.55$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>z</mi> <mi>t</mi> </msub> <mo>≈</mo> <mn>0.55</mn> </mrow> </math></EquationSource> </InlineEquation>, a present-day deceleration parameter <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$q_{0} \approx -0.55$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>q</mi> <mn>0</mn> </msub> <mo>≈</mo> <mo>-</mo> <mn>0.55</mn> </mrow> </math></EquationSource> </InlineEquation>, a Hubble constant in the range <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$67\; to- 74 kms^{-1} Mpc^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>67</mn> <mspace width="0.277778em" /> <mi>t</mi> <mi>o</mi> <mo>-</mo> <mn>74</mn> <mi>k</mi> <mi>m</mi> <msup> <mi>s</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>M</mi> <mi>p</mi> <msup> <mi>c</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, and a cosmic age <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$t_{0} \approx 13.7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>≈</mo> <mn>13.7</mn> </mrow> </math></EquationSource> </InlineEquation> Gyr, consistent with current observational constraints. This disparity highlights the ongoing Hubble tension-a discrepancy between locally measured values of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$H_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> (e.g., via supernovae) and values inferred from the early universe (e.g., CMB observations). Our findings are consistent with the larger trend of greater <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$H_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> values from late-universe observations compared with early-universe predictions, highlighting the need for more research into the underlying physical or systematic sources of this tension. We further evaluate the evolution of the cosmological parameters, including the equation of state parameter <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\omega (z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, energy density <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\rho (z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and pressure p(z). Statefinder diagnostics (r, s) demonstrate the versatility of the model, capturing its deviation from and convergence toward <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\Lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>CDM at specific epochs. This study emphasizes the importance of bulk viscosity in explaining the universe’s accelerated expansion and provides a strong framework for assessing various cosmological theories.

Role of viscous fluid in FLRW model with observational constraints

期刊论文
发表日期：

2026年7月4日

查看全文

Anirudh Pradhan,
G. K. Goswami

摘要

In this study, we investigate the role of bulk viscosity in the evolution of a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe dominated by dust. By parameterizing the bulk viscosity as $\tilde{p}= -3(l + m (H'(t) + H^2) + n H^2) H$ $\tilde{p} = - 3 (l + m (H^{'} (t) + H^{2}) + n H^{2}) H$ , we derived the modified Einstein field equations and reformulate them in terms of the redshift z. Using the Hubble datasets with 46 data points and Pantheon+ compilation with 1701 supernova measurements, we estimated the model parameters ( $H_0$ $H_{0}$ , l, m, n) through $\chi ^{2}$ $χ^{2}$ minimization and refined them using Markov chain Monte Carlo (MCMC) simulations. Our analysis reveals a transition redshift $z_t = 0.585$ $z_{t} = 0.585$ , marking the universe’s shift from decelerated to accelerated expansion, and a current deceleration parameter $q_0 =- 0.705$ $q_{0} = - 0.705$ consistent with the $\Lambda$ $Λ$ CDM model. The current age of the universe obtained from our model was $t_{0}$ $t_{0}$ = 14.5734 Gyrs. Additionally, we estimate the present Hubble constant, $H_0$ $H_{0}$ , to be 68 km/s/Mpc based on the Hubble datasets and approximately 73 km/s/Mpc using the Pantheon+ datasets. The best-fit cosmological parameters indicate a recent transition to accelerated expansion at $z_{t} \approx 0.55$ $z_{t} \approx 0.55$ , a present-day deceleration parameter $q_{0} \approx -0.55$ $q_{0} \approx - 0.55$ , a Hubble constant in the range $67\; to- 74 kms^{-1} Mpc^{-1}$ $67 t o - 74 k m s^{- 1} M p c^{- 1}$ , and a cosmic age $t_{0} \approx 13.7$ $t_{0} \approx 13.7$ Gyr, consistent with current observational constraints. This disparity highlights the ongoing Hubble tension-a discrepancy between locally measured values of $H_0$ $H_{0}$ (e.g., via supernovae) and values inferred from the early universe (e.g., CMB observations). Our findings are consistent with the larger trend of greater $H_0$ $H_{0}$ values from late-universe observations compared with early-universe predictions, highlighting the need for more research into the underlying physical or systematic sources of this tension. We further evaluate the evolution of the cosmological parameters, including the equation of state parameter $\omega (z)$ $ω (z)$ , energy density $\rho (z)$ $ρ (z)$ and pressure p(z). Statefinder diagnostics (r, s) demonstrate the versatility of the model, capturing its deviation from and convergence toward $\Lambda$ $Λ$ CDM at specific epochs. This study emphasizes the importance of bulk viscosity in explaining the universe’s accelerated expansion and provides a strong framework for assessing various cosmological theories.